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arXiv:quant-ph/0311151 · Journal of Physics A Mathematical and General

Husimi'sQ( ) function and quantum interference in phase space

Douglas F. Mundarain, Jorge Stephany

Published Mar 9, 2004• 14 citations
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Abstract

We discuss a phase-space description of the photon number distribution of non-classical states which is based on Husimi's Q (α) function and does not rely on the WKB approximation. We illustrate this approach using the examples of displaced number states and two photon coherent states and show it to provide an efficient method for computing and interpreting the photon number distribution. This result is interesting in particular for the two photon coherent states which, for high squeezing, have the probabilities of even and odd photon numbers oscillating independently.

DOI
10.1088/0305-4470/37/12/010

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We discuss a phase-space description of the photon number distribution of non-classical states which is based on Husimi's Q (α) function and does not rely on the WKB approximation. We illustrate this approach using the examples of displaced number states and two photon coherent states and show it to provide an efficient method for computing and interpreting the photon number distribution. This result is interesting in particular for the two photon coherent states which, for high squeezing, have the probabilities of even and odd photon numbers oscillating independently.
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.
Phase-space-interference approaches employ interference on the phase plane to explain a quantum state's photon-number distribution. This paper generalizes the approach so that knowledge of the photon-number distribution explains the state's quantum phase-plane distribution. The required generalization is as follows. The Bohr-Sommerfeld ``band'' or trajectory of the mixture of number states \ensuremath{\sim}(\ensuremath{\Vert}N〉〈N\ensuremath{\Vert}+\ensuremath{\Vert}M〉〈M\ensuremath{\Vert}) consists of two annuli centered on the origin. In contrast, the trajectory of the pure number state superposition \ensuremath{\sim}(\ensuremath{\Vert}N〉+\ensuremath{\Vert}M〉) (with M>N) contains additional interference structures, namely, (M-N) symmetrically positioned radial spokes connecting the annuli. These spokes intersect the circular band attributed to a photon-number state at (M-N) locations, with the resulting destructive interference eliminating states \ensuremath{\Vert}N+1〉 to \ensuremath{\Vert}M-1〉. These structures build state phase-plane quantum probability distributions, as we illustrate for the squeezed vacuum. This state is missing every odd-numbered photon number state, and thus its trajectory consists of a repeated pattern of two heavily emphasized bars eliminating every odd-numbered state. These bars sum to generate the traditional ``cigar'' shape of the squeezed state.
The relation between quantum interference and classical interference is discussed in terms of the Wigner function and an analogous classical expression. For two displaced coherent states and two displaced electric-field Gaussian pulses, both the quantum-mechanical and classical Wigner functions exhibit oscillatory behavior. Classical analogs of squeezing, photon-number oscillation, and the QQQ function are presented.
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.