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

F. Meier & D.D. Awschalom 2 New questions for optics & spin-physics J.M. Kikkawa and D.D. Awschalom, Nature 397, 139 (1999) J. A. Gupta et al., Science 292, 2412 (2001) optical spin injection detect spin coherence by time-resolved Faraday Rotation (TRFR) spin manipulation with optical pulses (fast) 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?

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

F. Meier & D.D. Awschalom 4 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 EcAEcA EcBEcB EvBEvB EvAEvA transfer via benzene ring QD B QD A Universal quantum computing with electron spins requires electron exchange interaction. coupled quantum dots Coupled QD´s

F. Meier & D.D. Awschalom 5 Coupled QD‘s: Absorption Molecularly coupled QD’s: 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. energy opt. absorption

F. Meier & D.D. Awschalom 6 Larmor frequency Energy 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.  - - EBEB B EAEA A Coupled QD‘s: TRFR

F. Meier & D.D. Awschalom 7 Coupled QD‘s: Theoretical Model Experimental results: absorption and TRFR imply delocalization of electrons over both coupled QD’s; transfer probability is of order 10-20% even at room T. Questions for theory: simple model which explains the exp. features electron exchange interaction? where (i) single-particle energy levels EcAEcA EcBEcB EvBEvB EvAEvA UBUB tctc (iii) transfer of electrons and holes, spin-conserving el. transfer hole transfer Coulomb interaction e-e repulsion h-h repulsion e-h attraction (ii)

F. Meier & D.D. Awschalom 8 The only unknown parameters of the model are t c and t v  0. Calculate exciton wave functions and eigenenergy: Coupled QD‘s: Tunnel matrix elements indirect exc. tctc EcAEcA EcBEcB EvBEvB EvAEvA en. red-shift find t c  0.08 eV

F. Meier & D.D. Awschalom 9 TRFR Signal: Theory mag. sample FF 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  - : EcBEcB EvBEvB  - : transition blocked  + : transition allowed ++ mathematically: with dipole transition operator for  circularly polarized light.

F. Meier & D.D. Awschalom 10 TRFR in Coupled QD’s: Theory all bi-exciton states initial one-exciton state Relevant bi-exciton states: (1) (2) Triplet state with parallel el. spins. Reached by absorption of  + photon. Triplet or singlet state with anti-parallel el. spins. Reached by absorption of  - photon. Transition matrix elements: reduced relative to 1 by probability p B  A that the electron injected in QD B has been transferred to QD A via the benzene ring. tctc EcAEcA EcBEcB EvBEvB EvAEvA

F. Meier & D.D. Awschalom 11 TRFR in Coupled QD’s: Theory From transition matrix elements to all bi-exciton states, find: where bi-exciton exchange splitting; probability for el. transfer from QD A to QD B (QD B to QD A); E X,A and  energy and linewidth of exciton-transition 1. TRFR signal depends on coupling via the transfer probabilities p. 2. Electron exchange coupling expected to show up in TRFR signal.

F. Meier & D.D. Awschalom 12 TRFR in Coupled QD’s: Results 1. Probability for electron transfer in coupled QD’s: Obtained with t c 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.

F. Meier & D.D. Awschalom 13 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 E[eV] FR [a.u.]  =20 meV  =50 meV  =80 meV Exchange interaction J  20 meV is too small compared to line- width   50 meV !

F. Meier & D.D. Awschalom 14 Coupled QD’s and Quantum Information 1.Coupled QD’s show strong delocalization of the electron wave function; spin is conserved. 2.Behavior well understood within a simple theoretical model. 3.Perspective: Detect electron exchange interaction spectroscopically or by exchange-governed dynamics.

F. Meier & D.D. Awschalom 15 QD’s in Cavities: Interface for Spin & Photon Qubits Motivation: Imamoglu, Zoller, Sham,...: QD σ 1 + -laser PL optical selection rules: spin dependent abs. and PL optical spin-readout Haroche, Kimble, Walther,....: atom Cavity QED: entanglement of atom and cavity SWAP atom state onto cavity Can one swap the spin state of a QD onto the cavity mode? Using a 2-mode cavity, can implement spin-photon entanglement; spin-photon SWAP gate.

F. Meier & D.D. Awschalom 16 2-mode Cavity and QD: The System propagating modes mode 1 mode 2 QD σ1+σ1+ y2y2 QD with excess electron, Two cavity modes with circular (mode 1) and linear (mode 2) polarization. Strong coupling. Dynamics if a photon is injected into mode 1? Dynamics depend on QD level scheme (hh or lh valence band maximum); one spin state of QD is always dark!

F. Meier & D.D. Awschalom 17 Spin-Photon Entanglement: The Hamiltonian QD with hh (|j z |=3/2) val. band maximum. Possible processes.... σ1+σ1+ s z =±1/2 j z =-3/2  hh lh σ 1 + or y 2 (b) Trion decays by photon emission into either  1 + or y 2 ; QD returns to its original spin state. (a)For spin state, transition to trion state by photon absorption: where g 1, g 2 are coupling constants for modes 1 and 2. (2-mode Jaynes-Cummings model)

F. Meier & D.D. Awschalom 18 Spin-Photon Entanglement: Dynamics σ1+σ1+ y2y2 Time evolution of σ1+σ1+ For g 1 =g 2 =g, Atmax. entangled states

F. Meier & D.D. Awschalom 19 Entanglement: Master Equation for Cavity Loss mode 1 mode 2 σ1+σ1+ y2y2 propagating modes t[  /g] Terminate time evolution here! Cavity loss is sufficient: with Liouville operator for cavity loss.

F. Meier & D.D. Awschalom 20 Entanglement: Von Neumann Entropy In which direction does the photon leave the cavity for spin state |  ? At least one oscillation between cavity modes. Cavity loss terminates coh. evolution exactly after one period. t[  /g] cav. loss from mode 2 cav. loss from mode 1 Prob. for cavity loss along 2: For photon loss into mode 2! Von Neumann entropy as fctn. of  2 :  2 [g/  ] E loss from mode 1 inefficient transfer to 2 (linewidth)

F. Meier & D.D. Awschalom 21 Entanglement: Robustness How sensitive are the above dynamics to experimental fine-tuning? 1.Coupling constants g 1  g 2 : 2.QD misalignment by angle  : 3.Detuning  of cavity modes relative to exciton transition: F g 1 /g 2 F  Resonance condition is crucial! σ1+σ1+ 

F. Meier & D.D. Awschalom 22 Spin-Photon SWAP: Hamiltonian QD with lh (|j z |=1/2) val. band maximum. Possible processes.... σ1+σ1+ s z =±1/2 j z =1/2 hh lh σ1+σ1+ (b) Trion decays by photon emission into either  1 + or z 2 ; QD spin can be flipped! (a)For spin state, transition to trion state by photon absorption: z2z2 Trion couples to two different spin states!

F. Meier & D.D. Awschalom 23 Spin-Photon SWAP: Dynamics σ1+σ1+ z2z2 Time evolution of σ1+σ1+ For g 1 =g 2 =g, At QD state swapped onto cavity state!

F. Meier & D.D. Awschalom 24 Experimental Implementation Main challenge: Scheme requires cavity with small mode volume of order 3 ; high Q-factor, Q>10 4 ; three degenerate modes, which are not all TE or TM; QD placed at mode maxima. Possible (at least in principle) with defect modes of a photonic crystal. K. Hennesy et al., APL 83, 3650 (2003)

F. Meier & D.D. Awschalom 25 Summary 1.Spin physics of molecularly coupled QD’s: delocalization of electron wave function; dynamics driven by electron exchange interaction? 2.QD’s in two-mode cavities: create spin-photon entanglement; implement spin-photon SWAP gate; system robust against experimental imperfections. y2y2 mode 1 mode 2 QD σ1+σ1+