## 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 [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.

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):

*Rabi frequencies*${\mathrm{\Omega}}_{S}(t)$ and ${\mathrm{\Omega}}_{P}(t)$, taken here to be real-valued, though not necessarily positive (cf. Subsection 2.2e and Appendix B). Subsection 2.2c defines the

*detunings*$\mathrm{\Delta}$ and $\delta $, 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 ${t}_{\text{in}}$. 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.

**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*.

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.

**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

*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 ${\tau}_{\mathrm{coh}}$ 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]:

- (i) The mixing angle $\vartheta (t)$ of Eq. (12) varies slowly (adiabatically).
- (ii) The $P$ pulse is negligible at the start of the pulse sequence ($|\vartheta |\to 0$ mod $\pi $).
- (iii) The $S$ pulse is negligible at the termination of the pulse sequence ($|\vartheta |\to \pi /2$ mod $\pi $).
- (iv) The two-photon detuning remains zero, i.e., $\delta =0$.
- (v) The single-photon detuning $\mathrm{\Delta}$ 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 ${C}_{2}(t)$ 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 $|\vartheta |\to 0$ and $|\vartheta |\to \pi /2$ (each mod $\pi $), there will occur the complete conversion of population in state 1, with $|{C}_{1}|=1$, to population entirely in state 3, with $|{C}_{3}|=1$.

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, $\delta =0$, 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.

#### 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.

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

### 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 ${\mathrm{\Delta}}_{S}/2\pi \approx 200\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}$. 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 ${P}_{2}(\infty )$.

## 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:

*eigenvalue*equation: the stationary-state wavefunctions ${\psi}_{n}(\mathbf{r})$ are

*eigenfunctions*of this Hamiltonian and the energies ${E}_{n}$ are its

*eigenvalues*. Solutions are obtained by replacing momentum components with partial derivatives, as in ${p}_{x}\to \mathrm{i}\hslash \partial /\partial x$ 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 ${\psi}_{n}(\mathbf{r})$. 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* $\u27e8A|B\u27e9$ between vectors $\mathbf{A}$ and $\mathbf{B}$. The two vectors are *orthogonal* (perpendicular) if $\u27e8A|B\u27e9=0$ and are parallel if $\u27e8A|B\u27e9=1$. It is useful to deal with vectors of unit length, meaning $\u27e8A|A\u27e9=1$. 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 $|{\psi}_{n}\u27e9$ or $|n\u27e9$) 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 $|{\psi}_{n}\u27e9$. The statevector $\mathbf{\Psi}(t)$ can be regarded as an $N$-component *column vector* $|\mathrm{\Psi}(t)\u27e9$ having $N$ elements ${C}_{n}(t){e}^{-\mathrm{i}{\zeta}_{n}(t)}$. Its Hermitian conjugate ${\mathbf{\Psi}}^{\u2020}(t)$ is a row vector $\u27e8\mathrm{\Psi}(t)|$ 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 $\u27e8{\psi}_{n}|\mathrm{\Psi}(t)\u27e9$ of the statevector onto the stationary Hilbert-space coordinate associated with quantum state $n$:

*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 ${\zeta}_{n}(t)$ 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* $\u27e8\mathrm{\Psi}(t)|\mathcal{M}|\mathrm{\Psi}(t)\u27e9$ 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:

*coherences*${C}_{m}{(t)}^{*}{C}_{n}(t)$. This phase has no effect on the probabilities ${P}_{n}(t)$, 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 ${C}_{n}(t)$ 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

The Hamiltonian matrix $\mathsf{H}(t)$ appearing here [the matrix representation of a Hamiltonian operator $\mathcal{H}(\mathbf{p},\mathbf{r},t)$] is the sum of two matrices: That for an undisturbed Hamiltonian ${\mathsf{H}}^{0}$ has the basis vectors ${\mathit{\psi}}_{n}$ as its eigenvectors, and its eigenvalues ${E}_{n}$ are the energies observable for the undisturbed system. The time dependence occurs through an interaction energy ${\mathsf{H}}^{\mathrm{int}}(t)$ 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

*diabatic eigenvalues*) are while the off-diagonal elements, responsible for transitions, are, without approximation,

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*:

*counter-rotating*term, is ${\mathrm{e}}^{\pm 2\mathrm{i}\omega t}$. We conclude the goal of obtaining a slowly varying $\mathsf{W}(t)$ by neglecting this rapid variation, thereby making the RWA [13,14,200,201], see Appendix C. The result is the formula

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:

Alternatively, $\phi (t)$ can be chosen to cancel any time variation of the energies ${E}_{n}$ (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 $\mathcal{E}(t)$ 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

where the minus sign on $\hslash {\omega}_{S}$ 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 matrixFor 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

in the basic STIRAP Eq. (2). When the Rabi frequencies are absent this revision leads to state-2 probabilities that decay exponentially: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

*Liouville equation*, where $[\mathsf{A},\mathsf{B}]\equiv \mathsf{A}\mathsf{B}-\mathsf{B}\mathsf{A}$ denotes the commutator of two matrices.

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

**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 $\mathcal{L}$ whose elements produce changes of coherences as well as transfers of population [29,60,121,214–217]:

**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 $\mathcal{L}$ in the density-matrix equation of Eq. (63). That equation, for constant $\mathsf{W}(t)$, has solutions for which, at large times, the density matrix undergoes no further change:

## 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*) $\mathbf{\Upsilon}(t)$ governs the motion of a vector $\mathbf{R}(t)$,

*adiabatically follows*the torque vector $\mathbf{\Upsilon}(t)$. Figure 21 shows an example of motion when the two vectors $\mathbf{R}$ and $\mathbf{\Upsilon}$ remain parallel.

Alternatively, when the two vectors are perpendicular, then the torque vector acts to rotate $\mathbf{R}(t)$ about the instantaneous axis of $\mathbf{\Upsilon}(t)$, at the rate $|\mathbf{\Upsilon}(t)|$. Such behavior appears as Rabi oscillations of two components of $\mathbf{R}$. Figure 24 shows an example when the two vectors $\mathbf{R}$ and $\mathbf{\Upsilon}$ 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 $\mathbf{R}(t)$ in an abstract vector space formed from elements of the density matrix, i.e., bilinear products of probability amplitudes, controlled by a torque vector $\mathbf{\Upsilon}(t)$ 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 $\mathbf{\Upsilon}(t)$ such that the Schrödinger equation takes the form of a torque equation, see Appendix E. When the three-state TDSE (2) is fully resonant (${\mathrm{\Delta}}_{S}={\mathrm{\Delta}}_{P}=\delta =0$) then it can be written, with some alteration of ordering, 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 $\mathbf{R}$ and $\mathbf{\Upsilon}$ 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 $\mathbf{R}$ and $\mathbf{\Upsilon}$ 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, ${C}_{3}(t)\to -1$, 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 [219–226] and Oreg [184], established a formalism for treating coherent $N$-state quantum dynamics by means of a coherence vector $\mathbf{S}(t)$ of dimension ${N}^{2}-1$ 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 $N\times N$ matrices ${\mathsf{s}}_{j}$, generators of the $SU(N)$ group and characterized by commutators

Here the ${f}_{jkl}$ are the structure constants of the chosen $SU(N)$ representation. We use these matrices to define a*coherence vector*$\mathbf{S}(t)$ and a torque vector $\mathbf{\Upsilon}(t)$, each of dimension ${N}^{2}-1$, having components

Choices of group representations for the matrices ${\mathsf{s}}_{j}$, fitting the symmetry of the Hamiltonian linkage, allow simplification of the description of $\mathbf{S}(t)$; 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 [224,227–231]. 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 ${\mathsf{s}}_{k}$ are then the Pauli matrices, angular momentum matrices for $j=\frac{1}{2}$.

This formalism was used in [184,219–226] for treating coherent $N$-state quantum dynamics by means of a coherence vector (here I suppress explicit time dependence)

whose real-valued elements were obtained from elements of the $N$-state density matrixThe 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 $\delta =0$ makes use of two-state superpositions that combine all of the transition strength into a so-called *bright state* ${\mathbf{\varphi}}_{B}(t)$ (also known as the *coupled state* [234]), leaving the orthogonal *dark state* ${\mathbf{\varphi}}_{D}(t)$ (also known as the *uncoupled state* [234]) immune to change, cf. Appendix D:

*trapped*or

*trapping*state [232–235]. Such states were detected by their lack of fluorescence, a property that led them to be termed “dark states” [236–240]. 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, $\mathbf{\Psi}(-\infty )={\mathbf{\varphi}}_{D}(-\infty )$, 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) [241–249].

#### 4.3. Insight: Diabatic Following

When the Rabi frequencies are negligible, the Hamiltonian produces no appreciable change in the populations ${P}_{n}(t)$. However, there may occur phase changes of the probability amplitudes ${C}_{n}(t)$. Consider a RWA Hamiltonian represented by a diagonal matrix:

*diabatic following*(DF). This behavior, wherein the statevector remains fixed in the basis of RW states ${\mathit{\psi}}_{n}^{\prime}(t)$, 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 $\mathsf{W}(t)$ by introducing instantaneous eigenvectors of that matrix that satisfy the eigenvalue equation

These are variously known as*dressed states*[250] [by contrast with the

*bare states*associated with basis vectors ${\mathit{\psi}}_{n}$ or the phased RW states ${\mathit{\psi}}_{n}^{\prime}(t)$] 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 $|{a}^{\nu}\u27e9$, $|{g}^{\nu}\u27e9$ and, most simply, $|\nu \u27e9$.

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

**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 $|+\u27e9$ and $|-\u27e9$ 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 $|+\u27e9$ has an eigenvalue larger than (or equal to) that of $|-\u27e9$. 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 ${\mathbf{\varphi}}_{\nu}(t)$ as an alternative Hilbert-space coordinate system to the $N$ unit vectors ${\mathit{\psi}}_{n}$ or ${\mathit{\psi}}_{n}^{\prime}(t)$, writing the vector of probability amplitudes as

*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 ${\mathit{\psi}}_{n}^{\prime}(t)$ 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

*chirp*). The two diabatic eigenvalues of this $\mathsf{W}(t)$ are zero (for state 1) and $\mathrm{\Delta}(t)$ (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

*mixing angle*$\theta (t)$ is defined by the relationships

### 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 ${\mathbf{\varphi}}_{\nu}(t)$ of the three-state RWA Hamiltonian matrix of Eq. (49). When the two-photon resonance condition $\delta =0$ is fulfilled the eigenvalues of $\mathsf{W}(t)$ are [252], in order of increasing value,

### 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:

The resulting three equations, with assumed two-photon resonance $\delta =0$, read where the three-state adiabatic RWA Hamiltonian is [7]The nonadiabatic coupling term $\frac{d}{dt}\phi (t)$ vanishes when $\mathrm{\Delta}=0$ (the eigenvalue separation is then ${\mathrm{\Omega}}_{\mathrm{rms}}$) 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),

but with the following correction [7]: Only in the limit of very slow change in mixing angle $\vartheta $ 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 $\mathsf{W}(t)$ be very slow. For the STIRAP system this becomes a requirement that the change in mixing angle $\vartheta (t)$ be much less than the eigenvalue separation, which for fully resonant excitation $(\mathrm{\Delta}=0,\delta =0)$ is the rms Rabi frequency:

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:

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 [9,15,254–257] (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

**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 $\pi /2$ (dominant $P$ field) is, for any form of intermediate time variation, $\pi /2$. The integral of a Rabi frequency ${\mathrm{\Omega}}_{n}(t)$ is a (temporal) pulse area ${\mathcal{A}}_{n}$. Thus by integrating Eq. (103) for a positive Rabi frequency we obtain the global condition that the rms pulse area be much larger than $\pi /2$:

#### 4.5. Insight: Statevector Rotation

The variation in the Hilbert-space structure of ${\mathbf{\varphi}}_{0}(t)$ during the course of a STIRAP process is most readily tracked by means of the mixing angle $\vartheta (t)$. By controlling the $P$ and $S$ fields an experimenter rotates the mixing angle from 0 to $|\pi /2|$ (all modulo $\pi $), thereby shifting the dominant component of ${\mathbf{\varphi}}_{0}(t)$ from state 1 to state 3. Figure 9 shows the Hilbert-space motion of the vectors ${\mathbf{\varphi}}_{\nu}(t)$ associated with STIRAP. [Here, and elsewhere, the coordinate vectors are RW unit vectors ${\mathit{\psi}}_{n}^{\prime}(t)$]. When the statevector $\mathbf{\Psi}(t)$ remains aligned with ${\mathbf{\varphi}}_{0}(t)$, 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 ${\mathbf{\varphi}}_{B}(t)$ and ${\mathbf{\varphi}}_{D}(t)$ rotate together in the 1,3 plane, while the third, independent unit vector ${\mathit{\psi}}_{2}^{\prime}(t)$ 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 ${\mathit{\psi}}_{n}^{\prime}(t)$ or adiabatic states ${\mathbf{\varphi}}_{\nu}(t)$ 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 $\mathsf{W}(t)$ 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.

**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)

*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 $\mathrm{\Delta}(t)$ sweeps from very large negative to very large positive values the two-state mixing angle of Eq. (93) rises from 0 to $\pi $, and with this the adiabatic state ${\mathbf{\varphi}}_{+}(t)$ varies from alignment with state ${\mathit{\psi}}_{1}^{\prime}(t)$ to alignment with ${\mathit{\psi}}_{2}^{\prime}(t)$. 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)

*b*) shows there will be no population transfer. For slow, adiabatic change there will be population transfer as the system point follows adiabatic state $|+\u27e9$. 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 [156–160] 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 ${\mathit{\psi}}_{n}^{\prime}(t)$ (or a fixed superposition of them). These states are associated with eigenvectors of an undisturbed RWA Hamiltonian ${\mathsf{W}}^{0}$ formed from the diagonal elements of the full RWA Hamiltonian $\mathsf{W}(t)$ of Eq. (49). The element ${W}_{11}^{0}(t)$ we have taken to be zero; the other elements of ${\mathsf{W}}^{0}$, the *diabatic eigenvalues*, are the detunings $\mathrm{\Delta}\equiv {\mathrm{\Delta}}_{P}$ and $\delta \equiv {\mathrm{\Delta}}_{S}-{\mathrm{\Delta}}_{P}$. Those numbers, times $\hslash $, are *diabatic energies*, associated with the RW (diabatic) states ${\mathit{\psi}}_{n}^{\prime}(t)$, 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

**Adiabatic following.** When changes of the RWA Hamiltonian occur sufficiently slowly (adiabatic motion) the statevector remains aligned with an adiabatic eigenvector ${\mathbf{\varphi}}_{\nu}(t)$, 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 ${\mathbf{\varphi}}_{0}(t)$ we track the continuing change of the statevector (and its associated populations) as the system point moves along the curve ${\u03f5}_{0}(t)$, an example of adiabatic following. The adiabatic coordinates are

### 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 $\mathrm{\Delta}=0$ and two-photon resonance, $\delta =0$. 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 (${\mathrm{\Delta}}_{3}={\mathrm{\Delta}}_{1}=0$) 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: $\delta =0$ and a positive $\mathrm{\Delta}$. The adiabatic eigenvalues ${\u03f5}_{\nu}(t)$ 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, $|0\u27e9$ or $|-\u27e9$ 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 $|0\u27e9$ 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 $|0\u27e9$ 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 $|0\u27e9$ 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 $|-\u27e9$, 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 $|-\u27e9$ 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 $|-\u27e9$ 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 $1\to 3$ population transfer and the return $3\to 1$ transfer.

As the static single-photon detuning $\mathrm{\Delta}$ 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 ${\mathrm{\Omega}}_{P}(t){\mathrm{\Omega}}_{S}(t)$ and the time-varying effective detuning is proportional to the difference of the two pulses, ${\mathrm{\Omega}}_{S}(t)-{\mathrm{\Omega}}_{P}(t)$. 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 $\mathrm{\Delta}={\mathrm{\Delta}}_{P}$ and $\delta ={\mathrm{\Delta}}_{P}-{\mathrm{\Delta}}_{S}$ 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 ${\mathrm{\Delta}}_{3}=2$, 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, ${\u03f5}_{-}(t)=0$, and so the initial conditions are $\mathbf{\Psi}(t)={\mathit{\psi}}_{1}^{\prime}(t)={\mathbf{\Psi}}_{-}(t)$. The system point initially follows the track of the eigenvalue ${\u03f5}_{-}(t)$ 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 $|0\u27e9$ until the time marked $A$ in the figure. At this moment there occurs a near-meeting of two adiabatic-eigenvalue curves, ${\u03f5}_{-}(t)\approx {\u03f5}_{0}(t)\approx 0$. 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 $|0\u27e9$. It adiabatically follows this dark state until the time marked $B$, when there occurs a near-meeting of adiabatic curves ${\u03f5}_{0}(t)\approx {\u03f5}_{+}(t)\approx {\mathrm{\Delta}}_{3}$. The statevector proceeds by DF through this brief interval, becoming aligned with $|+\u27e9$. 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 ${\u03f5}_{-}(t)$ initially. It remains so aligned, unaffected by the near-meetings at time $A$, until the near-meeting of eigenvalues ${\u03f5}_{-}(t)\approx {\u03f5}_{+}(t)\approx {\mathrm{\Delta}}_{3}$ at time $B$. There then occurs DF and the statevector becomes aligned with adiabatic state $|0\u27e9$. 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 ${\u03f5}_{0}(t)$ as the single variable $t$ changes, one can regard the two Rabi frequencies as variables that define a system-point surface ${\u03f5}_{0}[{\mathrm{\Omega}}_{P},{\mathrm{\Omega}}_{S}]$ 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 ${\u03f5}_{+}$, while the lowermost surface is that of ${\u03f5}_{-}$. For this example the two-photon detuning $\delta $ is nonzero, and so the eigenvalues are not obtained from the simple formulas shown earlier. In particular, ${\u03f5}_{0}$ is not always zero, as it is when $\delta =0$. 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.

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 ${\mathrm{\Omega}}_{P}={\mathrm{\Omega}}_{S}=0$ the diabatic energy for state 2 is $\hslash {\mathrm{\Delta}}_{P}<0$ while for state 3 it is $\hslash \delta >0$. (The labels on the vertical axis of Fig. 13 are shifted by $\delta /2$.)

Initially, when ${\mathrm{\Omega}}_{P}={\mathrm{\Omega}}_{S}=0$, the system point is on a corner of the ${\u03f5}_{0}$ 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 $\delta =0$. 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 $\vartheta =0$. In the final stage, here termed *Post*, there is only the $P$ interaction and so again the mixing angle is constant, now at $|\vartheta |=\pi /2$ for complete population transfer. At intermediate times, when both interactions are present, the statevector $\mathbf{\Psi}(t)$ must undergo adiabatic following of the dark state ${\mathbf{\varphi}}_{D}(t)$; motion of the other two complementary Hilbert-space coordinate vectors, either ${\mathbf{\varphi}}_{B}(t)$ and ${\mathit{\psi}}_{2}^{\prime}(t)$ or else ${\mathbf{\varphi}}_{+}(t)$ and ${\mathbf{\varphi}}_{-}(t)$, does not then affect the statevector.

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 [9,15,254–257].

## 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 [15–19].

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

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 $J=0\leftrightarrow 1\leftrightarrow 2$ in metastable neon [55,258,268–273]. 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 [55,268–270]. 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 [55,258,274–276]; see also [15].

#### 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 $J=0\leftrightarrow 1\leftrightarrow 2$ [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. [55,258,268–273].

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., $J=3$, $M=-2$.

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 ${p}_{\text{atom}}(m)$.

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 $\mathcal{E}(t)$, its duration, and its peak value. Let us denote the variable fields characteristics as $\nu $. In any experiment they have a probability distribution ${p}_{\text{field}}(\nu )$.

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

From a set of solutions to these TDSE we wish to evaluate the ensemble-average probability#### 5.5. Doppler Shifts

The field $\mathbf{E}(t)$ 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 $\mathbf{v}$ along the propagation axis of the electromagnetic field. (The unit vector $\mathbf{e}$ that interacts with the dipole moment $\mathbf{d}$ is perpendicular to this propagation direction.) The Doppler shift of the two beams alters their detunings:

*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:

#### 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 $\overline{\mathsf{W}}(t)$ such that its probability amplitudes ${\overline{C}}_{n}(t)$,

#### 5.8. STIRAP Errors

Manifestations of nonadiabatic evolution of the three-state system are evident in two ways: the failure of ${P}_{3}(t)$ to be unity at the termination time $T$ of the pulse sequence (when that is the goal) and the failure of ${P}_{2}(t)$ to be zero at all times. Let us define these errors as

### 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: ${\mathrm{\Delta}}_{P}={\mathrm{\Delta}}_{S}=0$. The figure shows that, based on the requirement of an error less than ${10}^{-2}$ in population transfer, a pulse area larger than $4\pi $ is needed for the Gaussian pulse shapes used. To maintain a state-2 population less than ${10}^{-2}$ it is necessary to have a pulse area less than around $15\pi $. Somewhat different values would be obtained if the error ${\mathit{\u03f5}}_{2}$ is defined as the integrated population in state 2.

The early use of STIRAP for preparation of preselected excited states found population transfer ${P}_{3}(T)=0.9$ satisfactory, and reviews [6,8,10] have suggested an area of at least 10 (or $3\pi $) as suitable for chemical applications. Applications in quantum information [36,112–116] require much smaller errors, typically less than ${10}^{-5}$ [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 [12,282–284]; 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 ${C}_{n}^{\text{ideal}}(t)$ of an ideal RWA Hamiltonian. The ensemble variations produce error measures such as

When the field envelope $\mathcal{E}(t)$ 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) [285–294]; see Subsection 6.6b and, for application to STIRAP, [281,289,290,293–296]. 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) [286,293,300–308] 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 $\mathsf{H}(a,b,\dots ;t)$ that is defined by a set of parameters $a,b,\dots $ such as static detunings and peak Rabi frequencies and which governs a statevector $\mathbf{\Psi}(a,b,\dots ;t)$ through a TDSE:

*fidelity*$f(a,b,\dots )$ with the actual statevector at the conclusion of a procedure, $t\to \infty $, fits some intended ideal result ${\mathbf{\Psi}}^{\text{ideal}}$:

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 [309–311] 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) [312–316] 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 [317–325].

### 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* [326–329]: 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 [257,330–336] and numerous papers have applied its algorithms to STIRAP [169,337–340]. 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) [341–344] 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 $\mathrm{\Delta}$, whether or not that varies with time, so long as the two-photon detuning $\delta $ 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

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 ${\mathit{\psi}}_{3}^{\prime}(t)$, the statevector will be frozen in a coherent superposition of RW states 1 and 3 [40]:

*fractional STIRAP*[65,249,280,345–348], 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 $\mathbf{\Psi}(t)$ adiabatically follows the dark adiabatic eigenvector ${\mathbf{\varphi}}_{0}(t)$. 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 $\mathrm{\Theta}=\pi /4$, then an equally weighted superposition of states 1 and 3 will be created, $\mathbf{\Psi}=({\mathit{\psi}}_{1}^{\prime}-{\mathit{\psi}}_{3}^{\prime})/\sqrt{2}$, as depicted in Fig. 20—so-called

*half STIRAP*. Moreover, if the final phases of the two fields differ by $\alpha $ then this phase will be mapped onto the created superposition, and the final statevector will be the superposition

Note that the vector-space coordinates appearing here are the RW (diabatic) states ${\mathit{\psi}}_{n}^{\prime}(t)$, carrying time-dependent phases ${\zeta}_{n}(t)$.

#### 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* [9,12,349–351].

Figure 21 presents a picture of the Hilbert-space motion of two of the adiabatic eigenvectors, ${\mathbf{\varphi}}_{0}(t)$ and ${\mathbf{\varphi}}_{-}(t)$, that serve as guides for the three-state adiabatic following illustrated in Fig 11. When the statevector is initially aligned with the dark state ${\mathbf{\varphi}}_{0}(t)$, 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 ${\mathbf{\varphi}}_{0}(t)$, 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 ${\mathbf{\varphi}}_{-}(t)$, the guide for adiabatic following when the pulse ordering is intuitive, $PS$.

As the single-photon detuning $\mathrm{\Delta}$ 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, $1\leftrightarrow 2\leftrightarrow 3\leftrightarrow \cdots \leftrightarrow N$ whose RWA Hamiltonian involves $N$ detunings and $N-1$ Rabi frequencies. The general RWA Hamiltonian for chain linkage is tri-diagonal, with the structure (for real-valued Rabi frequencies)

### 6.3a. Chain Timings

The possible timings of the $N-1$ pulsed Rabi frequencies ${\mathrm{\Omega}}_{n}(t)$ 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 $N-1$ resonantly adjusted frequencies [354–357]. With appropriate choice of the $N$ detunings and $N-1$ 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,\dots $ are stable or metastable and the even-integer states $2,4,\dots $ 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*).

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, ${\mathrm{\Delta}}_{1}={\mathrm{\Delta}}_{N}$. 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 ${\mathrm{\Omega}}_{2},{\mathrm{\Omega}}_{4},\dots $ 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 ${\mathrm{\Omega}}_{1},{\mathrm{\Omega}}_{3},\dots $. 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 [40,45,47,48,358–361]. 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 ${\mathrm{\Delta}}_{n}$ must be zero for odd-integer $n$:

**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 $\mathsf{W}(t)$ 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

### 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 $N-1$ transition, and has a “straddling” pulse (or pulses) that connects in chainwise fashion the remaining states $2,3,\dots ,N-1$. 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 $N-2$ dressed states that superpose all but states 1 and $N$ and that serve as intermediate states for a set of $N-2$ simultaneous three-state Lambda linkages [7,12,42]. The equations for states 1 and $N$ have the form

#### 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 $\mathbf{d}$ of the interaction $\mathbf{d}\xb7\mathbf{E}(t)$ becomes distorted by a strong electric field, as quantified by the frequency-dependent *polarizability tensor* $\mathit{\alpha}(\omega )$, 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 $|n\u27e9$ for $|{\psi}_{n}\u27e9$):

#### 6.5. Bloch-Vector Torque Equation

The populations ${P}_{n}(t)$ 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 ${C}_{n}(t)$. For an $N$-state system their real and imaginary parts require $2N$ real numbers for specification. Because the statevector has unit magnitude, only $2N-1$ 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 $\mathit{\rho}$):

*Bloch vector*[366], that moves in a three-dimensional abstract vector space: its tip moves on the surface of a unit sphere, the

*Bloch sphere*. The variables ${r}_{1}\equiv u$ and ${r}_{2}\equiv v$ are termed

*coherences*, the variable ${r}_{3}\equiv w$ is the

*population inversion*. Initially, with population entirely in state 1, the Bloch vector points to the south pole of the Bloch sphere:

*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]:

### 6.5c. Resonant Excitation

When the excitation is resonant, $\mathrm{\Delta}(t)=0$, 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

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 ${\zeta}_{2}(t)=\mathrm{i}\omega t$. Frame (*b*) shows the components in the Schrödinger picture, ${\zeta}_{n}(t)=0$, for the choice ${\omega}_{12}=\omega =8\mathrm{\Omega}$. The simple great-circle path of the RW picture incorporates the phase ${e}^{-\mathrm{i}\omega t}$ into coordinate ${\mathit{\psi}}_{2}^{\prime}(t)$. 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, $\mathbf{r}(t)\xb7\mathbf{\Upsilon}(t)=\pm \mathrm{\Upsilon}(t)$, so that $\mathbf{\Upsilon}(t)\times \mathbf{r}(t),=0$, 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 ${r}_{3}$ 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 $\mathbf{r}$ and $\mathbf{\Upsilon}$) or positive (for antiparallel $\mathbf{r}$ and $\mathbf{\Upsilon}$). 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 $\mathrm{\Omega}(t)$ is pulsed and $\mathrm{\Delta}(t)$ varies monotonically between large extreme values (a *linear chirp*). Both before and after the pulsed $\mathrm{\Omega}(t)$ the nonzero $\mathrm{\Delta}(t)$ 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, $\mathrm{\Delta}(t)$ 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 $\mathrm{\Omega}(t)$ and the detuning $\mathrm{\Delta}(t)$. The diabatic eigenvalues are 0 and $\mathrm{\Delta}(t)$, so a diabatic curve crossing occurs whenever $\mathrm{\Delta}(t)$ 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 ${r}_{2}(t)$ is not exactly zero, but nonetheless complete population inversion occurs.

The RWA Hamiltonian responsible for this behavior, shown in Fig. 26, is commonly idealized as an infinite time interval, $-\infty <t<+\infty $, 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 ${\mathit{\psi}}_{1}^{\prime}(t)\pm {\mathit{\psi}}_{2}^{\prime}(t)$, 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.

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

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, $w(-\infty )=-1$, becomes ${C}_{1}(-\infty )=-1$ in the three-state analog. The pulse ordering pertinent to STIRAP requires, for the two-state system, a time-dependent (pulsed) detuning $\mathrm{\Delta}(t)$ that precedes the Rabi frequency $\mathrm{\Omega}(t)$. 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 $|{C}_{3}|=1$, ${C}_{2}={C}_{1}=0$ 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 $u(t)$:

### 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

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 $\mathbf{R}(t)$ 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 $\mathbf{R}(t)$ of Eq. (150) does not, by itself, give any indication of the phase of component ${C}_{N}(t)$ when ${C}_{1}(t)$ 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

*a*) both of the Rabi frequencies are positive and when adiabatic following occurs the probability amplitudes are expressible as

With the customary choice of ${C}_{1}=1$ as the initial condition the final value of ${C}_{N}(t)$ 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

#### 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 $1\to 3$, as expressed by ${r}_{3}(t)$, but with differing results for coherence ${r}_{1}(t)$ at intermediate times, and with differing signs of final ${C}_{3}$. In each of these examples there occurs three complete population transfers,

*triple STIRAP*[260].

### 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 [14,48,166,353,370–375]—collectively forming a CP [285–294]. These can violate the slow-variation condition ($\mathrm{i}$) 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, $S-P-S-P-S-P\dots $. 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*)

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 ${\mathrm{sin}}^{2}(\pi /4N)$ for $N$ pulses. In the limit of $N\gg 1$, this technique reduces to PAP, while for small $N$, it is similar to generalized $\pi $ 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 [285,288,379–381]. 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.

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 ${P}_{3}(\infty )$ 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 ${P}_{3}>0.9999$. 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].

#### 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 $N=4$.

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 $\mathsf{W}(t)$. 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 [52,383–385], 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]

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

## 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* [112,345,386–393], 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 $\hslash \omega $ 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 [40,360,361,394–401]. 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* [397,398,402–409]. 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 [411–415].

**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* [7,417–421] 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 [422–427]; 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 [7,432–438]. 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 $\omega $, provides the $S$-field coupling $2g(t)\sqrt{n+1}$, 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.

The needed atom-field quantum states are taken as products $|\text{system}\u27e9=|\text{atom}\u27e9|n\u27e9$ of an $n$-photon Fock state $|n\u27e9$ 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

The dark atom-field state with energy ${E}_{n}=\hslash n\omega $ has the construction

#### 7.2. Correlated Center-of-Mass Motion

A particle moving with cm momentum $\mathbf{p}=\hslash \mathbf{k}$, upon gaining energy $\hslash \omega $ from a laser beam, will also gain momentum $\mathrm{\Delta}{\mathbf{p}}_{L}=\hslash (\omega /c){\mathbf{e}}_{L}$, where ${\mathbf{e}}_{L}$ is a unit vector along the laser-propagation axis (and thus perpendicular to the $\mathbf{E}$ and $\mathbf{B}$ fields that drive the energy changes). Figure 36 sketches the geometry of the resulting deflection; see [394].

This quantization of momentum increments underlies coherent deflection and splitting of atomic and molecular beams [40,360,361,394–401] and leads to “matter-wave optics” or “atom-optics” analogues of classical wave optics [397,398,402–409]. When combined with effects of spontaneous emission, which produce momentum decrements in random directions, it leads to procedures for cooling and trapping particles [411–415,439–441].

To treat cm motion of a particle acted on by a traveling electromagnetic field (e.g., a laser beam of frequency $\omega $ and propagation axis defined by a unit vector ${\mathbf{e}}_{L}$) we express the statevector as summed products of internal-excitation states ${\mathit{\psi}}_{n}$ or ${\mathit{\psi}}_{n}^{\prime}(t)$ and states of cm motion ${\mathit{\varphi}}_{m}$ 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 $\mathbf{p}=\hslash \mathbf{k}$ is expressible as a steady initial value ${\mathbf{p}}_{\perp}=\hslash {\mathbf{k}}_{\perp}$ perpendicular to the laser beam, to which are added $m$ discrete increments of laser-field momentum ${\mathbf{k}}_{L}=(\omega /c){\mathbf{e}}_{L}$:

The states $\mathit{\varphi}$ 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): The two indices $n$, $m$ are interdependent: when the internal states form a simple chain, with ordering ${E}_{n}<{E}_{n+1}$, then an increase of $n$ to $n+1$ will accompany a decrease of $m$ to $m-1$. 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

#### 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) [155,421,442–444]. With increasing strength of the control field, the group velocity ${v}_{g}$ 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 $\mathcal{N}$, the slowly varying envelope, expressed as a Rabi frequency ${\mathrm{\Omega}}_{P}(z,t)$, satisfies the equation

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 ${C}_{1}$ and ${C}_{2}$ are linked by the $P$ field, while the $S$ field links ${C}_{2}$ and ${C}_{3}$. Adiabatic elimination of these atomic probability amplitudes produces the equation

*dark-state polariton*[7,417–421] that is a coherent superposition of electric field (i.e., Rabi frequency) and macroscopic atomic-coherence components:

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].

#### 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 $\mathit{\rho}(t)$ formed from products of two complex-valued probability amplitudes ${C}_{n}(t)$. The concept is analogous to the formation, in the classical optics of polarized radiation, of the three-component Stokes vector $\mathbf{S}$ from bilinear products of two complex-valued electric field amplitudes ${\mathcal{E}}_{\mathrm{i}}(t)$ 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 [287,451–457], 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 ${n}_{0}$ 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) [459–461]. 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 ${f}_{n}(x,y)$:

### 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 [466–469] 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.

### 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 [312,313,315–320,322–325,482,483]

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

CP = Composite pulse (for composite adiabatic passage, CAP) [281,289,290,293–296]

CPR = Complete (coherent) population return [15,370,484–487]

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 [241–249]

EIT = Electromagnetically induced transparency [155,421,442–444]

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

LZMS = Landau–Zener–Majorana–Stueckelberg two-state model [156–160]

OC = Optimal control (theory) [330–333,338,488–491]

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

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

RAP = Rapid adiabatic passage [341–344]

RW = Rotating wave (picture)

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

RCAP = Raman chirped adiabatic passage [6,8,161–166]

SACS = Stark-assisted coherent superposition [71]

SAP = Spatial adiabatic passage [459–461]

SARP = Stark-induced adiabatic Raman passage [180–183]

SCRAP = Stark-chirped rapid adiabatic passage [8,171–179]

SQC = Semiconductor quantum circuit [145,493–495]

SRS = Stimulated Raman scattering [83–86]

STA = Shortcut to adiabaticity [286,293,300–308]

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

STIRAP = Stimulated Raman adiabatic passage [4,6–10,12,14,15]

## Appendix B: Rabi Frequency and Rabi Oscillations

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

*Rabi angle*),

## B.1. Pulse Area

When the integral of $\mathcal{A}(t)$ is over the complete duration of a pulse this Rabi angle is the (temporal) *pulse area* $\mathcal{A}$, 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 $\pi $ and with it complete population inversion, ${P}_{1}\to {P}_{2}$ for resonant excitation.

For constant Rabi frequency $\mathrm{\Omega}(t)\equiv \mathrm{\Omega}$ the probabilities for resonant excitation from initially populated state 1 are cyclic:

*Rabi oscillations*, and their frequency, the

*Rabi frequency*[201], is $\mathrm{\Omega}$. What appears in the TDSE, Eq. (B.1), is a

*half*Rabi frequency, $\mathrm{\Omega}/2$: the Rabi frequency is twice the value of the interaction energy divided by $\hslash $.

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 $\mathrm{\Delta}$,

*complete (coherent) population return*(CPR) [15,370,484–487]. By contrast, an abruptly changing pulse, such as a rectangular pulse, will induce population oscillations. As the detuning $\mathrm{\Delta}$ becomes large the permanent transfer of population becomes negligible. Figure 39 illustrates this behavior.

## 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 ${\mathit{\psi}}_{n}^{\prime}(t)$, cf. Subsection 2.2a.

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

*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 $\omega $, leaving the slower but larger changes induced at the Rabi frequency $\mathrm{\Omega}$, 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 $\omega $ relative to the change-inducing Rabi frequency $\mathrm{\Omega}$ can be regarded as the requirement of a photon energy much larger than the interaction energy: 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

## Appendix D: Dark-State: Eigenvalue and Eigenvector

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

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

## Appendix E: Vector Products and Torque Equations

The torque equation (68) $\text{relies on notation}\times \text{for the vector product}$ of two real-valued three-dimensional vectors,

and symbolizes three equations for components of vectors—cyclic permutations of the component equation Thus Eq. (E.1) can also be written in matrix form as## Appendix F: Pulse Shapes

Within the RWA it is the pulse envelope $\mathcal{E}(t)$ 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 ${\mathrm{\Omega}}_{0}$ and centers separated by $\tau /T$.

## F.1. Envelope Examples

**Infinite support.** These have support over the infinite interval $-\infty <t<+\infty $, 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 $\mathcal{A}=1.772\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\Omega}}_{0}T$, $\mathrm{FWHM}=0.833T$:

**Gauss six**: pulse area $\mathcal{A}=1.855\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\Omega}}_{0}T$, $\mathrm{FWHM}=0.815T$:

**Hyperbolic secant**: pulse area $\mathcal{A}=3.141\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\Omega}}_{0}T$, $\mathrm{FWHM}=1.317T$, see [209,499]:

**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 $\mathcal{A}=0.5\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{\Omega}}_{0}T$, $\mathrm{FWHM}=0.5T$:

**Shark fin**: pulse area $\mathcal{A}={\mathrm{\Omega}}_{0}T$, $\mathrm{FWHM}=0.523T$, see [9,15,254–257]:

**Rectangle**: pulse area $\mathcal{A}={\mathrm{\Omega}}_{0}T$, $\mathrm{FWHM}=T$:

## 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 $\pi $ 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*) [260,500–502]. The pulse of frame (*c*) can be regarded as a two-component *pulse train*.

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 $5\pi $ segments of opposite sign, as in frame (*b*). At the null of the pulse envelope, $t=0$, the Rabi angle is $5\pi $ 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 $1-i-2$ 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

When the intermediate-state detuning ${\mathrm{\Delta}}_{i}$ is sufficiently large the amplitude ${C}_{i}(t)$ 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 ${\overline{C}}_{i}(t)$ that average over these rapid oscillations. The averaging nullifies the time derivative of ${\overline{C}}_{i}(t)$, which is then expressible from Eq. (G.2b) as

*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]:

*dynamic Stark shifts*) proportional to the squares of Rabi frequencies (i.e., to radiation intensities)

In principle, one could redefine the phases ${\zeta}_{n}(t)$ to eliminate completely the diagonal elements of the matrix ${\mathsf{W}}^{\text{eff}}(t)$—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 $\mathbf{d}$ projected onto an electric field vector $\mathbf{E}(t)$. 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* $\mathit{\alpha}$, as in the replacement

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

## 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.

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

*adiabatic condition*

**The Demkov–Kunike model.**A more general form for pulsed and detuned excitation introduces the Demkov–Kunike model [209,499],

*Cayley–Klein parameters*[209,211,290,512]. An alternative parametrization uses

*Stückelberg*parameters [174,290]: two angles and the transition probability $p$,

## 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.

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 $|+\u27e9$. This adiabatic state pairs later with state 2. With negative chirp the adiabatic transfer state is $|-\u27e9$.

## 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

The simulations of this figure used $0,rt,2rt$ as the detunings (and diabatic eigenvalues) ${\mathrm{\Delta}}_{n}(t)$. 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 $|-\u27e9$. 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 $|+\u27e9$. 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 $\mathrm{\Omega}(t)$ involves a two-photon transition and is proportional to the product of two single-photon Rabi frequencies ${\mathrm{\Omega}}_{P}(t)$ and ${\mathrm{\Omega}}_{S}(t)$. The two-state effective detuning $\mathrm{\Delta}(t)$ 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 ${\mathrm{\Omega}}_{P}(t)$ and ${\mathrm{\Omega}}_{S}(t)$ permits construction of a pulsed detuning $\mathrm{\Delta}(t)$ that rises and falls, with consequent diabatic-curve crossings at two times.

When the pairing of the two pulses $\mathrm{\Omega}(t)$ and $\mathrm{\Delta}(t)$ 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 $|-\u27e9$. The two eigenvalue curves for those states coincide initially.

**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 $|-\u27e9$ 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 $|+\u27e9$ 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 $|+\u27e9$. There occurs an overall change both in the bare-state populations (1 to 2 at A) and the adiabatic-state populations ($|-\u27e9$ to $|+\u27e9$ 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 ($|-\u27e9$ to $|+\u27e9$ 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 $\mathrm{\Omega}(t)$ and $\mathrm{\Delta}(t)$ 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):

*probe*field and the $S$ field is a

*strong*control field, with $|{\mathrm{\Omega}}_{S}(t)|\gg |{\mathrm{\Omega}}_{P}(t)|$. We partition the states 2 and 3 that are linked by this strong transition as the matrix

## 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 ${\mathrm{\Delta}}_{P}=0$, two resonances, at the values

*Autler–Townes (AT) splitting*[516], an example of a dynamic Stark shift,

The form of $\tilde{\mathsf{W}}(t)$ in Eq. (I.7) requires modification when the RWA Hamiltonian ${\mathsf{W}}^{S}(t)$ of Eq. (I.2) includes losses ${\mathrm{\Gamma}}_{n}$ from states 2 and 3 [421]. The introduction of the states ${\mathbf{\varphi}}_{\pm}^{S}(t)$ defined in Eq. (I.5) produces then a matrix that is not diagonal. The eigenvalues ${\mathit{\epsilon}}_{\pm}^{S}(t)$ gain imaginary parts $-\mathrm{i}{\mathrm{\Gamma}}_{\pm}/2$ and the matrix acquires non-Hermitian off-diagonal elements $\mathrm{i}{\mathrm{\Gamma}}_{(+-)}/4$ [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\to \infty $ 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, ${\overline{\rho}}_{nm}$.

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 ${\overline{\rho}}_{nm}={\overline{C}}_{n}{\overline{C}}_{m}^{*}$, to the negative imaginary part of ${\overline{C}}_{2}$. A simple approach, used by [234,442], uses the TDSE by introducing loss rates ${\mathrm{\Gamma}}_{n}$ for states 2 and 3, setting ${C}_{1}(t)=1$, and seeking steady-state solutions for the other amplitudes by setting to zero all time derivatives, $\frac{d}{dt}{\overline{C}}_{n}=0$. The result for the steady-state excited-state amplitude ${\overline{C}}_{2}$ is

A plot of this steady excitation-amplitude ${\overline{C}}_{2}$, or the imaginary part of the $P$-field susceptibility, as a function of the $P$-field detuning $\mathrm{\Delta}$ for ${\mathrm{\Delta}}_{S}=0$, shows two absorption peaks [234,421], the *Autler–Townes doublet*, symmetric about ${\mathrm{\Delta}}_{P}=0$ and separated by $|{\mathrm{\Omega}}_{S}|$. At the midpoint of the pattern, where $\mathrm{\Delta}=0$, the absorption is zero. The result is EIT [155,421,442–444] 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 [517–519].

**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]

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

For any such separable state each of the subsystems $A,B,C,\dots $ 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]:

## 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 ${\mathbf{\varphi}}_{0}^{T}$. 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* [112,345,386–393]:

- 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 $\mathbf{d}$ onto the direction of the electric field, as parametrized by a unit vector $\mathbf{e}$, 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 $|j,m\u27e9$ 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* [266,522–528] provides formulas, summarized below, for evaluating the various discrete orientations of the two vectors that produce the dipole interaction Hamiltonian.

## K.1. Angular Momentum Operators and States

Any triplet of square, Hermitian matrices ${\mathsf{J}}_{1},{\mathsf{J}}_{2},{\mathsf{J}}_{3}$ of dimension $2j+1$ that have the commutators

**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),

*coupled*eigenstates of the combined angular momentum The $({j}_{1}{m}_{1},{j}_{2}{m}_{2}|JM)$ of Eq. (K.5) are

*Clebsch–Gordan coefficients*[266,522–528], real valued functions of the angular momentum quantum numbers. The coupled states $|{j}_{1},{j}_{2},JM\u27e9$ 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 ${j}_{1}={j}_{2}=1/2$.

## K.2. Irreducible Tensors

To evaluate the matrix elements of the dipole transition moment $\mathbf{d}$ we consider radiation whose electric field is specified by three amplitudes, corresponding to three basic spherical unit vectors, ${\mathbf{e}}_{q}$:

*irreducible tensor*${T}_{q}^{(k)}$ of order $k$ transform under rotation as

## 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

## 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 ${q}^{\prime}$:

The generalization of Eq. (K.10) for two-photon interaction in the RWA Hamiltonian, with ${E}_{2}>{E}_{1}$, is

## 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]:

*reduced matrix element*of the tensor operator ${T}^{(k)}$. It follows the Wigner

*three-j symbol*$(\text{: : :})$ 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]

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

The symmetry properties of the three-j symbols include## 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 ${d}_{q}$, only those sublevels for which