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Physical Review Letters

Resolution to the quantum phase problem

Gilad Gour

Published Feb 19, 2001• 0 citations
OpenAlex

Abstract

Defining the observable φ canonically conjugate to the number observable N has long been an open problem in quantum theory. The problem stems from the fact that N is bounded from below. Here we show how to define the absolute phase observable Φ ≡ |φ | by suitably restricting the Hilbert space of x and p like variables. This Φ is actually the absolute value of the phase and has the correct classical limit. A correction to the “cosine ” C and “sine ” S operators of Carruthers and Nieto is obtained.

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Annual Meeting Optical Society of America•Jan 1, 1992•0 citations
Defining the observable φ canonically conjugate to the number observable N has long been an open problem in quantum theory. The problem stems from the fact that N is bounded from below. Here we show how to define the absolute phase observable Φ ≡ |φ | by suitably restricting the Hilbert space of x and p like variables. This Φ is actually the absolute value of the phase and has the correct classical limit. A correction to the “cosine ” C and “sine ” S operators of Carruthers and Nieto is obtained.
Defining the observable ϕ{\bf ϕ}ϕ canonically conjugate to the number observable N{\bf N}N has long been an open problem in quantum theory. Here we show how to define the absolute phase observable Φ≡∣ϕ∣{\bf Φ}\equiv |{\bfϕ}|Φ≡∣ϕ∣ by suitably restricting the Hilbert space of xxx and ppp like variables. This Φ{\bf Φ}Φ is actually the absolute value of the phase and has the correct classical limit. A correction to the ``cosine'' C{\bf C}C and ``sine'' S{\bf S}S operators of Carruthers and Nieto is obtained.
We use a new limiting procedure, developed to study quantum-optical phase, to examine canonically conjugate operators in general. We find that Dirac's assumption that photon number and phase should be canonically conjugate variables, similar to momentum and position, is essentially correct. The difficulties with Dirac's approach are shown to arise through use of a form of the canonical commutator which, although the only possible form in the usual infinite Hilbert space approach, is not sufficiently general to be used as a model for a number-phase commutator. The approach in this paper unifies the theory of conjugate operators, which include photon number and phase, angular momentum and angle, and momentum and position as particular cases. The usual position-momentum commutator is regained from a more generally applicable expression by means of a domain restriction which cannot be used for the phase-number commutator.
Requirements of a conjugate operator are emphasized, especially in its role in uncertainty relations.It is argued that in many contexts it is necessary to extend the Hilbert space in order to define a conjugate operator as in gauge theories. Example of a particle in a box is analysed. This is closely related to the quantum oscillator through cosine states of Susskind and Glogower.It is used to justify London's phase wave functions albeit as part of a larger Hilbert space. A new definition phase uncertainty neccessiated by periodicity is proposed.It is close to the usual r.m.s. definition.Corresponding number- phase uncertainty relation is obtained and its implications are discussed. Hilbert space of an oscillator is identified with the Hilbert space of a planar rotor with a Z2Z_2Z2​ gauge invariance.This is used to construct states analogous to the cosine and sine states and to illustrate unitary equivalence of Hilbert spaces.
The well-known difficulties of defining a phase operator of an oscillator are considered from the point of view of the canonical transformation to action and phase-angle variables. This transformation turns out to be nonbijective, i.e., it is not a one-to-one onto mapping. In order to make possible the unitarity of its representations in quantum optics we should enlarge the Hilbert space of the problem. In this enlarged space we find a phase operator that, after projection, reproduces previous candidates to represent a well-behaved phase operator in the quantum domain.
To find an operator representation of the phase variable of a single-mode electromagnetic field, the Schrodinger representation is extended to a larger Hilbert space including states with infinite excitation by nonstandard analy- sis, so that the self-adjoint phase operator is acting on the extended Hilbert space.It is shown that this phase operator is a Naimark extension of the optimal probability operator-valued measure for the phase parameter already known in the quantum estimation theory, and is naturally considered as the controversial limit of the phase operators on finite dimensional spaces recently proposed by Pegg and Barnett.Eventually, two of recent promising attempts to obtain the correct statistics of the phase variable in quantum mechanics is synthesized in the present framework based on new mathematical ideas concerning the number system including infinite and infinitesimal numbers.
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.