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arXiv:quant-ph/0211196 · Physical Review A

Density-matrix operatorial solution of the non-Markovian master equation for quantum Brownian motion

F. Intravaia, Sabrina Maniscalco, ANTONINO MESSINA

Published Apr 22, 2003• 76 citations
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Abstract

An original method to exactly solve the non-Markovian master equation describing the interaction of a single harmonic oscillator with a quantum environment in the weak-coupling limit is reported. By using a superoperatorial approach, we succeed in deriving the operatorial solution for the density matrix of the system. Our method is independent of the physical properties of the environment. We show the usefulness of our solution deriving explicit expressions for the dissipative time evolution of some observables of physical interest for the system, such as, for example, its mean energy.

DOI
10.1103/physreva.67.042108

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Understanding the behavior of a quantum system coupled to its environment is of fundamental interest in the general field of quantum technologies. It also has important repercussions on foundational problems in physics, such as the process of decoherence and the so-called quantum measurement problem. There have been many approaches to explore Markovian and non-Markovian dynamics within the framework of open quantum systems, but the richness of the ensuing dynamics is still not fully understood. In this paper we develop a non-Markovian extension of the standard Caldeira-Leggett model, based on expanding the dynamics of the reduced system at high temperature in inverse powers of the high-frequency cutoff of the Ohmic spectral density of the environment and derive a non-Markovian master equation for the reduced density matrix for the case of a general potential. We also obtain a fully analytical solution in the free particle case. While the short-time behavior of this solution does not diverge substantially from the Markovian behavior, at intermediate times we find a resurgence of coherence, which we name lateral coherence. We identify this with a corresponding transient negative entropy production rate, which is understood to be characteristic of non-Markovian dynamics. We also analyze the positivity of the reduced density matrix and derive the corresponding Fokker-Planck equation in the classical limit.
Stochastic Schr\"odinger equations for quantum trajectories offer an alternative and sometimes superior approach to the study of open quantum system dynamics. Here we show that recently established convolutionless non-Markovian stochastic Schr\"odinger equations may serve as a powerful tool for the derivation of convolutionless master equations for non-Markovian open quantum systems. The most interesting example is quantum Brownian motion (QBM) of a harmonic oscillator coupled to a heat bath of oscillators, one of the most employed exactly soluble models of open system dynamics. We show explicitly how to establish the direct connection between the exact convolutionless master equation of QBM and the corresponding convolutionless exact stochastic Schr\"odinger equation.
We consider Quantum Brownian motion of a toy model, which is a harmonic oscillator that interacts with environment consisting of harmonic oscillators through both qQ and pP types of coupling. At first, we formally derive exact master equation following Unruh and Zurek's method. Subsequently we estimate the behavior of the coefficients in the master equation at initial stage and at late time stage under weak coupling approximation. We find that, as the model with qQ coupling only, the system in supraohmic environment is less dissipative and less diffusive beyond the cutoff time scale than the system in ohmic environment. The cause is due to the weakness of spectral densities of environment in infrared sector. We also find that the remarkable feature of the coexistence of qQ and pP coupling is steep dependence of dissipative coefficients on time at initial stage compared with the case of qQ coupling only or pP coupling only. Further we find that the sign of the dissipation coefficient at initial stage can be negative, that is, the system can be anti-dissipative.
We generalize the classical theory of Brownian motion so as to reckon with non-Markovian effects on both Klein-Kramers and Smoluchowski equations. For a free particle and a harmonic oscillator, it is shown that such non-Markovian effects account for the differentiability of the Brownian trajectories as well as the breakdown of the energy equipartition of statistical mechanics at short times in some physical situations. This non-Markovian approach is also extended to look at anomalous diffusion. Next, we bring in the dynamical-quantization method for investigating open quantum systems, which does consist in quantizing the classical Brownian motion starting directly from our non-Markovian Klein-Kramers and Smoluchowski equations, without alluding to any model Hamiltonian. Accordingly, quantizing our non-Markovian Klein-Kramers in phase space gives rise to a non-Markovian quantum master equation in configuration space, whereas quantizing our non-Markovian Smoluchowski equation in configuration space leads to a non-Markovian quantum Smoluchowski equation in phase space. In addition, it is worth noticing that non-Markovian quantum Brownian motion takes place in presence of a generic environment (e.g. a non-thermal quantum fluid). As far as the special case of a heat bath comprising of quantum harmonic oscillators is concerned, a non-Markovian Caldeira-Leggett master equation and a thermal quantum Smoluchowski equation are derived and extended to bosonic and fermionic heat baths valid for all temperatures.