Qubits by electron spins in quantum dot system Basic theory

> α θ Summer school of FIRST/Q-cybanetics Chinen, Okinawa, Aug. 21, 2010 > Qubits by electron spins in quantum dot system Basic theory Yasuhiro Tokura NTT Basic Research Laboratories FIRST project, theory subgroup Quantum Cybanetics, semiconductor qubits

Self-introduction NTT

Solid state qubits microscopic coherence- - - - I - - Macroscopic system + Ensemble measurement Nanostructures: Small ensemble + Single electron spectroscopy - - - - - - Mesoscopic physics: quantum - interference, low-dimensionality,. e e e System of just one or two electrons + Dynamics in single shot Control over microscopic nature of energy quanta, correlation Also, challenge to quantum information

Charge and spin: Tiny quantities to detect Charge A e =1.6X10-19 C I e Best resolution measurement with low-t and low-noise fa 10 4 electrons/sec Spin Ensemble measurement : 10 4-13 times/sec Magnetic moment µ µ B =gµ B S z = eh = 9.27x10-24 J/T for electron spin 2mc Standard measurement using a Hall device and a SQUID device 10-10 J/T Ensemble measurement: 10 14 spins or How to identify single e and µ?... Manipulate/Readout of quantum information

Orbital and spin degrees of freedom Energy relaxation T 1 ) Slow due to weak Nuclear spin coupling to environment ~> 1 msec Phonon +Spin-orbit Charge = Orbital T 1 ~T 2 ~ nsec due to strong phonon coupling Fast due to strong coupling to phonon bath ~ nsec Spin robust quantum number Spin qubits and quantum computing Use of Quantum Dots (QDs)

Part I, Aug. 21 Basic theory of spin qubits in QDs (Y. Tokura, NTT) Part II, Aug. 26 Experiments and future problems (Prof. S. Tarucha, Univ. Tokyo)

Single and double QDs holding a few electrons! Advent of one-electron single QDs Tarucha et al. PRL 96 Jung et al. APL05 Ciorga et al. PRB 02 Advent of two-electron double QDs nanotube Mason et al. Science 04 Hatano et al. Science 05 Petta et al. Science 04

Energy spectrum of a quantum dot -Hamiltonian: Quantum mechanical effect and interaction effect -Tunneling spectroscopy Conductance - Isolation of single electron

Eigen energy of two-dimensional harmonic QD 2p" H=h 2 k 2 /2m* + V(r) V(r)=(1/2)mω 02 r 2 (n, l)= (1, 0)(0, ±2) 1s" (n, l)=(0, ±1) (n, l)=(0, 0) n radial quantum number l angular momentum quantum number

Single-particle states in a 2D harmonic QD! electron! 0.20 hω 0 12" 0.15 Energy (mev)" 3d! 9" 3d! 6" 2p! 3" 1s! 3s! ev" V (V) 0.10 1s- 3s3d! 0" 2" 4" 6" Magnetic field (T)" c". B 1s- 2p! 0.05 0.0 0 3 6 1s- 1s! B(T)

Model Hamiltonian for an isolated QD Constant Interaction model: H CI = ε k c kσ c kσ + 1 2C (en Q 0) 2 kσ E N = Σ kσ c + kσ c kσ Q 0 =V g C g : offset charge V g, C g : Gate voltage, capacitance N=0 N=1 ε 1 N=2 2ε 1 N=3 2ε 1 +ε 2 U=e 2 /2C: Charging energy C: Total capacitance 0 e/c g 2e/C g 3e/C g V g S= 0 for even N S=1/2 for odd N Even-odd filling, Spin pairing Assumption: Δε k ~2ε F /N for large N and =0 for spin degeneracy U>> k B T (low temperature)

One and two electron states in QD N=1 φ (r) ε 1 = U(1)=ε 1 N=2 φ 2 φ 1 ε 2 ε 1 Δ 12 Hartree GS ES U(2)=2ε 1 + V intra U*(2)=ε 1 + ε 2 + V intra - V ex >, >, > + > Δ 12 >V ex Singlet Triplet Fock (exchange energy)

How do we detect the energy spectra? µ(1) µ(2) µ(3) Increment of µ(n) Single-particle level spacing + Interaction energy Current-sensitive measurement Source lead µ S >µ(n) e ΔN=+1 e I=ef tunnel-rate ΔN=-1 e Drain lead µ(n) >µ D

Probing electronic states in quantum Dot V g Conductance peaks appear every time when the cost of U+δε is paid: Coulomb oscillations : On-site repulsion : Level spacing (=0 for spin pairs) G Peak position: measure of µ(n) - V g. Spin paired states +

Single electron tunneling (SET) transistor Atom-like shell filling 2p! 1s! N=2" 6" 12" N=0" 1" Single electron tunnel=ensemble measurement I=fxe

Coulomb diamond: di/dv sd -V sd and V g Vg + N+1 E F d =µ(n+1) Source E Fd =µ(n) N+1 Drain N Δµ(N+1)=eV sd E F d =µ(n) E F s E Fs =µ(n+1) N E F d ΔE F =ev sd E s F =µ(n) I=0: Coulomb blockade N-1 E Fs =µ(n) E F s N N-1 0 Vsd

Coulomb diamond! -1.8 9! -2.0 Gate Voltage V) -2.2-2.4 4! -2.6-10 -5 0 5 10 Source-Drain Voltage(mV)

Evolution of Coulomb peaks (µ(n)) with B Energy (mev)" 12" 3d: =-2! 9 3s: =0! 3d: =2" 2p: =-1! 6" 2p: =1" 1s: =0 3" 0" 2" 4" 6" Magnetic field (T)" Gate voltage (V) -1.78" -0.78" 3rd shell 2nd shell 1st shell Spin pair! 0 Magnetic field (T) 3"

Excited states Vg + N+1 E F d =µ(n+1) E Fd =µ (N+1) E Fd =µ (N+1) µ(n) µ (N+1) µ (N) N E F d =µ(n) E Fs =µ(n+1) E F s =µ (N) E F s =µ (N) N-1 E Fs =µ(n) µ(n) 0 Vsd

Spin singlet-triplet transition Exact diagonalization! Experiment:! Excitation spectrum! Elextrochemical potential (mev)" Gate Voltage (V)" Magnetic field (T)"

Two-electrons in two quantum dots - Heitler-London state Dot 1 Dot 2 - Exchange coupling N 1 N2 V g1 V g2

Stability diagram of double dot system Vg2! Vg2! (2,0) (2,1) (2,2) (1,0) (1,1) (1,2) (0,0) (1,0) (2,0) (0,1) (0,0) (1,1) (1,0) µ(0,0)= µ(1,0)= µ(1,0) =µ(1,1) Vg1! N 1" N 2" Vg1! N 1" N 2" Vg1" Vg2" µ(n 1 ) and µ(n 2 ) independent Isolated two dots! Vg1" Vg2" Two dots are only coupled " through the gates."

Stability diagram of double dot system V g2" Electrostatic" V g2"electrostatic+quantum mechanical" V inter (0,1) (1,1) (0,0) (1,0) N 1" N 2" Vg1" Vg2" V intra1 V intra2 Capacitive coupling! between two dots! One-electron charging in one dot raises the electrostatic potential of the other dot by E c =e 2 /C inter, " V g1" V g1" (0,1) (0,1) 2t (0,0) (1,0) N 1" N 2" Tunnel coupling! between two dots! Vg1" Vg2" V g1" Degeneracies between different charge states are lifted by the tunnel coupling. The total electron number is only well defined."

Single electron states in coupled dots φ L (r) φ R (r) Anti-symmetric Symmetric N=1 U(1)=µ(N=1:S )=E s E L E A E R 2t E S A Potential offset between two dots: δ E S -E A = 4t 2 + δ

Two electron states: Hund-Mulliken approach ε Three singlet states: µ HM-S (2)=E HM-S (2)-E S (1) E HM S = 2ε + V int er + V ex 4 V int ra V int er V ex t 2 E HM T = 2ε + V int er V ex Exchange energy: E HM-T -E HM-S = 4t 2 /(V intra -V inter -V ex )-2V ex J

Exchange coupling in DQD J ~ 4t 2 /(V intra -V inter ) B ext H exch =JS 1 S 2 H exc =JS 1 S 2 S L S R V intra t, V inter V intra Center gate G. Burkard et al. PRB 00 t is controlled by center gate voltage. J=J(t : V g, B ext )

Offset dependence of exchange J Dot 1 N 1 Dot 2 N2 Two-electron spectra by Hund-Mulliken approach S** V g1 V g1 V g2 Offset δ V g2 Energy relative to E Triplet S* T S Level offset δ

Spin qubit using quantum dots Concept Initialization two-qubit operations

Use electron spin for making qubit Why!? Natural two level system Qubit = a > + b > Correlation of spin exchange H int JS 1 S 2 Robust quantum number Long T 1 and T 2 Scalable in solid state system Toward > 10 4-10 5 S L S R Possible information transfer Charge.useful for measurement Atom.useful for storage Photon.useful for communication Loss and DiVincenzo PRA (98)

Initialization Zeeman splitting E Zeeman = g dot µb ( g dot < g bulk =0.44 GaAs) Polarization =1 - exp [-E Zeeman /k B T] >99 pure state : > at 300mK for E Zeeman (B=8 T) >k B T For fast Initialization Γ Easy Initialization by waiting for a time longer than T 1 (ms) Spin exchange by tunneling between the QD and contact leads Initialization time < Γ -1 ~ nsec V G Γ -1

Universal logic gates Classical calculation completeness {AND, OR, NOT} {NAND} {XOR}.. Quantum calculation {Rotation, CNOT} input output 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 Bit A Bit C If Bit A input 1, then Invert Bit C {Rotation, SWAP 1/2 } S L S R CNOT = XOR can be prepared using SWAP 1/2 CNOT(Nonentangled state)=entangled state.swap 1/2 Entangler

CNOT using exchange coupling D. Loss and D. DiVincenzo, PRA98 U CNOT =Usw 1/2 exp[i(π/2) S 1z ] Usw 1/2 exp[i(π/2) S 1z ] exp[-i(π/2) S 2z ] Square root SWAP of U SW between spin 1 and 2 π/2 rotation of spin 1, 2 about z-axis z e π/2 H int = JS 1 S 2 z e

Effect of fluctuating nuclear field Mixing of singlet and triplet states in a double QD when J < gµ B ΔB nuc If J < gµ B ΔB nuc < E Zeeman, S(1,1)-T 0 (1,1) mixing due to ΔB nuc z ΔB nuc = B 1nuc -B 2nuc Johnson et al. Nature 05 Koppens et al. Sciece 05 Kodera, et al. Physica E 08 T - T 0 T + T - T 0 T + S gµ B ΔB nuc If J, E Zeeman < gµ B ΔB nuc, S(1,1)-T 0 (1,1) mixing due to ΔB nuc z S(1,1)-T ± (1,1) mixing due to ΔB nuc x,y gµ B ΔB nuc E(, ) <> E(, )

J manipulation: SWAP between > and > S(0, ) (, ) SWAP (, ) T 0 =( >+ >)/ 2 t SWAP =360 psec<< t Rabi control tunnel coupling to control SWAP 0> 1> Petta et al. Science 05

Single spin rotation using Global dc B field and local ac B field...on-chip coil...electrically driven spin resonance

Coherent manipulation of single electron spins Local ac magnetic field has been generated by various ways electrically by injecting an ac current to an on-chip coil Koppens et al. Science 2006 and by applying an ac electric field Nowack et al. Science 2007 Pioro-Ladriere et al. Nature Physics 2008 optically by applying an off-resonance laser pulse to induce optical Stark effect F. Jelezko et al. PRL2004 R. Hanson et al. PRL 2006; Science 2008 D. Press et al. Nature 2008 Note: First electrical control of spin qubit made out of > and > using SWAP in double QD Petta et al. Science 2005

Optical control of single electron spins N-V center in diamond F. Jelezko et al. PRL2004 R. Hanson et al. PRL 2006; Science 2008 Self-assembled InGaAs quantum dots D. Press et al. Nature 2008 Ramsey

Concept of spin rotation=esr ESR Hamiltonian Larmor precession µxb 0 > ω 0 =γb 0 Scalable qubits > B 0 ω 0 B AC To manipulate a single spin in a QD: Global DC B 0 + Local AC B AC To manipulate more spins in a multiple QD: Local DC B 0 + Local AC B AC

Paramagnetic defect located near Si/SiO2! Cavity ESR: An ESR peak in the shot noise current signal T2*= 100 nsec, due to spin-spin relaxation Note:T2 ~ 100 µsec for isolated paramagnetic defect T2* ~ 100 ns for Si/SiGe 2DEG

ESR of a single defect! T 2 derived from FWHM = 0.1 µsec, due to spin-spin relaxation Note:T 2 ~ 100 µsec for isolated paramagnetic defect Linear relationship of resonant magnetic field versus microwave frequency: g-factor ~2.0

Straightforward technique: on-chip coil Koppens et al Science 06 B ac induced by I ac flowing a coil ac B DC B 0 ac I I AC = 1 ma B ac ~ 1 mt π rotation: ~ 80 ns accompanied heating and difficulty in localizing the field.problem in qubit scalabilty f Rabi I ac Voltage driven ESR

Spin address with magnetic field gradient Direct NMR observation of local magnetic field generated by micromagnet S. Watanabe et al. Appl. Phys. Lett. b SL ~ -0.38 T/µm 92, 253116 (2008) Proposal of an all Silicon quantum computer T. D. Ladd et al. Phys. Rev. Lett. 89, 017901 (2002)

Selectivity: nano-mri Stray field parallel to external field B 0 50 nm misalignment M z x PNAS 106, 1313 (2009). Tobacco mosaic virus dot 1 dot 2 B x B 0 (mt) 0 d -50 Δ Z -100-0.5 x 1 x 2 x (µm) 0.5 d ~ 100 nm Δ Z ~ 20 mt δb 01 δb 02 E. A. Laird et al., Phys. Rev. Lett. (2007) M. Piore-Ladriere et al. Nat. Phys. 4, 776 (2008) Magnetic field gradient enables selective ESR addressing for each spins like MRI.

Standard spin resonance Time-dependent uniform magnetic field B x (t) Electric dipole Single ESR Electric driven resonance Non-uniform static magnetic field B x (z) Space z Space z B x + 0 - Time t ~20 GHz local B field: B x (t) Time t ~20 GHz local E field: z(t) No heating and better designed for making multiple qubits

Physical systems of non-uniform field g-tensor modulation Spin-orbit Hyperfine int. µ B N (x) Slanting Zeeman field µ-magnet S B x ( ) 0 e Y. Kato Science 2003 K. C. Nowack Science 2007 E. A. Laird PRL (2007) Y. Tokura PRL 2006 g-tensor engineering...optical control B loc =( Vxp)σ...realistic but not general and small in GaAs Δ not-coherent... Local, static B field Micro-magnet induced field gradient slanting Zeeman field

Spin-orbit Interaction (SOI) Rashba SOI HSOI = - Structural inversion symmetry e " 4m 2 c 2 " [ Ε x v ] s Dresselhaus SOI Bulk inversion asymmetry B 0 (001) H SOI = α(-p y σ x + p x σ y ) + β(-p x σ x +p y σ y )

Local B field generation by SOI y [010] [ 110] E ac B ext H spin = H Zeeman + H SOI H Zeeman = -(1/2)g B µb ext σ H SOI = α(-p y σ x + p x σ y ) + β(-p x σ x +p y σ y ) Rashba Dresselhaus [110] x [100] External B ext U = exp[ i m {(αx + βy)σ y (βx + αy)σ x }], H spin U H spin U = 1 2 gµ B[B ext σ x + B SOI (x)σ z ], B SOI (x) = B ext 2m -> position-dependent B loc (α ± β)x + Bext // [110] - Bext // [1-10] Golovach et al., PRB 06,, 44, 17 (2009).

Generic slanting Zeeman fields B SL =b SL x x Effective slanting field by SOI B SOI (x) = B ext 2m Slanting field by on-chip micro-magnet (α ± β)x B magnet = [δb + b SL z]ˆx + b SL xẑ e Effective slanting field by nuclear spin H HF = A Ψ 2 2 0(r j )I j σ j = H 0 HF + A 4 (r rj )Ψ 2 0(r j ){I + j σ + I j σ+ } j H 0 HF 1 2 gµ B{b + HF σ + b HF σ+ } r

AC electric field with a slanting Zeeman field E AC ~ B 0 B SL =b SL x x AC electric field E AC affects the electron charge: H el = ee AC (t) r Introduction of canonical transformation Ψ 0 (r,t) = e ik R(t) Ψ osc 0 (r,t), R(t) ee AC(t) mω0 2 which transforms position operator with a time-dependent displacement r e ik R(t) re ik R(t) = r + R(t) Hence, the slanting field become equivalent to timedependent transverse field: B SL (t) b SL R x (t) ESR Hamiltonian

EDSR with Slanting Zeeman Field by Micro-magnet M Proposal Y. Tokura et al., Phys. Rev. Lett. 96, 047202 (2006) Experiment M. Pioro-Ladrière et al., Nat. Phys. 4, 776 (2008). dc B 0 gate g µ Δ B B z B SL ΔB z E ac V ac Large gradient b SL = 1T/ µm z AC = 1nm Strong effective field B AC = 1mT ΔB z different in two dots. Address two spins independently. Small displacement (V AC ~ 1 mv) No need for spin-orbit coupling, hyperfine interaction or g-factor engineering.

Decoherence problem T 1, T 2, and T 2 * Spin-orbit and nuclear spin coupling

Energy Spin relaxation T 1 ) Nuclear spin Phonon +Spin-orbit Spin relaxation Spin scattering in nonmagnetic semiconductor SO coupling + coupling to phonons T 1 > 100 µsec Khaetskii and Nazarov, PRB (00) for GaAs; Golovach et al. PRL (04) for GaAs QD Slow due to weak coupling to environment > dc B field" Hyperfine coupling to nuclear spins T 1 >> µsec due to DOS discreteness Erlingsson et al. PRB (01); Khaetskii et al. PRL (02) Phonon only influences δb (t) along xy. > Larmor precession : f= gµb./h T 2 =2T 1 > msec

Electrical Pump & Probe Measurement! Fujisawa et al. Nature 419, 278 (02)

Measurement of Spin LifetimeT 1 : SO effect N=2! Triplet! N=1! 2p! 1s! phonon! Singlet! 1s! T 1 > 200 µs" 1s! Zeeman splitting" =gµ B B! B = 7T" 0.4 B // = 8 T T 1 ~ 0.85 ms 0.3 0.2 Fujisawa et al. Nature (2002) 0.0 0.5 1.0 1.5 Waiting time (ms) Elzerman et al. Nature (2004)

Spin-based Quantum Information Processing Energy relaxation time: field dependence T 1 = msec~sec >>Gate operation time ~ns S. Amasha et al. Phys. Rev. Lett. 100, 046803 (2008).

Nuclear Spin Bath Problem Contact interaction to nuclei: 69 Ga, 71 Ga, 75 As (I=3/2) in GaAs QD 50 nm Flip-flop e B ext N=10 5 to 10 6 nuclei in GaAs QD Overhauser shift (ΔE Zeeman ) Shift of ESR condition Nuclear spins are dynamically polarized because of the long lifetime (~min.). Usually very weak for ESR because of the large difference in the Zeeman energy Fal ko et al. J. Phys: Condense Matter 91; Khaetskii et al. ; Erlingsson et al. PRB 01 But can influence the ensemble measurement of ESR and Rabi e e

Decoherence by a nuclear spin bath Electron Zeeman states in the statistically fluctuating nuclear spin bath B tot z B ext B ext =a few T >> B nuc =A/ N ~ a few mt (~30 MHz) B ext µ B z nuc // B ext Fluctuate Larmor frequency by B z nuc /B ext (~0.1 %) but very slowly (~sec) B nuc. Influence of B xy nuc is small, because ΔB xy ~ (B ext 2 + B xy nuc 2 ) 1/2 B ext ~0.0001 % Inhomogeneous broadening of ESR condition Phase fluctuation (or dephasing) in the ensemble measurement T 2 *=10 to 30 ns Bracker et al. PRL 04; Petta et al. Science 05; Koppens et al. Nature 05 Pioro-Ladriere, et al..nature Physics 08, Tokura, Nature Physics 09. Decoherence mechanism by electron spin mediated spin diffusion T2 ~> 1µs L. Cywinski, et al., Phys. Rev. B 79, 245314 (2009). W. A. Coish, et al., Phys. Rev. B 81, 165316 (2010).

Summary Spin qubits with quantum dots Isolation of single electron spin in each quantum dots Exchange control Electrical modulation of exchange energy available for SWAP Exchange control EDSR with Spin-orbit+micromagnet useful for multiple qubits Decoherence problem Spin-orbit interaction Hyperfine coupling with nuclear spin bath