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Physica Scripta

Aharonov–Anandan phases for spin–spin coupling in a rotating magnetic field

Mai-Lin Liang, Shun-Lin Shu, Bing Yuan

Published Jan 3, 2007• 4 citations
Publisher

Abstract

Aharonov–Anandan (AA) phases for two coupled spins in a rotating magnetic field are obtained. The conditions for the adiabatic limit are also derived. It is found that if the rotating field is confined to a plane and the total spin is an integer, the adiabatic phase or the Berry phase will be zero, but the AA phase will not. Unlike adiabatic evolution, the AA phase is not the sum of geometric phases for the subsystems.

DOI
10.1088/0031-8949/75/2/003

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arXiv:1006.2528•Journal of Physics A Mathematical and Theoretical•Oct 28, 2010•1 citation
Aharonov–Anandan (AA) phases for two coupled spins in a rotating magnetic field are obtained. The conditions for the adiabatic limit are also derived. It is found that if the rotating field is confined to a plane and the total spin is an integer, the adiabatic phase or the Berry phase will be zero, but the AA phase will not. Unlike adiabatic evolution, the AA phase is not the sum of geometric phases for the subsystems.
The authors obtain general formulae to calculate exactly the orbital Aharonov-Anandan phase for a bounded system interacting with a precessing magnetic field in the case where the binding potential is rotationally invariant around the axis of rotation of the magnetic field. The particular case of a quadratic potential is considered in full detail, and the connection with the stability problem is established. The adiabatic limit of the Aharonov-Anandan phase to obtain the corresponding Berry phase is also considered.
We show the existence of a nonadiabatic geometric phase, i.e., an Aharonov-Anandan (AA) phase, in the Aharonov-Casher (AC) topological interference effect in one-dimensional mesoscopic rings. We find the AC phase is the phase accumulated by the spin wave function during a cyclic evolution, and show it is the sum of a geometric AA phase and a dynamical phase. In the adiabatic limit, the AA phase becomes the spin-orbit Berry phase introduced by Aronov and Lyanda-Geller. By solving exactly the model of a quasi-one-dimensional ring formed by the 2DEG on a semiconductor heterostructure, we discuss the observability of the AA phase in the AC effect.
We investigate how to obtain a non-trivial geometric phase gate for a two-qubit spin chain, with Ising interaction in different magnetic fields. Indeed, one of the spins is driven by a time-varying rotating magnetic field, and the other is coupled with a static magnetic field in the direction of the rotation axis. This is an interesting problem both for the purpose of measuring the geometric phases and in quantum computing applications. It is shown that the static magnetic field does not change the adiabatic states of the system, and it does not affect the geometric phases, whereas it may be used to control the dynamic phases. In addition, by considering the exact two-spin adiabatic geometric phases, we find that a non-trivial two-spin unitary transformation, purely based on Berry phases, can be obtained by using two consecutive cycles with opposite directions of the magnetic fields, opposite signs of the interaction constant and the phase shift of the rotating magnetic field. In addition, in the non-adiabatic case, starting with a certain initial state, a cycle can be achieved and thus the Aharonov–Anandan phase is calculated.
We derive the general formula giving Berry's phase for an arbitrary spin, having both magnetic-dipole and electric-quadrupole couplings with external time-dependent fields. We assume that the 'effective' electric and magnetic fields remain orthogonal during the quantum cycles. This mild restriction has many advantages. It provides simple symmetries leading to useful selection rules. Furthermore, the Hamiltonian parameter space coincides with the density matrix space for a spin S = 1. This implies a mathematical identity between Berry's phase and the Aharonov–Anandan phase, which is lost for S > 1. We have found that new physical features of Berry's phases emerge for integer spins ⩾2. We provide explicit numerical results of Berry's phases for S = 2, 3, 4. For any spin, one easily finds well-defined regions of the parameter space where instantaneous eigenstates remain widely separated. We have decided to stay within them and not deal with non-Abelian Berry's phases. We present a thorough and precise analysis of the non-adiabatic corrections with separate treatment for periodic and non-periodic Hamiltonian parameters. In both cases, the accuracy for satisfying the adiabatic approximation within a definite time interval is considerably improved if one chooses for the time derivatives of the parameters a time-dependence having a Blackman pulse shape. This has the effect of taming the non-adiabatic oscillation corrections which could be generated by a linear ramping. For realistic experimental conditions, the non-adibatic corrections can be kept below the 0.1% level. For a class of quantum cycles, involving as a sole periodic parameter the precession angle of the electric field around the magnetic field, the corrections odd upon the reversal of the associated rotation velocity can be canceled exactly if the quadrupole to dipole coupling is chosen appropriately ('magic values'). The even ones are eliminated by taking the difference of the Berry phases generated by two 'mirror' cycles. We end by indicating a possible application of the theoretical tools developed in this paper. We propose a way to perform a holonomic entanglement of N non-correlated 1/2-spins by performing adiabatic cycles governed by a Hamiltonian given as a nonlinear function of the total spin operator S , defined as the sum of the N individual spin operators. The basic idea behind this proposal is the mathematical fact that any non-correlated states can be expanded into eigenstates of S 2 and S z . The same eigenvalues appear several times in the decomposition when N > 2 but all these states differ by their symmetry properties under the N -spin permutations. The case N = 4 and S z = 1 is treated explicitly and a maximum entanglement is achieved.