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arXiv:1311.5714 · Canadian Journal of Physics

Relaxation of two coupled quantum oscillators to quasi-equilibrium states based on path integrals

Illarion Dorofeyev

Published Feb 18, 2014• 5 citations
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Abstract

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.

DOI
10.1139/cjp-2013-0708

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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.
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.
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 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.
Quantum systems are typically subject to various environmental noise sources. Treating these environmental disturbances with a system-bath approach beyond weak coupling, one must refer to numerical methods as, for example, the numerically exact quasi-adiabatic path integral approach. This approach, however, cannot treat baths which couple to the system via operators, which do not commute. We extend the quasi-adiabatic path integral approach by determining the time discrete influence functional for such non-commuting fluctuations and by modifying the propagation scheme accordingly. We test the extended quasi-adiabatic path integral approach by determining the time evolution of a quantum two-level system coupled to two independent baths via non-commuting operators. We show that the convergent results can be obtained and agreement with the analytical weak coupling results is achieved in the respective limits.
In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.
The non-Markovian Langevin approach is applied to study the dynamics of fermionic (bosonic) oscillator linearly coupled to a fermionic (bosonic) environment. The analytical expressions for occupation numbers in two different types of couplings (rotating-wave approximation and fully coupled) are compared and discussed. The weak-coupling and high- and low-temperature limits are considered as well. The conditions under which the environment imposes its thermal equilibrium on the collective subsystem are discussed. The sameness of the results, obtained with both the Langevin approach and the discretized environment method are shown. Short- and long-time nonequilibrium dynamics of fermionic and bosonic open quantum systems are analyzed both analytically and numerically.