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arXiv:quant-ph/9412001 · Physical Review A

Photon statistics of a two-mode squeezed vacuum

G. Schrade, V. M. Akulin, Vladimir I. Man’ko, W. P. Schleich

Published Sep 1, 1993• 38 citations
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Abstract

We investigate the photon distribution of a two-mode squeezed vacuum and show that the distribution of photons among the two modes depends on four parameters: two squeezing parameters, the relative phase between the two oscillators, and their spatial orientation. The distribution of the total number of photons depends only on the two squeezing parameters. We derive analytical expressions and present pictures for both distributions.

DOI
10.1103/physreva.48.2398

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We investigate the photon distribution of a two-mode squeezed vacuum and show that the distribution of photons among the two modes depends on four parameters: two squeezing parameters, the relative phase between the two oscillators, and their spatial orientation. The distribution of the total number of photons depends only on the two squeezing parameters. We derive analytical expressions and present pictures for both distributions.
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We derive the joint photon-number distribution for two-mode squeezed states with coherent amplitudes. For certain ranges of the parameters, the joint distribution exhibits nonclassical oscillations, which are the two-mode analog of the oscillations found in the single-mode case. These nonclassical oscillations are absent from the marginal distributions for each mode. The nonclassical features of the joint distribution may be explained using the interference--in--phase-space concept in four-dimensional phase space.
In experiments, pulses of two-mode squeezed light were generated by an optical parametric amplifier. For certain ranges of parameters the joint photon number distribution of two-mode squeezed coherent states exhibits nonclassical oscillations. The nonclassical features were explained using the interference-in-phase-space concept in four-dimensional phase space. We investigate joint phase distribution of two- mode squeezed coherent states for presence of peaks and attempt to relate the peaks to the manifest oscillations in the joint photon number distribution.
We investigate the statistical properties of the photon-subtractions from the two-mode squeezed vacuum state and its decoherence in a thermal environment. It is found that the state can be considered as a squeezed two-variable Hermite polynomial excitation vacuum and the normalization of this state is the Jacobi polynomial of the squeezing parameter. The compact expression for Wigner function (WF) is also derived analytically by using the Weyl ordered operators' invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of WF. The effect of decoherence on this state is then discussed by deriving the analytical time evolution results of WF. It is shown that the WF is always positive for any squeezing parameter and photon-subtraction number if the decay time exceeds an upper bound (}$κt>{1/2}\ln \frac{2\bar{n}+2}{2\bar{n}+1}).
Photon statistics and phase properties of two-mode squeezed number states are studied. It is shown that photon number distribution and Pegg-Barnett phase distribution for such states have similar (N + 1)-peak structure for nonzero value of the difference in the number of photons between modes. Exact analytical formulas for phase distributions based on different phase approaches are derived. The Pegg-Barnett phase distribution and the phase quasiprobability distribution associated with the Wigner function are close to each other, while the phase quasiprobability distribution associated with the Q function carries less phase information.
We investigate the statistical properties of photon subtractions from a two-mode squeezed vacuum state and its decoherence in a thermal environment. It is found that the state can be considered as a squeezed two-variable Hermite polynomial excitation vacuum, and the normalization of this state is the Jacobi polynomial of the squeezing parameter. A compact expression for the Wigner function (WF) is also derived analytically by using the Weyl-ordered operator invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of the WF. The decoherence effect on this state is then discussed by deriving the time evolution of the WF. It is shown that the WF is always positive for any squeezing parameter and any photon-subtraction number if the decay time exceeds an upper bound (κt>½ ln 2n¯+2/2n¯+1).