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The Journal of Chemical Physics

Relaxation in simple quantum systems

Abraham Nitzan, R. Silbey

Published May 15, 1974• 127 citations
Publisher

Abstract

The decay of an oscillator and a two level system into a general heat bath is studied using a truncated cumulant expression for the time evolution operator. Expressions for the decay rate in the weak coupling and rotating wave approximations are derived, and the temperature dependence of the rate examined in all cases. We find the temperature dependence to be strongly dependent on the nature of the system and bath.

DOI
10.1063/1.1680859

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The decay of an oscillator and a two level system into a general heat bath is studied using a truncated cumulant expression for the time evolution operator. Expressions for the decay rate in the weak coupling and rotating wave approximations are derived, and the temperature dependence of the rate examined in all cases. We find the temperature dependence to be strongly dependent on the nature of the system and bath.
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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.
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A quantum-mechanical treatment of the dynamics of relaxation of a twolevel mode coupled to a thermal bath is presented. The Zwanzig-Mori projection-operator formalism is employed to derive exact equations of motion for the correlation function and the excess population , where and are the coordinate and number operators associated with the two-level mode. In the van Hove weak-coupling limit it is shown that |G ν|2 and Δn decay exponentially with time constants τ and τ′, respectively. Explicit expressions for τ and τ′ are determined and the relationship between them clearly established. The application of the model to the specific problem of the effects of vibrational relaxation on isotropic Raman spectral lineshapes in dense media is emphasized.
We derive reduced equations of motion for simple quantum systems which are strongly driven by an external field and are modulated stochastically by a coupling to a bath. In the derivation we make use of the cumulant-expansion method of Kubo using two different time-ordering prescriptions. We demonstrate how the choice of the ordering prescription is related to the statistical properties of the bath, once the cumulant expansion is truncated. Our equations of motion are valid for arbitrary time scale for the motions of the bath relative to those of the system, and they change smoothly from the static to the Markov (motional narrowing) limit. As examples, we consider the problems of a randomly modulated and driven harmonic oscillator and a modulated and damped two-level system. In the Markovian limit both ordering prescriptions yield Bloch-type equations of motion; in general, however, the driving and modulation interfere and the different statistical properties of the bath, as determined by the two truncated ordering prescriptions, lead to different results.