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arXiv:1009.0411 · Communications in Theoretical Physics

Aharonov—Anandan Phases in Lipkin—Meskov—Glick Model

Da-Bao Yang, Jing‐Ling Chen

Published Aug 1, 2011• 2 citations
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Abstract

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.

DOI
10.1088/0253-6102/56/2/09

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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.
Operator decomposition approach is used to calculate the non-adiabatic geometric phase of anharmonic oscillator. As an example we focus on the isotonic oscillator, a type of anharmonic oscillator. The Aharonov-Anandan phase is derived when we choose the ground state and the first excited state as cyclic initial states. Then we generalize our result by choosing three states or more states as cyclic initial states. Finally, we give a general formula of Aharonov-Anandan phase for time-independent systems and discuss its applicability.
If the time evolution of a quantum state leads back to the initial state, a geometric phase is accumulated that is known as the Berry phase for adiabatic evolution or as the Aharonov-Anandan (AA) phase for nonadiabatic evolution. We evaluate these geometric phases using Floquet theory for systems in time-dependent external fields with a focus on paths leading through a degeneracy of the eigenenergies. Contrary to expectations, the low-frequency limits of the two phases do not always coincide. This happens as the degeneracy leads to a slow convergence of the quantum states to adiabaticity, resulting in a nonzero finite or divergent contribution to the AA phase. Steering the system adiabatically through a degeneracy provides control over the geometric phase as it can cause a π shift of the Berry phase. On the other hand, we revisit an example of degeneracy crossing proposed by AA. We find that, at suitable driving frequencies, both geometric-phase definitions give the same result and the dynamical phase is zero due to the symmetry of time evolution about the point of degeneracy, providing an advantageous setup for manipulation of quantum states.
The present paper analyses the relation between the theory of the time-dependent wave operator and the Berry phase concept. It is proved that the wave operator approach is consistent with the non-adiabatic (Aharonov–Anandan) Berry phase, given that the wave operator and the parallel transport commute. It is then demonstrated that the non-Abelian Aharonov–Anandan phase can be calculated by working inside a reduced active space in the framework of wave operator theory. Finally an adiabatic transport formula is derived in the wave operator context and the influence of this effective Hamiltonian theory on the Berry phase is analysed. The theoretical results concerning the non-adiabatic Berry phase are confirmed numerically by considering a photodissociation process in the framework of the generalized Floquet theory.