Entanglement generation between static and flying qubits Co-workers - - PowerPoint PPT Presentation

Entanglement generation between static and flying qubits

John Jefferson COQUSY06 Dresden July 24 - Oct 06 2006 Co-workers and Acknowledgements: Ljubljana Toni Ramsak, Tomaz Rejec Lancaster George Giavaras, Colin Lambert Oxford Daniel Gunlyke, David Pettifor, Andrew Briggs HP Labs Tim Spiller

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Outline

• Motivation and basic idea
• Possible realisations
• SWNT example
• SAW injection
• Summary and conclusions

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Motivation and basic idea

• Semiconductor quantum wires show spin-dependent conductance

anomalies near conduction edge.

• Can be explained in terms of effective spin interactions between bound

and propagating electrons

• Could this be used to demonstrate controlled entanglement?
• Major goal of UK IRC in QIP.

Can we choose tf and tnf? Is there a simple picture? What are the energy/time scales? How might it be realised in practise?

tnf + tf

Can ?

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Possible realisations

SE injector Spin analyser Spin-filter Spin-rotator Bound electron

Quantum wire: gated quantum well, nanotube, graphene strip (?)

• Magnetic contact
• Zeeman filter
• Turnstile
• SAW
• magnetic gates
• electric gates

(Rashba)

• quantum dot

Gates, fullerene..

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Scope of theory and modelling

• Studied Gated semiconductor 2DEGs and carbon

nanotubes

• Injection through turnstile or via a SAW
• Use simple effective-mass model but Coulomb repulsion

essential

• Solve 2-electron scattering problem exactly
• Interpretation of results

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Entanglement

• Schrödinger 1935 - Verschränkung
• EPR - spookey action at a distance
• Bohr - you shouldn’t ask
• Dirac (Penrose) “Philosophy does not help students

pass my quantum mechanics exams…..but Einstein was probably right”

|U〉 =| ↑〉 L | ↓〉 R

| E〉 = | ↑〉L | ↓〉 R− | ↓〉L | ↑〉 R 2

? ?

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Example 1 SWCNT kinetic injection

• All action near conduction

band edge

• Gates with positive bias

create potential well

• Effective mass approximation

with m*=Eg/2vF

2

H = − h2 2m * ∂ 2 ∂x1

2 + ∂ 2

∂x2

2

      + v(x1) + v(x2) + e2 4πε x1 − x2

( )

2 + λ2

Bohr radius typically~5-50A Strong Correlations

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Energy scales

• Well must bind one and only one electron
• No ionisation
• No inelastic scattering
• Solve 2-electron problem exactly for bound states

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Elastic scattering and spin entanglement

| k↑ ,↓〉 → r

nf |−k↑,↓〉 + rf |−k↓,↑〉 + tnf | k↑,↓〉 + t f | k↓,↑〉

• Solve scattering problem exactly
• Compute total transmission T=|tnf|2 + |tf|2
• Compute concurrence for transmitted electron:

Ct(k) = 2 |〈k↑ ,↓|ψ〉〈k↓,↑|ψ〉 | 〈ψ |ψ〉 = 2 | t f || tnf | T

• Similarly for reflected electron

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• Why two resonances?
• Why Tmax ~ 1/2 at resonances?
• Why is C~1 near resonances and ~0 between?

Typical results

V0 = 0.8V a=12nm

Tr

Transmission

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Physical picture

• Propagating electron sees double barrier
• Resonant scattering - spin dependent
• Singlet and triplet resonances
• 2-electron spin filter!
• Fully entangled
• Total transmission probability ~ 1/2

E0 U | k↑ ,↓〉 = | k,0,0〉+ | k,1,0〉 2 → | k,0,0〉 2

• r | k,1,0〉

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• n resonance

On resonance, the unentangled spin state splits into fully entangled components, one transmitted the other reflected

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Check - solve for singlet and triplet (eigenstates)

C=1 always

V0 = 0.8V a=12nm

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Transmission Probabilities -narrow well

E0 U

• Singlet resonances only

V0 =1.2 - 1.5 V a=4.8nm

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Transmission Probabilities - wider well

V0 = 0.4V a=19.2nm

• Singlet and triplet

resonances

• Strong-correlation regime
• Mean-field picture invalid
• Concurrence suppressed

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Electron density on resonance

V0 =1.5 V, a=4.8nm

• Intermediate correlation
• Singlet resonance only

V0 = 0.4V, a=19.2nm

• Strong correlation
• Singlet and triplet resonances

close in energy with similar charge densities

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• 2 electrons in well of width >> Bohr radius have low-lying singlet -triplet
• Bound states become resonances
• J reduces exponentially with well width
• Previous examples, J~30meV and 3 microeV respectively

Interpretation - Heisenberg exchange

Heff = Js1 •s2 J = ET − ES

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Example 2 SAW Entangler in semiconductor quantum wire

H = − h2 2m * ∂ 2 ∂x1

2 + ∂ 2

∂x2

2

      + v(x1,t) + v(x2,t) + e2 4πε x1 − x2

( )

2 + λ2

v(x,t) = vwell(x) + vSAW (x,t) vSAW (x,t) = v0 cos(kx −ωt)

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Preliminaries - Single-electron states

• Bound-electron must stay in well

– Adiabatic for small SAW amplitude – Landau-Zener transitions for quasi- bound state

• Electron must stay in SAW minima

v0=2meV, vwell=6meV, w=7.5nm

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Two-electron scattering

• Solve TD Schrödinger equation

numerically

• Concurrence

| Ψin〉 =|ψSAW↑,ψwell↓〉 →|ψSAW↑

R,NF ,ψwell↓〉+ |ψSAW↓ R,F ,ψwell↑〉+ |ψSAW↑ T,NF ,ψwell↓〉+ |ψSAW↓ T ,NF ,ψwell↑〉

W K Wooters, PRL, 1998 A Ramsak, I Sega and JHJ, PRA 2006.

CA,B(t) = 2 |〈SA

+SB − 〉 |=

2 | Φ*(x1,x2,t)Φ(x2,x1,t)dx1dx2 |

A,B

∫

|Φ(x1,x2,t) |2 + |Φ(x2,x1,t) |2

[ ]dx1dx2 |

A,B

∫

Depends on ‘measurement domain’ ‘heralded’ state

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Two-electron scattering - charge density

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Entanglement generation

• CASE 1 - singlet-triplet filter

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• CASE 2 - full transmission

ΦS(x1,x2,t) →ψS(x1,t)ψ0(x2) + (x1 ↔ x2) ΦT (x1,x2,t) →ψT (x1,t)ψ0(x2) − (x1 ↔ x2) ⇒ C =|Im〈ψS |ψT 〉 | If |ψS |=|ψT | then C =|sinδφ | else C <1

Phase regime - exchange Heff(t) = J(t)s1.s2 J(t) = ET (t) − ES(t) δφ = 1 h J(t)dt

∫

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Variation of C with well depth

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Summary and conclusions

• Controlled entanglement feasible
• Electron injected kinetically or via SAW
• Maximum entanglement induced near singlet and triplet

resonances -spin filter

• Electron correlations important, particularly in nanotubes
• Heisenberg spin exchange and phase-shift interpretation
• Other realisations plausible (graphene, peapods..)