PHYSICAL REVIEW X 5, 031038 (2015)
Quantum-Classical Correspondence Principle for Work Distributions
† ‡ Christopher Jarzynski,1,* H. T. Quan,2,3, and Saar Rahav4, 1Department of Chemistry and Biochemistry and Institute of Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA 2School of Physics, Peking University, Beijing 100871, China 3Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 4Schulich Faculty of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel (Received 14 May 2015; published 17 September 2015) For closed quantum systems driven away from equilibrium, work is often defined in terms of projective measurements of initial and final energies. This definition leads to statistical distributions of work that satisfy nonequilibrium work and fluctuation relations. While this two-point measurement definition of quantum work can be justified heuristically by appeal to the first law of thermodynamics, its relationship to the classical definition of work has not been carefully examined. In this paper, we employ semiclassical methods, combined with numerical simulations of a driven quartic oscillator, to study the correspondence between classical and quantal definitions of work in systems with 1 degree of freedom. We find that a semiclassical work distribution, built from classical trajectories that connect the initial and final energies, provides an excellent approximation to the quantum work distribution when the trajectories are assigned suitable phases and are allowed to interfere. Neglecting the interferences between trajectories reduces the distribution to that of the corresponding classical process. Hence, in the semiclassical limit, the quantum work distribution converges to the classical distribution, decorated by a quantum interference pattern. We also derive the form of the quantum work distribution at the boundary between classically allowed and forbidden regions, where this distribution tunnels into the forbidden region. Our results clarify how the correspondence principle applies in the context of quantum and classical work distributions and contribute to the understanding of work and nonequilibrium work relations in the quantum regime.
DOI: 10.1103/PhysRevX.5.031038 Subject Areas: Quantum Physics, Statistical Physics
I. INTRODUCTION end (t ¼ τ). If the system Hamiltonian is varied from ˆ ð0Þ¼ ˆ ˆ ðτÞ¼ ˆ Work is a familiar concept in elementary mechanics and H HA to H HB, then the measurement out- a central one in thermodynamics. In recent years, interest in comes will be eigenvalues of these operators. The work the nonequilibrium thermodynamics of small systems [1–4] performed during the process is then defined to be the has motivated careful examinations of how to define difference between these two values, e.g., quantum work [5–50]. In this context one often considers B A a process in which a quantum system evolves under the W ¼ En − Em; ð1Þ Schrödinger equation as its Hamiltonian is varied in time— for instance, a quantum particle in a piston undergoing if the measurements produce the mth and nth eigenvalues compression or expansion [51]. It is typically assumed that of the initial and final Hamiltonians. In this definition, work the system is initialized in thermal equilibrium, and the is inherently stochastic, with two sources of randomness: question becomes, how do we appropriately define the the statistical randomness associated with sampling an work performed on the system during a single realization of A initial energy Em from the canonical equilibrium distribu- this process? tion [Eq. (21)] and the purely quantal randomness asso- One answer involves two projective measurements of the “ ” jψ i ’ ¼ 0 ciated with the collapse of the final wave function τ system s energy, at the start of the process (t ) and at the upon making a projective measurement of the final energy [Eq. (22)] [52]. *[email protected] In an analogous classical process, a system is prepared † [email protected] in thermal equilibrium, then it evolves under Hamilton’s ‡ [email protected] equations as the Hamiltonian function is varied from H ðzÞ to H ðzÞ, where z denotes a point in the system’s Published by the American Physical Society under the terms of A B the Creative Commons Attribution 3.0 License. Further distri- phase space. Since the system is thermally isolated, it is bution of this work must maintain attribution to the author(s) and natural to define the work as the change in its internal the published article’s title, journal citation, and DOI. energy:
2160-3308=15=5(3)=031038(17) 031038-1 Published by the American Physical Society CHRISTOPHER JARZYNSKI, H. T. QUAN, AND SAAR RAHAV PHYS. REV. X 5, 031038 (2015)
SC Q W ¼ HBðzτÞ − HAðz0Þ; ð2Þ are included, P accurately captures the oscillations in P (Fig. 5). Thus, the oscillations in PQ can be understood as a for a realization during which the system evolves from z0 to quantum interference pattern superimposed on a classical zτ. As in the quantum case, W is a stochastic variable, but background. This picture breaks down around nmin and here the randomness arises solely from the sampling of the Q nmax, where P exhibits tails that tunnel into classically initial microstate z0 from an equilibrium distribution. forbidden regions. In Sec. V, we derive a semiclassical Much of the recent interest in quantum work has been approximation, expressed in terms of the Airy function, that stimulated by the discovery and experimental verification accurately describes these tails (Fig. 9). – of classical nonequilibrium work relations [53 58], which The analyses in Secs. IV and V rely on theoretical tools have provided insights into the second law of thermody- that are familiar in the field of semiclassical mechanics. To namics, particularly as it applies to small systems where the best of our knowledge, these tools have not previously fluctuations are important [2]. For thermally isolated been applied to study the relationship between classical and systems, the quantal counterparts of these relations follow quantum work distributions, although similar analyses have directly from the definition of work given by Eq. (1) been performed in the context of molecular scattering [6,7,9,15]. Moreover, experimental tests of quantum non- theory [65–67] and laser-pulsed atoms [68,69], among equilibrium work relations have recently been pro- other examples. In particular, our calculations in Sec. IV B – posed [59 62] and implemented [63,64], providing direct closely parallel those of Schwieters and Delos [69]. verification of the validity of these results. Despite the evident similarity between Eqs. (1) and (2), and despite the relevance of Eq. (1) to nonequilibrium work II. CLASSICAL AND QUANTUM relations, the definition of quantum work given by Eq. (1) TRANSITION PROBABILITIES might seem ad hoc. Indeed, given the absence of a broadly Here, we introduce notation and specify the problem we “ ” agreed-upon textbook definition of quantum work, one plan to study. In Sec. II A, we define a discretized classical might reasonably suspect that Eq. (1) has been introduced work distribution, Eq. (17), that can be compared directly to precisely because it leads to quantum nonequilibrium work the quantum work distribution, Eq. (20). These distribu- relations. Furthermore, the wave function collapse that tions involve classical and quantum transition probabilities, occurs upon measuring the final energy has no classical PCðnjmÞ and PQðnjmÞ, which are the central objects of counterpart. Because of this collapse, the quantum work study throughout the rest of the paper. has no independent physical reality until the measurement is performed at t ¼ τ. By contrast, the classical work can be viewed as a well-defined function of time, namely, the net A. Classical setup change in the value of the time-dependent Hamiltonian Consider a system with 1 degree of freedom, described along the trajectory zt. Our aim in this paper is to use the by a Hamiltonian tools of semiclassical mechanics, together with numerical simulations, to investigate the relationship between Eqs. (1) p2 and (2), and specifically to clarify how the correspondence Hðz; λÞ¼ þ Vðq; λÞ; ð3Þ 2M principle applies in this context. We restrict ourselves to systems with 1 degree of where z ¼ðq; pÞ denotes a point in phase space and λ is an freedom, for which there exist explicit semiclassical externally controlled parameter. We assume that the energy approximations of energy eigenstates. We focus on the shells (level surfaces) of the Hamiltonian form simple, transition probability PQðnjmÞ from the mth eigenstate of ˆ ˆ closed curves in phase space. We consider the evolution HA to the nth eigenstate of HB, and on its classical analog, of this system under the time-dependent Hamiltonian PCðnjmÞ , defined in Sec. II. In Sec. III, we investigate both Hðz; λtÞ, where λ is varied from λ0 ¼ A to λτ ¼ B.For quantities numerically for the example of a forced quartic compact notation, we define HAðzÞ ≡ Hðz; λ0Þ and Q oscillator. For high-lying initial energies P oscillates HBðzÞ ≡ Hðz; λτÞ. rapidly with n, whereas PC is a smooth function over a Suppose that prior to the start of the process the system finite range nmin 031038-2 QUANTUM-CLASSICAL CORRESPONDENCE PRINCIPLE … PHYS. REV. X 5, 031038 (2015) 1 ∂Ω W ¼ Eτ − E0 ≡ HBðzτÞ − HAðz0Þ: ð4Þ gλðEÞ¼ : ð9Þ h ∂E For an ensemble of realizations of this process, the ¯ C distribution of values of work performed on the system is The conditional probability distribution P ðEτjE0Þ is Z Z given by the expression C ¯ C ¯ C R P ðWÞ¼ dEτ dE0P ðEτjE0ÞP ðE0ÞδðW − Eτ þ E0Þ; A dz0δ½E0 − H ðz0Þ δ½Eτ − H ðzτðz0ÞÞ ¯ Cð j Þ¼ R A B P Eτ E0 δ½ − ð Þ ; ð5Þ dz0 E0 HA z0 ð10Þ ¯ Cð Þ where PA E0 is the probability distribution of initial ¯ C ð Þ energies, sampled from equilibrium, and P ðEτjE0Þ is where zτ z0 denotes the final conditions of a trajectory that the conditional probability distribution to end with a final evolves from initial conditions z0. It is useful to imagine an energy Eτ, given an initial energy E0. Let us consider these ensemble of initial conditions sampled microcanonically factors separately. from the energy shell E0 of the Hamiltonian HA [see ¯ C Fig. 1(a)]. The swarm of trajectories that evolves from these PA is simply the classical equilibrium energy distribution: initial conditions defines a time-dependent, closed curve At 1 in phase space. At t ¼ τ, the points of intersection between ¯ C −βE0 P ðE0Þ¼ e gAðE0Þ; ð6Þ Aτ and the energy shell Eτ of the Hamiltonian H [black A ZC B A dots in Fig. 1(b)] represent the final conditions of those trajectories that end with energy Eτ, having begun with where energy E0. If we consider two nearby energy shells Eτ and Z ¯ C 1 Eτ þ dE, then P ðEτjE0ÞdEτ is the fraction of trajectories C Zλ ¼ dz exp ½−βHðz; λÞ and whose final energies fall within this energy interval. h Z Let us define the mth energy interval of the Hamiltonian 1 gλðEÞ¼ dzδ½E − Hðz; λÞ ð7Þ HA to be the range of energy values E satisfying h mh ≤ ΩðE; AÞ < ðm þ 1Þh; ð11Þ are the classical partition function and density of states, −1 ¼ 0 1 2 … β ¼ðkBTÞ ,andh is Planck’s constant. We include the where m ; ; ; . Furthermore, let us use a midpoint −1 A factors h [which cancel in Eq. (6)] for later convenience. rule to assign a particular energy value Em to each interval: Let us also define I 1 Z I A ΩðEm;AÞ¼ pdq ¼ m þ h: ð12Þ A 2 ΩðE; λÞ¼ dzθ½E − Hðz; λÞ ¼ pdq; ð8Þ Em E Analogous definitions apply to the nth energy interval of B which is the phase space volume enclosed by the energy shell the Hamiltonian HB and the corresponding energy En. E of the Hamiltonian H, equivalently the integral of pdq From Eqs. (9) and (11), the width of the mth energy around this energy shell. We then have interval of HA is FIG. 1. An energy shell A0 of the initial Hamiltonian [panel (a), t ¼ 0] evolves under HðtÞ to the curve A ≡ Aτ [panel (b), t ¼ τ]. The curve B is an energy shell of the final Hamiltonian. 031038-3 CHRISTOPHER JARZYNSKI, H. T. QUAN, AND SAAR RAHAV PHYS. REV. X 5, 031038 (2015) 1 δ A ≈ ð Þ limit the density of states increases, as does the quantum Em A : 13 gAðEmÞ number at a given energy. In practice, the semiclassical limit is often simply identified with the condition m ≫ 1. For a smooth function of energy fðEÞ, Eq. (13) leads to Z B. Quantum setup A fðEmÞ dE0fðE0Þ ≈ ; ð14Þ In the quantum version of this problem, Hðz; λÞ is g ðEA Þ m A m replaced by the Hermitian operator where the integral is taken over the mth interval. ℏ2 ∂2 Readers will recognize Eq. (12) as the semiclassical ˆ ðλÞ¼− þ ð ; λÞ ð Þ H H 2 V q : 18 quantization condition pdq ¼½m þð1=2Þ h, which pro- 2M ∂q vides an excellent approximation for the mth eigenvalue of ψð Þ¼h jψ i ˆ ≫ 1 A wave function q; t q t evolves under the the quantum Hamiltonian HA [Eq. (18)] when m .For ˆ ˆ Schrödinger equation as Hðλ Þ is varied from Hðλ0Þ¼ convenience, we later use the notation EA to denote this t m ˆ ˆ ðλ Þ¼ ˆ ˆ eigenvalue, without making a distinction between the exact HA to H τ HB. The eigenstates and eigenvalues of HA eigenvalue and its semiclassical approximation. are specified as follows: The probability to obtain initial conditions z0 within the ϕA ðqÞ¼hqjϕA i; Hˆ jϕA i¼EA jϕA i; ð19Þ mth interval of HA, when sampling from equilibrium, is m m A m m m Z 1 ˆ −β A with similar notation for HB. Evolution in time is repre- Cð Þ¼ ¯ Cð Þ ≈ Em ð Þ PA m dE0PA E0 e ; 15 ˆ ˆ ðλ Þ ˆ ¼ ZC sented by the unitary operator Ut satisfying H t Ut m A ˆ ˆ ˆ ˆ iℏ∂Ut=∂t and U0 ¼ I, where I is the identity operator. using Eq. (14). Similarly, the conditional probability to end Using Eq. (1) and assuming thermal equilibration at in the nth energy interval, given a representative initial t ¼ 0, the work distribution is [15] energy in the mth interval, is X Qð Þ¼ Qð j Þ Qð Þδð − B þ A Þ ð Þ Z P W P n m PA m W En Em : 20 ¯ Cð Bj A Þ C ¯ C A P En Em n;m P ðnjmÞ¼ dEτP ðEτjE0 ¼ E Þ ≈ : m g ðEBÞ n B n Qð Þ ð Þ PA m is the probability of obtaining the mth eigenstate of 16 ˆ HA when making the initial energy measurement: We refer to this as the classical transition probability. The −β A C Em interpretation of P ðnjmÞ is straightforward. Imagine a Qð Þ¼e ð Þ PA m Q ; 21 swarm of trajectories evolving from initial conditions ZA sampled from a microcanonical ensemble with energy P EA t ¼ τ Q ¼ ð−β A Þ m [see Fig. 1]. At the end of the process, , the with ZA m exp Em . The quantum transition prob- fraction of these trajectories that fall into the energy ability PQðnjmÞ is the conditional probability to obtain the B B C window ½E ;E þ1 is equal to P ðnjmÞ [69]. The classical ˆ n n nth eigenstate of HB upon making the final measurement, work distribution can now be rewritten as a sum over initial ˆ given the mth eigenstate of HA at the initial measurement: and final energy intervals: Q B ˆ A 2 B 2 P ðnjmÞ¼jhϕ jUτjϕ ij ¼jhϕ jψ τij : ð22Þ PCðWÞ n m n Z Z X The classical and quantum work distributions given by ¼ ¯ Cð j Þ ¯ Cð Þδð − þ Þ dEτ dE0P Eτ E0 PA E0 W Eτ E0 Eqs. (17) and (20) can now be compared directly. We note n;m n m X Z Z that ≈ ¯ Cð j A Þ ¯ Cð Þδð − B þ A Þ Z dEτ dE0P Eτ Em PA E0 W En Em X −β −β A Q n;m n m C ¼ E ð Þ ≈ Em ¼ ; ð Þ X ZA dEe gA E e ZA 23 ¼ Cð j Þ Cð Þδð − B þ A Þ ð Þ m P n m PA m W En Em : 17 m;n Cð Þ ≈ Qð Þ hence, PA m PA m [see Eqs. (15) and (21)]. Thus, Finally, we point out that the approximations appearing what remains is to clarify the relationship between the in Eqs. (13)–(17) assume that the width of the mth energy classical and quantum transition probabilities, PCðnjmÞ and Qð j Þ interval δEm is very small on a classical scale. Formally, P n m . In the following section, we compare these this assumption can be justified by taking the semiclassical transition probabilities in a model system for which both limit ℏ → 0 with all classical quantities held fixed. In this the classical and quantum dynamics are simulated 031038-4 QUANTUM-CLASSICAL CORRESPONDENCE PRINCIPLE … PHYS. REV. X 5, 031038 (2015) PQ n 150 ,PC n 150 oscillator, for which analytical solutions are available. 0.06 Specifically, Deffner et al. [70,71] have studied an oscil- lator with a time-varying stiffness; Talkner et al. [72] have 0.05 obtained the work distribution for an oscillator driven by 0.04 time-dependent perturbations that are linear in q and p; Campisi [73] has studied changes in the Boltzmann entropy 0.03 for forced quantum and classical oscillators; Ford et al. [74] have investigated a harmonic oscillator with a cyclically 0.02 driven stiffness; and Talkner et al. have considered the 0.01 sudden quench of a two-dimensional oscillator [26]. While the harmonic oscillator has the advantage of being ana- n lytically tractable, it is somewhat special in that its quantum 100 120 140 160 180 200 220 dynamics can be reduced to its classical dynamics [75]. FIG. 2. Quantum [Eq. (22)] and classical [Eq. (16)] transition Here, we choose the more “generic” quartic oscillator, probabilities for the forced quartic oscillator. The solid black which requires numerical simulations. curve shows PQðnjmÞ, while the dashed red curve shows To evaluate PCðnjmÞ we simulate 104 Hamiltonian PCðnjmÞ, for m ¼ 150. trajectories evolving under Hðz; λtÞ. Initial conditions are sampled microcanonically, and final conditions are binned n n according to the energy intervals of HBðzÞ, as defined in PQ k 150 , PC k 150 k 0 k 0 Sec. II A. For our initial microcanonical ensemble, we A 1.0 choose m ¼ 150, corresponding to E0 ¼ Em ¼ 1749.23. The results are shown by the dashed line in Fig. 2. The 0.8 transition probability is a smooth function of n from ¼ 105 ¼ 215 nmin to nmax , and zero outside this range. 0.6 These sharp cutoffs reflect the fact that our initial conditions are sampled from a microcanonical distribution: 0.4 nmin and nmax correspond to the minimal and maximal final energies that can be reached by trajectories launched with 0.2 A initial energies E0 ¼ Em, as illustrated in Fig. 7. The characteristic U shape of the classical work distribution n 100 150 200 250 within the allowed region is related to the fact that the curves A and B shown in Fig. 1(b) become tangent to one FIG. 3. Accumulated transition probabilities for the forced ¼ ¼ another at E Emin and E Emax (again, see Fig. 7). For quarticP oscillator. The jagged black curve represents the quantum an exactly solvable model in which the same U shape case, n PQðkjmÞ, while the smooth red curve represents the k¼0 P appears, see Fig. 1 of Ref. [73]. classical case, n PCðkjmÞ. k¼0 For the quantum system, we expand a wave function evolving under Hˆ ðλ Þ¼pˆ 2=2M þ λ qˆ 4 as follows: C Q t t numerically. We find that P and P are manifestly X − γ ð Þ different (Fig. 2), but these differences mostly vanish after ψðq; tÞ¼ c ðtÞhqjϕ ðλ Þie i n t ; appropriate smoothing (Fig. 3). In Secs. IV and V,we n n t nZ develop a semiclassical theory to explain these features. t 1 0 γnðtÞ¼ Enðλt0 Þdt : ð25Þ ℏ 0 III. NUMERICAL CASE STUDY: THE FORCED QUARTIC OSCILLATOR ˆ Here, Hjϕni¼Enjϕni and the cn’s are expansion coef- We begin with a model system, the forced quartic ficients. The time-dependent Schrödinger equation iℏψ_ ¼ oscillator: Hˆ ψ then produces the set of coupled ordinary differential equations, p2 Hðz; λÞ¼ þ λq4: ð24Þ 2M X ∂ϕ _ k ðγ −γ Þ We set M ¼ 1=2 and ℏ ¼ h=2π ¼ 1, and we vary the work _ ¼ −λ ϕ i n k cn n ∂λ e ck parameter at a constant rate, λt ¼ λ0 þ vt, from λ0 ¼ 1 ¼ A k to λτ ¼ 5 ¼ B, taking v ¼ 50 and therefore τ ¼ 0.08. X hϕ j 4jϕ i _ n q k iðγ −γ Þ A number of groups have previously compared quantum ¼ −λ e n k ck: ð26Þ E − E and classical work distributions for a driven harmonic k≠n k n 031038-5 CHRISTOPHER JARZYNSKI, H. T. QUAN, AND SAAR RAHAV PHYS. REV. X 5, 031038 (2015) We integrate these equations numerically, from initial the final energy, for convenience. As discussed in Sec. II, conditions cnð0Þ¼δmn, to obtain we imagine a swarm of trajectories evolving in time under Hðz; λtÞ, with initial conditions sampled from the micro- Q 2 P ðnjmÞ¼jcnðτÞj : ð27Þ canonical phase space distribution, In Fig. 2 we plot the quantum transition probability as a δ½ − ð Þ δ½ − ð Þ η ð ; Þ¼R E0 HA z ¼ E0 HA z ð Þ function of the final quantum number n (solid line). A z E0 ; 28 dzδ½E0 − HAðzÞ hgAðE0Þ Although a correspondence between PQðnjmÞ and Cð j Þ P n m is visually evident, the quantum and classical where E0 is a parameter of the distribution. The quantity cases differ in two distinct ways: (i) the quantum proba- ¯ C P ðEjE0ÞdE is the fraction of these trajectories that end Qð j Þ bility oscillates rapidly with n and (ii) P n m shows tails with a final energy in the infinitesimal range ðE; E þ dEÞ. that “tunnel” into the classically forbidden regions n We begin with the conditional probability distribution A B ¯ C ðq ;pÞ¼ðq ;p ðq ÞÞ¼ðq ;p ðq ÞÞ: ð32Þ P ðEjE0Þ [Eq. (10)], where we drop the subscript τ from l l l bl l l bl l 031038-6 QUANTUM-CLASSICAL CORRESPONDENCE PRINCIPLE … PHYS. REV. X 5, 031038 (2015) A δ −1 With some abuse of notation, we write bl rather than b ¯ C A E l P ðEjE0Þ¼ρ ðq Þ · and bB, though in general the l’th intersection point may l l δq l occur at differently indexed branches of the two surfaces. ∂pA ∂pB −1 ¯ C ¼ ð ÞρAð ÞρBð Þ − To evaluate P ðEjE0Þ, we begin with the contribution hgB E ql ql : ∂q ∂q ¼ ¯ Cð j Þ q ql Pl E E0 from a single intersection point l. Figure 4 depicts this intersection, together with an intersection ð37Þ point involving an infinitesimally displaced energy shell Summing over all intersection points gives us HB ¼ E þ δE. The highlighted line segment connecting these two points represents a set of trajectories with final A B −1 þ δ XK ∂p ∂p energies between E and E E; hence, ¯ Cð j Þ¼ ð Þ ρAð ÞρB ð Þ bl − bl P E E0 hgB E b ql b ql : l l ∂q ∂q ¼ l¼1 q ql ρAð Þ jδ j¼ ¯ Cð j Þ jδ j ð Þ ql · q Pl E E0 · E : 33 ð38Þ A B Since we focus here on the contribution from a single If we now set E0 ¼ Em and E ¼ En , then Eq. (16) gives us A B intersection between and , we suppress the subscripts XK A B −1 bl indicating the branches of these surfaces. By construc- ∂p ∂p Cð j Þ¼ ρAð ÞρB ð Þ bl − bl ð Þ tion, the ratio δE=δq is the rate of change of H with P n m h b ql b ql : 39 B l l ∂q ∂q ¼ respect to q along the curve A: l¼1 q ql This is the main result of this subsection. δE d A ¼ H ½q; p ðqÞ j ¼ : ð34Þ We use the term classically allowed region to denote δ B q ql ≤ ≤ q dq the range Emin E Emax within which the function ¯ C P ðEjE0Þ does not vanish, representing all final energies B Since p ðqÞ specifies a surface of constant HB, we have that can be attained by trajectories with initial energy E0. Cð j Þ In the coarse-grained function P n m , the classically ∂ B ∂ ∂ ∂ allowed region is bracketed by the values n and n , p p HB= q min max ¼ ¼ − ; ð35Þ ∂q ∂q ∂H =∂p denoting the energy intervals of HB containing Emin and HB B Emax. As illustrated in Fig. 7, the values Emin and Emax correspond, respectively, to the largest energy shell of HB hence, that fits entirely within the closed curve A and the smallest energy shell of HB that surrounds the entire curve A.At d ∂H ∂H ∂pA these two energies, the curves A and B become tangent to H ½q; pAðqÞ ¼ B þ B dq B ∂q ∂p ∂q one another, ∂pA=∂q ¼ ∂pB =∂q, leading to divergences bl bl ∂H ∂pA ∂pB in Eq. (38), which are reflected in the two sharp peaks in ¼ B − : ð36Þ Cð j Þ ∂p ∂q ∂q P n m in Fig. 2. Similar divergences appear in a more familiar context, namely, the density ρBðqÞ that describes the projection of a fixed microcanonical ensemble H ¼ E Combining Eqs. (31), (33), (34), and (36), we get B onto the coordinate axis [see Eq. (31)]. In this case, the divergences occur at classical turning points at which the velocity ∂HB=∂p vanishes. In the example introduced in Sec. III, the reflection symmetry of the quartic potential implies that trajectories come in symmetry-related pairs: if ðqt;ptÞ is a solution of Hamilton’s equations under the time-dependent ð ; λ Þ ð− − Þ Hamiltonian H q; p t , then so is qt; pt . This means ð Þ A B that for every intersection point ql;pl between and there will be another intersection point at ð−ql; −plÞ, and the two will contribute equally to PCðnjmÞ. B. Semiclassical transition probabilities ¯ Qð j Þ FIG. 4. The filled circle indicates an intersection between the Let us now evaluate P n m in the semiclassical closed curves A and B; see Fig. 1. The open circle is an limit. Appendix A provides a brief introduction to time- intersection point between A and a nearby energy shell, dependent WKB theory, and for further details we refer the HB ¼ E þ δE. reader to the reviews by Delos [76] and Littlejohn [77]. 031038-7 CHRISTOPHER JARZYNSKI, H. T. QUAN, AND SAAR RAHAV PHYS. REV. X 5, 031038 (2015) ˆ A ∂2 A ∂2 B ∂ A ∂ B Using Eq. (A5), the wave function ψðq; tÞ¼hqjUtjϕmi S S p p κ ¼ bl ð Þ − bl ð Þ¼ bl ð Þ − bl ð Þ ˆ l 2 ql 2 ql ql ql : and the nth eigenstate of HB can be written as ∂q ∂q ∂q ∂q ð Þ X qffiffiffiffiffiffiffiffiffiffiffiffiffi 46 A i A A π ψðq; tÞ¼ ρ t ðqÞ S t ðqÞ − iμ t ; ð Þ b exp ℏ b b 2 40a b Performing the stationary phase integral then gives us qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi XK i π X i π hϕBjψ i ≈ ρAρB Δ − Δμ B B B B n τ b b exp Sl i l ϕn ðqÞ¼ ρ ðqÞ exp S ðqÞ − iμ ; ð40bÞ l l ℏ 2 b ℏ b b 2 l¼1 b Z iκ × dq exp l ðq − q Þ2 A 2ℏ l where the actions S t ðqÞ and SBðqÞ generate the momen- b b sffiffiffiffiffiffiffiffi tum fields XK qffiffiffiffiffiffiffiffiffiffiffi A i π 2πℏ B iσlπ=4 ¼ ρ ρ exp ΔSl − iΔμl e bl bl ℏ 2 jκ j ¼1 l A B l A ∂S t ∂S X t b B b θ p ðqÞ¼ ;pðqÞ¼ ; ð41Þ ¼ a ei l ; ð47Þ b ∂q b ∂q l l A and the integers μ t and μB are Maslov indices (see ρA;B ¼ Δμ ¼ μA − μB b b where b are evaluated at q ql; l b b ; and Appendix A). The densities and momentum fields in l l l σl ¼ sgnðκlÞ¼ 1. In Ref. [69], where similar calculations Eqs. (40) and (41) are the same as those appearing in are performed, a quantity equivalent to our πσl=4 is Sec. IVA. identified as a Maslov-like phase (Ref. [69], p. 1037). From Eq. (40) we get B Equation (47) expresses hϕn jψ τi as a sum over trajecto- X Z qffiffiffiffiffiffiffiffiffiffiffi ries, each contributing an amplitude and a phase: B A B sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hϕn jψ τi¼ dq ρ ρ 0 b b A B b;b0 ρ ρ 1 π π ¼ 2πℏ bl bl θ ¼ Δ − Δμ þ σ ð Þ π al ; l Sl l l: 48 i A B A B jκlj ℏ 2 4 × exp ðS − S 0 Þ − iðμ − μ 0 Þ ; ð42Þ ℏ b b b b 2 The semiclassical transition probability PSCðnjmÞ ≈ jhϕBjψ ij2 which appears as Eq. (37b) in Ref. [69]. Using the n τ is then given by stationary phase approximation to evaluate the integral, XK 2 we obtain a sum of contributions from points ql satisfying θ SCð j Þ¼ i l ð Þ the condition P n m ale : 49 l¼1 A B ∂S ∂S 0 The general form of Eq. (49) is not surprising. The path b ðq Þ¼ b ðq Þ; ð43Þ ∂q l ∂q l integral formulation of quantum mechanics tells us that solutions of the time-dependent Schrödinger equation can which is equivalent to be represented in terms of sums over classical paths decorated by phases expðiS=ℏÞ, and when ℏ → 0 the A B dominant contributions to the sums come from those paths p ðq Þ¼p 0 ðq Þ ≡ p : ð44Þ b l b l l that are solutions of the classical equations of motion [78]. We expect the semiclassical transition probability func- hϕBjψ i Hence, n τ reduces to a sum of contributions arising tion PSCðnjmÞ to provide a good approximation to the A B from the K intersection points of the surfaces and , quantum transition probability PQðnjmÞ, for large quantum A representing trajectories with initial and final energies Em numbers. For the forced quartic oscillator, Fig. 5 compares B and En , just as in the classical case [Eq. (32)]. these functions, taking m ¼ 150. PQ is computed as To determine the contribution from the lth intersection described in Sec. III, and PSC is evaluated directly from ¼ point, we perform a quadratic expansion around q ql: classical simulations of trajectories evolving from an initial microcanonical ensemble. (The evolution of the multi- 1 A A B 2 valued action S t is obtained by integrating pdq − Hdt S ðqÞ − S ðqÞ ≈ ΔSl þ κlðq − qlÞ ; ð45Þ bl bl 2 along each trajectory; see, e.g., Sec. 3.5 of Ref. [77].) We observe that the agreement is excellent, except in the with ΔS ¼ SAðq Þ − SB ðq Þ and vicinity of the boundaries between the classically allowed l bl l bl l 031038-8 QUANTUM-CLASSICAL CORRESPONDENCE PRINCIPLE … PHYS. REV. X 5, 031038 (2015) PQ n 150 ,PSC n 150 C. Interferences and symmetries 0.04 Let us now examine the interferences in more detail. Recall from Sec. IVA that the trajectories contributing to 0.03 Eq. (49) come in symmetry-related pairs. This allows us to PQ n 150 ,PSC n 150 0.02 write 0.07 0.01 XK XJ 0.06 B iθl iθl iθlþJ hϕn jψ τi¼ ale ¼ ðale þ alþJe Þ; ð52Þ 0.05 200 210 220 l¼1 l¼1 0.04 where J ¼ K=2 is an integer, and the lth and ðl þ JÞth 0.03 trajectories are related by symmetry: 0.02 ð Þ¼ð− − Þ ð Þ qlþJ;plþJ ql; pl : 53 0.01 Here, we index the points ðq ;pÞ from l ¼ 1 to K, n l l 100 120 140 160 180 200 220 according to the order in which they appear as we proceed clockwise around the energy shell B, first rightward along FIG. 5. A comparison of quantum and semiclassical transition the upper branch and then leftward along the lower branch. probabilities for our model system. The black solid curve shows PQðnjmÞ, while the red dashed curve shows PSCðnjmÞ,asgiven Equation (53) implies by Eq. (49). All of the parameters are the same as those in Fig. 2. alþJ ¼ al; σlþJ ¼ σl: ð54Þ and forbidden regions. We examine these boundaries in Using the fact that the points l and l þ J are located directly more detail in Sec. V. opposite one another on the closed curves A and B Now let us simplify Eq. (49) by simply ignoring the cross [Eq. (53)], we obtain terms in the double sum. This is the so-called diagonal I – 1 1 approximation [79 81], and it leads to A − A ¼ ¼ þ πℏ SlþJ Sl pdq m ; 2 A 2 I XK XK A B −1 1 1 diag ∂p ∂p B − B ¼ ¼ þ πℏ ð Þ SCð j Þ ≈ 2 ¼ 2πℏ ρAρB bl − bl SlþJ Sl pdq n ; 55 P n m al b b ; 2 B 2 l l ∂q ∂q ¼ l¼1 l¼1 q ql ð50Þ and μA − μA ¼ μB − μB ¼ 1; ð56Þ which is identical to the expression for the classical lþJ l lþJ l Cð j Þ transition probability, P n m [Eq. (39)]. P hence, Q SC iθl 2 The agreement between P and P ¼j ale j seen in Fig. 5 suggests that the rapid oscillations of the quantum ΔSlþJ ¼ ΔSl þðm − nÞπℏ; ΔμlþJ ¼ Δμl: ð57Þ probability distribution can be understood in terms of phase interference between different trajectories. When these [In Eq. (55) we invoke the semiclassical quantization interferences are ignored, we recover the classical proba- condition, Eq. (12), togetherH with Liouville’s theorem, bility distribution, as we see in Eq. (50). Hence, PSC acts as which guarantees that pdq remains constant with time. At a bridge between the quantum and classical transition In Eq. (56) we use the fact that the Maslov index is probabilities, incremented by þ2 as one proceeds around the closed curve A or B; see, e.g., Fig. 1 of Ref. [69].] Equations (48), diag (52), (54), and (57) give us PQðnjmÞ ≈ PSCðnjmÞ ≈ PCðnjmÞ; ð51Þ XJ B iθl iðm−nÞπ hϕn jψ τi¼ ale ½1 þ e : ð58Þ and thus between the quantum and classical work distri- l¼1 butions [Eqs. (20) and (17)]. The first approximation in Eq. (51) involves time-dependent WKB theory together When m − n is even, the symmetry-related trajectories with a stationary phase evaluation of integrals, and the interfere constructively, but when m and n have opposite second is the diagonal approximation in which interfer- parities, they interfere destructively and we get ences are ignored. PSCðnjmÞ¼0. This provides a semiclassical explanation 031038-9 CHRISTOPHER JARZYNSKI, H. T. QUAN, AND SAAR RAHAV PHYS. REV. X 5, 031038 (2015) for the observation, noted in Sec. III, that the quantum V. AIRY TAILS transition probability vanishes when m − n is odd: the most In Fig. 5 we see that our semiclassical approximation PQðnjmÞ rapid oscillations in arise from interference fails around the boundaries of the classical allowed region between symmetry-related trajectories. The same interpre- at n and n . Although the semiclassical transition tation was given by Miller to explain angular momentum min max probability derived in Sec. IV vanishes identically outside selection rules in the scattering matrix describing an atom the classically allowed region (since only classical trajec- impinging on a homonuclear diatomic molecule (see tories contribute to PSC), the quantum transition probability Sec. III C of Ref. [67]). tunnels into the classically forbidden region, as seen in The interference effects in PSC, due to cross terms in Fig. 5. In this section, we construct a semiclassical Eq. (49), include contributions not only from pairs of approximation that describes this behavior. Our central trajectories that are related by symmetry [Eq. (53)] but also results are given by Eqs. (64) and (69) and illustrated from those that are not related by symmetry: in Fig. 9. Similar tunneling behavior is observed in the wave functions of energy eigenstates, which exhibit tails that XK Xs Xn 2 ðθ −θ Þ ðθ −θ Þ reach into classically forbidden regions [82], as well as in SCð j Þ¼ þ i l k þ i l k P n m al alake alake ; semiclassical treatments of the S matrix in molecular l¼1 l≠k l≠k collisions [66]; in both cases the tails are approximated ð59Þ by Airy functions. The expressions that we obtain in this section are also expressed in terms of Airy functions. The failure of PSCðnjmÞ in the boundary regions P A B where s denotes a sum of symmetry-relatedP pairs of originates in the fact that the curves and become ð Þ¼−ð Þ n tangent to one another at E ¼ E and E ¼ E . In the trajectories, qk;pk ql;pl , and is a sum over min max ð Þ ≠−ð Þ example we study numerically, the curves are tangent nonsymmetry-related pairs, qk;pk ql;pl . To illus- simultaneously at two points, due to the symmetry of trate these contributions separately,P we construct the SC n the quartic potential, as shown in Fig. 7(b).However,itis quantity Psym by omitting the sum on the right-hand SC useful first to discuss the generic case, where there exists side of Eq.P (59), and we similarly construct Pnosym by s SC only a single point of tangency, depicted in Fig. 7(a).For omitting . The results are shown in Fig. 6. Psym displays specificity, we discuss the upper boundary of the classically a regular oscillation, as the symmetry-related trajectories allowed region at Emax, but similar comments apply to the are either exactly in phase or exactly out of phase, lower boundary at E . − min according to the parity of m n. The oscillations in Let us consider the function SC θ − θ Pnosym are less regular, as the phases l k are not necessarily integer multiples of π. When all cross terms Δ ð Þ¼ Að Þ − Bð ÞðÞ are included, both sets of oscillations combine to give S q; E S q S q; E 60 the pattern in Fig. 5. For instance, the gap observed near n ¼ 200 in Fig. 5 reflects the corresponding minimum in that appears inside the exponent in Eq. (42) and its SC Pnosym seen in Fig. 6(b). derivative SC C SC C Psym n 150 ,P n 150 Pnosym n 150 ,P n 150 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 n n 100 120 140 160 180 200 220 100 120 140 160 180 200 220 FIG. 6. Contributions to the interference pattern in PSCðnjmÞ. Panel (a) includes only interferences between symmetry-related trajectories, while (b) includes only interferences between nonsymmetry-related trajectories. 031038-10 QUANTUM-CLASSICAL CORRESPONDENCE PRINCIPLE … PHYS. REV. X 5, 031038 (2015) FIG. 7. Solid blue lines depict the surfaces A ¼ Aτ (as in Fig. 1). Dashed red lines are energy shells B of HB. Diamonds and dots A B indicate the points of tangency between and at Emin and Emax, respectively. (a) In the absence of symmetries, we generically expect a single tangent point at Emin and Emax. (b) For the Hamiltonian that we study numerically, Eq. (24), the symmetry of the quartic potential implies that both A and the energy shells B are symmetric under ðq; pÞ → ð−q; −pÞ; thus, points of tangency come in symmetry-related pairs. Z ∂ i Δpðq; EÞ¼pAðqÞ − pBðq; EÞ¼ ΔSðq; EÞ: ð61Þ dq exp ΔSðq; EÞ ∂q ℏ 2ℏ 1=3 i ¼ 2π exp ΔSðq ;EÞ We suppress the subscripts b and b0, and we explicitly jkj ℏ c B B indicate that S and p depend on theR final energy shell E. 21=3ν − ð − Þ ð Þ In Sec. IV, we evaluate the integral dq expðiΔS=ℏÞ by ×Ai E Emax ; 62 k1=3ℏ2=3 performing a quadratic expansion of ΔS around a single A B intersection point ql between the curves and , where Δ p vanishes (Fig. 4). As the energy E approaches Emax from below, two points of intersection between A and B where ð Þ coalesce at a point qc;pc , indicated in Fig. 7(a). Figure 8 illustrates the behavior of Δpðq; EÞ and ΔSðq; EÞ in the Δ vicinity of qc and Emax. In particular, S has two stationary 2 B points when E Δ Δ ≈ A B FIG. 8. Behavior of p and S for E Emax. The open circles correspond to intersection points between and when E 031038-11 CHRISTOPHER JARZYNSKI, H. T. QUAN, AND SAAR RAHAV PHYS. REV. X 5, 031038 (2015) Q SC SC;tail P n 150 ,P n 150 P1 ðnjmÞ 0.035 B 2 ¼jhϕ jψ τij n 0.030 2ℏ 2=3 21=3ν 2 0.025 ¼ 4π2ρAρB − ð B − Þ Ai E E ; Q SC 0.020 c c jkj k1=3ℏ2=3 n max P n 150 ,P n 150 0.015 ð64Þ 0.010 0.06 0.005 A ρ ;B ¼ ρA;Bð Þ 0.05 where c qc , and the subscript 1 emphasizes 200 210 220 A that we assume only a single point of tangency between 0.04 and B. At the lower boundary of the classically allowed 0.03 region, we obtain the same result but with Emax replaced by Emin in Eqs. (63) and (64). 0.02 The Airy function decays rapidly for positive values of its argument, and oscillates for negative values: 0.01 n 1 2 100120140160180200220 ðζÞ ∼ pffiffiffi ζ−1=4 − ζ3=2 Ai 2 π exp 3 ; FIG. 9. Comparison between the quantum transition probabil- 1 2 π ities evaluated by numerical integration of the Schrödinger ð−ζÞ ∼ pffiffiffi ζ−1=4 ζ3=2 þ ð Þ Ai π sin 3 4 ; 65 equation (black, solid line) and the semiclassical approximations given by Eq. (49) in the central region 110 031038-12 QUANTUM-CLASSICAL CORRESPONDENCE PRINCIPLE … PHYS. REV. X 5, 031038 (2015) PQðnjmÞ and PCðnjmÞ for systems with 1 degree of ACKNOWLEDGMENTS freedom, we obtain three main conclusions, and illustrate H. T. Q. acknowledges Meng-Li Du for helpful discus- them with simulations of a driven quartic oscillator: sions. C. J. gratefully acknowledges support from the (i) In the classically allowed region, the quantum National Science Foundation (U.S.) under Grants transition probability can be approximated in terms No. DMR-1206971 and No. DMR-1506969. H. T. Q. of interfering classical trajectories [Eq. (49), Fig. 5]. gratefully acknowledges support from the National (ii) When the interferences between these trajectories Science Foundation of China under Grant are ignored, the classical transition probability is No. 11375012, and The Recruitment Program of Global obtained [Eq. (50), Fig. 3]. Youth Experts of China. S. R. and C. J. are grateful for (iii) The tunneling of the quantum transition probability support from the U.S.-Israel Binational Science Foundation into the classically forbidden region is accurately (Grant No. 2010363). S. R. gratefully acknowledges the described by a decaying Airy function, whose support of the Israel Science Foundation (Grant No. 924/ arguments are expressed in terms of classical quan- 11). This work was partially supported by the COSTAction tities [Eqs. (64) and (69), Fig. 9]. MP1209. The second conclusion, and Fig. 3 in particular, clarifies the sense in which classical mechanics is “recovered” in the limit of large quantum numbers (in this context), whereas APPENDIX A: BRIEF REVIEW OF the first and third conclusions describe inherently quantal TIME-DEPENDENT WKB THEORY effects—interference, tunneling—that can nevertheless be In time-dependent WKB theory, approximate solutions understood and approximated in terms of classical trajec- of the time-dependent Schrödinger equation are expressed tories. In view of similar analyses in atomic and molecular in terms of classical constructions in phase space. In this contexts (see, e.g., Refs. [65–69]), these conclusions are brief summary, we restrict ourselves to a single degree of not surprising. However, given recent interest in quantum freedom and a Hamiltonian of the form Hðq; p; tÞ¼ nonequilibrium work relations, our results provide a timely 2 p =2M þ Vðq; tÞ. investigation of the correspondence principle as it applies A wave function ψðq; tÞ is said to be in WKB form when to the work performed on a system driven away from written as the product of a slowly varying amplitude and a equilibrium. As we show, semiclassical mechanics provides rapidly oscillating phase, a bridge between classical and quantal work distributions, and our conclusions provide some justification for the two- ψð Þ¼ ð Þ i ð Þ ð Þ point measurement definition of quantum work [Eq. (1)]. q; t A q; t exp ℏ S q; t ; A1 It will be interesting to study whether our results generalize to systems with more than 1 degree of freedom. ≫ 1 or else as a sum of such terms, as in Eq. (A5). Locally, For N-particle systems with N , we generically expect Eq. (A1) describes a wave train ψ ∝ expðikqÞ, with wave Cð j Þ that the classical transition probability P n m will be bell number k ¼ ℏ−1∂S=∂q. This implies a local momentum shaped rather than U shaped, with nmin and nmax found deep C in the tails of P ðnjmÞ. In this situation the quantum ∂S pðq; tÞ¼ℏk ¼ ðq; tÞ; ðA2Þ tunneling into the forbidden regions will be negligible, ∂q Cð j Þ Qð j Þ since both P n m and P n m will be very small at nmin and nmax. An obstacle to studying N particle systems and the Born rule implies a probability density analytically is the lack of semiclassical representations of energy eigenstates, analogous to Eq. (40b), for generic ρðq; tÞ¼jAðq; tÞj2: ðA3Þ systems with multiple degrees of freedom. Progress might be made by studying two limiting cases: integrable and These functions inherit their time dependence from ψðq; tÞ. fully chaotic systems [84]. For systems whose classical A standard calculation, in which Eq. (A1) is substituted dynamics are integrable, semiclassical expressions for into the Schrödinger equation and ℏ is treated as a small energy eigenstates can be constructed using classical parameter [77] produces the equations of motion action-angle variables [see, e.g., Eq. (11) of Ref. [85], Eq. (3.5) of Ref. [86], or Eq. (37) of Ref. [87]]. At the other ∂S ∂S þ H q; ;t ¼ 0; ðA4aÞ extreme, for fully chaotic systems, Berry’s conjecture [88] ∂t ∂q suggests that energy eigenstates can be treated as Gaussian random functions of the configuration q. In either case, the ∂ρ ∂ p þ ρ ¼ 0; ðA4bÞ expression for the energy eigenstate might be combined ∂t ∂q M with time-dependent WKB theory as a first step toward generalizing Eq. (42) and subsequent results to multidi- where terms of order ℏ2 are neglected. Equation (A4a) is mensional systems. the Hamilton-Jacobi equation. Equation (A4b) is the 031038-13 CHRISTOPHER JARZYNSKI, H. T. QUAN, AND SAAR RAHAV PHYS. REV. X 5, 031038 (2015) continuity equation for evolution under a velocity hence, by Eq. (A5), the wave function simply acquires a field pðq; tÞ=M. phase e−iEt=ℏ. Thus, in the semiclassical limit, energy To interpret these equations, let the function pðq; 0Þ eigenstates correspond to microcanonical ensembles. describe a curve A0 in phase space, at an initial time t ¼ 0. Indeed, Eq. (A5) in this case leads to the WKB approxi- If the wave function ψ is represented as a sum of terms of mation for energy eigenstates that is familiar from under- the form given by Eq. (A1), then pðq; 0Þ is multivalued, graduate textbooks on quantum mechanics, where it is and the curve A0 has multiple branches, but for the moment typically derived in a different manner. Moreover, a proper A we focus on a single branch. 0 is a Lagrangian manifold, treatment ofH the Maslov indices leads to the quantization determined by its generating function, Sðq; 0Þ, via condition pðqÞdq ¼½m þð1=2Þ h. Eq. (A2). Now imagine that this curve carries a probability density whose projection onto the coordinate axis is ρðq; 0Þ. It is convenient to picture a swarm of particles, APPENDIX B: SEMICLASSICAL TRANSITION sprinkled along A0 so that the fraction of particles between PROBABILITIES IN THE BOUNDARY AREA q and q þ dq is given by ρðq; 0Þdq. From these initial Δ ð Þ conditions, each particle in the swarm evolves under Expanding the function S q; E defined by Eq. (60) to Hamilton’s equations, giving rise to a time-dependent cubic order, we get curve At and density ρðq; tÞ. The Hamilton-Jacobi equation ΔSðq; EÞ ≈ ΔS ðEÞþΔS0 ðEÞðq − q Þ governs the evolution of At, through its generating function c c c Sðq; tÞ, and the continuity equation governs the evolution 1 1 þ Δ 00ð Þð − Þ2 þ Δ 000ð Þð − Þ3 ð Þ of the projected density ρðq; tÞ. 2 Sc E q qc 6 Sc E q qc ; B1 With these considerations in mind, approximate solu- tions of the time-dependent Schrödinger equation are where ΔScðEÞ¼ΔSðqc;EÞ and primes indicate deriva- written in the form tives with respect to q. Note that here we expand ΔS around ¼ A X the coalescence point q qc, where the surface becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i π B ψðq; tÞ¼ ρ ðq; tÞ exp S ðq; tÞ − iμ : ðA5Þ tangent toR the surface (see Fig. 8). We now evaluate the b ℏ b b 2 ð Δ ℏÞ b integral dq exp i S= , using Eq. (B1). Let us first consider the integral The sum is taken over the branches of a Lagrangian Z manifold A whose generating function Sðq; tÞ satisfies þ∞ i t I ¼ dq exp fðqÞ ; the Hamilton-Jacobi equation [Eq. (A4a)], and the pro- −∞ ℏ jected probability density for each branch satisfies the 1 1 continuity equation [Eq. (A4b)]. ð Þ¼α þ α þ α 2 þ α 3 ð Þ f q 0 1q 2 2q 6 3q : B2 The quantities μb are Maslov indices. These determine the relative quantum phases of the various branches of A , t Rewriting fðqÞ as and they differ from one another by integer values. (By convention, the overall phase of ψ is adjusted so that the 1 μ ’ ð Þ¼β þ β ð − Þþ β ð − Þ3 ð Þ b s themselves are integers.) See Ref. [77] for a more f q 0 1 q c 3 3 q c ; B3 detailed discussion of Maslov indices, as well as a careful A treatment of caustic points, where two branches of t meet where and the manifold becomes “vertical” in the ðq; pÞ plane. At the semiclassical level of approximation, Eq. (A5) 3 α2 α1α2 α2 connects the evolution of a quantum wave function to that c ¼ − ; β0 ¼ α0 − þ ; α α 3α2 of a swarm of classical particles “surfing” on a Lagrangian 3 3 3 2 manifold. This perspective provides a natural starting point α2 α3 β1 ¼ α1 − ; β3 ¼ ; ðB4Þ for a comparison between quantum and classical work 2α3 2 distributions. As an important example of a WKB wave function, we obtain consider a manifold A0 that is a single energy shell E of a Z ð Þ þ∞ fixed Hamiltonian H q; p , and consider a microcanonical i β3 iβ0=ℏ 3 distribution of initial conditions on this manifold. Under I ¼ e dy exp β1y þ y −∞ ℏ 3 time evolution, neither the manifold nor the probability 1 3 ℏ = β1 distribution changes, as each trajectory merely goes around ¼ 2πeiβ0=ℏ Ai ; ðB5Þ jβ j 2=3 1=3 the energy shell. The solution of Eq. (A4a) in this case is 3 ℏ β3 ð Þ¼ ð 0Þ − ; ð Þ Sb q; t Sb q; Et A6 using the integral representation of the Airy function, 031038-14 QUANTUM-CLASSICAL CORRESPONDENCE PRINCIPLE … PHYS. REV. X 5, 031038 (2015) Z 1 þ∞ t3 like ℏ2=3. Thus, we are interested in energies that have the AiðζÞ¼ dt exp i ζt þ : ðB6Þ 2π −∞ 3 following scaling relation: We now use this result to evaluate ϵ ∼ ℏ2=3; ðB18Þ Z i where ϵ ¼ E − E . To leading order in ℏ, and expressing IðEÞ¼ dq exp ΔSðq; EÞ ðB7Þ c ℏ quantities in terms of ϵ rather than E,wehave for small ℏ, applying the stationary phase approximation aϵ ∂β1 ζ ¼ ; where a ¼ ðϵ ¼ 0Þ; and treating E as a parameter of the integral. ℏ2=3b1=3 ∂ϵ The cubic expansion for ΔS given by Eq. (B1) gives us b ¼ β3ðϵ ¼ 0Þ: ðB19Þ the coefficients 2 Now recall that β1 ¼ α1 − α2=2α3. In the vicinity of α0ðEÞ¼ΔSðq ;EÞ; ðB8Þ c 2=3 ϵ ¼ 0 (where α1 ¼ α2 ¼ 0), we have α1 ∼ ϵ ∼ ℏ and 2 2 4=3 α ð Þ¼Δ 0ð Þ¼Δ ð Þ ð Þ α2=2α3 ∼ ϵ ∼ ℏ ; hence, the latter term can be ignored. 1 E S qc;E p qc;E ; B9 We thus write α ð Þ¼Δ 00ð Þ¼Δ 0ð Þ ð Þ 2 E S qc;E p qc;E ; B10 ∂α1 1 k a ¼ ð0Þ¼−ν;b¼ α3ð0Þ¼ ; ðB20Þ α ð Þ¼Δ 000ð Þ¼Δ 00ð Þ ð Þ ∂ϵ 2 2 3 E S qc;E p qc;E ; B11 A B 0 therefore, with Δp ¼ p − p ¼ ΔS [Eq. (61)]. Note that 1 3 α ð Þ¼α ð Þ¼0 ð Þ 2 = 1 Ec 2 Ec ; B12 ζ ¼ − νϵ: ðB21Þ kℏ2 since A and B are tangent at qc when E ¼ Ec. The integral IðEÞ is now given by Eq. (B5), with β0, β1, and β3 obtained Combining results and discarding terms that vanish as from α0, α1, α2, and α3 via Eq. (B4). This result simplifies if ℏ → 0, Eq. (B5) finally gives us β’ ¼ we consider how the s behave in the vicinity of E Ec as ℏ → 0. 2ℏ 1=3 i 21=3νϵ IðϵÞ¼2π exp α0ðϵÞ Ai − ; ðB22Þ Let us define jkj ℏ k1=3ℏ2=3 ¼ Að Þ¼ Bð Þ ð Þ pc p qc p qc;Ec ; B13 which is equivalent to Eq. 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