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Journal of Physics A Mathematical and General

Quantum treatment of the time-dependent coupled oscillators

M. Sebawe Abdalla

Published May 7, 1996• 32 citations
Publisher

Abstract

The Hamiltonian for a system of two coupled oscillators with time-dependent coupling parameter and the phase pump is considered. The wavefunction in Schrödinger picture and the Green function are calculated. The squeezing phenomena as well as the Glauber second-order correlation function is discussed. Statistical investigation for the quasi-probability distribution function (P-representation, W-Wigner and Q-functions) are given.

DOI
10.1088/0305-4470/29/9/015

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The Hamiltonian for a system of two coupled oscillators with time-dependent coupling parameter and the phase pump is considered. The wavefunction in Schrödinger picture and the Green function are calculated. The squeezing phenomena as well as the Glauber second-order correlation function is discussed. Statistical investigation for the quasi-probability distribution function (P-representation, W-Wigner and Q-functions) are given.
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In the present work we study the effects of squeezing on coherent states, number states, and on the thermal field states related to the time evolution operator, which is the result of the Hamiltonian describing the simultaneous non-degenerate parametric amplifier with mixing of two modes a and b via a rotation of their polarization. By using the Glauber second-order correlation function we examined the statistical properties of these various squeezed states. The quasi-probabilities of the W Wigner and Q functions are calculated. The Glauber P representation for the squeezed thermal state explicitly shows the limit of its applicability.
A quantum system described by a two-dimensional harmonic oscillator with a varying mass parameter plus the projection of the angular momentum in the z direction with a time-dependent mode coupling constant is considered to study the two-mode squeezing and photon generation phenomena. For some specific time dependences of the oscillator mass and an arbitrary function of time for the coupling constant exact results on the time evolution of the density matrix and the Wigner function are presented. An analytical expression of the photon distribution is obtained explicitly in terms of Hermite polynomials of two variables, and it shows the effect of oscillations characteristic of squeezing and correlation of the photon quadratures.
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.
At exact resonance we derive the linear and quadratic invariants for time-dependent coupled linear oscillators in the presence of second harmonic generation. Employing these invariants we introduce an accurate definition of the Dirac operators from which the wavefunctions in both coherent and Fock (number) states representations are calculated. We also derive the W-Wigner function of an arbitrary state at time t, evolving in the system under consideration. Moreover we calculate the correlation coefficient between position and momentum and we find that it is identical in both fields whatever the state used.