Quantum theory of dissipation of a harmonic oscillator coupled to a non-equilibrium bath; Wigner-Weisskopf decay and physical spectra (original) (raw)
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Annals of Physics, 1992
We discuss some aspects of dissipation in quantum field theory starting from the example of the quantum mechanical damped harmonic oscillator. We show that the set of states of the system splits into unitarily inequivalent representations of the canonical commutation relations. At quantum level the irreversibility of time evolution is expressed as tunneling among the unitarily inequivalent representations. Statistical and thermodynamical properties of the formalism are analysed and canonical quantization is shown to lead to time dependent SU( 1, 1) coherent states, well known in high energy physics as well as in quantum optics and thermal field theory. $0
The harmonic oscillator with dissipation within the theory of open quantum systems
1994
Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function, subject to either the Gaussian type or the δ-function type of initial conditions, is also solved by using the Wang-Uhlenbeck method. The obtained Wigner functions are two-dimensional Gaussians with different widths.
Classical and quantum dissipation in non-homogeneous environments
Physica A: Statistical Mechanics and its Applications, 1994
We generalize the oscillator model of a particle interacting with a thermal reservoir by introducing arbitrary nonlinear couplings in the particle coordinates. The equilibrium positions of the heat bath oscillators are promoted to space-time functions, which are shown to represent a modulation of the internal noise by the external forces. The model thus provides a description of classical and quantum dissipation in non homogeneous environments. In the classical case we derive a generalized Langevin equation with nonlinear multiplicative noise and a position-dependent fluctuation-dissipation theorem associated to non homogeneous dissipative forces. When time-modulation of the noise is present, a new force term is predicted besides the dissipative and random ones. The model is quantized to obtain the non homogenous influence functional and master equation for the reduced density matrix of the Brownian particle. The quantum evolution equations reproduce the correct Langevin dynamics in the semiclassical limit. The consequences for the issues of decoherence and localization are discussed.
Exact non-Markovian cavity dynamics strongly coupled to a reservoir
Physical Review A, 2010
The exact non-Markovian dynamics of a microcavity strongly coupled to a general reservoir at arbitrary temperature is studied. With the exact master equation for the reduced density operator of the cavity system, we analytically solve the time evolution of the cavity state and the associated physical observables. We show that the non-Markovian dynamics is completely determined by the propagating (retarded) and correlation Green functions. Compare the non-Markovian behavior at finite temperature with those at zero-temperature limit or Born-Markov limit, we find that the non-Markovian memory effect can dramatically change the coherent and thermal dynamics of the cavity. We also numerically study the dissipation dynamics of the cavity through the mean mode amplitude decay and the average photon number decay in the microwave regime. It is shown that the strong coupling between the cavity and the reservoir results in a long-time dissipationless evolution to the cavity field amplitude, and its noise dynamics undergoes a critical transition from the weak to strong coupling due to the non-Markovian memory effect.
Quantum decay rates for dissipative systems at finite temperatures
Physical Review B, 1987
The decay of a metastable state of a system coupled to a heat-bath environment is studied. A functional-integral method is presented allowing for the calculation of decay rates at finite temperatures and in the presence of dissipation. The theory is utilized to determine the rate for a wide range of parameters. The temperature extends from the region where the decay is thermally activated down to very low temperatures where the system decays by tunneling from its ground state in the metastable well. The range of damping parameters covers the region from weakly damped to heavily overdamped motions. It is found that the transition between thermally activated decay and tunneling occurs near a crossover temperature To which decreases with increasing damping strength. Well above To the rate follows the classical Arrhenius law where the preexponential factor is aff'ected by the frequency-dependent damping. As To is approached, quantum corrections to the classical rate formula become increasingly important. In the vicinity of To the rate follows a scaling law describing the crossover between thermally activated and quantum-mechanical decay. In the region below To the decay rate can be determined analytically only in limiting cases. For a system with Ohmic dissipation and a cubic potential, accurate numerical calculations are presented exhausting the range of parameters not covered by analytical results.
An Approach To Quantum Dissipation
1993
We show that the extension to dissipative situations of the usual quantization method, based on a formal analogy between the classical and quantal Hamiltonian dynamics, cannot be considered plausible since the Heisenberg principle is violated. Alternatively, we demonstrate that, by dealing with two oscillators, a quantum and classical one, coupled through a linear term, we can mimic the dissipating behaviour of a quantum system, without recourse to any quantization of classical equations and without violation of any quantum rule.
Non-Markovian Quantum Dissipation in the Presence of External Fields
Progress in Theoretical Chemistry and Physics, 2003
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.
Dissipation in a quantum-mechanical system
Optical Materials, 2008
We consider a model of a dissipative quantum-mechanical system consisting of weakly coupled quantum and classical subsystems. The classical subsystem is assumed to be infinite, and thus serves as a means to transfer the energy of the quantum subsystem to the infinity (actually, to dissipate the energy). The quantum-classical coupling is treated in the spirit of the mean-field approximation. Solving the equations for the classical subsystem explicitly an effective dissipative Schrö dinger equation for the quantum subsystem is obtained. The proposed method is illustrated by calculating the shape of the nonlinear resonance.
Quantum dissipation in unbounded systems
Physical Review E, 2002
In recent years trajectory based methodologies have become increasingly popular for evaluating the time evolution of quantum systems. A revival of the de Broglie-Bohm interpretation of quantum mechanics has spawned several such techniques for examining quantum dynamics from a hydrodynamic perspective. Using techniques similar to those found in computational fluid dynamics one can construct the wave function of a quantum system at any time from the trajectories of a discrete ensemble of hydrodynamic fluid elements ͑Bohm particles͒ which evolve according to nonclassical equations of motion. Until very recently these schemes have been limited to conservative systems. In this paper, we present our methodology for including the effects of a thermal environment into the hydrodynamic formulation of quantum dynamics. We derive hydrodynamic equations of motion from the Caldeira-Leggett master equation for the reduced density matrix and give a brief overview of our computational scheme that incorporates an adaptive Lagrangian mesh. Our applications focus upon the dissipative dynamics of open unbounded quantum systems. Using both the Wigner phase space representation and the linear entropy, we probe the breakdown of the Markov approximation of the bath dynamics at low temperatures. We suggest a criteria for rationalizing the validity of the Markov approximation in open unbound systems and discuss decoherence, energy relaxation, and quantum/classical correspondence in the context of the Bohmian paths.
Phenomenological-operator approach to dissipation in cavity quantum electrodynamics
Physical Review A, 2000
We present a phenomenological-operator approach to describe energy dissipation in cavity QED phenomena. This approach, developed for an absolute-zero and a thermal environment, considerably simplifies the introduction of the inevitable errors due to the environmental degrees of freedom when describing processes involving dispersive atom-field interactions. The main result in the present work consists in furnishing a straightforward technique to estimate the fidelity resulting from dispersive atom-field interactions, precluding the necessity of performing the usually extensive ab initio calculations. Furthermore, we expect that the present work can help us account for dissipation in resonant atom-field interactions and even help us achieve a general phenomenological approach to estimate the effects of dissipation in whichever system. To illustrate the universal applicability of the present technique, we calculate the fidelity of a mesoscopic quantum superposition state engineered in a lossy cavity, considering also the excited-state spontaneous decay of the required atom. For the case of a stable atomic excited state, the fidelity computed here is in agreement with a recently announced exact calculation. PACS number͑s͒: 42.50.Dv, 03.65.Bz FIG. 1. Sketch of the experimental setup for engineering the MQSS.