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Europhysics Letters (EPL)

The Berry phase of spin-j particle in q-deformed magnetic field and Wigner D-function

Guijiao Du, K. Xue, Chengcheng Zhou

Published Jan 1, 2016• 3 citations
Publisher

Abstract

We investigate the Berry phase for quantum spin system in q -deformed magnetic field. Through the q -deformed Lie algebra, we found a set of magnetic field expressions which contain the parameter q . The Berry phase of the system could be adjusted by the parameter q . The corresponding general formula of the q -deformed Berry phase is presented for an arbitrary spin system. The same Hamiltonian can also be generated from the Wigner D -function which satisfies the Yang-Baxter equation.

DOI
10.1209/0295-5075/113/20002

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We investigate the Berry phase for quantum spin system in q -deformed magnetic field. Through the q -deformed Lie algebra, we found a set of magnetic field expressions which contain the parameter q . The Berry phase of the system could be adjusted by the parameter q . The corresponding general formula of the q -deformed Berry phase is presented for an arbitrary spin system. The same Hamiltonian can also be generated from the Wigner D -function which satisfies the Yang-Baxter equation.
We study the properties of the Berry phase in quantum systems where momentum and position operators satisfy the R -deformed Heisenberg relations and the noninteger dimension d and angular momentum ℓ appear as parameters. We focus on the dynamics of the generalized time-dependent harmonic oscillator which satisfies the Schrödinger wave mechanics formulated in arbitrary dimensions. We also examine quantum systems which can be expressed as linear combination of generators with time-dependent coefficients in arbitrary dimensions. We show the explicit dependence of the Berry phase on d and the parity ℓ of quantum systems which satisfy the R -deformed Heisenberg relations. Using an explicit relation of the Fubini–Study metric in fractional dimensions, we highlight a geometrical interpretation of d and ℓ of systems which are quantized in arbitrary dimensions.
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.
The time-dependent Schr\"odinger equation for a spin particle in a rotating magnetic field is solved analytically by the cranking method, and the exact solutions are employed to study the nonadiabatic Berry's phase. An alternative expression for Berry's phase is given, which shows that Berry's phase is related to the expectation value of spin along the rotating axis and gives Berry's phase a physical explanation besides its gauge geometric interpretation. This expression also presents a simple algorithm for calculating the nonadiabatic Berry's phase for Hamiltonians that are nonlinear functions of the SU(2) generators. It is shown that nonadiabaticity alters the time evolution ray and in turn changes its Berry's phase. For the SU(2) dynamical group, the nonadiabatic effect on Berry's phase manifests itself as spin alignment (a phenomenon in nuclear physics), and spin-alignment quantization (observed recently in high-spin nuclear physics) is related to Berry's-phase quantization.
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.