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arXiv:1303.0127 · Physical Review A

Measuring the canonical phase with phase-space measurements

Juha-Pekka Pellonpää, Jussi Schultz

Published Jul 22, 2013• 8 citations
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Abstract

We study measurements of single-mode phase observables with emphasis on the canonical phase. We focus on the class of phase-shift covariant phase-space observables, which then yield the phase observables as the angle margins after integrating over the radial part. We consider the possibility of measuring such observables by using a double homodyne detector and its modification. We show that, in principle, the canonical phase distribution of the signal state can be measured via double homodyne detection by first processing the state using a two-mode unitary channel.

DOI
10.1103/physreva.88.012121

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We study measurements of single-mode phase observables with emphasis on the canonical phase. We focus on the class of phase-shift covariant phase-space observables, which then yield the phase observables as the angle margins after integrating over the radial part. We consider the possibility of measuring such observables by using a double homodyne detector and its modification. We show that, in principle, the canonical phase distribution of the signal state can be measured via double homodyne detection by first processing the state using a two-mode unitary channel.
We discuss the possibility of sampling exponential moments of the canonical phase from the s-parametrized phase-space functions. We show that the sampling kernels exist and are well-behaved for any $s>\ensuremath{-}1,$ whereas for $s=\ensuremath{-}1$ the kernels diverge in the origin. In spite of that, we show that the phase-space moments can be sampled with any predefined accuracy from the Q function measured in the double-homodyne scheme with perfect detectors. We discuss the effect of imperfect detection and address sampling schemes using other measurable phase-space functions. Finally, we discuss the problem of sampling the canonical phase distribution itself.
This thesis addresses the use of covariant phase space observables in quantum tomography. Necessary and sufficient conditions for the informational completeness of covariant phase space observables are proved, and some state reconstruction formulae are derived. Different measurement schemes for measuring phase space observables are considered. Special emphasis is given to the quantum optical eight-port homodyne detection scheme and, in particular, on the effect of non-unit detector efficiencies on the measured observable. It is shown that the informational completeness of the observable does not depend on the efficiencies. As a related problem, the possibility of reconstructing the position and momentum distributions from the marginal statistics of a phase space observable is considered. It is shown that informational completeness for the phase space observable is neither necessary nor sufficient for this procedure. Two methods for determining the distributions from the marginal statistics are presented. Finally, two alternative methods for determining the state are considered. Some of their shortcomings when compared to the phase space method are discussed.
Abstract Covariant phase observables constitute a simple solution of the quantum phase problem of a single‐mode optical field. They share three important properties: their range of values is the phase interval [0,2π), they are covariant under shifts generated by the number operator (which is necessary for coherent state phase measurements), and they have the uniform phase distribution in number states. Moreover, some phase observables have been measured (e.g. the phase observable associated to the Q‐function). The canonical phase observable has some additional properties which distinguish it from other covariant phase observables: it generates number shifts, it is uniquely associated to the polar decomposition of the lowering operator (Dirac's idea [1]), it has a projection valued covariant dilation (Newton's extension [2]), and it has a projection valued discretization (the Pegg‐Barnett formalism [3]). The single‐mode covariant phase theory can easily be extended to the two‐mode theory of covariant phase difference observables. Finally, most of all phase theories have connections to the covariant phase theory.
We show that the phase shift of $\frac{\ensuremath{\pi}}{2}$ is crucial for the phase-space translation covariance of the measured high-amplitude limit observable in eight-port homodyne detection. However, for an arbitrary phase shift $\ensuremath{\theta}$ we construct explicitly a different nonequivalent projective representation of R2{\mathbb{R}}^{2}R2 such that the observable is covariant with respect to this representation. As a result we are able to determine the measured observable for an arbitrary parameter field and phase shift. Geometrically the change in the phase shift corresponds to the tilting of one axis in the phase space of the system.
It is well known that the Husimi Q-function of the signal field can actually be measured by the eight-port homodyne detection technique, provided that the reference beam (used for homodyne detection) is a very strong coherent field so that it can be treated classically. Using recent rigorous results on the quantum theory of homodyne detection observables, we show that any phase space observable, and not only the Q-function, can be obtained as a high amplitude limit of the signal observable actually measured by an eight-port homodyne detector. The proof of this fact does not involve any classicality assumption.