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Journal of Modern Optics

Coherent Phase State and Displaced Phase State in a Finite-dimensional Basis and Their Light Field Limits

Gautam Gangopadhyay

Published Mar 1, 1994• 5 citations
Publisher

Abstract

Using the Pegg-Barnett phase operator formalism we have introduced the coherent phase state and the displaced phase state in a finite dimension. Using the definition of phase creation and annihilation operators we have constructed a displacement operator which yields coherent phase states and displaced phase states within a finite-dimensional basis and numerically evaluated observables as a function of the size of the basis set. We have shown that, when the dimension of the state space is much larger than the average excitation of phase quanta, then these states are the coherent phase state and displaced phase state of a harmonic oscillator which has an infinite number of eigenstates and exhibits phase squeezing up to 100%. The coherent phase state is a minimum-uncertainty state with respect to the phase quadrature operators but not the minimum-uncertainty state with respect to number and phase operators. The properties of coherent phase states for a two-state system are also investigated. It is interesting to realize that these coherent phase states and displaced phase states are exactly the same as the phase vacuum state and different phase states respectively for the single-mode light field case.

DOI
10.1080/09500349414550511

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Using the Pegg-Barnett phase operator formalism we have introduced the coherent phase state and the displaced phase state in a finite dimension. Using the definition of phase creation and annihilation operators we have constructed a displacement operator which yields coherent phase states and displaced phase states within a finite-dimensional basis and numerically evaluated observables as a function of the size of the basis set. We have shown that, when the dimension of the state space is much larger than the average excitation of phase quanta, then these states are the coherent phase state and displaced phase state of a harmonic oscillator which has an infinite number of eigenstates and exhibits phase squeezing up to 100%. The coherent phase state is a minimum-uncertainty state with respect to the phase quadrature operators but not the minimum-uncertainty state with respect to number and phase operators. The properties of coherent phase states for a two-state system are also investigated. It is interesting to realize that these coherent phase states and displaced phase states are exactly the same as the phase vacuum state and different phase states respectively for the single-mode light field case.
The Pegg-Barnett phase-operator formalism utilizes a finite basis set to represent operators of the harmonic oscillator; this enables the phase to be represented by a Hermitian operator, but rests on taking the dimensionality of the basis set to infinity for observable quantities. Simultaneously, in their approach Pegg and Barnett consider quantum states of a harmonic oscillator which are normalized in the Fock space, i.e., the dimensionality of the basis set in which the states of the harmonic oscillator are defined is supposed to be infinite, while the phase operator is defined in the finite-dimensional basis set. In this paper we address the problem of a consistent definition of a coherent state within a finite state basis. We employ displacement operators to define such coherent states and numerically evaluate observables as a function of the size of the basis set. We investigate phase properties of these coherent states. We find that if the dimensionality of the state space is much larger than the mean occupation number of the coherent states, then the results obtained in the finite-dimensional basis are applicable in the case of a ordinary quantum-mechanical harmonic oscillator. These coherent states are minimum uncertainty states with respect to quadrature operators (i.e., the position and momentum operators) and do not exhibit quadrature squeezing. A weakly excited (compared with the dimensionality of the state space) coherent state in finite-dimensional basis is not strictly speaking a minimum uncertainty state with respect to the number and phase operators. We give definitions of amplitude and phase squeezing and show that weakly excited coherent states can be amplitude squeezed. In the high-intensity limit (again compared with the dimensionality of the state space) these states exhibit phase squeezing.
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.
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.
Even and odd phase coherent states associated with the Hermetian phase operator theory are introduced in terms of the creation operation of the phase quanta defined in a finite-dimensional phase state space. Some mathematical and physical properties of these quantum states are studied in some detail. It is shown that the even phase coherent states together with the odd ones build an overcomplete Hilbert space. Even and odd coherent-state formalism of the Pegg - Barnett phase operator is given in terms of the projection operator in the even and odd phase coherent-state space. The number - phase uncertainty relation is investigated for these quantum states. It is shown that even and odd phase coherent states are not minimum uncertainty and intelligent states for the number and phase operators.
Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)] have proposed a formalism for a Hermitian optical phase operator, constructed from the so-called phase states. In this paper we give an alternative derivation. We begin by defining a unitary phase operator ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$ from the relationship $\stackrel{^}{a}={e}^{i\stackrel{^}{\ensuremath{\varphi}}}R(\stackrel{^}{N})$ ($\stackrel{^}{a}$ is an annihilation operator for a single-mode boson field), but borrow the idea of a finite- (but arbitrarily large) dimensional space from Pegg and Barnett. Finally, we comment on an experiment to determine quantum phase uncertainty. Our analysis uses the operator ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$, which we feel is a more natural choice than $\stackrel{^}{\ensuremath{\varphi}}$. The use of ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$ also avoids the multivalued problems associated with $\stackrel{^}{\ensuremath{\varphi}}$.