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Acta Sinica Quantum Optica

Geometric and Dynamic Phases for a Spin-1 Particle in a Rotating Magnetic Field

Xuehua Yi

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

The geometric and dynamic phase for spin-1 particle in a rotating magnetic field are studied.We derive formulae of how to calculate the geometric and dynamic phases for spin-1 particle in adiabatic and nonadiabatic evolutions,and calculate the phases by using these formulae.Finally,we discuss three kinds.Berry phase of When considering ω1ω,the system is in adiabatic approximation,at this time geometric phase is Berry phase.

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•Jan 1, 2008•0 citations
The geometric and dynamic phase for spin-1 particle in a rotating magnetic field are studied.We derive formulae of how to calculate the geometric and dynamic phases for spin-1 particle in adiabatic and nonadiabatic evolutions,and calculate the phases by using these formulae.Finally,we discuss three kinds.Berry phase of When considering ω1ω,the system is in adiabatic approximation,at this time geometric phase is Berry phase.
The geometric phase, in particular the Berry phase, in an entangled state of five spin-1/2 particles is studied. A time-dependent magnetic field is applied to control the time evolution of the cluster. Using the method of algebraic dynamics, we calculate the non-adiabatic geometric phase or Berry phase and the degeneracy energy levels when the magnetic rotates around Z axis. Based on the exact analytical solutions, we show how the Berry phase of the entangled state of this cluster depends on the external magnetic field parameters ω (the angular velocity of the rotating magnetic field) and θ (the angle between the magnetic field and Z axis).
The time-dependent Schr\"odinger equation for a spin particle in a rotating magnetic field is solved analytically by the cranking method, and the exact solutions are employed to study the nonadiabatic Berry's phase. An alternative expression for Berry's phase is given, which shows that Berry's phase is related to the expectation value of spin along the rotating axis and gives Berry's phase a physical explanation besides its gauge geometric interpretation. This expression also presents a simple algorithm for calculating the nonadiabatic Berry's phase for Hamiltonians that are nonlinear functions of the SU(2) generators. It is shown that nonadiabaticity alters the time evolution ray and in turn changes its Berry's phase. For the SU(2) dynamical group, the nonadiabatic effect on Berry's phase manifests itself as spin alignment (a phenomenon in nuclear physics), and spin-alignment quantization (observed recently in high-spin nuclear physics) is related to Berry's-phase quantization.
In this paper we define a non-dynamical phase for a spin-1/2 particle in a rotating magnetic field in the non-adiabatic non-cyclic case, and this phase can be considered as a generalized Berry phase. We show that this phase reduces to the geometric Berry phase, in the adiabatic limit, up to a factor independent of the parameters of the system. We could add an arbitrary phase to the eigenstates of the Hamiltonian due to the gauge freedom. Then, we fix this arbitrary phase by comparing our Berry phase in the adiabatic limit with the Berry's result for the same system. Also, in the extreme non-adiabatic limit our Berry phase vanishes, modulo 2π2π2π, as expected. Although, our Berry phase is in general complex, it becomes real in the expected cases: the adiabatic limit, the extreme non-adiabatic limit, and the points at which the state of the system returns to its initial form, up to a phase factor. Therefore, this phase can be considered as a generalization of the Berry phase. Moreover, we investigate the relation between the value of the generalized Berry phase, the period of the states and the period of the Hamiltonian.
Based on the homotrinuclear spin cluster having SU(2)⊗SU(2) symmetry with twofold orbital degeneracy τ = 1/2) and the SU(2) algebraic structures of both ŝ and τ subspaces in the external magnetic field, we calculate exactly the non-adiabatic energy levels and the cyclic and non-cyclic non-adiabatic geometric phase of the homotrinuclear spin cluster by making use of the method of algebraic dynamics. The solution will show that the Berry phase is much influenced by the parameters N = γs/γτ (γs and γτ are the magnetic momentums of ŝ and τ subspaces, respectively) in addition to ω/Ω in a rotating magnetic field. The change of the Berry phase in the basis state of the system is demonstrated from the changing diagram.
The geometric phase for states of two spin-1/2 particles in rotating magnetic field is calculated, in particular, the noncyclic and cyclic non-adiabatic phases for the general case are explicitly derived and discussed. We find that the cyclic geometric phase for the entangled state can always be written as a sum of the phases of the two particles respectively; the same cannot be said for the noncyclic phase. We also investigate the geometric phase of mixed state of one particle in a biparticle system, and we find that the geometric phase for one subsystem of an entangled system is always affected by another subsystem of the entangled system.
Based on he homotrinuclear spin cluster having SU(2)circle times SU(2) symmetry With twofold orbital degeneracy tau = 1/2) and the SU(2) algebraic structures of both (s) over cap and (tau) over cap subspaces in the external magnetic field. We calculate exactly the non-adiabatic energy levels and the cyclic and non-cyclic non-adiabatic geometric phase of the homotrinuclear spin cluster by making use of the method of algebraic dynamics. The solution Will show that the Berry phase is much influenced by the parameters N = gamma(s)/gamma(tau) (gamma(s) and gamma(tau) are the magnetic momentums of (s) over cap and (tau) over cap subspaces, respectively) in addition to omega/Omega in a rotating magnetic field. The change of the Berry phase in the basis state of the system is demonstrated from the changing diagram.