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Journal of Optics B Quantum and Semiclassical Optics

Complementary coherent state for measuring theQ-function: generation and properties

Ardiley Torres Avelar, B. Baseia, N. G. de Almeida

Published Oct 20, 2003• 8 citations
Publisher

Abstract

In a previous paper a proposal for measuring the Husimi Q -function based on the projection synthesis method (Barnett and Pegg 1996 Phys. Rev. Lett. 76 4148) was presented. The scheme required the presence of an unavailable state, named the complementary coherent state. This paper shows how to generate this state in travelling waves and studies its properties.

DOI
10.1088/1464-4266/6/1/007

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In a previous paper a proposal for measuring the Husimi Q -function based on the projection synthesis method (Barnett and Pegg 1996 Phys. Rev. Lett. 76 4148) was presented. The scheme required the presence of an unavailable state, named the complementary coherent state. This paper shows how to generate this state in travelling waves and studies its properties.
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.
We present an extension of the projection synthesis technique which allows, at least in principle, the determination of the probability distribution of any physical observable of a quantized optical field mode. This involves directly measuring a quantity proportional to the squared modulus of the projection of an input pure state onto the eigenstate associated with the observable to be measured. It is found that we need to synthesize a specific reference state conditioned upon the property under investigation. A general expression for these reference states is given. We provide the specific reference states needed to synthesize a measurement of canonical phase. A variant on homodyne detection is investigated using projection synthesis and we form the probability operator measure (POM) required to characterize the measurement. The POM is not that which is formed in conventional balanced homodyne detection. As an application, we utilize these results to form the POM and a quasi probability distribution function associated with a measurement scheme used to distinguish between nonorthogonal coherent states.