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Journal of Modern Optics

Properties of the Measured Phase Operators in the Squeezed Number States

Hsi-teh Tu, Chang-De Gong

Published Jan 1, 1993• 13 citations
Publisher

Abstract

The phase properties of the squeezed number states have been studied from the lower order to the higher order via the measured phase operators (MPO) introduced by Barnett and Pegg and compared with those obtained via the Hermitian and Susskind-Glogower phase operators. We have found many new, interesting results. For the lower order, the phase fluctuations of the MPO are exactly independent of the photon number n. For the higher order, the corresponding fluctuations are a decreasing function of n.

DOI
10.1080/09500349314550091

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•May 1, 1994•0 citations
The phase properties of the squeezed number states have been studied from the lower order to the higher order via the measured phase operators (MPO) introduced by Barnett and Pegg and compared with those obtained via the Hermitian and Susskind-Glogower phase operators. We have found many new, interesting results. For the lower order, the phase fluctuations of the MPO are exactly independent of the photon number n. For the higher order, the corresponding fluctuations are a decreasing function of n.
It is shown that there exist two kinds of operational (or measured) phase operators, those of the Hradil type [Phys. Rev. A 47 (1993), 4532], defined on the entire Hilbert space, and those of the Noh-Fougères-Mandel type [Phys. Rev. A 45 (1992), 424], defined only on a subspace. As a consequence, strictly speaking, the Noh-Fougères-Mandel measured phase cannot be defined in the case of, for example, coherent states, squeezed states and the vacuum. The difference between these operators is made clear by using the concept of the Bargmann distance between a quantum state and its “reduced” state. Since Noh et al. did not use the correct variances of the Susskind-Glogower and Pegg-Barnett phase operators when they compared their data with the results predicted through use of these operators, the correct expressions in that case are presented. Comparing the measured operators with the Hermitian phase operator in the extended Susskind-Glogower formalism, we show that the quantum phase given by the Hermitian phase operator is more appropriate than the measured phases. The variance of the Hermitian operator in the vacuum is consistent with the complementarity of the photon number and phase, which implies that the vacuum has a uniform phase distribution. Hradil's measured operator does not have such a property. Although the Noh-Fougères-Mandel operator has such a property in a particular limit, this operator does not represent, in principle, a meaningful measure of the phase measurement in a typical quantum region, because the Bargmann distance becomes large in such a region.
The phase properties of the self-squeezed states generated during the evolution of the anharmonic oscillator are discussed from the point of view of the new phase formalism of Pegg and Barnett. The phase distribution, the expectation values and the variances of the Hermitian phase operator are obtained and their evolution illustrated graphically. The mean values for the phase cosine and sine functions as well as their variances are also calculated. The results are compared to the Susskind-Glogower formalism results and the results based on the measured phase concept. The relation between squeezing and the phase properties of the field is discussed briefly.
Photon statistics and phase properties of two-mode squeezed number states are studied. It is shown that photon number distribution and Pegg-Barnett phase distribution for such states have similar (N + 1)-peak structure for nonzero value of the difference in the number of photons between modes. Exact analytical formulas for phase distributions based on different phase approaches are derived. The Pegg-Barnett phase distribution and the phase quasiprobability distribution associated with the Wigner function are close to each other, while the phase quasiprobability distribution associated with the Q function carries less phase information.