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arXiv:1511.06095 · Journal of Statistical Mechanics Theory and Experiment

Computation of transient dynamics of energy power for a dissipative two state system

Matteo Carrega, Paolo Solinas, Alessandro Braggio, Maura Sassetti

Published May 1, 2016• 4 citations
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Abstract

Abstract We consider a two-level system coupled to a thermal bath and we investigate the variation of energy transferred to the reservoir as a function of time. The physical quantity under investigation is the time-dependent quantum average power. We compare quantum master equation approaches with the functional influence method. Differences and similarities between the methods are analysed, showing deviations at low temperature between the functional integral approach and the predictions based on master equations.

DOI
10.1088/1742-5468/2016/05/054014

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Abstract We consider a two-level system coupled to a thermal bath and we investigate the variation of energy transferred to the reservoir as a function of time. The physical quantity under investigation is the time-dependent quantum average power. We compare quantum master equation approaches with the functional influence method. Differences and similarities between the methods are analysed, showing deviations at low temperature between the functional integral approach and the predictions based on master equations.
We study the quantum dissipative dynamics of a particle coupled linearly to a set of two-level systems (the heat bath) via the master equation method, which we extract from the path integral formalism independently from the form of the bath spectral density. We compare our results with the standard models based on bosonic heat baths, showing their main differences and similarities. In particular, we study special forms for the spectral density of the bath which give results quite different from the standard models.
A master equation for a quantum system coupled to a heat bath with stochastically fluctuating energy levels is derived by making use of the ensemble averaging and the averaging with respect to a stochastic process in the bath. Relaxation terms are determined in the Born approximation with respect to the system-bath interaction and the damping parameters related to a relaxation kernel are specified. In parallel with the spectral strength of the bath, the damping parameters determine the transient times for the Markovian description creating the physical origin of the slippage [A. Suarez, R. Sibey, and I. Oppenheim, J. Chem. Phys. 97, 5101 (1992)]. The influence of energy fluctuations of the bath is analyzed for a two-level system including the solution of the corresponding non-Markovian equation for the level population difference. The conditions for the formation of Boltzmann's thermal ratio between steady-state populations are evaluated as well.
We investigate a system of two interacting qubits having one of them isolated and the other coupled to a thermal reservoir. We consider two different models of system-reservoir interaction: i) a model, in which the master equation is derived taking into account the interaction between the two subsystems (qubits), ii) a naive model, in which the master equation consists of a dissipative term added to the unitary evolution term. We study the dynamics of quantities such as bipartite entanglement, quantum discord and the linear entropy of the isolated qubit for both strong and weak coupling regimes (qubit-qubit interaction) as well as for different temperatures of the reservoir. We find significant disagreements between the results obtained from the two models even in the weak coupling regime. For instance, we show that according to the phenomenological model, the isolated qubit would approach a maximally mixed state more slowly for higher temperatures (unphysical result), while the microscopic model predicts the opposite behaviour (correct result).
We establish the path integral approach for the time-dependent heat exchange of an externally driven quantum system coupled to a thermal reservoir. We derive the relevant influence functional and present an exact formal expression for the moment generating functional which carries all statistical properties of the heat exchange process for general linear dissipation. The method is applied to the time-dependent average heat transfer in the dissipative two-state system (TSS). We show that the heat can be written as a convolution integral which involves the population and coherence correlation functions of the TSS and additional correlations due to a polarization of the reservoir. The corresponding expression can be solved in the weak-damping limit both for white noise and for quantum mechanical coloured noise. The implications of pure quantum effects are discussed. Altogether a complete description of the dynamics of the average heat transfer ranging from the classical regime down to zero temperature is achieved.
The methods of nonequilibrium density-matrix and coarse-temporal conception are used to obtain the kinetic equation for the parameters ${\ensuremath{\gamma}}_{\mathit{n}\mathit{m}}$(t)=Sp[\ensuremath{\rho}^(t)\ensuremath{\Vert}n〉〈m\ensuremath{\Vert}] of a quantum-dynamic system (QDS) interacting with a thermal bath and external stochastic field. It is important that the stochastic field is taken exactly into consideration. For diagonal QDS parameters ${\ensuremath{\gamma}}_{\mathit{n}\mathit{n}}$(t) this equation is reduced to the generalized Pauli equation (GPE) with stochastic time-dependent coefficients wnm{\mathit{w}}_{\mathit{n}\mathit{m}}wnm​(t). Special attention is given to the procedure of averaging over stochastic processes. It is shown that after averaging over energy fluctuations affected by the stochastic field, in the first cumulant approximation in terms of stochastic processes wnm{\mathit{w}}_{\mathit{n}\mathit{m}}wnm​(t), the GPE is transformed to the Pauli equation for the QDS state population Pn{\mathit{P}}_{\mathit{n}}Pn​(t)=〈${\ensuremath{\gamma}}_{\mathit{n}\mathit{n}}$(t)〉f{\mathrm{〉}}_{\mathit{f}}〉f​. As an example, the relaxation behavior of a two-level system interacting with a dichotomous field (dichotomous Markovian process of kangaroo type) and a harmonic oscillator coupled with a thermal bath is considered. It is shown that the probability of relaxation transitions between energy levels may be changed by several orders of magnitude under the influence of the dichotomous field.
The dynamics of a two-level system (TLS) coupled to an external heat bath has been studied. Correlation functions are obtained by combining analytical and numerical renormalization-group methods. It is shown that perturbation theory can be applied to obtain the low-energy, long-time behavior of the system. Analytical expressions are given for 〈${\ensuremath{\sigma}}_{z}$(t)${\ensuremath{\sigma}}_{z}$(0)〉T=0{〉}_{T=0}〉T=0​ valid in the long- and short-time limits (as compared with the renormalized frequency of the TLS). It is shown that other correlation functions, like 〈${\ensuremath{\sigma}}_{x}$(t)${\ensuremath{\sigma}}_{x}$(0)〉, depend sensitively on the nature of the high-energy cutoff of the environment. Thermodynamic functions, like the susceptibility to a bias potential and the correction to the specific heat of the environment, are also calculated. Finally, it is discussed how, in these systems, the experimental setup may influence the quantity being measured.