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arXiv:0803.2094

Inverse operator representations of quantum phase

G. M. Saxena

Published Mar 14, 2008• 0 citations
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Abstract

We define quantum phase in terms of inverses of annihilation and creation operators. We show that like Susskind - Glogower phase operators, the measured phase operators and the unitary phase operators can be defined in terms of the inverse operators. However, for the unitary phase operator the Hilbert space includes the negative energy states. The quantum phase in inverse operator representation may find the applications in the field of quantum optics particularly in the squeezed states.

DOI
10.48550/arxiv.0803.2094

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We define quantum phase in terms of inverses of annihilation and creation operators. We show that like Susskind - Glogower phase operators, the measured phase operators and the unitary phase operators can be defined in terms of the inverse operators. However, for the unitary phase operator the Hilbert space includes the negative energy states. The quantum phase in inverse operator representation may find the applications in the field of quantum optics particularly in the squeezed states.
We define a unitary phase operator for photons in a single momentum mode. The operator acts on a Hilbert space with basis consisting of all number states in both polarizations. The Susskind Glogower operator, E^=eiϕ^{\hat{E}} = e^{i \hatϕ}E^=eiϕ^​, thus defined is unitary and acts as a true ladder operator over all the states. It satisfies the commutation relation [E^,n^]=E^[{\hat{E}}, {\hat{n}}] = {\hat{E}}[E^,n^]=E^; consequently, the phase ϕ^\hatϕϕ^​ satisfies the canonical commutation relation [n^,ϕ^]=i[{\hat{n}}, {\hatϕ}] = i[n^,ϕ^​]=i. The eigenstates of the Susskind Glogower operator are found. Our model is able to directly account for phase measurements when polarization changes cannot be neglected and at the same time gives a well defined phase operator.
In this paper, we define the eigenvalue equation of the squeezed coherent state using recently introduced inverse annihilation and creation operators. The salient feature of the eigenvalue equation of the squeezed coherent state is that it consistently yields the eigenvalue equation of the squeezed vacuum when the coherent amplitude is reduced to zero. The squeezed coherent state defined by others, without using the inverse operators, cannot be reduced to the squeezed vacuum state. The time evolution of the squeezed states is also discussed. The squeezed states have phase-dependent quantum noise. The phase operators have an important role in the squeezing experiments. We show that the inverse operators are also useful in defining the phase operators. We also discuss the eigenvalue equation of the squeezed coherent state defined by Yuen vis-à-vis that is introduced in this paper using inverse annihilation and creation operators for bosons.
It is shown that there exist two kinds of operational (or measured) phase operators, those of the Hradil type [Phys. Rev. A 47 (1993), 4532], defined on the entire Hilbert space, and those of the Noh-Fougères-Mandel type [Phys. Rev. A 45 (1992), 424], defined only on a subspace. As a consequence, strictly speaking, the Noh-Fougères-Mandel measured phase cannot be defined in the case of, for example, coherent states, squeezed states and the vacuum. The difference between these operators is made clear by using the concept of the Bargmann distance between a quantum state and its “reduced” state. Since Noh et al. did not use the correct variances of the Susskind-Glogower and Pegg-Barnett phase operators when they compared their data with the results predicted through use of these operators, the correct expressions in that case are presented. Comparing the measured operators with the Hermitian phase operator in the extended Susskind-Glogower formalism, we show that the quantum phase given by the Hermitian phase operator is more appropriate than the measured phases. The variance of the Hermitian operator in the vacuum is consistent with the complementarity of the photon number and phase, which implies that the vacuum has a uniform phase distribution. Hradil's measured operator does not have such a property. Although the Noh-Fougères-Mandel operator has such a property in a particular limit, this operator does not represent, in principle, a meaningful measure of the phase measurement in a typical quantum region, because the Bargmann distance becomes large in such a region.