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Physical Review A

Quantum description of nonlinearly interacting oscillators via classical trajectories

G. Drobný, A. Bandilla, Igor Jex

Published Jan 1, 1997• 34 citations
Publisher

Abstract

We investigate systems of few harmonic oscillators with mutual nonlinear coupling. Using classical trajectories---the solutions of Hamiltonian equations of motion for a given nonlinear system---we construct the approximate quasiprobability distribution function in phase space that enables a quantum description. The nonclassical effects (quantum noise reduction) and their scaling laws can be so studied for high excitation numbers. In particular, the harmonic oscillators represent modes of the electromagnetic field and the Hamiltonians under consideration describe representative nonlinear optical processes (multiwave mixings). The range of the validity of the approximation for Wigner and Husimi functions evolved within the classical Liouville equation is discussed for a diverse class of initial conditions, including those without classical counterparts, e.g., Fock states.

DOI
10.1103/physreva.55.78

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We investigate systems of few harmonic oscillators with mutual nonlinear coupling. Using classical trajectories---the solutions of Hamiltonian equations of motion for a given nonlinear system---we construct the approximate quasiprobability distribution function in phase space that enables a quantum description. The nonclassical effects (quantum noise reduction) and their scaling laws can be so studied for high excitation numbers. In particular, the harmonic oscillators represent modes of the electromagnetic field and the Hamiltonians under consideration describe representative nonlinear optical processes (multiwave mixings). The range of the validity of the approximation for Wigner and Husimi functions evolved within the classical Liouville equation is discussed for a diverse class of initial conditions, including those without classical counterparts, e.g., Fock states.
Systems of interacting harmonic oscillators have recently received considerable attention as models for describing a variety of physical problems. We have investigated the validity of the rotating-wave approximation which constitutes the traditional approach to the solution of the dynamical problem by comparing it with the exact solution. A numerical comparison has been made and the limits of validity of the rotating-wave approximation has been established in terms of the strength of oscillator interaction.In particular, the time development of dynamical operators and certain transition probabilities have been compared. In the region where the rotating-wave approximation is valid, the time evolution of the quasiprobability distribution $P(\ensuremath{\alpha}, t)$ of one oscillator is given for several initial conditions. A counting scheme similar to the argument given by Feynman for the driven harmonic oscillator, is proposed for the interpretation of the time-dependent transition amplitudes between number states.
Quantum mechanics has been formulated in phase space, with the Wigner function as the representative of the quantum density operator, and classical mechanics has been formulated in Hilbert space, with the Groenewold operator as the representative of the classical Liouville density function. Semiclassical approximations to the quantum evolution of the Wigner function have been defined, enabling the quantum evolution to be approached from a classical starting point. Now analogous semiquantum approximations to the classical evolution of the Groenewold operator are defined, enabling the classical evolution to be approached from a quantum starting point. Simple nonlinear systems with one degree of freedom are considered, whose Hamiltonians are polynomials in the Hamiltonian of the simple harmonic oscillator. The behavior of expectation values of simple observables and of eigenvalues of the Groenewold operator are calculated numerically and compared for the various semiclassical and semiquantum approximations.
We numerically investigate the classical limit of quantum trajectories in optical second-harmonic generation. This is a dissipative system of two nonlinearly coupled harmonic oscillators, corresponding to the fundamental and second-harmonic optical cavity modes. Classically it is described by a system of nonlinear differential equations which have a range of attractors: fixed points, limit cycles, and chaotic attractors. We consider the field-amplitude mean values corresponding to the quantum trajectory wave functions. We find that in the classical limit of large photon number the trajectories of these mean values approach the corresponding classical attractors. This is because the mean values of operator products factorize into products of mean values in the classical limit.
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The statistical properties of systems of coupled quantum-mechanical harmonic oscillators are analyzed. The Hamiltonian for the system is assumed to be an inhomogeneous quadratic form in the creation and annihilation operators, and is allowed to have an explicit time dependence. The relationship to classical theory is emphasized by expressing pure states in terms of the coherent-state vectors, and density operators by means of the PPP representation and an analogous representation involving the Wigner function. The state which evolves from an initially coherent state of the system is found, and equations governing the time evolution of the Wigner function and the weight function for the PPP representation are derived, in differential and integral form, for arbitrary initial states of the system. The results remain valid for couplings which do not preserve the vacuum state, and for cases in which the time dependence of the coupling parameters gives rise to large-scale amplification of the initial field intensities. The analysis is performed by first treating general linear inhomogeneous canonical transformations on the oscillator variables, and then specializing to the case in which these transformations represent the solutions for the Heisenberg operators in terms of their initial values. The results are illustrated within the context of a model of parametric amplification.
Abstract The response of classical or quantum harmonic oscillators coupled linearly to a classical field is strictly linear; all nonlinear response functions vanish identically. We show that, if the oscillators interact with quantum modes of the radiation field, they acquire nonlinear susceptibilities. The effective third order susceptibility induced by four interactions with quantum modes contains collective resonances involving pairs of oscillators. All nonlinearities are missed by the conventional approximate treatment based on the quantum master equations.