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arXiv:1207.6497 · Journal of Physics A Mathematical and Theoretical

Spin in a general time-varying magnetic field: generalization of the adiabatic factorization of time evolution

J. Chee

Published Jul 19, 2010• 2 citations
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Abstract

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.

DOI
10.1088/1751-8113/43/34/345301

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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.
The quantum adiabatic theorem incorporating the Berry phase phenomenon can be characterized as a factorization of the time evolution operator into a path-dependent geometric factor, a usual dynamical factor and a non-adiabatic factor that approaches the identity in the adiabatic limit. We study a case where all these three factors can be constructed explicitly and where the instantaneous Hamiltonian has infinitely degenerate energy eigenstates associated with magnetic translation symmetry. Significance of the non-adiabatic factor in terms of transition probabilities is discussed.
For a spin subjected to an adiabatically changing magnetic field, the solid angle result as embodied by a rotation operator is the only path-dependent factor in the quantum evolution operator. For a charged particle, the infinite degeneracy calls for a rigorous investigation. We find that in this case, it is the product of the rotation operator and a magnetic translation operator that is the path-dependent factor in the quantum evolution operator. This result agrees with the fact that the instantaneous hamiltonian is invariant under magnetic translation as well as rotation. Simple experimental verification is proposed.
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.
A precise definition of an adiabaticity parameter \ensuremath{\nu} of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator U(\ensuremath{\tau})=${\ensuremath{\sum}}_{\mathrm{\ensuremath{\ell}}}$${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) with ${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) being at least of the order ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\ell}}}$. In particular, U(0){\mathrm{U}}^{(0)}U(0)(\ensuremath{\tau}) corresponds to the adiabatic approximation and yields Berry's adiabatic phase. It is shown that this series expansion has nothing to do with the 1/\ensuremath{\tau} expansion of U(\ensuremath{\tau}). It is also shown that the nonadiabatic part of the evolution operator is generated by a transformed Hamiltonian which is off-diagonal in the eigenbasis of the initial Hamiltonian. This suggests the introduction of an adiabatic product expansion for U(\ensuremath{\tau}) which turns out to yield exact expressions for U(\ensuremath{\tau}) for a large number of quantum systems. In particular, a simple application of the adiabatic product expansion is used to show that for the Hamiltonian describing the dynamics of a magnetic dipole in an arbitrarily changing magnetic field, there exists another Hamiltonian with the same eigenvectors for which the Schr\"odinger equation is exactly solvable. Some related issues concerning geometric phases and their physical significance are also discussed.
We examine spinor evolutions in a general time-dependent magnetic field B. We derive a condition the B variation must satisfy in order that the effective field direction ${\mathbf{r}\mathbf{\^{}}}_{\mathrm{B}}^{\ensuremath{'}}$ for a frame rotating with B becomes stationary in that frame. This condition is both sufficient and necessary for B to bring about a spinor evolution which exactly maintains a fixed angle between the spin and B. The adiabatic evolution used by Berry is just a special case, albeit only approximate, of these fixed-angle evolutions. To the other extreme of the fixed-angle evolutions, the nontrivial parallel transportation occurs, which is exact, necessarily nonadiabatic, and yields a pure geometric phase. Hence an experiment aimed at observing the geometric phase should seek to effect not an adiabatic evolution but a parallel transportation.