The 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 one 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.

Maximum Efficiency of Heat Engines Based on a Small System: Carnot Cycle at the Nanoscale

We study the maximum efficiency of a Carnot cycle heat engine based on a small system. It is revealed that due to the finiteness of the system, irreversibility may arise when the working substance contacts with a heat bath. As a result, there is a working-substance-dependent correction to the usual Carnot efficiency, which is valid only when the working substance is in the thermodynamic limit. We derives a general and simple expression for the maximum efficiency of a Carnot cycle heat engine in terms of the relative entropy. This maximum efficiency approaches the usual Carnot efficiency asymptotically when the size of the working substance increases to the thermodynamic limit. Our study extends the Carnots result to cases with arbitrary size working substance and demonstrates the subtlety of thermodynamics in small systems.

Validity of nonequilibrium work relations for the rapidly expanding quantum piston

Recent work by Teifel and Mahler [Eur. Phys. J. B 75, 275 (2010)] raises legitimate concerns regarding the validity of quantum nonequilibrium work relations in processes involving moving hard walls. We study this issue in the context of the rapidly expanding one-dimensional quantum piston. Utilizing exact solutions of the time-dependent Schru007fodinger equation, we find that the evolution of the wave function can be decomposed into static and dynamic components, which have simple semiclassical interpretations in terms of particle-piston collisions. We show that nonequilibrium work relations remains valid at any finite piston speed, provided both components are included, and we study explicitly the work distribution for this model system.

Testing quantum adiabaticity with quench echo

Adiabaticity of quantum evolution is important in many settings. One example is the adiabatic quantum computation. Nevertheless, up to now, there is no effective method to test the adiabaticity of the evolution when the eigenenergies of the driven Hamiltonian are not known. We propose a simple method to check adiabaticity of a quantum process for an arbitrary quantum system. We further propose a operational method for finding a uniformly adiabatic quench scheme based on Kibble-Zurek mechanism for the case when the initial and the final Hamiltonians are given. This method should help in implementing adiabatic quantum computation.

Finite-Temperature Scaling of Magnetic Susceptibility and Geometric Phase in the XY Spin Chain

We study the magnetic susceptibility of 1D quantum XY model, and show that when the temperature approaches zero, the magnetic susceptibility exhibits the finite-temperature scaling behavior. This scaling behavior of the magnetic susceptibility in 1D quantum XY model, due to the quantum-classical mapping, can be easily experimentally tested. Furthermore, the universality in the critical properties of the magnetic susceptibility in quantum XY model is verified. Our study also reveals the close relation between the magnetic susceptibility and the geometric phase in some spin systems, where the quantum phase transitions are driven by an external magnetic field.

Quantum Thermodynamic Cycles and Quantum Heat Engines (II)

We study the quantum mechanical generalization of force or pressure, and then we extend the classical thermodynamic isobaric process to quantum mechanical systems. Based on these efforts, we are able to study the quantum version of thermodynamic cycles that consist of quantum isobaric process, such as quantum Brayton cycle and quantum Diesel cycle. We also consider the implementation of quantum Brayton cycle and quantum Diesel cycle with some model systems, such as single particle in 1D box and single-mode radiation field in a cavity. These studies lay the microscopic (quantum mechanical) foundation for Szilard-Zurek single molecule engine.

Quantum Fidelity and Thermal Phase Transitions

We study the quantum fidelity approach to characterize thermal phase transitions. Specifically, we focus on the mixed-state fidelity induced by a perturbation in temperature. We consider the behavior of fidelity in two types of second-order thermal phase transitions (based on the type of non-analiticity of free energy), and we find that usual fidelity criteria for identifying critical points is more applicable to the case of $lambda$ transitions (divergent second derivatives of free energy). Our study also reveals limitations of the fidelity approach: sensitivity to high temperature thermal fluctuations that wash out information about the transition, and inability of fidelity to distinguish between crossovers and proper phase transitions. In spite of these limitations, however, we find that fidelity remains a good pre-criterion for testing thermal phase transitions, which we use to analyze the non-zero temperature phase diagram of the Lipkin-Meshkov-Glick model.