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Journal of Modern Optics

Quantum phase distribution by operator synthesis

K. L. Pregnell, David Thomas Pegg

Published Jun 1, 2002• 5 citations
Publisher

Abstract

Methods proposed so far for measuring the canonical phase probability distribution of light are very difficult to implement. In particular the method of projection synthesis, which synthesizes a phase state projector, involves the use of a beam splitter and a specially engineered reference state. Here we extend this method by use of an additional beam splitter to synthesize phase sine and cosine operators. This allows the measurement of all the moments needed to construct the phase distribution. The reference state required for operator synthesis is, remarkably, just an easily prepared coherent state.

DOI
10.1080/09500340110108611

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Methods proposed so far for measuring the canonical phase probability distribution of light are very difficult to implement. In particular the method of projection synthesis, which synthesizes a phase state projector, involves the use of a beam splitter and a specially engineered reference state. Here we extend this method by use of an additional beam splitter to synthesize phase sine and cosine operators. This allows the measurement of all the moments needed to construct the phase distribution. The reference state required for operator synthesis is, remarkably, just an easily prepared coherent state.
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.
We present an extension of the projection synthesis technique which allows us, at least in principle, to determine the probability distribution for any physical observable associated with a quantized optical field mode. This probability can be inferred from the photodetection statistics obtained by causing the field mode to be measured to interfere with a second mode prepared in a suitable reference state. We give an explicit expression for the required reference state. Weak-field homodyne detection using photon counting is, perhaps, the simplest system to which projection synthesis can be applied. We find the complete set of synthesized projectors for this system and show that they form a probability operator measure. We apply this set of projectors to study the discrimination between coherent states and to measure a quasiprobability distribution.
The results of experiments designed to measure the operational phase cosine and sine variances of weak states of light disagree with the variances predicted by canonical phase formalisms. As these variances are fundamental manifestations of the quantum nature of phase, it is important to be able to measure the canonical variances also. A recent suggestion to do so, based on the use of a two-component probe, involves the difficult preparation of exotic states of light which have not yet been produced. In this paper we show how the variances can be measured with simple coherent state inputs. The retrodictive formalism of quantum mechanics provides useful insight into the physics involved.