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Homotrinuclear Spin Cluster with Orbital Degeneracy in a Magnetic Field: Algebraic Dynamic Studies of the Geometric Phase

Ji-Wen Chenga, Qin-Sheng Zhua, Xiao-Yu Kuanga, Shi-Xun Zhanga, Cai-Xia Zhanga

Published Jan 1, 2008• 0 citations
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Abstract

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.

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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 τˆ 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 =γ s /γ τ (γ s and γ τ are the magnetic momentums of ŝ and τ̂ 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.
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.
Using the method of dynamical algebras, the solution of a homotriparticle linear spin cluster (each particle with S=1∕2S=1∕2S=1∕2) in a rotating magnetic field is obtained. We derive degeneracy energy levels and each level's Berry phase of this system. The Berry phase as a function of $\ensuremath{\omega}$ and $\ensuremath{\theta}$ has been determined by using the relation between the Berry phase and the angular velocity $\ensuremath{\omega}$ of the rotating magnetic field as well as the angle $\ensuremath{\theta}$ between the magnetic field and ZZZ axis. We obtain the changing diagram of the Berry phase of the basis state.
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).
A homonuclear linear [Formula: see text] coupling spin cluster with the middle particle driven by an external time-dependent magnetic field is investigated by using the method of algebraic dynamics. The exact analytical solutions of the time-dependent Schrodinger equation of the spin cluster system are derived and employed to study the geometric phase. An alternative expression of the geometric phase in each eigenstate is obtained. It is shown that the geometric phase is related to the external magnetic-field parameter θ (the angle between the magnetic field and the Z axis) and the effective coupling strength Jn. Based on the relation, how the geometric phase depends on the coupling strength Jn in different reducible subspace is discussed.PACS Nos.: 33.20.Wr, 03.65.Fd, 03.65.Vf