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:cond-mat/0409754

Geometrical phase for a three-dimensional anisotropic quantum well

Vladimir A. Geyler, A. V. Shorokhov

Published Sep 30, 2004• 0 citations
arXivHTMLPDF

Abstract

A three-dimensional anisotropic quantum well placed in an adiabatically precessing uniform magnetic field is considered and an explicit formula for the Berry phase is obtained. To get the Berry phase, a purely algebraic algorithm of reducing a quadratic Hamiltonian to the canonical form via symplectic transformations of the phase space is presented.

DOI
10.48550/arxiv.cond-mat/0409754

Related papers

Geometric phase of a spin-1 2 particle coupled to a quantum vector operator

Pedro Aguilar, Chryssomalis Chryssomalakos, Edgar Guzmán-González

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.

arXiv:1609.03578•Modern Physics Letters A•May 21, 2016•2 citations

Berry phases for 3D Hartree-type equations with a quadratic potential and a uniform magnetic field

F. N. Litvinets, Alexander V. Shapovalov, Andrey Yu. Trifonov

A countable set of asymptotic space-localized solutions is constructed for a 3D Hartree-type equation with a quadratic potential by the complex germ method in the adiabatic approximation. The asymptotic parameter is 1/ T , where T ≫ 1 is the adiabatic evolution time. A generalization of the Berry phase of the linear Schrödinger equation is formulated for the Hartree-type equation. For the solutions constructed, the Berry phases are found in an explicit form.

arXiv:math-ph/0610076•Journal of Physics A Mathematical and Theoretical•Aug 21, 2007•0 citations

Orbital Aharonov-Anandan geometric phase for confined motion in a precessing magnetic field

David J. Fernández C., M. A. del Olmo, Mariano Santander

The authors obtain general formulae to calculate exactly the orbital Aharonov-Anandan phase for a bounded system interacting with a precessing magnetic field in the case where the binding potential is rotationally invariant around the axis of rotation of the magnetic field. The particular case of a quadratic potential is considered in full detail, and the connection with the stability problem is established. The adiabatic limit of the Aharonov-Anandan phase to obtain the corresponding Berry phase is also considered.

Journal of Physics A Mathematical and General•Dec 7, 1992•13 citations

Berry phases for quadratic spin Hamiltonians taken from atomic and solid state physics: examples of Abelian gauge fields not connected to physical particles

Claude C. Bouchiat

This paper contains an evaluation of the Berry phases associated with the following class of nonlinear spin Hamiltonians : H (B, n ) = γS S · B + γQ((S · n) 2 - S2/3), with B · n = 0. Examples of these Hamiltonians are given in Atomic and Solid State Physics. We compute exactly the Berry phases for S = 1, 3/2 and for S > 3/2, to second order with respect to the ratio γQ/(γS B ). The results involve loop integrals of an Abelian gauge field which can be considered as a generalization of the magnetic monopole field, the projective complex plane P2(C) playing the role of the sphere S2. In the case of the spin-1 the relation between the Aharonov and Anandan cyclic quantum phase and the Berry adiabatic phase is made explicit. The connection between the semiclassical limit of the Berry phase and the classical adiabatic Hannay angle is analyzed using the classical spin model recently proposed by Nielsen and Rohrlich.

Journal de physique•Jan 1, 1989•6 citations

Geometric phase related to point-interaction transport on a magnetic Lobachevsky plane

Sergio A Albeverio, Pavel Exner, Vladimir A. Geyler

We consider a charged quantum particle living in the Lobachevsky plane and interacting with a homogeneous magnetic field perpendicular to the plane and a point interaction which is transported adiabatically along a closed loop C in the plane. We show that the bound-state eigenfunction acquires at that the Berry phase equal to 2 times the number of the flux quanta through the area encircled by C. 1 Introduction The phenomena arising from a geometric phase called Berry phase [Ber] have been put in evidence in many quantum mechanical systems. Recently such a "Berry phase effect" has been observed in some magnetic systems [LSG, MHK]. Moreover, it was shown that the geometric phase can emerge even in a time-independent homogeneous magnetic field when a potential 1 well trapping a two-dimensional particle is transported along a closed loop. An example in which the potential is of zero range is worked out in [EG], where a formula was proved showing that in the absence of an additional conf...

•Jan 1, 2000•7 citations

1 A New Class of Adiabatic Cyclic States and Geometric Phases for Non-Hermitian Hamiltonians

For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field. Since the publication of Berry’s paper [1] on the adiabatic geometrical phase, the subject has undergone a rapid development. Berry’s adiabatic geometric phase for periodic Hermitian Hamiltonians with a discrete nondegenerate spectrum has been generalized to arbitrary changes of a quantum state. In particular, the conditions on the adiabaticity [2] and cyclicity [3, 4, 5] of the evolution, Hermiticity of the Hamiltonian [6], and degeneracy [7] and discreteness [8] of its spectrum have been lifted. Moreover, the classical [9] and relativistic [10] analogues of the geometric phase have been considered. The purpose of this note is to offer an alternative generalization of Berry’s treatment of the adiabatic geometric phase for a non-Hermitian parametric Hamiltonian H[R] with a nondegenerate discrete spectrum. The parameters R = (R1, · · ·, Rn) are assumed to be real ∗E-mail address:

•Jan 1, 1999•0 citations
A three-dimensional anisotropic quantum well placed in an adiabatically precessing uniform magnetic field is considered and an explicit formula for the Berry phase is obtained. To get the Berry phase, a purely algebraic algorithm of reducing a quadratic Hamiltonian to the canonical form via symplectic transformations of the phase space is presented.
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 authors obtain general formulae to calculate exactly the orbital Aharonov-Anandan phase for a bounded system interacting with a precessing magnetic field in the case where the binding potential is rotationally invariant around the axis of rotation of the magnetic field. The particular case of a quadratic potential is considered in full detail, and the connection with the stability problem is established. The adiabatic limit of the Aharonov-Anandan phase to obtain the corresponding Berry phase is also considered.
This paper contains an evaluation of the Berry phases associated with the following class of nonlinear spin Hamiltonians : H (B, n ) = γS S · B + γQ((S · n) 2 - S2/3), with B · n = 0. Examples of these Hamiltonians are given in Atomic and Solid State Physics. We compute exactly the Berry phases for S = 1, 3/2 and for S > 3/2, to second order with respect to the ratio γQ/(γS B ). The results involve loop integrals of an Abelian gauge field which can be considered as a generalization of the magnetic monopole field, the projective complex plane P2(C) playing the role of the sphere S2. In the case of the spin-1 the relation between the Aharonov and Anandan cyclic quantum phase and the Berry adiabatic phase is made explicit. The connection between the semiclassical limit of the Berry phase and the classical adiabatic Hannay angle is analyzed using the classical spin model recently proposed by Nielsen and Rohrlich.
We consider a charged quantum particle living in the Lobachevsky plane and interacting with a homogeneous magnetic field perpendicular to the plane and a point interaction which is transported adiabatically along a closed loop C in the plane. We show that the bound-state eigenfunction acquires at that the Berry phase equal to 2 times the number of the flux quanta through the area encircled by C. 1 Introduction The phenomena arising from a geometric phase called Berry phase [Ber] have been put in evidence in many quantum mechanical systems. Recently such a "Berry phase effect" has been observed in some magnetic systems [LSG, MHK]. Moreover, it was shown that the geometric phase can emerge even in a time-independent homogeneous magnetic field when a potential 1 well trapping a two-dimensional particle is transported along a closed loop. An example in which the potential is of zero range is worked out in [EG], where a formula was proved showing that in the absence of an additional conf...
For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field. Since the publication of Berry’s paper [1] on the adiabatic geometrical phase, the subject has undergone a rapid development. Berry’s adiabatic geometric phase for periodic Hermitian Hamiltonians with a discrete nondegenerate spectrum has been generalized to arbitrary changes of a quantum state. In particular, the conditions on the adiabaticity [2] and cyclicity [3, 4, 5] of the evolution, Hermiticity of the Hamiltonian [6], and degeneracy [7] and discreteness [8] of its spectrum have been lifted. Moreover, the classical [9] and relativistic [10] analogues of the geometric phase have been considered. The purpose of this note is to offer an alternative generalization of Berry’s treatment of the adiabatic geometric phase for a non-Hermitian parametric Hamiltonian H[R] with a nondegenerate discrete spectrum. The parameters R = (R1, · · ·, Rn) are assumed to be real ∗E-mail address: