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arXiv:quant-ph/0206004

Aharanov Anandan Phase for the Quasi Exactly Solvable Bose Systems

Anirban Pathak

Published Jun 2, 2002• 0 citations
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Abstract

An extended notion of quasi-exactly solvable potential model is used here to treat quasi exactly solvable (QES) Bose systems. We report an analytic expression for the Ahoronov Anandan non-adiabatic geometric phase for the QES Bose system in general. The generalized expression is then used to study some particular cases of physical interest and we observe that the geometric phase can be tuned.

DOI
10.48550/arxiv.quant-ph/0206004

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An extended notion of quasi-exactly solvable potential model is used here to treat quasi exactly solvable (QES) Bose systems. We report an analytic expression for the Ahoronov Anandan non-adiabatic geometric phase for the QES Bose system in general. The generalized expression is then used to study some particular cases of physical interest and we observe that the geometric phase can be tuned.
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-adiabatic counterpart (Aharonov and Anandan phase). In this paper, we analyze both the non-degenerate and degenerate geometric phase of Lipkin—Meskov—Glick type model, which has many application in Bose—Einstein condensates and entanglement theory. Furthermore, in order to calculate degenerate geometric phases, the Floquet theorem and decomposition of operator are generalized. And the general formula is achieved.
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.
Quasi-energy spectral series ( epsilon nu (h(cross)), Psi ( epsilon nu )) which, in the limit h(cross) to 0, correspond to stable motions of a classical system along closed phase trajectories are built up in terms of a quasi-classical approximation for the Schrodinger equation with an arbitrary T-periodic h(cross) -1 (pseudo)differential Hamilton operator. Using the procedure of splitting the quantum-mechanical phase into dynamic and geometric components, the "geometric" contribution of the Aharonov-Anandan phase gamma epsilon ( nu ) to the quasi-energy spectrum is calculated. It is shown that the gamma epsilon ( nu ) phase, in the adiabatic approximation, coincides with the Berry phase that corresponds to a cyclic evolution of a stable rest-point of a classical system. Some examples are considered.
By analyzing an exactly solvable model in the second quantized formulation which allows a unified treatment of adiabatic and non-adiabatic geometric phases, it is shown that the topology of the adiabatic Berry's phase, which is characterized by the singularity associated with possible level crossing, is trivial in a precise sense. This topology of the geometric phase is quite different from the topology of the Aharonov-Bohm effect, where the topology is specified by the external local gauge field and it is exact for the slow as well as for the fast motion of the electron.