Abstract

The procedure of stimulated-Raman adiabatic passage (STIRAP), one of many well-established techniques for quantum-state manipulation, finds widespread application in chemistry, physics, and information processing. Numerous reviews discuss these applications, the history of its development, and some of the underlying physics. This tutorial supplies material useful as background for the STIRAP reviews as well as related techniques for adiabatic manipulation of quantum structures, with emphasis on the theory and simulation rather than on experimental results. It particularly emphasizes the picturing of behavior in various abstract vector spaces, wherein torque equations offer intuition about adiabatic changes. Appendices provide brief explanations of related coherent-excitation topics and useful evaluations of relative strengths of coherent transitions—the Rabi frequencies—involving Zeeman sublevels.

© 2017 Optical Society of America

1. Introduction

The technique of stimulated Raman adiabatic passage (STIRAP), combining coherent, two-field, three-state pulsed stimulated Raman scattering (SRS) with adiabatically (slowly) changing interaction strengths, finds increasing applications well beyond its first use in producing transfer of population [1] between rovibrational energy levels of molecules in beams [24]. It now offers a tool for general quantum-state manipulation, as described in various reviews [59] and, most recently, [1012]. Its popularity comes in part because, like other adiabatic procedures, it is relatively insensitive to details of the excitation-producing pulsed interactions and in part because it minimizes the presence of radiatively lossy excited states, two characteristics that the present article examines in some detail.

The theory here presented complements some of the discussion of experimental results in the reviews, and notes some of the most common applications, but with emphasis on the mathematical theory—the depiction of motions in an abstract vector space—and on numerical simulation, rather than on the confirming results of experiments [10,12]. The topics are chosen primarily for instruction rather than completeness, and include discussions of general coherent-excitation theory found in textbooks and monographs [1315].

1.1. Overview

The original STIRAP concept [4] dealt with time-varying adiabatic change of three molecular quantum-state probability-amplitude variables induced by a Lambda linkage of two pulsed and counterintuitively ordered single-photon interactions, aimed at producing nearly complete population transfer [24]. Like many other techniques for quantum-state manipulation, it relied on coherent excitation [9,13,15] rather than changes induced by broad-bandwidth radiation and described by rate equations [15,1519]. Its connection with preceding work has been presented in some detail [9] and its extensive experimental applications have been regularly reviewed [68,10,12]. As the several reviews have pointed out, and this tutorial will explain, the original concept has been extended to treat a variety of more general situations that have aspects in common with that original proposal.

All of these procedures complement alternative techniques for quantum-state manipulation that rely on diabatic rather than adiabatic change (e.g., generalized pi pulses, cf. Subsection 4.3) or which rely on adiabatic changes of detunings (e.g., chirped pulses, cf. Appendix H), each of which has an extensive literature, cf. the reviews [7,8,12]. Techniques based on optical pumping [2025] require, as one step, incoherent change induced by spontaneous emission, and so they are excluded from the present discussion of fully coherent processes.

Over the years, as advancing technology has created new opportunities and incentives for quantum-state manipulation the term STIRAP has gained something of the generic usage that befell “cola.” The present article distinguishes basic notions that attended the original three-state STIRAP concept (cf. Subsection 2.3) from more involved multistate procedures that evolved quite naturally (one might say adiabatically) from those origins. The topics chosen for discussion are those I have found particularly interesting from the extensive review of [12], augmented by reviewer suggestions.

To understand how the STIRAP mechanism, and other adiabatic-manipulation techniques, accomplish their quantum-state changes, and to devise new related procedures for future work, it is useful to view graphical displays of simulations that guide intuition, as was recognized from the outset of the STIRAP concept. Such pictures require some helpful geometric foundation. A variety of abstract vector spaces serve this purpose well, and will be discussed.

1.2. Outline

This article follows, in part, the general structure set by the review [12], with material from [9,1315], but it has little to say about the many experimental results that have confirmed the relevance of the STIRAP concepts to demonstrations [7,8,10,12]. It has more to say about torque equations and the insight they offer into the STIRAP process, as viewed in various abstract vector spaces, and it includes tutorial material the reviews assume to be understood.

Section 2 Basics. The concept of STIRAP, as originally envisioned [4], relies on three equations of motion—three coupled differential equations that describe time-changing probability amplitudes. Subsection 2.2 presents the equations, with an illustration of solutions in Subsection 2.2a. Subsections 2.2c2.2e detail the experimenter-controllable parameters of the equations—termed Rabi frequencies and detunings. Subsection 2.3 then discusses the requirements and constraints that distinguish STIRAP, as here defined, from alternative solutions to the three basic equations and from other techniques of quantum-state manipulation (see Appendix A for acronyms of some of these alternative protocols).

Section 3 Background. All quantum-state manipulations rest upon general principles of quantum theory (e.g., [26]) that have been discussed in treatments of coherent excitation [9,13,15] and, more generally, quantum optics [2739]. The original STIRAP procedure described in Section 2 relied on a statevector representation of the quantum theory, as do many of the subsequent extensions to other quantum systems. Section 3 first presents a brief definition of wavefunctions and the Hilbert space of statevectors and explains the origin of the three basic equations, constituents of the time-dependent Schrödinger equation (TDSE), that underlies all descriptions of coherent changes to quantum systems.

Section 4 Insights. This section introduces the notion of a torque equation, in Subsection 4.1, and offers insights into the adiabatic time evolution associated with the dynamics of STIRAP as defined in Subsection 2.3. These include population-trapping states, Subsection 4.2; adiabatic states, Subsection 4.4; three-dimensional pictures of statevector rotation, Subsection 4.5; and following the system point along a curve of eigenvalues, Subsection 4.6a.

Section 5 Ensembles. In practice, experiments often must deal with ensembles of structured quantum particles and fields—collections that are defined by common procedures of preparation. Section 5 discusses how this alteration of the physics is reflected in the mathematics. It also comments on the nature of errors in the STIRAP procedure and some of the possibilities for reducing those errors.

Section 6 Related. The original three-state two-field concept of STIRAP has been extended to treat two pulsed interactions acting adiabatically on multistate chains [4048], to treat a three-field loop-linkage of three states [4951], to treat a three-field tripod-linkage of four states [5261], and as so-called “straddle”-STIRAP in odd-N multistate chain linkages [41,42,62,63]. Section 6 discusses some of the procedures such as these for quantum-state manipulation that share many, but not all, of the characteristics of the original STIRAP.

Section 7 Composites. The three probability amplitudes of basic STIRAP refer to a single degree of freedom, traditionally the discrete internal energy states of a structured quantum particle (e.g., an atom, molecule, or ion). The induced alteration of those three states by STIRAP often accompanies changes to correlated degrees of freedom. Section 7 discusses three examples of systems in which controlled changes of atomic excitation are associated with correlated changes to electromagnetic fields.

Section 8 Inspired. The notion of adiabatic following that underlies STIRAP has also been applied to treat three variables, and two pulsed interactions, that have no quantum-state association, and for which position rather than time serves as the independent variable. Section 8 discusses some examples that use the three basic STIRAP equations, and their requirements and conditions, in contexts that are well removed from the original application to atomic and molecular excitation.

Sections 9. A brief summary and some acknowledgments conclude the main portion of this article.

Appendices. A set of appendices provide supportive details for some of the discussions, included to make the present article more self-contained. The topics are those I have found useful when discussing STIRAP. These include discussions of excitation involving variable detuning, Appendix H; formulas for evaluating Rabi frequencies, Appendices K and L; and discussions of the Maxwell equations for the excitation-producing fields, Appendices M and N.

References. The ready Internet availability of Wikipedia and Google Scholar greatly simplifies needed searches for clarification of technical terms and publications relevant to any topic. The citations I have provided are some that I have collected over the years.

2. Basic STIRAP

This section presents the basic equations that govern the time dependence of the three probability amplitudes used in the mathematical description of the original concept of the STIRAP process [4,6] and whose origin rests on the principles of coherent excitation to be discussed in Section 3. It then places these equations, and their solutions, within the context of STIRAP. Section 4 presents further details of the STIRAP process and its operation.

2.1. Three-State Two-Interaction Linkages

In its simplest version [4], STIRAP produces nearly complete transfer (i.e., passage) of population through a three-state chain of excitations 1–2–3 among quantum states with energies E1,E2,E3, from an initially populated quantum-state 1 to a target quantum-state 3, induced by two pulsed coherent-radiation fields that couple the intermediate state 2 to states 1 and 3. (Subsequent investigations have considered more general quantum-state changes involving initial and/or final coherent superpositions of these states, cf. Subsection 5.2.) Figure 1 shows the possible transition-linkage patterns of three-state chains affected by two fields: a ladder (or cascade, Ξ), a Lambda Λ (or bent linkage), and a letter-Vee (or inverted Lambda). The patterns differ only in the connections made between a set of three fixed energies En. Although the usual linkage pattern for STIRAP has been the Lambda, the ladder has also been considered [6773]. Allowance for a third field makes possible a loop (or triangle or Delta) linkage, considered elsewhere in connection with STIRAP [4951]. The figure also shows the detunings of Eq. (2): the one-photon detuning Δ and the two-photon detuning δ. These are mismatches between system energy differences (Bohr transition frequencies) and photon energies, i.e., field carrier frequencies.

 

Figure 1 Three-state two-interaction linkage patterns (and detunings Δ and δ) of (a) Ladder (or Cascade, Ξ), (b) Lambda (Λ), and (c) letter-Vee, showing chained P and S linkages 1–2–3. Inclusion of a third interaction 1-3 turns the linkages into a loop (or triangle or Delta) [4951]. Horizontal lines mark unperturbed-state (bare-atom) energies En. [In (b) and (c) the energies E1 and E3 need not differ.] Dashed vertical arrows for the ladder linkage (a) show possible radiative decays associated with the two radiative transitions, not repeated in frames (b) and (c). The numbering of the states is here defined by their ordering in the three-state chain 1–2–3, not by the ranking of their energies En: The relative ordering of energies does not matter when using the rotating-wave approximation (RWA); see Subsection 3.2. In the letter-Vee linkage the initial population is traditionally in the middle of the chain, state 2 [6466], whereas in the other patterns state 1 has the initial population. See also Fig. 5 of [7], Fig. 1.2 of [9], Fig. 1 of [66].

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The basic traditional Raman process [7478], long used in molecular spectroscopy [7982], is an inelastic light-scattering process in which a photon from a pump (or primary) field P scatters into a Stokes (or secondary) field S, with the difference in photon energies going to an energy change of a molecule, most commonly rotational or vibrational excitation. When the S field is supplied, rather than created from spontaneous emission, the process is termed stimulated Raman scattering [8386]. In the originally studied context of SRS the P field linked initially populated ground state 1 to an electronically excited state 2, and the S field linked state 2 to a lower-energy, metastable target state 3, in a Lambda linkage pattern, as shown in Fig. 1(b).

The following Subsection 2.2 presents the three basic differential equations that describe the STIRAP process and comments on their mathematical properties. Subsection 2.2a exhibits a common simulation of STIRAP. As Section 8 will illustrate, the three basic STIRAP equations (2) need not have any association with systems for which there are undisturbed energies En associated with the parameters Δ and δ. For such alternative systems the patterns of Fig. 1, and their interpretation of atomic or molecular excitations involving “pump” and “Stokes” (or “dump”) radiation, have no relevance.

Noteworthy in STIRAP. What makes STIRAP particularly notable is that the two pulses must act in a seemingly counterintuitive (or anti-intuitive) ordering [87,88]: the S field, although acting on the unpopulated states 2 and 3, must occur before (but overlapping) the P field, which acts on the populated state 1. (As discussed in Subsection 6.2, the intuitive ordering of pulses, P preceding S, succeeds equally well in moving population when the single-photon detuning Δ is sufficiently large.)

2.2. Basic Differential Equations of STIRAP

The principal observables of a multistate quantum system are the probabilities Pn(t) that the system be found in state n at time t, given a specified earlier description. When the system is unaffected by randomizing phase-disrupting influences, as will be assumed in the present article (see Subsection 2.2f), these probabilities, known also as populations, are expressible as absolute squares of complex-valued probability amplitudes Cn(t):

Pn(t)=|Cn(t)|2.

For a system undergoing coherent changes the behavior of the probability amplitudes is governed by the TDSE, traditionally treated within the rotating-wave approximation (RWA), cf. Subsection 3.2, Appendix C, and [13,15]. For three states the defining equations for complex-valued probability amplitudes Cn(t) at time t are typically written as three coupled ordinary differential equations (ODEs):

iddtC1(t)=12ΩP(t)C2(t),
iddtC2(t)=12ΩP(t)C1(t)+ΔC2+12ΩS(t)C3(t),
iddtC3(t)=12ΩS(t)C2(t)+δC3(t),
supplemented by three specified initial values at some initial time tin, cf. Subsection 2.2. In the differential equations of motion, Eq. (2), the strengths of the excitation-producing interactions are parametrized by two time-dependent Rabi frequencies ΩS(t) and ΩP(t), taken here to be real-valued, though not necessarily positive (cf. Subsection 2.2e and Appendix B). Subsection 2.2c defines the detunings Δ and δ, most often taken to be zero as occurs with excitation by resonant radiation—the desired population changes occur most rapidly under resonant conditions.

Initial conditions. Traditionally, for the Lambda and ladder linkages, the population is taken to be in state 1 initially, at time tin. With the traditional assumption of an initial real-valued and positive probability amplitude the initial conditions are then

Cn(tin)=δn,1{1if  n=10otherwise.
More generally, after some manipulation has taken place (as it does with the pulse trains of Subsection 6.6), the system will start in a coherent superposition as it begins further excitation; cf. Subsection 3.3g and [15].

When dealing with pulses of finite support (cf. Appendix F) it is convenient to start the time interval with the onset of the first pulse, setting tin=0. Alternatively, when one takes time t=0 to mark the midpoint of the pulse sequence this leads to the conventional assignment of tin for pulses such as Gaussians.

In systems described by the Vee linkage (or inverted Lambda) of Fig. 1(c), the initially populated state 2 passes population to two excited states [6466]. This excitation exhibits noteworthy quantum-beat interference patterns between the two excitation branches [8991].

Equation structure. A key characteristic of these three first-order ODEs, in addition to their linearity, is the presence of the imaginary unit i1. When the coupling coefficients—the Rabi frequencies—are constant (so that Laplace transforms provide solutions [9295]) this has the effect of producing solutions that are trigonometric functions rather than the exponentials that occur with the rate equations of incoherent excitation: the solutions oscillate indefinitely rather than steadily approaching fixed final values, cf. Appendix B. (The presence of imaginary components of the diagonal elements of the coefficient matrix, i.e., complex-valued Δ and δ, damps these oscillations and brings the probability amplitudes to definite final values; see Subsection 3.3f).

A second significant property is that the matrix of coefficients is symmetric, cf. Subsection 3.3b. This gives the solutions properties analogous to those of torque equations, cf. Subsection 4.1.

Finally, the Rabi frequencies depend upon time; solutions are not then obtainable by Laplace transforms, except by piecewise construction.

It is these several mathematical characteristics that ultimately underlie the physics associated with STIRAP.

Solutions. There exist a variety of known closed-form expressions for solutions to these equations for particular choices of the parameters—the Rabi frequencies, detunings, and initial conditions—that distinguish different uses of the equations, cf. [15,9698]. The numerical solution of coupled ODEs poses no great challenge [99108] for any choice of the parameters, thanks particularly to readily available packaged computer routines, e.g., Mathematica (used for the simulations created for this article), MATLAB, and Maple. The present article aims to offer insights into how the solutions depend qualitatively upon the several parameters, and how the parameter choices relate to possible experimental procedures.

Coherences; qubits. Although populations Pn(t) remain important observables, there is increasing interest in creating and manipulating specified coherent superpositions of quantum states, as expressed by bilinear products of probability amplitudes, the coherences ρmn(t)=Cm(t)*Cn(t)—the off-diagonal elements of the density matrix ρ(t) [109111], see Subsection 3.3h. These require careful attention to phases of complex numbers.

The basic element of quantum information [36,112116], the qubit [117], is a controlled superposition of two quantum states defined by amplitude and phase. The ability of STIRAP to provide such superpositions with high accuracy (fidelity) has made it an important tool for dealing with quantum information processing (QIP) [113119].

Analysis versus synthesis. As with other equations of motion, two general uses of these equations are to be found. With analysis the intent is to evaluate the behavior that follows from a given set of equation parameters—the Rabi frequencies and detunings. With synthesis the object is to determine Rabi frequencies and detunings that will produce a specified solution. Often this objective is approached by systematic trial and error, for example evaluating the most satisfactory of a set of functional forms. Such optimization is particularly suited to the determination of final population values, cf. Subsection 5.8d. More general inverse-engineering procedures, as in designer evolution of quantum systems by inverse engineering (DEQSIE) [120], can provide prescriptions for obtaining predetermined time dependence for any variables.

2.2a. Archetype Simulations

Figure 2 repeats the most common example of a STIRAP simulation, one in which two Gaussian pulses of equal peak value and equal temporal width, suitably offset in time, serve as Rabi frequencies in a lossless, fully resonant two-state system, i.e., Δ=δ=0. As in other figures, these plots of Rabi frequencies are normalized to unit peak value. One sees here the characteristic complete population transfer of population from state 1 to state 3, with negligible population in state 2 at any time. As will be discussed in following sections, these particular solutions of the basic Eq. (2) rely on slowly (adiabatically) varying Rabi frequencies; see Subsections 2.3 and 4.4.

 

Figure 2 Example of STIRAP dynamics for fully resonant Raman linkages, Δ=δ=0. (a) The pulsed Rabi frequencies ΩS(t) and ΩP(t), here Gaussians of equal width and height, and temporal pulse area (integrated Rabi frequency, see Appendix B.1) of A=5π, are normalized to unit peak value for display. (b) The populations Pn(t), for n=1,2,3. Times are in units of the pulse-width T. When the time evolution is adiabatic, as in this simulation, there is complete population transfer from state 1 to state 3 and negligible population in state 2 at any time, a signature of traditional STIRAP. At the midpoint of the pulse pairs the system is in an equal-amplitude coherent superposition of states 1 and 3. Similar plots have appeared in Refs. [121123] and in the reviews [69,12,14,15].

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Pulse ordering. The time-ordering of the two resonant pulses is an essential characteristic of traditional STIRAP: the S field acts first, followed by and overlapping the P field. However, as evidenced by the structure of Eq. (2), it is the P field that has direct action on the initially populated state 1. Were the excitation to be produced by incoherent light sources, rather than by coherent pulses, it would be necessary for the P pulse to arrive first, an intuitive ordering contrasted with the counterintuitive ordering of STIRAP.

The importance of pulse ordering can be seen from alternatives shown in Fig. 3. Figure 3 (left side) shows an example of population transfer produced by coherent excitation with intuitively ordered resonant pulses. In this example there occurs first a series of population inversions (resonant Rabi oscillations, cf. Appendix B) between states 1 and 2 that leave the population in state 2. There next occurs a series of inversions between states 2 and 3 that leave the population in state 3. The results of each pulse depend on the pulse duration, quantified by the value of the integrated Rabi frequency (the temporal pulse area A, see Appendix B.1). The two pulse areas, here taken as A=5π, must be carefully adjusted to produce the desired complete transfer of population, first into state 2 and then into state 3. By contrast, the population transfer of the STIRAP procedure is not sensitive to the pulse areas; it is said to be robust.

 

Figure 3 Pulses and populations as in Fig. 2 but for pulses intuitively ordered, PS (left side) or simultaneous (right side). All pulses have identical Gaussian shape and pulse area A=5π, as in Fig. 2, and are fully resonant (Δ=δ=0). For intuitively ordered pulses (left side) the transfer of population from state 1 to state 3 takes place in two stages, first 12 and then 23, each of which is sensitive to the value of the relevant pulse area. For simultaneous (and resonant) pulses (right side) the population undergoes Rabi oscillations that include state 2 as an intermediary. With the chosen areas of 5π and zero delay, state 2 receives half the population after the pulses.

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When the two resonant pulses overlap fully, and have identical Rabi frequencies, as shown in Fig. 3 (right side) for the resonant 5π pulses of the two preceding examples, the populations undergo oscillatory transfer (three-state Rabi oscillations) 12321. The final population distribution depends sensitively on the pulse areas and on the delay between pulse maxima. With careful control of these parameters (not done for this figure) one can obtain complete population transfer. Subsection 2.4b discusses the effect of pulse delay upon population transfer.

The following subsections present definitions of the Rabi frequencies and detunings that control the solutions to the basic STIRAP Eq. (2). Section 3 explains the origin of the equations themselves, in the time-dependent Schrödinger equation that underlies all descriptions of coherent changes to quantum systems.

2.2b. Electric-Dipole Interaction

In the original STIRAP work each state-changing interaction came from the energy of a molecular dipole moment d in the electric field E(t) of a laser beam, idealized as

E(t)=ReeE(t)eiωt+iφ,
with unit vector e, amplitude envelope E(t), carrier frequency ω, and phase φ. The Rabi frequencies then derived from matrix elements [cf. Eq. (19) for notation] of the electric-dipole interaction (see Subsection 3.3c),
ΩP(t)=ψ1|d·eP|ψ2EP(t),ΩS(t)=ψ2|d·eS|ψ3ES(t).
This electric-dipole origin of the Rabi frequencies allows them to be complex valued; the absolute square of a Rabi frequency is proportional to the instantaneous intensity of the radiation (cf. Appendix N.1):
I(t)=cε02|E(t)|2.
By expressing |E(t)|2 in terms of laser intensity, I(t), we obtain for the Rabi frequency the formula
|Ω(t)|=8παa02I(t)|dnm·e|ea0=2.2068×106  sec1I(t)[Wm2]|dnm·e|ea0,
where α1/137 is the Sommerfeld fine-structure constant, a0 is the Bohr radius, and ea0 is the atomic unit of dipole moment,
ea0=2.542  debye=8.478×1031  C  m.
Thus an atomic-unit dipole moment exposed to a laser intensity of 102  W/cm2 has a Rabi frequency of 2×1011  rad/s (or approximately 30 GHz). By comparison, the frequency of optical transitions is around 1014  rad/s. This comparison, |Ω|ω, underlies the rotating-wave approximation (cf. Subsection 3.3d): the optical field will complete many cycles during one Rabi cycle.

As the formulas of Eq. (5) show, the dipole interactions are proportional to the projection of a dipole moment vector d onto a unit vector e that parametrizes the electric field direction. Because this is a quantum system the possible values of this scalar product form a discrete set, and consequently a given transition may have different Rabi frequencies depending on the polarization direction of the electric field. Appendix K discusses the evaluation of these discrete values for quantum states of angular momentum. Subsection 5.1 discusses the management of Rabi frequencies between degenerate Zeeman sublevels.

Other electromagnetic interactions share properties of the formulas of Eq. (5); they are products of two factors: a constant transition moment that is fixed by the choice of specific transitions in a specific system, and a time-varying field amplitude that an experimenter controls. In all such cases it is possible to choose phases such that any single Rabi frequency is real at a given reference time. (With inclusion of angular momentum, arrays of Rabi frequencies are phase related, cf. Appendix K.)

2.2c. Detunings

With the electric-dipole (or other single-photon) interaction, the individual-state energies En and field carrier frequencies appear in the basic Eq. (2) only indirectly, as constituents of detunings ΔP and ΔS of carrier frequencies ωP, ωS from Bohr frequencies. For the Lambda linkage these are

ΔP=E2E1ωP,ΔS=E2E3ωS.
The detunings that appear in the equations of motion, Eq. (2), are [124]
Δ=ΔP
δ={ΔPΔS,E1,E3<E2,Lambda,ΔP+ΔS,E1,<E2<E3,Ladder.

Thus the equations apply equally well to the Lambda linkage of traditional SRS, cf. Fig. 1(b), for which the middle state has highest energy, and to the ladder (or cascade) linkage of Fig. 1(a), for which the energies increase along the linkage chain. The two-photon detuning δ appearing in Eq. (2c) is either the sum (for the ladder linkage) or the difference (for the Lambda linkage) of the single-photon detunings ΔP and ΔS. Although a variety of pulsed interactions have subsequently been considered as underlying the Rabi frequencies of Eq. (2), some of which have no association with a carrier frequency (cf. Section 8), the terminology “detuning” persists.

In the early STIRAP investigations [6] the two detunings were constant (and small, optimally zero) but the success of adiabatic population transfer only required that at all times they combine to satisfy the two-photon resonance condition,

δ=0,
for the 13 transition, meaning: ωPωS=E3E1 for the Lambda and ωP+ωS=|E3E1| for the ladder. Subsections 2.3e and 4.6e discuss excitation with nonzero two-photon detuning.

2.2d. Selectivity

The numerous energy states of an individual atom or molecule offer many opportunities for radiative transitions. The polarization directions of the fields (the directions of the unit vectors eP and eS) provide some selectivity among these, see Appendix K. The choice of carrier frequencies offer further selectivity: there should be a close match between a carrier frequency and one Bohr transition frequency ωnm=|EnEm|/. Generally the separation between allowed transitions must be larger than the Rabi frequencies, so that each field is uniquely associated with a single transition. For steady fields this selectivity requires that the detuning be small compared with the Rabi frequency so that excitation can be appreciable. Ambiguity in assignment of laser pulses to unique transitions leads to beat frequency contributions to Rabi frequencies [125].

2.2e. Pulsed Rabi Frequencies

The time variation of the Rabi frequencies may occur in various ways. When the interaction is with electromagnetic fields, as is traditional but by no means necessary (cf. Section 8), the variation may arise in two ways, illustrated schematically in Fig. 4.

 

Figure 4 Schematic picture of field intensity of two pulses usable for STIRAP. (a) Particle with velocity v moves transversely across electromagnetic fields whose axes are in the z direction and whose centers are offset in the direction of particle motion, x. Trajectories that differ in y encounter different peak values of the fields, and hence different peak Rabi frequencies. Their pulse durations also differ. (b) Pulsed fields, idealized as plane waves in the x, y plane, moving in the z direction past a stationary particle. For both examples the electric field vectors lie in the x, y plane, the propagation vectors are in the z direction.

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Moving particles. Early works dealt with excitation of ensembles of molecules that moved steadily across monochromatic, continuous-wave (CW) laser beams. Later work dealt with atoms that dropped through single-mode standing-wave cavities (cf. Subsection 7.1). In all such situations a particle center of mass, moving across a spatially varying field, encounters time-varying fields. Figure 4(a) shows a schematic diagram of an example. A particle of steady velocity v in the x direction, and whose transverse offset in y is adjustable, passes transversely across laser beams whose axes lie in the z direction. A field amplitude with x, y variation E(x,y) appears to a particle having a moving center of mass at xcm(t)=vt as the pulsed field E(vt,y). An experimenter, by crafting a transverse spatial variation of a stationary laser beam, traveling or standing, creates a shaped pulse of interaction. For example, the most common lowest-order Gaussian beam produces a Gaussian pulse (cf. Appendix F). More elaborate beam profiles can include nodes [126128], so that a particle moving across such a beam experiences a null in the pulse envelope, e.g., a zero-area pulse, cf. Appendix F.2.

Pulsed fields. Other work has directed pulses of radiation onto ensembles of trapped particles: ions held in magneto-optical traps [129131] or impurity atoms in transparent crystals [57,132134], or other discrete-state structures. The experimenter directly controls either the temporal variation of a traveling wave or controls the Fourier components of that field [135139]. Typically one idealizes such fields as pulses of finite support; see Appendix F. Figure 4(b) shows a schematic representation of this situation. It is also possible to construct trains of pulses, cf. Subsection 6.6.

The mixing angle. The pulse sequence required for STIRAP, in which the S pulse precedes but overlaps the P pulse, is most simply expressed in terms of the time-varying mixing angle ϑ(t), defined, for real-valued Rabi frequencies, by the formula

tanϑ(t)=ΩP(t)/ΩS(t).

During the course of complete population-transfer from state 1 to state 3 (the adiabatic passage) via STIRAP the mixing angle traditionally undergoes a monotonic adiabatic change from ϑ=0 to |ϑ|=π/2, as befits positive Rabi frequencies, but these limits should be understood as being mod π, cf. Fig. 29 of Subsection 6.5h.

2.2f. Coherence

Just as the study of physics relies on simplifying idealizations to form its separate disciplines, so too does the notion of STIRAP rest upon a few idealizations, most notably the notion of an isolated, quantum-state system that is under complete control of an experimenter and uninfluenced by random changes of its environment. It is this idealization that allows the use of wavefunctions and statevectors, and their governing time-dependent Schrödinger equation, cf. Section 3.

The uncontrollable, detrimental effects are quantified by a coherence time τcoh that bounds the interval during which uncontrollable environmental phase alterations can be neglected [140]. The time required to produce controlled, coherent system change, say by radiation pulses, must be much shorter than τcoh. This idealization of a relatively long coherence time, conveniently treated here as a binary “true or false” judgment on the presence of decoherence, is a limiting condition, discussed in Subsection 5.8, that rests upon criteria for defining successful change and the continuum of dephasing errors. The STIRAP reviews [610,12,14] mention numerous works that have (using density matrices) examined the effects of decoherence upon such procedures as STIRAP.

Bandwidth. The distribution of frequencies present in a pulsed electric field E(t), or its associated Rabi frequency, can be expressed by the Fourier transform

E˜(ω˜)=dt  eiω˜tE(t),
a complex-valued function of frequency ω˜ centered around the carrier frequency. The width of this spectral distribution (its bandwidth) receives contributions from temporal variations of the carrier frequency, the envelope and the polarization direction. A bandwidth-limited pulse (a transform-limited pulse) is a pulse that has the minimum possible temporal width for a given spectral bandwidth. Enlargement of this minimum bandwidth indicates undesirable fluctuations (noise) that tend to introduce decoherence in the quantum system. The pulses used as explicit examples in the present article are transform limited.

2.2g. Probability Loss

In all the linkage patterns of Fig. 1 the excited states can undergo spontaneous emission to lower-lying states; those that lie outside the three-state system constitute an undesirable probability-loss mechanism. This is usually incorporated into the TDSE by adding an imaginary term to the appropriate diagonal element of the RWA Hamiltonian; see Subsection 3.3f.

One of the popularizing characteristics of STIRAP is that, although it induces population transfer from state 1 to state 3, it places negligible population at any time into state 2, from which it might be lost by radiative decay. This property, not present for adiabatic transfer by intuitively ordered pulses (cf. Subsection 6.2), allows the coherent population-transfer process to proceed during a time interval much longer than the radiative lifetime of state 2.

2.2h. Rabi Frequency Bounds

The Rabi frequencies are responsible for inducing coherent changes of the quantum system, and so they must be large enough to produce the desired alterations in a time that is shorter than any time interval τcoh for losing coherence (i.e., introducing decoherence). But the Rabi frequencies cannot be too large. The validity of the RWA underlying these equations (cf. Subsection 3.3b and Appendix C) requires, among other things, that the Rabi frequencies be much smaller than the Bohr transition frequencies,

|ΩP(t)||E2E1|,|ΩS(t)||E3E2|.

2.2i. Appropriate Systems

Traditionally the three variables Cn(t) have been taken as probability amplitudes for discrete quantum states of atoms, molecules, or ions. These may either be moving freely as beams or may be trapped, either in vacuum by arranged electromagnetic fields or as impurities in transparent solid matrices. But numerous other systems offer similar discrete energy structures and associated variables [12]. Among the most notable for possible use with STIRAP are the following examples from [12]. Detailed discussions will be found there. The review of STIRAP in [11] deals specifically with application to such artificial atoms.

Quantum dots (QDs). Quantum dots are single-crystal semiconductor structures, typically around 100 nm in diameter—a mezoscopic size intermediate between that of bulk matter and the atoms of which they are composed. As with bulk matter, their fundamental optical excitation consists of an electron in the conduction band and a hole in the valence band, but spatial confinement (a three-dimensional “box”) discretizes the energy structure of the bound states. Their spectroscopic properties—relatively sharp spectral lines indicative of discrete energy levels—are determined largely by composition, impurities, size, shape, and shell material. Because these are, in principle, controllable during construction the structures are regarded as artificial atoms. Their properties can subsequently be changed in a controlled way by electrostatic gates or applied magnetic fields.

Although the purely quantum-mechanical properties of QDs have become of interest only in recent decades, such particles have been created and used for millenia [141]. Nowadays they have industrial application in light-emitting devices and solar cells [141] and have use in analytic chemistry, biology, and medicine [142]. QDs for use in biology and medicine are typically freely moving, luminescent, colloid particles formed from solution by chemical precipitation [141,143]. Stationary QDs for use in physics as photon sources or quantum information processing are commonly created by molecular-beam epitaxy deposition onto a substrate [144].

Superconducting quantum circuits (SQCs). Superconducting circuits that involve the nonlinear effects of Josephson-tunnel junctions can exhibit many of the characteristics of artificial atoms [145]. They are classified into three types, based on the degree of freedom that is responsible for the discreteness of the circuit behavior: electric charge [146148], magnetic flux [148151], and phase [152]. Three-state SQCs offer opportunities for implementing various quantum-state manipulations such as STIRAP [11,12,145,153].

2.3. Basic Conditions and Characteristics

Although the STIRAP-describing equations (2) had their origin in modeling probability amplitudes of stimulated Raman excitation, they have subsequently found application in many diverse areas of chemistry, physics, and engineering, cf. reviews [9,10,12]. The dependent variables Cn(t) of such STIRAP-inspired work need have no quantum properties, and there need be no underlying model associated with one of the linkage patterns of Fig. 1; see Section 8. Accompanying this broadening of the original application (to Raman transitions in molecules) has come an appreciation of ways in which solutions of a set of N coupled equations with slowly varying amplitudes lead to behavior that is relatively insensitive to details of the pulses (i.e., it is robust, cf. Subsection 5.8), not only when directed toward producing complete population transfer but also for dealing with coherent superpositions of quantum states (cf. Subsection 6.1). With this diversity of application, and the diversity of quantum systems and procedural objectives, the appellation “STIRAP” has become associated with a variety of techniques that differ appreciably from the original work but retain a few common elements; see Sections 6 and 8. The terms “STIRAP-like,” “STIRAP extension,” and “STIRAP inspired” have been used [12] to distinguish some of these from the more restricted original STIRAP, as that designation is defined by the following characteristics [9].

2.3a. Requirements

The traditional requirements for STIRAP, in addition to the three-state two-field coherent TDSE of Eq. (2) and the initial conditions of Eq. (3), can be summarized by the following list [154]:

  • (i) The mixing angle ϑ(t) of Eq. (12) varies slowly (adiabatically).
  • (ii) The P pulse is negligible at the start of the pulse sequence (|ϑ|0 mod π).
  • (iii) The S pulse is negligible at the termination of the pulse sequence (|ϑ|π/2 mod π).
  • (iv) The two-photon detuning remains zero, i.e., δ=0.
  • (v) The single-photon detuning Δ is constant (optimally zero).

These requirements are equivalent to mandating that the statevector remain aligned with the population-trapping dark-state instantaneous-eigenvector of the RWA Hamiltonian, as discussed in Subsections 4.2 and 4.4b, and that the two pulses arrive with counterintuitive ordering.

2.3b. Consequences

The results of the STIRAP procedure defined by these requirements have the following characteristics:

  • (a) The values of C2(t) are negligible at all times.
  • (b) The results are relatively insensitive to details of pulse shape and timing—the process is robust.
  • (c) As the mixing angle traces a course between |ϑ|0 and |ϑ|π/2 (each mod π), there will occur the complete conversion of population in state 1, with |C1|=1, to population entirely in state 3, with |C3|=1.

The requirement for relatively slow (adiabatic) time evolution is essential for traditional STIRAP, though not for other protocols that produce specifiable quantum changes. As with other adiabatic procedures it makes the process relatively insensitive to changes in the controlling parameters—the shapes and timings of the two Rabi frequencies and the single-photon detunings (see Subsection 5.8).

The requirements and consequences listed here are complementary, and the traditional STIRAP procedure can be defined, in part, by either set alone: the consequences follow from the requirements and the requirements are necessary for the consequences. However, these consequences, though requiring resonant two-photon detuning, δ=0, do not require single-photon resonance, Δ=0, nor even a constant detuning. With these requirements it is the pulsed variation of Rabi frequencies, rather than any manipulation of detuning, that produces the population transfer associated with STIRAP; see Appendix H.2.

Portions of Section 3 offer insights into the behavior of the STIRAP process as defined above. Section 5 discusses examples of systems of three degenerate or nearly degenerate sets of states that have been regarded as demonstrating the essence of STIRAP. Although other three-state procedures share some of these characteristics, such as complete population transfer and adiabatic evolution, they are distinguishable by lacking other characteristics; see Section 6.

2.3c. Included as STIRAP

In addition to having the general characteristics presented above, the original STIRAP-defining work [4] dealt with numerous specifics: electric-dipole vibrational transitions involving excitation of degenerate ground levels in a Lambda linkage of molecular-beam particles transiting Gaussian-profile linearly polarized CW laser beams and equal-peak Rabi frequencies, characteristics that are not usually considered essential in defining STIRAP.

Although Eq. (2) that here is taken to define STIRAP does not depend directly on radiation wavelengths—only detunings appear—some experimenters find it helpful to precede the name STIRAP by a modifier such as RF STIRAP, IR STIRAP, or X-RAY STIRAP to indicate the spectral regime of interest. In principle the P and S fields can come from very different regimes. The equations apply equally to ladder and Lambda linkages, but here too a prefix may be useful, e.g., ladder STIRAP or Lambda STIRAP.

2.3d. Not Included as STIRAP

Various procedures, noted in Sections 6 and 8, are often treated as forms of STIRAP though they do not fit the restricted definition presented in the preceding sections. The following paragraphs mention some of these.

Not three states. Even within the earliest studies of STIRAP the limit to three states required revision: the rotation-vibration structure of molecules, degenerate Zeeman sublevels and hyperfine structure, require treatment of multistate systems [155]. However, in these situations it is often possible to extract excitation dynamics that fits the three-state pattern of STIRAP; see Subsection 5.1

Not constant detuning. In presenting the definitions of STIRAP above I have followed the underlying concepts of the original work on STIRAP [4] and have required that the detunings be constant. The present article restricts attention primarily to procedures that maintain constant detunings. Numerous successful schemes for adiabatically manipulating quantum systems rely on frequency-swept (chirped) detunings, a concept that originated in the early days of quantum mechanics with the Landau–Zener–Majorana–Stuckelberg (LZMS) two-state model [156160], see Appendix H. Three-state and four-state excitation chains subject to chirped detunings have been discussed in some detail in [67]. Generalizations to three-state systems have been discussed as Raman chirped adiabatic passage (RCAP) [6,8,161166] or chirped Raman adiabatic passage (CHIRAP) [167,168] [169,170], mentioned in reviews [9], and as chirped adiabatic passage by two-photon absorption (CAPTA) [63,163].

The detuning variations need not be from carrier-frequency modulation but may originate with pulse-induced dynamic Stark shifts of the Bohr frequencies that arise when the S and P fields are detuned from single-photon resonance (though maintaining two-photon resonance), so that a two-state effective Hamiltonian serves as the excitation model, cf. Appendix G.1. Examples include what has been termed Stark-chirped rapid adiabatic passage (SCRAP) [8,171179] in which a pulsed P field induces a two-photon transition and a more briefly pulsed, intense far-off-resonance S field, offset in time from the P-field peak, produces a varying Stark shift (a briefly chirped detuning) that can lead to population transfer, cf. Appendix H.3. The creation of superpositions assisted by Stark shifts has been termed Stark-assisted coherent superposition (SACS) [71]. A procedure in which the P and S fields together produce not only the two-photon Rabi frequency but Stark-shifted detunings has been termed Stark-induced adiabatic Raman passage (SARP) [180183].

Prior to the first STIRAP demonstrations the possibility for adiabatic passage in a three-state stimulated Raman system was discussed by [184], based on counterintuitive ordering of diabatic-curve crossings, cf. Appendix H.2. Although this was the first suggestion of adiabatic passage with stimulated Raman transitions by means of counterintuitive interactions, it relied on swept frequencies rather than pulsed amplitudes, and so it does not fit the definition of STIRAP as presented above in Subsection 2.3a.

Not counterintuitive ordering. As will be discussed in Subsections 4.6d and 6.2, there exist possibilities for successful adiabatic passage with three-state stimulated Raman linkages that rely on intuitive, PS, pulse ordering. Though these are robust and offer complete population transfer, they do not fit the narrow definition of STIRAP presented above.

Not adiabatic. A principal requirement for STIRAP is for adiabatic time variation of the RWA Hamiltonian, see Subsection 4.4c. Typically the condition for such change is expressed as a need for sufficiently large values of time-integrated Rabi frequencies—the temporal pulse area of Appendix B.1; see Subsection 4.4c. When two-photon detuning is present, as discussed in Subsection 4.6e, satisfactory population transfer can be produced by a combination of adiabatic and diabatic changes; see Fig. 12. When the excitation is by means of pulse trains, it can be robustly successful despite abrupt changes in constituent pulselets: the pulses are then not locally adiabatic; see Subsection 6.6.

2.3e. Two-Photon Resonance Condition

The theory for STIRAP has most often been considered for S and P pulses that have the same shapes and peak values, differing only in their arrival times, and in which the carrier frequencies were resonant with the Bohr frequencies. The traditional STIRAP mechanism relies on adiabatic changes of a population-trapping dark state (see Subsection 4.2) and for such pulses the maintenance of two-photon resonance is desirable: departure from two-photon resonance δ=0 dilutes the dark state and thereby brings nonadiabatic change that diminishes the population transfer into state 3 after the conclusion of the pulse pair. However, as pointed out by [185], when the Rabi frequencies of the P and S pulses differ significantly—in peak height, pulse area, or pulse duration—the optimum conditions for population transfer in an ensemble no longer center on two-photon resonance δ=0 and it may become desirable to select a nonzero value of δ in order for the population transfer to be most effective [12,186,187].

Figure 5 illustrates this result [185]. The figure shows population transfer for three choices of pulse pairs, distinguished by differing ratios of peak Rabi frequencies. For the solid lines the peak values are the same, whereas for the red, short-dashed lines the peak S is larger and for the blue, long-dashed lines the peak P field is larger. Frame (a) shows the results for single-photon resonance, Δ=0. Here the inequality of peak Rabi frequencies merely acts to narrow the range of two-photon detunings that produce satisfactory population transfer, i.e., STIRAP success. Frame (b) shows the results when the single-photon detuning is relatively large. Here not only is the range of STIRAP-supporting two-photon detunings smaller, but there is a shift of the transfer profile away from δ=0. For small values of Δ the traditional STIRAP requirement of two-photon resonance is valid.

 

Figure 5 Population transfer P3() versus two-photon detuning δ for Gaussian pulses of width T. (a) Single-photon resonance, Δ=0. (b) Large single-photon detuning. Solid black line: maximum Rabi frequencies are equal for S and P pulses. Red short-dashed line: maximum ΩS is 2.4 times the peak ΩP. Blue long-dashed line: maximum ΩP is 2.4 times the peak ΩS. In frame (a) the two dashed curves overlap. Figure 3 reprinted with permission from Boradjiev and Vitanov, Phys. Rev. A 81, 053415 (2010) [185]. Copyright 2010 by the American Physical Society.

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2.4. STIRAP Signature

Numerous procedures can accomplish the complete population transfer offered by STIRAP, but no others possess all of the characteristics listed above. It is also possible to devise adiabatic procedures, involving only two states [133] or more than three states [41,42,62], that share the robustness of the complete population transfer offered by the STIRAP process, and to employ these with systems whose initial conditions differ from Eq. (3), cf. Subsection 6.2. To identify an observed population transfer as an example of traditional STIRAP, as defined above, it is necessary to show that it has the several distinguishing properties of those lists. The following paragraphs discuss the three traditional observable requirements—the signature of traditional three-state STIRAP. These are aspects of the sensitivity of STIRAP to detunings, to pulse delay and to pulse area, discussed at length in reviews [7,12].

2.4a. Insensitivity to Pulse Details

The original STIRAP investigations, motivated by chemical interests, considered final-state populations satisfactory that differed by a few percent from the ideal of P3=1. Because the STIRAP process relies on an adiabatically changing RWA Hamiltonian its success—transfer of population from state 1 to state 3—was relatively insensitive to details of the Hamiltonian such as variations in detuning, pulse duration, pulse delay, pulse area, and pulse shape. Typically simulations have illustrated STIRAP induced by pulses that have equal peak Rabi frequencies and equal pulse durations—pulses that differ only in their temporal offset. These conditions work well but are by no means necessary. Subsection 5.8c discusses quantification of the traditional claim of robustness for the STIRAP procedure.

By contrast, protocols that rely on pulse details to produce population inversion (e.g., pi pulses of a two-state system, cf. Appendix B) will exhibit errors that increase with the mismatch between intent and realization. They require careful control of pulse areas.

Advances in technology, and correspondingly more stringent requirements for successful STIRAP, have led to protocols that adjust details of pulse shapes to achieve population transfers potentially as high as P3>0.9999, a requirement for quantum-information processing [12]. Such goals require careful attention to the crafting of suitable pulses; see Subsection 6.6.

2.4b. Dependence on Pulse Delay

Although STIRAP does not require an exact value for the delay between the two pulses, it does require that the S pulse precede, and overlap, the P pulse. Any other pulse sequence will fail to produce a robust population transfer with resonant fields (but see Subsection 6.2 for discussion of successful reversed pulse sequences with large single-photon detuning). The next two figures illustrate the effect of pulse delay in the three-state system.

Lossy state 2. Figure 6 illustrates predicted population transfer P3() as a function of delay time between two resonant Gaussian pulses of equal peak values and widths, for several values of pulse area. For this simple system, in which the single-photon detunings are zero, the two principle parameters are the pulse areas (they are both the same here) and the delay between the two pulses. The frames on the left are for a lossless state 2, the frames on the right are for a state 2 that undergoes loss.

 

Figure 6 Population transfer into state 3 versus delay for pulse areas, top to bottom, 5π, 10π, 15π, and 20π. S and P pulses are Gaussians of equal peak value and widths, resonantly tuned (Δ=δ=0). Left: lossless state 2. Right: loss from state 2. In each frame the left-hand portions, where S precedes P, exhibit the robust population transfer of STIRAP. Toward the right, where P precedes S, transient population passes through state 2 from which it is lost before it can move into state 3; population transfer 13 is not successful. Other depictions of this behavior appear in Fig. 10 of [7].

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As is evident, the behavior for negative delay (sequence SP) differs qualitatively from that of positive delay (sequence PS). In all cases there is an optimum pulse delay for which the population transfer is essentially complete and relatively insensitive to the delay. This is the regime of STIRAP, wherein the pulse sequence is SP. By contrast, pulse sequences in which the P pulse precedes the S pulse lead to three-state population oscillations, from state 1 through state 2 to state 3. When state 2 is lossy such oscillations will be incomplete, and population transfer will accordingly be incomplete. With STIRAP negligible population reaches the lossy state, and so nearly complete population transfer is possible.

The Rabi oscillations visible for positive detuning (PS pulse sequence) in the left-hand frames of Fig. 6 occur because there are here precisely defined pulse areas. The oscillations will be damped significantly when state 2 undergoes appreciable loss during the pulse sequence. When observations are made of ensembles that have a range of pulse areas (cf. Subsection 5.4), as will occur when particles move at differing speeds across a laser beam, then averages will tend to wash out the oscillations. The averaged population will be 1/2 rather than the null value that occurs when there is loss.

Single-photon detuning. The simulations displayed in Fig. 6 have set the single-photon detuning Δ to zero. The left-hand portions of the figures are little affected when that detuning is nonzero: there the STIRAP mechanism, based on adiabatic following of the dark adiabatic state (see Subsections 4.2 and 4.4), is independent of detuning. By contrast, the right-hand portions of the frames, the “intuitive” pulse ordering where P precedes S, are notably sensitive to the detuning. Figure 7 shows examples, again for lossless excitation that maintains two-photon resonance, of the effect of pulse delay when Δ is nonzero. When the pulses completely overlap (zero delay) there occur population oscillations, whatever the detuning, and the final transfer ranges between 0 and 1 depending on the pulse area and detuning. Viewing the right-hand side we see that as the detuning increases the oscillations become less intrusive. For the large detuning of the lowest frame there occurs a symmetric pattern of robust population transfer: for either SP or PS pulse ordering there is a delay for which there is complete population transfer (by adiabatic passage of a stimulated Raman linkage) that is insensitive to delay or to pulse area, just as with conventional STIRAP. This large-detuning regime does not have the counterintuitive pulse sequence nor the consequent absence of state-2 population of traditional STIRAP, as defined in Subsection 2.3; it is an example of reverse, backward, or bright-state STIRAP; see Subsection 6.2

 

Figure 7 Population transfer probability, from state 1 to state 3, by SP pulse sequence, versus delay between S and P pulses, for four choices of the single-photon detuning Δ (in units of the peak Rabi frequency) and pulse areas of 10π. From top to bottom the detunings are 0, 5, 10, 15. The relatively large detuning of the lowest frame makes possible successful adiabatic population transfer (adiabatic passage) for either pulse ordering, SP or PS; see Subsection 6.2.

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2.4c. Bright and Dark Resonances

The absence of population in state 2 that characterizes STIRAP is associated with a population-trapping dark state, cf. Subsection 4.2. This presence is maximum at two-photon resonance, δ=0. Thus any probe of populations as a function of detuning will, for STIRAP, reveal a sharp resonance. At the center of this resonance, where δ=0, there will be a pronounced increase in state-3 population observable by an increase of spontaneous emission (a bright resonance) and a corresponding decrease in state-2 population, observable by decrease of probing absorption (a dark resonance) [68,12].

Figure 8 shows STIRAP signatures obtained in experiments with transitions in metastable neon [188]. Frames (a) and (b) show plots of a spontaneous emission signal D versus P-field detuning ΔP for fixed S-field detuning ΔS/2π200  MHz. The accompanying diagrams at the right show the linkage patterns associated with the signal. In each plot the broad emission feature centered around ΔP=0 originates with single-photon excitation of state 2, followed by spontaneous emission into state 3. Upon each of these background curves there is a narrow resonance feature, a peak in (a) and a dip in (b), each centered at the P-field detuning that gives two-photon resonance, ΔP=ΔS. In frame (a) the D signal, a bright resonance, measures population in state 3 and originates from an auxiliary state 4 after a probing transition of state 3. In frame (b) the D signal, a dark resonance, originates from decays of state 2 and thereby measures its population. Any claim of STIRAP should exhibit both these resonance features: the peak in P3() and the dip in P2().

 

Figure 8 Final populations versus P-field detuning ΔP/2π in metastable neon [188]. (a) Target state population P3(). (b) Intermediate excited-state population P2(). The peak in P3(), frame (a), and the dip in P2(), frame (b), are typical signatures of STIRAP. Diagrams at the right show the linkage patterns associated with the detected field D: spontaneous emission from state 2 in frame (b) or from a probing transition out of state 3 into an ancillary fluorescing state in frame (a). Adapted from Fig. 8 of [188] with permission. Copyright 2006 by the American Physical Society. See also Fig. 18 of [6], Fig. 16 of [7], Figs. 54 and 55 of [14], Figs. 14.13 and 14.14 of [15], and Fig. 4 of [12].

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Figure 9 Example of Hilbert-space rotation into final location induced by STIRAP [shown in the RW picture, with RW vectors ψn(t) as coordinate vectors]. (a) The three adiabatic eigenvectors rotate as shown: the eigenvectors ϕ+(t) and ϕ(t) rotate onto the equatorial (1,2) plane, the eigenvector ϕ0(t)=ϕD(t) rotates in the 1,3 plane from the 1 axis to the 3 axis. [The unit vectors ϕB(t) and ϕD(t) remain in the 1,3 plane.] (b) The statevector Ψ(t), maintaining alignment with ϕ0(t)=ϕD(t), rotates from the 1 axis to the 3 axis, remaining always in the 1,3 plane that passes through the two poles: the statevector adiabatically follows the dark-statevector. Other depictions of the adiabatic states appear in Fig. 4 of [251], Fig. 4 of [6], Fig. 9 of [7], Fig. 53 of [14], Fig. 3 of [258], and Fig. 14.11 of [15].

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3. Theoretical Background

The three differential equations of Eq. (2) that underlie STIRAP occur in a variety of contexts, but they first drew attention as STIRAP from application to alteration of molecular and atomic quantum states. The following sections summarize the mathematics that underlies that quantum-mechanical context of the equations, as it would be applied to a single isolated atom or other discrete-state quantum system. Section 5 discusses the extension to treatment of ensembles of independent quantum systems.

3.1. Wavefunction

The presentation of quantum theory relevant to controlled changes typically begins with discussion of a single confined particle, say a particle localized by a box of bounding walls, a particle constrained by spring-like forces to an equilibrium position (an harmonic oscillator), or the single electron bound by Coulomb attraction to the nucleus of a hydrogen atom—model systems treated in all standard textbooks on quantum theory. In these and other single-particle systems all measurable properties are obtainable from a wavefunction: a complex-valued function Ψ(r) of position r whose absolute square gives the probability P(r) of finding the particle at r:

P(r)=|Ψ(r)|2.

The wavefunction therefore contains a full description of the size and shape of a particle—its quantum state.

Quantized energies. The confinement of a particle, by whatever means, implies that its stationary states of motion can only have discrete (quantized) energy values, E1,E2,. Associated with each discrete energy En is a wavefunction ψn(r) whose absolute square gives the probability Pn(r) that, if the particle is known to be in state n it will be found at position r:

Pn(r)=|ψn(r)|2.

These basis wavefunctions are obtainable as solutions to the time-independent Schrödinger equation:

H0(p,r)ψn(r)=Enψn(r),
where H0(p,r) is the Hamiltonian energy operator, a function of position r and momentum p expressing the sum of kinetic and potential energies. This equation is an example of an eigenvalue equation: the stationary-state wavefunctions ψn(r) are eigenfunctions of this Hamiltonian and the energies En are its eigenvalues. Solutions are obtained by replacing momentum components with partial derivatives, as in pxi/x for the x component, thereby turning the time-independent Schrödinger Eq. (17) into a second-order partial differential equation. Localized, bound-state solutions obtain by imposing the boundary condition that the wavefunction must become negligibly small at large distances from the coordinate origin. From this constraint comes the restriction of energies to discrete, quantized values, labeled here by integers n. (Unconstrained, scattering solutions have no such discretizing restriction on energy.)

For many simple idealized systems, such as the particle in a box, a harmonically bound particle, or the electron of hydrogen, it is possible to obtain simple analytic expressions for the basis wavefunctions ψn(r). The eigenfunctions of the hydrogen electron (electron orbitals) describe the size, shape, and orientation of the hydrogen atom.

More complicated systems, comprising many individual particles, have wavefunctions whose arguments comprise sets of positions for each constituent: for these the argument r appearing above denotes a complete set of position values and the Hamiltonian is a sum of kinetic and potential energies of all the particles. When the particles are all spatially confined, as are the electrons of a molecule, the possible quantum states form a discrete set, with discrete energies. The energies need not all be different: states with equal energy are said to be degenerate. As long as the Hamiltonian remains fixed, a particle known to have nondegenerate energy En will remain associated with the stationary motion described by ψn(r): it will be in a specific quantum state.

Our interest is with changes that will occur to such a discrete-state quantum system when it is acted upon not only by the time-independent Hamiltonian H0 that dictates stationary structure but by an additional, controlled, time-dependent energy operator Hint(t). The needed mathematics is presented most succinctly by treating the discrete basis wavefunctions ψn(r) as unit vectors in an abstract vector space, a Hilbert space [189192] in which lengths and angles are defined and quantified by scalar products A|B between vectors A and B. The two vectors are orthogonal (perpendicular) if A|B=0 and are parallel if A|B=1. It is useful to deal with vectors of unit length, meaning A|A=1. For a single particle the geometry of this Hilbert space involves orthogonality integrals as scalar products,

ψn|ψmn|m=dr  ψn(r)*ψm(r),
and matrices of operators M(p,r) whose elements are obtainable as
ψn|M|ψmMnm=dr  ψn(r)*M(p,r)ψm(r).

In defining this mathematical structure it is useful to suppress the arguments of position and momentum and to indicate that suppression by typography: the basis wavefunctions ψn(r) become basis vectors ψn and the system wavefunction becomes a vector, the statevector Ψ(t), now with time dependence [193]. We take the basis vectors to provide a complete orthonormal set of Hilbert-space coordinates, a relationship expressed by the Kronecker delta:

ψn|ψm=δnm{1if  n=m0if  nm.

The Hamiltonian operators thereby become represented by matrices:

H0H0,Hint(t)Hint(t).

States and vectors. The symbol ψn (or |ψn or |n) has two meanings. Physically, it represents a quantum state, one of N distinguishable states needed to describe the quantum system. Mathematically, it stands for a unit vector, one of N orthogonal coordinates in an abstract vector space. The words “state” and “vector” are commonly used interchangeably, e.g., eigenstate and eigenvector for an eigenfunction of an operator.

3.2. Statevector; Pictures

In descriptions of coherently changing quantum-state systems, as contrasted with systems affected by randomizing thermal or stochastic interactions, all of the information is, in principle, extractable from mathematical properties of a wavefunction. For systems that can be described by N discrete quantum states, as I shall assume in this article, the information comes from a statevector Ψ(t), an element of an N-dimensional Hilbert space that has complex-valued probability amplitudes Cn(t) as coordinate components. That is, we express any statevector Ψ(t) as a construct of N fixed Hilbert-space unit vectors (basis vectors) ψn, each associated with a distinct quantum state, with time-varying coefficients Cn(t):

Ψ(t)=n=1NCn(t)  eiζn(t)ψn.

The statevector, rather than a wavefunction, becomes the repository of information about a quantum system. The ψn are often denoted, with Dirac notation, as |ψn. The statevector Ψ(t) can be regarded as an N-component column vector |Ψ(t) having N elements Cn(t)eiζn(t). Its Hermitian conjugate Ψ(t) is a row vector Ψ(t)| whose elements are Cn(t)*e+iζn(t).

Pictures. Here, as explained below, the prescribed functions ζn(t) are phases chosen for subsequent mathematical simplification of the equations governing the probability amplitudes. They define a picture: the choice ζn(t)=0 is the Schrödinger picture, the choice ζn(t)=Ent/ is the Dirac or interaction picture. The choice taken in the present work, in which the ζn(t) are related to carrier frequencies, is the rotating-wave (RW) picture. Figure 24 shows a comparison of the RW and Schrödinger pictures. The RW picture facilitates, but is not identical to, the RW approximation; cf. Subsection 3.3d.

Essential states. The restriction of the summation in Eq. (22) to a finite (relatively small) number of quantum states, N, must include all the states that, during the course of controlled, pulsed excitation, will receive appreciable population. This restriction, the essential states approximation, is made by including all those states whose energies are brought into near-resonant linkage with the initially populated state (or states), cf. Appendix G. The remaining states affect the dynamics through such secondary effects as dynamic Stark shifts, cf. Appendices D and G.2.

Probabilities. The probability Pn(t) of finding the system in state n at time t is defined to be the absolute square of the projection ψn|Ψ(t) of the statevector onto the stationary Hilbert-space coordinate associated with quantum state n:

Pn(t)=|ψn|Ψ(t)|2=|Cn(t)|2.
The requirement that the quantum-state probabilities must, at all times, sum to unity, means that the statevector has unit length (unit norm):
n=1NPn(t)=n=1N|Cn(t)|2=|Ψ(t)|2.
This equation is the mathematical statement of the assumption that at all times a suitable measurement will reveal the system to be in one of the N quantum states: the set of quantum states corresponding to the unit vectors ψn is complete for the purpose of describing the quantum changes of interest.

3.2a. Basis Vectors; RW Phases

Equation (22) offers two choices of basis vectors, differing by phases ζn(t) that are chosen for subsequent convenience: a static bare basis ψn and a RW or diabatic basis ψn(t):

Ψ(t)=n=1NCn(t)ψn(t),ψn(t)=eiζn(t)ψn.
It is in the rotating-coordinate system of ψn(t) that traditional portrayals of statevector motion have been displayed (cf. Fig. 24 for an alternative), though usually without noting explicitly the time dependence of the Hilbert-space coordinate system. Subsections 4.2 and 4.4 define two other moving coordinate systems—a dark-bright basis and an adiabatic basis—that offer other useful pictures of Hilbert-space motion. By using the ψn(t) as basis coordinates the statevector construction of Eq. (25) becomes
Ψ(t)=C(t),with[10]=ψ1(t),[01]=ψ2(t),.
The simple formula Ψ(t)=C(t) carries with it the assumption of an underlying time-dependent coordinate system ψn(t).

Spatial phases. When one deals with changes induced in the electromagnetic fields by the particles that are subject to the TDSE it is useful to incorporate spatial positions into the phases. This allows treatment of statevectors at different positions:

Ψ(r,t)=n=1NCn(r,t)  ψn(r,t),ψn(r,t)=eiζn(r,t)ψn.
Such phases are used when one deals with the coupled Maxwell–Schrödinger equation, cf. Appendix M.3.

3.2b. Expectation Values; Phase Observables

Particles in a beam can, in principle, be distinguished by their excitation-dependent size, shape, magnetic moment, and polarizability, as well as by the internal energy they can convey upon impacts. For these, as well as stationary systems, the traditional measures of excitation probability draw upon either the detection of spontaneous emission or upon the absorption of a weak, resonant probe field. More generally, quantum-system observables are expressible as expectation values Ψ(t)|M|Ψ(t) of some operator M, for example a mean-square radius or a dipole moment. These expectation values are, in turn, expressible as bilinear products of probability amplitudes and phase differences:

M(t)Ψ(t)|M|Ψ(t)=mnCm(t)*Cn(t)eiζn(t)+iζm(t)ψm|M|ψn.
The phases ζn(t) are defined, for mathematical convenience, in setting up the TDSE for probability amplitudes Cn(t); together these underlie the experimentally observable expectation values. For given choice of ζn(t) what affects the expectation value is the relative phase of paired probability amplitudes, the coherences Cm(t)*Cn(t). This phase has no effect on the probabilities Pn(t), which involve the absolute square of a single probability amplitude. Other observables, such as the dipole moment expectation values that serve as probe-field altering-elements of propagation equations (cf. Appendix M.2), are sensitive to those phase differences. Such observables provide measures of the relative signs of the Cn(t) and provide incentive for manipulating the phases of probability amplitudes.

3.3. Time-Dependent Schrödinger Equation

Changes to a statevector are governed by the TDSE, expressed in statevector form as

iddtΨ(t)=H(t)Ψ(t).
The Hamiltonian matrix H(t) appearing here [the matrix representation of a Hamiltonian operator H(p,r,t)] is the sum of two matrices:
H(t)=H0+Hint(t).
That for an undisturbed Hamiltonian H0 has the basis vectors ψn as its eigenvectors,
H0ψn=Enψn,
and its eigenvalues En are the energies observable for the undisturbed system. The time dependence occurs through an interaction energy Hint(t) that is controlled by an experimenter. It is this matrix that is responsible for controlled changes to the quantum-state structure of the system.

3.3a. Coherent Hamiltonian

The quantum-state manipulations considered in this article are “coherent,” meaning that they do not introduce irregular unpredictable changes to statevector phases. In turn, this means that the Hamiltonian can have no such “incoherent” contributions. Pulsed fields must have phases that follow a well-defined course, either by simple expressions for envelopes and carrier frequencies or by Fourier transforms of controlled frequency components. The environment may comprise an ensemble of neighbors, but these must remain static, introducing no stochastic changes.

3.3b. Coupled ODEs

The substitution of construction (22) into the TDSE of Eq. (29) produces a set of coupled ODEs of the form

iddtCn(t)=mWnm(t)Cm(t)oriddtC(t)=W(t)C(t),
where C(t) is a column vector with N components Cn(t) and W(t) is an N×N matrix representation of the Hamiltonian. The elements of W(t), dimensionally frequencies, depend explicitly on the choice of phases ζn(t): the diagonal elements (diabatic eigenvalues) are
Wnn(t)=Enddtζn(t),
while the off-diagonal elements, responsible for transitions, are, without approximation,
Wnm(t)=exp[iζm(t)+iζm(t)]  Hnmint(t).

When losses are not present, the matrix W(t) is Hermitian, meaning its transpose is equal to its complex conjugate, Wnm(t)=Wmn(t)*, and its eigenvalues are real valued. With suitable phase choices it can be made symmetric, Wnm(t)=Wmn(t), as was done in Eq. (2).

Linkage patterns. The nonzero off-diagonal elements of the matrix W(t)—the Rabi frequencies—define a linkage pattern: in the mathematics of graph theory [194196] a set of nodes (the N quantum states 1,2,) and edges [the connections m, n of Wnm(t) that are nonzero for some time t]. Applied to a chain linkage, i.e., one with nonzero elements nn±1, the W(t) can be presented as a tridiagonal matrix, having a graph in which edges connect only nearest neighbors. The diagonal elements of W(t)—the detunings—have no relevance for the linkage pattern, although they are often indicated on displays of linkage patterns; cf. Fig. 1.

Resonance. When two diagonal elements of the RWA Hamiltonian are equal and constant, say Wnn=Wmm, there will occur population transfer between the states n and m: the two states are said to be in resonance. When the interaction is radiative, as discussed in the following sections, this resonance can occur because the carrier frequency of a field matches the Bohr transition frequency, i.e., the energy of a single photon ω equals the energy difference between the two quantum states, |EnEm|. A multiphoton resonance occurs when the sum of two or more carrier frequencies matches a Bohr frequency. It does not matter which pair of states produce the energy matching Wnn=Wmm, nor do these elements need to be zero. It is the combination of resonance and nonzero linkage that produces quantum-state change. Resonance between an excited state and the initially populated state is needed for excitation to take place.

3.3c. Dipole Interaction

The interaction of visible, infrared, and radio-frequency radiation with atoms and molecules is best treated by expressing the distribution of charges and currents within the particle as a succession of multipole moments centered at the particle center of mass, r=0. The most important of these are the electric dipole moment d, interacting with the electric field E(r,t), and the magnetic-dipole moment m, interacting with the magnetic field B(r,t). The interaction energies of these two moments are

HE1(t)=d·E(0,t),HM1(t)=m·B(0,t).
Each of these interactions is proportional to the projection of a vector of the particle, d or m, upon the direction of a field. The interaction energy is a scalar, although it acts as an operator (a matrix) upon the statevector.

In much of the discipline of quantum optics, and in the use of lasers for inducing coherent quantum change at optical frequencies, the interaction parametrized by a Rabi frequency is the electric-dipole interaction. For definiteness, that will be the choice assumed in the present article: Hint(t)=HE1(t).

Typically the electric field is expressed by means of a complex-valued unit vector e (defining the field polarization direction) and a complex-valued field amplitude [197] E(t) that may vary slowly with time (i.e., changing only over many cycles of the carrier angular frequency ω):

E(0,t)=12eE(t)eiωt+12e*E(t)*eiωtReeE(t)eiωt.
A zero-frequency pulse, ω=0, corresponds to a quasi-static interaction, and is an option compatible with the mathematics that follows.

3.3d. Two-State RWA Equations

Consider the action of this near-monochromatic field upon two quantum states, having energies E1 and E2 and linked by dipole transition moments but lacking a permanent dipole moment, so that Hnnint(t)=0. The TDSE of Eq. (32) then comprises two coupled ODEs for complex-valued probability amplitudes C1(t) and C2(t),

iddt[C1(t)C2(t)]=[Δ1(t)W12(t)W21(t)Δ2(t)][C1(t)C2(t)],
in which the diagonal elements of W(t) are
Δ1(t)=E1ddtζ1(t),Δ2(t)=E2ddtζ2(t),
and the off-diagonal elements are
W12(t)=H12(t)exp[iζ2(t)+iζ1(t)]=W21(t)*.
Although there is, in principle, no difficulty in obtaining numerical solutions to these coupled ODEs for arbitrary models of time-varying interactions, in practice it is desirable to avoid situations in which there occur two or more very different time scales. Such “stiff” ODEs [104,198,199] pose challenges for numerical techniques, because they require integration over many short-time-scale intervals (e.g., optical cycles of the carrier frequency) in order to obtain slow-time-scale changes (e.g., cycles of the Rabi frequency). Inherent inaccuracies in numerical techniques may cause unphysical exponential growth of solutions. Thus it is desirable to choose the phases ζn(t) to eliminate rapid variations from the matrix W(t). An added benefit is that the slow variation of W(t) makes the probability amplitudes of the RW picture more easily depicted; see Subsection 6.5c and Fig. 24.

The RW picture. The common choice for excitation by a pulsed field having a fixed carrier frequency ω is the RW picture:

ζ2(t)=ζ1(t)±ωt+φ(t),
where the upper, plus sign is used when E2>E1 and φ(t) is a slowly varying phase still to be defined. This choice allows us to write, for arbitrary ζ1(t), the off-diagonal element of W(t) as
W12(t)=121|d·e|2E(t)ei(ω±ω)t+iφ(t)121|d·e*|2E(t)*e+i(ωω)t+iφ(t).
The two terms of this exact expression each have an exponential function of frequency. One of these is e0=1, the other, associated with the so-called counter-rotating term, is e±2iωt. We conclude the goal of obtaining a slowly varying W(t) by neglecting this rapid variation, thereby making the RWA [13,14,200,201], see Appendix C. The result is the formula
W12(t)=12{1|d·e*|2E(t)*eiφ(t)if  E2>E1,1|d·e|2E(t)eiφ(t)if  E2<E1.
It is convenient to choose the phase ζ1(t) to nullify the element W11(t) of the RWA Hamiltonian:
ζ1(t)=E1t,
a choice that is equivalent to taking E1 as the zero-point of energies. When E2>E1 this choice produces the a two-state TDSE with the RWA Hamiltonian matrix
W(t)=[012Ω(t)*12Ω(t)Δ(t)],
where the matrix elements are
W22(t)Δ(t)=(E2E1)ωφ˙(t),
W12(t)12Ω(t)*=12ψ1|d·e*|ψ2E(t)*eiφ(t).
The dipole moment factor of the Rabi frequency,
d12=ψ1|d·e*|ψ2,d21=ψ2|d·e|ψ1,
depends not only on the two quantum states but on the polarization direction of the field, here parametrized by the unit vector e; cf. Appendix K.

The phase φ(t) is available to cancel any phase change of the field envelope E(t) and give a real-valued Rabi frequency. If the phase φ(t) is taken to be constant the Rabi frequency can be made real valued at t=0. The detuning is then the difference between the Bohr transition frequency and the carrier frequency:

Δ=(E2E1)ω.
Alternatively, φ(t) can be chosen to cancel any time variation of the energies En (e.g., dynamic Stark or Zeeman shifts) or variations of the carrier frequency (e.g., chirped frequency), thereby making the detuning constant (even zero) and the Rabi frequency explicitly complex-valued and time dependent.

Beyond the RWA: Floquet theory. When the envelope E(t) is idealized as constant, the interaction Hamiltonian that underlies the RWA is periodic. A systematic approach to constructing solutions to the TDSE for periodic Hamiltonians rests upon Floquet theory [202204]. Refinement to treat slowly varying envelopes leads to adiabatic Floquet theory [205].

3.3e. Three-State RWA Equations

The conventional phase choice for the three states used for STIRAP is

ζ1(t)=E1t,
ζ2(t)=(E1+ωP)t+φP(t),
ζ3(t)=(E1+ωPωS)t+φP(t)±φS(t),
where the minus sign on ωS goes with the Lambda linkage. This phase choice facilitates the RWA, which neglects terms that vary with twice the carrier frequencies and leads to the rotating-wave representation of the RWA Hamiltonian by the matrix
W(t)=[012ΩP(t)*012ΩP(t)Δ(t)12ΩS(t)012ΩS(t)*δ(t)].

For the Lambda linkage (E2>E1 and E2>E3) the Rabi frequencies are, by analogy with Eq. (45b),

ΩP(t)*=ψ1|d·eP*|ψ2EP(t)*eiφP(t),ΩS(t)*=ψ3|d·eS*|ψ2ES(t)*eiφS(t).

The diagonal elements of this RWA Hamiltonian (the diabatic eigenvalues),

Δ(t)=ΔP(t),δ(t)=ΔP(t)ΔS(t),
contain the single-photon and two-photon detunings of Eq. (10):
ΔP(t)=E2E1ωPddtφP(t),ΔS(t)=E2E3ωSddtφS(t).
For application to STIRAP the detunings are assumed constant, meaning the phases φ(t) are constants. It is also common to assume the Rabi frequencies are real valued. (This can always be done for a specific time for at least one of the Rabi frequencies.)

3.3f. Probability Loss and Complex-Valued Detuning

Although the TDSE only describes excitation in the absence of any incoherent processes, it is easy to include the possibility of probability loss from state n at a rate Γ by making the replacement EnEniΓ/2 in the bare-state energies and consequent detunings, cf. [206208]. This involves the replacement

ΔΔiΓ/2
in the basic STIRAP Eq. (2). When the Rabi frequencies are absent this revision leads to state-2 probabilities that decay exponentially:
C2(t)=eΓt/2C2(0),P2(t)=eΓtP2(0).

As long as the excitation is adiabatic, the presence of a complex-valued detuning has no effect on the STIRAP process, because population never is found in state 2. However, as the detuning and the loss rate increase, adiabaticity deteriorates, eventually reducing the population transfer.

In practice, a part of the spontaneous emission acts to repopulate states 1 and 2. Such effects cannot be treated within the Schrödinger equation; they require a density matrix treatment [15,109111]; see Subsection 3.3h.

3.3g. Propagator

As interest in quantum-state manipulation turns away from the early demonstrations of population transfer to consider the formation and alteration of coherent superpositions the formalism uses a propagator matrix U(t2,t1) [15] such that changes to probability amplitudes in a time interval t1t2 are obtainable by multiplication as

Cn(t2)=mUnm(t2,t1)Cm(t1)orC(t2)=U(t2,t1)C(t1).
The propagator, or time-evolution operator, satisfies the TDSE and initializes as the unit matrix:
iddtU(t,t1)=W(t)U(t,t1),U(t1,t1)=1.
The propagator formalism allows treatment of multiple pulses, each with its own Hamiltonian W(e;t):
U=(tn,t1)=U(en;tn,tn1)U(e3;t3,t2)U(e2;t2,t1).
Analytic expressions for propagators exist for a number of two-state RWA Hamiltonians [209212]. The propagator is particularly useful as a numerical technique in which the RWA Hamiltonian is approximated as a succession of piecewise-constant matrices.

3.3h. Density Matrix; Steady State

The formula of Eq. (28) for an expectation value simplifies with the introduction of two matrices: the density matrix ρ(t) descriptive of the statevector components, and a static matrix M that incorporates the properties of the system. These matrices have the elements

ρnm(t)=Cm(t)*Cn(t),Mmn=ψm|M|ψn,
and allow the expectation value to be written as
M(t)=mneiζn(t)+iζm(t)ρnm(t)Mmn.
From the definition of Eq. (58) it follows that when they system undergoes purely coherent change the density matrix ρ(t) satisfies the quantum Liouville equation,
iddtρ(t)=[W(t),ρ(t)],
where [A,B]ABBA denotes the commutator of two matrices.

It should be noted that an often-used alternative definition of a density matrix,

ρ˜nm(t)=Cm(t)*Cn(t)eiζn(t)+ζm(t),
simplifies the expression of an expectation value to the trace of a matrix product [213],
M(t)=Trρ˜(t)M,
but the equation satisfied by ρ˜(t) does not have the simplification allowed by the rotating-wave approximation and its Hamiltonian W(t).

Decoherence. The Liouville equation (60) expresses the same coherent dynamics as does the TDSE. However, it allows the incorporation of incoherent influences, such as spontaneous emission, with the addition of an operator L whose elements produce changes of coherences as well as transfers of population [29,60,121,214217]:

iddtρ(t)=[W(t),ρ(t)]iLρ(t).
When the density matrix is used with this equation it is no longer possible to express it in the simple form of probability-amplitude products, as in Eq. (58): it has incorporated decoherence that is not treated with the TDSE.

Steady state. The solutions to the TDSE for excitation by steady fields (constant Rabi frequencies and detunings) oscillate indefinitely; they do not approach any fixed value. To attain a steady state it is necessary for the system to experience decoherence and relaxation, as is parametrized by the numbers of L in the density-matrix equation of Eq. (63). That equation, for constant W(t), has solutions for which, at large times, the density matrix undergoes no further change:

ddtρ(t)0for  [W,ρ(t)]=iLρ(t).
As with other linear ODEs that have constant coefficients, the solutions are obtainable by Laplace transform.

4. Insights

This section presents various views of the STIRAP dynamics, augmenting the briefer presentation of Section 2, starting with a discussion of torque equations and proceeding to displays of the statevector motion in Hilbert space and the use of the system point to follow diabatic and adiabatic time evolution.

4.1. Insight: Torque Equations; Adiabatic Following

Three-dimensional torque equations (cf. Appendix E), in which an angular velocity vector (or torque vector) ϒ(t) governs the motion of a vector R(t),

ddtR(t)=ϒ(t)×R(t),
can offer a very simple and intuitive picture of adiabatic motion, including STIRAP: when the two vectors are aligned, either parallel or antiparallel, then the torque vector ϒ(t) produces no change in the vector R(t). The two vectors will remain aligned if the torque changes adiabatically: the vector R(t) then adiabatically follows the torque vector ϒ(t). Figure 21 shows an example of motion when the two vectors R and ϒ remain parallel.

Alternatively, when the two vectors are perpendicular, then the torque vector acts to rotate R(t) about the instantaneous axis of ϒ(t), at the rate |ϒ(t)|. Such behavior appears as Rabi oscillations of two components of R. Figure 24 shows an example when the two vectors R and ϒ are at right angles. Figure 25 shows examples when the two vectors are at some other angle.

Torque equations have a long history in the classical mechanics of rotating objects. They have famously been applied to the description of two-state quantum systems (see Subsection 6.5) and to three-state quantum systems [184,218,219], using vectors R(t) in an abstract vector space formed from elements of the density matrix, i.e., bilinear products of probability amplitudes, controlled by a torque vector ϒ(t) whose elements are taken from the RWA Hamiltonian, see Subsection 4.1b.

4.1a. Resonant STIRAP as a Torque Equation

Conversion of the TDSE into a torque equation requires that the elements of the RWA Hamiltonian be organized into an angular velocity vector ϒ(t) such that the Schrödinger equation takes the form of a torque equation, see Appendix E. When the three-state TDSE (2) is fully resonant (ΔS=ΔP=δ=0) then it can be written, with some alteration of ordering, as

ddt[C3(t)iC2(t)C1(t)]=12[0ΩS(t)0ΩS(t)0ΩP(t)0ΩP(t)0][C3(t)iC2(t)C1(t)].
This matrix equation can be put into the form of a torque equation (cf. Appendix E) for a population-amplitude vector R(t) having components [15]
R1(t)=C3(t),R2(t)=iC2(t),R3(t)=C1(t)),
where the torque vector has the elements
ϒ1(t)=ΩP(t),ϒ2(t)=0,ϒ3(t)=ΩS(t).
If the vectors R(t) and ϒ(t) are parallel (or antiparallel) at some initial time, then the torque equation (66) predicts the vector R(t) will undergo no change:
ddtR(t)0.
This will happen if C2(t)=0 and the following relationships hold:
C3(t)=ΩP(t)Ωrms(t)=sinϑ(t),C1(t)=ΩS(t)Ωrms(t)=cosϑ(t),
where ϑ(t) is the mixing angle of Eq. (12) and Ωrms(t) is the root-mean-square (rms) Rabi frequency,
Ωrms(t)=ΩP(t)2+ΩS(t)2.
If the subsequent change in the torque vector is sufficiently slow (i.e., adiabatic change) then the two vectors will remain aligned, a condition known as adiabatic following (AF).

When there is no population in state 2 the conditions (70) for adiabatic following will be recognized as those responsible for the STIRAP process: population initially in state 1 acted on by the S field alone (so that both R and ϒ lie along the 1 axis), will be carried by adiabatic following to state 3 as this pulse gives way to the P field (and both R and ϒ lie along the 3 axis). The motions in Hilbert space are two dimensional in this simplification, requiring only the 1,3 plane for presentation; see Fig. 9(b). The sign associated with the final value of the state-3 probability amplitude, C3(t)1, has no effect on the probability, and is often ignored. It becomes observable, and must be considered, when the three-state system is embedded in a larger system [15].

This torque form of the fully resonant three-state Schrödinger equation provides a basis for useful analogies between STIRAP and various processes in classical physics that involve three degrees of freedom or three variables [9,10]. Any (three-dimensional) torque equation in which there is adiabatic following offers a potential analogy with STIRAP; see Subsection 8.4a.

4.1b. Coherence Vector

A series of papers by Hioe, some with colleagues Eberly and Carroll [219226] and Oreg [184], established a formalism for treating coherent N-state quantum dynamics by means of a coherence vector S(t) of dimension N21 and unit length, whose real-valued elements are obtained from elements of the N-state density matrix. The formalism recast the quantum Liouville equation (60) for the density matrix as a generalized torque equation.

Their approach can be presented as follows. Take a complete set of traceless antisymmetric N×N matrices sj, generators of the SU(N) group and characterized by commutators

[sj,sk]=2ilfjks.
Here the fjkl are the structure constants of the chosen SU(N) representation. We use these matrices to define a coherence vector S(t) and a torque vector ϒ(t), each of dimension N21, having components
Sj(t)=Trρ(t)sjϒj(t)=TrW(t)sj.
Then the unit-length coherence vector undergoes a generalized rotation described by the generalized torque equation
ddtSj(t)=klfjklϒk(t)Sl(t).
Adiabatic following occurs when the vectors S(t) and ϒ(t) are parallel or antiparallel and ϒ(t) changes sufficiently slowly.

Choices of group representations for the matrices sj, fitting the symmetry of the Hamiltonian linkage, allow simplification of the description of S(t); with proper choice of basis the evolution arising from a particular initial condition lies at all times within a subspace of the full Hilbert space [224,227231]. In particular, for N=3 there can occur motion in a two-dimensional space spanned by the two unit vectors that constitute the dark population-trapping state. The basis matrices sk are then the Pauli matrices, angular momentum matrices for j=12.

This formalism was used in [184,219226] for treating coherent N-state quantum dynamics by means of a coherence vector (here I suppress explicit time dependence)

S={u12,,v12,,w1,,wN1},
whose real-valued elements were obtained from elements of the N-state density matrix
ujk=ρjk+ρkj,vjk=i[ρjkρkj],wl=2/l(l+1)[ρ11,,ρlllρl+1,l+1],
a generalization of the variables used by Feynman, Vernon, and Hellwarth (FVH) in treating two-state behavior, cf. Subsection 6.5.

The coherence-vector formalism offers a description of STIRAP as well as various modulated-detuning mechanisms of adiabatic following. Hioe and Eberly [220] showed that, under appropriate conditions, the eight-dimensional space of the coherence vector for a three-state system can be decomposed into three independent subspaces, of dimension one, three and four, with three independent coherence vectors. The squares of the lengths of these vectors are constants of the motion. For a detailed review of these and related results of dynamic symmetries, see [98].

Although the development of the coherence vector preceded the discovery of STIRAP as an effective technique for population transfer, the papers presenting its use did not explicitly predict the original STIRAP concept, as defined in Subsection 2.3; see [9].

4.2. Insight: Trapping States; Bright and Dark States

Understanding of the STIRAP mechanism can follow from several approaches. Prior to the advent of STIRAP, the works [232,233] pointed out that a three-state chain-linked system, acted upon by two coherent fields whose carrier frequencies maintain two-photon resonance, permits a superposition of states 1 and 3 that is immune from excitation into intermediate state 2—a state that is made observable by spontaneous emission. At that time they and others were not concerned with slowly varying superpositions, such as STIRAP requires. One long-established treatment of the Lambda linkage when there is two-photon resonance δ=0 makes use of two-state superpositions that combine all of the transition strength into a so-called bright state ϕB(t) (also known as the coupled state [234]), leaving the orthogonal dark state ϕD(t) (also known as the uncoupled state [234]) immune to change, cf. Appendix D:

ϕB(t)=1Ωrms(t)[ΩP(t)ψ1(t)+ΩS(t)ψ3(t)],ϕD(t)=1Ωrms(t)[ΩS(t)ψ1(t)ΩP(t)ψ3(t)].
In this bright-dark basis, useful when the RWA Hamiltonian varies sufficiently slowly (adiabatically), the original three equations become a pair of equations for coupled amplitudes CB(t) and C2(t) that incorporate all the state-2 excitation, plus a dark-state amplitude CD(t) that remains unchanged by the interaction:
iddtCB(t)=12Ωrms(t)C2(t),
iddtC2(t)=Δ+12Ωrms(t)CB(t),
iddtCD(t)=0.
Hence the state ϕD(t) traps the population as a superposition of states 1 and 3; it is known as a trapped or trapping state [232235]. Such states were detected by their lack of fluorescence, a property that led them to be termed “dark states” [236240]. In discussing extensions of the three-state STIRAP I shall regard a “bright state” to be any state, or superposition of states, that includes an excited, fluorescing state. Any superposition of states that has no such component is a “dark state,” comprising only stable or metastable states.

Adiabatic following. To implement STIRAP one devises a pulse sequence that orients the initial statevector with the dark state, Ψ()=ϕD(), and ensures by enforcing adiabatic conditions (see Subsection 4.4c) that the alignment continues; the statevector then adiabatically follows the dark state, even as the composition of the dark state changes.

Decoherence-free subspace. The common treatment of Lambda-linkage excitation involves two stable or metastable states 1 and 3 linked to an excited state 2 that can undergo spontaneous emission. Even when those emissions lead only to a return to the two lower-energy states, their effect is to introduce randomization of the phases of the statevector components—a form of decoherence. By avoiding placement of population into this radiatively decaying excited state the dark state deals with a two-dimensional subspace of the full three-dimensional Hilbert space: its motion is in a decoherence-free subspace (DFS) [241249].

4.3. Insight: Diabatic Following

When the Rabi frequencies are negligible, the Hamiltonian produces no appreciable change in the populations Pn(t). However, there may occur phase changes of the probability amplitudes Cn(t). Consider a RWA Hamiltonian represented by a diagonal matrix:

W(t)=[Δ1(t)000Δ2(t)000Δ3(t)].
The detunings Δn(t) are the diabatic eigenvalues of this W(t). With such a Hamiltonian the TDSE becomes a set of uncoupled equations:
iddtCn(t)=Δn(t)Cn(t).
The produced probability amplitudes change only by a phase, leaving the populations unchanged:
Cn(t)=exp[i0tdtΔn(t)]Cn(0),Pn(t)=Pn(0),
and so a system that is describable as a single quantum state at time t=0 will remain in that quantum state as long as the Rabi frequencies are negligible. With more general initial conditions the statevector will be the superposition
Ψ(t)=nexp[i0tdtΔn(t)]Cn(0)  ψn(t),
and will undergo only phase alteration of the RW (or diabatic) states ψn(t) that define it—a change known as diabatic following (DF). This behavior, wherein the statevector remains fixed in the basis of RW states ψn(t), will occur so long as the Rabi frequencies are much smaller than the differences of detunings.

Diabatic curve crossings. For laser-induced changes the N diagonal elements of the RWA Hamiltonian of Eq. (79) represent detunings of carrier frequencies from Bohr frequencies. These are all independent parameters that can be positive, negative, or zero. Thus there is no inhibition that prevents the individual curves—the diabatic eigenvalues—from crossing.

4.4. Insight: Adiabatic States

The simplicity of the mathematical description of diabatic following, which occurs when the RWA Hamiltonian is dominated by its diagonal elements, can be adapted to a multistate RWA Hamiltonian matrix W(t) by introducing instantaneous eigenvectors of that matrix that satisfy the eigenvalue equation

W(t)ϕν(t)=ϵν(t)ϕν(t).
These are variously known as dressed states [250] [by contrast with the bare states associated with basis vectors ψn or the phased RW states ψn(t)] or as adiabatic states (implying that their structure changes only slowly with time). When two eigenvalues are degenerate, then any orthogonal, normalized superposition of the two eigenvectors can be used as a basis. Alternative Dirac notation for these eigenstates includes |aν, |gν and, most simply, |ν.

Equation (83) does not provide a complete definition of the eigenvectors: it is necessary to specify, in addition, not only their normalization,

ϕν(t)|ϕν(t)=δνν,
but also to set the phase at some reference time. That is, the vectors ϕν(t) and ϕν(t) equally satisfy Eqs. (83) and (84). When one has an algebraic expression for the eigenvectors, as occurs for two-state systems and three-state systems (cf. [251]), then these can provide the needed complete definition. If one relies on numerical procedures, such as are provided in algorithms by Mathematica and MATLAB, then the phase remains an adjustable option for the user.

Adiabatic labels and curve noncrossings. The labeling of adiabatic eigenvectors also remains at the discretion of the user. In situations when algebraic expressions are used it is possible to assign labels such as |+ and | to specific expressions valid for all time, independent of the instantaneous ordering of their eigenvalues. When constructions rely on numerical algorithms then one must accept the numerical ordering when assigning labels. That is what I do in the present article: by definition |+ has an eigenvalue larger than (or equal to) that of |. With this numerical basis for defining adiabatic eigenvectors plots of their eigenvalues can intersect but can never cross. Alternative definitions involving algebraic expressions for eigenvectors (e.g., [251]) in which the assignment of labels depends on continuity of eigenvalues, can allow crossing.

Adiabatic following. We can use the N adiabatic eigenvectors ϕν(t) as an alternative Hilbert-space coordinate system to the N unit vectors ψn or ψn(t), writing the vector of probability amplitudes as

C(t)=νAν(t)ϕν(t).
The coordinate coefficients Aν(t) of this vector satisfy the equations
iddtAν(t)=ϵν(t)Aν(t)iνGνν(t)Aν(t),
where the nonadiabatic coupling terms Gνν(t) are obtained from time derivatives of the eigenvectors:
Gνν(t)=ϕν(t)|ddtϕν(t).
When these terms are negligible the time-induced alteration of the adiabatic coefficients is merely a phase change:
Aν(t)=exp[i0tdtϵν(t)]Aν(0).
Thus if, under these conditions, the system is initially in a single adiabatic state, it will remain in that state, undergoing only a change of phase. The statevector will adiabatically follow an adiabatic state. When adiabatic following produces a complete transfer of population it is termed adiabatic passage (AP).

The usefulness of adiabatic states is that for simple systems, their relationship to the RW unit vectors ψn(t) is readily expressed with simple formulas. Whenever adiabatic following occurs a construction of adiabatic states provides a construction of the statevector.

4.4a. Two-State Adiabatic Basis

Control of a coherent two-state system follows from the two-dimensional RWA Hamiltonian matrix, which can always be written as

W(t)=[012Ω(t)eiφ12Ω(t)eiφΔ(t)],
and involves a controllable Rabi frequency Ω(t) and a controllable detuning Δ(t) as well as a possible phase φ(t). In principle, the Rabi frequency may vary arbitrarily with time, through positive and negative values, and the detuning may typically sweep between large values, positive or negative (a chirp). The two diabatic eigenvalues of this W(t) are zero (for state 1) and Δ(t) (for state 2).

Eigenvectors. Simple algebraic expressions are available for the eigenvalues and eigenvectors of a two-state RWA Hamiltonian. These apply at any given time, and so they provide formulas for constructing the adiabatic states: it is only necessary to interpret the Rabi frequency and detuning as dependent on the time t and to include that argument when needed. The eigenvalues of this matrix are

ϵ±(t)=12[Δ(t)±ϒ(t)],ϒ(t)Δ(t)2+|Ω(t)|2,
and the two adiabatic states obey the equation
W(t)ϕ±(t)=ϵ±(t)ϕ±(t).
The two eigenvectors can be expressed in trigonometric form as
ϕ+(t)=sin(θ/2)ψ1(t)+cos(θ/2)eiφψ2(t),ϵ+(t)=ϒ(t)cos2(θ/2),
ϕ(t)=cos(θ/2)ψ1(t)sin(θ/2)eiφψ2(t),ϵ(t)=ϒ(t)sin2(θ/2),
where the two-state mixing angle θ(t) is defined by the relationships
cosθ(t)=Δ(t)/ϒ(t),sinθ(t)=Ω(t)/ϒ(t).
The inverse of relationships (92) provides the connections
ψ1(t)=sin(θ/2)ϕ+(t)+cos(θ/2)ϕ(t),
ψ2(t)=cos(θ/2)ϕ+(t)sin(θ/2)ϕ(t),
that allow one, for any time t, to express the RW diabatic states ψn(t) in terms of adiabatic states ϕ±(t) and, in particular, to express initial and final states in an adiabatic basis. The time dependence of the adiabatic eigenvectors has two sources: that of the RW unit vectors ψn(t) and that from the variation of the RWA Hamiltonian elements in this rotating frame.

4.4b. Adiabatic States for STIRAP

Although the dark state alone is sufficient to describe the STIRAP process, with bright state and excited state as the two additional Hilbert-space coordinates, the required adiabatic change can also be depicted by introducing the three instantaneous eigenvectors ϕν(t) of the three-state RWA Hamiltonian matrix of Eq. (49). When the two-photon resonance condition δ=0 is fulfilled the eigenvalues of W(t) are [252], in order of increasing value,

ϵ(t)=12[Δ(t)Δ(t)2+Ωrms(t)2]=12Ωrms(t)tanφ(t),
ϵ0(t)=0,
ϵ+(t)=12[Δ(t)+Δ(t)2+Ωrms(t)2]=12Ωrms(t)cotφ(t),
where
tan2φ(t)=Ωrms(t)/Δ(t),Ωrms(t)=ΩP(t)2+ΩS(t)2.
The corresponding adiabatic eigenvectors are expressible in terms of bright and dark states of Eq. (77) as
ϕ(t)=ϕB(t)cosφ(t)ψ2(t)sinφ(t),
ϕ0(t)=ϕD(t),
ϕ+(t)=ϕB(t)sinφ(t)+ψ2(t)cosφ(t).
That is, the adiabatic state ϕ0(t) is the dark state of Subsection 4.2 and the two other adiabatic states are superpositions of the bright state ϕB(t) and the excited state ψ2(t). Expressed in terms of RW states these expressions read [4,7]
[ϕ(t)ϕ0(t)ϕ+(t)]=[sinϑ(t)cosφ(t)sinφ(t)cosϑ(t)cosφ(t)cosϑ(t)0sinϑ(t)sinϑ(t)sinφ(t)cosφ(t)cosϑ(t)sinφ(t)][ψ1(t)ψ2(t)ψ3(t)],
where the time-dependent three-state mixing angles ϑ(t) and φ(t) are defined by Eqs. (12) and (96). The adiabatic state ϕ0(t) is particularly significant for STIRAP because, in the Lambda linkage, it has no component of the (possibly lossy) excited state 2: it is a population-trapping state [233,235,240,253] as well as a dark state [4,237]. The other two adiabatic states each have a component of state 2 and so they are termed “bright” states.

4.4c. Adiabatic Condition

Numerous articles and texts discuss the conditions needed for adiabatic evolution (and adiabatic following). The following discussion is taken from [9]. The adiabatic conditions can be found by expressing the original probability amplitudes in an adiabatic basis:

C(t)=ν=+,0,Aν(t)ϕν(t).
The resulting three equations, with assumed two-photon resonance δ=0, read
iddtA(t)=Wad(t)A(t),
where the three-state adiabatic RWA Hamiltonian is [7]
Wad(t)=[12Ωrms(t)cosφ(t)iddtϑ(t)sinφ(t)iddtφ(t)iddtϑ(t)sinφ(t)0iddtϑ(t)cosφ(t)iddtφ(t)iddtϑ(t)cosφ(t)12Ωrms(t)tanφ(t)].
The condition for adiabatic following is that the matrix Wad(t) be close to diagonal, meaning that the nonadiabatic off-diagonal elements, here involving φ(t) and time derivatives of ϑ(t) and φ(t), should be much smaller than the separation of the diagonal elements, i.e., the adiabatic eigenvalues. In the limit of vanishing off-diagonal elements there occurs “perfect adiabatic evolution” (an idealization akin to the frictionless plane), with Aν(t) changing only by a phase.

The nonadiabatic coupling term ddtφ(t) vanishes when Δ=0 (the eigenvalue separation is then Ωrms) but it couples A+(t) and A(t) when there is nonzero single-photon detuning. Thus in the limit of large single-photon detuning this adiabatic basis is inappropriate.

When pulses rather than steady fields are used, the appropriate approximation to the probability amplitudes used for adiabatic following are, as in Eq. (70),

C1(t)=cosϑ(t),C3(t)=sinϑ(t),
but with the following correction [7]:
C2(t)=2iΩrms(t)ddtϑ(t).
Only in the limit of very slow change in mixing angle ϑ is there no population in the intermedia state 2.

The condition for adiabatic evolution in STIRAP has been expressed in three ways, which I shall call: “local,” “worst-case,” and “global”; see [7,12].

Local condition. The first of these adiabatic conditions comes from the requirement that change of the RWA Hamiltonian W(t) be very slow. For the STIRAP system this becomes a requirement that the change in mixing angle ϑ(t) be much less than the eigenvalue separation, which for fully resonant excitation (Δ=0,δ=0) is the rms Rabi frequency:

|Ωrms(t)||ddtϑ(t)|.

This is a “local condition” for adiabatic evolution, and it must be satisfied at all times. The pulses must therefore be smooth, with no rapid variations. For fully resonant excitation the adiabatic condition translates into a condition on changes of the Rabi frequencies:

|Ωrms(t)|3|ΩP(t)ddtΩS(t)ΩS(t)ddtΩP(t)|.

When either field is very small, there can be abrupt changes in the other field without violating this condition. This observation justifies the use of “shark-fin” pulses, cf. Appendix F.1 and [9,15,254257] (see also [184]), when analyzing the STIRAP process.

Worst-case condition. The early papers on STIRAP [3,4] dealt with Gaussian pulses of equal peak Rabi frequency and Gaussian width T. They examined the requirement that at the midpoint of the pulse sequence, t=0, where the adiabatic condition was most restrictive, the evolution would satisfy the local condition (103). They found, for optimally delayed pulses, an inequality that can be written

Ωrms(0)T1.
This is a “worst-case condition” of Eq. (103). It was expected to be applicable not only for Gaussian pulses but for other optimally delayed smooth shapes.

Global condition. A “global condition” on the Rabi frequencies is obtainable by integrating Eq. (103). The integral of the mixing angle from its initial value of 0 (dominant S field) to its final value of π/2 (dominant P field) is, for any form of intermediate time variation, π/2. The integral of a Rabi frequency Ωn(t) is a (temporal) pulse area An. Thus by integrating Eq. (103) for a positive Rabi frequency we obtain the global condition that the rms pulse area be much larger than π/2:

Armsπ/2.
This inequality has often been replaced by the statement that the individual pulse areas must be larger than 10 (or 3π), see [6,8]. Such a numerically specific inequality carries with it an implicit assumption about the permissible error allowable with the process; see Subsection 5.8: generally the larger the pulse area the smaller will be any measure of error.

4.5. Insight: Statevector Rotation

The variation in the Hilbert-space structure of ϕ0(t) during the course of a STIRAP process is most readily tracked by means of the mixing angle ϑ(t). By controlling the P and S fields an experimenter rotates the mixing angle from 0 to |π/2| (all modulo π), thereby shifting the dominant component of ϕ0(t) from state 1 to state 3. Figure 9 shows the Hilbert-space motion of the vectors ϕν(t) associated with STIRAP. [Here, and elsewhere, the coordinate vectors are RW unit vectors ψn(t)]. When the statevector Ψ(t) remains aligned with ϕ0(t), following its Hilbert space motion [an example of adiabatic following] the population transfers from state 1 to state 3. Note that with the alternate coordinate system of Subsection 4.2 the two vectors ϕB(t) and ϕD(t) rotate together in the 1,3 plane, while the third, independent unit vector ψ2(t) remains fixed along the 2 axis.

4.6. Insight: Adiabatic and Diabatic Following; AF and DF

Two limiting types of statevector motion in Hilbert space allow particularly simple descriptions. These are (a) adiabatic, when the elements of the RWA Hamiltonian change very slowly, and (b) diabatic, when those elements change rapidly. Plots of adiabatic and diabatic eigenvalues are often used as guides to the behavior of the system when changes fit one of these two limits. The following subsections present examples.

4.6a. System Point

Under appropriate circumstances the statevector may be aligned initially with a single bare state (a single diabatic state) or a with a single dressed state (a single adiabatic state). Under those circumstances one can place onto one of the energy-eigenvalue curves a system point that denotes this alignment of the statevector. As time increases the statevector changes. If it remains aligned in one of the two coordinate systems [i.e., either RW, diabatic states ψn(t) or adiabatic states ϕν(t) as reference coordinates], then the system point will remain attached to a definite curve of the unchanging coordinates. At very late times the location of the system point will identify the final state. When the statevector remains aligned in the RW diabatic basis the motion of the system point is termed diabatic following. There is then no transfer of population between bare states. Alternatively, when the statevector remains aligned in the adiabatic (dressed) basis the system point follows an adiabatic curve; the statevector undergoes adiabatic following, and there will often occur a transfer of population between bare states. Which type of following occurs, if any, depends on the rate with which the elements of the matrix W(t) change. One must understand that at any crossing of diabatic-eigenvalue curves or avoided crossings of adiabatic-eigenvalue curves the population divides into two channels: only in limiting cases will a single channel dominate, to produce DF or AF.

4.6b. Two-State Adiabatic Following; Diabatic Curve Crossing

For a two-state system the adiabatic states are particularly useful when the RWA Hamiltonian includes time-varying detuning that sweeps between large negative to large positive values, or vice versa. Such variable detunings, discussed in Appendix H, fall outside the definition of STIRAP presented in Subsection 2.3, but they have had widespread interest as creators of adiabatic following related to or inspired by STIRAP, see Sections 6 and 8. Figure 10 illustrates dynamics of two examples of RWA Hamiltonians that involve variable detuning.

 

Figure 10 Examples of two-state eigenvalues for a system that has time-varying detuning. Left: for swept frequency, W(t) from Eq. (107). Right: for pulsed two-photon interaction and dynamic Stark shifts, W(t) from Eq. (108). This is an example of the SARP process. (a) Elements of the RWA Hamiltonian W(t): on left, pulsed Rabi frequency and swept carrier frequency; on right, two Rabi frequencies. (b) Diabatic eigenvalues 0 and Δ(t). The system point (thick arrow) follows the diabatic curve (a straight line) of state 1 when the changes are fast. (c) Eigenvalues for adiabatic states | and |+. The system point (thick arrow) follows the curve of adiabatic state |+ when the changes are slow.

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Figure 11 Examples of three-state two-pulse adiabatic passage with two-photon resonance and single-photon detuning. Left: the S pulse precedes and overlaps the P pulse (counterintuitive ordering). Population is transferred from state 1 to state 3, with brief nonadiabatic presence of state 2. Right: the P pulse precedes and overlaps the S pulse (intuitive ordering). Appreciable population resides transiently in state 2 before passing to state 3. (a) Relative Rabi frequencies. (b) Adiabatic eigenvalues |+,|0 and | and diabatic eigenvalues 1, 2, and 3. Thick arrow shows course of system point. State 2 is detuned positively, states 1 and 3 are degenerate, with zero eigenvalues. (c) Adiabatic components of statevector. Here with counterintuitive ordering of the pulses the statevector remains aligned with |0, whereas with intuitive ordering the statevector is aligned with |. (d) Populations Pn(t). With the counterintuitive pulse ordering there occurs complete population transfer from state 1 to state 3, with negligible population in state 2. With the intuitive ordering there also occurs complete population transfer, but with appreciable intermediate population in state 2. These simulations use identical Gaussian pulses, of areas 20π, and detunings Δ=0.1Ωmax, δ=0. Similar figures appear in Fig. 3 of [251] and Fig. 1.5 of [9].

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Figure 12 Examples of three-state two-pulse population transfer with nonzero single-photon and two-photon detunings. The detunings (the diabatic eigenvalues) are Δ1=0, Δ2=1, Δ3=2. Left: counterintuitive pulse ordering. Right: intuitive pulse ordering. The frames are as in Fig. 11. (a) Relative Rabi frequencies. (b) Adiabatic and diabatic eigenvalues. There are two intersections of adiabatic curves, labeled A and B. (c) Adiabatic components of the statevector. Here with counterintuitive ordering the statevector undergoes a realignment from | to |0 at the intersection A, and a second realignment, from |0 to |+ at intersection B. With intuitive ordering there is a realignment only at crossing B, where it changes from | to |0. (d) Populations Pn(t). With the counterintuitive pulse sequence there occurs complete population transfer from state 1 to state 3, with negligible intermediate population. With the intuitive sequence there occurs complete population transfer from state 1 to state 2, with intermediate population in state 3. Similar figures appear as Fig. 2 of [262], Fig. 3 of [251], and Fig. 9 of [12].

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Swept detuning. The left-side frames of Fig. 10 illustrate the most common example of swept (chirped) detuning: a detuning that varies linearly with time and that passes through zero at the moment (t=0) of maximum pulsed Rabi frequency, as described by a two-state RWA Hamiltonian having the elements (see Appendix H)

W11(t)=0,W22(t)=ω21ωrt,W12(t)=12Ω(t).
When changes to the RWA Hamiltonian occur rapidly it is frame (b) that provides instruction: the system point follows the null diabatic eigenvalue associated with state 1 and no population transfer occurs. When changes are slower, instruction comes from frame (c). As the detuning Δ(t) sweeps from very large negative to very large positive values the two-state mixing angle of Eq. (93) rises from 0 to π, and with this the adiabatic state ϕ+(t) varies from alignment with state ψ1(t) to alignment with ψ2(t). Thus such a swept detuning, if adiabatic, will transfer of population from state 1 to state 2. This population inversion occurs whether the detuning changes positively or negatively; the only requirement on the Rabi frequency and detuning is that their changes be adiabatic.

Two-photon interaction with dynamic Stark shifts. The swept detuning model has been the traditional approach to adiabatic passage in two-state systems, but the essential physics also occurs with the effective two-state Hamiltonian used when there is large single-photon detuning of a three-state chain, cf. Appendix G.1. A simple RWA Hamiltonian that embodies this physics—a two-photon Rabi frequency and dynamic Stark shifts, as in SARP—has elements of the form (here S(t) and P(t) are just scaled pulse values)

W11(t)=0,W22(t)=ω21ωS(t)+P(t),W12(t)=W21(t)=S(t)P(t).
In this model what matters is a finite interval when the effective detuning sweeps monotonically through zero—a crossing of diabatic curves at t=0. The right side of Fig. 10 illustrates the motion of the system point: for fast, diabatic change frame (b) shows there will be no population transfer. For slow, adiabatic change there will be population transfer as the system point follows adiabatic state |+. With such model RWA Hamiltonians population changes can occur when there is a crossing of the diabatic curves. At such points there is an avoided crossing of the adiabatic curves [259], and if there is adiabatic following the system point will have changed its diabatic-curve attachment: a LZMS transition [156160] will have occurred, see Subsection 4.6b. Although such curve crossings are the most common characteristic of adiabatic passage, in principle it is possible, with pulsed detuning variations, to achieve population transfer without such crossings [260].

4.6c. Three-State Following; AF and DF

The two limiting cases of diabatic and adiabatic evolution underlie the dynamics of traditional STIRAP and of related quantum-state manipulation techniques. The next paragraphs discuss these cases.

Diabatic following. When changes of the RWA Hamiltonian occur very rapidly (diabatic motion) the statevector remains aligned with a single RW basis-statevector ψn(t) (or a fixed superposition of them). These states are associated with eigenvectors of an undisturbed RWA Hamiltonian W0 formed from the diagonal elements of the full RWA Hamiltonian W(t) of Eq. (49). The element W110(t) we have taken to be zero; the other elements of W0, the diabatic eigenvalues, are the detunings ΔΔP and δΔSΔP. Those numbers, times , are diabatic energies, associated with the RW (diabatic) states ψn(t), and they will change with time only if the detunings change, as happens if there are dynamic Stark shifts or changes in the carrier frequencies of the two fields. The diabatic coordinates are

Diabatic state:ψ1(t)ψ2(t)ψ3(t)Diabatic eigenvalue:0Δ(t)δ(t).
The ordering of these states on an energy scale depends on the relative values of the detunings, which may be positive, negative, or zero. There is no other ordering of these values, and the curves may cross. For STIRAP the requirement of δ=0 gives a twofold degeneracy of the diabatic eigenvalues. When the three-state system is fully resonant, with Δ=δ=0, there is a threefold degeneracy in the absence of Rabi frequencies.

Adiabatic following. When changes of the RWA Hamiltonian occur sufficiently slowly (adiabatic motion) the statevector remains aligned with an adiabatic eigenvector ϕν(t), changing only with an adiabatic phase factor. This situation is of particular interest for understanding STIRAP. Starting from the initial alignment of the statevector with the adiabatic state ϕ0(t) we track the continuing change of the statevector (and its associated populations) as the system point moves along the curve ϵ0(t), an example of adiabatic following. The adiabatic coordinates are

Adiabatic state:ϕ(t)ϕ0(t)ϕ+(t)Adiabatic eigenvalue:ϵ(t)ϵ0(t)ϵ+(t).
By construction, the ordering of these eigenvalues is always ϵ, ϵ0, ϵ+. A degeneracy (a meeting of curves) can occur only when Ωrms(t)=0.

4.6d. Two-Photon Resonance; STIRAP and B-STIRAP

Tracking of the system-point motion on a plot of eigenvalues becomes particularly useful as a tool for understanding the consequences of detuning. Figure 11 shows two simulations of situations in which there is static, single-photon detuning Δ=0 and two-photon resonance, δ=0. Initially the system is in state 1, which has zero as its diabatic energy (the 1,1 element of the RWA Hamiltonian). In these simulations the third diabatic eigenvalue is set to zero (Δ3=Δ1=0) so diabatic states 1 and 3 are degenerate: there is a two-photon resonance between these. Diabatic state 2 is offset from these; there is no single-photon resonance. Frame (b) shows the constant diabatic eigenvalues as thin, horizontal gray lines: δ=0 and a positive Δ. The adiabatic eigenvalues ϵν(t) appear as thicker, colored lines. Initially and finally these are degenerate with specific diabatic eigenvalues. As time progresses the degenerate adiabatic curves separate. The initial ratio of Rabi frequencies, parametrized by the mixing angle, determines which adiabatic state, |0 or | in this example, is associated with the initial state 1. The following paragraphs discuss these two possibilities.

Counterintuitive pulse sequence: STIRAP. The frames on the left-hand side of Fig. 11 show results for a counterintuitive SP pulse sequence, in which the first pulse to act, the S pulse, has no direct link with the initially populated state. This is the pulse sequence of traditional STIRAP. Adiabatic state |0 has initial alignment with state 1, and hence it is to this adiabatic state that adiabatic evolution will tie the statevector. At no time does |0 have any component of the excited state 2, and so there is no opportunity for the system to undergo spontaneous emission from this state—a process that would be visible as fluorescence: this adiabatic state is a dark state. At the conclusion of the pulse sequence the system point remains associated with |0 but now this adiabatic state has become target state 3 and so the population has undergone transfer.

The success of the counterintuitive sequence, used by traditional STIRAP, imposes no requirement on single-photon detuning. As in other adiabatic processes, it takes place for any pulse shape, so long as the pulse areas are sufficiently large that there occurs adiabatic following.

Intuitive pulse sequence: B-STIRAP. The frames on the right-hand side of Fig. 11 show results for an intuitively ordered, PS pulse sequence. Under these conditions the statevector is initially aligned with adiabatic state |, and if the changes to pulse envelopes are sufficiently slow, as they are for this simulation, the statevector retains that alignment during the course of the pulse sequence. As the two pulses vary, and with them the mixing angle, the adiabatic state | temporarily acquires a component of the excited state 2, before ultimately becoming aligned with diabatic state 3. Because fluorescence may occur from the excited state, the adiabatic state | is termed a bright state. Population transfer that relies on adiabatic following of a bright state, as in this example, is sometimes known as B-STIRAP; see Subsection 6.2. Unlike STIRAP, which proceeds most effectively when there is single-photon resonance, B-STIRAP requires appreciable single-photon detuning. However, it has the property that it enables the use of a single pulse sequence for both the 13 population transfer and the return 31 transfer.

As the static single-photon detuning Δ becomes very large, adiabatic elimination of state 2 becomes useful, leading to a two-state effective RWA Hamiltonian as discussed in Appendix G.1. In this model the effective Rabi frequency is proportional to the instantaneous product ΩP(t)ΩS(t) and the time-varying effective detuning is proportional to the difference of the two pulses, ΩS(t)ΩP(t). Figure 10 shows an example of such a situation, an example of SARP. Both pulse sequences, SP and PS, are then capable of producing adiabatic following and consequent adiabatic passage, although the state-2 populations differ. The article [261] has a long discussion of pulse ordering with detuning.

4.6e. Population Transfer Despite Two-Photon Detuning

Although two-photon resonance is needed for the success of traditional STIRAP, population transfer can proceed successfully when there is nonzero two-photon detuning. General discussion of three-state adiabatic following with two pulses and static detunings will be found in [65]. The following paragraphs examine specific examples. Figure 12 shows results in which the two detunings Δ=ΔP and δ=ΔPΔS are both positive. These constant values, the diabatic eigenvalues, are the light-gray horizontal lines. Colored curves show the time-varying adiabatic eigenvalues. Arrows mark the course of the system point with time.

In Fig. 12 the third diabatic eigenvalue is Δ3=2, and there is no degeneracy of the diabatic states: there is neither a one-photon resonance nor a two-photon resonance. Because the diabatic eigenvalues are constant, there are no crossings of the diabatic curves. However, there occur two intersections of the adiabatic curves, labeled A and B, and at each of these the statevector may undergo a realignment that corresponds to DF. The remainder of the history involves AF.

Initially the system is in state 1, which has zero as its diabatic energy (the 1,1 element of the RWA Hamiltonian). With the present choice of detunings this is the adiabatic state associated with the smallest of the three adiabatic eigenvalues, ϵ(t)=0, and so the initial conditions are Ψ(t)=ψ1(t)=Ψ(t). The system point initially follows the track of the eigenvalue ϵ(t) by AF. Its subsequent behavior depends on the pulse ordering.

Counterintuitive pulse ordering. With counterintuitive pulse ordering, shown at the left of Fig. 12, the system point follows the track of |0 until the time marked A in the figure. At this moment there occurs a near-meeting of two adiabatic-eigenvalue curves, ϵ(t)ϵ0(t)0. In the simulation behind this figure the system follows a diabatic path through this avoided crossing, after which the statevector becomes aligned with the adiabatic state |0. It adiabatically follows this dark state until the time marked B, when there occurs a near-meeting of adiabatic curves ϵ0(t)ϵ+(t)Δ3. The statevector proceeds by DF through this brief interval, becoming aligned with |+. After these two DF interruptions the statevector continues by AF to a final alignment with diabatic state 3.

The population histories here are indistinguishable from population transfers associated with traditional STIRAP, which requires two-photon resonance and proceeds without any diabatic following. However, the result of the pulse sequence shown here, though using counterintuitive pulse ordering to produce completed population transfer with negligible population in state 2, has two intervals of nonadiabatic time evolution, marked A and B, and so the procedure, though robust, is not the traditional STIRAP defined in Subsection 2.3.

Intuitive pulse ordering. With intuitive pulse ordering, shown at the right of Fig. 12, the system point again follows the track of ϵ(t) initially. It remains so aligned, unaffected by the near-meetings at time A, until the near-meeting of eigenvalues ϵ(t)ϵ+(t)Δ3 at time B. There then occurs DF and the statevector becomes aligned with adiabatic state |0. This superposition evolves into state 2 upon completion of the pulse sequence. The final state, and most of the history, differs qualitatively from that of STIRAP.

4.7. Insight: The Topology of Adiabatic Following

Rather than follow the system point on the curve of ϵ0(t) as the single variable t changes, one can regard the two Rabi frequencies as variables that define a system-point surface ϵ0[ΩP,ΩS] in a two-dimensional parameter space. One is led to construct displays of 3D surfaces rather than 2D curves as time causes the two Rabi frequencies to change. Such plots of eigenvalue surfaces (or quasi-energy surfaces), with their attendant topology, have been presented and described by [65,173,263] as guides to visualizing the population transfer based on adiabatic following and its concluding stage of adiabatic passage.

Figure 13 shows one of the examples discussed by [173] as pertinent to Lambda or ladder linkages. The uppermost quasi-energy surface is, by definition, that of ϵ+, while the lowermost surface is that of ϵ. For this example the two-photon detuning δ is nonzero, and so the eigenvalues are not obtained from the simple formulas shown earlier. In particular, ϵ0 is not always zero, as it is when δ=0. The places where two surfaces meet are known as conical intersections, and the behavior of the system point nearby is determined by whether the evolution is adiabatic, diabatic, or neither.

 

Figure 13 Surfaces of eigenenergies (in units of δ) as functions of ΩP/δ and ΩS/δ. The paths (a) and (b) constructed with delayed pulses of the same length and peak amplitude correspond, respectively, to the intuitive and counterintuitive pulse sequences in Lambda or ladder systems for which the initial population resides in state 1. The surfaces have been shifted by δ/2. Figure 3 reprinted with permission from [173]. Copyright 2002 by the American Physical Society. See also [65].

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The three quasi-energy surfaces of eigenvalues are separated, along the vertical axis by the detunings associated with the three diabatic energies. Along the vertical line where ΩP=ΩS=0 the diabatic energy for state 2 is ΔP<0 while for state 3 it is δ>0. (The labels on the vertical axis of Fig. 13 are shifted by δ/2.)

Initially, when ΩP=ΩS=0, the system point is on a corner of the ϵ0 surface. In path (a) the P field is the first to occur (the intuitive pulse sequence), while for path (b) the S field occurs first (the counterintuitive sequence). With either path the system point encounters a conical intersection where its subsequent path depends on whether the evolution there is adiabatic (in which case the system point remains on the lower of the two intersecting surfaces) or is diabatic (in which case the system point continues on its trajectory and changes surfaces). The choice will determine the final destination of the system point, either to state 3 by diabatic change or returning to state 1 by adiabatic change. With the intuitively ordered path (a) the conical intersection is encountered before the S field is present, as the P pulse is increasing. With the counterintuitively ordered path (b) the intersection occurs after the S field has completed its pulse, and the P pulse is decreasing. With either path, i.e., either pulse sequence, it is possible to obtain complete population transfer by allowing a diabatic change—a contrast to traditional STIRAP, which requires fully adiabatic change but also requires δ=0. The population transfer shown here is not STIRAP as defined in Subsection 2.3. As such displays make clear, the result of a pulse sequence does not depend on details of the individual pulse shapes; only in the region of the conical intersection (in this example occurring when the S pulse is absent) is the distinction between adiabatic and diabatic change particularly critical.

4.8. Insight: Excitation Stages

Because the STIRAP process relies on adiabatic time evolution it is relatively insensitive to details of the two pulses. It makes no requirement upon pulse shape (cf. Appendix F) nor upon precise value of the delay between pulse maxima, nor even upon the peak values of the two pulses, although such details do have a measurable effect, see Subsection 5.8.

When the pulses are idealized as having finite support (cf. Appendix F), as befits excitation of stationary systems by pulsed laser fields, one can regard the overall STIRAP process as occurring in three stages [264], shown schematically in Fig. 14. In the first stage, here termed Prior, only the S interaction is present, and so the mixing angle remains fixed at ϑ=0. In the final stage, here termed Post, there is only the P interaction and so again the mixing angle is constant, now at |ϑ|=π/2 for complete population transfer. At intermediate times, when both interactions are present, the statevector Ψ(t) must undergo adiabatic following of the dark state ϕD(t); motion of the other two complementary Hilbert-space coordinate vectors, either ϕB(t) and ψ2(t) or else ϕ+(t) and ϕ(t), does not then affect the statevector.

 

Figure 14 Three-stage description of the STIRAP process driven by pulses of finite support: Prior (P absent), Interaction (both S and P), and Post (S absent). The success of adiabatic passage does not depend on details of the pulse shapes in the Interaction interval, so long as the mixing angle varies adiabatically between the two required limits. The S pulse in the Prior interval, and the P pulse in the Post interval, can be idealized by any convenient time dependence, including zero [15,255,256]. Figure 1.9 reprinted with permission from [9]. See also [15,255,256].

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In both the Prior and Post segments of the process the single, acting interaction may have arbitrary time variation (it is only necessary that this field have no effect on the state with which it is unconnected by the linkage pattern). Indeed, it is possible to deal only with the interaction interval, and replace the Post and Prior intervals with segments in which the two interactions are set to zero. These then are idealized as “shark-fin” pulses, that have an abrupt start or finish; cf. Appendix F.1 and [9,15,254257].

5. Ensembles and Averages

The TDSE applies to individual quantum systems, e.g., single isolated quantum particle. Inevitably any experiments upon field-excited atoms deal with ensembles, meaning samples of identically prepared atoms and fields that differ in some uncontrolled respect. When we deal with coherent excitation and the TDSE each distinct sample must be treated separately—unlike incoherent excitation whose rate equations deal with averaged attributes [1519].

Within a few years of the original demonstrations of STIRAP with molecular beams researchers extended the technique of counterintuitive pulse pairs to achieve population transfer in bulk matter [155], cf. Appendix M.2. They also dealt with hyperfine structure in molecules [155]; see Appendix L.1. Treatment of ensembles and degeneracies, discussed in the following subsections, is essential for such applications.

5.1. Degeneracy Treatment

Observations of STIRAP have often dealt with unresolved transitions between magnetic sublevels of three degenerate energy levels [265]. Because the Rabi frequencies depend on the relative orientation of the dipole transition moment and the electric field vector (i.e., the field polarization direction) each sublevel transition has a different Rabi frequency, with a value dependent on the two magnetic quantum numbers, see Appendix K. Unlike the rate equations that describe the incoherent absorption, stimulated emission and spontaneous emission of radiation, the TDSE does not deal with sets of states as a single unit; it deals with individual quantum-state probability amplitudes. To treat degenerate states, for example an initial situation in which there is angular momentum degeneracy (i.e., quantum states are Zeeman sublevels), it is necessary to include each quantum state individually and explicitly in the description: a degenerate level having angular momentum quantum number J has 2J+1 distinguishable sublevels and hence it requires 2J+1 quantum states for its description; see Appendix K.

With allowance for angular momentum the linkage pattern of the Hamiltonian matrix depends on the polarizations of the P and S fields [13,15,266]; see Appendix K. When the field polarizations are chosen appropriately, the linkage pattern of a multistate system may be reducible to a set of independent three-state and two-state chains (and possibly unlinked dark states or spectator states [267]). Figure 15 illustrates the linkage complication introduced by angular momentum degeneracy. Frame (a) is the depiction used for treating incoherent excitation by rate equations or coherent excitation of nondegenerate levels. Frame (b) shows the separate linkages needed when successive levels have J values that diminish along the chain, starting from J=3. To treat this system we require a set of 2J+1 equation sets, one for each of the initial sublevels. In turn, each set comprises one, two, or three equations. For the pattern of Fig. 15 there are seven of these equation sets, each of the form

iddtC(m;t)=W(m;t)C(m;t),m=J,,+J,
where the label m identifies one of the 2J+1 Zeeman sublevels of the degenerate ground level. The set of equations comprise two that have a single state (unaffected by the radiation and hence dark), two that link a pair of states, and three that link three states, a total of 15 states. If there has been no prior disturbance of the system then we can assume that initially the sublevels are equally populated and each of these equations has the initial value C1(m;0)=1. For comparison with experiment we require a sum of separate probabilities for each m, equally weighted:
P¯n(t)=12J+1m|Cn(m;t)|2.

 

Figure 15 (a) A three-state Lambda linkage 123 as used in STIRAP. (b) The same linkage when there is angular momentum degeneracy, for J=3J=2J=1, requires multiple independent TDSEs. Numbers on the arrows show the relative values of the Rabi frequencies. Encircled sublevels do not have connections to further excitation; they are dark states. In this example only three of the initially populated seven sublevels of level 1 will be transferable by STIRAP to the degenerate target level 3.

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For this particular sequence of angular momenta only three of the seven initially populated sublevels can undergo a full three-state transfer by STIRAP. To achieve complete transfer of the ground-level population into level 3 it is necessary to have a sequence of increasing J values for integer J and nondecreasing J for half-integer J [15]. As was pointed out in [4], if the adiabaticity conditions for STIRAP apply to the weakest chain, i.e., that with the smallest dipole transition moments, then in all three-state chains there will be complete transfer of all magnetic-sublevel populations.

5.2. Superpositions with STIRAP

The degeneracy of Zeeman sublevels that occurs in the absence of an appreciable static magnetic field makes possible not only adiabatic population transfer between preselected sublevels but also the creation of preselected coherent superpositions of sublevels. These possibilities have been extensively studied for linkages based on the angular momentum sequence J=012 in metastable neon [55,258,268273]. Depending on the polarizations of the S and P fields, as many as nine sublevels may become linked. Figure 16 shows examples of the possible linkage patterns, drawn with allowance for single-photon detuning but without Zeeman shifts. To describe adiabatic evolution of this degenerate system we replace the traditional two-component dark state of STIRAP with constructions that involve superpositions of sublevels [55,268270]. Starting from the single state of J=0, a generalization of the STIRAP procedure will lead to a final predetermined superposition of the five sublevels of J=3 [258,272,273]. Procedures that will characterize prepared superpositions have been discussed by [55,258,274276]; see also [15].

 

Figure 16 Linkage pattern for degenerate J=012 excitation. Colors identify components of P or S that occur together. Suitable choice of polarizations will produce, by adiabatic passage, a superposition of J=2 sublevels. Figure 1 reprinted with permission from [270]. Copyright 2005 by the American Physical Society. See also Fig. 1 of [271], Fig. 1 of [273], and Fig. 2 of [272].

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5.3. Near Degeneracy; Adiabatic Transfer State

The basic three-state model of traditional STIRAP provides an adequate description of excitation involving degenerate Zeeman sublevels, and generalizes to treat degenerate superpositions, but a static magnetic field introduces Zeeman shifts that complicate the linkage patterns and alter the simplicity of the three-state STIRAP model of adiabatic population transfer. No longer does one have the conditions listed in Subsection 2.3 for defining STIRAP, although adiabatic population transfer can occur between preselected states. These effects were examined in detail during the early studies of STIRAP [7,188,277,278]. Vibrational and rotational structure, as well as hyperfine structure, also complicate the simple three-state linkages of traditional STIRAP: any one of the three states may be embedded in a manifold of nonresonant states.

Figure 17 shows examples of linkage patterns that occur with linearly polarized P and S fields when there are Zeeman shifts in the angular momentum sequence J=012 [87,188,258,269,277,279]. The states are labeled with integers 1–9, as shown on the left. The window-pane schematics on the right are perpendicular to the common propagation axes and show the direction of the B field that produces the Zeeman shifts and defines the quantization axis. They also show the directions of the S and P polarizations. The early aim of producing complete population transfer between Zeeman sublevels has subsequently been enlarged to consider creation of coherent superpositions of Zeeman sublevels, cf. [55,258,268273].

 

Figure 17 Examples of linkage patterns for linearly polarized P and S fields acting on the nine-state system of Zeeman sublevels having ladder linkages J=0J=1J=2, as studied for metastable neon [87,188]. The states are labeled with integers 1–9, as shown on the left. The window-pane diagrams on the right show the propagation directions (arrows normal to the panes) and the polarization directions of the P and S fields (in-plane arrows). The static B field defines the (vertical) quantization axis. (a) P field linearly polarized along the B field, S field at right angle to P. (b) P and S fields linearly polarized at right angles to B field. (c) S field linearly polarized at 45° to P. Figure 5 reprinted with permission from [278]. Copyright 1995 by the American Physical Society.

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Figure 18 shows two examples of population dynamics for the linkages of Fig. 17(c), wherein the P field is polarized along the quantization axis and the S field is at 45° to this direction. By convention, state 1 (J=0,M=0) is associated with the zero adiabatic eigenvalue (see Appendix D). The choice of Zeeman splittings and single-photon detunings select a particular state of J=2 for two-photon resonance and consequent population transfer. In the left-hand frames this resonance is to state 7, i.e., J=3, M=0. In the right-hand frames the resonance is to state 9, i.e., J=3, M=2.

 

Figure 18 Examples of time evolution for lossless nine-state system of Fig. 17(c) starting from state 1. Top frames show Rabi frequencies. Middle frames show adiabatic eigenvalues. Heavy lines are states between which population transfer occurs. Bottom frames show populations. Left: two-photon resonance occurs between states 1 and 7. Right: two-photon resonance occurs between states 1 and 9. Figures 7 and 9 reprinted with permission from [87]. Copyright 1995. Reproduced by permission of Taylor and Francis Group, LLC, a division of Informapic.

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These and other examples show that the possibility of multiple linkage paths between specified quantum states brings opportunity for constructive and destructive interference that are not present with three states. As pointed out in [188], when there occur such multiple paths the success of adiabatic population transfer relies on the possibility of an (adiabatic) transfer state: an adiabatic state that coincides initially with the populated bare state 1 and that, after the pulse-pair sequence, coincides with the target state 3. The conditions for occurrence of such AT states in multistate chains was examined by [7,41,42,280].

5.4. Ensemble Averages

The prescription of Eq. (112) for evaluating sublevel averages requires only slight alteration to deal with more general ensembles. Stationary quantum particles of an ensemble can differ by their orientation along an arbitrary quantization axis, as specified by a magnetic quantum number m for the ground state. Particles that are moving will differ by their velocity, as parametrized by Doppler shifts, and by their exposure times in a fixed field, as parametrized by pulse areas. Let us denote the set of such particle attributes by an index m. In any experiment they have a probability distribution patom(m).

Classical traveling-wave fields that produce the quantum changes of an experiment can differ in frequency, polarization direction, propagation direction and intensity, thereby affecting the detunings and the Rabi frequencies. Pulsed traveling-wave fields can differ in these ways and by the design of the pulses, i.e., the time dependence of E(t), its duration, and its peak value. Let us denote the variable fields characteristics as ν. In any experiment they have a probability distribution pfield(ν).

All of these characteristics of the atoms and the interactions affect the TDSE. When we show explicitly the dependence of the RWA Hamiltonian on these quantities the TDSE takes the form

iddtC(m,ν;t)=W(m,ν;t)C(m,ν;t).
From a set of solutions to these TDSE we wish to evaluate the ensemble-average probability
P¯n(t)=mpatom(m)νpfield(ν)|Cn(m,ν;t)|2.
The summation signs are to be interpreted as summations over discrete parameters and integration over continuous ones. It is this ensemble average that must be used for interpreting experiments. Both the particle environment and the pulsed field preparation affect the averaging, and both of these require attention for best results.

5.5. Doppler Shifts

The field E(t) that drives coherent excitation of an atom or molecule is the electric field evaluated at the center of mass for the particle. When the particle is moving with respect to the laboratory reference frame in which we typically describe a traveling-wave field, there will occur a Doppler shift of the carrier frequency as seen in the moving reference frame tied to the particle. The Doppler shift depends, in the nonrelativistic regime, on the component v of the particle velocity v along the propagation axis of the electromagnetic field. (The unit vector e that interacts with the dipole moment d is perpendicular to this propagation direction.) The Doppler shift of the two beams alters their detunings:

ΔPΔP+v/ƛP,ΔPΔS+v/ƛS,
where ƛ=λ/2π=c/ω is the reduced wavelength of the field. When the individual particles are part of an ensemble that undergoes thermal motion at temperature T the observable excitation probability averages over a Maxwellian velocity distribution:
p(v)dv=1v¯πexp[(v/v¯)2]  dv,
where v¯2=2kT/M for particles of mass M. The variation of Doppler shifts is negligible when the magnitude of the Rabi frequency is much larger than the mean shift ω(v¯/c).

5.6. Homogeneous Relaxation

In the study of spectral-line profiles (the frequency distribution of radiative absorption or emission) ensemble averaging leads to a broadening of the profile that originates with the variety of environments and is termed inhomogeneous broadening. A second contribution to spectral-line widths originates in the act of spontaneous emission, which occurs with the same decay rate for all similar atoms and is termed homogeneous broadening.

The same two forms of relaxation are present with STIRAP or other coherent-excitation procedures. Whatever the environmental variations may be, whatever irregularities may be present in the laser fields, there is present an irreducible source of decoherence in the spontaneous emission from excited states. This homogeneous relaxation contrasts with the inhomogeneous relaxation that is treated with ensemble averaging. Its detrimental effect on population transfer or other quantum-state manipulation can only be avoided by ensuring that the system remains in a decoherence-free subspace (e.g., a dark state) or that the decoherence time is longer than the time required for completion of the quantum-state change (i.e., “rapid” passage).

5.7. Average Atom

It is generally not possible to find an “average atom” Hamiltonian W¯(t) such that its probability amplitudes C¯n(t),

iddtC¯(t)=W¯(t)C¯(t),
give the required average probabilities
P¯n(t)=|C¯n(t)|2.
However, adiabatic following offers an exception. The first STIRAP experiments [2,4] dealt with rotational states of molecules, i.e., a system with degenerate Zeeman sublevels. The two laser fields were linearly polarized and so the system could be described by a set of independent three-state STIRAP equations, each labeled by the magnetic quantum number of the ground state as in Fig. 15. By ensuring that the subsystem with weakest links underwent STIRAP the experimenters induced population transfer of all the linked sublevels (M=0 is unlinked). Their published results were thus of measured sublevel averages.

5.8. STIRAP Errors

Manifestations of nonadiabatic evolution of the three-state system are evident in two ways: the failure of P3(t) to be unity at the termination time T of the pulse sequence (when that is the goal) and the failure of P2(t) to be zero at all times. Let us define these errors as

ε3=1P3(T),ε2=MaxP2(t).
An important experimentally controllable measure of interaction strength (and adiabaticity) is the temporal pulse area A, cf. App. B.1. As the pulse area becomes larger, each of these errors diminishes. By choosing some acceptable error criteria an experimenter is led to a requirement on minimum pulse area A.

5.8a. Error and Pulse Area

Figure 19 shows an example of the connection between error and pulse area for two identically shaped Gaussian pulses, suitably offset in time, and a lossless RWA Hamiltonian that has both single-photon and two-photon resonance: ΔP=ΔS=0. The figure shows that, based on the requirement of an error less than 102 in population transfer, a pulse area larger than 4π is needed for the Gaussian pulse shapes used. To maintain a state-2 population less than 102 it is necessary to have a pulse area less than around 15π. Somewhat different values would be obtained if the error ϵ2 is defined as the integrated population in state 2.

 

Figure 19 (a) The error measure ε3 and (b) the error measure ε2, both as a function of temporal pulse area for two near-optimally overlapping Gaussian pulses. Dashed horizontal lines mark errors of 102. Errors will differ with other pulse shapes. Figure 1.7 reprinted with permission from [9]. See also Fig. 2 of [281] and Fig. 13 of [12].

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The early use of STIRAP for preparation of preselected excited states found population transfer P3(T)=0.9 satisfactory, and reviews [6,8,10] have suggested an area of at least 10 (or 3π) as suitable for chemical applications. Applications in quantum information [36,112116] require much smaller errors, typically less than 105 [12], and hence these uses typically need larger pulse areas.

5.8b. Error and Pulse Shape

The plots of Fig. 19 rely on Gaussian pulses of equal widths and heights, a common choice for simulating the effect on particles that cross a CW laser beam. But other pulse shapes offer opportunity for reducing these errors, while maintaining a given pulse area or pulse energy, by better satisfying the adiabatic condition (cf. Subsection 4.4c) throughout the pulse sequence [12,282284]; see Subsection 5.8d.

The challenge posed by an experiment may be presented in the following way. Say we wish to obtain results that are modeled by some ideal probability amplitudes Cnideal(t) of an ideal RWA Hamiltonian. The ensemble variations produce error measures such as

ε¯n(t)=mpatom(m)νpfield(ν)|Cn(m,ν;t)Cnideal(t)|2.
We wish this, or other error measures, to be small. Optimization of STIRAP excitation requires considering the effect of a pair of pulses. The minimization of error is particularly important in protocols of QIP, where both amplitude and phase require control.

When the field envelope E(t) originates in a pulsed traveling wave then the experimenter has the opportunity to design the time dependence: it need not be a simple Gaussian, nor a rectangular pulse, but can incorporate a train of pulselets that together form a composite pulse (CP) [285294]; see Subsection 6.6b and, for application to STIRAP, [281,289,290,293296]. By crafting the time dependence of the interaction, as embodied in the Rabi frequency, an experimenter can create a designer pulse that is relatively insensitive to some field-ensemble characteristic or some environmental effect. The use of such a CP makes the ensemble average less sensitive to the ensemble characteristics—for example, to unavoidable pulse-to-pulse variation of peak Rabi frequency or of pulse area.

Because the failure of adiabatic evolution is a readily identifiable detriment, various researchers have proposed pulse forms that are well suited to maintaining adiabatic change [12,282,297,298]. Other suggestions include use of additional fields [12]. Systematic studies termed transitionless quantum driving [299] or shortcut to adiabaticity (STA) [286,293,300308] aim to design pulses that will ensure adiabatic evolution for the propagator between a given initial state and a specified target state.

5.8c. Robustness

The STIRAP process is often said to be “robust,” meaning that its desirable results (typically production of complete population transfer) are relatively insensitive to small, inadvertent (or uncontrollable) alterations of operating conditions, as embodied in the responsible Hamiltonian—such things as single-photon detunings, pulse shapes, pulse durations, and peak Rabi frequencies or temporal pulse areas.

To quantify this notion we may consider a Hamiltonian H(a,b,;t) that is defined by a set of parameters a,b, such as static detunings and peak Rabi frequencies and which governs a statevector Ψ(a,b,;t) through a TDSE:

iddtΨ(a,b,;t)=H(a,b,;t)Ψ(a,b,;t).
As a useful measure of success we may take the fidelity f(a,b,) with the actual statevector at the conclusion of a procedure, t, fits some intended ideal result Ψideal:
f(a,b,)=|Ψideal|Ψ(a,b,;)|2.
For parameters that take a continuum of values, as do pulse durations, the partial derivatives f/a, f/b, provide a measure of system sensitivity, and we can regard the process as robust if these are small.

The pulse shapes are often taken from a discrete set of families, such as Gaussians or sine-squared pulses, and robustness can mean that the choice of shape does not appreciably affect the fidelity. Alternatively, we can introduce pulse shapes that depend upon a discrete set of alterable parameters, such as B-splines of given order [309311] whose defining elements (piecewise-continuous polynomials) offer flexible descriptors of very general forms. Partial derivatives again measure robustness of a particular shape.

Elements of the Hamiltonian must also include measures of detuning, either static (as describes Doppler shifts) or time varying (as describes carrier-frequency chirps), and these too are part of the parameter set whose variation affects the outcome of a process.

Procedures termed coherent control (CC) [312316] sought control over molecular reactions, as examples of quantum-mechanical change, by using coherent properties of fields to control interfering routes between initial and final states, rather than use frequency selection alone. That terminology subsequently found wider application in manipulation of quantum states [317325].

5.8d. Optimization

Once a Hamiltonian has been parametrically defined it is natural to ask whether fidelity can be improved by suitably choosing the parameters. Inevitably we must consider such constraints as the need to carry out the process in a time that is shorter than some incoherence time and with field intensities that will not damage equipment. Thus our task is an example of constrained optimization [326329]: minimization or maximization of some function that is subject to restrictions. A very considerable literature exists on this mathematical problem applied to quantum-state manipulation under the title of optimal control (OC) theory [257,330336] and numerous papers have applied its algorithms to STIRAP [169,337340]. Optimization of composite pulses can fit into such a framework for designing procedures that are robust as well as economical and brief.

6. STIRAP-Related Processes

A number of procedures, whose names indicate a connection with the original STIRAP, will produce results similar to those of Subsection 2.3b without satisfying all of the requirements of Subsection 2.3a. In selecting examples for discussion here I have, with a few exceptions, ignored excitation schemes that rely on time-varying detunings; see Appendix H. The techniques of SCRAP, SARP, RCAP, and CHIRAP mentioned in Subsection 2.3d are examples of such variable-detuning processes, typically involving monotonically swept detuning [9]. Such procedures, though offering robust population-transfer opportunities, draw on the heritage of rapid adiabatic passage (RAP) [341344] and its reliance on diabatic curve crossings and avoided adiabatic crossings [9] rather on the dark-state adiabatic following and counterintuitively sequenced pulses that distinguish traditional STIRAP. (The STIRAP dark state is, of course, independent of the single-photon detuning Δ, whether or not that varies with time, so long as the two-photon detuning δ remains zero.)

6.1. Fractional STIRAP

With the conventional STIRAP process, in which the S pulse precedes but overlaps the P pulse, the population changes are not controlled by pulse areas alone, as they are for resonant two-state excitation (cf. Appendix B). Instead, when the RWA Hamiltonian changes only slowly (adiabatically) subsequent to initiation at t=0 the populations follow the formulas

P1ad(t)=cos2[ϑ(t)],P3ad(t)=sin2[ϑ(t)],
where the mixing angle ϑ(t) is defined by Eq. (12) of Subsection 2.2e. That is, the final population is fixed by the final ratio of the two Rabi frequencies, independent of the intermediate history of population change.

It was recognized soon after the discovery of STIRAP that if the time evolution is halted and the mixing angle remains fixed, then instead of reaching alignment with the RW basis vector ψ3(t), the statevector will be frozen in a coherent superposition of RW states 1 and 3 [40]:

Ψ(t)=ψ1(t)cosΘψ3(t)sinΘ,
where Θ is the value of the mixing angle ϑ(t) at the termination of the evolution. Thereby, instead of STIRAP, we have fractional STIRAP [65,249,280,345348], in which only a controlled fraction of the population is transferred to state 3. As in STIRAP, state 2 remains unpopulated in the adiabatic limit because the statevector Ψ(t) adiabatically follows the dark adiabatic eigenvector ϕ0(t). The required terminating mixing angle can be produced experimentally either by a sudden interruption of the changing P and S pulses or by allowing them to vanish simultaneously, in a smooth fashion [280], so that the populations evolve smoothly from their initial values to their finite ratio. In particular, if this ratio is 1, meaning Θ=π/4, then an equally weighted superposition of states 1 and 3 will be created, Ψ=(ψ1ψ3)/2, as depicted in Fig. 20—so-called half STIRAP. Moreover, if the final phases of the two fields differ by α then this phase will be mapped onto the created superposition, and the final statevector will be the superposition
Ψ(t)=ψ1(t)cosΘψ3(t)eiαsinΘ.

 

Figure 20 Example of fractional-STIRAP. Top: time dependences of the P and S Rabi frequencies. Bottom: populations Pn. As in STIRAP, the S pulse arrives before the P, but unlike traditional STIRAP here the two pulses vanish simultaneously while maintaining a fixed ratio. Consequently, instead of complete population transfer 13, the pulses create a coherent superposition of states 1 and 3. In this example the two states have equal final probabilities, an example of half-STIRAP. Adapted from Fig. 2 of [280]. Copyright IOP Publishing. Reproduced with permission. All rights reserved. See also [12].

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Figure 21 Depiction of three-state adiabatic following for large single-photon detuning. (a) Motion of ϕ0(t), the guide for statevector motion when the pulse ordering is counterintuitive, SP. (b) Motion of ϕ(t), the guide for the statevector when the pulse ordering is intuitive, PS.

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Note that the vector-space coordinates appearing here are the RW (diabatic) states ψn(t), carrying time-dependent phases ζn(t).

6.2. Bright-State Adiabatic Following: B-STIRAP

The definition of traditional STIRAP requires the intuitively acting P pulse to arrive first. As was shown in Subsection 2.4b with Fig. 7 and in Subsection 4.6d with Fig. 11, successful adiabatic passage can occur with either pulse ordering if there is appreciable single-photon detuning. When the P pulse occurs first the transfer, though adiabatic and therefore robust, is not by means of the dark state, and so it is not the traditional STIRAP of Section 2. It has been termed variously reverse STIRAP [4] and bright-state STIRAP or b-STIRAP [9,12,349351].

Figure 21 presents a picture of the Hilbert-space motion of two of the adiabatic eigenvectors, ϕ0(t) and ϕ(t), that serve as guides for the three-state adiabatic following illustrated in Fig 11. When the statevector is initially aligned with the dark state ϕ0(t), whose motion is shown in frame (a), it undergoes a simple Hilbert-space rotation that corresponds to a transfer of population from state 1 to state 3 without passing through state 2. The motion of ϕ0(t), unlike that of the other adiabatic eigenvectors, is unaffected by the single-photon detuning, when two-photon resonance is present. This insensitivity to detuning provides robustness to the STIRAP process. Frame (b) shows the more elaborate Hilbert-space motion of vector ϕ(t), the guide for adiabatic following when the pulse ordering is intuitive, PS.

As the single-photon detuning Δ increases in magnitude the transient population in state 2 decreases. When that detuning becomes much larger than the peak Rabi frequencies one has a situation in which adiabatic elimination applies, cf. Appendix G. The dynamics then is that of a two-state system governed by an effective Hamiltonian, see Appendix G.1. Successful population transfer does not then depend on the pulse ordering.

6.3. Chain STIRAP

The original STIRAP provided a mechanism for passing population between two states, a complement to other procedures such as pi pulses or chirped pulses that also provide such action. One can extend, in various ways, such basic two-state excitation to a chain of linked states, 123N whose RWA Hamiltonian involves N detunings and N1 Rabi frequencies. The general RWA Hamiltonian for chain linkage is tri-diagonal, with the structure (for real-valued Rabi frequencies)

W(t)=12[2Δ1Ω1(t)000Ω1(t)2Δ2Ω2(t)000Ω2(t)2Δ3Ω3(t)000Ω3(t)2Δ4Ω4(t)000Ω4(t)2Δ5].
With suitable choice of RW phases we nullify the first detuning, Δ1=0; there is then resonance between state 1 and each state for which Δn=0. As with the three states of traditional STIRAP, the ordering of the energies En does not affect the structure of the RWA Hamiltonian: they may form an simple ladder, En1<En, a letter-M, a letter-W, an enlarged Lambda (hyper-Raman linkage, cf. Subsection 6.4), or any other arrangement. Such a Hamiltonian has also been examined for modeling the motion of a single electron along a chain of quantum dots [44].

6.3a. Chain Timings

The possible timings of the N1 pulsed Rabi frequencies Ωn(t) fit into the following three classes, cf. [44].

Sequential (stepwise) transfer: Multiple photon. The simplest procedure for moving population along a chain is by a succession of separate, independent two-state transfers, as illustrated on the left-hand side of Fig. 3. Although the excitation shown there is produced by Rabi oscillations, other procedures, such as RAP or STIRAP, would accomplish the same result. A succession of such independent pulses (or STIRAP pulse pairs) can be designed to produce complete transfer of population along a chain of any length. It is only necessary that each individual pulse produce complete transfer and that there is no loss of population or coherence during the succession of pulses (not always easy in practice). Such “one-at-a-time” or “step-by-step” procedures have been called multiple-photon processes. When the individual steps occur by STIRAP the process has been termed sequential or stepwise STIRAP [352,353].

Simultaneous pulses: Multiphoton. An alternative procedure deals with a RWA Hamiltonian in which all the pulses are simultaneous. Early in the ongoing studies of coherent excitation came examination of ladder-like chains of N nondegenerate energy levels driven by simultaneous pulses, either of a single frequency or of N1 resonantly adjusted frequencies [354357]. With appropriate choice of the N detunings and N1 Rabi frequencies there occur Rabi oscillations between chain ends, as shown on the right-hand side of Fig. 3. Such “all-at-once” processes have been called multiphoton processes.

Holistic transfer. The third possibility, generalizing STIRAP, is an example of what might be termed holistic excitation: a set of two-state interaction links that are neither sequentially independent nor identical but have partial temporal overlap. It is with such protocols that this section deals. Various detuning-varying procedures, such as RAP, CHIRAP and RCAP, offer other examples.

6.3b. Alternating STIRAP

Following the initial development of STIRAP subsequent work extended the notion of three-state, holistic, adiabatic passage to longer chains. The first such extension [62,358] dealt with a model in which the odd-integer states 1,3, are stable or metastable and the even-integer states 2,4, are excited and undergo radiative decay (bright states). For odd-integer N the linkage pattern is then a succession of Lambda patterns, the simplest of which is the five-state letter-M linkage pattern of Fig. 22(a).

 

Figure 22 (a) Letter-M linkage pattern for five-state chain and four pulsed fields. When implemented among hyperfine levels or Zeeman sublevels the odd-numbered states may be degenerate, as may be the even-numbered states. (b) Linkage pattern for straddle version (or triple Lambda) [359] of frame (a). States 2,3,4 together with their interactions P,S, have been replaced by their adiabatic states 2,3,4. The linkage pattern is now a set of three simultaneous Lambda linkages, between states 1 and 3 via separate intermediate states.

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For any odd-N chain it is possible to have adiabatic passage between state 1 and state N when there is resonant excitation between these states, Δ1=ΔN. The required model replaces the early-arriving S-field Rabi frequency of STIRAP with a set of S-field interactions, the even-n Rabi frequencies Ω2,Ω4, of Eq. (126). It replaces the late-arriving STIRAP P-field Rabi frequency with a matching, delayed set of Rabi frequencies, the odd-n set Ω1,Ω3,. That is, alternating linkages along the chain are P and S-type pulses.

The dynamics of such linkages, driven by two pulsed interactions that have STIRAP timing, have been discussed in some detail by [40,45,47,48,358361]. This model has been termed alternating STIRAP or A-STIRAP [362] to distinguish it from the straddling timings of Subsection 6.3c below. As with traditional STIRAP, the individual pulses have no restriction on shape or peak value and they can, when suitably adiabatic, produce complete population transfer without introducing transient population into even-n states of the chain.

Chain dark state. Given a tridiagonal matrix it is a straightforward matter to find the eigenvectors and eigenvalues. Expressions for the needed eigenvectors—the adiabatic states—are available for arbitrary N [41,230]. For an extension of the STIRAP procedure to the RWA Hamiltonian of Eq. (126) we require components of a dark, adiabatic-transfer state—a state that connects the end states 1 and N and has no even-n excited, lossy-state components, cf. [358] and Appendix D. For such an adiabatic state to exist the detunings Δn must be zero for odd-integer n:

Cn(t)=0,n=2,4,,Δn=0,n=1,3,.
As with the original STIRAP, we can use this dark state to transfer population 1N by requiring that initially all the odd-n Rabi frequencies are negligible and that finally all the even-n Rabi frequencies are negligible. Symbolically we require
initially:|Ωodd||Ωeven|,finally:|Ωodd||Ωeven|.
General multistate chains have been discussed by [4143] and numerous experimental realizations are cited in [12]. As with the nearly degenerate Zeeman sublevels discussed in Subsection 3.3, successful population transfer requires that there be an adiabatic transfer state, one that coincides with the populated initial state 1 prior to the pulses and that coincides (apart from a phase) with the chain-ending state N after the passage of the two pulses [4143].

Example: N=5. Equation (D.2) of Appendix D gives the construction of the null-eigenvalue eigenvector of the letter-M RWA Hamiltonian of Eq. (126). This eigenstate of W(t) has no components of excited states 2 and 4; it is a dark state. For adiabatic passage between states 1 and 5 we require that initially there be only a combination of Rabi frequencies 2 and 4, and that finally there be only the combination 1 and 3. For the labelings of Fig. 22 the initial and final nonzero Rabi frequencies are

initial:  Ω2(t)Ω4(t)=ΩS(t)ΩS(t),final:  Ω1(t)Ω3(t)=ΩP(t)ΩP(t).
This is the generalization of the counterintuitive pulse sequence of STIRAP. Population transfer is readily accomplished if, like the original STIRAP, all of the time dependence originates with just two fields: the even-n Rabi frequencies share a common S-field pulse s(t), the odd-n Rabi frequencies share a common P-field pulse p(t), and the two overlapping pulses have the typical SP ordering found in the original STIRAP. With the phase choice W11(t)=0 and the resonance condition W11(t)=W55(t) the RWA Hamiltonian for the letter-M linkage then becomes
W(t)=12[0ΩP(t)000ΩP(t)2Δ2ΩS(t)000ΩS(t)0ΩP(t)000ΩP(t)2Δ4ΩS(t)000ΩS(t)0].
Figure 23 shows simulations appropriate to adiabatic evolution for the population histories of this model. As with the original three-state STIRAP, there occurs complete population transfer between the end states of the chain, here 1 and 5, and the excited states 2 and 4 remain unpopulated throughout the pulse sequence. The additional low-lying (but not excited) intermediate state 3 receives transient population: the dynamics is that of two two-photon transitions that drive the transfer 135 of a three-state system.

 

Figure 23 Populations resulting from adiabatic passage with the system of Fig. 22. The STIRAP-like pulse sequence produces complete population transfer between end states of the chain, 1 and 5, and allows no population into excited states 2 and 4. Adapted from Fig. 1 of [358] with permission. Copyright 1991 by the American Physical Society.

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Figure 24 Bloch-vector presentation of two-state excitation by resonant (ω21=ω) pi pulse in the RWA. Population is initially in state 1, so the Bloch vector (blue) starts at the south pole of the Bloch sphere and ends at the north pole. (a) In the RW picture, with rotating coordinate vectors ψn(t), the Bloch vector moves smoothly along a great-circle path, under the action of a torque vector (red) that remains fixed, in the equatorial plane, along the 1 axis. Coordinate vector ψ2(t) contains the phase eiωt. The RW picture is used in this article for all other displays of system properties. (b) In the Schrödinger picture, with stationary coordinate vectors ψn, the individual probability amplitudes undergo rotations, as does the Bloch vector constructed from them. The computations of this figure are for ω=8Ω.

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Figure 25 Motion of Bloch vector (with RW coordinates) subject to constant Rabi frequency and constant detuning. The Bloch vector (blue) rotates about the fixed torque vector (red). With the chosen time duration the Bloch vector reaches the equatorial plane, a coherent superposition of the two states having equal populations but different relative phases. (a) Negative detuning. (b) Positive detuning.

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6.3c. Straddle STIRAP

A variant on the chain system driven by two offset pulses uses the P pulse only on the 1–2 transition and the S pulse only on the N to N1 transition, and has a “straddling” pulse (or pulses) that connects in chainwise fashion the remaining states 2,3,,N1. The primed linkages of Fig. 22 are examples of the straddling pulses, to extend in duration beyond the S and P pulses. It is possible with this linkage to produce transfer from state 1 to state N by what has been termed straddle STIRAP [7,12,41,42,44,62,63,363]. The mechanism for successful adiabatic population transfer involves N2 dressed states that superpose all but states 1 and N and that serve as intermediate states for a set of N2 simultaneous three-state Lambda linkages [7,12,42]. The equations for states 1 and N have the form

iddtC1(t)=νW1νP(t)Aν(t),
iddtAν(t)=Δν(t)+Wν1P(t)C1(t)+WνNS(t)CN(t),
iddtCN(t)=νWNνS(t)Aν(t),
where the elements of WS(t) precede but overlap those of WP(t). By appropriate adjustment of the carrier frequencies the detuning Δν(t) can be nullified for one adiabatic state, say ν=0. When the remaining detunings are large, we have an example of the basic three-state excitation used by traditional STIRAP, with
C2(t)=A0(t),ΩP(t)=2W10P(t),ΩS(t)=2W0NS(t).
For further discussion, see [12].

6.4. Hyper-Raman Transitions and STIHRAP

Traditional STIRAP deals, as the name indicates, with Raman transitions—two single-photon transitions that are driven by a Hamiltonian whose elements are linearly proportional to electric field amplitudes. When fields become strong they can induce a variety of multiphoton transitions that originate with Hamiltonian matrix elements that are proportional to products of several field amplitudes: an n-photon process depends upon an interaction that varies as n products of electric fields. The simplest of these are two-photon transitions. These occur, for example, when the electric dipole moment d of the interaction d·E(t) becomes distorted by a strong electric field, as quantified by the frequency-dependent polarizability tensor α(ω), whose matrix elements between states n and m involve the product of two dipole moment components summed over all possible intermediate states q [255] (for typographical simplification these formulas use the notation |n for |ψn):

n|αij(ω)|m=q[n|di|qq|dj|m(EqEnω)+n|dj|qq|di|m(EqEn+ω)].
The interaction energy associated with the polarizability involves the product of two electric field amplitudes leading, for linearly polarized light, to the altered expressions for the Rabi frequencies [255]:
ΩP(t)=12cε01|αzz(ωP)|2IP(t),ΩS(t)=12cε02|αzz(ωS)|3IS(t).
Each of these is proportional to the instantaneous intensity of the light, Ia(t), rather than the amplitude of the electric field, Ea(t). In addition to this change there occurs a dynamic Stark-shift,
Sn(a;t)=12cε0n|αzz(ωa)|nIa(t),
that increments the diagonal elements of the RWA Hamiltonian matrix W(t) by
WnnStark(t)=Sn(P;t)+Sn(S;t).
These shifts too are proportional to the intensities; both the Rabi frequencies and the detunings necessarily scale together proportional to intensities. It is these inevitable dynamic Stark shifts that diminish the usefulness of stimulated hyper-Raman adiabatic passage—the so-called STIHRAP [205,255,364,365]: the customary approach to improving adiabaticity by increasing the field intensities introduces detrimental two-photon detuning, needed for the formation of the dark state. Nevertheless, it has been possible to achieve population transfer by a combination of adiabatic and diabatic evolution, albeit with transient population in state 2, as in the technique of SCRAP or SARP; see Appendix H.3.

6.5. Bloch-Vector Torque Equation

The populations Pn(t) give only a partial picture of the statevector motion. Particularly when our interest lies with coherent superpositions of quantum states we require further information. This is available from the phases of the complex-valued probability amplitudes Cn(t). For an N-state system their real and imaginary parts require 2N real numbers for specification. Because the statevector has unit magnitude, only 2N1 real-valued numbers are needed to specify the statevector motion.

6.5a. Three Bloch Variables

A useful choice of variables with which to describe a two-state statevector, proposed by FVH [15,366] is constructed from bilinear products of probability amplitudes (i.e., elements of the density matrix ρ):

u(t)r1(t)=2ReC1(t)C2(t)*=2Reρ12(t),
v(t)r2(t)=2ImC1(t)C2(t)*=2Imρ12(t),
w(t)r3(t)=|C2(t)|2|C1(t)|2=ρ22(t)ρ11(t).
The three real numbers rj(t) serve as coordinates of a vector r(t) of unit length, the Bloch vector [366],
|r(t)|2=r1(t)2+r2(t)2+r3(t)2=1,
that moves in a three-dimensional abstract vector space: its tip moves on the surface of a unit sphere, the Bloch sphere. The variables r1u and r2v are termed coherences, the variable r3w is the population inversion. Initially, with population entirely in state 1, the Bloch vector points to the south pole of the Bloch sphere:
initial:  r1=0,r2=0,r3=1.
Complete population transfer (inversion) corresponds to a Bloch vector pointing to the north pole of the Bloch sphere. In picturing motion of the Bloch vector it is often useful to introduce Cartesian-coordinate labels for its the three dimensions, with x,y,z for indices 1, 2, 3. Note that the probability amplitudes and density matrix used in defining the Bloch vector are themselves defined in the rotating-wave picture, meaning that their time variation originates with Rabi frequencies and detunings, omitting variation at the carrier frequency of the field.

6.5b. Bloch Equation

From the TDSE it follows that the Bloch vector, when undergoing coherent change, obeys the optical Bloch equation [13,366,367]:

ddt[r1(t)r2(t)r3(t)]=[0Δ(t)Ω(t)sinφΔ(t)0Ω(t)cosφΩ(t)sinφΩ(t)cosφ0][r1(t)r2(t)r3(t)].
Here the angle φ is the phase of the electromagnetic field, which is taken to be varying as cos(ωt+φ). Because of the antisymmetric structure of the coefficient matrix of the coupled equations, one can write this as a torque equation for the Bloch vector (cf. Appendix E),
ddtr(t)=ϒ(t)×r(t),|ϒ(t)|=ϒ(t)=Δ(t)2+Ω(t)2,
where the torque vector ϒ(t) incorporates the instantaneous properties of the RWA Hamiltonian [15,366]:
ϒ(t)=[Ω(t)cosφΩ(t)sinφΔ(t)],for  r(t)=[r1(t)r2(t)r3(t)].
For simplicity, the laser phase is commonly taken as zero, φ=0. The torque vector then has only two nonzero components, and will move always in the 1,3 plane:
ϒ1(t)=Ω(t),ϒ2(t)=0,ϒ3(t)=Δ(t).
Like the adiabatic eigenvectors, whose components are constructed from elements of the RWA Hamiltonian, the torque vector incorporates the same potential dynamics. The behavior of a particular system depends not only on the Hamiltonian and its torque vector but on the initial conditions of the Bloch vector.

6.5c. Resonant Excitation

When the excitation is resonant, Δ(t)=0, the torque vector of the FVH model is orthogonal to the polar-pointing Bloch vector of an unexcited state: it lies in the 1, 2 plane (the equatorial, x, y plane of the Bloch sphere) and has components

ϒ1(t)ϒx(t)=Ω(t)cosφ(t),ϒ2(t)ϒy(t)=Ω(t)sinφ(t),ϒ3(t)ϒz(t)=0.
When the phase φ is zero, as is customarily assumed, the torque vector lies along the 1 axis (the x-axis), while the initial Bloch vector points to the south pole of the Bloch sphere. The torque vector induces rotation of the Bloch vector about the 1 axis at the instantaneous Rabi frequency Ω(t). When the Rabi frequency remain constant and, in addition, φ(t)=0, the tip of the Bloch vector traces a great-circle path on the Bloch sphere, away from the south pole and passing eventually through the north pole. The Bloch vector remains always in the 2, 3 plane and the system behavior appears as Rabi oscillations of the components r2(t) and r3(t).

Figure 24 provides two representations of this resonant behavior, produced by resonant excitation by a constant Rabi frequency. Frame (a) shows the slowly varying components of the Bloch vector in the RW picture ζ2(t)=iωt. Frame (b) shows the components in the Schrödinger picture, ζn(t)=0, for the choice ω12=ω=8Ω. The simple great-circle path of the RW picture incorporates the phase eiωt into coordinate ψ2(t). Apart from frame (b) of this figure, all other plots use the RW picture.

6.5d. Detuning

The x, z (or 1,2) plane of the Bloch sphere is, by definition, the default location of the torque vector for a real-valued Rabi frequency and any detuning. Figure 25 shows two examples of the motion of the Bloch vector produced by a constant Rabi frequency and constant detuning. When the population is initially in state 1 and the detuning is nonzero the path of the Bloch vector traces a cone whose generator initially lies at the south pole and which can never reach the north pole. This simple geometric picture illustrates the fact that, for constant Rabi frequency, complete population transfer is only possible for resonant excitation. However, various pulse sequences that rely on alteration of detuning and Rabi frequency can produce complete population transfer.

6.5e. Adiabatic Following

Adiabatic following occurs when the Bloch vector is either parallel or antiparallel to the torque vector, r(t)·ϒ(t)=±ϒ(t), so that ϒ(t)×r(t),=0, and the torque vector changes slowly: the two vectors then remain aligned as the Bloch vector adiabatically follows the torque vector. The initial Bloch vector has r3 as its only nonzero element, and so to move this vector adiabatically the torque vector must also lie initially along the 3 axis. This will occur only with negligible Rabi frequency and nonzero detuning, either negative (for parallel r and ϒ) or positive (for antiparallel r and ϒ). Subsequent adiabatic following will carry the Bloch vector along with the torque vector on a path in the 1,3 plane. Because both the torque vector and its linked Bloch vector remain in the 1,3 plane, it is not necessary to display the full three-dimensional Bloch sphere, as in the preceding figures: it is only necessary to show the 1,3 plane. The next figures make that simplification.

6.5f. Adiabatic Passage

The most common example of two-state adiabatic following occurs when Ω(t) is pulsed and Δ(t) varies monotonically between large extreme values (a linear chirp). Both before and after the pulsed Ω(t) the nonzero Δ(t) will cause the torque vector to point toward one of the two Bloch-sphere poles, and hence be parallel or antiparallel to the initial Bloch vector, poised to induce adiabatic following. If, with the cessation of the pulsed Rabi frequency, Δ(t) has changed sign, then the final torque vector will have reversed its alignment and point to the opposite pole of the Bloch sphere. The Bloch vector will consequently have been brought, by adiabatic following, from the south pole to the north pole, and the population will have been inverted—it will have undergone an adiabatic passage between quantum states. The complete population transfer produced by a monotonic chirp is robust: unlike a pi pulse (cf. Appendix B.1) it is insensitive to the Rabi frequency once the conditions for adiabatic following are satisfied.

Figure 26 shows an example of a linearly chirped pulse that produces complete population transfer by adiabatic following. On the left the upper frame shows the Rabi frequency Ω(t) and the detuning Δ(t). The diabatic eigenvalues are 0 and Δ(t), so a diabatic curve crossing occurs whenever Δ(t) changes sign, here at t=0. The middle frame shows the adiabatic eigenvalues. The bottom frame shows the three Bloch-vector components evaluated from a numerical solution to the Bloch equations. The dynamics of this simulation is not perfectly adiabatic, and so component r2(t) is not exactly zero, but nonetheless complete population inversion occurs.

 

Figure 26 Two-state excitation by positively chirped detuning to produce complete population transfer. The torque vector ϒ, moving always in the 1,3 plane of the Bloch sphere, remains parallel to the Bloch vector r as it induces adiabatic following. Left: top frame: the relative Rabi frequency Ω(t)/ϒ(t) (black) and relative detuning Δ(t)/ϒ(t) (dashed red). Middle frame: the adiabatic eigenvalues ε±(t)/ϒ(t). Bottom frame: Bloch-vector components: coherences r1(t) (blue), r2(t) (red), and population inversion r3(t) (green). Right: parametric plots of Bloch vector (red) in r1,r3 coordinates and torque vector (blue) in Δ,Ω coordinates. The adiabatic motion of the Bloch vector is entirely within the 1,3 plane shown here.

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The RWA Hamiltonian responsible for this behavior, shown in Fig. 26, is commonly idealized as an infinite time interval, <t<+, but in practice, and in simulations, it is only necessary to treat a suitably large interval around t=0.

6.5g. Two-State STIRAP

The most common temporal variations considered for the detunings that appear in the TDSE or the Bloch equation have been simple sweeps (frequency chirps), idealized as linear variations with time and designed to produce complete population inversion by adiabatic passage. On occasion it is desirable to produce not complete transfer of population but a superposition of two quantum states. The desired combination is typically ψ1(t)±ψ2(t), a 50:50 superposition in which each RW state has equal population of 1/2. To accomplish this the torque vector must move the Bloch vector from the south pole of the Bloch sphere to a point on the equator. Such a final state can be produced by using a pulsed detuning rather than a value that varies linearly. Figure 27 shows an example.

 

Figure 27 Two-state excitation by pulsed positive detuning to produce a coherent superposition. Layout is as in Fig. 26. The torque vector ϒ remains parallel to the Bloch vector r as it induces adiabatic following to produce the superposition ψ1(t)+ψ2(t).

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When one considers more general, pulsed variations of detuning, it is instructive to compare the torque equation of the two-state Bloch variables, Eq. (143), to the torque equation (66) for the probability amplitudes of the resonant three-state system [174,368]. The two-state Bloch-vector components correspond to the three-state probability amplitudes through the formulas

r1(t)=u(t)=C3(t),r2(t)=v(t)=iC2(t),r3(t)=w(t)=C1(t),
as illustrated in Fig. 28. Apart from a factor of 2 the two-state detuning Δ(t) corresponds to the S pulse and the two-state Rabi frequency Ω(t) corresponds to the P pulse. With both sets of variables the existence of a torque equation implies the possibility of adiabatic passage.

 

Figure 28 Analogy between resonant STIRAP, frame (a) and the two-state Bloch equations, frame (b). STIRAP-like adiabatic passage will occur in the two-state system if the detuning precedes, and overlaps, the Rabi frequency, and all changes occur adiabatically.

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Figure 29 Example of conventional, fully resonant STIRAP produced by Gaussian pulses and expressed with Bloch variables rk(t). The mixing angle increases monotonically with time, between values that produce complete population transfer. Top frame: S and P Rabi frequencies Ωk(t). Middle frame: mixing angle ϑ(t). Bottom frame: Bloch variables rk(t). (a) Left-hand frames: two positive Rabi frequencies. Initial ϑ=0, final C1=0, C3=1. (b) Right-hand frames: opposite signs of Rabi frequencies. Initial ϑ=π, final C1=0, C3=+1

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The procedure of STIRAP for a resonant three-state system has an analog, two-state STIRAP, for the two-state system. The condition for initial population in the lower state of the two-state system, w()=1, becomes C1()=1 in the three-state analog. The pulse ordering pertinent to STIRAP requires, for the two-state system, a time-dependent (pulsed) detuning Δ(t) that precedes the Rabi frequency Ω(t). This sequence will rotate the tip of the Bloch vector from the south pole to the equator, creating thereby a final state described by the values |u|=1, v=w=0. These values correspond to the three-state amplitudes |C3|=1, C2=C1=0 appropriate to the adiabatic passage of traditional STIRAP. States with w=0 are states of maximal coherence, wherein the two states have equal magnitude.

The “dark” superposition used by two-state STIRAP is not a quantum state of the system, it is the sum of the inversion w(t) and the coherence u(t):

D(t)=w(t)cosϑ(t)+u(t)sinϑ(t),ϑ(t)=arctan[Ω(t)/Δ(t)].
When the detuning pulse Δ(t) precedes the pulsed Rabi frequency Ω(t) the mixing angle ϑ(t) has the same asymptotic limits as in STIRAP; hence the initial and final conditions are D()=w() and D()=u(). Because the adiabatic passage is robust, the Bloch-vector rotation is also robust: it depends only weakly on the extent of the overlap of the two pulses and the peak values of Δ(t) and Ω(t). The adiabatic condition is similar to the one in STIRAP. It requires large coupling pulse area and large detuning area:
|Ω(t)dt|1,|Δ(t)dt|1.
For further discussion, see [12].

6.5h. Insight: Three-State Bloch Vector for STIRAP

The FVH vector model [366] of two-state dynamics provides a simple three-dimensional geometric picture of the excitation of a two-state system, one that allows a ready interpretation of the importance of the phase of the envelope function and the associated Rabi frequency. This model generalizes to treatments of STIRAP behavior in a resonant three-state system. The needed Bloch-like vector, an example of the coherence vector of Subsection 4.1b, has the components

R1(t)=2ReC1(t)CN(t)*,
R2(t)=2ImC1(t)CN(t)*,
R3(t)=|CN(t)|2|C1(t)|2,
where N is the dimension of the full Hilbert space within which the two-state behavior occurs; the present discussion considers N=2 and N=3, but the model is applicable to larger N [369]. For N=2 the vector R(t) is the traditional Bloch vector. For larger N the generalized Bloch vector of Eq. (150) is appropriate only for two-dimensional motion, and is a special case of the more general multidimensional coherence vector of Subsection 4.1b. Note that the vector R(t) of Eq. (150) does not, by itself, give any indication of the phase of component CN(t) when C1(t) is zero, as occurs at the end of a pulse sequences that produces complete population transfer.

This presentation of the equation of motion permits various simple geometrical images of the excitation dynamics associated with adiabatic following. When the traditional STIRAP procedure takes place the dynamics occurs in a two-dimensional subspace of the full three-dimensional Hilbert space, N=3. When the Rabi frequencies are real-valued the vector of Eq. (150) obeys a torque Eq. (68) whose angular velocity vector has the components

ϒ1(t)=ΩP(t),ϒ2(t)=0,ϒ3(t)=ΩS(t),
and thus all adiabatic motion occurs in the 2,3 plane. Figure 29 shows two examples of conventional STIRAP produced by Gaussian pulses of equal peak value, as it is described by the variables of Eq. (150). In frame (a) both of the Rabi frequencies are positive and when adiabatic following occurs the probability amplitudes are expressible as
C1(t)cosϑ(t),CN(t)sinϑ(t).

With the customary choice of C1=1 as the initial condition the final value of CN(t) is 1. In frame (b) the two Rabi frequencies, though real-valued, are assumed to have opposite signs. The needed identification for adiabatic following is

C1(t)cosϑ(t),CN(t)sinϑ(t).
Thus this pulse pair produces the final value CN=+1. This sign alteration has no effect on probabilities, nor on R3(t), but it can be inferred from a plot of the mixing angle.

6.6. Structured-Pulse and Pulse-Train STIRAP

The original STIRAP work idealized the pulses as simple Gaussians but complete population transfer can also be produced by pulses that have multiple maxima or minima, and even null values. These can be constructed mathematically from a succession of pulselets that form an overall pulse. Figure 30 shows three examples of pulses that are each constructed from two Gaussian pulselets [260]. Those pulses that undergo a sign change are examples of zero-area pulses; see Appendix F.2. Each of these pulses produces the same, complete population transfer 13, as expressed by r3(t), but with differing results for coherence r1(t) at intermediate times, and with differing signs of final C3. In each of these examples there occurs three complete population transfers,

1SP3,3PS1,1SP3,
separated by mixing-angle plateaus in the course of changing ϑ(t), and so they may be termed examples of triple STIRAP [260].

 

Figure 30 Example of triple STIRAP produced with four pulselets, in the sequence SPSP. Frames are as in Fig. 29. (a) Left-hand frames: all pulselets have the same sign. Final ϑ=0.5π, C1=0, C3=1. (b) Middle frames: the S pulse has zero area. Final ϑ=0.5π, C1=0, C3=1. (c) Right-hand frames: both S and P pulses have zero area. Final ϑ=1.5π, C1=0, C3=+1. The population histories and r3(t) are identical for each choice of pulses but the intermediate coherences r1(t) differ. Reprinted from Figs. 9–11 of [260]. Copyright 2010, with permission from Elsevier.

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6.6a. Discretized-Pulse STIRAP

Although the original concept of STIRAP required continually adiabatic changes to the Hamiltonian, it is possible to produce the principle characteristics of STIRAP with pulses that form a train of impulses [14,48,166,353,370375]—collectively forming a CP [285294]. These can violate the slow-variation condition (i) listed in Subsection 2.3a but still produce complete population transfer. The action of such pulses to produce population transfer has been termed piecewise adiabatic passage (PAP) [353,376,377]. This section discusses some examples.

Figure 31 shows examples of three such pulse trains. The first of these, frames (a), replaces the traditional continuous S and P pulses by successive on-off segments, SPSPSP. The mixing angle associated with each “on” segment increases steadily. The second example-train, frames (b), uses a succession of simultaneous short, smooth pulses whose two constituents, S and P, change from pulse to pulse to produce a steadily increasing mixing angle. The third train, frames (c), uses a succession of ultrashort offset pairs, a smoothed version of the discontinuous pulse trains of frames (a)

 

Figure 31 Examples of piecewise adiabatic passage (PAP). Left: piecewise pulses. Right: populations resulting from these pulses. (a) Seven fragments of smooth curves, turned abruptly on and off. The S and P pulses are simultaneous. (b) Five simultaneous smooth pulses, each with sine-squared shape (cf. Appendix F). (c) Thirteen alternating sin2 pulses. In all three examples mixing angle increases, from pulse to pulse, from 0 to π/2. All examples produce nearly complete population transfer, 13. Figure 1 reprinted with permission from [353]. Copyright 2007 by the American Physical Society.

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Such pulse trains can produce the principal characteristics of STIRAP—the robust and complete population transfer without intermediate-state population—though the individual pulses are too brief to induce adiabatic change; they can be ultrashort pulses rather than the lengthier ones required for continual maintenance of adiabaticity.

Coincident SP pulses, considered also by [291], will place some population transiently into state 2 as the system undergoes a portion of a Rabi oscillation. In that work this population had a maximum value of sin2(π/4N) for N pulses. In the limit of N1, this technique reduces to PAP, while for small N, it is similar to generalized π pulses. Reference [378] proposed a similar scheme, with rectangular pulses, which they named digital adiabatic passage (DAP).

6.6b. Composite-Pulse STIRAP

Pulse trains (composite pulses) have long been used in nuclear magnetic resonance to minimize errors in achieving a desired objective [285,288,379381]. Such work uses a sequence of identical pulselets, separated by a fixed time interval and with controlled relative phases that serve as adjustable parameters. The review [12] discusses applications of this technique in quantum optics and adiabatic passage.

When applied to the two pulses of a STIRAP-like procedure the flexibility of multiparameter optimization offered by composite pulses can permit quantum-state changes that fit any desired goal with very high fidelity. Figure 32 shows an example: a train of simultaneous S and P pulses whose fixed mixing angle varies along the pulse sequence [291]. The succession of discrete interactions produce the complete population transfer to be found with conventional STIRAP but without the traditional adiabatic change of the mixing angle, which here changes by discrete increments.

 

Figure 32 Examples of trains of n simultaneous Gaussian S and P pulses. Plots are time dependences of the following. Top row: Rabi frequencies. Middle row: populations Pn. Bottom row: log Pn. Adapted from Fig. 1 of [291] with permission. Copyright 2012 by the American Physical Society.

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Another example [293], shown in Fig. 33, used a sequence of offset S and P pulse pairs to accomplish the complete STIRAP-like population transfer. Each of the two frames show plots of the final state-3 population P3() versus peak Rabi frequency and delay between S and P pulselet for a fully resonant scenario. Frame (a) shows the results with a single pulse pair, and frame (b) shows the results with a composite of five optimized SP pulselets. In each frame one can see, along a vertical band at the left-hand side, the Rabi oscillations between states 1 and 3 that characterized simultaneous pulses. In vertical bands above a delay around 0.35 one can see the robust population transfer of STIRAP. A green-shaded region marks the regime of ultrahigh efficiency, where P3>0.9999. That region is significantly larger in the right-hand frame, where it forms a relatively large plateau. The availability of such a region of high fidelity makes this procedure attractive for quantum-information processing [113,114,118,119].

 

Figure 33 Contours of final population P3() in desired final state 3 as a function of peak Rabi frequency and delay between S and P pulses. Left: single pulse. Right: composite of five pulselets. For these simulations the pulses have sin2 shapes of equal peak value. Green shaded region is where P3>0.9999. Adapted from Fig. 3 of [293] with permission. Copyright 2013 by the American Physical Society.

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6.7. Branches: Tripod STIRAP

When the number of linked quantum states N exceeds the three of traditional STIRAP it becomes possible to have branches or “star” patterns in which at least one state has links with more than two states. Figure 34 shows examples of such patterns for N=4.

 

Figure 34 Examples of four-state three-field “star”-linkage patterns with state 2 linked to three states. (a) Fan, (b) letter Y, (c) inverted Y, (d) tripod. The energy labels 1,2,3,4 are adapted from those used, in the absence of the Q interaction, for the three-state linkages of Fig. 1. All of these linkage patterns have the same structure for their RWA Hamiltonian. Patterns (b), (c), and (d) are therefore essentially equivalent. In pattern (a) the initially populated state, 2, has multiple links and so the dynamics differs from that of the other linkages where the branching of population flow occurs after one excitation.

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The RWA Hamiltonian is basically the same for each of these four-state linkages when the three fields are each resonant with their transition, although the ordering of the states affects the appearance of the matrix W(t). However, the dynamics of the fan linkage of frame (a) differs from that of the other three linkages because in the fan it is the initial state that has multiple links, whereas in the other linkages it is an excited state that has multiple links.

The addition of a branch onto an existing chain can have dramatic effects on the population flow along the chain, and can act, in effect, to break the chain or enhance it, depending on the number of states in the added branch [13,15,382]. The effects of pulsed branches excitation have also been discussed by [52,383385], and [292] has proposed a method for suppression of unwanted branches. The application of such a branched four-state Hamiltonian to dark-state adiabatic passage, extending the concept of STIRAP, was first considered by [52,53]; see [7].

For these branched linkages a third field, denoted Q here (also termed a control or C field), introduces a branch into each of the three-state patterns of Fig. 1 previously linked by P and S fields. Letting that branch be from state 2 to state 4, and taking each field to be resonant with its associated Bohr frequency, we write the fully resonant RWA Hamiltonian as [52,53]

W(t)=12[0ΩP(t)00ΩP0ΩS(t)ΩQ(t)0ΩS(t)000ΩQ(t)00].
This RWA Hamiltonian allows three independent Rabi frequencies. When one allows nonzero detunings then the tripod, letter Y and inverted Y linkages can fit excitation in which a single pulsed laser fulfills the interaction of a pair of fields. For example, a Lambda linkage, when generalized to replace state 3 by a pair of nondegenerate states, becomes a tripod linkage with detuning: the S and Q fields of the tripod share a common time dependence.

The RWA Hamiltonian of Eq. (155) has two dark states, i.e., states that lack component of state 2, rather than the single dark state of STIRAP. With the present phase choices they have null eigenvalues. Their specification requires two angles, rather than one. The construction used by [7,52,53], differing from that of [12], is

ϕD1(t)=[cosϑ(t)0sinϑ(t)0],ϕD2(t)=[sinφ(t)sinϑ(t)0sinφ(t)cosϑ(t)cosφ(t)],
where the two dynamical angles satisfy the formulas
tanϑ(t)=ΩP(t)ΩS(t),tanφ(t)=ΩQ(t)ΩP(t)2+ΩS(t)2.
The remaining (bright) eigenvectors are
ϕB1(t)=12[cosφsinϑ1cosφcosϑsinφ],ϕB1(t)=12[cosφsinϑ1cosφcosϑsinφ].
The two eigenvalues associated with these are ±12Ωrms(t), where
Ωrms(t)ΩP(t)2+ΩS(t)2+ΩQ(t)2.
Because the two dark states are degenerate the nonadiabatic coupling between them cannot be neglected, even in the adiabatic limit. Instead of identifying the statevector with a single dark state we must write
Ψ(t)=BD1(t)ϕD1(t)+BD2(t)ϕD2(t),
where the dark-state amplitudes satisfy the coupled equations [52]
ddt[BD1(t)BD2(t)]=[0ddtϑ(t)sinφ(t)ddtϑ(t)sinφ(t)0][BD1(t)BD2(t)].
There occur generalized Rabi oscillations between the two dark states, with Rabi angle
AD(t)=tdtddtϑ(t)sinφ(t).
If the system starts in state 1 and the evolution is adiabatic, it will end in a superposition of the two dark states and, in turn, a coherent superposition of RW states. The final Rabi angle (counterpart of the pulse area) depends on the pulse ordering and so by controlling this one can create any desired superposition of the RW states 1, 3, and 4 with a procedure that remains decoherence free. The review [7] gives four examples of varied superpositions produced by different pulse orderings.

7. STIRAP in Composite Systems

Many situations occur in which the quantum system (an atom, say) that undergoes controlled manipulation—by STIRAP or other means—is part of a larger system whose properties change along with those of the manipulated system: the two subsystems form a composite of correlated parts (see Appendix J). This section discusses three examples, in which STIRAP produces alteration of electromagnetic fields. All of these topics are examples of correlations (or entanglement [112,345,386393], cf. Appendix J.2) created by the STIRAP process (or any other coherent-excitation process) between two distinguishable subsystems: a degree of freedom associated with internal excitation and an external degree of freedom, e.g., that of a field.

Altering field energy. A quantum particle undergoing traditional STIRAP excitation gains or loses increments of electromagnetic energy ω from the two driving fields. When a field is relatively weak its incremental changes of energy are observable as photons. STIRAP provides a procedure for manipulating photon states of a field: the STIRAP process, in transferring populations from state 1 to state 3, also transfers photons, from the P field to the S field. These increments become apparent when one of the fields is quantized. Subsection 7.1 and Appendix N describe the formalism for such applications.

Altering field momentum. The fields of traveling waves carry momentum and so the incremental internal-energy change induced by radiation beams acting on moving particles will also accompany incremental alterations of the particle momentum [40,360,361,394401]. A quantum particle undergoing STIRAP excitation of internal degrees of freedom may thereby undergo a discrete, measurable change of center-of-mass (cm) momentum. Subsection 7.2 discusses some of the effects that accompany the coherent energy changes induced by STIRAP in freely moving particles. These momentum manipulations have had application in matter-wave (atom) optics [397,398,402409]. Other examples occur when the excitable particle is trapped in a harmonic potential in which discrete momentum changes occur between the particle and the trapping lattice [129,130,410]. With the inclusion of randomizing effects of spontaneous emission these offer possibilities for cooling and trapping particles [411415].

Altering pulse shape. For many purposes the fields that alter quantum states can be regarded as given, controlled quantities, parametrized by Rabi frequencies [416]. However, a radiation pulse that has passed by an atom, leaving excitation in its wake, will necessarily have undergone change itself. And if it encounters additional atoms its altered properties will affect its next interactions. To treat a spatial distribution of atoms it is necessary to treat fields and atoms in a unified manner, using coupled Maxwell–Schrödinger (or Maxwell–Bloch) equations. Appendix M.3 sketches the traditional Maxwell equations for one-dimensional propagation that augment the TDSE for atom structure, as discussed in the literature of quantum optics. Subsection 7.3 presents one example of these coupled equations, in which dark-state polaritons [7,417421] appear as coherent superpositions of field and bulk-atom states.

7.1. Cavity STIRAP

The fields that contribute to the Rabi frequencies of a TDSE are typically treated as experimentally controlled traveling waves. However, when the system is confined by conducting surfaces, either within an enclosure or between mirrors, or by abrupt changes by refractive index, then the surface boundaries restrict the fields to discrete modes in at least one direction. Such a field must be treated using the quantum theory of radiation, with appropriate operators of creation and annihilation of photons [422427]; see Appendix N. When an excitable quantum particle is enclosed within a cavity whose allowable electromagnetic-field modes are long-lasting and well-separated in frequency, and one of these standing-wave modes is near resonance with an atomic transition, this mode field can induce a single-photon transition for an atom that passes through the cavity [428,429]. The dynamics is that of the Jaynes-Cummings model (JCM) [13,430,431], see Appendix N.6. With the allowance of a second, unquantized laser-beam field, it is possible to create a system that undergoes STIRAP dynamics [7,432438]. The following paragraphs, based on the reviews [7,12], describe a scenario for such an application of STIRAP.

Figure 35(a) shows a schematic layout of the needed apparatus for an atom falling, say from a magneto-optical trap, through a standing wave (the S field) that is constrained along the axis of a cavity bounded by two mirrors. The quantized field of the single-mode cavity, occupied by n photons of frequency ω, provides the S-field coupling 2g(t)n+1, where 2g(t) is the coupling strength in vacuum (n=0). This vacuum Rabi frequency varies with time as the atom moves through the fixed mode structure. Slightly below the cavity axis the atom encounters the P field of a CW laser beam, focused inside the cavity, through which the atom also passes. In this cavity-STIRAP (or vacuum-STIRAP) the two fields form the branches of a single-atom Raman transition, as shown in Fig. 35(b). With the geometry of frame (a) an atom falling through the cavity experiences the S and P fields as a counterintuitive pulse sequence, thereby offering a possible implementation of the STIRAP procedure.

 

Figure 35 (a) Schematic of experimental setup used for cavity STIRAP: an atom drops through a resonant cavity to encounter, below the cavity axis, a laser beam. The atom encounters first the cavity field and then the laser field. (b) Linkage pattern for vacuum-STIRAP. The labels g,e,x refer to atomic levels and 0,1 refer to the cavity photon number. Here the initial state is an atom in state e and a cavity vacuum, |e,0. The final state, producing the outgoing photon of frequency ω, is the ground atomic state plus a cavity photon, |g,1. The pulsed P field is Ω(t), the S field is the coupling 2g(t) between the atom and the standing-wave cavity mode of frequency ω. For similar figures see [12,434438].

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The needed atom-field quantum states are taken as products |system=|atom|n of an n-photon Fock state |n with three atom states, a ground state g, an excited state e, and an auxiliary state x. For given n the three states used for STIRAP are

ψ1(t)=|e,n,ψ2(t)=|x,n,ψ3(t)=|g,n+1.
The Rabi frequency Ω(t) of the laser field P induces transitions between atomic states e and x but has no influence on the cavity field. The cavity field S produces atom excitation ex and loss of a cavity photon nn1. It also induces atomic de-excitation xe along with creation of a cavity photon nn+1. Cavity photons eventually leak out of their enclosure (to become freely traveling photons) at a rate γ. Atoms in state x can also undergo spontaneous emission, at rate γ, returning to atomic states e and g by radiation into noncavity modes. When the P and S fields are resonant with their respective Bohr frequencies the three-state RWA Hamiltonian has the matrix
W(t)=12[0Ω(t)0Ω(t)2iγ2g(t)02g(t)2iκ].

The dark atom-field state with energy En=nω has the construction

ϕD(t)=cosϑ(t)|e,nsinϑ(t)|g,n+1,
where the mixing angle is
tanϑ(t)=ΩP(t)/2g(t)n+1.
In the adiabatic limit, the STIRAP procedure induces complete transfer |e,n|g,n+1, allowing negligible population in the decaying excited state |x,n at any time. Because the S field is a quantized cavity mode, the usual adiabatic condition becomes |ΩPTP|1 for the P field, and 2gmaxT(n+1)1 for the S field. When the cavity is initially empty (n=0) the passage of an atom through the cavity leaves a single-photon field: the STIRAP mechanism converts a single P photon of the laser beam into a single S photon in the cavity, with accompanying atom excitation. This photon eventually tunnels through one of the bounding surfaces of the cavity, at a rate κ, to become a traveling-wave photon.

7.2. Correlated Center-of-Mass Motion

A particle moving with cm momentum p=k, upon gaining energy ω from a laser beam, will also gain momentum ΔpL=(ω/c)eL, where eL is a unit vector along the laser-propagation axis (and thus perpendicular to the E and B fields that drive the energy changes). Figure 36 sketches the geometry of the resulting deflection; see [394].

 

Figure 36 Geometry of particle-beam deflection. Particle with initial momentum p=k gains increments of momentum ΔpL=(ω/c)eL from traveling-wave laser beam and is thereby deflected by angle ϑ.

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This quantization of momentum increments underlies coherent deflection and splitting of atomic and molecular beams [40,360,361,394401] and leads to “matter-wave optics” or “atom-optics” analogues of classical wave optics [397,398,402409]. When combined with effects of spontaneous emission, which produce momentum decrements in random directions, it leads to procedures for cooling and trapping particles [411415,439441].

To treat cm motion of a particle acted on by a traveling electromagnetic field (e.g., a laser beam of frequency ω and propagation axis defined by a unit vector eL) we express the statevector as summed products of internal-excitation states ψn or ψn(t) and states of cm motion ϕm with discrete well-defined values of linear momentum along the propagation axis of the field—viewable either as a classical Fourier series expansion or photon-momentum-state expansion. When only a single field is present the particle-cm momentum p=k is expressible as a steady initial value p=k perpendicular to the laser beam, to which are added m discrete increments of laser-field momentum kL=(ω/c)eL:

k=k+m(ω/c)eL.
The states ϕ needed for describing the cm motion can be labeled by an integer m (positive, negative, or zero) in the construction of a statevector that describes the combined internal excitation and cm motion of a particle (an atom-cm system):
Ψ(t)=nCn(m;t)ψn(t)ϕm.
The two indices n, m are interdependent: when the internal states form a simple chain, with ordering En<En+1, then an increase of n to n+1 will accompany a decrease of m to m1. Thus the restriction of the statevector to a description of a system of fixed total energy limits the summation to n alone.

When there are two traveling-wave fields, P and S (as with pump and Stokes fields but also with counterpropagating contributors to a standing wave [394,396]), then the particle cm momentum is expressible in the form

k=k+mP(ωP/c)eP+mS(ωS/c)eS,
and the statevector requires a sum involving triple products, of the form [394]
Ψ(t)=nCn(mP,mS;t)ψ1(t)ϕmPϕmS.
Again the field indices, mP and mS, are correlated with the excitation label n. For example, the three states required for a conventional STIRAP process (Lambda linkage) are
|atom,Pfield,Sfield:  |1,mp,mS,|2,mp1,mS,|3,mp,mS+1.
The 3×3 matrix of the RWA Hamiltonian for this atom-cm system mimics that of STIRAP. Each change of internal energy accompanies a discrete change of cm momentum. Population transfer between system states 1 and 3 will accompany two momentum increments, one from each field. When the two laser beams are counterpropagating (eP=eS) these increments add constructively and thereby produce a deflection of the particle beam (a grazing-incidence mirror) [40,361].

7.3. STIRAP and Dark-State Polaritons

When a weak probe-field pulse (P) passes into a medium in which it resonantly couples states 1 and 2, and that medium is subjected to a strong control field (S) that resonantly couples state 2 to a third state 3 (either in a Lambda or ladder linkage), the propagation of the P field is strongly affected. A strong and steady S field will alter the resonance condition of the weak P field and lead to electromagnetically induced transparency (EIT) [155,421,442444]. With increasing strength of the control field, the group velocity vg of the P-field pulse diminishes and its electric field transforms into a spatial-distribution field of atomic coherences. The required slow adjustment of the S field, and the use of the dark state, has features in common with STIRAP, and so the procedure has been included in reviews of that process [7,12]. The text below repeats that discussion.

The simplest starting point for a description of this behavior is an idealization of the electric field envelopes as pulses that move in one direction, taken to be z; see Appendix M.3. For the P field traveling through a constant and uniform number density N, the slowly varying envelope, expressed as a Rabi frequency ΩP(z,t), satisfies the equation

(z+1ct)ΩP(z,t)=iαpΓ2C2(z,t)C1*(z,t),
where Γ denotes the decay rate of state 2 and the P-field resonant absorption coefficient,
αp=ωpN|d12|22ε0cΓ,
parametrizes the effect of the atoms on the P field. A similar equation describes the S field, which is taken to be sufficiently strong to be relatively unaffected during its travel at speed c.

The probability amplitudes that serve in Eq. (172) as sources to spatial changes in Rabi frequency satisfy the usual three-state TDSE: the probability amplitudes C1 and C2 are linked by the P field, while the S field links C2 and C3. Adiabatic elimination of these atomic probability amplitudes produces the equation

(z+1vgt)ΩP(z,t)=0,with  vg=c1+ngandng=αPΓcΩrms2.
By adjusting the intensity of the S field, and thereby controlling the rms Rabi frequency, it is possible to control the propagation velocity vp of the P pulse, to bring it to a full stop, and to re-accelerate it on demand. This behavior is associated with the existence of a quasi-particle called a dark-state polariton [7,417421] that is a coherent superposition of electric field (i.e., Rabi frequency) and macroscopic atomic-coherence components:
F(z,t)=cosθ(t)Ωp(z,t)sinθ(t)αcΓC3(z,t)C1*(z,t).
The angle θ (not to be confused with the mixing angle ϑ used earlier) is defined by
tanθ(z,t)=αcΓΩs(z,t).
The dark-state polariton obeys the simple propagation equation
[t+ccos2θ(z,t)z]F(z,t)=0.
If ΩS (and hence θ) is approximately uniform in z, Eq. (177) describes a form-invariant propagation of the quasi-particle with propagation velocity:
v(t)=ccos2θ(t).
By causing the S-field Rabi frequency to adiabatically rotate θ(t) from 0 to π/2 one can transform an initially pure electromagnetic polariton (F=Ωp) into a pure atomic polarization (F=αcΓC3C1*). Concurrently the polariton propagation velocity slows from the vacuum speed of light to zero. The P pulse is thus “stopped”: its coherent information is transferred to collective atomic states. The atomic polarization can be extracted by reversing the transfer process, i.e., rotating θ back from π/2 to 0 and recreating the P pulse.

The stopping and release of the incident P pulse makes possible storage of the phase as well as the amplitude of a field, unlike conventional photographic techniques or radiation detectors, which record only intensities. Thus the technique has drawn interest from those who wish to store photons [134,445,446].

8. STIRAP-Inspired Examples

The basic principles of Subsection 2.3 that define the traditional STIRAP process need not be limited to excitation of atoms or molecules, or even to quantum systems. The Rabi frequencies and detunings need not derive from laser pulses and the three dependent variables need not be probability amplitudes. The following section illustrates some of the possibilities for adapting the basic STIRAP concept to such applications [10,12].

8.1. Tunneling STIRAP

The basic TDSE can be applied to describing the localization of a particle within adjacent potential wells (traps) whose separation is adjustable. When neighboring traps are sufficiently close together the particle, treated as a matter wave, can tunnel between them, in a manner described by an ODE equivalent to a TDSE in which the Rabi frequency is the tunneling rate. By diminishing the trap separation, or by lowering the barrier between traps, an experimenter can increase the Rabi frequency, thereby enhancing the rate at which probability passes between traps.

A chain of three potential wells, as shown schematically in Fig. 37, provides an opportunity for STIRAP-like probability transfer that has been termed coherent tunneling by adiabatic passage (CTAP) [447]. Timed counterintuitive ordering of S and P interactions moves the particle between ends of the trapping-well chain without allowing either thermal excitation of confinement vibration or localization in the intermediate well. The model of frame (a) was implemented by [448,449], using potentials of optical lattices and trapped neutral atoms. The schematic model of frame (b) was implemented by [447] in a construction of quantum dots between which tunneling was adjustable by gates. For further examples, see [12].

 

Figure 37 Examples of a chain of three potential wells separated by two barriers. Alteration of barrier widths or heights adjusts Rabi-frequency rates of tunneling, ΩP and ΩS. (a) Rabi frequencies are altered by changing the separation of wells, and with that the barrier width. (b) Rabi frequencies are altered by changing the barrier heights.

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8.2. Laser-Beam Polarization Optics; Propagating the Stokes Vector

The description of two-state excitation formulated by [366] relied on three real-valued combinations of the complex-valued elements of the density matrix ρ(t) formed from products of two complex-valued probability amplitudes Cn(t). The concept is analogous to the formation, in the classical optics of polarized radiation, of the three-component Stokes vector S from bilinear products of two complex-valued electric field amplitudes Ei(t) for a collimated beam of radiation; cf. [15,450]. The three components of this vector define a point on a unit sphere, the Poincaré sphere. The effect of one-dimensional beam propagation through optical material that induces negligible reflection can be regarded as motion of a point on this sphere, and can therefore be presented in the form of a torque-like equation [287,451457], analogous to the motion of the Bloch vector on the Bloch sphere. Here the elements of the torque vector are taken from the birefringence vector of the medium. Various authors have noted the analogy and have examined the application of STIRAP-like principles to treat controlled alteration of laser-beam polarization [287,456,457].

8.3. Fiber-Waveguide Analogs

A waveguide for electromagnetic radiation is a structure that allows free propagation along one axis, taken to be z, while confining the field in the transverse x, y plane. An optical fiber is one example, wherein the confinement in the transverse (x,y) plane originates with a transverse variation of the refractive index n(x,y,z) of the fiber cladding and n0 of the core; see Appendix M.5. Outside of the fiber the field is evanescent, falling exponentially in magnitude with separation distance from the fiber axis. This evanescent field allows a coupling between the radiation flowing in adjacent fibers, a tunneling between confinement regions. The strength of this coupling, and its consequent transfer of radiation energy from one fiber to its neighbor, increases as the fibers become closer together, and so by suitably laying out a network of fibers it is possible, in principle, to control the flow of radiation among the fibers. The equations that govern the transfer of electric field between adjacent fibers as a function of distance z are analogous to the ODEs that govern the variation with time t of coupled probability amplitudes [458]. This analogy makes possible the implementation of adiabatic passage procedures that transfer light between fibers in a manner reminiscent of the STIRAP transfer of quantum-state populations, a procedure that has been termed spatial adiabatic passage (SAP) [459461]. The review [12] discusses a number of fiber-waveguide systems that have dynamics analogous to STIRAP and to related adiabatic procedures. The next two subsections summarize the physics of such systems.

8.3a. Coupled-Mode Equations

The multiple quantum states of an excitable quantum particle have an optical counterpart in sets of identical single-mode optical waveguides, most simply forming a planar layout of nearly parallel and independent propagation channels. When the waveguides are well separated and straight, they serve as independent modes of lossless field propagation, see Appendix M.3. The counterpart of the basis-state expansion (22) of the statevector that underlies the derivation of the coupled ODEs for probability amplitudes from the TDSE is the expansion of an electric field amplitude (for a monochromatic traveling wave in the paraxial approximation) in terms of discrete transverse-field modes fn(x,y):

E(x,y,z)=nCn(z)exp[iβnz]fn(x,y),
where βn is the propagation constant. When separation between waveguides becomes sufficiently small, there will exist transverse evanescent waves that allow the propagating radiation to tunnel coherently between adjacent waveguides. This concept was applied to three-waveguide systems by several researchers [462469]. Their basic equations are an adaptation of the resonant equations used for STIRAP but with distance z in place of time t:
iddz[C1(z)C2(z)C3(z)]=12[0ΩP(z)0ΩP(z)0ΩS(z)0ΩS(z)0][C1(z)C2(z)C3(z)].
The two Rabi frequencies ΩS(z) and ΩP(z) appearing here quantify the tunneling rate between adjacent fibers 1–2 and 2–3, respectively.

8.3b. Waveguide STIRAP Analog

The possibility of a dark mode of Eq. (180) was noted by several papers [463,464,466,467,470]. The analogy with STIRAP, made possible by spatial positioning of the evanescent couplings and ensuring the tunneling rates vary adiabatically with distance, was noted by [466469] and was demonstrated experimentally by [164].

The demonstration system comprised three waveguides lying on a flat surface, the x, z plane, with axes predominantly along the z direction and closely spaced in the x direction. As with the timing of Rabi frequencies for STIRAP, the spatial proximity of adjacent waveguides was adjusted so that the couplings occur sequentially but overlapping. With increasing z beyond the point of radiation injection the waveguides 3 and 2 draw closer together in the x direction, thereby establishing a S interaction. Further along waveguide 1 approaches waveguide 2, initiating the P interaction. Still further along the three waveguides separate, first waveguide 3 (the S interaction) and then waveguide 1 (the P interaction). Waveguides 1 and 3 remain well separated and so they have no direct coupling.

To mimic the counterintuitive pulse sequence of STIRAP the field is injected into waveguide 1. When the changes to field amplitudes occur adiabatically there occurs complete transfer of intensity between waveguides 1 and 3. Little field energy occurs in waveguide 2. Figure 38 shows a simulation of the predicted behavior, along with plots of relevant experimental confirmation, both for the STIRAP analogy (S preceding P) and for Rabi-oscillating analogy of intuitively ordered interactions (PS). Because the tunneling rates are relatively insensitive to wavelength, the behavior is achromatic.

 

Figure 38 Transfer of light intensity in a three-waveguide optical system. (a) Numerical simulation of the light intensity distribution versus propagation distance z. Locations of S and P interactions are marked. (b) Measured fluorescence pattern as recorded on a CCD camera above the sample. (c) Corresponding fractional beam power trapped in each waveguide versus propagation distance z. Left: initial intensity in waveguide 1, with counterintuitive ordering of SP interactions appropriate for STIRAP analogy. Right: initial intensity in waveguide 3, with intuitive interaction ordering PS conducive to Rabi oscillations. Adapted from Figs. 4 and 5 of [164]. Copyright 2002 by the American Physical Society. Also Fig. 60 of [12], Fig. 8 of [471], and Fig. 6.8 of [9].

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8.3c. Beam Splitters and Fractional STIRAP in Waveguides

Beam splitters are an important element in the assortment of tools used in optical-beam manipulation, and they were an early application of STIRAP in atom optics, as discussed in Subsection 7.2. In the context of waveguides beam splitters are devices that can divide a single propagating mode into two or more modes, each in a separate waveguide. Several designs based on adiabatic passage have been proposed to create such devices [348,472,473]. The analogy between three coupled waveguides and STIRAP provides a means of accomplishing the splitting of an incoming beam into two outgoing waveguides, by means of fractional STIRAP. The principle is simple and has been demonstrated for the half-STIRAP protocol [347]. In a layout that would transfer field from waveguide 1 to waveguide 3 by means of a STIRAP-like interaction sequence, cut off the central waveguide 2 midway through what would be a STIRAP sequence, at a position where the P and S amplitudes are equal. The cessation of the interactions fixes the subsequent values of the fields and creates an achromatic 50:50 beam splitter. An alternative procedure uses the two-state STIRAP of Subsection 6.5g, made possible by creating a waveguide array in which the propagation constants vary with propagation distance [474].

8.4. Classical Analogs of STIRAP

Systems whose behavior is governed by classical equations of motion (i.e., Newtonian mechanics) and whose description requires only three variables offer opportunity for applying the conditions of Subsection 2.3a that define STIRAP. The review [12] discusses a number of these, some of which the following paragraphs mention as illustrative of possibilities for analogs of STIRAP.

8.4a. Classical Torque Equations

The three-dimensional torque equation of Eq. (68) occurs commonly in the classical mechanics of rotating objects, and these examples offer opportunities for introducing adiabatic-following procedures analogous to STIRAP. The Lorentz force of a magnetic field acting on a moving charged particle is one example. The STIRAP analogy applied to motion subject to the Lorentz force allows one to change the direction of a moving particle by 90° in a robust way, independently of its mass and charge [475].

8.4b. Coupled Classical Oscillators

Harmonic oscillators have a long and distinguished history, in quantum-mechanical descriptions of electromagnetic fields and molecular vibrations and in the classical description of electric circuits and gently swinging pendulums. All such classical systems offer opportunity for creating analogies with STIRAP or other examples of adiabatic following by controlling the timing of slowly changing parameters.

Coupled pendula. In seeking a mechanical model that would offer understanding of the population-trapping state, [476] built a device comprising a row of three pendulums, hung from a support and coupled by springs, with damping of the center-pendulum motion—a “triple pendulum.” Later [477] examined the coupled equations that describe such a set of three classical coupled pendulums in order to model the stimulated Raman interaction. Although this model provided a simple physical description of the resonance Raman process for steady radiation fields, the strength of the springs in such a construction are necessarily unchanging: the model could not accommodate the time-varying interactions needed for STIRAP. A possible revision of the mechanism to allow the slow variation of spring constants needed for analogy STIRAP was proposed by [478]. Such a construction could demonstrate the transfer of oscillating motion between the two pendulums at the ends of a support.

Coupled induction loops. A classical system of capacitatively loaded current loops, inductively coupled through an intermediate loop, has a description involving three coupled equations for the loop currents [479]. Appropriate slow variation of the loop geometry brings linkage changes analogous to the slowly changing Rabi frequencies of STIRAP. The analogy has been proposed as a procedure for achieving wireless energy transfer [479].

9. Summary

The STIRAP procedure has migrated from a technique for inducing molecular excitation to a tool for manipulating general quantum states or other systems described by discrete variables and coupled, linear first-order ODEs. It has engendered an increasing number of applications in physics, chemistry, and engineering, reviewed in [10,12]. The basic three-state equations (2), with their restrictions and consequences of Subsection 2.3, have been extended to a variety of multistate systems [7,8,10,12] and the observables of interest now extend beyond quantum-state populations to include coherences and phase relationships. (Still further extensions, largely unmentioned here, deal with procedures that involve time-varying detunings to produce desired changes.) Depictions of system dynamics discussed in this review offer insights into how extensions of, and alternatives to, the original STIRAP prove useful for contemporary tasks. As new technology becomes available, further diversity of usage is to be expected, limited only by the imagination of users.

Appendix A: Acronyms

For further examples related to STIRAP see [9].

AF = Adiabatic following [184,342,367,480481]

AT = Adiabatic transfer (state) [7,41,42,188,280]

ARP = Adiabatic rapid passage (also AP, adiabatic passage), see RAP

CAPTA = Chirped adiabatic passage by two-photon absorption [63,163]

CC = Coherent control [312,313,315320,322325,482,483]

CHIRAP = Chirped Raman adiabatic passage [167,168] [169,170]

CP = Composite pulse (for composite adiabatic passage, CAP) [281,289,290,293296]

CPR = Complete (coherent) population return [15,370,484487]

CTAP = Coherent tunneling by adiabatic passage [447]

DAP = Digital adiabatic passage [378]

DEQSIE = Designer evolution of quantum systems by inverse engineering [120]

DF = Diabatic following

DFS = Decoherence-free subspace [241249]

EIT = Electromagnetically induced transparency [155,421,442444]

JCM = Jaynes–Cummings model [13,430,431]

LZMS = Landau–Zener–Majorana–Stueckelberg two-state model [156160]

OC = Optimal control (theory) [330333,338,488491]

PAP = Piecewise adiabatic passage [353,376,377]

QIP = Quantum information processing [113,114,118,119]

RAP = Rapid adiabatic passage [341344]

RW = Rotating wave (picture)

RWA = Rotating-wave approximation [15,200,492]

RCAP = Raman chirped adiabatic passage [6,8,161166]

SACS = Stark-assisted coherent superposition [71]

SAP = Spatial adiabatic passage [459461]

SARP = Stark-induced adiabatic Raman passage [180183]

SCRAP = Stark-chirped rapid adiabatic passage [8,171179]

SQC = Semiconductor quantum circuit [145,493495]

SRS = Stimulated Raman scattering [8386]

STA = Shortcut to adiabaticity [286,293,300308]

STIHRAP = Stimulated hyper-Raman adiabatic passage [205,255,364,365]

STIRAP = Stimulated Raman adiabatic passage [4,610,12,14,15]

Appendix B: Rabi Frequency and Rabi Oscillations

The TDSE for a lossless, resonant two-state system,

iddtC1(t)=12Ω(t)C2(t),iddtC2(t)=12Ω(t)C1(t),
has, for initial values Cn(0)=δn,1, the solutions
C1(t)=cos[A(t)/2],C2(t)=sin[A(t)/2],
where A(t) is the definite integral, to time t, of the Rabi frequency (i.e., the Rabi angle),
A(t)=0tdtΩ(t).

B.1. Pulse Area

When the integral of A(t) is over the complete duration of a pulse this Rabi angle is the (temporal) pulse area A, a count of the number of completed two-state population transfers. A so-called pi pulse is one, of whatever temporal form, that produces a Rabi angle of π and with it complete population inversion, P1P2 for resonant excitation.

For constant Rabi frequency Ω(t)Ω the probabilities for resonant excitation from initially populated state 1 are cyclic:

P1(t)=12[1+cos(Ωt)],P2(t)=12[1cos(Ωt)].
Such probability oscillations are traditionally known as Rabi oscillations, and their frequency, the Rabi frequency [201], is Ω. What appears in the TDSE, Eq. (B.1), is a half Rabi frequency, Ω/2: the Rabi frequency is twice the value of the interaction energy divided by .

Oscillatory solutions to the N-state TDSE occur for a variety of constant Rabi frequencies, e.g., [356,357]. In such situations the pulses required for complete population transfer are said to be generalized pi pulses.

B.2. Population Return: CPR

The two-state TDSE with constant detuning Δ,

iddtC1(t)=12Ω(t)C2(t),iddtC2(t)=ΔC2(t)+12Ω(t)C1(t),
has more complicated solutions than does the resonant equation, Eq. (B.2). For very smooth pulses, such as a Gaussian, the statevector will undergo adiabatic following and return population to initial values: there occurs complete (coherent) population return (CPR) [15,370,484487]. By contrast, an abruptly changing pulse, such as a rectangular pulse, will induce population oscillations. As the detuning Δ becomes large the permanent transfer of population becomes negligible. Figure 39 illustrates this behavior.

 

Figure 39 Final excitation probability P2() as a function of (static) detuning Δ and temporal pulse area A for two-state system. White areas are complete population return (CPR). (a) Gaussian pulse, (b) rectangular pulse. Adapted from Fig. 15 of [155]. With permission from Springer.

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Appendix C: Rotating-Wave Approximation (RWA)

The RWA allows, in the coupled ODEs of the time-dependent Schrödinger equation, the replacement of Hamiltonian elements that vary with the relatively rapid carrier frequencies of the pulsed fields by elements that vary with the more slowly changing pulse amplitudes. In so doing it introduces a rotating set of Hilbert-space coordinates ψn(t), cf. Subsection 2.2a.

Two states. Specifically, for a two-state system the RWA introduces replacements of the form

1+exp[±i2ωt]1.
The omitted exponentials are known as counter-rotating terms; various articles discuss their effect [255,496]. The resulting probability amplitudes represent behavior that has been averaged over the relatively rapid but small oscillations induced at the carrier frequency ω, leaving the slower but larger changes induced at the Rabi frequency Ω, cf. [15,478]. This simplification can become invalid for ultrashort pulses that comprise only a few cycles of the optical field [497] or when the peak Rabi frequency becomes relatively large. The assumption of a large carrier frequency ω relative to the change-inducing Rabi frequency Ω can be regarded as the requirement of a photon energy much larger than the interaction energy:
|Ω|ω.
However, when the carrier frequency becomes very large, the consequent diminution of the wavelength can lead to breakdown of the dipole approximation. Momentum change will also become important.

Multiple states. When the two fields of the three-state system have different polarizations then the dipole selection rules provide a unique assignment of the fields to the transitions; see Appendix K. However, when the two polarizations are the same, then one must rely on frequency selectivity to remove the possible ambiguity of the association of a given pulsed field with a particular two-state linkage. This the multifield RWA does. In a multistate system driven by fields of various carrier frequencies the RWA makes replacements of the form

sum:  1+exp[±i(ωj+ωk)t]1,beat: 1+exp[±i(ωjωk)t]1.
The first of these, applicable to sum frequencies, merely extends the two-state RWA. The second of these replacements, applicable to beat frequencies, requires that when selection rules allow more than one field to contribute to a particular linkage, these fields should have sufficiently different carrier frequencies that all but one of them can be neglected: each transition is driven, unambiguously, by a specific field. When this assumption fails the RWA Hamiltonian will contain terms that vary with beat frequencies of the fields, ωjωk [125].

Appendix D: Dark-State: Eigenvalue and Eigenvector

The existence of an adiabatic state—an instantaneous eigenstate of a RWA Hamiltonian W(t)—that lacks state-2 component is readily seen from matrix multiplication WC [9]. For the three-state chain with two-photon resonance the relevant result is

[aP*0PDS0S*a][S0P]=a[S0P].
In general all of the elements a,b,P,S may be time dependent. As can be seen, the 1,1 and 3,3 elements of the matrix W must be equal in order for the dark state to be an eigenvector. This is the condition for two-photon resonance. The common association of the dark state with the null-value eigenstate rests on a choice of phases ζn that nullifies the diagonal element a.

The dark state for any tri-diagonal matrix can be found [358] by carrying out similar matrix multiplication WC and setting the result to zero. For the five-state letter-M linkage with real Rabi frequencies the matrix multiplication gives

[0P000PDS000S0P000PDS000S0][C10C30C5]=[0PC1+SC30PC3+SC50]=0,ϕD=1N[SS0PS0PP].
Here N is a normalizing factor. The proper choice of amplitudes Cn to make this result the null vector yields the dark state ϕD(t) [40,63,67,358,260,498]. The occurrence of products of two Rabi frequencies as components of the dark state show that its structure relies on two-photon transitions along the three-state chain 1–3–5.

Appendix E: Vector Products and Torque Equations

The torque equation (68) relies on notation×for the vector product of two real-valued three-dimensional vectors,

ddtR=ϒ×R,
and symbolizes three equations for components of vectors—cyclic permutations of the component equation
ddtR1=ϒ2R3ϒ3R2.
Thus Eq. (E.1) can also be written in matrix form as
ddtR=MR,where  R=[R1R2R3],M=[0ϒ3ϒ2ϒ30ϒ1ϒ2ϒ10].
Conversely, any real antisymmetric matrix M can serve to define a torque vector ϒ:
ϒ1=M23,ϒ2=M13,ϒ3=M12.
The most general such vector has three components, as would be appropriate for a loop-linkage pattern. The more limited chain linkage is associated with a torque vector that has only two nonzero elements, say ϒ1 and ϒ3, and that therefore guides adiabatic following of R in two dimensions, the 1,3 plane.

Appendix F: Pulse Shapes

Within the RWA it is the pulse envelope E(t) that determines the system dynamics, through the TDSE. Figure 40 shows two examples of pulse envelopes, covering underlying field oscillations at carrier frequencies. The envelopes shown here, and other often-used examples, are of two classes, distinguished by their support (the range of independent variable t for which the envelope is nonzero): infinite support and finite support. Following are defining formulas for common examples of pulse pairs with peak value Ω0 and centers separated by τ/T.

 

Figure 40 Pulse envelopes (black solid lines) covering carrier-frequency field oscillations (red curves). (a) Rectangular pulse, an example of a pulse that has finite support. (b) Gaussian pulse, an example having infinite support.

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F.1. Envelope Examples

Infinite support. These have support over the infinite interval <t<+, with time t=0 defined as the instant of peak value. When used in numerical simulation these must be limited to a finite interval: at the start and end of this interval they have discontinuities, although those may be negligibly small if the interval is large.

Gaussian: pulse area A=1.772  Ω0T, FWHM=0.833T:

ΩP(t)=Ω0exp[(tτ/2T)2],<t<+,
ΩS(t)=Ω0exp[(t+τ/2T)2],<t<+.
Gauss six: pulse area A=1.855  Ω0T, FWHM=0.815T:
ΩP(t)=Ω0exp[(tτ/2T)6],<t<+,
ΩS(t)=Ω0exp[(t+τ/2T)6],<t<+.
Hyperbolic secant: pulse area A=3.141  Ω0T, FWHM=1.317T, see [209,499]:
ΩP(t)=Ω0sech(tτ/2T),<t<+,
ΩS(t)=Ω0sech(t+τ/2T),<t<+.
Finite support. These have support over an interval of duration T. Typically the time t=0 is taken to be the start of the first pulse. The sine-squared pulse has no discontinuity in value, the rectangular pulse has a discontinuity at the start and end.

Sine squared: pulse area A=0.5  Ω0T, FWHM=0.5T:

ΩP(t)=Ω0sin2(πtτT),τtT+τ,
ΩS(t)=Ω0sin2(πtT),0tT.
Shark fin: pulse area A=Ω0T, FWHM=0.523T, see [9,15,254257]:
ΩP(t)=Ω0sin(πt/T),0tT,
ΩS(t)=Ω0cos(πt/T),0tT.
Rectangle: pulse area A=Ω0T, FWHM=T:
ΩP(t)=Ω0,τtT+τ,
ΩS(t)=Ω0,0tT.

F.2. Zero-Area Pulses

Although it is common to deal with simple pulses that have a single maximum, as does a Gaussian, more complicated shapes may include envelopes with multiple maxima and both positive and negative values, separated by a node. The left side of Fig. 41 shows examples of such pulses, each with the same carrier frequency and the same intensity centered at time t=0, but with different envelopes (solid lines) before and after t=0. In frame (a) the envelope has a single maximum. In frame (b) there occurs a phase change of π at t=0. The subsequent envelope is negative, and its integral exactly reverses the accumulation of pulse area that occurred prior to t=0. When the positive values exactly balance the negative values the pulse area will be zero (a zero-area pulse) [260,500502]. The pulse of frame (c) can be regarded as a two-component pulse train.

 

Figure 41 Left: pulse envelopes (and carrier fields) illustrating zero-area pulses. (a) A simple Gaussian pulse, (b) a zero-area pulse, (c) a two-component pulse train, with phase change between components. Right: dynamics of two-state resonant excitation by a zero-pi pulse, composed of two 5π segments, as in frame (c). (d) The Rabi frequency Ω(t) for a zero-area pulse. (e) The adiabatic eigenvalues ϵ±(t). The two curves meet, but do not cross, at t=0. (f) The populations Pn(t). There is complete population transfer at the midpoint, followed by complete population return.

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A resonant zero-area pulse, acting on a two-state system, will produce temporary excitation but will, upon conclusion of the pulse, return all population to the initial state. The right side of Fig. 41 shows an example, composed of two sequential 5π segments of opposite sign, as in frame (b). At the null of the pulse envelope, t=0, the Rabi angle is 5π and so there occurs complete population transfer at t=0. Upon completion of the pulse there occurs CPR.

Appendix G: Adiabatic Elimination

Justification for a simple two- or three-state model of coherent excitation relies on the assumption that only two or three quantum states are near resonant with the applied radiation fields. In the multistate TDSE all other states appear with detunings that are large compared with Rabi frequencies and so these states never acquire appreciable populations. The simplest example is that of a set of three-state chains 1i2 in which the intermediate state i is far from resonance and there is no direct link between states 1 and 2. In this extension of the three-state Lambda linkage we have just two fields, P and S, and we write the off-diagonal elements of the RWA Hamiltonian in the form

W1i(t)=12ΩP(t)pi*,Wi2(t)=12ΩS(t)si,
where ΩP(t) and ΩS(t) are the Rabi frequencies for some convenient index i, say i=3, and the constant factors pi and si are the values of dipole transition moments relative to those of this reference state. The TDSE then reads [280]
iddtC1(t)=Δ1C1(t)+12ΩP(t)ipi*Ci(t),
iddtCi(t)=12ΩP(t)piC1+ΔiCi(t)+12ΩS(t)siC2(t),
iddtC2(t)=12ΩS(t)isi*Ci(t)+Δ2C2(t).

When the intermediate-state detuning Δi is sufficiently large the amplitude Ci(t) varies rapidly but with small magnitude. The effects of interest (the complete transfer of population) take place over many such oscillations. To describe them we introduce amplitudes C¯i(t) that average over these rapid oscillations. The averaging nullifies the time derivative of C¯i(t), which is then expressible from Eq. (G.2b) as

C¯i(t)=ΩP(t)pi2ΔiC1(t)ΩS(t)si2ΔiC2(t).
We substitute this approximation into the remaining two equations, an example of adiabatic elimination, to obtain a pair of coupled equations for states 1 and 2.

G.1. Effective Hamiltonian; Dynamic Stark Shifts

The pair of equations obtained by adiabatic elimination of nonresonant states provide an example of a two-state effective Hamiltonian [172,174,280,487,503]:

Weff(t)=[012Ω12(t)12Ω21(t)ΔS2(t)+S1(t)].
Here the 1,1 element has been nullified by the RW phase choices
ζ1(t)=E1t0tdtS1(t),ζ2(t)=ζ1(t)+ωt.
The matrix Weff(t) involves a two-photon Rabi frequency,
Ω12(t)=Ω21(t)*=ΩP(t)*ΩS(t)ipi*si2Δi,
proportional to the product of two Rabi frequencies and time-dependent frequency shifts (dynamic Stark shifts) proportional to the squares of Rabi frequencies (i.e., to radiation intensities)
S1(t)=|ΩP(t)|2i|pi|24Δi,S2(t)=|ΩS(t)|2i|si|24Δi,
that augment the original two-photon detuning Δ. Note that such dynamic Stark shifts necessarily accompany a nonzero two-photon transition and vice versa; the two contributions to the RWA Hamiltonian cannot be considered as independent. The two dynamic shifts combine to produce, during the presence of the two-photon Rabi frequency, a chirped detuning of the sort that underlies the traditional RAP. Subsection 2.3d referred to examples of such pulsed two-state models as SCRAP and SARP.

In principle, one could redefine the phases ζn(t) to eliminate completely the diagonal elements of the matrix Weff(t)—a generalization of the Dirac picture. However, that redefinition would place an exponential time dependence into the off-diagonal element, and the resulting two-photon Rabi frequency would not be slowly varying.

G.2. Induced Dipoles; Polarizability

The basic dipole interaction is proportional to the product of an electric dipole moment d projected onto an electric field vector E(t). As the fields grow stronger the dipole moments alter in response. This distortion of atomic structure can be taken into account by introducing an effective Hamiltonian in which there appears a dipole interaction that is nonlinear in the electric field. The changes amount to induced-dipole moments. The lowest-order nonlinearity is expressed by means of a polarizability tensor α, as in the replacement

de*·α·eE(t).ord·E(t)=ReidiEi(t)ReijαijEi(t)Ej(t).
Such replacements form the foundation of nonlinear optics [504508].

The notion of polarizability complements the N-state modeling of the quantum system, with its TDSE that links N essential states, by introducing additional interactions based on induced moments. Those states that are not included in the essential states—because they are relatively far from resonant—are treatable by adiabatic elimination and give rise to dynamic Stark shifts and two-photon Rabi frequencies. For a single field these shifts can be written

S1(t)=14|E(t)/|21|d·e*Q^e·d|1,S2(t)=14|E(t)/|22|d·e*Q^e·d|2,
where [509]
Q^=i|ii|Δi.
They accompany the two-photon Rabi frequencies:
Ω12(t)=12|E(t)/|21|d·e*Q^e·d|2.
Generalizations of these formulas provide the basis for describing n-photon transitions and associated dynamic Stark shifts. These involve products of n field amplitudes.

Appendix H: Chirped Detuning

The earliest treatments of quantum-state adiabatic evolution were of two-state systems subject to linearly swept energies. In the context of a RWA Hamiltonian this means detunings that vary linearly with time (a frequency chirp) and that thereby have a crossing of the diabatic eigenvalues. When adiabatic conditions prevail, this crossing induces a population transfer. Figure 42 shows an example of adiabatic two-state population transfer produced by a linearly chirped pulse.

 

Figure 42 Example of two-state population transfer by a chirped pulse. (a) The electric field amplitude, modulated by a chirped carrier frequency. (b) Populations Pn(t). Pattern at left shows linkage between states sweeping across resonance.

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The LZMS model. The basic RWA Hamiltonian for two states subject to variable detuning and pulsed Rabi frequency is

W(t)=[012Ω(t)12Ω(t)Δ(t)].
Numerous functional forms of the time dependences have closed-form analytic solutions [60,206,209,211,227,279,290,510,511]. One of the first of these to be published dealt with an idealization of the Rabi frequency as a constant and the detuning as a linear ramp, the LZMS model [156160]:
Ω(t)=Ω0,Δ(t)=rt.
The solutions to the resulting TDSE are expressible as parabolic cylinder functions [280], over the infinite interval <t<. For population that starts in state 1 as t the probability of remaining in that state, unexcited, at t+ is expressible as an exponential, eξ. Variants of the simple LZMS model that allow pulsed variation of the Rabi frequency and monotonic change to the detuning give population changes that fit the LZMS exponential formula but with generalized definition of the exponent ξ, cf. [7,8]:
P2(t)=1P1(t),P1()=eξ,ξ=πΩ(t0)22|Δ˙(t0)|.
Here t0 is the time at which the detuning passes through zero, Δ(t0)=0. Population transfer, P2()1, requires a combination of large peak Rabi frequency and slowly changing detuning, as expressed by the adiabatic condition
πΩ(t0)22|ddtΔ(t0)|.
The Demkov–Kunike model. A more general form for pulsed and detuned excitation introduces the Demkov–Kunike model [209,499],
Ω(t)=Ω0sech(t/T),Δ(t)=Δ0+Btanh(t/T),
involving three free parameters, Ω0, Δ0, B and a time scale T. The effect of the associated RWA Hamiltonian over an infinite interval is best encapsulated by the two-state propagator:
U(,)=[abb*a*],with  |a|2+|b|2=1.
Here the complex numbers a and b are Cayley–Klein parameters [209,211,290,512]. An alternative parametrization uses Stückelberg parameters [174,290]: two angles and the transition probability p,
a=eiθ1p,b=eiϕp.
The figures of Appendix H.1 show examples in which Δ(t) varies linearly with time and the pulsed Rabi frequency is timed to have peak value at the moment when Δ(t) is zero and the diabatic curves cross.

H.1. Single Diabatic Curve Crossing; RAP

The left side of Fig. 43 shows an example of two-state behavior with a positive chirp and population initially in state 1. The diabatic-eigenvalue curves are straight lines, crossing at t=5. The adiabatic-eigenvalue curves have a bulge (an avoided crossing) around the moment of diabatic curve crossing at t=5. The dynamics is that of complete population transfer, from state 1 to state 2, a transition known as rapid adiabatic passage (RAP) or adiabatic rapid passage (ARP) that occurs during the pulse interval around t=5. The adverb “rapid” infers that the population change occurs during a time interval that is shorter than decoherence times such as the lifetime for spontaneous emission. During the depicted change the time evolution is adiabatic and the statevector remains aligned with an adiabatic state.

 

Figure 43 Examples of two-state rapid adiabatic passage induced by a linearly varying detuning. Left: positive chirp. Right: negative chirp. (a) The Rabi frequency, relative to peak value. (b) Diabatic eigenvalues (dashed) 1, 2 and adiabatic eigenvalues (solid) | and |+. Arrows show motion of the system point. The evolution is adiabatic in this simulation, resulting in complete population transfer, by RAP, from state 1 to state 2. (c) Populations Pn(t).

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Adiabatic population transfer can occur for either slope of the detuning curve; it is only necessary that there be a crossing of diabatic curves. When the chirp is positive (left-hand frames of Fig. 43) there is an initial pairing of RW state 1 with adiabatic state |+. This adiabatic state pairs later with state 2. With negative chirp the adiabatic transfer state is |.

H.2. Chirping in a Three-State Chain

The chirped-detuning adiabatic passage of population between two linked quantum states can occur also in N-state systems, cf. [67]. The simplest of those are the three-state chains described by the basic Eq. (2) with allowance of variable detunings. Figure 44 shows examples in which the two Rabi frequencies are taken to be equal valued and simultaneous, while the two detunings vary linearly with time, as described by the RWA Hamiltonian

W(t)=[Δ1(t)12Ω(t)012Ω(t)Δ2(t)12Ω(t)012Ω(t)Δ3(t)].

 

Figure 44 Results for three-state chain with chirped detuning. Frames as in Fig. 43. Three diabatic-curve crossings are marked as A,B,C. Left: the two detunings each have positive chirp and the population transfer occurs with the diabatic crossing 1,3: the crossings 1,2 and 2,3 occur with counterintuitive ordering, a possibility (“nonintuitive” or backward-sweeping) first proposed by [184]. Right: the detunings have negative chirp and the diabatic crossings occur with intuitive ordering, 1,2 and then 2,3. See also figures in [63,168,514].

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The simulations of this figure used 0,rt,2rt as the detunings (and diabatic eigenvalues) Δn(t). The dynamics, though not the eigenvalue plots, remain unchanged if we use the common choice rt,0,+rt for the detunings [63,168]. There are three crossings of diabatic curves [513]:

  • A between the initially populated state 1 and the excited intermediate state 2,
  • B between states 1 and 3 (a two-photon linkage), and
  • C between the intermediate state 2 and the final target state 3.

The chosen pulse envelope overlaps each of these crossings, under conditions that produce adiabatic following at each crossing. Thus in these simulations the system point always follows an adiabatic curve. The followed curve differs in the two frames.

Intuitive. With negative chirp, the frame to the right, the statevector remains aligned with state |. Population in state 1 meets first crossing A with state 2, into which it transfers, and subsequently it meets crossing C with state 3, again producing complete population transfer. With the negative chirp the population transfer is by two sequential one-photon transitions, an “intuitive” or “natural” ordering [184,513] that places appreciable population temporarily in state 2.

Counterintuitive. With positive chirp, the frame to the left, the statevector remains aligned with adiabatic state |+. Population initially in state 1 meets only a single diabatic crossing, B, at which time it transfers to state 3 by means of a two-photon transition. The diabatic crossings 1,2 and 2,3 that produce the population transfer arrive with “counterintuitive” ordering [8,63,168,184,513,514].

STIRAP comparison. These population transfers, applicable to stimulated Raman transitions, rely on adiabatic passage (including statevector alignment with a “dark” adiabatic state that has minimal transient state-2 population) [251] and so they have the robustness associated with that procedure. They were noted prior to the development of STIRAP [184,515]. But the dynamics is that of AF through diabatic curve crossings associated with chirped detunings rather than with resonant timing of pulsed Rabi frequencies that were the essence of the original STIRAP as defined in Subsection 2.3.

H.3. Asymmetric Curve Crossings; SCRAP and SARP

In a three-state chain that has large single-photon detuning it is possible to adiabatically eliminate the intermediate state from the dynamics and obtain effectively a two-state description; see Appendix G.1. The two-state effective Rabi frequency Ω(t) involves a two-photon transition and is proportional to the product of two single-photon Rabi frequencies ΩP(t) and ΩS(t). The two-state effective detuning Δ(t) is the difference between two dynamic Stark shifts, each proportional to the square of a single-photon Rabi frequency. The controlled variation of the dynamic Stark shifts by crafted pulses ΩP(t) and ΩS(t) permits construction of a pulsed detuning Δ(t) that rises and falls, with consequent diabatic-curve crossings at two times.

When the pairing of the two pulses Ω(t) and Δ(t) is not symmetric in time (i.e., one of them is very large) it becomes possible to have the system point undergo DF at one crossing and AF at the other. Such situations are encountered with the SCRAP and SARP mechanisms mentioned in Subsection 2.3d as associated with two-photon processes and dynamic Stark shifts.

Figure 45 shows examples of such asymmetry at two diabatic crossings, A and B. In both examples there is an initial alignment of the statevector with state 1 and adiabatic state |. The two eigenvalue curves for those states coincide initially.

 

Figure 45 Examples of pulsed effective detuning (dynamic Stark shift) Δ(t) and effective Rabi frequency Ω(t) in an effective two-state system. Frames as in Fig. 43. The pulsed detuning leads to two diabatic-curve crossings, A and B. Here they are asymmetrically located with respect to the center of the pulsed Rabi frequency. Left: in the left-hand frames there occurs AF at crossing A and DF at crossing B. These produce complete population transfer, 12. Right: in the right-hand frames the sequence is reversed, and population transfer occurs by DF and AF, again producing complete transfer 12.

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AF first. In the frames on the left side the two-photon Rabi frequency is timed to peak near the first crossing time, A, and to be negligible at crossing B. The adiabatic curves have an avoided crossing at time A and intersect at time B. The statevector remains aligned with the adiabatic eigenvector | during the A crossing, but at the B crossing, where the evolution is diabatic, there occurs an abrupt change of the adiabatic-state construction such that state 1 becomes aligned with the |+ state. The system point undergoes AF at crossing A and then DF at crossing B, to finally follow the overlapping eigenvalue curves of states 2 and |+. There occurs an overall change both in the bare-state populations (1 to 2 at A) and the adiabatic-state populations (| to |+ at B).

DF first. In the right-hand frames the sequence is reversed, and population transfer occurs by DF and then AF. There again occurs an overall change in both the adiabatic-state populations (| to |+ at A) and bare-state populations (1 to 2 at B).

Summary. Just as with simpler systems, the possibilities of AF and DF do not depend on signs of detunings and Rabi frequencies. Any combination of pulses Ω(t) and Δ(t) that produces an asymmetry of AF and DF at diabatic crossings will make possible a complete transfer of population.

Appendix I: Three States with Weak and Strong Fields; AT Splitting and EIT

When some of the Rabi frequencies are much larger than the others the system behavior can be most simply understood and treated by introducing eigenstates of the strongly coupled subsystem and linking the weaker Rabi frequencies to this [9,15]. In a two-field system such as STIRAP the simplest implementation of this approach is by the RWA Hamiltonian that nullifies the detuning of intermediate state 2 (shown here without accounting for the fluorescent loss from state 2):

W(t)=[ΔP12ΩP(t)012ΩP(t)012ΩS(t)012ΩS(t)ΔS].
A resonance between states 1 and 2 occurs when ΔP=0, and a resonance between states 2 and 3 occurs when ΔS=0. The STIRAP requirement for two-photon resonance is ΔP=ΔS. Let us suppose the P field is a weak probe field and the S field is a strong control field, with |ΩS(t)||ΩP(t)|. We partition the states 2 and 3 that are linked by this strong transition as the matrix
WS(t)=[012ΩS(t)12ΩS(t)ΔS].
Using the two eigenvectors and eigenvalues for this 2×2 matrix,
WS(t)ϕ±S(t)=ϵ±S(t)ϕ±S(t),
we express the original probability amplitudes as
C(t)=C1(t)ψ1(t)+C(t)ϕ(t)+C+(t)ϕ+(t).
The eigenvectors and eigenvalues of WS(t) are expressible with the formulas (see Subsection 4.4a)
ϕS(t)=[cosβ(t)sinβ(t)],ϵS(t)=ϒ(t)sin2β(t)=12[ΔSϒ(t)],
ϕ+S(t)=[sinβ(t)cosβ(t)],ϵ+S(t)=+ϒ(t)cos2β(t)=12[ΔS+ϒ(t)],
where the S-field mixing angle β(t) is defined by the formulas
sin[2β(t)]=ΩS(t)/ϒ(t),cos[2β(t)]=ΔS/ϒ(t),ϒ(t)ΩS(t)2+ΔS2.
When expressed in this new basis the full RWA Hamiltonian of Eq. (I.1) reads
W˜(t)=12[2ΔPΩ(t)Ω+(t)Ω(t)2ϵ(t)0Ω+(t)02ϵ+(t)],
in which the links between initially populated state 1 and the S-field dressed states are by the Rabi frequencies
Ω(t)=cosβ(t)ΩP(t),Ω+(t)=sinβ(t)ΩP(t).
These are equal when ΔS=0. The two dressed states linked to state 1 in Eq. (I.2) are then equivalent to the single state 2 of Eq. (I.1).

I.1. Autler–Townes (AT) Splitting

When the Rabi frequencies are constant, as occurs with unpulsed, CW radiation, there will occur, instead of the original resonance between state 1 (the initial state) and state 2, at detuning ΔP=0, two resonances, at the values

ΔP=12[ΔS+ϒ],ΔP=12[ΔSϒ].
In particular, when the S field is resonant with the 2–3 transition, so that ΔS=0 and β=π/2, the two resonant detunings ΔP of the probe field differ by the S-field Rabi frequency, a difference known as the Autler–Townes (AT) splitting [516], an example of a dynamic Stark shift,
ΔP=12|ΩS|,ΔP=+12|ΩS|.
These resonances are made visible by fluorescence from the excited state 2, which is a component of each of the two dressed states. When ΔS=0 the dressed states of the S-coupled system are constant equal-amplitude superpositions of states 2 and 3, independent of the strength of the S field. The AT splitting occurs for all values of the mixing angle, though it can be obscured by the range of detunings in an ensemble of Doppler shifts [517519].

The form of W˜(t) in Eq. (I.7) requires modification when the RWA Hamiltonian WS(t) of Eq. (I.2) includes losses Γn from states 2 and 3 [421]. The introduction of the states ϕ±S(t) defined in Eq. (I.5) produces then a matrix that is not diagonal. The eigenvalues ε±S(t) gain imaginary parts iΓ±/2 and the matrix acquires non-Hermitian off-diagonal elements iΓ(+)/4 [421]. The predicted probe-photon loss shows an interference-produced null at the center of the AT splitting [421].

I.2. Electromagnetically Induced Transparency (EIT)

The flow of CW radiation through bulk matter is governed by the steady-state response of the illuminated atoms to the electromagnetic field, typically expressed, for each frequency component, in terms of an absorption coefficient and a refractive index or by their combination into a complex-valued susceptibility. To evaluate such steady-state quantities it is necessary to extend the modeling of the quantum system well beyond the short-time behavior provided by the TDSE, to a time t when the time derivatives of the density matrix are negligible, and to include the various incoherent relaxation phenomena that produce steady-state populations and coherences for the density matrix, ρ¯nm.

The absorption coefficient for the P field is proportional to the negative imaginary part of the relevant susceptibility [see Appendix M.4] and hence, in the approximation ρ¯nm=C¯nC¯m*, to the negative imaginary part of C¯2. A simple approach, used by [234,442], uses the TDSE by introducing loss rates Γn for states 2 and 3, setting C1(t)=1, and seeking steady-state solutions for the other amplitudes by setting to zero all time derivatives, ddtC¯n=0. The result for the steady-state excited-state amplitude C¯2 is

C¯2=ΩPX+iY(|ΩS|2+Γ2Γ34Δδ)2+4(Γ2δ+Γ3Δ)2,
where
X=2δ(|ΩS|24Δδ)2ΔΓ3Γ3,Y=Γ3(|ΩS|2+Γ2Γ3)+4δ2Γ2].
Here ΔΔP and δ is the two-photon detuning, equal to ΔSΔP for the Lambda linkage. The nonzero value of Γ3 is needed to give a bound to C¯2 when the detuning Δ picks one of the AT doublets. In the limit of negligible Γ3, as is appropriate for the usual Lambda linkage, these expressions become
X=2δ(|ΩS|24Δδ),Y=4δ2Γ2.
These steady-state values both vanish when the two-photon detuning vanishes, δ=0, for any value of the S-field Rabi frequency ΩS.

A plot of this steady excitation-amplitude C¯2, or the imaginary part of the P-field susceptibility, as a function of the P-field detuning Δ for ΔS=0, shows two absorption peaks [234,421], the Autler–Townes doublet, symmetric about ΔP=0 and separated by |ΩS|. At the midpoint of the pattern, where Δ=0, the absorption is zero. The result is EIT [155,421,442444] for the P field. This null in the P-field absorption can be regarded as destructive interference between two dressed-state amplitudes [421,443,520]. The formulas above show that the steady-state value of the state-2 probability amplitude is proportional to a detuning, and so it vanishes when that detuning vanishes. In treating ensembles that include a range of Doppler shifts, the resulting distribution of detunings fill in the Autler–Townes gap in transparency [517519].

EIT, AT splitting and STIRAP. EIT is a steady-state phenomena based inseparably on dynamic Stark shifts contributing to AT splitting. These observable effects, and their theoretical description, apply to atomic systems after relaxation processes have established steady values of density matrices. By contrast, the time intervals appropriate for STIRAP and other adiabatic processes must be shorter than relaxation times—as indicated by the use of “rapid” in their naming. Both EIT and STIRAP draw their interesting behavior from the properties of population-trapping dark states.

Appendix J: Composites Systems

Situations arise when we need to consider several degrees of freedom, either for a single particle (e.g., spin and spatial motions of an electron) or for separate particles (e.g., an atom and a photon or two atoms in a molecule). Each degree of freedom requires its own multidimensional Hilbert space, in which basis vectors describe the possible states of that degree of freedom. A possible quantum state of the complete system might therefore have the form of a product, written variously (with suppression hereafter of time arguments) as [15]

ψabc=ψaAψbBψcC|aA|bB|cC|a,b,c,ABC,
where the labels A,B,C, identify the various degrees of freedom (e.g., the various particles), and the subscripts a,b,c, specify particular basis states within the subspaces. More generally the statevector will be some superposition of such products, say [521]
Ψ=a,b,c,Cabcψabc=a,b,c,Cabc|a,b,c,ABC.
When applied to treatments of identical particles (e.g., electrons within an atom or molecule) the superposition must take account Fermion or Boson properties, most simply exhibited as
Ψ=a,b,c,CabcS^ψabc=a,b,c,CabcS^|a,b,c,ABC,
where S^ denotes any necessary symmetrizing (or antisymmetrizing) operator.

Superpositions such as Eq. (J.2) are said to be separable if they can be expressed as the unsummed product of several factors, one for each degree of freedom, as

Ψ=ϕAϕBϕC[acaAψaA][bcbBψbB][cccCψcC].
Separability is possible if, and only if, the coefficients Cabc of Eq. (J.2) can be written as products:
Cabc=CaACbBCcC
When the quantum statevector is not separable then in general it is not possible to assign a single statevector to any of the subsystems.

For any such separable state each of the subsystems A,B,C, have definite properties, known because the statevector is known. In particular, we can deduce properties of B from measurements of A. The separate subsystems are said to be correlated.

J.1. Bipartite System

The eigenstates of a Hamiltonian that links two subsystems—a bipartite system—can be classified by their correlation properties. Let the full Hamiltonian for a two-part system be the sum of the following parts [15]:

H(t)=HApartA,eigenstatesψnA+HBpartB,eigenstatesψmB+VA(t)+VB(t)+VABinteraction.
For simplification, suppose that each subsystem has just two quantum states, as occurs with a spin one-half particle or a two-state atom:
ψ+A,ψAandψ+B,ψB.
When the interaction has the form VA·VB, or
HAB=VAV+B+V0AV0BV+AVBwhere  Vjψk=vj,kψj+k,
then the eigenstates of HAB are a triplet
ϕ+1T=ψ+Aψ+B,
ϕ0T=(ψ+AψB+ψAψ+B)/2correlated,
ϕ1T=ψAψB,
and a singlet
ϕ0S=(ψ+AψBψAψ+B)/2correlated.
The terminology of singlet and triplet comes from the spectroscopic description of the collective spin states for two electrons.

J.2. Correlation and Entanglement

Under appropriate conditions the degrees of freedom can become correlated, so that a measurement of one part of the system gives information about another part. In the bipartite example above, suppose the system is known (from its energy) to be in the collective state described by ϕ0T. Then the individual parts may be in either plus or minus states, with equal probability. But if a measurement shows part A to be in the plus state then we know, with certainty and without further ado, that a measurement of part B will reveal it to be in the minus state. This collective state is therefore one in which the two subsystems are correlated.

Generally two subsystems (i.e., two degrees of freedom) A and B are termed correlated if a priori neither part is known to be in a definite state but by learning the state of one part the state of the other part becomes known. When one of the following conditions holds [15] such correlation is known as entanglement [112,345,386393]:

  • I. One degree of freedom represents internal structure while another represents center of mass motion (e.g., of a trapped particle).
  • II. The degrees of freedom are associated with distinct particles (or photons) which, though initially together, are observed physically separated.

Appendix K: Rabi Frequency with Zeeman Sublevels

The Rabi frequency associated with electric-dipole interactions involves the projection of the system dipole transition moment d onto the direction of the electric field, as parametrized by a unit vector e, i.e., the interaction involves the scalar product of two vectors. When the states of the system, say a single freely moving atom or molecule, are those of angular momentum |j,m the undisturbed energy levels comprise degenerate Zeeman sublevels, identified by angular momentum j and 2j+1 values of the magnetic quantum number m. The presence of a static magnetic field introduces m-dependent Zeeman shifts of the energies. With the introduction of angular momentum, and negligible Zeeman shifts, a two-state system becomes a two-level system, and the interaction, i.e., the Rabi frequency, becomes a matrix, as indicated in Fig. 46. The theory of angular momentum [266,522528] provides formulas, summarized below, for evaluating the various discrete orientations of the two vectors that produce the dipole interaction Hamiltonian.

 

Figure 46 Structure of the resonant RWA Hamiltonian matrix W showing the ordering of arguments needed for the definition of the matrices Ω12 and Ω21 in an angular momentum basis. For resonant excitation the unshaded diagonal blocks are filled with zeros. When the excitation is not resonant, ω|E2E1|, the lower-right block is a diagonal matrix Δ with detunings as elements. Although the four constituent blocks of W are here shown as squares, as would occur when j1=j2, generally they are rectangles.

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K.1. Angular Momentum Operators and States

Any triplet of square, Hermitian matrices J1,J2,J3 of dimension 2j+1 that have the commutators

[J1,J2]=iJ3,[J2,J3]=iJ1,[J3,J1]=iJ2,
and
[Ji,J2]=0,i=1,2,3,where  J2=J12+J22+J32.
can serve as examples of angular momentum operators. (Often the labels x,y,z are used in place of 1, 2, 3.) From the commutation relations one obtains the properties of the orthonormal angular momentum eigenstates:
J2|jm=j(j+1)|jm,J3|jm=m|jm.
The matrix representations of J2 and J3Jz are diagonal,
j,m|J2|j,m=j(j+1)  δjjδmm,
j,m|J3|j,m=m  δjjδmm,
while those of J1Jx and J2Jy are tridiagonal,
j,m|J1|j,m=12j(j+1)m(m±1)  δjjδm,m±1,
j,m|J2|j,m=12ij(j+1)m(m±1)  δjjδm,m±1.
Thus these matrices offer an opportunity for expressing certain (but not all) chain-linked N-state quantum systems in a manner that can allow closed-form expressions drawn from angular momentum theory, for the probability amplitudes at arbitrary time t, and to portray these as simple rotations [356].

Coupled states. Superpositions of independent sets of eigenstates, for vector spaces A and B (e.g., two particles or two degrees of freedom for one particle),

|j1,j2,JM=m1m2(j1m1,j2m2|JM)  |j1m1A|j2m2B,
provide coupled eigenstates of the combined angular momentum
Ji=Ji(A)+Ji(B).
The (j1m1,j2m2|JM) of Eq. (K.5) are Clebsch–Gordan coefficients [266,522528], real valued functions of the angular momentum quantum numbers. The coupled states |j1,j2,JM are examples of correlated relationships (entanglement) between the constituent subspaces A and B; see Appendix J.2 for examples of coupled states for angular momenta j1=j2=1/2.

K.2. Irreducible Tensors

To evaluate the matrix elements of the dipole transition moment d we consider radiation whose electric field is specified by three amplitudes, corresponding to three basic spherical unit vectors, eq:

e0=ez,e±1=12[ex±iey],with  eq*=(1)qeq.
The three spherical unit vectors eq are examples of components of unit irreducible tensors of order one, recognizable because a rotation R of the coordinate system produces the unit vectors:
eq(R)=qDqq(1)(R)eq,
where the coefficients Dqq(1) are elements of the rotation matrix of order 1; cf. [266,522,527,528]. More generally the 2k+1 components q=k,k+1,,+k of an irreducible tensor Tq(k) of order k transform under rotation as
Tq(k)(R)=qDqq(k)(R)Tq(k).
Such irreducible tensors fit well into the use of angular momentum states. The dipole moment d and the magnetic moment m offer examples of order-one irreducible tensors. Components of the electric quadrupole moment are expressible as irreducible tensors of order 2.

K.3. Dipole Interaction Matrix Elements

In an angular momentum basis the expression for the elements of the dipole interaction RWA Hamiltonian, a generalization of Eq. (45b), reads

2Wm1,m2(t)=qα1j1m1|dq|α2j2m2×{(1)qEq(t)*if  E1<E2Eq(t)if  E1>E2,
where the αn abbreviate the list of additional quantum numbers needed to identify the quantum state. To treat magnetic-dipole interactions we make the replacements
dqmq,EqBq=Eq/c.

K.4. Two-Photon Interaction

The products of dipole moments appearing in the two-photon interaction form a constant, nine-component tensor labeled by indices q and q:

Tq,q=dqQ^dq.

The generalization of Eq. (K.10) for two-photon interaction in the RWA Hamiltonian, with E2>E1, is

2Wm1,m2(t)=qqα1j1m1|Tq,q|α2j2m2Eq(t)Eq(t)/2.
From Tq,q we can form irreducible tensors of orders k=0,1,2 with the use of Clebsch–Gordan coefficients:
TQ(k)=qq(1q,1q|kQ)Tq,q,Tq,q=K(1q,1q|kQ)TQ(k).
Here Q=q+q. Specifically, the connection between an “uncoupled” tensor component Tq,q and irreducible tensors TQ(k) is
T1,1=T2(2),T1,0=12T1(2)+12T1(1),T1,1=16T0(2)+12T0(1)+12T0(0),T0,1=12T1(2)12T1(1),T0,0=23T0(2)13T0(0),T0,1=12T1(2)+12T1(1),T1,1=16T0(2)12T0(1)+12T0(0),T1,0=12T1(2)12T1(1),T1,1=T2(2).
Formula (K.13) then reads
2Wm1,m2(t)=qqEq(t)Eq(t)/2K(1q,1q|kQ)α1j1m1|TQ(k)|α2j2m2,Q=q+q.

K.5. Wigner–Eckart Theorem

To evaluate matrix elements of irreducible tensors we use the Wigner–Eckart theorem (WET), as presented for components of irreducible tensors of order k on p. 181 of [266] or p. 75 of [522]:

α1j1m1|Tq(k)|α2j2m2=(1)j1m1(j1kj2m1qm2)(α1j1T(k)α2j2).
Here the final factor (T(k)) is the reduced matrix element of the tensor operator T(k). It follows the Wigner three-j symbol (: : :) that contains all dependence on the magnetic quantum numbers (and thereby all dependence on the orientation of the atom). The three-j symbol is related to the Clebsch–Gordan coefficient by the formula [266,522]
(j1m1,j2m2|j3m3)=(1)j1j2+m32j3+1(j1j2j3m1m2m3).
Both the three-j symbol and the Clebsch–Gordan coefficient are available as standard mathematical functions in programs such as Mathematica; with their use it is a simple matter to obtain numerical values for all matrix elements, relative to the reduced matrix elements.

The general three-j symbol incorporates a number of selection rules, most notably the requirement

m1+m2=m3.
The symmetry properties of the three-j symbols include
(j1kj2m1qm2)=(j2kj1m2qm1).
Applied to the dipole moment, a tensor of order 1, the Wigner–Eckart theorem gives the formula
α1j1m1|dq|α2j2m2=(1)j1m1(j11j2m1qm2)(α1j1dα2j2),
with the requirement that q=m1m2 for specified m1,m2 or, conversely, m1=m2+q for specified m1 and q. Thus the specification of magnetic sublevel pairs m1,m2 fixes the single allowable value of q in the summations. Conversely, a given q limits the pairs of quantum states that have nonzero couplings. For example, if the angular momentum component increases with excitation, m2=m1+1, then q=1. Conversely a decrease of angular momentum occurs with q=+1. With this proviso the formula of Eq. (K.10) for the RWA Hamiltonian generalizes to
2Wm1,m2(t)=(1)j1m1(j11j2m1qm2)(α1j1dα2j2)×{(1)qEq(t)*if  E2>E1Eq(t)if  E2<E1,
with the understanding that q=m1m2.

K.6. Geometric Selection Rules

The factors appearing in the Wigner–Eckart theorem incorporate various selection rules that select, from all possible pairs of quantum states that satisfy the near-resonance condition, those that meet additional criteria. The three-j symbol has nonzero values, for a given dipole component dq, only those sublevels for which