Driven Brownian oscillator

Abstract. This article reviews from both theoretical and numerical aspects three non-equivalent complete second-order formulations of quantum dissipation theory, in which both the reduced dynamics and the initial canonical thermal equilibrium are properly treated in the weak system-bath coupling limit. Two of these formulations are rather familiar as the time-local and the memory-kernel prescriptions, while another which can be termed as correlated driving-dissipation equations of motion will be shown to have the combined merits of the two conventional formulations. By exploiting the exact solutions to the driven Brownian oscillator system, we demonstrate that the time-local and correlated driving-dissipation equations of motion formulations are usually better than their memory-kernel counterparts, in terms of their applicability to a broad range of system-bath coupling, non-Markovian, and temperature parameters. Numerical algorithms are detailed for an efficient evaluation of both the reduced canonical thermal equilibrium state and the non-Markovian evolution at any temperature, in the presence of arbitrary time-dependent external fields. 

Recently we have constructed a complete second-order QDT (CS-QDT), in which all excessive approximations, except that of weak system-bath interaction, are removed [38]. Besides two forms of CS-QDT corresponding to the memory-kernel COP [Eq. (1.2)] and the time-local POP [Eq. (1.3)] formulations, respectively, we have also constructed a novel CS-QDT that is particularly suitable for studying the effects of correlated non-Markovian dissipation and external time-dependent field driving. This paper constitutes a review of the three nonequivalent CS-QDT formulations [38] from both theoretical and numerical aspects. Concrete comparisons will be carried out in connection with the exact results for driven Brownian oscillator systems, so that sensible comments on various forms of CS-QDT can be reached. Note that QDT shall describe not only the evolution of p(t), but also the reduced thermal equilibrium system as p t oo) = peq(7 )-... 

In this section, we shall consider an exactly solvable model, that of driven Brownian oscillator (DBO) systems. The exact formulations will be presented in this subsection, while the POP-CS-QDT counterpart will be outlined in the next. We will show in Sec. 5 that the POP-CS-QDT and the CODDE agree with the exact QDT very well however, the COP-CS-QDT results in contrary and unphysical behaviors, for example, on its temperature dependence of the equilibrium properties. 

Analysis of various approximation schemes using driven Brownian oscillator systems... 

Except for special cases such as the driven Brownian oscillator (Sec. 4), the direct evaluation of SQa t) as Eq. (B.3) is in general numerically formidable. The parameterization of spectral density [Eq. (2.24)] that leads to the form of Cab t) in Eq. (2.25) provides a mean to overcome this problem. The 6pt in Eq. (2.17) in this case assumes... 

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