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arXiv:0711.3914 · Physical Review E

Nonequilibrium work distribution of a quantum harmonic oscillator

Sebastian Deffner, Eric Lutz

Published Feb 27, 2008• 167 citations
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Abstract

We calculate analytically the work distribution of a quantum harmonic oscillator with arbitrary time-dependent angular frequency. We provide detailed expressions for the work probability density for adiabatic and nonadiabatic processes, in the limits of low and high temperature. We further verify the validity of the quantum Jarzynski equality.

DOI
10.1103/physreve.77.021128

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We calculate analytically the work distribution of a quantum harmonic oscillator with arbitrary time-dependent angular frequency. We provide detailed expressions for the work probability density for adiabatic and nonadiabatic processes, in the limits of low and high temperature. We further verify the validity of the quantum Jarzynski equality.
We consider a thermal quantum harmonic oscillator weakly coupled to a heat bath at a different temperature. We analytically study the quantum heat exchange statistics between the two systems using the quantum-optical master equation. We exactly compute the characteristic function of the heat distribution and show that it verifies the Jarzynski-W\'ojcik fluctuation theorem. We further evaluate the heat probability density in the limit of long thermalization times, both in the low- and the high-temperature regimes, and investigate its time evolution by calculating its first two cumulants.
A result of great theoretical and experimental interest, the Jarzynski equality predicts a free energy change $\mathrm{\ensuremath{\Delta}}F$ of a system at inverse temperature $\ensuremath{\beta}$ from an ensemble average of nonequilibrium exponential work, i.e., $\ensuremath{\langle}{e}^{\ensuremath{-}\ensuremath{\beta}W}\ensuremath{\rangle}={e}^{\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}F}$. The number of experimental work values needed to reach a given accuracy of $\mathrm{\ensuremath{\Delta}}F$ is determined by the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, denoted $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$. We discover in this work that $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ 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 $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ at sufficiently low temperatures, markedly different from the classical behavior. The divergence of $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ 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.
We present a general scheme to obtain work distribution in closed systems under continuous quantum histories of corresponding "power" operator. The scheme is tested by analytically calculating the quantum work distribution for a prototype model of a center-shifted one-dimensional quantum harmonic oscillator. Furthermore, its relationship to the path integral formalism with the Wick rotation is immediately noticed. We find that, based on the continuous measurement, generally Jarzynski equality in quantum regime is not valid, though it is recovered in high-temperature limit. Also we briefly explore the extension to open systems, in which the work done and heat transfer can be calculated in the same set of quantum histories. Our results for this model can be directly verified by single-molecule force spectroscopy.
The quantum harmonic oscillator with time-dependent frequency is a paradigmatic model of driven quantum dynamics and one of the few nontrivial systems that admits an exact analytical solution. In this review paper, we present a unified treatment of the time-dependent oscillator based on the Lewis-Riesenfeld invariant method, Bogoliubov transformations and the Ermakov-Pinney equation. We show how these approaches naturally connect to squeezing for the description of excitations production, and to the breakdown of adiabaticity under generic frequency protocols. Exact results for sudden quenches and smooth ramps are discussed in detail. By explicitly bridging invariant methods and squeezing formalism, this review is meant to provide a comprehensive framework for understanding nonequilibrium dynamics in quadratic potentials, with applications ranging from thermodynamics and condensed matter to quantum control theory.
We derive a Markovian master equation for a linearly driven dissipative quantum harmonic oscillator, valid for generic driving beyond the adiabatic limit. We solve this quantum master equation for arbitrary Gaussian initial states and investigate its departure from the adiabatic master equation in the regime of fast driving. We concretely examine the behavior of dynamical variables, such as position and momentum, as well as of thermodynamic quantities, such as energy and entropy. We additionally study the influence of the nonequilibrium driving on the quantum coherence of the oscillator in the instantaneous energy eigenbasis. We further analyze the approach to the adiabatic limit and the relaxation to the instantaneous steady state as a function of the driving speed.
We combine the formalisms of diagonal entropy and Jarzynski Equality to study the thermodynamic properties of closed quantum systems. Applying this approach to a quantum harmonic oscillator, the diagonal entropy offers a notion of temperature for closed systems away from equilibrium, and allows computing free-energy profiles. We also apply this approach to a hard-core boson lattice model, and discuss measures how to estimate temperature, entropy and measure work distribution functions. This technique offers a path to investigate the non-equilibrium quantum dynamics by means of performing work in a series of quenches.