Physical Review A

Number-phase uncertainty product for generalized squeezed states arising from the Pegg-Barnett Hermitian phase operator formalism

Istok P. Mendaš, Duska B Popovic

Published Dec 1, 1995• 12 citations

Abstract

The number-phase uncertainty relation based on the Pegg-Barnett Hermitian phase operator formalism is discussed for generalized squeezed states of the harmonic oscillator. The corresponding number-phase uncertainty product is calculated for the magnitudes of the squeeze and displacement parameters ranging from 0 to 3/2 in the former case and from 0 to 4 in the latter case for the first few classes of generalized squeezed states (m=0, 1, and 2) and for different values of their combined phases. It is found that for a given magnitude of the squeeze parameter, the number-phase uncertainty product tends to the fixed limiting value m+1/2 when the magnitude of the displacement parameter tends to infinity. On the other hand, for a fixed magnitude of the displacement parameter, the uncertainty product grows indefinitely as the magnitude of the squeeze parameter increases. It is also observed that the number-phase uncertainty product tends to zero for few-photon generalized squeezed states (when the magnitudes of both squeeze and displacement parameters tend to zero) so that, according to the Pegg-Barnett Hermitian phase formalism, it is possible to have generalized squeezed states with a number-phase uncertainty product smaller than 1/2.

DOI: 10.1103/physreva.52.4356

Related papers

Number-phase uncertainty product for displaced number states

Istok P. Mendaš, Duska B Popovic

The number-phase uncertainty product for displaced number states of the harmonic oscillator is examined within the framework of three different approaches: (i) the Carruthers-Nieto number-phase uncertainty relation in terms of the Susskind-Glogower sine and cosine phase operators, (ii) a similar relation to this calculated with the Pegg-Barnett unitary phase operator, and (iii) the number-phase uncertainty relation arising from the Pegg-Barnett Hermitian phase operator. The corresponding number-phase uncertainty product is calculated extending from average photon numbers 30 down to m for the first few classes of displaced number states (m=0,1,...,5). It is found that, for a displaced number state with a reasonable average number of excited quanta, all three rival phase formalisms yield similar number-phase uncertainty products, tending, for increasingly large magnitude of the displacement parameter, to the constant value m+1/2. On the other hand, for a small number of excited quanta it is found that, according to the first two formalisms, the number-phase uncertainty product for a given class of displaced number states tends to the maximum value \ensuremath{\surd}2m+1 /(\ensuremath{\surd}m+1 - \ensuremath{\surd}m ), while the third phase formalism predicts an entirely different behavior; with decreasing magnitude of displacement parameter, after passing through a maximum, the number-phase uncertainty product falls off eventually to zero making, in particular, the search for minimum number-phase uncertainty states futile. It is argued in favor of this last result, and the possibility of experimental verification, in the realm of quantum optics, is briefly considered.

Physical Review AAug 1, 19947 citations

Coherent states in a finite-dimensional basis: Their phase properties and relationship to coherent states of light

Vladimír Bužek, A. D. Wilson‐Gordon, P. L. Knight +1 more

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.

Physical Review AJun 1, 199289 citations

Number-Phase Uncertainty Relations for Optical Fields

R. Tanaś

The Hermitian phase formalism of Pegg and Barnett allows for direct calculations of the phase variance and, consequently, the number-phase uncertainty product. This gives us a unique opportunity, inaccessible before, to study the number-phase uncertainty relations for optical fields in a direct way within a consistent quantum formalism. A few examples of fields generated in nonlinear optical processes are studied from the point of view of their number-phase uncertainty relations.

Jan 1, 19960 citations

Phase-coherent states and their squeezing properties

Le‐Man Kuang, Xin Chen

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.

Physical Review ANov 1, 199419 citations

Fluctuation of Phase in the Displaced Number States

Zhengfeng Hu

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.

Journal of Modern OpticsJun 1, 19928 citations

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

Gautam Gangopadhyay

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.

Journal of Modern OpticsMar 1, 19945 citations