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Topological structures of adiabatic phase

Zheng-Xin Liu, Chen, J L, Xin Liu, Xiong-Jun Liu, Zhou, X T

Published Jul 19, 2006• 0 citations
OpenAlex

Abstract

The topological properties of adiabatic gauge fields for multilevel systems are analyzed. It's known that the adiabatic gauge field for two level systems have a monopole structure. We mainly study the curvature of the adiabatic holonomy for three level systems and find that they have monopole (for non-degenerate level) or instanton-like (for degenerate level) structures.

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The topological properties of adiabatic gauge fields for multilevel systems are analyzed. It's known that the adiabatic gauge field for two level systems have a monopole structure. We mainly study the curvature of the adiabatic holonomy for three level systems and find that they have monopole (for non-degenerate level) or instanton-like (for degenerate level) structures.
The topological properties of adiabatic gauge fields for multi-level (three-level in particular) quantum systems are studied in detail. Similar to the result that the adiabatic gauge field for SU (2) systems (e.g. two-level quantum system or angular momentum systems, etc) has a monopole structure, the curvature 2-forms of the adiabatic holonomies for SU (3) three-level and SU (3) eight-level quantum systems are shown to have monopole-like (for all levels) or instanton-like (for the degenerate levels) structures.
We analyze the geometric aspects of unitary evolution of general states for a multilevel quantum system by exploiting the structure of coadjoint orbits in the unitary group Lie algebra. Using the same method in the case of SU(3) we study the effect of degeneracies on geometric phases for three-level systems. This is shown to lead to a highly nontrivial generalization of the result for two-level systems in which degeneracy results in a "monopole" structure in parameter space. The rich structures that arise are related to the geometry of adjoint orbits in SU(3). The limiting case of a two-level degeneracy in a three-level system is shown to lead to the known monopole structure.
We analyze the geometric aspects of unitary evolution of general states for a multilevel quantum system by exploiting the structure of coadjoint orbits in the unitary group Lie algebra. Using the same methods in the case of SU(3) we study the effect of degeneracies on geometric phases for three-level systems. This is shown to lead to a highly nontrivial generalization of the result for two-level systems in which degeneracy results in a "monopole" structure in parameter space. The rich structures that arise are related to the geometry of adjoint orbits in SU(3). The limiting case of a two-level degeneracy in a three-level system is shown to lead to the known monopole structure.
Recently, an effective non-Abelian magnetic field with a topology of a monopole was shown to emerge from the adiabatic motion of multilevel atoms in spatially varying laser fields [J. Ruseckas, Phys. Rev. Lett. 95, 010404 (2005)10.1103/PhysRevLett.95.010404]. We study this monopole in a Bose-Einstein condensate of degenerate dressed states and find that the topological charge of the pseudospin cancels the monopole charge resulting in a vanishing gauge invariant charge. As a function of the laser wavelength, different stationary states are classified in terms of their effect to the monopole part of the magnetic field and a crossover to vortex ground state is observed.
It is well known that the Berry phase of a cyclic adiabatic dynamical system appears formally as the flux of a magnetic field in the control parameter manifold. In this electromagnetic picture a level crossing appears as a Dirac magnetic monopole in this manifold. We make an extensive study of the magnetic monopole model of eigenvalue crossings. We show that the properties of the monopole magnetic field in the control manifold are determined by the immersion of the control manifold in a space given by the universal classifying theorem of fiber bundles. We give a detailed illustrative study of the simple but instructive case of a two level crossing of a system controlled by a two-dimensional manifold.
The second quantized approach to geometric phases is reviewed. The second quantization generally induces a hidden local (time-dependent) gauge symmetry. This gauge symmetry defines the parallel transport and holonomy, and thus it controls all the known geometric phases, either adiabatic or non-adiabatic, in a unified manner. The transitional region from the adiabatic to non-adiabatic phases is thus analyzed in a quantitative way. It is then shown that the topology of the adiabatic Berry's phase is trivial in a precise sense and also the adiabatic phase is rather fragile against the non-adiabatic deformation. In this formulation, the notion such as the projective Hilbert space does not appear.