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Journal of the Optical Society of America B

Squeezed states, photon-number distributions, and U(1) invariance

Biswajit Dutta, Narasimhaiengar Mukunda, R. Simon, Ajay Subramaniam

Published Feb 1, 1993• 27 citations
Publisher

Abstract

We study the photon-number distribution in squeezed states of a single-mode radiation field. A U(1)-invariant squeezing criterion is compared and contrasted with a more restrictive criterion, with the help of suggestive geometric representations. The U(1) invariance of the photon-number distribution in a squeezed coherent state, with arbitrary complex squeeze and displacement parameters, is explicitly demonstrated. The behavior of the photon-number distribution for a representative value of the displacement and various values of the squeeze parameter is numerically investigated. A new kind of giant oscillation riding as an envelope over more rapid oscillations in this distribution is demonstrated.

DOI
10.1364/josab.10.000253

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We study the photon-number distribution in squeezed states of a single-mode radiation field. A U(1)-invariant squeezing criterion is compared and contrasted with a more restrictive criterion, with the help of suggestive geometric representations. The U(1) invariance of the photon-number distribution in a squeezed coherent state, with arbitrary complex squeeze and displacement parameters, is explicitly demonstrated. The behavior of the photon-number distribution for a representative value of the displacement and various values of the squeeze parameter is numerically investigated. A new kind of giant oscillation riding as an envelope over more rapid oscillations in this distribution is demonstrated.
Two-mode squeezed coherent states with complex squeeze and displacement parameters are studied, taking advantage of the SU(2) dynamical symmetry underlying two-mode systems. Expressions for the photon distributions in such states are derived using an SU(2) identity and the fact that the two-mode squeeze operator can be viewed as a rotated version of the product of reciprocal single-mode squeeze operators. An important U(1)\ifmmode\times\else\texttimes\fi{}U(1) invariance of this photon distribution is established. As a consequence, the three phases of the complex squeeze and displacement parameters enter the photon distribution through just one U(1)\ifmmode\times\else\texttimes\fi{}U(1) invariant combination. An associated Gouy effect is noted. Numerical examples of two-mode photon distributions are shown, and interesting new features demonstrated. Second-order coherence properties and their nonclassical nature are briefly studied.
From the normally ordered form of the density operator of a squeezed coherent state(SCS),we directly derive the compact expression of the SCS’s photon-number distribution(PND).Besides the known oscillation characteristics,we find that the PND is a periodic function with a period of π and extremely sensitive to phase.If the squeezing is strong enough,and the compound phase which is relevant to the complex squeezing and displacement parameters are assigned appropriate values,different oscillation behaviours in PND for even and odd photon numbers appear,respectively.
From the normally ordered form of the density operator of a squeezed coherent state (SCS), we directly derive the compact expression of the SCS's photon-number distribution (PND). Besides the known oscillation characteristics, we find that the PND is a periodic function with a period of π and extremely sensitive to phase. If the squeezing is strong enough, and the compound phase which is relevant to the complex squeezing and displacement parameters are assigned appropriate values, different oscillation behaviours in PND for even and odd photon numbers appear, respectively.
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.
We compute the photon-number distribution, the Q (α) distribution function and the wavefunctions in the momentum and position representation for a single mode squeezed number state using generating functions which allow one to obtain any matrix element in the squeezed number state representation from the matrix elements in the squeezed coherent state representation. For highly squeezed number states we discuss the previously unnoted oscillations which appear in the Q (α) function. We also note that these oscillations can be related to the photon-number distribution oscillations and to the momentum representation of the wavefunction.