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arXiv:1409.8238 · Journal of Physics Conference Series

Supersymmetry and non-Abelian geometric phase for a free particle on a circle with point-like interactions

Satoshi Ohya

Published Nov 26, 2014• 0 citations
arXivHTMLPDF

Abstract

Though not so widely appreciated in the literature, supersymmetric quantum mechanics provides an ideal playground for studying non-Abelian geometric phase, because supersymmetry always guarantees degeneracies in energy levels. In this paper we first present a simple supersymmetric model for a free particle on a circle with point-like interactions that exhibits N = 2 supersymmetry and doubly degenerate energy levels. We then show that Berry's connection in this model is given by the Wu-Yang-like magnetic monopole in SU (2) Yang-Mills gauge theory. This article is largely based on our recent work [1].

DOI
10.1088/1742-6596/563/1/012021

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Though not so widely appreciated in the literature, supersymmetric quantum mechanics provides an ideal playground for studying non-Abelian geometric phase, because supersymmetry always guarantees degeneracies in energy levels. In this paper we first present a simple supersymmetric model for a free particle on a circle with point-like interactions that exhibits N = 2 supersymmetry and doubly degenerate energy levels. We then show that Berry's connection in this model is given by the Wu-Yang-like magnetic monopole in SU (2) Yang-Mills gauge theory. This article is largely based on our recent work [1].
We generalise the study of constraints imposed by supersymmetry on the Berry connection to transformations with component fields in representations of an internal symmetry group G. Since the fields act as co-ordinates of the underlying space one finds a non-trivial extension to its structure and, correspondingly, there are new non-abelian constraints on the Berry connection. The specific case of G=SU(2) is shown to constrain the connection to behave as a magnetic monopole over su(2), its Lie algebra.
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.
Abstract This work revisit on the intrinsic connection between Dirac monopole theory and Berry geometric phases by extending Dirac's theory to the parameter space. Using the simplest two-mode Hamiltonian model, we explicitly visualize Dirac strings with endpoints in the parameter space, demonstrating that these endpoints correspond to accidental degeneracy points of energy eigenvalues in Hermitian systems. We show that non-integrable phase factors, induced by such Dirac strings, directly give rise to the well-known Berry connection and curvature, which can be re-derived by following the spirit of Dirac's monopole theory with the aid of the quantum adiabatic theorem. Our results indicate that the Berry geometric phase is essentially the non-integrable phase factor induced by Dirac strings with endpoints in the parameter space. These results further deepen our understanding on monopole and geometric phases, and would inspire more studies on them especially when Berry's framework is not applicable.
For pt.I see ibid., vol.25, p.1745 (1992). In the first paper the authors studied certain crucial mathematical properties of Abelian Berry's phases in two quantum systems with supersymmetrically related Hamiltonians. They follow up such investigation by extending their analysis to nonAbelian Berry's phases in this present study. They present the derivation of an explicit expression for the difference in the relevant nonAbelian Berry's connections. Also they have derived an expression for a connection one form, which contain the non-Abelian Berry's connection, and which is invariant in the two supersymmetrically related quantum systems. To illustrate their findings mentioned above, they take the example of a Hamiltonian expressible in spin quadrupole. In this example, with a particular choice of supersymmetric partner, they show that the two Berry's connection one forms of the two supersymmetrically related systems may be related by gauge transformation.
The Hamiltonians of two- and three-level atomic systems interacting with nearly resonant electro-magnetic fields are described. The geometric phase related to the σ z operator of a two-level system is analysed and it is shown that this phase is well defined also for an open circuit. For a closed circuit this phase becomes equivalent to twice the Berry phase. For three-level atomic system the eigenenergies of the Hamiltonian are obtained by solving a cubic equation. Simplified forms for these eigenenergies and a condition for degeneracy are obtained. For non-degenerate three-level system geometric phases related to Cartan subalgebra are analysed under the adiabatic approximation and their relations with the three-level Berry phases are clarified. We show how to use the general theory of non-Abelian connections to the Hamiltonian of a three-level atomic system which includes a degeneracy.
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.