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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Exact quantum statistics of a nonlinear dissipative oscillator evolving from an arbitrary state

Vlasta Peřinová, A. Lukš

Published Jul 1, 1990• 0 citations
Publisher

Abstract

Recently the statistical properties of the third-order nonlinear dissipative oscillator have been determined exactly to an approximation respecting non-zero reservoir temperatures for a most general input state.1 Later the assumption of zero-'temperature heat bath has been removed, but the initial coherent state has been considered. In this coinmunication we present the statistical characteristics of the third-order non linear dissipative oscillator responsive to the requirements of quantum optics and established without any restrictions on the initial state.

DOI
10.1117/12.22199

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Recently the statistical properties of the third-order nonlinear dissipative oscillator have been determined exactly to an approximation respecting non-zero reservoir temperatures for a most general input state.1 Later the assumption of zero-'temperature heat bath has been removed, but the initial coherent state has been considered. In this coinmunication we present the statistical characteristics of the third-order non linear dissipative oscillator responsive to the requirements of quantum optics and established without any restrictions on the initial state.
The statistical properties of radiation passing through a nonlinear medium modelled as a third-order nonlinear dissipative oscillator interacting with squeezed light have been investigated. Whereas the photon statistics, being insensitive to the nonlinearity, are determined exactly, an approximative approach has been adopted in the description of squeezing in order to involve fluctuations. As a consequence of the general treatment, new results for a damped linear oscillator with initial squeezed light are provided.
The statistical properties of nonlinear oscillators modeling dissipative forward four-wave mixing are investigated, adopting the Fokker-Planck equation approach. The photon statistics are shown to be insensitive to the nonlinearities and are formulated in terms of the superposition of coherent and chaotic fields. The only approximation in describing the squeezing properties consists in the way of including thermal noise. The effect of the initial coherent amplitudes, that of the interplay of the coupling constants, and that of the reservoir on the squeezing properties are studied. An interesting connection between the revivals of coherent amplitudes and squeezing is pointed out.
We investigate the low-temperature statistical properties of a harmonic oscillator coupled to a heat bath, where the low-frequency spectrum vanishes. We obtain the exact result of the zero point energy. Due to the low frequency shortage of environmental oscillators' spectral density, the coordinate and momentum correlation functions decay as τ−4 and τ−6 respectively at zero temperature, where τ is the correlation time. The low-temperature behavior of the mean energy does not violate the third law of thermodynamics, but differs largely from the Ohmic spectrum case.
The focus of the study is the free quantum evolution of Duffing oscillators which are nonlinearly coupled to bosonic environments. The work evolves from a particular system of an undamped, nonlinear graphene resonator mode to a more generalized framework of oscillator modes interacting with two different configurations of bosonic environments. In a model with no dissipation, the nonlinear Duffing oscillator allows an initial coherent state to evolve into a macroscopic superposition state, a Schrödinger Cat state. By further subjecting a Duffing mode to nonlinear damping, the parity conservation due to the two-quanta system-bath exchange, brings the system to a nonclassical steady state - an equilibrium state very much different from the ground state. The quantum features of this state are analysed for temperatures above zero and in a more realistic scenario where the interplay of linear and nonlinear decay is taken into account. The scope of the study is then extended to bipartite systems of nonlinear interacting oscillators, each nonlinearly coupled to a bosonic environment. The generation of entanglement in initially separable states and entanglement's asymptotic behaviour are investigated. One of the outcomes is the Duffing nonlinearity not affecting the entanglement asymptote. Therefore, finally a bipartite system of harmonic oscillators, with nonlinear system-bath coupling is initiated with entangled, squeezed states, and the asymptotic behaviour is evaluated in the parameter space of temperature, squeezing and dissipation rate. This is done for common and individual reservoir configurations, and either zero or non-zero inter-mode coupling.