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International Journal of Modern Physics B

PATH INTEGRAL APPROACH TO A DAMPED AND DRIVEN OSCILLATOR COUPLED WITH ANOTHER ONE

Emmanouil George Thrapsaniotis

Published Feb 20, 2009• 0 citations
Publisher

Abstract

We approach the case of two coupled oscillators where the first one may correspond to a photonic field, while the second one is damped and driven. We model the oscillator's damping via a bath and consider the relevant master equation. We use perturbation theory to handle it. We then path integrate over the effective Hamiltonian of the two oscillators and derive the path integrated density matrix. We suppose that initially both of the oscillators are in coherent states and study the quadrature squeezing effect of the second oscillator.

DOI
10.1142/s021797920904998x

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We approach the case of two coupled oscillators where the first one may correspond to a photonic field, while the second one is damped and driven. We model the oscillator's damping via a bath and consider the relevant master equation. We use perturbation theory to handle it. We then path integrate over the effective Hamiltonian of the two oscillators and derive the path integrated density matrix. We suppose that initially both of the oscillators are in coherent states and study the quadrature squeezing effect of the second oscillator.
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.
The time evolution of a system of bilinearly coupled bosonic modes is investigated using the real-time path integral technique in the coherent-state representation. 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 present treatment is applied to a dissipative harmonic oscillator within the Caldeira–Leggett model. 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 contrast to many perturbative approaches, the bath degrees of freedom are treated explicitly. 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 can the classical trajectories for the system coordinate be recovered.
The present review is devoted to the study of the problem of coupled anharmonic oscillators. A perturbative solution is obtained for the system of an undamped and the damped coupled anharmonic oscillators in the coherent state representation. The solution does not contain the vicious secular terms and shows, explicitly, the anharmonic effects of a coupled system. In order to derive the perturbative solution of a damped coupled system a new frame of time, called quasi time τ, has been exploited. The large quantum number behaviour of coupled anharmonic oscillators has been derived using the path integral method.
We investigate in detail the role of a reservoir concerning the squeezing effect in a driven coupled-oscillator system. In our model, a harmonic oscillator (single cavity mode) is coupled to a two-state oscillator (two-level atom) driven by a classical field. The oscillators are coupled to their respective reservoirs, by which the oscillators are damped. The degree of squeezing in the short-time limit is found to be independent of the damping rate of the harmonic oscillator in a perturbative regime. More remarkably, the harmonic oscillator, which is in a coherent state in the absence of its reservoir, gets squeezed in the steady state when it is coupled to the reservoir. The reservoir assumes two competing roles of assisting and suppressing the squeezing effect induced by the nonlinear interaction of the coupled oscillators. As a result, the squeezing effect is increased to some extent with the damping rate of the harmonic oscillator. In addition, the ``matching condition'' for the optimal squeezing is obtained such that the pumping field intensity should be inversely proportional to the damping rate.