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Journal of Modern Optics

Statistical Properties of the Time Evolution Operator for Two Coupled Oscillators

M. Sebawe Abdalla

Published Jul 1, 1993• 12 citations
Publisher

Abstract

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.

DOI
10.1080/09500349314551411

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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.
The statistical properties of the time evolution operator for the Hamiltonian describing the simultaneous non-degenerate parametric amplifier, and mixing of two modes in the rotator frame is considered. Here we study the effects of this operator on the photon number sum and difference. The Glauber second-order correlation function is calculated, and the squeezing related to the frequency converter and the parametric amplifier has been discussed. The statistical investigations for the quasiprobability distribution functions are carried out.
Much attention has been given in the past to two classes of squeezed states: the squeezed vacuum and coherent states. Here we study the effects of squeezing on number states and on thermal field states. The statistical properties of these various squeezed states are discussed using the second-order correlation functions. The quasiprobabilities of the Wigner, Q, and positive P representations are calculated and compared for the squeezed states. The Glauber P representation for the squeezed thermal state explicitly shows the limit of its applicability. The photon number distributions of the squeezed number and squeezed thermal states are extensively discussed and new interference effects in phase space are shown to lead to highly structured number distributions.
We have investigated the properties of two-mode squeezed states of various kinds: squeezed coherent states, squeezed number states, squeezed thermal states, and squeezed number-thermal states. The properties studied include characteristic functions, photon number distributions, and high-order moments. We show that a two-mode squeezed coherent state appears in either of its modes as a Glauber-Lachs superposition of a coherent signal with thermal noise; this generalizes the well-known thermofield result that a two-mode squeezed vacuum state behaves like a pure thermal state in either of its modes. We investigate the higher-order squeezing (in the sense of Hong and Mandel) of squeezed coherent-thermal and squeezed number-thermal states. Although the squeezing is found to increase with order in both cases, there are nevertheless interesting differences that can be traced to the fact that the Wigner function has a Gaussian form for the former but not for the latter.