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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Optical measurements using projection synthesis

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

Published Jul 6, 1998• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.312645

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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.
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.
Experiments have been performed to determine the Wigner distribution and the density matrix (and for pure states the wave function) of a light mode, by using tomographic inversion of a set of measured probability distributions for quadrature amplitudes.From these measurements the quantum distributions of optical phase and photon number have been obtained.The measurements of quadrature-amplitude distributions for a temporal mode of the electromagnetic field are carried out using balanced homodyne detection.We refer to this new method as optical homodyne tomography.Given the measured density matrix, one can experimentally infer any of the various quantum distributions of optical phase, in particular the Pegg-Barnett (or, equivalently, Shapiro-Shepard) phase distribution, the marginal Wigner distribution, and the Vogel-Schleich operational phase distribntion.We have used this approach to make measurements of the number-plase uncertainty relation for coherent-state fields.The coherent states do not attain the minimum value for the number-phase uncertainty product, as set by the expectation value of the commutator of the number and phase operators; this is true theoretically and experimentally.
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.