scholarly journals TWO COUPLED HARMONIC OSCILLATORS ON NONCOMMUTATIVE PLANE

2005 ◽  
Vol 20 (07) ◽  
pp. 1515-1529 ◽  
Author(s):  
AHMED JELLAL ◽  
MICHAEL SCHREIBER ◽  
EL HASSAN EL KINANI
Keyword(s):  
Ground State ◽  
Uncertainty Relation ◽  
Harmonic Oscillators ◽  
Mixing Angle ◽  
Coupled Harmonic Oscillators ◽  
First Case ◽  
Magnetic Length ◽  
Spatial Coordinates ◽  
Basic Features ◽  
Noncommutative Plane

We investigate a system of two coupled harmonic oscillators on the noncommutative plane [Formula: see text] by requiring that the spatial coordinates do not commute. We show that the system can be diagonalized by a suitable transformation, i.e. a rotation with a mixing angle α. The obtained eigenstates as well as the eigenvalues depend on the noncommutativity parameter θ. Focusing on the ground state wave function before the transformation, we calculate the density matrix ρ0(θ) and find that its traces Tr (ρ0(θ)) and [Formula: see text] are not affected by the noncommutativity. Evaluating the Wigner function on [Formula: see text] confirms this. The uncertainty relation is explicitly determined and found to depend on θ. For small values of θ, the relation is shifted by a θ2 term, which can be interpreted as a noncommutativity correction. The calculated entropy does not change with respect to the normal case. We consider the limits α = 0 and α = π/2. In the first case, by identifying θ to the squared magnetic length, one can recover basic features of the Hall system.

NONCOMMUTATIVE DESCRIPTION OF SPIN HALL EFFECT

2009 ◽  
Vol 06 (02) ◽  
pp. 343-360 ◽  
Author(s):  
AHMED JELLAL ◽  
RACHID HOUÇA
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
Hall Effect ◽  
High Frequency ◽  
Hall Conductivity ◽  
Spin Hall Effect ◽  
Harmonic Oscillators ◽  
Frequency Regime ◽  
Basic Features ◽  
Noncommutative Plane

We propose an approach based on a generalized quantum mechanics to deal with the basic features of the intrinsic spin Hall effect. This can be done by considering two decoupled harmonic oscillators on the noncommutative plane and evaluating the spin Hall conductivity. Focusing on the high frequency regime, we obtain a diagonalized Hamiltonian. After getting the corresponding spectrum, we show that there is a Hall conductivity without an external magnetic field, which is noncommutativity parameter θ-dependent. This allows us to make contact with the spin Hall effect and also give different interpretations. Fixing θ, one can recover three different approaches dealing with the phenomenon.

scholarly journals Ground state entanglement entropy for discrete-time two coupled harmonic oscillators

2017 ◽  
Vol 901 ◽  
pp. 012168
Author(s):  
W Kantayasakun ◽  
S Yoo-Kong ◽  
T Deesuwan ◽  
M Tanasittikosol ◽  
W Liewrian
Keyword(s):  
Discrete Time ◽  
Ground State ◽  
Entanglement Entropy ◽  
Harmonic Oscillators ◽  
Coupled Harmonic Oscillators

TIGHT-BINDING MODEL FOR THE QUANTUM LADDER IN A MAGNETIC FIELD

1996 ◽  
Vol 10 (28) ◽  
pp. 3827-3856 ◽  
Author(s):  
KAZUMOTO IGUCHI
Keyword(s):  
Magnetic Field ◽  
Magnetic Flux ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Tight Binding ◽  
Critical Value ◽  
Tight Binding Model ◽  
Binding Model ◽  
First Case

A tight-binding model is formulated for the calculation of the electronic structure and the ground state energy of the quantum ladder under a magnetic field, where the magnetic flux at the nth plaquette is given by ϕn. First, the theory is applied to obtain the electronic spectra of the quantum ladder models with particular magnetic fluxes such as uniform magnetic fluxes, ϕn=0 and 1/2, and the staggered magnetic flux, ϕn= (−1)n+1ϕ0. From these, it is found that as the effect of electron hopping between two chains—the anisotropy parameter r=ty/tx—is increased, there are a metal-semimetal transition at r=0 and a semimetal–semiconductor transition at r=2 in the first case, and metal-semiconductor transitions at r=0 in the second and third cases. These transitions are thought of as a new category of metal-insulator transition due to the hopping anisotropy of the system. Second, using the spectrum, the ground state energy is calculated in terms of the parameter r. It is found that the ground state energy in the first case diverges as r becomes arbitrarily large, while that in the second and third cases can have the single or double well structure with respect to r, where the system is stable at some critical value of r=rc and the transition between the single and double well structures is associated with whether tx is less than a critical value of txc. The latter cases are very reminiscent of physics in polyacetylene studied by Su, Schrieffer and Heeger.

Adaptive cluster synchronisation of coupled harmonic oscillators with multiple leaders

2013 ◽  
Vol 7 (5) ◽  
pp. 765-772 ◽  
Author(s):  
Housheng Su ◽  
Hongwei Wang ◽  
Michael Z. Q. Chen ◽  
Najl V. Valeyev ◽  
Xiaofan Wang
Keyword(s):  
Harmonic Oscillators ◽  
Coupled Harmonic Oscillators ◽  
Multiple Leaders

scholarly journals Group synchronization of diffusively coupled harmonic oscillators

Kybernetika ◽  
2016 ◽  
pp. 629-647 ◽  
Author(s):  
Liyun Zhao ◽  
Jun Liu ◽  
Lan Xiang ◽  
Jin Zhou
Keyword(s):  
Harmonic Oscillators ◽  
Coupled Harmonic Oscillators ◽  
Group Synchronization

scholarly journals ENTANGLEMENT AND FRUSTRATION IN ORDERED SYSTEMS

2003 ◽  
Vol 01 (04) ◽  
pp. 465-477 ◽  
Author(s):  
MICHAEL M. WOLF ◽  
FRANK VERSTRAETE ◽  
J. IGNACIO CIRAC
Keyword(s):  
Ground State ◽  
State Energy ◽  
Nearest Neighbor ◽  
Bell Inequalities ◽  
Harmonic Oscillators ◽  
Neighbor Interaction ◽  
Gaussian States ◽  
Classical Systems ◽  
Entanglement Of Formation ◽  
Starting Point

This article reviews and extends recent results concerning entanglement and frustration in multipartite systems which have some symmetry with respect to the ordering of the particles. Starting point of the discussion are Bell inequalities: their relation to frustration in classical systems and their satisfaction for quantum states which have a symmetric extension. We then discussed how more general global symmetries of multipartite systems constrain the entanglement between two neighboring particles. We prove that maximal entanglement (measured in terms of the entanglement of formation) is always attained for the ground state of a certain nearest neighbor interaction Hamiltonian having the considered symmetry with the achievable amount of entanglement being a function of the ground state energy. Systems of Gaussian states, i.e. quantum harmonic oscillators, are investigated in more detail and the results are compared to what is known about ordered qubit systems.

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