Optical Manipulation of an Electron Spin in Quantum Dots

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1 Optical Manipulation of an Electron Spin in Quantum Dots Al. L. Efros Naval Research Laoratory, Washington DC, USA Acknowledgements: DARPA/QuIST and ONR Kavli Institute for Theoretical Physics, UC Santa Barara, May 4, 2006.

2 Electron Spin Manipulation for Quantum Computer General physical interest to this prolem (R. Feynman, 1983) Shor s algorithm for large numer factorization (1994) The major difference from a computer is replacement of a it ( on or off states) y a quit ( a coherent superposition of on and off states) A single electron spin in a QD is a natural quit [Loss and DiVincenzo, PRA(1998)] Universal quantum computation requires: Aritrary 1-quit rotations (single spin rotation) Performed y turning on local magnetic field A single 2-quit gate operation (spin-spin interaction) spin-spin exchange interaction controlled y gates T(t) Coulom repulsion U in each dot. Tunneling T(t) etween the dots. Effective coupling J(t) ~ T 2 (t)/u. Tunneling T(t) controlled y varying gate voltage. Fast operations in order to keep a coherent superposition

3 Optical Manipulation of Electron Spin Initialization Critical Steps for Implementation of a Solid State Based Quantum Computer One quit operation Entanglement of 2 quits Read-out Optical methods provide high speed techniques of spin control and manipulation and allow to access an electron spin locally. Non-resonant optical pumping of an electron spin in negatively charged quantum dots [Phys. Rev. Lett. 94, (2005)] Optical initialization of electron spins y resonant π-pulses of σ ± -polarized light [Phys. Rev. B 68, (R) (2003)] Optical control of spin coherence in charged QDs [condmat/ ]

4 Optical excitations in singly charged QD X h e z X - e - AlGaAs GaAs AlGaAs σ + σ Polarized laser light X - +3/2> X - 3/2> Disk-shaped QDs σ + r r σ z e - +1/2> e - 1/2> The ground electron and hole states are ±1/2> and ±3/2>, respectively. PL polarization is controlled y the hole spin projection

Single Charge-Tunale QD Spectroscopy Exp: on non-resonant pumping interface

B. V.B. AlGaAs Schottky diode heterostructure GaAs e - AlGaAs E F Energy PL

2-25 μm Al 800Å QDs charge state control charged exciton h + neutral

5 Single Charge-Tunale QD Spectroscopy Exp: on non-resonant pumping interface fluctuation QDs E Charge control: -1V C.B. V.B. AlGaAs GaAs AlGaAs E F 0V C.B. V.B. AlGaAs Schottky diode heterostructure GaAs e - AlGaAs E F Energy PL LASER 100 nm μm Al 800Å QDs charge state control charged exciton h + neutral exciton V X gnd e - X - Ti / Al AlGaAs 4nmthickGaAs QW AlGaAs n-gaas 1 mm rel. PL Energy (mev) X + X X Diode Bias (V)

6 Photoluminescence Polarization Memory Effect (initial expectations) Nonresonant excitation in GaAs quantum well: initial charge state CB VB CB VB CB VB σ + excitation X 0 CB VB CB VB CB VB luminescence σ + positive polarization X + σ + positive polarization X - σ + positive polarization

Polarization (%) 40 30 20 10 0-10 -20 1.655 PL Polarization Memory Effect: Experiment X + X - X 0 +? X + σ + Long electron spin memory X Anisotropic e-h exchange X - hole spin?

7 Polarization (%) PL Polarization Memory Effect: Experiment X + X - X 0 +? X + σ + Long electron spin memory X Anisotropic e-h exchange X - hole spin?... π + 0 σ + σ - PL Energy (ev) X 0 X + X Bias (V) Polarization (%) X - - polarization Power (mw)

8 Negative PL Polarization Degree of X - in QD Why is X - polarization degree negative, why does it grow with pumping intensity and changes sign with ias? Negative polarization was oserved in ensemles of charged QDs: Dzhioev et al. Phys. Solid State 40, 1587 (1998) Cortez et al. Phys Rev. Lett. 89, (2002) Kalevich et al. phys. stat. sol. 238, 250 (2003) Spectroscopy of a single charge QD [Bracker et al. Phys. Rev. Lett. 94, (2005)] allows us to suggest a model that 1. Descries unusual PL polarization properties of charged quantum dots. 2. Explains mechanism responsile for optical pumping of electron spins

9 Mechanism for X - polarization e-h pair creation σ + Hole flip in "hot" excitons > Bright exciton recomination QD ound states σ + right X + X - G σ + T Freezing of hole spins Trap exciton in charged QD Trion recomines T + - n n d G d + σ - dark X X - Dark exciton can only form trion X - polarization: = d e - e - + d

10 ( ) ( ) ( ) ( ) ( ) ( ) = = = = = Ω = + + Ω = + + Ω = + = + = T T n n N d d Wn G dt dd d d Wn G dt dd d Wn G dt d d Wn G dt d d d S W S n n dt ds n n T d Wn S dt dn n n T d Wn S dt dn T d n n W dt dt T d n n W dt dt H d H d EX H EX H x S x x S T x S T x T T 2 2 These Processes are Descried:

11 Steady State Solution If the hole spin relaxation time H is very long the steady state concentrations of Bright,, and Dark, d, excitons is determined: where is the rate of exciton capture on charged quantum dots The concentrations of spin and spin electrons (n and n ) depends on the pumping intensity. At low intensity: n = n G d where is the electron spin relaxation time SE

12 Electron population right X Optical Pumping of Electron Spin X - e - e - X - e - dark X Power (mw) Bright exciton leaves electron unchanged n Dark exciton leaves ehind electron with spin down Optical pumping n Polarization (%) Pumping loop: excitons capture rate ~ n DE recomination decreases n leading to DE accumulation Power (mw)

13 X - polarization: ias dependence Electrons in QW: few many - many excitons per charged QD -pumping of electron spins -accumulation of Dark Excitons -negative PL polarization X - Polarization (%) few excitons per charged QD - no pumping of electron spins - no accumulation of Dark Excitons -positive PL polarization Diode Bias (V)

14 Erase Electron Spin Polarization Hanle effect: Erase spin polarization through precession in magnetic field B X Z Y X - Polarization (%) Hanle effect B x = 0.5 T 100 W/cm W/cm 2 σ + S (Holes are not affected y B x ) Diode Bias (V)

15 Hanle effect: electron spin lifetimes B X Z Y Depolarize electrons ut not holes with magnetic field in the QD plane Smaller depolarization field implies longer electron spin lifetime σ + S Polarization ρ z (%) X + X B X (Tesla) SE 160 psec (trion recomination) SE 14 nsec Ground state electron

16 Theory of Erasing Electron Spin Polarization X - - polarization measurement Polarization (%) 10 in magnetic field Hanle effect: 5 Power (mw) <S z > B X Z Y σ + s Ω B = g e μb x Average electron spin: Polarization (%) Ω B 7 mw Ω B 45% polarization 7 mw

17 Summary Optical orientation is oserved in a single, charge-tunale QD. Theory explains the negative optical polarization in negatively charged QDs resulting from the electron spin pumping, which leads to accumulation of dark excitons. Theory quantitatively descries amplitude and sign of Hanle effect, as well as the polarization power dependence in negatively charged QDs. However the non-resonant optical pumping is not the est way of the electron spin initialization.

18 Resonant Optical Spin Orientation: Atoms vs. QD s In atoms +1/2> 1/2> In artificial atoms (quantum dots) +3/2> 3/2> σ (-) r linear polarization r σ (+) r Ω T +1/2> 1/2> (Brossel and Kastler, Comp. Rend. 229, 1213 (1949).) Selection rules allow luminescence in oth spin states. It leads to accumulation of S z = -1/2. +1/2> 1/2> Ω s Luminescence returns electron to the same initial spin state. Optical initialization is not possile!! Electron spin can change due to either electron or hole spin relaxation.

19 Spin flip transitions fluctuation of nuclear spin directions in a finite size QD I.A. Merkulov, Al. L. Efros, and M. Rosen Phys. Rev. B 65, (2002). precession of electron spin in effective hyperfine B-field of nuclei 1/Ω N 1-10 ns +3/2> s h 3/2> σ + r r Net electron spin of trion = 0 Hole is not affected y nuclei. +1/2> Ω N 1/2> Two-phonon process flips the spin of hole. It requires phonons: 1/ sh 0 at low temperature. T. Takagahara Phys. Rev. B 62, (2000).

20 Spin initialization y polarized light Steady state electron spin polarization at t is independent of initial spin. ρ =(P 1/2 P -1/2 ) / (P 1/2 + P -1/2 ) 1.0 s h +3/2> 3/2> 0.5 h s / r = 105 h s / r = 104 σ (+) Ω R r r ρ low Intensity Ω R r h s / r = 103 h s / r = 102 high Intensity +1/2> 1/2> Ω N Ω N r = 0.1 Ω R ~ (Intensity) 1/2

21 Effect of σ + Polarized Optical π- Pulses The intense (+) polarized light drives the +1/2 electron into the +3/2 trion states and ack with Rai frequency: Ω R = 2 (E d/ ) +3/2> population Electron P 1/2 and trion P 3/2 population: 1 0 P 1/2 P 3/2 π 2π t Ω R σ (+) Ω R 1 r +1/2> Optical -pulse of (+) polarization creates the +3/2 trion from the 100% polarized S z = +1/2 electron, ut it does not affect the 100% polarized S z = -1/2 electron. σ (+) Δt = π / Ω R

22 Effect of π- pulses of optical and magnetic field If an electron is unpolarized S z = 0 optical -pulse of (+) polarization creates a coherent superposition of the +3/2 trion and the optically passive S z = -1/2 electron. σ (+) Δt = π / Ω R coherent superposition S z = 0 50% 50% z B x Transverse magnetic field rotates electron spin: Ω s = B g e B x / B x (t) B x (t) Δt = π / Ω s Δt = π / Ω s

23 Initialization y optical and magnetic -pulses Short magnetic -pulse flips the electron spin ut it does not affect the trion. σ (+) 50% B x (t) 50% 1 r S z = 0 Δt = π / Ω R 50% Δt = π / Ω s σ (+) z B x 50% Luminescence coherent superposition coherent superposition 100% electron spin polarization can e achieved! Prolem: It is difficult to generate short magnetic pulses ( r >> Δt ).

24 Summary Electron spin in QDs can e optically initialized y intense polarized light. We propose optical initialization of an electron spin that using a comination of optical and transverse magnetic field π-pulses.

25 Controlled Initialization of Spin Coherence in Ensemle of Singly charged QDs. M. Bayer, Dortmund Pump-proe Faraday and Kerr rotation measurements * resonant optical excitation of charged QDs Charged exciton or "trion" self-assemled QDs GaAs InAs QDs Samples: 20 layers of QDs separated y 60 nm wide arriers, QD density ~ cm -2, n -modulation doped 20nm elow each layer with the same Si density. B X QD σ + light z y 3 2 σ Semiconductor Spintronics and Qauntum Computation Eds. D.Awschalom, D. Loss (2002), J. Kikkawa & D. Awschalom, Science 287, 473 (2000), J. Gupta & D. Awschalom PRB 59, (1999). Ground state electron

26 Pump-Proe Faraday Rotation in (In,Ga)As/GaAs QDs Traces of FR signal, excited at T= 2K, pulses duration of 1ps resonant excitation. FR amplitude (ar. units) (a) B=0 T PL intensity QD emission laser Energy (ev) 1. Pronounced oscillations 2. Duration longer than r = 400 ps 3. Oscillation frequencies increases with magnetic field, B 4. Decay increases with B 5. Additional modulations at high B Time (ps)

27 Long-Lived Electron Spin Coherent State Increase of frequencies A FR Ω * ~ exp( t / T 2 )cos( Ω t) e e = g μ B e B Ω (THz) 0.4 (d) hole 0.2 electron B (T) Additional modulations: Ω g h =0.66 at B= 5 T with T 2* =170 ps vs r = 400 ps h = g h μ B B g-factor dispersion Δ g e T * 2 1 = ( B) T * 2 1 (0) Δgeμ B B + 2 est fit: Δ g e =0.004 The optical intialization of LL-ESC was oserved in GaAs/AlGaAs interface QDs: Gurudev Dutt et al. PRL 94, (05) Theory of this effect: S. E. Economou et al., Phys. Rev. B 71, (2005)

28 Pump Power Dependence of FR Amplitude Nonmonotonic dependence Pulse area: Θ = (2 / ) ( d E( t)) dt FR amplitude (ar. units) (a) B=1 T 150mW B X σ + z y FR amplitude (ar. units) π 2π 3π 60% Time (ps) Pulse area (ar.units) Rai-like oscillations of the FR amplitude: origin of the spin coherence

29 Coherent Spin Superposition A short pulse of resonant circularly polarized light with external transverse magnetic field creates a CS of the electron and trion states. If pulse length Δt << r, sh, and s e : coherent spin state α 0 + β 0 electron ground state QD x σ + light z y short pulse Δt p electron/trion coherent superposition α 0 cos(θ/2) + β 0 electron ground state - i α 0 sin(θ/2) 2d E / Δt pulse area Θ = ( ) p trion state S x z x z y S z y α = 1, β = 0, S x = S y =0, S z = 1/2 Spin polarization vector Aritrary spin orientation: ψ =α +β, where α 2 + β 2 =1 in the ground state, and α 2 + β 2 <1 in the excited state. spin vector: S x = Re(αβ * ), S y = -Im(αβ * ), S z = ( α 2 - β 2 )/2 x S z y α = β = 1/2 1/2, S x = 1/2, S y =S z = 0

30 Optical Control of Electron Spin σ + polarized pulse decreases S z : S z -S 0 z =0.5 α 0 2 sin 2 (Θ/2) and it reaches maximum at Θ=(2n +1)π S x and S y component change sign with period 2π, Θ=2nπpulses can e used

31 Spin Dynamics After Pulse After the pulse the electron and trion spin vectors in a single QD: Trion spin vector: J = (J x,j y,j z ), descries polarization of the trion state: ψ tr =α tr +β tr J x =Re(α tr β * tr ), J y =-Im(α tr β * tr ), J z =-(1/2)( α tr 2 - β tr 2 ). The long lived electron spin polarization at t >> r :S z (t) = S z cos[(ω e + Ω N,x ) t], excited state trion time ground state electron time If Ω e >> 1/ r, S z =S z (0) is the electron polarization created y the pulse ONLY

32 Faraday Rotation Amplitude in a QD Ensemle In a QD ensemle J(t) and S(t) should e averaged over Δ g e =0.004 Optically induced FR amplitude: Proportional to the population difference of states involved in σ + and σ - transitions: Δn + =n -n and Δn - =n -n. The FR angle: φ( t) ~ ( n Δn ) / 2 = S ( t) J ( t). Δ + + z z Theoretical time dependence of FR signal for ensemle of QDs, which use parameters from experiment.

33 Summary 1. We have shown experimentally and theoretically that short pulses of circularly polarized light with transverse magnetic field allow initialization a complete control of electron spin coherence in a single quantum dot. 2. For resonant excitation, the pulse area uniquely determines the electron spin coherence. The spontaneous decay of the trion does not affect the spin coherence at Ω e r >> 1.

34 Acknowledgement A. Shaaev, A.S. Bracker, E.A. Stinaff, D. Gammon, J.G. Tischler, D. Park Naval Research Laoratory Washington, DC D. Gershoni Technion-Israel Institute of Technology, Haifa, Israel V.L. Korenev, I.A. Merkulov A. F. Ioffe Institute, St. Petersurg, Russia A. Greilich, R. Oulton, D.R. Yakovlev, M. Bayer University of Dortmund, Dortmund, Germany V. Stavarache, D. Reuter, A. Wieck Ruhr-University of Bochum, Bochum, Germany Phys. Rev. Lett. 94, (2005); Phys Rev. B, 68, R (2003), Sumitted to Phys. Rev. Lett. [condmat/ ]

Ultrafast optical rotations of electron spins in quantum dots. St. Petersburg, Russia

Ultrafast optical rotations of electron spins in quantum dots A. Greilich 1*, Sophia E. Economou 2, S. Spatzek 1, D. R. Yakovlev 1,3, D. Reuter 4, A. D. Wieck 4, T. L. Reinecke 2, and M. Bayer 1 1 Experimentelle