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Physical Review A

Phase statistics and phase-space measurements

Paul Busch, Marian Grabowski, PEKKA J. LAHTI

Published Oct 1, 1994• 11 citations
Publisher

Abstract

The statistics of quantum optical phase observables are recovered as patterns from both number as well as phase-space measurement statistics. A model for the measurement of generalized Q functions studied recently by Leonhardt and Paul [Phys. Rev. A. 47, R2460 (1993); 48, 3265 (1993)] is shown to yield measurements of arbitary phase-space observables.

DOI
10.1103/physreva.50.2881

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The statistics of quantum optical phase observables are recovered as patterns from both number as well as phase-space measurement statistics. A model for the measurement of generalized Q functions studied recently by Leonhardt and Paul [Phys. Rev. A. 47, R2460 (1993); 48, 3265 (1993)] is shown to yield measurements of arbitary phase-space observables.
Generalizing a recent theoretical result obtained by Freyberger and Schleich [Phys. Rev. A 47, R30 (1993)], we show that the phase-measurement scheme proposed and realized by Noh, Foug\`eres, and Mandel [Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992)] amounts to measuring the Q function for the light under investigation, provided the reference beam (used for homodyne detection) is a very strong coherent field so that it can be described classically. The desired phase distribution follows from the Q function by averaging over the field amplitude. Since an analysis of an earlier proposal by Bandilla and Paul [Ann. Phys. (Leipzig) 23, 323 (1969)] to measure phase distributions via amplification led to just the same result, a perfect physical equivalence of those two approaches has thus been established.
In the two papers [T. Kiss, U. Herzog, and U. Leonhardt, Phys. Rev. A 52, 2433 (1995); U. Herzog, Phys. Rev. A 53, 1245 (1996)] with titles similar to the one given above, the authors assert that in some cases it is possible to compensate a quantum efficiency $\ensuremath{\eta}<~1/2$ in quantum-state measurements, violating the lower bound 1/21/21/2 proved in a preceding paper [G. M. D'Ariano, U. Leonhardt, and H. Paul, Phys. Rev. A 52, R1801 (1995)]. Here we reestablish the bound for homodyning any quantum state, and show how the proposed loss-compensation method would fail in a real measurement outside the $\ensuremath{\eta}>1/2$ regime.