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Chinese Physics

Berry phase of coupled two arbitrary spins in a time-varying magnetic field

许长谭, 贺明明, Chen Gang

Published May 1, 2006• 4 citations
Publisher

Abstract

In this paper, we investigate the Berry phase of two coupled arbitrary spins driven by a time-varying magnetic field where the Hamiltonian is explicitly time-dependent. Using a technique of time-dependent gauge transform the Berry phase and time-evolution operator are found explicitly in the adiabatic approximation. The general solutions for arbitrary spins are applied to the spin-1/2 system as an example of explanation.

DOI
10.1088/1009-1963/15/5/006

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•Jun 13, 2010•0 citations
In this paper, we investigate the Berry phase of two coupled arbitrary spins driven by a time-varying magnetic field where the Hamiltonian is explicitly time-dependent. Using a technique of time-dependent gauge transform the Berry phase and time-evolution operator are found explicitly in the adiabatic approximation. The general solutions for arbitrary spins are applied to the spin-1/2 system as an example of explanation.
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.
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 Berry phase in a composite system induced by the time-dependent interaction is discussed. We choose two coupled spin-1/2 systems as the composite system: one of the subsystems is subjected to a static magnetic field, and the coupling parameters between two spins are controllable in time. We show that the time-dependent interaction can induce the Berry phase in a similar way as that a spin-1/2 system (qubit) is driven by an effective time-dependent magnetic field.Furthermore, using two consecutive cycles with opposite directions of both the static magnetic field as well as opposite signs of the coupling parameters, a nontrivial two-qubit unitary transformation purely based on Berry phases can be constructed.
The Berry phase in a composite system induced by the time-dependent interaction is discussed. We choose two coupled spin-1/2 systems as the composite system: one of the subsystems is subjected to a static magnetic field, and the coupling parameters between two spins are controllable in time. We show that the time-dependent interaction can induce the Berry phase in a similar way as that a spin-1/2 system (qubit) is driven by an effective time-dependent magnetic field. Furthermore, using two consecutive cycles with opposite directions of both the static magnetic field as well as opposite signs of the coupling parameters, a nontrivial two-qubit unitary transformation purely based on Berry phases can be constructed.
We study the dynamics of a quantum system with a time-dependent Hamiltonian which is given by a linear combination of SU(1,1) generators and coupled with a heat bath. An effective Hamiltonian of this dissipative system is derived. With the help of a time-dependent gauge transformation, we obtain the exact solutions of the time-dependent Schrödinger equation, from which the time-evolution operator and the non-adiabatic and adiabatic Berry phases, which depend on time, are calculated explicitly. In the weak dissipation limit, an additional term besides the original Berry phase is found. The additional phase does not have a geometrical meaning due to the dissipation.
We derive the general formula giving the Berry phase for an arbitrary spin, having both magnetic-dipole and electric-quadrupole couplings with external time-dependent fields. We assume that the effective E and B fields remain orthogonal during the quantum cycles. This mild restriction has many advantages. It provides simple symmetries leading to selection rules and the Hamiltonian-parameter and density-matrix spaces coincide for S=1. This implies the identity of the Berry and Aharonov-Anandan phases, which is lost for S>1. We have found that new features of Berry phases emerge for integer spins>2. We provide explicit numerical results of Berry phases for S=2,3,4. We give a precise analysis of the non-adiabatic corrections. The accuracy for satisfying adiabaticity is greatly 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 < 0.1%. For quantum cycles,involving as sole periodic parameter the precession angle of E around B, the corrections odd upon the reversal of the angular velocity can be cancelled exactly if the quadrupole to dipole coupling ratio takes a "magic" value. The even ones are cancelled by subtraction of the phases relative to opposite velocities. As a possible application of the results of this paper we suggest a route to holonomic entanglement of N non-correlated 1/2-spins by performing adiabatic cycles governed by a Hamiltonian which is a non-linear function of the total spin operator S defined as the sum of the N spin operators. The case N=4 and Sz=1 is treated explicitly and maximum entanglement is achieved.