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Physical Review A

Non-Abelian phase and geometric force in a quantum-classical hybrid system

Yang Liu, Yanan Zhang, X. X. Yi, H. D. Liu

Published May 2, 2024• 3 citations
Publisher

Abstract

It is well known that the dynamics of a hybrid system, which consists of a quantum subsystem and a classical subsystem, can be studied by transforming the quantum subsystem into a classical one. Based on our previous work [Phys. Rev. A 102, 032213 (2020)], we present the formalism of the quantum-classical hybrid system containing a quantum subsystem with energy degeneracy. By simulating the non-Abelian dynamics and geometric phases for the quantum subsystem with classical non-Abelian dynamics and geometric angles, the effect of the non-Abelian gauge field and the geometric phase of the quantum (sub)system on the dynamics of the classical subsystem is revealed by a simple canonical transformation. Remarkably, the non-Abelian angle geometric vector potential is found not only to induce the Wilczek-Zee phase in the dynamics of the quantum subsystem but also to give rise to a Lorentz-like non-Abelian geometric force. To illustrate the dynamics of the hybrid system, we examine the interaction between a spin-half particle and a magnetic field coupled with a magnetic particle. The explicit expression of the non-Abelian magneticlike gauge field is presented, introducing a different form of non-Abelian geometric force in the realm of quantum-classical dynamics.

DOI
10.1103/physreva.109.052203

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It is well known that the dynamics of a hybrid system, which consists of a quantum subsystem and a classical subsystem, can be studied by transforming the quantum subsystem into a classical one. Based on our previous work [Phys. Rev. A 102, 032213 (2020)], we present the formalism of the quantum-classical hybrid system containing a quantum subsystem with energy degeneracy. By simulating the non-Abelian dynamics and geometric phases for the quantum subsystem with classical non-Abelian dynamics and geometric angles, the effect of the non-Abelian gauge field and the geometric phase of the quantum (sub)system on the dynamics of the classical subsystem is revealed by a simple canonical transformation. Remarkably, the non-Abelian angle geometric vector potential is found not only to induce the Wilczek-Zee phase in the dynamics of the quantum subsystem but also to give rise to a Lorentz-like non-Abelian geometric force. To illustrate the dynamics of the hybrid system, we examine the interaction between a spin-half particle and a magnetic field coupled with a magnetic particle. The explicit expression of the non-Abelian magneticlike gauge field is presented, introducing a different form of non-Abelian geometric force in the realm of quantum-classical dynamics.
We present a general theoretical framework for a hybrid system that is composed of a quantum subsystem and a classical subsystem. We approach such a system with a simple canonical transformation which is particularly effective when the quantum subsystem is dynamically much faster than the classical counterpart, which is commonly the case in hybrid systems. Moreover, this canonical transformation generates a vector potential which, on one hand, gives rise to the familiar Berry phase in the fast quantum dynamics and, on the other hand, yields a Lorentz-like geometric force in the slow classical dynamics.
It is known that the dynamics and geometric phase of a quantum system could be mapped mathematically into a classical system of coupled oscillators without loss of physics. In this work, we show that this quantum-classical mapping can also be used to simulate non-Abelian dynamics and phase of the quantum system with energy degeneracy by interpreting the quantum degeneracy as the resonance between the eigenfrequencies of the classical oscillators. The quantum evolution and Wilczek-Zee phase in the degenerate subspace can also be described by the classical non-Abelian evolution and a non-Abelian geometric angle in the resonant subspace. By studying the adiabatic evolution in this classical non-Abelian case, our results can also be used to generalize the concept of averaging principle and Hannay's angle into the classical system with resonant eigenfrequencies. The quantum-classical mapping of a tripod-scheme Hamiltonian with two degenerate dark states is employed to illustrate the classical non-Abelian dynamics and geometric angles.
The vector potential and Berry-phase-induced force in a dissipative hybrid system are studied in this paper. The hybrid system is a compound of a classical magnetic particle and a quantum single spin. Two cases are considered. In the first case, we investigate the effect of the dissipative quantum subsystem on the motion of its classical partner, whereas in the second case, we show how the dynamics of the quantum single spin are affected by the dissipation of the classical particle. Extension to general dissipative hybrid systems is discussed.
Abstract Quantum states can acquire a geometric phase called the Berry phase after adiabatically traversing a closed loop, which depends on the path not the rate of motion. The Berry phase is analogous to the Aharonov–Bohm phase derived from the electromagnetic vector potential, and can be expressed in terms of an Abelian gauge potential called the Berry connection. Wilczek and Zee extended this concept to include non-Abelian phases—characterized by the gauge-independent Wilson loop—resulting from non-Abelian gauge potentials. Using an atomic Bose–Einstein condensate, we quantum-engineered a non-Abelian SU(2) gauge field, generated by a Yang monopole located at the origin of a 5-dimensional parameter space. By slowly encircling the monopole, we characterized the Wilczek–Zee phase in terms of the Wilson loop, that depended on the solid-angle subtended by the encircling path: a generalization of Stokes’ theorem. This observation marks the observation of the Wilson loop resulting from a non-Abelian point source.
We calculate Berry’s phase when the driving field, to which a spin-[Formula: see text] is coupled adiabatically, rather than the familiar classical magnetic field, is a quantum vector operator, of noncommuting, in general, components, e.g. the angular momentum of another particle, or another spin. The geometric phase of the entire system, spin plus “quantum driving field”, is first computed, and is then subdivided into the two subsystems, using the Schmidt decomposition of the total wave function — the resulting expression shows a marked, purely quantum effect, involving the commutator of the field components. We also compute the corresponding mean “classical” phase, involving a precessing magnetic field in the presence of noise, up to terms quadratic in the noise amplitude — the results are shown to be in excellent agreement with numerical simulations in the literature. Subtleties in the relation between the quantum and classical case are pointed out, while three concrete examples illustrate the scope and internal consistency of our treatment.
The dynamics of observables which are matrices depending on ℏ and taking values in classical phase-space is defined by retaining the terms up to the first order in ℏ of the Moyal bracket. Within this semiclassical approach a first-order Lagrangian involving gauge fields is studied as a constrained Hamiltonian system. This provides a systematic study of spin dynamics in the presence of non-Abelian Berry gauge fields. We applied the method to various types of dynamical spin systems and clarified some persisting discussions. In particular employing the Berry gauge field which generates the Thomas precession, we calculated the force exerted on an electron in the external electric and magnetic fields. Moreover, a simple semiclassical formulation of the spin Hall effect is accomplished.