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arXiv:quant-ph/0004059 · Physical Review A

Sampling the canonical phase from phase-space functions

Jaromı́r Fiurášek, Mohammed Dakna, Tomáš Opatrný, D.‐G. Welsch

Published Nov 14, 2000• 2 citations
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Abstract

We discuss the possibility of sampling exponential moments of the canonical phase from the s-parametrized phase-space functions. We show that the sampling kernels exist and are well-behaved for any $s>\ensuremath{-}1,$ whereas for $s=\ensuremath{-}1$ the kernels diverge in the origin. In spite of that, we show that the phase-space moments can be sampled with any predefined accuracy from the Q function measured in the double-homodyne scheme with perfect detectors. We discuss the effect of imperfect detection and address sampling schemes using other measurable phase-space functions. Finally, we discuss the problem of sampling the canonical phase distribution itself.

DOI
10.1103/physreva.62.063811

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We discuss the possibility of sampling exponential moments of the canonical phase from the s-parametrized phase-space functions. We show that the sampling kernels exist and are well-behaved for any s>\ensuremath−1,s>\ensuremath{-}1,s>\ensuremath−1, whereas for s=\ensuremath−1s=\ensuremath{-}1s=\ensuremath−1 the kernels diverge in the origin. In spite of that, we show that the phase-space moments can be sampled with any predefined accuracy from the Q function measured in the double-homodyne scheme with perfect detectors. We discuss the effect of imperfect detection and address sampling schemes using other measurable phase-space functions. Finally, we discuss the problem of sampling the canonical phase distribution itself.
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 study measurements of single-mode phase observables with emphasis on the canonical phase. We focus on the class of phase-shift covariant phase-space observables, which then yield the phase observables as the angle margins after integrating over the radial part. We consider the possibility of measuring such observables by using a double homodyne detector and its modification. We show that, in principle, the canonical phase distribution of the signal state can be measured via double homodyne detection by first processing the state using a two-mode unitary channel.
We show that the exponential phase moments associated with the s-ordered generalized Wigner function can be sampled from homodyne data for s 0. The tomographic sampling functions are evaluated for non-unit homodyne quantum efficiency, and the statistical reliability of the reconstruction is tested by Monte Carlo simulated experiments. The reconstruction of the s-ordered phase distribution from sampled moments is discussed, and the reduction of statistical fluctuations by adaptive techniques is also analysed.
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.