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Chinese Physics Letters

Aharonov-Anandan Phase in the Degenerate Parameteric Amplifier

Tiezheng Qian, Jing‐Bo Xu, Gao Xiaochun

Published Oct 1, 1991• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0256-307x/8/10/002

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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.
In this paper, the exact evolving state for a two-level quantum system is found by making use of the Lewis-Riesenfeld invariant theory. In principle, the correction to an arbitrary order in the adiabatic approximation parameter can be obtained from this exact evolving state. To the -th order in , it is shown that the exact evolving state reduces to the superadiabatic basis introduced by Berry recently. The Aharonov-Anandan phase and its adiabatic limit (Berry phase for the system) are also calculated.
The dynamics of a generalized non-degenerate optical parametric down-conversion interaction whose Hamiltonian includes an arbitrary time-dependent driving part and a two-mode coupled part is studied by adopting the Lewis–Riesenfeld invariant theory. The closed formulae for the evolution of the quantum states and the evolution operators of the system are obtained. It is shown that various generalized squeezed states arise naturally in the process, and the two-mode squeezed effect is independent of the driving part. An explicitly analytical solution of the Schrödinger equation is further derived as the classical generalized force acting on each mode and the coupling of the two modes both have harmonic time dependences. This solution is found to be in agreement with previous research in special cases.
In the system of several interacting spins,geometric phases have been researched intensively.However,the studies are mainly focused on the adiabatic case (Berry phase),so it is necessary for us to study the non-adiabaticcounterpart (Aharonov and Anandan phase).In this paper,we analyze both the non-degenerate and degenerate geometricphase of Lipkin-Meskov-Glick type model,which has many application in Bose-Einstein condensates and entanglementtheory.Furthermore,in order to calculate degenerate geometric phases,the Floquet theorem and decomposition ofoperator are generalized.And the general formula is achieved.