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Physical Review A

Nonclassical photon statistics of two-mode squeezed states

Ching Tsung Lee

Published Aug 1, 1990• 73 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.42.1608

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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.
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.
A sufficient condition for nonclassical two-mode states in terms of three probabilities, (n2+1)p(n1-1, n2+1)2n1p(n1, n2)+(n1+1)p(n1+1, n2-1)2n2p(n1, n2)<1, is established as a new type II criterion. This criterion is applied to the two-mode coherent state, the two-mode squeezed vacuum state, and the pair coherent state to show how easily it can be used. It is also applied to the photon-added two-mode thermal state and the mixing of two two-photon states by a symmetric beam splitter to show that the type II criterion is superior to its type I counterpart in at least these two examples.
The stimulated Compton scattering of photons between two oppositely propagating coherent lights by a relativistic electron moving along the same line is studied analytically. The system is described in a moving frame so that the electron becomes nonrelativistic and the frequencies of the two lights coincide. Except for the replacement of the static wiggler by a propagating light, our system is essentially the same as a free-electron laser working in the Compton regime and described in the usual Bambini-Renieri frame. The Q representation of two-mode coherent states is adopted to describe the radiation fields, and the electron states are expressed in terms of plane waves. A perturbative solution, with the quantum recoil of the electron as the perturbation parameter, of the time evolution of the system is carried out far beyond the first order. It demonstrates that two nonclassical effects, i.e., squeezing and photon antibunching, occur as results of the scattering. It also confirms once more the fact that quantum recoils play the exclusive role in generating such nonclassical photon states. A very important improvement in the present treatment is that the electron part of the density matrix is traced out before the calculation of the squeezing effect. As a result, it is found that there is no squeezing if the initial state is a vacuum. This is, perhaps, a significant new discovery. Many-electron effects are ignored in the present study.
In this paper, we study the nonclassical properties and find out the effect of photon addition on these properties as well as the process of teleportation in the two-mode photon-added displaced squeezed (TMPADS) states. We derive the analytic expressions of the Wigner function, the photon number distribution and the intermode photon antibunching for these states. We show that photon addition operation not only makes the Wigner function become negative but also leads to increase the degree of antibunching. The peak of the photon number distribution becomes flatter and shifts to the greater number of photons by adding photons to both modes simultaneously. Furthermore, it is proved that the degree of intermodal entanglement becomes bigger and bigger through increasing the number of photons added to both modes. As expected, when using these states as an entanglement resource to teleport a state, the average fidelity of teleportation process is also improved by increasing the number of added photons.