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Progress of Theoretical Physics

Hermitian Phase Operator and Approximate Quantum Measurement in Extended Number-State Space

K. Kakazu

Published Apr 1, 2002• 2 citations
Publisher

Abstract

Considering an approximate quantum phase measurement, we show that the extended Pegg-Barnett formalism with dimension (2 s + 1) is not plagued by Vorontsov and Rembovsky problem whereby unphysical states can appear after the phase measurement. The approximate phase measurement in this formalism gives, in the large s limit, the same result as that in the Susskind-Glogower formalism. The difficulty of defining a Hermitian phase operator, caused by the periodicity of the phase angle, is overcome by introducing an extended number-state space and by making use of periodic generalized functions (distributions) and their Fourier transforms. This space is the large s limiting case of the extended Pegg-Barnett formalism. Similarly, with the help of periodic generalized functions, Hermitian phase operators can be defined in other infinite-dimensional extended theories. It is also shown that approximate phase measurements give the same results in all these extended formalisms.

DOI
10.1143/ptp.107.655

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Considering an approximate quantum phase measurement, we show that the extended Pegg-Barnett formalism with dimension (2 s + 1) is not plagued by Vorontsov and Rembovsky problem whereby unphysical states can appear after the phase measurement. The approximate phase measurement in this formalism gives, in the large s limit, the same result as that in the Susskind-Glogower formalism. The difficulty of defining a Hermitian phase operator, caused by the periodicity of the phase angle, is overcome by introducing an extended number-state space and by making use of periodic generalized functions (distributions) and their Fourier transforms. This space is the large s limiting case of the extended Pegg-Barnett formalism. Similarly, with the help of periodic generalized functions, Hermitian phase operators can be defined in other infinite-dimensional extended theories. It is also shown that approximate phase measurements give the same results in all these extended formalisms.
Vorontsov and Rembovsky pointed out recently a problem that unphysical states appear after an approximate quantum measurement of the Pegg-Barnett phase [Phys. Lett. A 254 (1999), 7; 262 (1999), 486]. To give a solution to this problem, we introduce a quantum phase in the extended number-state space and show that there is no Vorontsov-Rembovsky problem in the extended space. In the infinite-dimensional limit, the present extended formalism gives the same results in almost all cases as the Pegg-Barnett formalism. Also, it is shown that unphysical states can be suppressed by using the projection operator to the physical number space.
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.
A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ and so the correponding limit points are ordinary operators on Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.