Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0607130 · Journal of Physics A Mathematical and Theoretical

Topological structures of adiabatic phase for multi-level quantum systems

Zheng-Xin Liu, Xiao-Ting Zhou, Xin Liu, Xiong-Jun Liu, Jing-Ling Chen

Published Jan 30, 2007• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/40/7/015

Related papers

Topological structures of adiabatic phase

Zheng-Xin Liu, Chen, J L, Xin Liu +2 more

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.

•Jul 19, 2006•0 citations

Geometry of mixed states and degeneracy structure of geometric phases for multi-level quantum systems. A unitary group approach

Elisa Ercolessi, Giuseppe Marmo, G. Mor +1 more

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.

•May 3, 2001•22 citations

GEOMETRY OF MIXED STATES AND DEGENERACY STRUCTURE OF GEOMETRIC PHASES FOR MULTI-LEVEL QUANTUM SYSTEMS: A UNITARY GROUP APPROACH

E. ERCOLESSI, G. MARMO, G. MORANDI +1 more

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.

arXiv:quant-ph/0105007•International Journal of Modern Physics A•Dec 20, 2001•25 citations

Quantum simulation of gravitational-like waves in minisuperspace with an artificial qubit

Ze-Lin Zhang, Mingfeng Chen, Huai‐Zhi Wu +1 more

On the background of the Born-Oppenheimer (adiabatic) approximation, we investigate the geometrical and topological structure in the theory of quantum gravity by using the path integral method and half-classical approximation. As we know, Berry curvature can be extracted from the linear response of a driven two-level system to nonadiabatic manipulations of its Hamiltonian. In parameter space of the Hamiltonian, magnetic monopoles can be artificially simulated. Ripples occur in Hilbert space when the monopole travels from inside to outside the surface of energy manifold spanned by system parameters. From this point of view, we set up the connection between the ripples characterized by the fidelity of quantum states in Hilbert space and gravitational-like waves in minisuperspace. This might open a window for the study of geometrical and topological properties of quantum gravity with the help of quantum physical systems.

arXiv:1611.01592•Physical review. D/Physical review. D.•Feb 21, 2017•7 citations

Magnetic monopoles in quantum adiabatic dynamics and the immersion property of the control manifold

David Viennot

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.

Journal of Mathematical Physics•Sep 1, 2006•5 citations

Time reversal symmetry and the geometric phase

Mattias Johnsson

This thesis examines the quantum-mechanical geometric phase with a view toward time reversal symmetry considerations. The idea of time reversal in quantum mechanics is investigated, disagreements and inconsistencies in the literature are examined, and the action of the time reversal operator is extended to time-dependent Hamiltonians. With this background, and using a definition of time reversal symmetry based on the evolution operator, I demonstrate that the existence of a non-zero geometric phase can in all cases be attributed to a breakdown of time reversal symmetry in some form. This result holds for both adiabatic and nonadiabatic evolutions, and for arbitrary dimensional parameter spaces. I explore the role of the geometric phase in a two-level Kramers system described by a parameter-dependent Hamiltonian such that the two levels can become degenerate for some value of the parameters, and discuss, from a mathematical point of view, the monopole geometric potential that results. I then extend this analysis by considering a pair of Kramers doublets, each doublet degenerate due to time reversal symmetry, where the parameters can be chosen so that each of the pair of doublets becomes degenerate with the other. I find the explicit functional forms for the two resulting nonabelian geometric gauge potentials and show that they can be identified exactly with the only two gauge-inequivalent SU(2) monopole potentials of Yang. Furthermore, following a conformal transformation these potentials can be mapped to those of the SU(2) instanton/anti-instanton pair. Finally I examine the relevance of the geometric phase to the molecular physics of time-odd systems. Time-odd coupling in molecular physics is a much under-studied area, with many potentially interesting results. Specifically I study time-odd coupling in Jahn-Teller systems under the Born-Oppenheimer approximation, where the electronic position states are coupled to the lattice momentum rather than the usual time-even lattice position. As an example I solve the E ⊗ (b₁ ⊕ b₂ ⊕ a₂) Jahn-Teller system exactly, showing that once again monopole-like geometric potentials arise, and comment on how this affects the angular momentum of the lattice subsystem.

•Jan 1, 1998•0 citations
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.
On the background of the Born-Oppenheimer (adiabatic) approximation, we investigate the geometrical and topological structure in the theory of quantum gravity by using the path integral method and half-classical approximation. As we know, Berry curvature can be extracted from the linear response of a driven two-level system to nonadiabatic manipulations of its Hamiltonian. In parameter space of the Hamiltonian, magnetic monopoles can be artificially simulated. Ripples occur in Hilbert space when the monopole travels from inside to outside the surface of energy manifold spanned by system parameters. From this point of view, we set up the connection between the ripples characterized by the fidelity of quantum states in Hilbert space and gravitational-like waves in minisuperspace. This might open a window for the study of geometrical and topological properties of quantum gravity with the help of quantum physical systems.
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.
This thesis examines the quantum-mechanical geometric phase with a view toward time reversal symmetry considerations. The idea of time reversal in quantum mechanics is investigated, disagreements and inconsistencies in the literature are examined, and the action of the time reversal operator is extended to time-dependent Hamiltonians. With this background, and using a definition of time reversal symmetry based on the evolution operator, I demonstrate that the existence of a non-zero geometric phase can in all cases be attributed to a breakdown of time reversal symmetry in some form. This result holds for both adiabatic and nonadiabatic evolutions, and for arbitrary dimensional parameter spaces. I explore the role of the geometric phase in a two-level Kramers system described by a parameter-dependent Hamiltonian such that the two levels can become degenerate for some value of the parameters, and discuss, from a mathematical point of view, the monopole geometric potential that results. I then extend this analysis by considering a pair of Kramers doublets, each doublet degenerate due to time reversal symmetry, where the parameters can be chosen so that each of the pair of doublets becomes degenerate with the other. I find the explicit functional forms for the two resulting nonabelian geometric gauge potentials and show that they can be identified exactly with the only two gauge-inequivalent SU(2) monopole potentials of Yang. Furthermore, following a conformal transformation these potentials can be mapped to those of the SU(2) instanton/anti-instanton pair. Finally I examine the relevance of the geometric phase to the molecular physics of time-odd systems. Time-odd coupling in molecular physics is a much under-studied area, with many potentially interesting results. Specifically I study time-odd coupling in Jahn-Teller systems under the Born-Oppenheimer approximation, where the electronic position states are coupled to the lattice momentum rather than the usual time-even lattice position. As an example I solve the E ⊗ (b₁ ⊕ b₂ ⊕ a₂) Jahn-Teller system exactly, showing that once again monopole-like geometric potentials arise, and comment on how this affects the angular momentum of the lattice subsystem.