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 <n<nmax (Fig. 2). After integrating over has come to equilibrium with a thermal reservoir at the oscillations, the two quantities are nearly identical temperature T and the reservoir has been removed. (Fig. 3), which provides some justification for viewing Therefore, at t ¼ 0 the microstate of the system will be Eq. (1) as the quantal counterpart of Eq. (2). In Sec. IV,we treated as a random sample from a canonical distribution SCð j Þ Q derive P n m , a semiclassical approximation for P corresponding to the Hamiltonian HA. From t ¼ 0 to t ¼ τ, expressed as a sum over classical trajectories. Each the system is described by a trajectory zt evolving under trajectory in this sum carries a quantal phase, giving rise Hamilton’s equations, and the work performed on the to coherent interferences between trajectories.