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Physical Review A

Photon-counting distribution in squeezed states

Apostol Vourdas, Richard M. Weiner

Published Dec 1, 1987• 96 citations
Publisher

Abstract

We calculate the counting (multiplicity) distribution ${P}_{N}=〈N|\ensuremath{\rho}|N〉$ where $\ensuremath{\rho}$ is (i) the density matrix corresponding to a (pure) squeezed state and (ii) the density matrix of a superposition of squeezed states with thermal light. In the first case, we find that ${P}_{N}$ is an oscillating function of $N$. In the second case, this behavior, characteristic for squeezed states, disappears even for small amounts of noise.

DOI
10.1103/physreva.36.5866

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We calculate the counting (multiplicity) distribution PN=〈N∣\ensuremathρ∣N〉{P}_{N}=〈N|\ensuremath{\rho}|N〉PN​=〈N∣\ensuremathρ∣N〉 where \ensuremathρ\ensuremath{\rho}\ensuremathρ is (i) the density matrix corresponding to a (pure) squeezed state and (ii) the density matrix of a superposition of squeezed states with thermal light. In the first case, we find that PN{P}_{N}PN​ is an oscillating function of NNN. In the second case, this behavior, characteristic for squeezed states, disappears even for small amounts of noise.
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 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.
For continuous-wave (cw) squeezed light that has a monochromatic mean field and a rectangular squeezing spectrum covering bandwidth B, we calculate the photocount distribution for a counting time T. The nonclassical oscillations in the photon-number distribution of single-mode squeezed light have reduced visibility in the cw photostatics and often disappear entirely. The even-odd oscillations associated a with single-mode squeezed vacuum are absent for BT\ensuremath{\ll}1, but for BT\ensuremath{\gtrsim}1 they appear in the cw photocount distribution with reduced visibility under appropriate conditions. The oscillations associated with a single-mode squeezed state that has a mean complex amplitude disappear from the cw photostatistics for all counting times.