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

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

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

3 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. ? + t f Can t nf Can we choose t f and t nf ? Is there a simple picture? What are the energy/time scales? How might it be realised in practise?

Possible realisations 4 Quantum wire: gated quantum well, nanotube, graphene strip (?) SE injector Spin-rotator Spin-filter Bound electron Spin analyser • magnetic gates • quantum dot • Magnetic contact • Turnstile • electric gates Gates, fullerene.. • Zeeman filter • SAW (Rashba)

5 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

6 Entanglement • Schrödinger 1935 - Verschränkung | U 〉 = | ↑〉 L | ↓〉 R | E 〉 = | ↑〉 L | ↓〉 R − | ↓〉 L | ↑〉 R ? ? 2 • 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”

Example 1 • All action near conduction 7 band edge SWCNT • Gates with positive bias kinetic injection create potential well • Effective mass approximation 2 with m*=E g /2v F   H = − h 2 ∂ 2 2 + ∂ 2 e 2  + v ( x 1 ) + v ( x 2 ) + 2 + λ 2  2 2 m * ∂ x 1 ∂ x 2   ( ) 4 πε x 1 − x 2 Bohr radius typically~5-50A Strong Correlations

Energy scales 8 • Well must bind one and only one electron • No ionisation • No inelastic scattering • Solve 2-electron problem exactly for bound states

9 Elastic scattering and spin entanglement • Solve scattering problem exactly | k ↑ , ↓〉 → r nf | − k ↑ , ↓〉 + r f | − k ↓ , ↑〉 + t nf | k ↑ , ↓〉 + t f | k ↓ , ↑〉 • Compute total transmission T =| t nf | 2 + | t f | 2 • Compute concurrence for transmitted electron: = 2 | t f || t nf | C t ( k ) = 2 | 〈 k ↑ , ↓ | ψ 〉〈 k ↓ , ↑ | ψ 〉 | 〈 ψ | ψ 〉 T • Similarly for reflected electron

Typical results 10 Tr Transmission V 0 = 0.8V a=12nm • Why two resonances? • Why T max ~ 1/2 at resonances? •Why is C~1 near resonances and ~0 between?

11 Physical picture • Propagating electron sees double barrier • Resonant scattering - spin dependent • Singlet and triplet resonances • , ↓〉 = | k ,0,0 〉 + | k ,1,0 〉 → | k ,0,0 〉 or | k ,1,0 〉 | k ↑ 2 2 2 U on resonance E 0 • 2-electron spin filter! • Fully entangled •Total transmission probability ~ 1/2 On resonance, the unentangled spin state splits into fully entangled components, one transmitted the other reflected

12 Check - solve for singlet and triplet (eigenstates) V 0 = 0.8V a=12nm C=1 always

13 Transmission Probabilities -narrow well V 0 =1.2 - 1.5 V a=4.8nm U E 0 • Singlet resonances only

Transmission Probabilities - wider well 14 V 0 = 0.4V a=19.2nm • Singlet and triplet resonances • Strong-correlation regime • Mean-field picture invalid • Concurrence suppressed

Electron density on resonance 15 V 0 =1.5 V, a=4.8nm • Intermediate correlation • Singlet resonance only V 0 = 0.4V, a=19.2nm • Strong correlation • Singlet and triplet resonances close in energy with similar charge densities

Interpretation - Heisenberg exchange 16 • 2 electrons in well of width >> Bohr radius have low-lying singlet -triplet H eff = J s 1 • s 2 • Bound states become resonances J = E T − E S • J reduces exponentially with well width • Previous examples, J~30meV and 3 microeV respectively

17 Example 2 SAW Entangler in semiconductor quantum wire   H = − h 2 ∂ 2 2 + ∂ 2 e 2  + v ( x 1 , t ) + v ( x 2 , t ) + 2 + λ 2  2 2 m * ∂ x 1 ∂ x 2   ( ) 4 πε x 1 − x 2 v ( x , t ) = v well ( x ) + v SAW ( x , t ) v SAW ( x , t ) = v 0 cos( kx − ω t )

18 Preliminaries - Single-electron states • Bound-electron must stay in well – Adiabatic for small SAW amplitude – Landau-Zener transitions for quasi- bound state v 0 =2meV, v well =6meV, w=7.5nm • Electron must stay in SAW minima

19 Two-electron scattering • Solve TD Schrödinger equation numerically R , NF , ψ well ↓ 〉 + | ψ SAW ↓ R , F , ψ well ↑ 〉 + | ψ SAW ↑ T , NF , ψ well ↓ 〉 + | ψ SAW ↓ T , NF , ψ well ↑ 〉 | Ψ in 〉 = | ψ SAW ↑ , ψ well ↓ 〉 → | ψ SAW ↑ • Concurrence W K Wooters, PRL, 1998 A Ramsak, I Sega and JHJ, PRA 2006. ∫ Φ * ( x 1 , x 2 , t ) Φ ( x 2 , x 1 , t ) dx 1 dx 2 | 2 | − 〉 | = + S B A , B C A , B ( t ) = 2 | 〈 S A | Φ ( x 1 , x 2 , t ) | 2 + | Φ ( x 2 , x 1 , t ) | 2 [ ] dx 1 dx 2 | ∫ A , B Depends on ‘measurement domain’ ‘heralded’ state

Two-electron scattering - charge density 20

21 Entanglement generation • CASE 1 - singlet-triplet filter

22 • CASE 2 - full transmission Φ S ( x 1 , x 2 , t ) → ψ S ( x 1 , t ) ψ 0 ( x 2 ) + ( x 1 ↔ x 2 ) Φ T ( x 1 , x 2 , t ) → ψ T ( x 1 , t ) ψ 0 ( x 2 ) − ( x 1 ↔ x 2 ) ⇒ C = |Im 〈 ψ S | ψ T 〉 | If | ψ S | = | ψ T | then C =|sin δφ | else C < 1 Phase regime - exchange H eff ( t ) = J ( t ) s 1 .s 2 J ( t ) = E T ( t ) − E S ( t ) δφ = 1 ∫ J ( t ) dt h

23 Variation of C with well depth

24 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..)