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arXiv:1210.0357

A rigorous derivation of the transition rate for the dissipative two-state system

Marco Merkli, Haifeng Song

Published Oct 1, 2012• 0 citations
arXivHTMLPDF

Abstract

The transition rate for a two-state system interacting with a bosonic heat bath, from the initial state `up' at time t=0 to `down' at time t>0, was derived formally in the seminal paper [12] by Leggett, Chakravarty, Dorsey, Fisher, Garg and Zwerger. The expression was obtained for small tunneling parameter of the two-level system using time-dependent perturbation theory to second order. It contains all orders in the parameter of coupling between the system and the bath. We give a rigorous derivation of the transition rate, using a quantum resonance approach.

DOI
10.48550/arxiv.1210.0357

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The transition rate for a two-state system interacting with a bosonic heat bath, from the initial state up' at time t=0 to down' at time t>0, was derived formally in the seminal paper [12] by Leggett, Chakravarty, Dorsey, Fisher, Garg and Zwerger. The expression was obtained for small tunneling parameter of the two-level system using time-dependent perturbation theory to second order. It contains all orders in the parameter of coupling between the system and the bath. We give a rigorous derivation of the transition rate, using a quantum resonance approach.
The recently developed theory for tunneling in dissipative systems is rederived using quantal transition-state theory. As predicted by Caldeira and Leggett, we find an exponential damping of the tunneling rate at 0 K. The exponential rate enhancement at low temperatures as well as the crossover temperature are also obtained with this approach. Moreover, the rate enhancement is given explicitly in terms of energy transfer from the bath to the dissociative mode. The present derivation includes memory effects.
In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.
We present a systematic renormalization-group analysis of the transition occurring in a symmetric two-state system with a linear dissipative coupling to a heat bath. We define the theory as a double expansion in $\ensuremath{\Delta}$ (tunneling frequency) and $\ensuremath{\delta}$ (distance from the critical coupling to the heat bath at $\ensuremath{\Delta}=0$), following a similar strategy as in the sine-Gordon theory. We reproduce previous results and obtain higher-order terms and a new universal correction in the flow equations.
A variational calculation of the dynamics of a two level system interacting with a bath is presented. The effective tunneling matrix element is renormalized by the interaction with the bath. For the case of a bath with ohmic dissipation (or infrared divergence), the variational calculation gives a vanishing tunneling at T=0 for critical values of the coupling and gives a highly unusual temperature dependence for the tunneling rate for large kT, all of which agree with recent path integral calculations. The present method also yields results for intermediate temperatures and coupling. This indicates that a judicious choice of the zeroth order Hamiltonian to include the major part of the coupling can lead to correct results even in low order perturbation theory.
: The Brownian dynamics of the density operator for a quantum system interacting with a classical heat bath is described using a stochastic, non-linear Liouville equation obtained from a variational principle. The rate of the noise-induced transitions is expressed as a function of the environmental spectral density, and is discussed for the case of white noise and blackbody radiation. The time-scale of decoherence and dissipation is obtained numerically for a system of two quantum states. These are the ground and first excited state of the center of mass vibrations for a single ion confined in a harmonic trap. PACS numbers: 42.50.Lc,05.40.+j,32.80.Pj 1 Introduction The evolution of a quantum system surrounded by a thermal environment represents a fundamental problem of the quantum theory, increasingly important for applications. In the classical mechanics, a particle interacting with a heat bath has a Brownian evolution towards thermalization, which reduces to a purely dissipative moti...
In this letter, we introduce a novel method for investigating dissipation (gain) and thermalization in an open quantum system. In this method, the quantum system is coupled linearly with a copy of itself or with another system described by a finite number of bosonic operators. The time-dependent coupling functions play a fundamental role in this scheme. To demonstrate the efficiency and significance of the method, we apply it to some ubiquitous open quantum systems. Firstly, we investigate a quantum oscillator in the presence of a thermal bath at the inverse temperature βββ, obtaining the reduced density matrix, the Husimi distribution function, and the quantum heat distribution function accurately. The results are consistent with existing literature by appropriate choices for the time-dependent coupling function. To illustrate the generalizability of this method to systems interacting with multiple thermal baths, we study the interaction of a quantum oscillator with two thermal baths at different temperatures and obtain compatible results. Subsequently, we analyze a two-level atom with energy or phase dissipation and derive the spontaneous emission and the pure dephasing processes consistently using the new method. Finally, we investigate the Markovianity in a dissipative two-level system.