Geometric phase in quantum mechanics and calculation for spin one-half in a rotating magnetic field

2012 ◽  
Vol 90 (7) ◽  
pp. 605-609
Author(s):  
Paul Bracken
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
Quantum System ◽  
Geometric Phase ◽  
Rotating Magnetic Field ◽  
Mathematical Formalism ◽  
Geometric Phases

A mathematical formalism for describing geometric phases is presented. A development of the geometric phase is given, which is valid for noncyclic nonadiabatic processes. The result is used to calculate exactly the geometric phase for a quantum system that is made up of a spin one-half system in a rotating magnetic field.

scholarly journals Design of geometric phase gates and controlling the dynamic phase for a two-qubit Ising model in magnetic fields

2013 ◽  
Vol 469 (2153) ◽  
pp. 20120743 ◽  
Author(s):  
M. Amniat-Talab ◽  
H. Rangani Jahromi
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Static Magnetic Field ◽  
Geometric Phase ◽  
Rotation Axis ◽  
Interaction Constant ◽  
Rotating Magnetic Field ◽  
Initial State ◽  
Geometric Phases ◽  
Phase Gate

We investigate how to obtain a non-trivial geometric phase gate for a two-qubit spin chain, with Ising interaction in different magnetic fields. Indeed, one of the spins is driven by a time-varying rotating magnetic field, and the other is coupled with a static magnetic field in the direction of the rotation axis. This is an interesting problem both for the purpose of measuring the geometric phases and in quantum computing applications. It is shown that the static magnetic field does not change the adiabatic states of the system, and it does not affect the geometric phases, whereas it may be used to control the dynamic phases. In addition, by considering the exact two-spin adiabatic geometric phases, we find that a non-trivial two-spin unitary transformation, purely based on Berry phases, can be obtained by using two consecutive cycles with opposite directions of the magnetic fields, opposite signs of the interaction constant and the phase shift of the rotating magnetic field. In addition, in the non-adiabatic case, starting with a certain initial state, a cycle can be achieved and thus the Aharonov–Anandan phase is calculated.

scholarly journals A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics

2022 ◽  
Author(s):  
Eric J. Pap ◽  
◽  
Daniël Boer ◽  
Holger Waalkens ◽  
◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Geometric Phase ◽  
Principal Bundle ◽  
Natural Generalization ◽  
Energy Bands ◽  
Geometric Framework ◽  
Geometric Phases ◽  
Dynamical Phase ◽  
Natural Connection ◽  
Finite Dimensional

We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection yielding the (generalized) geometric phase. This bundle provides in addition a natural generalization of the quantum geometric tensor and derived tensors, and we show how it can incorporate the non-geometric dynamical phase as well. We finish by demonstrating how the bundle can be recast as a principal bundle, so that both the geometric phases and the permutations of eigenstates can be expressed simultaneously by means of standard holonomy theory.

Homotrinuclear Spin Cluster with Orbital Degeneracy in a Magnetic Field: Algebraic Dynamic Studies of the Geometric Phase January 25, 2008

2008 ◽  
Vol 63 (7-8) ◽  
pp. 405-411
Author(s):  
Ji-Wen Cheng ◽  
Qin-Sheng Zhu ◽  
Xiao-Yu Kuang ◽  
Shi-Xun Zhang ◽  
Cai-Xia Zhang
Keyword(s):  
Magnetic Field ◽  
Geometric Phase ◽  
Berry Phase ◽  
Energy Levels ◽  
Rotating Magnetic Field ◽  
Algebraic Dynamics ◽  
Orbital Degeneracy ◽  
Spin Cluster ◽  
Dynamic Studies ◽  
Adiabatic Energy

Based on the homotrinuclear spin cluster having SU(2)⊗SU(2) symmetry with twofold orbital degeneracy τ = 1/2) and the SU(2) algebraic structures of both ŝ and τˆ subspaces in the external magnetic field, we calculate exactly the non-adiabatic energy levels and the cyclic and non-cyclic non-adiabatic geometric phase of the homotrinuclear spin cluster by making use of the method of algebraic dynamics. The solution will show that the Berry phase is much influenced by the parameters N =γs/γτ (γs and γτ are the magnetic momentums of ŝ and τ̂ subspaces, respectively) in addition to ω/Ω in a rotating magnetic field. The change of the Berry phase in the basis state of the system is demonstrated from the changing diagram.

THE LINEAR NON-IRON INDUCTOR WITH ROTATING MAGNETIC FIELD

2018 ◽  
Vol 2018 (49) ◽  
pp. 39-50
Author(s):  
О. Karlov ◽  
◽  
I. Kondratenko ◽  
R. Kryshchuk ◽  
A. Rashchepkin ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Rotating Magnetic Field
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