Formally exact solution and geometric phase for the spin-j system

Paul Bracken

A mathematical formalism for describing geometric phases is presented. A development of the geometric phase is given, which is valid for noncyclic nonadiabatic processes. The result is used to calculate exactly the geometric phase for a quantum system that is made up of a spin one-half system in a rotating magnetic field.

Bharath Hebbe Madhusudhana, Matthew Boguslawski, Maryrose Barrios +1 more

Xiaoyan Deng

Objective The algebriaic dynamics is applied to investigate the geometric phase of a Heisenberg-chain,in order to find out how the coulping strengths between spins influence the geometric phase of this system.Method An interacting anisotropic 3-spin 1/2 system,which is driven by a rotating magnetic field,is considered.The system has a su(2) algebraic structure.By an unitary gauge transformation,this nonautonomous system becomes an autonomous system.The Schrodinger equation of the autonomous system is solved.By virtue of the inverse transformation of the original gauge transformation,the exact analytical solution to the time dependent Schrodinger equation is obtained in total spin 1/2 subspace.Resutlt The geometric phases in this subspace are calculated.Conclusion The result shows that(i)Coulping strengths don't influence the geometric phases of this system;(ii) The geometric phases of the system is the same as that of one spin 1/2 system in a rotating magnetic field.

A. R. Kessel, Raoul Rashidovich Nigmatullin, Airat A. Khamzin +1 more

Yu You

Quantum invariant theory is an effective method to study geometric phase problems. By making use of the Lewis Riesenfeld invariant theory, a particle with spin j in an oscillating magnetic field is studied and the exact solutions for two different cyclic periods are found. It is then used to obtain two different geometric phases corresponding to two different cyclic periods. It is quite clear from our derivation that the geometric phases corresponding to two different cyclic periods are closely associated with the initial states of the system.

David J. Fernandez C., Óscar Rosas-Ortiz