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arXiv:quant-ph/9603027

Homodyne detection for measuring coherent phase state distributions

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

Published Mar 25, 1996• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/9603027

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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.
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.
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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.
We use the field-strength eigenstates, that is, the quadrature eigenstates rotated by an angle ccphi, to define a phase distribution of a single mode of the radiation field. A measurement procedure lies at the heart of this operational phase distribution: A balanced homodyne-detection scheme measures, in principle, the field-strength probability curve in its dependence on ccphi. The probability of finding a zero electric field plotted versus ccphi constitutes the proposed distribution. For a wide class of quantum states, this curve is in good agreement with the abstract phase probability curve obtained from phase states, but it is free of the familiar problems accompanying the notion of a Hermitian phase operator. The phase distribution of a quantum state has been achieved without using phase states; this can be summarized as ``phase without phase.''