Single Spin Qubits, Qubit Gates and Qubit Transfer with Quantum Dots

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1 International School of Physics "Enrico Fermi : Quantum Spintronics and Related Phenomena June 22-23, 2012 Varenna, Italy Single Spin Qubits, Qubit Gates and Qubit Transfer with Quantum Dots Seigo Tarucha Dep. of Appl. Phys. The Univ. of Tokyo

2 Outline Micro-magnet technique for qubits and two qubit gates Two-qubit gate for controlling entanglement Fast Z-rotation and CPHASE Qubit transfer Multiple QDs array Electron transfer between between distant QDs Joint project with Dr. Bauerle and Munier, Grenoble

3 Double QD for Qubit Gates Individual two qubits M. Pioro-Ladrier et al. Nat. Phys. 2008; T. Obata et a. PRB 2010 Two qubit gate for controlling entanglement R. Brunner et al. PRL > = >, 1>= > Universal set of quantum gates Time control of spin rotation single qubit 0 1 Time control of spin exchange coupling entangled state 01 10

4 Qubit Hamiltonian Well-defined two states 0> and 1> Hybridization of two states H qubit = ( z /2)[ 0><0-1><1 ] + ( x (t)/2)[ 0><1 + 1><0 ] = + = ( z /2) z + ( x /2) x z Define two eigen states x Mix up two eigen states z 1> z /2 0> - z /2 0>= > 1>= >

5 0>= S> 1>= T 0 > S-T 0 Qubit Hamiltonian H qubit = (-J/2)[ S><S - T 0 ><T 0 ] + g B B nz [ S><T 0 + T 0 ><S ] = (-J/2) z + g B B z nuc x S> J Temporal J manipulation with inter-dot detuning: J >> B z nuc > > > = > z B nuc > = > J << B z nuc S> T 0 > T 0 >

6 Charge Qubit with a Tunnel-coupled Double QD 0 e 1 e E L E 0 E R Non-adiabatic 2t Microwave E 0 = (E L + E R )/2 = 0 E L =- /2, E R = /2 H = (-1/2)[ ( 0><0-1><1 )-2t( 0><1 + 1><0 )] =(-1/2)[ z +2t x ] where B 0 = (,0, ) and B AC =(2t,0,0) Fictitious magnetic fields for universal rotation z B 0 /ħ =ħ B 0 2t= ħ B 1 cos( t) y x B AC 2t/ħ

7 Single Spin Qubit with QD 0>= > 1>= > > Qubit concept = Electron spin resonance Loss and DiVincenzo PRA 1998 B = B 0 z + B 1 xcos t H ESR = -(1/2) ħ 0 z (1/2) ħ 1 x cos t B 1 B 0 E Zeeman =hf AC B AC > Apply ESR for single dot with a single electron using a local AC B 1 field Single spin qubits: (GaAs) Koppens et al. Science 2006 Nowack et al. Science 2007 Pioro-Ladriere, et al. Nat. Phys Obata et al. PRB 2010 Brunner et al. PRL 2011 (InAs) Nadj-Perge et al. Nature 2010

8 Qubit Hamiltonian for Spin Rotation Time evolution H qubit = (- z /2) z + (- x /2) x Larmor precession

9 Qubit Gates ROTATION R x ( ) =2 f Rabi t z R z > R y ( ) = R x ( : t t- /2) =2 f Rabi t x y R z ( ) = R x ( /2)R y ( )R x ( /2) R x > R y NOT : R x ( ) HADAMARD : R x ( /2) Temporal detuning of E Zeeman 2 t f L ( E Zeeman =h f L )

10 Local AC B Field Driven by AC Current or AC Voltage AC current to on-chip coil ac B Our proposal : Micro-magnet technique Tokura et al. PRL 2006 DC B 0 ac I -magnet I AC = 1 ma B ac ~ 1 mt rotation: ~ 80 ns Koppens et al. Science 2006 B x (x) 0 e From voltage to B field Usig spin-orbit interaction Pioro-Ladriere et al. Nat. Phys B loc =(Exp) Oscillation of an electron under a stray field by ac voltage S Nowack et al. Science 2007

11 Spin Qubit with a Micromagnet Technique

12 Local B Field Generation by SO Effect y [010] [110] x [100] [ 110] E ac B ext H spin = H Zeeman + H SO H Zeeman = (1/2)g B B ext H SO = (-p y x + p x y ) + (-p x x +p y y ) Rashba Dresselhaus l SO -1 =m*/( + )h for B ext, E ac //[110] =m*/( - )h for B ext, E ac //[110] f Rabi =(g B B eff )/2h E ac B eff Levitov and Rashba, PRB 2003 Golovach et al. PRB 2006

13 Spin Qubit Using SO Effect : GaAs QD [110] Nowack et al. Science 2007 f Rabi E ac B ext Rabi oscillations f Rabi E ac E ac increase B ext

14 Micro-magnet for Spin Qubit with Quantum Dot x B ext Stray field magnet B 0 Simulation 300 nm M Co = 1.8 T M 70 nm 80 nm z Out-of plane B x (z) f ac ~T/ m B ac = B 1 xcos2 f ac In-plane B z = excess local Zeeman field 10 to50 mt/0.1 m B 0 = B ext + B z f ESR B x (mt) dot 1 dot b SL z ( m) 0.3 m b SL ~ 0.6 T/ m (saturation) 90 nm

15 Addressing Two Individual Spin Qubits To manipulate more spins in a multiple QD: Local DC B 0 + Local AC B AC B ext = B ESR B z1 =B ESR B z2 ESR at two different B ext Local Zeeman field B 0 =B ext + B z g B B ESR =hf MW MW In-plane stray field B z at each dot depends on the micro-magnet geometry M M M M

16 ESR/Qubit Readout using Pauli Effect Formation of a triplet state blocks electron transition Pauli spin blockade Ono and ST, Science 2002 PRL 2004 I QPC E Zeeman = hf P-SB is lifted by spin flip most sensitive spin information detector QPC current External field

17 Two Spin Qubits in Double Quantum Dot QPC charge sensing to detect to charge change in the double dot Left dot Right 1T CW EDSR Left spin B 0L Right spin Right dot 15 mt B 0R B 0R B 0L Left dot -17 dbm, f Rabi 0.85 MHz -18 dbm, f Rabi 0.75 MHz -19 dbm, f Rabi 0.70 MHz -21 dbm, f Rabi MHz T. Obata et al. PRB 2010 R. Brunner et al. PRL 2011

18 Rabi Frequencies vs. MW Field Rabi frequency ~ P MW 1/2 = E MW Left spin Right spin Brunner et al. PRL2010 Obata et al. PRB 2010

19 Entanglement Control

20 Temporal Operation of Spin Exchange Coupling Exchange = (-J/4) 1 2 H exc = (-J/4) 1 2 = -J(1/4)(I ) + (J/4)I = (-J/2)U SWAP + (J/4)I Time evolution exp[-ih exc t/ħ] = exp[iju SWAP t/2ħ]exp[-ijt/4ħ] = Icos(Jt/2ħ) + iu SWAP sin(jt/2ħ) = iu SWAP for Jt=h/2 U SWAP for Jt=h/4

21 How to control exchange coupling? S(0,2) S(1,1) S(0,2) (0,1) (1,2) (2,2) Detuning N 2 S(1,1) Detuning (2,1) S(1,1) (N 1,N 2 )=(0,0) (1,0) S(2,0) S(1,1) S(2,0) N 1

22 Two-electron States in Double QD Tokura et al. Springer 2009 Small detuning : J=4t 2 /U H Large detuning: J E triplet =0 S(2,0) S(0,2)-S(1,1) T(1,1) S(1,1) S(0,2) S(2,0)-S(1,1) T - T 0 (1,1) T + S(1,1), S(0,2) T +, T -, T 0 (1,1) Energy detuning = E 1 -E 2

23 Control of Pauli Blockade and Exchange Coupling K. Ono et al. Science 2002 Pauli spin blockade J = E triplet -E singlet S(0,2) S(0,2)-S(1,1) S(0,2) S(0,2)-S(1,1) (1,2) (0,1) (2,2) Detuning S(1,1) (2,1) Energy T 0 J~ 2t T - T + J>0 J S(1,1) J~0 T(1,1) (N 1,N 2 )=(0,0) (1,0) S(2,0) S(0,2) Detuning 0

24 Quantum Gate Operation with Double QD Pauli spin blockade J = E triplet -E singlet S(0,2) S(0,2)-S(1,1) Use for initialize and readout Energy T 0 J~ 2t T - T + J>0 J switch for entanglement control J~0 Spin rotation at J=0 S(1,1) (2,1) Exchange control at J 0 S(0,2) 0 Detuning

25 Concurrence of Two Qubit Entanglement Magic basis: Bell states Hill and Wooters PRL

26 Two-qubit Gate to Control and Detect Single State Initialization > > /2 rotation Exchange control Preparation > > with J=0 Exchange operation J ex = ħ ex with finite J ex = /2 for SWAP = for SWAP ex cos ex sin ex exp ex 1 Readout of S 0 using Pauli effect Change of charge state from (1,1) to (0.2) What we measure is P: (1,1) to (0,2) P:(1,1) (0,2) Concurrence= sin2 ex Hill and Wooters, PRL ex / SWAP

27 Controlled Manipulation of the Two Spin Exchange exc :SWAP 2 > = (1/2)(T + - T - - 2S 0 ) R. Brunner et al. PRL 2011 Single weight: P bright ( ex ) 1/2 : simple product SWAP 2 > = SWAP 2n 2 > = T + 1 [(2+i)T+ -it - +i 2S 0 )] 2 3/2 1/4: partially entangled 0 : simple product 0 P:(1,1) (0,2)=P(S 0 ) Concurrence ex / SWAP

28 R. Brunner et al. PRL 2011 Exchange + Spin rotation Probability of finding the singlet in the output 2 > Calculation of concurrence using parameters derived from experiments SWAP fidelity ~ 98% Rabi fidelity ~ 50%

29 CNOT with rotations of 1 spin and SWAP/SWAP 1/2 Loss & DiVincenzo, PRA 1998 Quantum circuit for CNOT > > > > H z z z H > > > > Input Control-PHASE Output z R z > Rotation about z R z (θ) = XR y (θ) X = Y R x ( θ) Y Preparation of 4 logical basis states X 2 +SWAP x R x y R y >

30 Connection between Distant Qubit Systems Multiple qubit system in multiple QDs Connection between distant quantum systems with qubit state transfer a > + b > Photons:Quantum cryptography, communication

31 Inhomogenous B Field Induced by Micro-magnet Multiple quantum dots in an inhomogeneous Zeeman field 150 nm 13 bits 2DEG depth -1 m 1 m 200 nm 50 nm T. Takakura et al. APL 2011

32 Electron Transfer by Surface Acoustic Wave (SAW) IDT piezoelectric (GaAs) 2ef SAW 1 nsec for 3 m << spin T 2 I = ef SAW

33 SAW Induced Electron Transport d d Gate electrode Energy I = nef

34 Single Electron Transfer between Distant QDs QD 3 m QD Electron loading in source QD Source QD Charge states of two QDs probed by two QPC charge sensors Detector QD High I QPC Low I QPC

35 Single Electron Transfer between Distant QDs Taking out an electron in SAW Source QD Detector QD High I QPC High I QPC

36 Single Electron Transfer between Distant QDs Putting an electron in SAW onto detector QD Source QD Detector QD Low I QPC High I QPC

37 Single Electron Transfer between Distant QDs 16m-D 2 3 Before SAW SAW > 94 % SAW Before SAW After SAW After SAW Before SAW Before SAW After SAW After SAW Transfer time < T 2 * (=25 nsec )

38 McNeil et al. Nature 477, 439 (2011)

39 Summary Developed a micro-magnet technique for making single spin qubits, which is applicable to non-magnetic material systems, i.e. Si, C-based. Realized a two-qubit gate of combined spin rotation and exchange control, which enables control and detect the partial entanglement. Proposed micro-magnet techniques to implement fast z-rotation and CPHASE. Proposed a SAW technique of transferring qubit state between distant QDs.