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Journal of the Optical Society of America B

Phase operator formalism for a two-mode photon system

Masashi Ban

Published Jul 1, 1992• 36 citations
Publisher

Abstract

The phase operator for a two-mode photon system is defined in terms of the relative-number states, and its property is investigated in detail. The phase operator thus defined has a unitary exponential form. It is shown that the average value and fluctuation of the phase operator, where one mode is in some physical state, such as a coherent or a squeezed state, and the other is in a vacuum state, are equivalent to those obtained by means of the Pegg–Barnett phase operator for a single-mode photon. The average value and fluctuation of the phase operator are calculated for coherent and squeezed states of a two-mode photon system.

DOI
10.1364/josab.9.001189

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Physical Review A•Nov 1, 1994•19 citations
The phase operator for a two-mode photon system is defined in terms of the relative-number states, and its property is investigated in detail. The phase operator thus defined has a unitary exponential form. It is shown that the average value and fluctuation of the phase operator, where one mode is in some physical state, such as a coherent or a squeezed state, and the other is in a vacuum state, are equivalent to those obtained by means of the Pegg–Barnett phase operator for a single-mode photon. The average value and fluctuation of the phase operator are calculated for coherent and squeezed states of a two-mode photon system.
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.
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.
Using Pegg-Barnett formalism the author has studied the phase properties of displaced number states. It has been found that the phase probability distribution exhibits the same oscillations as the photon number distribution. Using the Hermitian phase operator the fluctuations of the phase of the displaced number states have been studied. The dependence of the fluctuations on the value of the reference phase have been analysed in detail. It has been found that phase of displaced number states exhibit squeezing in the high-intensity limit of the driving field. These states are approximately equivalent to phase states with precisely defined phase. It has been shown that displaced number states are not minimum uncertainty states with respect to the phase and the number operators.
Phase-coherent states associated with the Pegg-Barnett (PB) phase operator formalism are introduced in terms of the algebraic properties of the PB phase theory. The phase-coherent-state representation of the PB phase states and the PB phase operators is obtained. Squeezing properties of these coherent states with respect to the phase quadrature operators are studied. The number-phase squeezing and number-phase uncertainty relations are also investigated in detail for a two-state system. Some new number-phase minimum uncertainty states are found.