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Physical Review A

Relaxation of quantum systems weakly coupled to a bath. I. Total-time-ordering-cumulant and partial-time-ordering-cumulant non-Markovian theories

J. Bretón, Arturo Hardisson, F. Mauricio, Santiago Velasco

Published Jul 1, 1984• 38 citations
Publisher

Abstract

The present paper studies the formal aspects of the total-time-ordering-cumulant (TTOC) and the partial-time-ordering-cumulant (PTOC) relaxations of a quantum system, of few degrees of freedom, in weak interaction with a bath, both within and outside the Markovian limit. To this end, the general expressions connecting the matrix elements of the TTOC and PTOC relaxation superoperators with the bath correlation functions are determined. Special attention is paid to two particular cases: a system with a nonequidistant energy spectrum and a system with an equidistant energy spectrum. Discussions revolve mainly around the possibility of applying the secular approximation to the TTOC and PTOC master equations for the off-diagonal matrix elements of the reduced-density operator of the system.

DOI
10.1103/physreva.30.542

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The present paper studies the formal aspects of the total-time-ordering-cumulant (TTOC) and the partial-time-ordering-cumulant (PTOC) relaxations of a quantum system, of few degrees of freedom, in weak interaction with a bath, both within and outside the Markovian limit. To this end, the general expressions connecting the matrix elements of the TTOC and PTOC relaxation superoperators with the bath correlation functions are determined. Special attention is paid to two particular cases: a system with a nonequidistant energy spectrum and a system with an equidistant energy spectrum. Discussions revolve mainly around the possibility of applying the secular approximation to the TTOC and PTOC master equations for the off-diagonal matrix elements of the reduced-density operator of the system.
Fourth-order quantum master equations (FQMEs) are derived in both time nonlocal and local forms for a general system Hamiltonian, with new detailed expressions for the fourth-order kernel, where the bath correlation functions are explicitly decoupled from the system superoperators. Further simplifications can be made for the model of linearly coupled harmonic oscillator bath. Consideration of the high temperature Ohmic bath limit leads to a general Markovian FQME with compact forms of time independent superoperators. Two examples of this equation are then considered. For the system of a quantum particle in a continuous potential field, the equation reduces to a known form of the quantum Fokker–Planck equation, except for a fourth-order potential renormalization term that can be neglected only in the weak system-bath interaction regime. For a two-level system with off-diagonal coupling to the bath, fourth-order corrections do not alter the relaxation characteristics of the second-order equation and introduce additional coherence terms in the equations for the off-diagonal elements.
For sufficiently low reservoir temperatures, it is known that open quantum systems subject to decoherent interactions with the reservoir relax toward their ground state in the weak coupling limit. Within the framework of quantum master equations, this is formalized by the Born-Markov-secular (BMS) approximation, where one obtains the system Gibbs state with the reservoir temperature as a stationary state. When the solution to some problem is encoded in the (isolated) ground state of a system Hamiltonian, decoherence can therefore be exploited for computation. The computational complexity is then given by the scaling of the relaxation time with the system size nnn. We study the relaxation behavior for local and nonlocal Hamiltonians that are coupled dissipatively with local and nonlocal operators to a bosonic bath in thermal equilibrium. We find that relaxation is generally more efficient when coherences of the density matrix in the system energy eigenbasis are taken into account. In addition, the relaxation speed strongly depends on the matrix elements of the coupling operators between initial state and ground state. We show that Dicke superradiance is a special case of our relaxation models and can thus be understood as a coherence-assisted relaxation speedup.
A general procedure is given for finding the equation of motion of the density matrix of a system in contact with a thermal bath, as for example a nuclear spin system weakly coupled to a crystal lattice. The thermal bath is treated both classically and quantum mechanically, and the theory is similar to, and a generalization of, conventional theories of time proportional transition probabilities. Relaxation of the system by the thermal bath is expressed by a linear matrix operator, and it is stressed that elements of this operator can be regarded as secular or nonsecular perturbations on the equation of motion and can be treated accordingly. When the motion of the system is slow compared to that of the thermal bath, the equation of motion can be expressed in an operator form which is independent of representation. If the system has a time-dependent Hamiltonian which varies slowly compared to the motion of the thermal bath, the same equation of motion is obeyed and the system is relaxed by the bath toward a Boltzmann distribution with respect to its instantaneous Hamiltonian. If the time variation of the Hamiltonian is more rapid, higher order corrections to the equation of motion must be applied. The theory is applied to spin-lattice relaxation of a coupled nuclear spin system in a metal, for arbitrary externally applied fixed magnetic field.
Using Terwiel's cumulants the Markovian approximation to arrive to the Quantum Master Equation, for a system interacting with a thermal bath, is revisited.The second order weak coupling approximation is analyzed, then a Kossakowski-Lindblad form for the generator is written in terms of the position and momentum operators.A weak coupling approximation for the stochastic non-Markovian wave function is worked out.A free particle model interacting with a thermal quantum bath is studied in the context of Schr ödinger-Langevin picture.A phenomenological point of view is introduced in order to overcome certain difficulties in the time evolution -in the second order approximation -for the free particle Hamiltonian.
We introduce simple diagrams and rules to calculate the exact time-local master equation for an arbitrary open quantum system interacting with a bath. In it, the time-convolutionless generator is expanded in the system-bath coupling to all orders, and the nthn\mathrm{th}nth-order contribution is expressed by the superoperators generated from the system operator in the system-bath coupling and the associated nnn-point nonequilibrium correlation functions of the environment operators. This theory generalizes previous attempts to obtain the time-convolutionless generator and provides a general framework to study the non-Markovian, nonequilibrium dynamics of open quantum systems.