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Chinese Physics C

Calculation and Generalization of Non-Adiabatic Geometric Phase of Isotonic Oscillator with Operator Decomposition

Nan An

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

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.

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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.
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.
We construct an operator for the Aharonov-Anandan phase for time independent Hamiltonians. This operator is shown to generate the motion of cyclic quantum systems through an equation of evolution involving only geometric quantities, i.e. the distance between quantum states, the geometric phase and the total length of evolution. From this equation, we derive an operator for the Samuel and Bhandari phase (SB-phase) for non cyclic evolutions. Finally we show how the SB-phase can be used to construct an operator corresponding to a quantum clock which commutator with the Hamiltonian has a canonical expectation value.
A formalism is developed for calculating Berry phases for nonadiabatic time-periodic quantum systems when a dynamical invariant is known. It is found that, when the invariant is periodic and has a nondegenerate spectrum, this method allows a convenient way to obtain generalized Berry phases and the proper cyclic initial states. The method is applied to the generalized harmonic oscillator and the two-level system, where the invariant operators are explicitly constructed. Formulae for the conventional Berry's phases are readily obtained by taking the adiabatic limit of the exact results.