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Communications in Theoretical Physics

Phase State as a Limiting Case of SU(1,1) Coherent State

Zhisong Yu, Fan Hong‐Yi

Published Jan 1, 2009• 1 citation
Publisher

Abstract

We show that the Susskind—Glogower phase state is a limiting case of a kind of SU(1,1) coherent states. By analogy, based on the bipartite entangled state representation (ESR) we demonstrate that an appropriate SU(1,1) coherent state composed of the two-mode unitary phase operator also leads to a new phase state in two-mode Fock space. is diagonalized in the ESR.

DOI
10.1088/0253-6102/51/1/14

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We show that the Susskind—Glogower phase state is a limiting case of a kind of SU(1,1) coherent states. By analogy, based on the bipartite entangled state representation (ESR) we demonstrate that an appropriate SU(1,1) coherent state composed of the two-mode unitary phase operator also leads to a new phase state in two-mode Fock space. is diagonalized in the ESR.
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We have constructed a new kind of two-mode bosonic realization of SU (1,1) Lie algebra, on the basis of which the SU (1,1) generalized coherent states in the two-mode Fock space are derived. These two-mode SU (1,1) coherent states, which are called uncorrelated two-mode SU (1,1) coherent states, include three special cases. For these states, we study the mean photon number distribution and their non-classical properties, which are photon anti-bunching, violations of Cauchy-Schwarz inequality and two-mode squeezing.
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.
We define a Susskind–Glogower operator that is unitary and acts as a true ladder operator for photons of a given frequency and any polarization. The operator acts on a Hilbert space consisting of all number states in both polarizations. The phase operator derived from this ladder operator satisfies the canonical commutation relations with the number operator. The eigenstates of this ladder operator are also found. Our method based on all the observable states enables us to write the operators for objects such as beam splitters, phase shifters or polarizers, as constant matrices; those used to study classical polarized beams. The Pegg–Barnett operator and the two-mode operator of Ban are special cases of our operator. We compared the phase state defined here with that obtained from the original Susskind–Glogower case, as well as that obtained by the method of Pegg and Barnett in the limit of a very small number of photons being present, and found that there are differences which may enable a direct experimental comparison of these operators.