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 [2–4]. It now offers a tool for general quantum-state manipulation, as described in various reviews [5–9] and, most recently, [10–12]. 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 [13–15].

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 [2–4]. 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 [15,15–19]. Its connection with preceding work has been presented in some detail [9] and its extensive experimental applications have been regularly reviewed [6–8,10,12]. 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 [20–25] 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 [9,13–15], 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 2.2c–2.2e 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 [27–39]. 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 [40–48], to treat a three-field loop-linkage of three states [49–51], to treat a three-field tripod-linkage of four states [52–61], 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 $E_1,E_2,E_3$, 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, $\mathrm{\Xi }$), a Lambda $\mathrm{\Lambda }$ (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 $E_n$. Although the usual linkage pattern for STIRAP has been the Lambda, the ladder has also been considered [67–73]. Allowance for a third field makes possible a loop (or triangle or Delta) linkage, considered elsewhere in connection with STIRAP [49–51]. The figure also shows the detunings of Eq. (2): the one-photon detuning $\mathrm{\Delta }$ and the two-photon detuning $\delta$. 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 $\mathrm{\Delta }$ and $\delta$) of (a) Ladder (or Cascade, $\mathrm{\Xi }$), (b) Lambda ($\mathrm{\Lambda }$), 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) [49–51]. Horizontal lines mark unperturbed-state (bare-atom) energies $E_n$. [In (b) and (c) the energies $E_1$ and $E_3$ 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 $E_n$: 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 [64–66], 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 [74–78], long used in molecular spectroscopy [79–82], 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 [83–86]. 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 $E_n$ associated with the parameters $\mathrm{\Delta }$ and $\delta$. 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 $\mathrm{\Delta }$ is sufficiently large.)

2.2. Basic Differential Equations of STIRAP

The principal observables of a multistate quantum system are the probabilities $P_n(t)$ 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 $C_n(t)$:

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 $C_n(t)$ at time $t$ are typically written as three coupled ordinary differential equations (ODEs):

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

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 $t_{\text{in}}=0$. Alternatively, when one takes time $t=0$ to mark the midpoint of the pulse sequence this leads to the conventional assignment of $t_{\text{in}}\to -\infty$ 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 [64–66]. This excitation exhibits noteworthy quantum-beat interference patterns between the two excitation branches [89–91].

Equation structure. A key characteristic of these three first-order ODEs, in addition to their linearity, is the presence of the imaginary unit $\mathrm{i}\equiv \sqrt{-1}$. When the coupling coefficients—the Rabi frequencies—are constant (so that Laplace transforms provide solutions [92–95]) 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 $\mathrm{\Delta }$ and $\delta$, 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. [15,96–98]. The numerical solution of coupled ODEs poses no great challenge [99–108] 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 $P_n(t)$ 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 ${\rho }_{mn}(t)=C_m{(t)}^{*}{C}_n(t)$—the off-diagonal elements of the density matrix $\mathit{\rho }(t)$ [109–111], see Subsection 3.3h. These require careful attention to phases of complex numbers.

The basic element of quantum information [36,112–116], 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) [113–119].

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., $\mathrm{\Delta }=\delta =0$. 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, $\mathrm{\Delta }=\delta =0$. (a) The pulsed Rabi frequencies ${\mathrm{\Omega }}_S(t)$ and ${\mathrm{\Omega }}_P(t)$, here Gaussians of equal width and height, and temporal pulse area (integrated Rabi frequency, see Appendix B.1) of $\mathcal{A}=5\pi$, are normalized to unit peak value for display. (b) The populations $P_n(t)$, 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. [121–123] and in the reviews [6–9,12,14,15].

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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 $\mathcal{A}$, see Appendix B.1). The two pulse areas, here taken as $\mathcal{A}=5\pi$, 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 $\mathcal{A}=5\pi$, as in Fig. 2, and are fully resonant ($\mathrm{\Delta }=\delta =0$). For intuitively ordered pulses (left side) the transfer of population from state 1 to state 3 takes place in two stages, first $1\to 2$ and then $2\to 3$, 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 $5\pi$ 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 $5\pi$ pulses of the two preceding examples, the populations undergo oscillatory transfer (three-state Rabi oscillations) $1\to 2\to 3\to 2\to 1\to \cdots$. 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 $\mathbf{d}$ in the electric field $\mathbf{E}(t)$ of a laser beam, idealized as

As the formulas of Eq. (5) show, the dipole interactions are proportional to the projection of a dipole moment vector $\mathbf{d}$ onto a unit vector $\mathbf{e}$ 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 $E_n$ and field carrier frequencies appear in the basic Eq. (2) only indirectly, as constituents of detunings ${\mathrm{\Delta }}_P$ and ${\mathrm{\Delta }}_S$ of carrier frequencies ${\omega }_P$, ${\omega }_S$ from Bohr frequencies. For the Lambda linkage these are

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 $\delta$ appearing in Eq. (2c) is either the sum (for the ladder linkage) or the difference (for the Lambda linkage) of the single-photon detunings ${\mathrm{\Delta }}_P$ and ${\mathrm{\Delta }}_S$. 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,

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 ${\mathbf{e}}_P$ and ${\mathbf{e}}_S$) 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 ${\omega }_{nm}=|E_n-E_m|/\hslash$. 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 $\mathbf{v}$ 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 $\mathbf{v}$ 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 $\mathcal{E}(x,y)$ appears to a particle having a moving center of mass at $x_{cm}(t)=vt$ as the pulsed field $\mathcal{E}(vt,y)$. 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 [126–128], 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 [129–131] or impurity atoms in transparent crystals [57,132–134], 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 [135–139]. 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 $\vartheta (t)$, defined, for real-valued Rabi frequencies, by the formula

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 $\vartheta =0$ to $|\vartheta |=\pi /2$, as befits positive Rabi frequencies, but these limits should be understood as being mod $\pi$, 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 ${\tau }_{\mathrm{coh}}$ 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 ${\tau }_{\mathrm{coh}}$. 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 [6–10,12,14] 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 $\mathbf{E}(t)$, or its associated Rabi frequency, can be expressed by the Fourier transform

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 ${\tau }_{\mathrm{coh}}$ 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,

2.2i. Appropriate Systems

Traditionally the three variables $C_n(t)$ 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 [146–148], magnetic flux [148–151], 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 $C_n(t)$ 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]:

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:

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, $\delta =0$, do not require single-photon resonance, $\mathrm{\Delta }=0$, 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 [156–160], 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) [6,8,161–166] 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) [8,171–179] 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) [180–183].

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 $\delta =0$ 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 $\delta =0$ and it may become desirable to select a nonzero value of $\delta$ 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, $\mathrm{\Delta }=0$. 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 $\delta =0$. For small values of $\mathrm{\Delta }$ the traditional STIRAP requirement of two-photon resonance is valid.

 

Figure 5 Population transfer $P_3(\infty )$ versus two-photon detuning $\delta$ for Gaussian pulses of width $T$. (a) Single-photon resonance, $\mathrm{\Delta }=0$. (b) Large single-photon detuning. Solid black line: maximum Rabi frequencies are equal for $S$ and $P$ pulses. Red short-dashed line: maximum ${\mathrm{\Omega }}_S$ is 2.4 times the peak ${\mathrm{\Omega }}_P$. Blue long-dashed line: maximum ${\mathrm{\Omega }}_P$ is 2.4 times the peak ${\mathrm{\Omega }}_S$. 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 $P_3=1$. 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 $P_3>0.9999$, 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 $P_3(\infty )$ 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, $5\pi$, $10\pi$, $15\pi$, and $20\pi$. $S$ and $P$ pulses are Gaussians of equal peak value and widths, resonantly tuned ($\mathrm{\Delta }=\delta =0$). 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 $1\to 3$ 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 $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$ (in units of the peak Rabi frequency) and pulse areas of $10\pi$. 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, $\delta =0$. Thus any probe of populations as a function of detuning will, for STIRAP, reveal a sharp resonance. At the center of this resonance, where $\delta =0$, 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) [6–8,12].

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 ${\mathrm{\Delta }}_P$ for fixed $S$-field detuning ${\mathrm{\Delta }}_S/2\pi \approx 200\mathrm{MHz}$. The accompanying diagrams at the right show the linkage patterns associated with the signal. In each plot the broad emission feature centered around ${\mathrm{\Delta }}_P=0$ 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, ${\mathrm{\Delta }}_P={\mathrm{\Delta }}_S$. 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 $P_3(\infty )$ and the dip in $P_2(\infty )$.

 

Figure 8 Final populations versus $P$-field detuning ${\mathrm{\Delta }}_P/2\pi$ in metastable neon [188]. (a) Target state population $P_3(\infty )$. (b) Intermediate excited-state population $P_2(\infty )$. The peak in $P_3(\infty )$, frame (a), and the dip in $P_2(\infty )$, 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 ${\mathit{\psi }}_n^{\prime }(t)$ as coordinate vectors]. (a) The three adiabatic eigenvectors rotate as shown: the eigenvectors ${\mathbf{\varphi }}_{+}(t)$ and ${\mathbf{\varphi }}_{-}(t)$ rotate onto the equatorial (1,2) plane, the eigenvector ${\mathbf{\varphi }}_0(t)={\mathbf{\varphi }}_D(t)$ rotates in the 1,3 plane from the 1 axis to the $-3$ axis. [The unit vectors ${\mathbf{\varphi }}_B(t)$ and ${\mathbf{\varphi }}_D(t)$ remain in the 1,3 plane.] (b) The statevector $\mathbf{\Psi }(t)$, maintaining alignment with ${\mathbf{\varphi }}_0(t)={\mathbf{\varphi }}_D(t)$, 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 $\mathrm{\Psi }(\mathbf{r})$ of position $\mathbf{r}$ whose absolute square gives the probability $P(\mathbf{r})$ of finding the particle at $\mathbf{r}$:

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, $E_1,E_2,\dots$. Associated with each discrete energy $E_n$ is a wavefunction ${\psi }_n(\mathbf{r})$ whose absolute square gives the probability $P_n(\mathbf{r})$ that, if the particle is known to be in state $n$ it will be found at position $\mathbf{r}$:

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

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 ${\psi }_n(\mathbf{r})$. 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 $\mathbf{r}$ 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 $E_n$ will remain associated with the stationary motion described by ${\psi }_n(\mathbf{r})$: 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 ${\mathcal{H}}^0$ that dictates stationary structure but by an additional, controlled, time-dependent energy operator ${\mathcal{H}}^{\mathrm{int}}(t)$. The needed mathematics is presented most succinctly by treating the discrete basis wavefunctions ${\psi }_n(\mathbf{r})$ as unit vectors in an abstract vector space, a Hilbert space [189–192] in which lengths and angles are defined and quantified by scalar products $⟨A|B⟩$ between vectors $\mathbf{A}$ and $\mathbf{B}$. 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,

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 ${\psi }_n(\mathbf{r})$ become basis vectors ${\mathit{\psi }}_n$ and the system wavefunction becomes a vector, the statevector $\mathbf{\Psi }(t)$, 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:

The Hamiltonian operators thereby become represented by matrices:

States and vectors. The symbol ${\mathit{\psi }}_n$ (or $|{\psi }_n⟩$ 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 $\mathbf{\Psi }(t)$, an element of an $N$-dimensional Hilbert space that has complex-valued probability amplitudes $C_n(t)$ as coordinate components. That is, we express any statevector $\mathbf{\Psi }(t)$ as a construct of $N$ fixed Hilbert-space unit vectors (basis vectors) ${\mathit{\psi }}_n$, each associated with a distinct quantum state, with time-varying coefficients $C_n(t)$:

The statevector, rather than a wavefunction, becomes the repository of information about a quantum system. The ${\mathit{\psi }}_n$ are often denoted, with Dirac notation, as $|{\psi }_n⟩$. The statevector $\mathbf{\Psi }(t)$ can be regarded as an $N$-component column vector $|\mathrm{\Psi }(t)⟩$ having $N$ elements $C_n(t)e^{-\mathrm{i}{\zeta }_n(t)}$. Its Hermitian conjugate ${\mathbf{\Psi }}^{†}(t)$ is a row vector $⟨\mathrm{\Psi }(t)|$ whose elements are $C_n{(t)}^{*}{e}^{+\mathrm{i}{\zeta }_n(t)}$.

Pictures. Here, as explained below, the prescribed functions ${\zeta }_n(t)$ are phases chosen for subsequent mathematical simplification of the equations governing the probability amplitudes. They define a picture: the choice ${\zeta }_n(t)=0$ is the Schrödinger picture, the choice ${\zeta }_n(t)=E_{n}t/\hslash$ is the Dirac or interaction picture. The choice taken in the present work, in which the ${\zeta }_n(t)$ 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 $P_n(t)$ of finding the system in state $n$ at time $t$ is defined to be the absolute square of the projection $⟨{\psi }_n|\mathrm{\Psi }(t)⟩$ of the statevector onto the stationary Hilbert-space coordinate associated with quantum state $n$:

3.2a. Basis Vectors; RW Phases

Equation (22) offers two choices of basis vectors, differing by phases ${\zeta }_n(t)$ that are chosen for subsequent convenience: a static bare basis ${\mathit{\psi }}_n$ and a RW or diabatic basis ${\mathit{\psi }}_n^{\prime }(t)$:

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:

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 $⟨\mathrm{\Psi }(t)|\mathcal{M}|\mathrm{\Psi }(t)⟩$ of some operator $\mathcal{M}$, 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:

3.3. Time-Dependent Schrödinger Equation

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

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

When losses are not present, the matrix $\mathsf{W}(t)$ is Hermitian, meaning its transpose is equal to its complex conjugate, $W_{nm}(t)=W_{mn}{(t)}^{*}$, and its eigenvalues are real valued. With suitable phase choices it can be made symmetric, $W_{nm}(t)=W_{mn}(t)$, as was done in Eq. (2).

Linkage patterns. The nonzero off-diagonal elements of the matrix $\mathsf{W}(t)$—the Rabi frequencies—define a linkage pattern: in the mathematics of graph theory [194–196] a set of nodes (the $N$ quantum states $1,2,\dots$) and edges [the connections $m$, $n$ of $W_{nm}(t)$ that are nonzero for some time $t$]. Applied to a chain linkage, i.e., one with nonzero elements $n\leftrightarrow n\pm 1$, the $\mathsf{W}(t)$ can be presented as a tridiagonal matrix, having a graph in which edges connect only nearest neighbors. The diagonal elements of $\mathsf{W}(t)$—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 $\hslash W_{nn}=\hslash W_{mm}$, 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 $\hslash \omega$ equals the energy difference between the two quantum states, $|E_n-E_m|$. 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 $\hslash W_{nn}=\hslash W_{mm}$, 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, $\mathbf{r}=0$. The most important of these are the electric dipole moment $\mathbf{d}$, interacting with the electric field $\mathbf{E}(\mathbf{r},t)$, and the magnetic-dipole moment $\mathbf{m}$, interacting with the magnetic field $\mathbf{B}(\mathbf{r},t)$. The interaction energies of these two moments are

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: ${\mathsf{H}}^{\mathrm{int}}(t)={\mathsf{H}}^{E1}(t)$.

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

3.3d. Two-State RWA Equations

Consider the action of this near-monochromatic field upon two quantum states, having energies $E_1$ and $E_2$ and linked by dipole transition moments but lacking a permanent dipole moment, so that $H_{nn}^{\mathrm{int}}(t)=0$. The TDSE of Eq. (32) then comprises two coupled ODEs for complex-valued probability amplitudes $C_1(t)$ and $C_2(t)$,

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

The phase $\phi (t)$ is available to cancel any phase change of the field envelope $\mathcal{E}(t)$ and give a real-valued Rabi frequency. If the phase $\phi (t)$ 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:

Beyond the RWA: Floquet theory. When the envelope $\mathcal{E}(t)$ 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 [202–204]. 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

For the Lambda linkage ($E_2>E_1$ and $E_2>E_3$) the Rabi frequencies are, by analogy with Eq. (45b),

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

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 $\mathrm{\Gamma }$ by making the replacement $E_n\to E_n-\mathrm{i}\mathrm{\Gamma }/2$ in the bare-state energies and consequent detunings, cf. [206–208]. This involves the replacement

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 [15,109–111]; 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 $\mathsf{U}(t_2,t_1)$ [15] such that changes to probability amplitudes in a time interval $t_1\le t_2$ are obtainable by multiplication as

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 $\mathit{\rho }(t)$ descriptive of the statevector components, and a static matrix $\mathsf{M}$ that incorporates the properties of the system. These matrices have the elements