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arXiv:quant-ph/9606014 · Physical Review A

Direct sampling of the Susskind-Glogower phase distributions

Mohammed Dakna, L. Knöll, D.‐G. Welsch

Published Mar 1, 1997• 3 citations
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Abstract

Coarse-grained phase distributions are introduced that approximate to the Susskind-Glogower cosine and sine phase distributions to any desired degree of accuracy. The integral relations between the phase distributions and the phase-parametrized field-strength distributions observable in balanced homodyning are derived and the integral kernels are analyzed. It is shown that the phase distributions can be directly sampled from the field-strength distributions which offers the possibility of measuring the Susskind-Glogower cosine and sine phase distributions with sufficiently high precision. Numerical simulations are performed to demonstrate the applicability of the method.

DOI
10.1103/physreva.55.2360

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Coarse-grained phase distributions are introduced that approximate to the Susskind-Glogower cosine and sine phase distributions to any desired degree of accuracy. The integral relations between the phase distributions and the phase-parametrized field-strength distributions observable in balanced homodyning are derived and the integral kernels are analyzed. It is shown that the phase distributions can be directly sampled from the field-strength distributions which offers the possibility of measuring the Susskind-Glogower cosine and sine phase distributions with sufficiently high precision. Numerical simulations are performed to demonstrate the applicability of the method.
The problem of direct sampling of the London phase distribution in balanced homodyning is studied. For this purpose a continuous sequence of parametrized phase distributions tending to the London phase distribution as the sequence parameter goes to zero is considered. From the integral relation of the parametrized phase distributions to the field-strength distributions measurable in homodyne detection, it is seen that for any non-vanishing sequence parameter the associated phase distribution can be obtained by direct sampling from the measured data. This offers a possibility of asymptotically measuring the London phase distribution. Numerical simulations show that it can be obtained with sufficiently high accuracy.
Using coherent phase states, parameterized phase state distributions for a single-mode radiation field are introduced and their integral relation to the phase-parameterized field-strength distributions is studied. The integral kernel is evaluated and the problem of direct sampling of the coherent phase state distributions using balanced homodyne detection is considered. Numerical simulations show that when the value of the smoothing parameter is not too small the coherent phase state distributions can be obtained with sufficiently well accuracy. With decreasing value of the smoothing parameter the determination of the coherent phase state distributions may be an effort, because both the numerical calculation of the sampling function and the measurement of the field-strength distributions are required to be performed with drastically increasing accuracy.
We investigate exponential phase moments of the s-parametrized quasidistributions (smoothed Wigner functions). We show that the knowledge of these moments as functions of s provides, together with photon-number statistics, a complete description of the quantum state. We demonstrate that the exponential phase moments can be directly sampled from the data recorded in balanced homodyne detection and we present simple expressions for the sampling kernels. The phase moments are Fourier coefficients of phase distributions obtained from the quasidistributions via integration over the radial variable in polar coordinates. We performed Monte Carlo simulations of the homodyne detection and we demonstrate the feasibility of direct sampling of the moments and subsequent reconstruction of the phase distribution.
A similarity between the photon annihilation and creation operators and the Susskind-Glogower exponential phase operators motivates the introduction and study of quasidistributions of the cosine and sine of phase. These quasidistributions are related to the standard and antistandard orderings of trigonometric phase operators and to the normal, antinormal, and symmetrical orderings of the exponential phase operators. The symmetrical ordering is connected to an optical ideal tomography of the appropriate quasidistribution from the rotated Susskind-Glogower cosine operator.