Quantum-Classical Correspondence Principle for Work Distributions

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.

Popular Summary

Introductory physics textbooks define work as the energy required to move an object against an opposing force. Work also plays a central role in the field of thermodynamics, but the concept is notably absent from most texts on quantum mechanics. In recent years, there has been growing interest in properly defining the amount of work performed on a quantum system that is manipulated by externally applied forces. This interest is motivated both by theoretical progress describing how the laws of thermodynamics apply to small systems and by experimental advances in the control of individual atoms and molecules. Even so, defining quantum work has proven to be challenging given the strange characteristic of quantum mechanics in which the very act of observing a system can dramatically change its state.

Quantum work is often defined as the difference between the results of initial and final energy measurements. We show that this definition leads to quantum work distributions that can be understood as interference patterns between classical trajectories. In effect, if there are multiple pathways over which a classical particle can arrive at a particular final energy given an initial energy, then the quantum work includes contributions from all these paths and gives rise to patterns similar to those arising from the interference between diffracted waves. If the interference patterns are ignored, then a classical result is recovered. We additionally obtain an accurate description of the “tails” of quantum work distributions, which correspond to values of work that are classically forbidden.

Our results, which are illustrated by numerical simulations of a driven quartic oscillator, clarify the correspondence between classical and quantum work and provide intuition for the patterns observed in quantum work distributions. We expect that our findings will motivate studies of systems with additional degrees of freedom.

Figure 1

An energy shell ${\mathcal{A}}_0$ of the initial Hamiltonian [panel (a), $t=0$] evolves under $H(t)$ to the curve $\mathcal{A}\equiv {\mathcal{A}}_{\tau }$ [panel (b), $t=\tau$]. The curve $\mathcal{B}$ is an energy shell of the final Hamiltonian.

Figure 2

Quantum [Eq. (22)] and classical [Eq. (16)] transition probabilities for the forced quartic oscillator. The solid black curve shows $P^Q(n|m)$, while the dashed red curve shows $P^C(n|m)$, for $m=150$.

Figure 3

Accumulated transition probabilities for the forced quartic oscillator. The jagged black curve represents the quantum case, ${\sum }_{k=0}^n{P}^Q(k|m)$, while the smooth red curve represents the classical case, ${\sum }_{k=0}^n{P}^C(k|m)$.

Figure 4

The filled circle indicates an intersection between the closed curves $\mathcal{A}$ and $\mathcal{B}$; see Fig. 1. The open circle is an intersection point between $\mathcal{A}$ and a nearby energy shell, $H_B=E+\delta E$.

Figure 5

A comparison of quantum and semiclassical transition probabilities for our model system. The black solid curve shows $P^Q(n|m)$, while the red dashed curve shows $P^{\mathrm{SC}}(n|m)$, as given by Eq. (49). All of the parameters are the same as those in Fig. 2.

Figure 6

Contributions to the interference pattern in $P^{\mathrm{SC}}(n|m)$. Panel (a) includes only interferences between symmetry-related trajectories, while (b) includes only interferences between nonsymmetry-related trajectories.

Figure 7

Solid blue lines depict the surfaces $\mathcal{A}={\mathcal{A}}_{\tau }$ (as in Fig. 1). Dashed red lines are energy shells $\mathcal{B}$ of $H_B$. Diamonds and dots indicate the points of tangency between $\mathcal{A}$ and $\mathcal{B}$ at $E_{\min }$ and $E_{\max }$, respectively. (a) In the absence of symmetries, we generically expect a single tangent point at $E_{\min }$ and $E_{\max }$. (b) For the Hamiltonian that we study numerically, Eq. (24), the symmetry of the quartic potential implies that both $\mathcal{A}$ and the energy shells $\mathcal{B}$ are symmetric under $(q,p)\to (-q,-p)$; thus, points of tangency come in symmetry-related pairs.

Figure 8

Behavior of $\mathrm{\Delta }p$ and $\mathrm{\Delta }S$ for $E\approx E_{\max }$. The open circles correspond to intersection points between $\mathcal{A}$ and $\mathcal{B}$ when $E<E_{\max }$. These coalesce into a single point of tangency at $E=E_{\max }$ (solid circle), and for $E>E_{\max }$, $\mathcal{A}$ and $\mathcal{B}$ do not intersect.

Figure 9

Comparison between the quantum transition probabilities evaluated by numerical integration of the Schrödinger equation (black, solid line) and the semiclassical approximations given by Eq. (49) in the central region $110<n<200$ (red, dashed line) and by Eq. (69) in the boundary regions $90<n<110$ and $200<n<230$ (green, dashed line).