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/9608031 · Journal of Physics A Mathematical and General

Non-Abelian geometric phase for general three-dimensional quantum systems

Ali Mostafazadeh

Published Nov 7, 1997• 9 citations
arXivHTMLPDF

Abstract

Adiabatic U(2) geometric phases are studied for arbitrary quantum systems with a three-dimensional Hilbert space. Necessary and sufficient conditions for the occurrence of the non-Abelian geometrical phases are obtained without actually solving the full eigenvalue problem for the instantaneous Hamiltonian. The parameter space of such systems which has the structure of is explicitly constructed. The results of this article are applicable for arbitrary multipole interaction Hamiltonians and their linear combinations for spin j = 1 systems. In particular it is shown that the nuclear quadrupole Hamiltonian does actually lead to non-Abelian geometric phases for j = 1. This system, being bosonic, is time-reversal invariant. Therefore, it cannot support Abelian adiabatic geometrical phases.

DOI
10.1088/0305-4470/30/21/023

Related papers

CONDITIONS FOR THE EXISTENCE OF NON-ZERO GEOMETRIC PHASE AND ITS FORMULA

Chun-Fang Li, Guo Guang-Can, 中国科学技术大学物理系,合肥230026

First we put forward an essential condition, for the existence of the non-zero adiabatic geometric phase of a quantum-mechanical Hamiltonian system, that the Hamiltonian operators at different times do not commute. Then it is shown that constraint =0 determines completely the phase relation of normalized eigenstate vectors │n' (t)> at different times. According to this property, we advance the sufficient and necessary condition for the existence of the non-zero adiabatic geometric phase, which is |n′(T)>≠|n′(0)> when =0. And also we derive a general time-integral formula for the adiabatic geometric phase along the line. Finally, as an application of it, we calculate the geometric phase of a spin 1/2 system once discussed by Solem and Biedenharn. It is pointed out that the problem encountered by them lies in the multi-valuedness of the eigenstate vector in parameter space.

Acta Physica Sinica•Jan 1, 1996•1 citation

Geometric Phases for Three State Systems

Mark Byrd

The adiabatic geometric phases for general three state systems are discussed. An explicit parameterization for space of states of these systems is given. The abelian and non-abelian connection one-forms or vector potentials that would appear in a three dimensional quantum system with adiabatic characteristics are given explicitly. This is done in terms of the Euler angle parameterization of SU(3) which enables a straight-forward calculation of these quantities and its immediate generalization.

arXiv:quant-ph/9902061•Feb 17, 1999•8 citations

Non-Abelian geometric phase, Floquet theory and periodic dynamical invariants

Ali Mostafazadeh

For a periodic Hamiltonian, periodic dynamical invariants may be used to obtain non-degenerate cyclic states. This observation is generalized to the degenerate cyclic states, and the relation between the periodic dynamical invariants and the Floquet decompositions of the time-evolution operator is elucidated. In particular, a necessary condition for the occurrence of cyclic non-adiabatic non-Abelian geometrical phase is derived. Degenerate cyclic states are obtained for a magnetic dipole interacting with a precessing magnetic field.

arXiv:quant-ph/9810064•Journal of Physics A Mathematical and General•Dec 11, 1998•17 citations

Noncyclic geometric phase and its non-Abelian generalization

Ali Mostafazadeh

We use the theory of dynamical invariants to yield a simple derivation of noncyclic analogues of the Abelian and non-Abelian geometric phases. This derivation relies only on the principle of gauge invariance and elucidates the existing definitions of the Abelian noncyclic geometric phase. We also discuss the adiabatic limit of the noncyclic geometric phase and compute the adiabatic non-Abelian noncyclic geometric phase for a spin-1 magnetic (or electric) quadrupole interacting with a precessing magnetic (electric) field.

arXiv:quant-ph/9909093•Journal of Physics A Mathematical and General•Nov 4, 1999•47 citations

Quaternionic gauge fields and the geometric phase

Péter Lévay

The quaternionic representation of the SU(2) non-Abelian, nonadiabatic geometric phase for Fermi systems with time reversal invariance is investigated. The underlying differential geometric structure originating from the Riemannian metric on HPn (the quaternionic projective space) is studied in detail. For two simple model Hamiltonians corresponding to the cases of adiabatic, and nonadiabatic cyclic evolutions, the gauge fields are shown to be identical with Yang’s SU(2) monopole solutions. This example of nonadiabatic cyclic evolution turns out to be useful in the context of Polyakov’s spin factors also. Employing bosonic degrees of freedom interacting with the fermionic ones, it is found that the gauge structures are also present in the bosonic effective action. However, this topological part of the effective action cannot solely be interpreted as a Wess–Zumino term unlike the one in the complex case.

Journal of Mathematical Physics•Sep 1, 1991•12 citations

Abelian and non-Abelian geometric phases in adiabatic open quantum systems

M. S. Sarandy, Daniel A. Lidar

We introduce a self-consistent framework for the analysis of both Abelian and non-Abelian geometric phases associated with open quantum systems, undergoing cyclic adiabatic evolution. We derive a general expression for geometric phases, based on an adiabatic approximation developed within an inherently open-systems approach. This expression provides a natural generalization of the analogous one for closed quantum systems, and we prove that it satisfies all the properties one might expect of a good definition of a geometric phase, including gauge invariance. A striking consequence is the emergence of a finite time interval for the observation of geometric phases. The formalism is illustrated via the canonical example of a spin-$1∕2$ particle in a time-dependent magnetic field. Remarkably, the geometric phase in this case is immune to dephasing and spontaneous emission in the renormalized Hamiltonian eigenstate basis. This result positively impacts holonomic quantum computing.

arXiv:quant-ph/0507012•Physical Review A•Jun 5, 2006•82 citations
Adiabatic U(2) geometric phases are studied for arbitrary quantum systems with a three-dimensional Hilbert space. Necessary and sufficient conditions for the occurrence of the non-Abelian geometrical phases are obtained without actually solving the full eigenvalue problem for the instantaneous Hamiltonian. The parameter space of such systems which has the structure of is explicitly constructed. The results of this article are applicable for arbitrary multipole interaction Hamiltonians and their linear combinations for spin j = 1 systems. In particular it is shown that the nuclear quadrupole Hamiltonian does actually lead to non-Abelian geometric phases for j = 1. This system, being bosonic, is time-reversal invariant. Therefore, it cannot support Abelian adiabatic geometrical phases.
First we put forward an essential condition, for the existence of the non-zero adiabatic geometric phase of a quantum-mechanical Hamiltonian system, that the Hamiltonian operators at different times do not commute. Then it is shown that constraint =0 determines completely the phase relation of normalized eigenstate vectors │n' (t)> at different times. According to this property, we advance the sufficient and necessary condition for the existence of the non-zero adiabatic geometric phase, which is |n′(T)>≠|n′(0)> when =0. And also we derive a general time-integral formula for the adiabatic geometric phase along the line. Finally, as an application of it, we calculate the geometric phase of a spin 1/2 system once discussed by Solem and Biedenharn. It is pointed out that the problem encountered by them lies in the multi-valuedness of the eigenstate vector in parameter space.
For a periodic Hamiltonian, periodic dynamical invariants may be used to obtain non-degenerate cyclic states. This observation is generalized to the degenerate cyclic states, and the relation between the periodic dynamical invariants and the Floquet decompositions of the time-evolution operator is elucidated. In particular, a necessary condition for the occurrence of cyclic non-adiabatic non-Abelian geometrical phase is derived. Degenerate cyclic states are obtained for a magnetic dipole interacting with a precessing magnetic field.
We use the theory of dynamical invariants to yield a simple derivation of noncyclic analogues of the Abelian and non-Abelian geometric phases. This derivation relies only on the principle of gauge invariance and elucidates the existing definitions of the Abelian noncyclic geometric phase. We also discuss the adiabatic limit of the noncyclic geometric phase and compute the adiabatic non-Abelian noncyclic geometric phase for a spin-1 magnetic (or electric) quadrupole interacting with a precessing magnetic (electric) field.
The quaternionic representation of the SU(2) non-Abelian, nonadiabatic geometric phase for Fermi systems with time reversal invariance is investigated. The underlying differential geometric structure originating from the Riemannian metric on HPn (the quaternionic projective space) is studied in detail. For two simple model Hamiltonians corresponding to the cases of adiabatic, and nonadiabatic cyclic evolutions, the gauge fields are shown to be identical with Yang’s SU(2) monopole solutions. This example of nonadiabatic cyclic evolution turns out to be useful in the context of Polyakov’s spin factors also. Employing bosonic degrees of freedom interacting with the fermionic ones, it is found that the gauge structures are also present in the bosonic effective action. However, this topological part of the effective action cannot solely be interpreted as a Wess–Zumino term unlike the one in the complex case.
We introduce a self-consistent framework for the analysis of both Abelian and non-Abelian geometric phases associated with open quantum systems, undergoing cyclic adiabatic evolution. We derive a general expression for geometric phases, based on an adiabatic approximation developed within an inherently open-systems approach. This expression provides a natural generalization of the analogous one for closed quantum systems, and we prove that it satisfies all the properties one might expect of a good definition of a geometric phase, including gauge invariance. A striking consequence is the emergence of a finite time interval for the observation of geometric phases. The formalism is illustrated via the canonical example of a spin-1∕21∕21∕2 particle in a time-dependent magnetic field. Remarkably, the geometric phase in this case is immune to dephasing and spontaneous emission in the renormalized Hamiltonian eigenstate basis. This result positively impacts holonomic quantum computing.