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Physical Review A

Analytic solution for the dissipative anharmonic quantum oscillator and semiclassical analysis

Franz X. Kärtner, Axel Schenzle

Published Aug 1, 1993• 14 citations
Publisher

Abstract

The analytic solution of the time evolution of a dissipative nonlinear quantum oscillator in the generalized s-ordered representation is derived. The analytical solution allows for comparsion of the various representations with the classical and semiclassical dynamics. It is shown by the eigenvalue analysis of the evolution operators that in the case of a small nonlinearity a semiclassical analysis is an excellent approximation to the exact quantum dynamics. This is further demonstrated by using the Green's function of the anharmonic oscillator to solve the initial-value problem for a superposition of coherent states.

DOI
10.1103/physreva.48.1009

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The analytic solution of the time evolution of a dissipative nonlinear quantum oscillator in the generalized s-ordered representation is derived. The analytical solution allows for comparsion of the various representations with the classical and semiclassical dynamics. It is shown by the eigenvalue analysis of the evolution operators that in the case of a small nonlinearity a semiclassical analysis is an excellent approximation to the exact quantum dynamics. This is further demonstrated by using the Green's function of the anharmonic oscillator to solve the initial-value problem for a superposition of coherent states.
We present the general solutions for the classical and quantum dynamics of the anharmonic oscillator coupled to a purely diffusive environment. In both cases, these solutions are obtained by the application of the Baker-Campbell-Hausdorff (BCH) formulas to expand the evolution operator in an ordered product of exponentials. Moreover, we obtain an expression for the Wigner function in the quantum version of the problem. We observe that the role played by diffusion is to reduce or to attenuate the the characteristic quantum effects yielded by the nonlinearity, as the appearance of coherent superpositions of quantum states (Schr\\"{o}dinger cat states) and revivals.
Quantum systems interacting with a dissipative environment can effectively be described by a nonlinear modification of the Schrodinger equation. For some standard problems that possess analytical solutions, e.g. the free motion and the harmonic oscillator, also the nonlinear equation can be solved in a closed form and has Gaussian-shaped wave packet solutions. The equation determining the time-evolution of the wave-packet width is - already in the case of the usual conservative Schrodinger equation - nonlinear and provides the quantities that are necessary to construct the Green’s function for the respective system. The Green’s functions for the conservative and dissipative cases will be given explicitly in position and momentum space.
We develop a semiclassical method to determine the nonlinear dynamics of dissipative quantum optical systems in the limit of large number of photons N; it is based on the 1/N-expansion and the quantum-classical correspondence. The method is used to tackle two problems: the study of the dynamics of nonclassical state generation in higher order anharmonic dissipative oscillators and the establishment of the difference between the quantum and classical dynamics of the second-harmonic generation in a self-pulsing regime. In addressing the first problem, we obtain an explicit time dependence of the squeezing and the Fano factor for an arbitrary degree of anharmonism in the short-time approximation. For the second problem, we analytically find a characteristic time scale at which the quantum dynamics differs insignificantly from the classical one.
In this Master's thesis, we give a pedagogical introduction to the semiclassical analysis of wave dynamics in single-band and multi-band systems. We show how the semiclassical evolutions of Wigner functions can approximate the dynamics of wave packet centers and wave packet deformations. Three systems of increasing complexity are considered: the quantum harmonic oscillator and the quantum anharmonic oscillator as single-band systems, and the Dirac oscillator as a two-band system. We explain the origins of the first-order semiclassical corrections (Moyal, Littlejohn-Flynn, and Berry) in the equations of motion and examine their effects on predictive accuracy. Another focus is set on the conditions for validity of the semiclassical analysis. For both the quantum anharmonic and the Dirac oscillators, a dimensionless parameter controlling the accuracy of the approximation is identified. In multi-band systems, the formalism applies only to states within a single adiabatic subspace, which, for the Dirac oscillator, corresponds to an energy branch. This motivates the construction of single-branch almost coherent states: approximately Gaussian wave packets formed by superpositions of states from a single energy branch. Their centers undergo oscillations with an aperiodic envelope. These states enable comparison between semiclassical Wigner function evolution and the commonly used Sundaram-Niu formalism. We show that semiclassical Wigner function evolution captures deformation-related dynamics, including the initial decay of oscillations, which the Sundaram-Niu formalism cannot describe by construction. Additionally, exact quantum evolution reveals partial relocalization of these single-branch almost coherent states, a phenomenon that cannot be captured by the semiclassical evolution.
The time-dependent WKB approximation for a coherent state is expanded to third order around a guiding real trajectory, allowing for the novel treatment of nonlinearity in its semiclassical dynamics, which is generally present in all physical systems far from the classical regime. The result is a closed-form solution consisting of a linear combination of Airy functions and their derivatives multiplied by an exponential. The expression's ability to capture nonlinearity is demonstrated by examining the autocorrelation of initial coherent states in anharmonic systems with few to many contributing periodic orbits. Its accuracy is compared to the quadratic expansion and found to be superior in regimes of ℏ where the curvature begins to be significant, as expected. Moreover, the expression is shown to be a real-trajectory uniformization over two coalescing saddle points that are emblematic of significant curvature. This extends real-trajectory time-dependent wave-packet semiclassical methods to highly anharmonic systems for the first time and establishes their regime of validity.
We develop a method for the determination of the dynamics of dissipative quantum systems in the limit of large number of quanta N , based on the 1/ N -expansion of Heidmann et al. (1985 Opt. Commun. 54 189) and the quantum-classical correspondence. Using this method, we find analytically the dynamics of nonclassical states generation in the higher-order anharmonic dissipative oscillators for an arbitrary temperature of a reservoir. We show that the quantum correction to the classical motion increases with time quadratically up to some maximal value, which is dependent on the degree of nonlinearity and a damping constant, and then it decreases. Similarities and differences with the corresponding behavior of the quantum corrections to the classical motion in the Hamiltonian chaotic systems are discussed. We also compare our results obtained for some limiting cases with the results obtained by using other semiclassical tools and discuss the conditions for validity of our approach.