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arXiv:1304.6286

Work distribution under continuous quantum histories

Huanan Li, Jian‐Sheng Wang

Published Apr 23, 2013• 3 citations
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Abstract

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.

DOI
10.48550/arxiv.1304.6286

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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.
A central topic in the emerging field of quantum thermodynamics is the definition of thermodynamic work in the quantum regime. One widely used solution is to define work for a closed system undergoing non-equilibrium dynamics according to the two-point energy measurement scheme. However, due to the invasive nature of measurement the two-point quantum work probability distribution cannot describe the statistics of energy change from the perspective of the system alone. We here introduce the quantum histories framework as a method to characterise the thermodynamic properties of the unmeasured , closed dynamics. Constructing continuous power operator trajectories allows us to derive an alternative quantum work distribution for closed quantum dynamics that fulfils energy conservation and is time-reversal symmetric. This opens the possibility to compare the measured work with the unmeasured work, contrasting with the classical situation where measurement does not affect the work statistics. We find that the work distribution of the unmeasured dynamics leads to deviations from the classical Jarzynski equality and can have negative values highlighting distinctly non-classical features of quantum work.
By introducing novel concepts of work and heat functionals along individual pathpathpath, we reformulate quantum Jarzynski equality based on the path integral formulation of quantum mechanics. When applied to an open quantum system described by the quantum Brownian motion model, we establish a consistent framework of quantum thermodynamics in the strong coupling regime. Using the work and heat functionals, we derive a path-integral expression for the work and heat statistics. This formalism provides an effective way to calculate the work and heat in open quantum systems by utilizing various path integral techniques. By performing the ℏ\hbarℏ expansion, we analytically prove the quantum-classical correspondence of the work and heat statistics. In addition, we obtain the nnn-th order quantum correction to the classical work.
In this paper we present a first-principles analysis of the nonequilibrium work distribution and the free energy difference of a quantum system interacting with a general environment (with arbitrary spectral density and for all temperatures) based on a well-understood microphysics (quantum Brownian motion) model under the conditions stipulated by the Jarzynski equality [Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)] and Crooks' fluctuation theorem [Crooks, Phys. Rev. E 60, 2721 (1999)] (in short, fluctuation theorems, FTs). We use the decoherent histories conceptual framework to explain how the notion of trajectories in a quantum system can be made viable and use the environment-induced decoherence scheme to assess the strength of noise that could provide sufficient decoherence to warrant the use of trajectories to define work in open quantum systems. From the solutions to the Langevin equation governing the stochastic dynamics of such systems we were able to produce formal expressions for these quantities entering in the FTs and from them prove explicitly the validity of the FTs at the high temperature limit. At low temperatures our general results would enable one to identify the range of parameters where FTs may not hold or need be expressed differently. We explain the relation between classical and quantum FTs and the advantage of this microphysics open-system approach over the phenomenological modeling and energy-level calculations for substitute closed quantum systems.
We present a generalization of Jarzynski's equality, applicable to quantum systems, that is related to discretized mechanical work and free-energy changes. The theory is based on a stepwise pulling protocol. We find that work distribution functions can be constructed from fluctuations of a reaction coordinate along a reaction pathway in the stepwise pulling protocol. We also propose two sets of equations to determine the two possible optimal pathways that provide the most significant contributions to free-energy changes. We find that the transitions along these most optimal pathways, satisfying both sets of equations, follow the principle of detailed balance. We then test the theory by explicitly computing the free-energy changes for a one-dimensional quantum harmonic oscillator. This approach suggests a feasible way of measuring the fluctuations to experimentally test Jarzynski's equality in many-body systems, such as Bose-Einstein condensates.
In open quantum systems, a clear distinction between work and heat is often challenging, and extending the quantum Jarzynski equality to systems evolving under general quantum channels beyond unitality remains an open problem in quantum thermodynamics. In this Letter, we introduce well-defined notions of guessed quantum heat and guessed quantum work, by exploiting the one-time measurement scheme, which only requires an initial energy measurement on the system alone. We derive a modified quantum Jarzynski equality and the principle of maximum work with respect to the guessed quantum work, which requires the knowledge of the system only. We further show the significance of guessed quantum heat and work by linking them to the problem of quantum hypothesis testing.