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arXiv:1407.0129 · Canadian Journal of Physics

Quasi-equilibrium relaxation of two identical quantum oscillators with arbitrary coupling strength

Illarion Dorofeyev

Published Dec 9, 2014• 4 citations
arXivHTMLPDF

Abstract

This paper deals with the problem of open systems out of equilibrium. An analytical expression for a time-dependent density matrix of two arbitrarily coupled identical quantum oscillators interacting with separate reservoirs is derived using path integral methods. The temporal behavior of spatial variances and of covariance from given initial values up to stationary values is investigated. It is shown that at comparatively low coupling strengths, the asymptotic variances in the long-time limit achieve steady states independently of initial values. Stationary variances differ from the case of total equilibrium due to their coupling with separate thermal reservoirs of different temperatures. The larger the difference in temperatures of thermal baths, the larger is the difference of the stationary values of variances of coupled oscillators compared with values given by the fluctuation dissipation theorem. At strong couplings the variances have divergent character in the framework of the accepted model. Otherwise, in the weak coupling limit the asymptotic stationary variances are always in accordance with the fluctuation dissipation theorem with some accuracy.

DOI
10.1139/cjp-2014-0376

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This paper deals with the problem of open systems out of equilibrium. An analytical expression for a time-dependent density matrix of two arbitrarily coupled identical quantum oscillators interacting with separate reservoirs is derived using path integral methods. The temporal behavior of spatial variances and of covariance from given initial values up to stationary values is investigated. It is shown that at comparatively low coupling strengths, the asymptotic variances in the long-time limit achieve steady states independently of initial values. Stationary variances differ from the case of total equilibrium due to their coupling with separate thermal reservoirs of different temperatures. The larger the difference in temperatures of thermal baths, the larger is the difference of the stationary values of variances of coupled oscillators compared with values given by the fluctuation dissipation theorem. At strong couplings the variances have divergent character in the framework of the accepted model. Otherwise, in the weak coupling limit the asymptotic stationary variances are always in accordance with the fluctuation dissipation theorem with some accuracy.
This paper addresses the problem of relaxation of open quantum systems. Using the path integral methods, we found an analytical expression for the time-dependent density matrix of two coupled quantum oscillators interacting with different baths of oscillators. The expression for density matrix was found in the linear regime with respect to the coupling constant between selected oscillators. Time dependent spatial variances and covariances were investigated analytically and numerically. It was shown that asymptotic variances in the long-time limit are always in accordance with the fluctuation dissipation theorem despite their initial values. In the weak coupling approach there is good reason to believe that subsystems are asymptotically in equilibrium at their own temperatures despite the arbitrary difference in temperatures within the whole system.
We consider a problem of description of quantum correlations and dispersions of subsystems of complex open systems. Based on our previous results we proposed a method to evaluate pure quantum contributions from total statistical characteristics of two coupled oscillators interacting with thermal reservoirs. The natural way to extract pure quantum characteristic of the system under study is a calculation of difference between the total value and its pure classical part. A numerical study of temporal behavior of quantum variances and covariances from given initial states up to states in the infinite time limit is given using the proposed method. It is shown that at comparatively high temperatures the steady state total variances are almost classical, because their pure quantum parts tend to zero in this regime. Otherwise, at comparatively low temperatures, the pure quantum parts of variances are not zeroing in steady states manifesting a finite contribution to total values. The same is true for the covariance. The larger the temperature, the lower is the quantum contribution to the total covariance. As well as entanglement measures, quantum discord, the pure quantum contributions in the covariance matrix elements are important characteristics of the temporal dynamics of complex quantum systems.
We study the quantum fluctuational properties of a parametric oscillator with and without coupling to an Ohmic environment. After considering the momentum and coordinate variances as a function of initial squeezing for the undamped dynamics, we invoke the functional integral method to derive the fully exact reduced density matrix for parametric dissipative quantum Brownian motion, covering the whole temperature regime from T=0 up to the classical limit at room temperatures. Moreover, we present the exact result for the quantum master equation for both the density matrix and the corresponding Wigner function. The time evolution of the covariance matrix elements of damped quantum fluctuations is studied numerically. These variances undergo within the regime of global stability asymptotic, periodic oscillations. As an interesting result, we find that the minima of these oscillations fall below the corresponding thermal equilibrium values.
We study the nonequilibrium dynamics of two mechanical oscillators with general linear couplings to two uncorrelated thermal baths at temperatures T1{T}_{1}T1​ and T2{T}_{2}T2​, respectively. We obtain the complete solution of the Heisenberg-Langevin equations, which reveal a coherent mixing among the normal modes of the oscillators as a consequence of their off-diagonal couplings to the baths. Unique renormalization aspects resulting from this mixing are discussed. Diagonal and off-diagonal (coherence) correlation functions are obtained analytically in the case of strictly Ohmic baths with different couplings in the strong- and weak-coupling regimes. An asymptotic nonequilibrium stationary state emerges for which we obtain the complete expressions for the correlations and coherence. Remarkably, the coherence survives in the high-temperature, classical limit for ${T}_{1}\ensuremath{\ne}{T}_{2}$. This is a consequence of the coherence being determined by the difference of the bath correlation functions. In the case of vanishing detuning between the oscillator normal modes both coupling to one and the same bath, the coherence retains memory of the initial conditions at long times. An out-of-equilibrium setup with small detuning and large $|{T}_{1}\ensuremath{-}{T}_{2}|$ produces nonvanishing steady-state coherence in the high-temperature limit of the baths.
We develop an exact analytical framework for studying the nonequilibrium dynamics of interacting quantum harmonic oscillators coupled to a modelled thermal reservoir and driven by external classical fields. The total Hamiltonian is diagonalized via successive Bogoliubov transformations, enabling a nonperturbative treatment of dissipation and driving. For two coupled oscillators with an external source applied to the first, we derive explicit energy expressions and analyze their dependence on coupling and bath parameters. The phase-space structure is characterized through Husimi QQQ-functions for initial separable coherent and number states. We obtain the reduced density matrix elements in the number-state basis and demonstrate that, in the absence of driving and at zero temperature, the reduced density matrix of the main system satisfies the Lindblad master equation. A generating function for the Husimi function is introduced to facilitate computation of higher-order correlations. The analysis is then generalized to nnn interacting oscillators, with analytical energy expressions provided and the case n=3n=3n=3 examined in detail. Our results establish a versatile and exact framework for driven-dissipative quantum systems, with potential applications in quantum thermodynamics, open quantum systems, and many-body physics.
Abstract The non‐equilibrium state of two oscillators with a mutual interaction and coupled to separate heat baths is discussed. Bosonic baths are considered, and an exact spectral representation for the elements of the covariance matrix is provided analytically. A wide class of spectral densities for the relevant bath modes is allowed for. The validity of the fluctuation‐dissipation relation is established for global equilibrium (both baths at the same temperature) in the stationary state. Spectral measures of entanglement are suggested by comparing to the equilibrium spectrum of zero‐point fluctuations. No rotating‐wave approximation is applied, and anomalous heat transport from cold to hot bath, as reported in earlier work, is demonstrated not to occur.