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arXiv:1610.03161 · Journal of Statistical Mechanics Theory and Experiment

Quantum violation of fluctuation-dissipation theorem

Akira Shimizu, Kyota Fujikura

Published Feb 1, 2017• 16 citations
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Abstract

Abstract We study quantum measurements of temporal equilibrium fluctuations in macroscopic quantum systems. It is shown that the fluctuation-dissipation theorem, as a relation between observed quantities, is partially violated in quantum systems, even if measurements are made in an ideal way that emulates classical ideal measurements as closely as possible. This is a genuine quantum effect that survives on a macroscopic scale. We also show that the state realized during measurements of temporal equilibrium fluctuations is a ‘squeezed equilibrium state’, which is macroscopically identical to the pre-measurement equilibrium state but is squeezed by the measurement. It is a time-evolving state, in which macrovariables fluctuate and relax. We also explain some of subtle but important points, careless treatments of which often lead to unphysical results, of the linear response theory.

DOI
10.1088/1742-5468/aa5a67

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Abstract We study quantum measurements of temporal equilibrium fluctuations in macroscopic quantum systems. It is shown that the fluctuation-dissipation theorem, as a relation between observed quantities, is partially violated in quantum systems, even if measurements are made in an ideal way that emulates classical ideal measurements as closely as possible. This is a genuine quantum effect that survives on a macroscopic scale. We also show that the state realized during measurements of temporal equilibrium fluctuations is a ‘squeezed equilibrium state’, which is macroscopically identical to the pre-measurement equilibrium state but is squeezed by the measurement. It is a time-evolving state, in which macrovariables fluctuate and relax. We also explain some of subtle but important points, careless treatments of which often lead to unphysical results, of the linear response theory.
For macroscopic quantum systems, we study what is measured when equilibrium fluctuations of macrovariables are measured in an ideal way that mimics classical ideal measurements as closely as possible. We find that the symmetrized time correlation is always obtained for such measurements. As an important consequence, we show that the fluctuation-dissipation theorem is partially violated as a relation between observed quantities in macroscopic quantum systems even if measurements are made in such an ideal way.
For equilibrium states of macroscopic quantum systems, we study what are measured when fluctuations of macrovariables are measured. To mimic classical ideal measurements, we consider `quasi-classical measurements' which are minimally-disturbing, homogeneous, and unbiased quantum measurements with moderate magnitudes of errors. For all such measurements, we find that the symmetrized time correlation is obtained among many quantum correlations that reduce, in the classical limit, to the same classical time correlation. As an important consequence of this universal result, we show that the fluctuation-dissipation theorem (FDT) and Onsager's hypothesis are partially violated in macroscopic quantum systems. The post-measurement state is squeezed by backactions of the measurements, but is macroscopically identical to the pre-measurement equilibrium state. This state evolves with time, in which macrovariables fluctuate and relax, unlike other microstates (such as the Gibbs state) that represent the same equilibrium state.
We examine the Hall conductivity of a macroscopic two-dimensional quantum system and show that the observed quantities can sometimes violate the fluctuation dissipation theorem (FDT), even in the linear response (LR) regime infinitesimally close to equilibrium. At low temperature and in strong magnetic field, which are experimentally accessible, the violation can be by an order of magnitude larger than the Hall conductivity itself. We further generalize the results and obtain a necessary condition for such large-scale violation to happen. This violation is a genuine quantum phenomenon that appears on a macroscopic scale when the time-reversal symmetry is broken (by, e.g., a magnetic field). Our results are not only bound to the development of the fundamental issues of nonequilibrium physics, but the idea is also meaningful for practical applications, since the FDT is widely used for the estimation of noises from the LRs. We give an alternative formula evaluating precisely the fluctuations from the LRs even in the quantum systems with large FDT violation.
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolution, as given by quantum maps. Third, it clarifies the connection between the FDT and quantum metrology in systems with a non-equilibrium steady state.
This article traces the development of fluctuation theory and its deep connection to irreversibility, from equilibrium to near-equilibrium, and finally to far-from-equilibrium systems. Classical fluctuation theorems, which capture the statistical behaviour of thermodynamic systems far from equilibrium, are now well established. Their quantum counterparts, however, remain an active area of research. In this review, we highlight recent advances by linking quantum fluctuation theorems with linear response theory, offering new insights into the nature of quantum fluctuations and irreversibility in the near-equilibrium regime. Particular emphasis is placed on dissipated work in quantum systems as a pathway to observing non-classical effects in quantum thermodynamics. Understanding quantum fluctuations is not only essential for clarifying the foundations of irreversibility but also crucial for the development of novel quantum technologies, including quantum computers, sensors, and metrological devices.
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolutions, as given by quantum maps. Third, it brings out the full connection between the FDT and the Quantum Fisher information, the figure of merit in quantum metrology.