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Physical Review A

Number-phase uncertainty products and minimizing states

A. Lukš, Vlasta Peřinová

Published May 1, 1992• 21 citations
Publisher

Abstract

The similarity between the system of the plane rotator and that of the optical oscillator (mode) has been exposed. In both the systems special states have been examined, which minimize the usual and unusual uncertainty relations. Number-sine-cosine uncertainty relations for these systems have been derived and the number-sine minimum-uncertainty states have been investigated in greater detail than in the literature.

DOI
10.1103/physreva.45.6710

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The similarity between the system of the plane rotator and that of the optical oscillator (mode) has been exposed. In both the systems special states have been examined, which minimize the usual and unusual uncertainty relations. Number-sine-cosine uncertainty relations for these systems have been derived and the number-sine minimum-uncertainty states have been investigated in greater detail than in the literature.
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.
The minimization problem of finding the number-phase minimum uncertainty states (MUS) is considered and its solutions are found either numerically or, under some special conditions, analytically. The phase uncertainty measure is based on the Bandilla-Paul dispersion. The problem is treated (i) in a finite-dimensional Hilbert space and (ii) for a countably infinite-dimensional Hilbert space (i.e. the standard quantum harmonic oscillator), with the constraint of a given mean photon number. The MUS relations between the photon number uncertainty and phase uncertainty are presented. Connections to some other minimization problems are discussed.
The recently introduced Hermitian phase operator allows a phase-state representation of the single-mode light field. We find the requirement that a light field is in a physical state, that is, has finite energy moments, imposes strict and simple continuity conditions on the phase-amplitude distribution. We exploit these conditions to examine the physical intelligent and minimum-uncertainty states associated with the number-phase uncertainty relations, including those involving the Hermitian phase operator and its sine and cosine forms. The single number-state is found to be the only physical exact intelligent state and also the only physical exact minimum-uncertainty state for all the uncertainty relations considered. We construct states which are both physical states and approximately intelligent states. Under certain conditions coherent states, ideal squeezed states and the number-phase intelligent states associated with the Susskind-Glogower cosine and sine operators are found to be both physical states and approximately number-phase intelligent states for the phase operators considered here.