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Physical Review A

Berry’s phase in the degenerate parametric amplifier

Christopher C. Gerry

Published Mar 1, 1989• 59 citations
Publisher

Abstract

We discuss the manifestation of Berry's phase in the degenerate parametric amplifier of quantum optics. Mainly we derive the Berry phase in the simplest case when the phase of the classical pump field varies adiabatically. Also we show that the effect of the phase may be discerned at all times in the photon-counting statistics of the output field.

DOI
10.1103/physreva.39.3204

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Berry's phase in the degenerate parametric amplifier

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We discuss the manifestation of Berry’s phase1 in connection with the degenerate parametric amplifier (DPA). Previously,2 Berry’s phase has been discussed in connection with a two-level atom interacting with a resonant electromagnetic field whose phase is varied adiabatically. In the present work we are basically replacirtg the compact system [SU(2) for the two-level atom] by a noncompact system [SU(1,1) for the DPA3].

Quantum Electronics and Laser Science Conference•Apr 24, 1989•0 citations

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As a non-adiabatic generalization of a recent work of Gerry, the manifestation of the Aharonov-Anandan phase in the degenerate parametric amplifier of quantum optics is discussed by making use of the Lewis-Riesenfeld quantum theory. We find the Lewis invariant for the system and study its evolution. The Aharonov-Anandan phase is then calculated and the connection between the Aharonov-Anandan phase difference and the average photon-number of the output field is obtained. Finally, the observable effect of the phase difference is investigated.

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Soviet Journal of Quantum Electronics•Dec 31, 1977•0 citations
We discuss the manifestation of Berry's phase in the degenerate parametric amplifier of quantum optics. Mainly we derive the Berry phase in the simplest case when the phase of the classical pump field varies adiabatically. Also we show that the effect of the phase may be discerned at all times in the photon-counting statistics of the output field.
As a non-adiabatic generalization of a recent work of Gerry, the manifestation of the Aharonov-Anandan phase in the degenerate parametric amplifier of quantum optics is discussed by making use of the Lewis-Riesenfeld quantum theory. We find the Lewis invariant for the system and study its evolution. The Aharonov-Anandan phase is then calculated and the connection between the Aharonov-Anandan phase difference and the average photon-number of the output field is obtained. Finally, the observable effect of the phase difference is investigated.
The Berry phases most often observed in experiments are produced by rotations. Berry phases produced by Lorentz transformations are almost as simple mathematically. They have been discussed at length, but there has been no suggestion of observing them in an experiment. Indeed, it has been shown that in one context they can be removed by a canonical transformation; thus they appear to be a property of the mathematical description that could not be observed physically. Here we show that the essential element of these Berry phases could be observed as a change of phase of the electromagnetic field in squeezed states of light or microwaves, as could be produced by degenerate parametric amplifiers. It could be seen as a fringe shift in interference experiments. In this way, a physical effect of the squeezing could be measured without measuring anything like photon statistics or noise.
We study the non-degenerate parametric amplifier problem from an algebraic approach of the SU(1,1) group. We write the Hamiltonian of this problem in terms of the boson generators of the SU(1,1) group and the difference operator. We apply the tilting transformation to our results to exactly solve this Hamiltonian and obtain its energy spectrum and eigenfunctions. Then, by assuming that our Hamiltonian is an explicit function of time, we calculate its Berry phase. Finally we obtain the Mandel Q -parameter of the photon numbers n a and n b .