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Physical Review A

Two-mode squeezed states with thermal noise

Ching Tsung Lee

Published Oct 1, 1990• 31 citations
Publisher

Abstract

The thermofield introduced by Takahashi and Umezawa [Collect. Phenom. 2, 55 (1975)] is used to represent the thermal noise; it is generated from the absolute vacuum by a so-called thermalizing operator and serves as the initial state for generating two-mode squeezed states by squeeze and displacement operators. The ones obtained by applying the squeeze operator first followed by the displacement operator are called coherent squeezed states, while the others obtained by applying the displacement operator first are called squeezed coherent states. A criterion for the existence of nonclassical photon statistics in a two-mode radiation introduced by us in a previous paper [Phys. Rev. A 42, 1609 (1990)] is then used to study the conditions for the two types of squeezed states to be nonclassical states. Several three-dimensional plots are presented to illustrate the effects of thermal noise on squeezed states.

DOI
10.1103/physreva.42.4193

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The thermofield introduced by Takahashi and Umezawa [Collect. Phenom. 2, 55 (1975)] is used to represent the thermal noise; it is generated from the absolute vacuum by a so-called thermalizing operator and serves as the initial state for generating two-mode squeezed states by squeeze and displacement operators. The ones obtained by applying the squeeze operator first followed by the displacement operator are called coherent squeezed states, while the others obtained by applying the displacement operator first are called squeezed coherent states. A criterion for the existence of nonclassical photon statistics in a two-mode radiation introduced by us in a previous paper [Phys. Rev. A 42, 1609 (1990)] is then used to study the conditions for the two types of squeezed states to be nonclassical states. Several three-dimensional plots are presented to illustrate the effects of thermal noise on squeezed states.
The concept of photon antibunching as a manifestation of nonclassical character in single-mode radiation is extended to two-mode radiation. A simple criterion for the existence of nonclassical effects in two-mode radiation is established. It implies that the intermode photon bunching, rather than antibunching, can play a key role in rendering two-mode radiation nonclassical. The two-mode squeezed vacuum states are used as simple examples to illustrate the essential point of the criterion. The criterion is then used to study the nonclassical properties of two types of two-mode squeezed states. A state generated by applying the squeeze operator to the two-mode vacuum first, followed by the displacement operator, is called a two-mode coherent squeezed state. If the order of the two operators is reversed, a two-mode squeezed coherent state is obtained. It is found that the latter has a much stronger tendency to become a nonclassical state than the former, assuming that the parameters involved are all the same. A measure of nonclassical depth of radiation is adopted. It is found that, according to this measure, the effect of squeezing in making two-mode coherent squeezed states nonclassical is not monotonic. Such phenomena do not occur in squeezed coherent states.
We consider a single-mode radiation field initially in a displaced squeezed thermal state. The weak interaction of such a field with a heat bath of arbitrary temperature is shown to preserve the Gaussian form of the characteristic function. Accordingly, the study of the time development of the density operator reduces to our previous description [P. Marian and T. A. Marian, preceding paper, Phys. Rev. A 47, 4474 (1993)] of the initial quantum state. As examples, photon statistics and squeezing properties of the damped field are analyzed. Based on the close relation between field dissipation and photon detection, we derive simple analytic formulas for the counting distribution and its factorial moments. Nonclassical features of a displaced squeezed thermal state, such as oscillations of the photon-number distribution, survive in the counting process, provided that the quantum efficiency of the detector is high enough.
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.
We investigate a free monochromatic electromagnetic field which is the superposition of a squeezed thermal radiation and a coherent one. The main tool in our analysis is the characteristic function that has a Gaussian form. We establish an analytic formula for an arbitrary correlation function, as well as its strong-squeezing limit. Besides the usual quasiprobability densities, the coherent-state, number-state, coordinate, and momentum representations of the density operator are derived. We point out the nonclassical oscillations of the photon-number distribution and find its generating function. Collaterally, displaced thermal states and squeezed thermal states are revisited as nontrivial particular cases. We examine finally the squeezing properties of the field using the distribution functions of the quadratures.
The properties of correlated two-mode SU(1, 1) coherent states are studied. These states are a generalization of the two-mode squeezed vacuum states and are generated by applying the two-mode squeeze operator to a state in which there are initially q photons in one mode, |q, 0〉, the squeezed vacuum (q = 0) obviously being a special case. For these states the photon-number distribution and the nonclassical properties of photon antibunching, enhanced phase fluctuations, violations of the Cauchy–Schwarz inequality, two-mode squeezing, and sum squeezing are studied. A Q-function quasi-probability distribution for these states is also presented.