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Entropy production and correlation spreading in Gaussian quantum mechanics.

Hao Xu, Si Yu Chen

Published Nov 7, 2021• 0 citations
OpenAlex

Abstract

Entropy production is the key quantity behind the second law of thermodynamics, and it is well defined by considering a joint unitary evolution of a system $S$ and a thermal environment $E$. However, due to the diversity of initial state and Hamiltonian of $S$ and $E$, it is hard to evaluate the characterization of the entropy production. In this Letter we propose that the evolution of $S$ and $E$ can be solved non-perturbatively in the framework of Gaussian quantum mechanics (GQM) for Gaussian states and quadratic Hamiltonian. We study the entropy production and correlation spreading in two models of $E$: one-dimensional free massless scalar quantum field theory in a cavity and an oscillator chain with adjacent interactions. The result is not what one might naively expect, $S$ being thermalized and keeps stable, but is all the physical quantities, such as entropy production, mutual information, and the effective temperature of $S$, are fluctuating. In general the interaction between $S$ and $E$ creates correlation, and this correlation could spread in $E$ and cause the physical quantities to fluctuate. The detailed information depends on the coupling strength, number of modes, boundary condition and other parameters. This analysis can be extended to any other models in the framework of QGM.

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Entropy production is the key quantity behind the second law of thermodynamics, and it is well defined by considering a joint unitary evolution of a system SSS and a thermal environment EEE. However, due to the diversity of initial state and Hamiltonian of SSS and EEE, it is hard to evaluate the characterization of the entropy production. In this Letter we propose that the evolution of SSS and EEE can be solved non-perturbatively in the framework of Gaussian quantum mechanics (GQM) for Gaussian states and quadratic Hamiltonian. We study the entropy production and correlation spreading in two models of EEE: one-dimensional free massless scalar quantum field theory in a cavity and an oscillator chain with adjacent interactions. The result is not what one might naively expect, SSS being thermalized and keeps stable, but is all the physical quantities, such as entropy production, mutual information, and the effective temperature of SSS, are fluctuating. In general the interaction between SSS and EEE creates correlation, and this correlation could spread in EEE and cause the physical quantities to fluctuate. The detailed information depends on the coupling strength, number of modes, boundary condition and other parameters. This analysis can be extended to any other models in the framework of QGM.
According to the first and second laws of thermodynamics and the definitions of work and heat, microscopic expressions for the non-equilibrium entropy production have been achieved. Recently, a redefinition of heat has been presented in [Nature Communications volume 8, Article number: 2180 (2017)]. Since thermal operations play an important role in the resource theory of thermodynamics, it would be very interesting to find out the effect of the above-mentioned definition on the expression of the entropy production for these kind of operations. This is one of the aims of the present paper. Using the new definition of heat, it is shown that the entropy production is the same as the mutual information between a system and a bath both for thermal operations and, if the system-bath initial state is factorized, for entropy-preserving operations. It is also discussed that how one can recognize the type of the correlation between a system and a bath through knowledge of the initial state of the system only. It is shown that if the initial state of a system is diagonal in the energy basis, the thermal operations cannot create a quantum correlation between the system and the bath, however, if the system initial state is coherent Gibbs state, there cannot be classical correlation due to the thermal operations.
As stated by the second law of thermodynamics, the irreversible entropy of a system keeps being produced until it reaches the equilibrium, and the entropy production rate is always non-negative. But when some exotic quantum properties, e.g., quantum coherence or squeezing, exist in the bath, it has been reported that the 2nd law does not apply. Correspondingly, the previous description of the entropy production rate does not apply either, and it is not always non-negative in these cases. We notice that the entropy production rate can be generalized to be the increasing rate of the mutual information between the open system and its environment. For thermal baths, this mutual information production rate equals to the previous entropy production rate. For non-thermal baths, in an example of a single boson mode interacting with multiple squeezed thermal baths, we proved that the mutual information production rate is always non-negative. The monotonic increasing mutual information means the system and its environment are becoming more and more correlated during the evolution.
Abstract The entropy production rate is a key quantity in nonequilibrium thermodynamics of both classical and quantum processes. No universal theory of entropy production is available to date, which hinders progress toward its full grasping. By using a phase space-based approach, here we take the current framework for the assessment of thermodynamic irreversibility all the way to quantum regimes by characterizing entropy production—and its rate—resulting from the continuous monitoring of a Gaussian system. This allows us to formulate a sharpened second law of thermodynamics that accounts for the measurement back action and information gain from a continuously monitored system. We illustrate our framework in a series of physically relevant examples.
The Markovian time evolution of the entropy production rate is studied as a measure of irreversibility generated in a bipartite quantum system consisting of two coupled bosonic modes immersed in a common thermal environment. The dynamics of the system is described in the framework of the formalism of the theory of open quantum systems based on completely positive quantum dynamical semigroups, for initial two-mode squeezed thermal states, squeezed vacuum states, thermal states and coherent states. We show that the rate of the entropy production of the initial state and nonequilibrium stationary state, and the time evolution of the rate of entropy production, strongly depend on the parameters of the initial Gaussian state (squeezing parameter and average thermal photon numbers), frequencies of modes, parameters characterising the thermal environment (temperature and dissipation coefficient), and the strength of coupling between the two modes. We also provide a comparison of the behaviour of entropy production rate and Rényi-2 mutual information present in the considered system.
Accessing the physical mechanisms behind non-Markovian phenomena in open quantum dynamics requires the study of the statistical properties of the joint system-environment dynamics. This is impossible at the level of the reduced dynamics of the open system alone as the latter is obtained by suitably eliminating the environment. The task is instead made possible by considering multi-time correlation functions involving observables of the open system, only: the open system-environment interactions turn them into global ones thus building up correlations between the two systems. Multi-time correlations form the basis of both the theory of quantum stochastic processes and of the Alicki-Lindblad-Fannes dynamical entropy (ALF entropy for short). This latter quantity provides for quantum systems a measure of the dynamical entropy production as the Kolmogorov-Sinai entropy does for classical systems. In the case of a collisional model whereby the dissipative dynamics of a finite-level system is obtained by its coupling to an infinite classical spin chain, the ALF entropy can be explicitly computed. It turns out to depend on the parameters characterizing the statistical properties of the environment and can be related to the activation and super-activation of memory effects in the open quantum system.
We study the time evolution of the entanglement entropy in bosonic systems with time-independent, or time-periodic, Hamiltonians. In the first part, we focus on quadratic Hamiltonians and Gaussian initial states. We show that all quadratic Hamiltonians can be decomposed into three parts: (a) unstable, (b) stable, and (c) metastable. If present, each part contributes in a characteristic way to the time dependence of the entanglement entropy: (a) linear production, (b) bounded oscillations, and (c) logarithmic production. In the second part, we use numerical calculations to go beyond Gaussian states and quadratic Hamiltonians. We provide numerical evidence for the conjecture that entanglement production through quadratic Hamiltonians has the same asymptotic behavior for non-Gaussian initial states as for Gaussian ones. Moreover, even for nonquadratic Hamiltonians, we find a similar behavior at intermediate times. Our results are of relevance to understanding entanglement production for quantum fields in dynamical backgrounds and ultracold atoms in optical lattices.