• Study Entanglement Between Two Electrons Study Entanglement Between Two Electrons 

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inQuantum

Dots

By:Hossien Hossieni

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Additionspin

of

2electron

For2electronadditionofspinsyields1,0

Spin1istrip et s=1,m=1,0,1

= = ,

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QuantumDot

Quantumdots,alsocalledartificialatoms,are

smallregions(from1toabout100nm;1nm)

ofonesemiconductormaterialburiedin

energygap E.Theyaremadeupbyroughly1

millionatoms

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TwoQuantum

Dot

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ConfinementPotentialModelsinQuantumDots

bythefollowingsimpleforms:

2 Parabolic(harmonicoscillator)potential 3 Gaussianpotentialwell

4 Powerex onential otentialwell

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RectangularPotentialWell

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ParabolicPotentialWell

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GaussianPotential

Well

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PowerExponentialPotentialWell

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PowerExponentialPotentialinCoupledQuantum

Dots

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TheHamiltonianofthesystem

Coulombpotential(betweentwoelectron)

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GroundStateEnergy

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Bychoosing

Weget

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Groundstateeigen valueenergyE0vs.confinementpotentialrangeRfor

,

=

,

=

=

.

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GroundstateenergyE0vs.confinementpotentialrangeRfordifferent

,

=

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Coulomb

interaction

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LinearEntropyandEntanglementMeasure

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Groundstateeigen valueenergyE0,CoulombenergyCandLinearEntropy

LofthereduceddensitymatrixfortwoelectronsconfinedincoupledQDstructures

versusvaryingconfinementpotentialrangeRfordifferentvaluesofp.

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TimeDependenceinQuantumMechanics

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SuddenChangeintheHamiltonian

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TimeDependentPerturbationTheory

Inthatcase,theSchrdingerequationfortheHamiltonian

isequiva enttot eseto coup e equations

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TransitionAmplitudesandProbabilities

Thesetofcoupledequations

isnoteasiertosolvewemultiply

theperturbationV(t)bytheunphysicalparameter (0 1)andexpand

theam litudes

Thetermslinearin yield

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Electronlocalizationandentanglementinatwoelectron

uantum

dot

ThelastterminEq.istheZeemaninteractionwithgbeingtheeffectivegfactor, B the

Bohrmagneton,andsthespinofanindividualelectron.

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Entanglemententropyofexcitedstates

consideringtheXYspinchainoflengthNwith

Hamiltonianisgivenby

.isthetransversemagneticeld and theanisotro arameter.

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QuantumEntanglementandSymmetry

Bipartitesymmetricstates

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1DanielR.Bes,QuantumMechanicsAModernandConciseIntroductoryCourse 2ShirwanMuhamad Abdullah,EffectofConfinementPotentialGeometryonEntanglement

BetweenTwoElectronsinQuantumDotsMSc.Thesis 3DavidJ.Griffiths IntroductiontoQuantumMechanics

,

twoelectronquantumdot

5VincenzoAlbaetal,Entanglemententropyofexcitedstates

6DariuszChruscinski etal,QuantumEntanglementandSymmetry