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Physica Scripta

Geometric Phases and Probabilities for an Alkali Atom in a Strong Magnetic Field

Jing‐Bo Xu, Dao‐Xin Yao

Published Jun 1, 1999• 0 citations
Publisher

Abstract

A general formalism of time evolution and geometric phase for a system with a dynamical semisimple Lie algebra is presented. The formalism is then used to study an alkali atom in a strong time-dependent magnetic field. The non-adiabatic transition probability among the angular momentum states is also calculated. Finally, explicit expressions for the geometric phase and probability in particular case of a rotating magnetic field are given.

DOI
10.1238/physica.regular.059a00411

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A general formalism of time evolution and geometric phase for a system with a dynamical semisimple Lie algebra is presented. The formalism is then used to study an alkali atom in a strong time-dependent magnetic field. The non-adiabatic transition probability among the angular momentum states is also calculated. Finally, explicit expressions for the geometric phase and probability in particular case of a rotating magnetic field are given.
We study the valence electron of an alkaline atom or a general charged particle with arbitrary spin and with magnetic moment moving in a rotating magnetic field. By using a time-dependent unitary transformation, the Schr\"odinger equation with the time-dependent Hamiltonian can be reduced to a Schr\"odinger-like equation with a time-independent effective Hamiltonian. Eigenstates of the effective Hamiltonian correspond to cyclic solutions of the original Schr\"odinger equation. The nonadiabatic geometric phase of a cyclic solution can be expressed in terms of the expectation value of the component of the total angular momentum along the rotating axis, regardless of whether the solution is explicitly available. For the alkaline atomic electron and a strong magnetic field, the eigenvalue problem of the effective Hamiltonian is completely solved, and the geometric phase turns out to be a linear combination of two solid angles. For a weak magnetic field, the same problem is solved partly. For a general charged particle, the problem is solved approximately in a slowly rotating magnetic field, and the geometric phases are also calculated.
An extension of the adiabatic factorization of the time evolution operator is studied for spin in a general time-varying magnetic field B ( t ). When B ( t ) changes adiabatically, such a factorization reduces to the product of the geometric operator which embodies the Berry phase phenomenon and a usual dynamical operator. For a general time variation of B ( t ), there should be a non-adiabatic operator in the factorization that is responsible for non-adiabatic transitions. A simple and explicit expression for the instantaneous angular velocity of the non-adiabatic operator is derived. This is done in a way that is independent of any specific representation of spin. Two classes of simple conditions are given under which the non-adiabatic operator itself can be made explicit. These conditions are used to find new expressions for B ( t ) that allow the explicit construction of the time evolution operator.
It is well known that any cyclic solution of a spin 1/2 neutral particle moving in an arbitrary magnetic field has a nonadiabatic geometric phase proportional to the solid angle subtended by the trace of the spin. For neutral particles with higher spin, this is true for cyclic solutions with special initial conditions. For more general cyclic solutions, however, this does not hold. As an example, we consider the most general solutions of such particles moving in a rotating magnetic field. If the parameters of the system are appropriately chosen, all solutions are cyclic. The nonadiabatic geometric phase and the solid angle are both calculated explicitly. It turns out that the nonadiabatic geometric phase contains an extra term in addition to the one proportional to the solid angle. The extra term vanishes automatically for spin 1/2. For higher spin, however, it depends on the initial condition. We also consider the valence electron of an alkaline atom. For cyclic solutions with special initial conditions in an arbitrary strong magnetic field, we prove that the nonadiabatic geometric phase is a linear combination of the two solid angles subtended by the traces of the orbit and spin angular momenta. For more general cyclic solutions in a strong rotating magnetic field, the nonadiabatic geometric phase also contains extra terms in addition to the linear combination.
The algebriaic dynamics is applied to investigate the geometric phase of a Heisenberg-chain in a rotating magnetic field.It is a su(2) algebraically symmetric 3-spin12 system with partially anisotropic coupling.The exact analytical solution of the time dependent Schrdinger equation and the geometric phases of this system are obtained in terms of gauge transformation and algebriaic dynamics.The partially anisotropic coupling is then extended to a more general coupling and the adiabatic geometric phases of the extended system are calculated.We find that the coupling strengths have no impact on the adiabatic geometric phases of the system.