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Discrete Photodetection and Susskind-Glogower Phase Operators

Y. Ben-Aryeh

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

State reduction processes in different types of photodetection experiments are described by using different kinds of ladder operators. A special model of discrete photodetection is developed by the use of superoperators which are based on the Susskind-Glogower raising and lower operators. The possibility to realize experimentally the discrete photodetection scheme in a micromaser is discussed.

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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.

Physica Scripta•Oct 1, 2012•0 citations
State reduction processes in different types of photodetection experiments are described by using different kinds of ladder operators. A special model of discrete photodetection is developed by the use of superoperators which are based on the Susskind-Glogower raising and lower operators. The possibility to realize experimentally the discrete photodetection scheme in a micromaser is discussed.
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 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.
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.