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Physical Review A

Compatibility of the cosine and sine operators

A. Luk, V. Peinová

Published Nov 1, 1990• 8 citations
Publisher

Abstract

By modifying the phase formalism of Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)], we illustrate the idea of Susskind and Glogower [Physics 1, 49 (1964)] concerning the zero uncertainty of their cosine and sine operators in the phase state.
DOI
10.1103/physreva.42.5805

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We have previously developed a formalism for identifying quantum operators corresponding to the sine and cosine of the phase differences between two quantum fields.1 Our approach is based on what is actually measured in an experiment, and it leads to different operators for different measurement schemes. Predictions based on our theory differ from those based on the Susskind-Glogower and the Pegg-Barnett operators. We now present the results of new experiments designed to test the validity of our approach, including measurements of the higher moments of the sine and cosine for different quantum states. The results of all measurements confirm our formalism in every case.
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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.