Quantum information processing with electron spins

1. Quantum information processing with electron spins  Optics & Spin Physics Florian Meier and David D. Awschalom Funding from:

2. spin manipulation in an extended system • spin dynamics in QD’s (one qubit)? • spin dynamics in an external B-field • el. spin interactions (CNOT)? • Laser pulse (many photons) for spin read-out or manipulation • single photon as qubit? New questions for optics & spin-physics • optical spin injection • detect spin coherence by time-resolved • Faraday Rotation (TRFR) J.M. Kikkawa and D.D. Awschalom, Nature 397, 139 (1999) • spin manipulation with • optical pulses (fast) J. A. Gupta et al., Science 292, 2412 (2001)

3. mode 1 QD mode 2 σ1+ Outline • Towards electron interactions (coupled quantum dots): • M. Ouyang and D. D. Awschalom, Science 301, 1074 (2003) [exp.] • F. Meier et al., PRB 69, 195315 (2004) [thy.] • Cavity QED as interface for spin and photon quantum states: • F. Meier and D. D. Awschalom, • cond-mat/0405342. PRB (in press) [thy.]

4. Universal quantum computing with electron spins requires electron exchange interaction. coupled quantum dots transfer via benzene ring EcA EcB EvB EvA QD B QD A Coupled QD´s • pair of coupled QD´s with one exciton • spin dynamics probed by TRFR • Results: • strong delocalization of spin via conjugated molecule • electron exchange interaction relevant for TRFR

5. Absorption spectroscopy: • Coupled QD’s with different radii • 1.7 nm (A) and 3.5 nm (B) • Difference in quantum size levels • allows one to selectively address • both QD’s of a coupled pair. • Absorption peaks in the coupled • system are red-shifted. • Consistent with a coherent • delocalization of the electron • or hole over the coupled system. opt. absorption energy Coupled QD‘s: Absorption Molecularly coupled QD’s:

6. Energy Larmor frequency Coupled QD‘s: TRFR • Faraday rotation: • g-factor size-dependent: distinguish • spin in QD A from spin in QD B. • Pump at low energy: inject • exciton into QD B. • Measure TRFR signal • at varying probe energies: • Find spin in QD A with probability • 10-20% even at T=300 K. EA EB  - B A

7. absorption and TRFR imply delocalization of electrons • over both coupled QD’s; • transfer probability is of order 10-20% even at room T. • Experimental results: tc EcA EcB where UB Questions for theory: • simple model which explains the exp. features • electron exchange interaction? EvB EvA single-particle energy levels (i) (ii) Coulomb interaction e-h attraction h-h repulsion e-e repulsion transfer of electrons and holes, spin-conserving (iii) el. transfer hole transfer Coupled QD‘s: Theoretical Model

8. EcA tc EcB EvB EvA Coupled QD‘s: Tunnel matrix elements The only unknown parameters of the model are tc and tv0. Calculate exciton wave functions and eigenenergy: en. red-shift indirect exc. find tc 0.08 eV

9. mag. sample F dipole transition with operator for  circularly polarized light. TRFR Signal: Theory • FR “macroscopically”: magnetization M • rotates the polarization direction of a • linearly polarized Laser beam. • FR “microscopically”: Because of Pauli blocking, dielectric response • is different for + and -: mathematically: EcB -: transition blocked +: transition allowed EvB +

10. where • bi-exciton exchange splitting; • probability for el. transfer from QD A to QD B • (QD B to QD A); • EX,A and  energy and linewidth of exciton-transition TRFR in Coupled QD’s: Theory From transition matrix elements to all bi-exciton states, find: • TRFR signal depends on coupling via the transfer probabilities p. • Electron exchange coupling expected to show up in TRFR signal.

11. TRFR in Coupled QD’s: Results 1. Probability for electron transfer in coupled QD’s: • Obtained with tc calculated from absorption data. • Comparable to exp. spin transfer probability 10%. 2. TRFR signal amplitude as a function of probe energy and Larmor frequ.: theory experiment Reentrant behavior is well reproduced by theory.

12. =50 meV FR [a.u.] =80 meV =20 meV 2.3 2.4 2.5 E[eV] TRFR: What about Exchange Interaction? 3. Electron exchange interaction is expected to show up in TRFR signal amplitude: Expect several zeroes in F(E). linecut at fixed Larmor frequency Exchange interaction J  20 meV is too small compared to line- width   50 meV !

13. Coupled QD’s and Quantum Information • Coupled QD’s show strong • delocalization of the electron • wave function; spin is conserved. • Behavior well understood within • a simple theoretical model. • Perspective: Detect electron exchange interaction spectroscopically • or by exchange-governed dynamics.

14. QD PL σ1+-laser Cavity QED: optical selection rules: • entanglement of atom and cavity • SWAP atom state onto cavity • spin dependent abs. and PL • optical spin-readout QD’s in Cavities: Interface for Spin & Photon Qubits Motivation: Imamoglu, Zoller, Sham, ...: Haroche, Kimble, Walther, ....: atom Can one swap the spin state of a QD onto the cavity mode? • spin-photon entanglement; • spin-photon SWAP gate. Using a 2-mode cavity, can implement

15. mode 1 y2 QD • QD with excess electron, • Two cavity modes with circular (mode 1) and linear (mode 2) polarization. • Strong coupling. mode 2 σ1+ • QD level scheme (hh or lh valence band maximum); • one spin state of QD is always dark! 2-mode Cavity and QD: The System propagating modes Dynamics if a photon is injected into mode 1? Dynamics depend on

16. For spin state , transition • to trion state by photon • absorption: σ1+ sz=±1/2 hh jz=-3/2  lh σ1+ or y2 Spin-Photon Entanglement: The Hamiltonian QD with hh (|jz|=3/2) val. band maximum. Possible processes .... (b) Trion decays by photon emission into either 1+ or y2; QD returns to its original spin state. where g1, g2 are coupling constants for modes 1 and 2. (2-mode Jaynes-Cummings model)

17. Time evolution of y2 σ1+ σ1+ At max. entangled states Spin-Photon Entanglement: Dynamics For g1=g2=g,

18. propagating modes t[/g] mode 1 y2 Cavity loss is sufficient: mode 2 with σ1+ Liouville operator for cavity loss. Entanglement: Master Equation for Cavity Loss Terminate time evolution here!

19. cav. loss from mode 2 photon loss into mode 2! For cav. loss from mode 1 t[/g] E inefficient transfer to 2 (linewidth) loss from mode 1 2[g/] Entanglement: Von Neumann Entropy In which direction does the photon leave the cavity for spin state |? Cavity loss terminates coh. evolution exactly after one period. At least one oscillation between cavity modes. Von Neumann entropy as fctn. of 2: Prob. for cavity loss along 2:

20. F   σ1+ Entanglement: Robustness • How sensitive are the above dynamics to experimental fine-tuning? • Coupling constants g1g2: • QD misalignment by angle : • Detuning  of cavity modes relative to exciton transition: F g1/g2 Resonance condition is crucial!

21. For spin state , transition • to trion state by photon • absorption: Spin-Photon SWAP: Hamiltonian QD with lh(|jz|=1/2) val. band maximum. Possible processes .... (b) Trion decays by photon emission into either 1+ or z2; QD spin can be flipped! sz=±1/2 σ1+ σ1+ z2 jz=1/2 lh hh Trion couples to two different spin states!

22. Time evolution of z2 σ1+ σ1+ QD state swapped onto cavity state! At Spin-Photon SWAP: Dynamics For g1=g2=g,

23. Experimental Implementation • Main challenge: Scheme requires • cavity with • small mode volume of order 3; • high Q-factor,Q>104; • three degenerate modes, which are • not all TE or TM; • QD placed at mode maxima. K. Hennesy et al., APL 83, 3650 (2003) Possible (at least in principle) with defect modes of a photonic crystal.

24. mode 1 QD mode 2 σ1+ Summary • Spin physics of molecularly coupled QD’s: • delocalization of electron wave function; • dynamics driven by electron exchange • interaction? • QD’s in two-mode cavities: • create spin-photon entanglement; • implement spin-photon SWAP gate; • system robust against experimental • imperfections. y2