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Results in Physics

On the path of spin- 1/2 Berry phase

De‐Hone Lin, Dah-Wei Chiou

Published Aug 5, 2024• 0 citations
Publisher

Abstract

This paper investigates the relation between the Berry phase of a spinor accumulated along a path corresponding to an adiabatically varying magnetic field and the conformal deformation along the path in the parameter space. The first part demonstrates that the accumulated phase can be controlled by continuous deformation characterized by a single conformal parameter. The source of the corresponding field strength responsible for the deformed phase path is shown to be a conformally deformed monopole located at the origin in the parameter space with the fundamental strength g=±1/2. In the second part, we model a rotating magnetic field coupled to a charged spinor with a time-varying amplitude subjected to the conformal deformation, realizing a single-variable deformation of the evolution process of the phase path.

DOI
10.1016/j.rinp.2024.107907

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This paper investigates the relation between the Berry phase of a spinor accumulated along a path corresponding to an adiabatically varying magnetic field and the conformal deformation along the path in the parameter space. The first part demonstrates that the accumulated phase can be controlled by continuous deformation characterized by a single conformal parameter. The source of the corresponding field strength responsible for the deformed phase path is shown to be a conformally deformed monopole located at the origin in the parameter space with the fundamental strength g=±1/2. In the second part, we model a rotating magnetic field coupled to a charged spinor with a time-varying amplitude subjected to the conformal deformation, realizing a single-variable deformation of the evolution process of the phase path.
The effects of the conformal deformation of physical parameters on the measurement, Berry phase, and resonance of a spin are investigated. We first construct a form-invariant yet measurement-tunable representation of a spinor from the deformed parameter surface. Then, it is shown that the evolution path of the Berry phase is tunable by the conformal index. Third, the source of the path-tunable Berry phase is shown to be the conformal monopole revealed by the recent investigation of the transformation design of quantum states. The dual fields produced by the dual conformal monopole are presented. It is demonstrated that the strengths of the pole and its dual partner are determined by the reciprocally quantized conformal index. Finally, we discuss the influence of the deformation on the resonance of the spin, and show that the resonant frequency follows the quantization rule ω¯R=ωRconformalfactorn/n with ωR as the Rabi frequency without deformation.
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.
Berry's phase, which is associated with the slow cyclic motion with a finite period, looks like a Dirac monopole when seen from far away but smoothly changes to a dipole near the level crossing point in the parameter space in an exactly solvable model. This topology change of Berry's phase is visualized as a result of the lensing effect; the monopole supposed to be located at the level crossing point appears at the displaced point when the variables of the model deviate from the precisely adiabatic movement. The effective magnetic field generated by Berry's phase is determined by a simple geometrical consideration of the magnetic flux coming from the displaced Dirac monopole.
We investigate the dynamics of a two-component Bose-Einstein condensate with spin-orbit coupling numerically and analytically. Under the drive of a weak segmented rotational external field, we observe that the system exhibits cyclic soliton motion; however, in contrast to the predictions of quasi-particle theory, the trajectory of the soliton center shows a distinct drift. The underlying mechanism of this anomalous drift is revealed: the moving soliton experiences a Lorentz force induced by a virtual magnetic monopole field in momentum space. We further calculate the phase evolution of the soliton during this cyclic motion and find that its geometric component comprises both an adiabatic Berry phase and a nonadiabatic Aharonov-Anandan phase. Notably, the Berry phase can be expressed in terms of the magnetic flux of the aforementioned virtual monopole field. Our findings hold implications for geometric phase theory and experiments on two-component Bose-Einstein condensates, and may establish a novel link between quantum geometry and soliton dynamics.
We investigate the geometric phase or Berry phase acquired by a spin half which is both subject to a slowly varying magnetic field and weakly coupled to a dissipative environment (either quantum or classical). We study how this phase is modified by the environment and find that the modification is of a geometric nature. While the original Berry phase (for an isolated system) is the flux of a monopole field through the loop traversed by the magnetic field, the environment-induced modification of the phase is the flux of a quadrupolelike field. We find that the environment-induced phase is complex, and its imaginary part is a geometric contribution to dephasing. Its sign depends on the direction of the loop. Unlike the Berry phase, this geometric dephasing is gauge invariant for open paths of the magnetic field.