Quantum work fluctuations in connection with the Jarzynski equality

Abstract

A result of great theoretical and experimental interest, the Jarzynski equality predicts a free energy change ΔF of a system at inverse temperature β from an ensemble average of nonequilibrium exponential work, i.e., 〈e^{-βW}〉=e^{-βΔF}. The number of experimental work values needed to reach a given accuracy of ΔF is determined by the variance of e^{-βW}, denoted var(e^{-βW}). We discover in this work that var(e^{-βW}) in both harmonic and anharmonic Hamiltonian systems can systematically diverge in nonadiabatic work protocols, even when the adiabatic protocols do not suffer from such divergence. This divergence may be regarded as a type of dynamically induced phase transition in work fluctuations. For a quantum harmonic oscillator with time-dependent trapping frequency as a working example, any nonadiabatic work protocol is found to yield a diverging var(e^{-βW}) at sufficiently low temperatures, markedly different from the classical behavior. The divergence of var(e^{-βW}) indicates the too-far-from-equilibrium nature of a nonadiabatic work protocol and makes it compulsory to apply designed control fields to suppress the quantum work fluctuations in order to test the Jarzynski equality.

Deformed Jarzynski Equality

“…The deviation of Q * from unity describes the non-adiabaticity [46] of a work protocol. Note that the case of g = 1 reproduces our previous result [29] for the standard quantum work characteristic function.…”

“…As shown recently [29], this quantum second moment can also diverge in systems with an infinitedimensional Hilbert space. As a matter of fact, the divergence in the quantum case occurs more frequently than in the classical case.…”

“…For example, a work protocol could still lead to divergence in quantum var e −βW even if its classical counterpart has a finite var e −βW . In particular, as the temperature characterizing a quantum system initially at thermal equilibrium decreases, quantum effects become more appreciable and then var e −βW tends to diverge purely due to nonclassical effects [29]. This finding should have an important impact on future experimental studies of the quantum JE.…”

Quantum-Heat Fluctuation Relations in Three-Level Systems Under Projective Measurements

“…As shown in Ref. [23], even when the quantum Jarzynski equality holds and the average of the exponential of the work equals the free energy difference, the variance of the energy difference may diverge for continuum systems, an exception being provided by finite-level quantum systems.…”

“…The evaluation of the relevant work originated by using a coherent modulation of the system Hamiltonian has been the subject of intense investigation [18][19][20][21][22][23][24]. A special focus was devoted to the study of heat and entropy production, obeying of the second law of thermodynamics, the interaction with one or more external bodies, and/or the inclusion of an observer [25][26][27][28][29][30][31][32][33][34].…”