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Journal of Physics A Mathematical and General

Noh-Fougères-Mandel operational phase operator as an entangling operator and the corresponding squeezed state

Hong-Yi Fan, Zhihu Sun

Published Feb 19, 2001• 12 citations
Publisher

Abstract

We construct a new number-difference and operational phase entangled state by operating the Noh-Fougères-Mandel (NFM) phase operator on the two-mode twin-photon state. Its Schmidt decomposition is derived by virtue of the previously constructed Einstein-Podolsky-Rosen eigenstate. We reach the conclusion that the NFM phase operator is essentially an entangling operator, based on which a new type of number-difference and operational phase squeezed state can be introduced.

DOI
10.1088/0305-4470/34/8/309

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We construct a new number-difference and operational phase entangled state by operating the Noh-Fougères-Mandel (NFM) phase operator on the two-mode twin-photon state. Its Schmidt decomposition is derived by virtue of the previously constructed Einstein-Podolsky-Rosen eigenstate. We reach the conclusion that the NFM phase operator is essentially an entangling operator, based on which a new type of number-difference and operational phase squeezed state can be introduced.
For the two-mode photon number-difference operator D = a†a -b†b and the Noh, Fougers and Mandel (NFM) operational phase operator [Formula: see text], we introduce the concept of number-difference-phase squeezing. We then find a new minimum-uncertainty state for number-difference-phase squeezing, which turns out to be a special combination of generalized SU(1,1) coherent state. As a by-product, a new orthonormal and complete representation in two-mode Fock space made up of R†n|m, m> is found.
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.