Quantum Work Fluctuations in connection with Jarzynski Equality

Abstract

A result of great theoretical and experimental interest, Jarzynski equality predicts a free energy change $ΔF$ of a system at inverse temperature $β$ from an ensemble average of non-equilibrium exponential work, i.e., $\langle e^{-βW}\rangle =e^{-βΔF}$. The number of experimental work values needed to reach a given accuracy of $ΔF$ is determined by the variance of $e^{-βW}$, denoted ${\rm var}(e^{-βW})$. We discover in this work that ${\rm var}(e^{-βW})$ in both harmonic and an-harmonic Hamiltonian systems can systematically diverge in non-adiabatic 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 non-adiabatic work protocol is found to yield a diverging ${\rm var}(e^{-βW})$ at sufficiently low temperatures, markedly different from the classical behavior. The divergence of ${\rm var}(e^{-βW})$ indicates the too-far-from-equilibrium nature of a non-adiabatic work protocol and makes it compulsory to apply designed control fields to suppress the quantum work fluctuations in order to test Jarzynski equality.