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Journal of Modern Optics

Quantum phase distribution by projection synthesis

David Thomas Pegg, Stephen Mark Barnett, Lee S. Phillips

Published Nov 1, 1997• 46 citations
Publisher

Abstract

We show how the recently-introduced method of projection synthesis can be applied to find the phase probability distribution for a single-mode field. The realization of such a measurement scheme requires the production of reciprocal-binomial states of light or suitable alternative states. We describe how such states might be produced in the laboratory.

DOI
10.1080/09500349708231874

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We show how the recently-introduced method of projection synthesis can be applied to find the phase probability distribution for a single-mode field. The realization of such a measurement scheme requires the production of reciprocal-binomial states of light or suitable alternative states. We describe how such states might be produced in the laboratory.
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 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.
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.
The state projection associated with the familiar (von Neumann) measurement of an observable does not describe the most general means of interrogating a quantum system. Two problems requiring a more general description are the discrimination between non–orthogonal quantum states and the simultaneous measurement of incompatible observables. We discuss both of these problems within the context of measurements of optical polarization. We illustrate the technique of field measurement by projection synthesis by describing a possible means for determining the optical phase probability distribution.
Quantum projection synthesis can be used for phase-probability-distribution measurement and optical-state truncation and preparation. The method relies on interfering optical light beams, which is a major challenge in experiments performed by pulsed light sources. In the pulsed regime, the time frequency overlap of the interfering light beams has a major impact on the efficiency of the method. In this paper, the pulse-mode projection-synthesis approach is developed, the mode structures of interfering light beams are characterized, and the effect of this overlap on the fidelity of optical-state truncation and preparation is investigated. By introducing the positive-operator-valued measure for the detection events in the scheme, the effect of mode mismatch between the photon-counting detectors and the incident light beams is also presented.