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Mitglied der Helmholtz-Gemeinschaft Quantum Computing with Quantum Dots IFF Spring School 18. March 2009 Mitglied der Helmholtz-Gemeinschaft Quantum Computing with Quantum Dots IFF Spring School 18. March 2009 | Carola Meyer

Why a quantum computer? 18. March 2009 IFF Spring School Folie 2 Why a quantum computer? 18. March 2009 IFF Spring School Folie 2

Quantum computing quantum-bit (qubit) 0 a a 1 0 + a 2 1 = Quantum computing quantum-bit (qubit) 0 a a 1 0 + a 2 1 = a 1 2 1 classical bit 1 ON 3. 2 – 5. 5 V 0 OFF -0. 5 – 0. 8 V calculation decoherence preparation Y 0 H U read-out H-1 time Y|A|Y time exponentially faster for Fourier transformation (Shor algorithm) 18. March 2009 IFF Spring School Folie 3

„Di. Vincenzo“ Criteria Di. Vincenzo: Fortschr. Phys. 48 (2000) 9 -11, pp. 771 -783 „Di. Vincenzo“ Criteria Di. Vincenzo: Fortschr. Phys. 48 (2000) 9 -11, pp. 771 -783 A scalable system with well characterized qubits A qubit-specific measurement capability A („read-out“) The ability to initialize the state of the qubits to a simple fiducial state, e. g. |00. . . 0> A „universal“ set of quantum gates U Long relevant decoherence times, much longer than the gate operation time 18. March 2009 IFF Spring School Folie 4

Outline Part I § Brief introduction to quantum dots and transport - How can Outline Part I § Brief introduction to quantum dots and transport - How can this be used to build a quantum computer? § Measurement of spin states - Fast charge measurement - Spin to charge conversion Part II § Manipulation of single qubits § SWAP: implementation of a two-qubit gate § Relaxation 18. March 2009 IFF Spring School Folie 5

Quantum dots single molecule self-assembled QD 1 nm metallic (superconducting) nanoparticle 18. March 2009 Quantum dots single molecule self-assembled QD 1 nm metallic (superconducting) nanoparticle 18. March 2009 IFF Spring School 10 nm lateral QD 100 nm vertical QD nanotube 1 mm nanowire Folie 6

Confining Electrons in a Semiconductor 18. March 2009 IFF Spring School Folie 7 Confining Electrons in a Semiconductor 18. March 2009 IFF Spring School Folie 7

From 3 D to 0 D 3 D 2 D 0 D D(E) EF From 3 D to 0 D 3 D 2 D 0 D D(E) EF EF E D(E) E 1 D 18. March 2009 IFF Spring School Folie 8

Gate fabrication 18. March 2009 IFF Spring School Folie 9 Gate fabrication 18. March 2009 IFF Spring School Folie 9

Real Quantum Dot structures • Ohmic contacts by RTA of Ni/Au/Ge (diffusion from surface Real Quantum Dot structures • Ohmic contacts by RTA of Ni/Au/Ge (diffusion from surface to 2 DEG) • Electrical control of dot potential and tunnel barriers • Electron spins can be polarized with large B and low T P| 18. March 2009 IFF Spring School = 99. 9% Tel = 150 m. K, B = 7 T, g = 0. 44 Folie 10

transport measurements source drain Vg Kouwenhoven et al. , Science 278, 1788 (`97) 18. transport measurements source drain Vg Kouwenhoven et al. , Science 278, 1788 (`97) 18. March 2009 IFF Spring School Folie 11

Loss & Di. Vincenzo Proposal Loss & Di. Vincenzo, Phys. Rev. A 57, 120 Loss & Di. Vincenzo Proposal Loss & Di. Vincenzo, Phys. Rev. A 57, 120 (1998) gates e e 2 DEG • Quantum dots defined in 2 DEG by gates • Coulomb blockade used to fix number of electrons at one per dot • Electron spin used as Qubit 18. March 2009 IFF Spring School Folie 12

Loss & Di. Vincenzo: Qubit Manipulation Loss & Di. Vincenzo, Phys. Rev. A 57, Loss & Di. Vincenzo: Qubit Manipulation Loss & Di. Vincenzo, Phys. Rev. A 57, 120 (1998) J-gates B e e Bac 2 DEG back gates A-gates high-g layer • Qubit manipulation using spin resonance • Addressing of single qubits by manipulation of g-factor • 2 Qubit operations using J coupling 18. March 2009 IFF Spring School Folie 13

„Di. Vincenzo“ Criteria Di. Vincenzo: Fortschr. Phys. 48 (2000) 9 -11, pp. 771 -783 „Di. Vincenzo“ Criteria Di. Vincenzo: Fortschr. Phys. 48 (2000) 9 -11, pp. 771 -783 A(scalable)system with well characterized qubits A qubit-specific measurement capability A („read-out“) The ability to initialize the state of the qubits to a simple fiducial state, e. g. |00. . . 0> A „universal“ set of quantum gates U Long relevant decoherence times, much longer than the gate operation time 18. March 2009 IFF Spring School Folie 14

Read-Out of Electron Spin Requirements • Charges are measured → Spin to charge conversion Read-Out of Electron Spin Requirements • Charges are measured → Spin to charge conversion • Read-out has to be fast enough → Shorter than T 1 (spin energy relaxation) • Back-action on qubit system should be small → decouple read-out from qubit system 18. March 2009 IFF Spring School Folie 15

QPC as charge detector T Working point: max. sensitivity to electrostatic environment IQPC RESERVOIR QPC as charge detector T Working point: max. sensitivity to electrostatic environment IQPC RESERVOIR 200 nm DRAIN Q MP R SOURCE • Define QPC by negative voltage on R and Q • Tune S-D conductance to last plateau at working point • Change number of electrons in dot: make VM more negative 18. March 2009 IFF Spring School Folie 16

QPC as charge detector T IQPC RESERVOIR 200 nm DRAIN N Q MP R QPC as charge detector T IQPC RESERVOIR 200 nm DRAIN N Q MP R N-1 N-2 SOURCE Reduce number of electrons in dot: Change in charge lifts the electrostatic potential at the QPC constriction, resulting in a step-like feature in IQPC Enhance sensitivity 18. March 2009 IFF Spring School Folie 17

QPC as charge detector T IQPC RESERVOIR 200 nm DRAIN Q MP R SOURCE QPC as charge detector T IQPC RESERVOIR 200 nm DRAIN Q MP R SOURCE Measure differential conductance in IQPC Coulomb oscillations in dot can be detected by QPC highly sensitive charge detector (1/8 e) 18. March 2009 IFF Spring School allows to study QD even when isolated from reservoirs (s. Qu. Bits) Folie 18

Read-Out of Electron Spin Requirements • Charges are measured → Spin to charge conversion Read-Out of Electron Spin Requirements • Charges are measured → Spin to charge conversion • Read-out has to be fast enough → Shorter than T 1 (spin energy relaxation) • Back-action on qubit system should be small → decouple read-out from qubit system 18. March 2009 IFF Spring School Folie 19

RESERVOIR How fast is the charge detection? T DRAIN (a) IQPC Q G 200 RESERVOIR How fast is the charge detection? T DRAIN (a) IQPC Q G 200 nm M P R (b) SOURCE • VSD = 1 m. V • IQPC ~ 30 n. A • ∆IQPC ~ 0. 3 n. A • shortest steps ~ 8 µs 18. March 2009 IFF Spring School • Observation of singel tunneling events • Spontaneous back and forth tunneling between dot and reservoir (a) electron predominantly in reservoir (b) electron predominantly in dot Folie 20

Pulsed-induced tunneling response to electron tunneling DIQPC (n. A) 0. 8 response to pulse Pulsed-induced tunneling response to electron tunneling DIQPC (n. A) 0. 8 response to pulse 0. 4 0. 0 -0. 4 0 18. March 2009 IFF Spring School 0. 5 1. 0 Time (ms) 1. 5 Real time single electron tunneling Folie 21

Histograms tunnel time G ~ (60 ms)-1 G ~ (230 ms)-1 Increase tunnel barrier Histograms tunnel time G ~ (60 ms)-1 G ~ (230 ms)-1 Increase tunnel barrier 18. March 2009 IFF Spring School Folie 22

Spin read out principle: convert spin to charge SPIN UP 0 -e time N=1 Spin read out principle: convert spin to charge SPIN UP 0 -e time N=1 charge SPIN DOWN 0 -e N=1 18. March 2009 IFF Spring School N=0 N=1 ~G-1 time Folie 23

Initialization Energy selective tunneling • spin-up will stay in dot • spin down will Initialization Energy selective tunneling • spin-up will stay in dot • spin down will tunnel • wait a few tunneling processes (high polarization in state) • fast initialization process 18. March 2009 IFF Spring School Folie 24

Read-Out of Electron Spin Requirements • Charges are measured → Spin to charge conversion Read-Out of Electron Spin Requirements • Charges are measured → Spin to charge conversion • Read-out has to be fast enough → Shorter than T 1 (spin energy relaxation) • Back-action on qubit system should be small → decouple read-out from qubit system 18. March 2009 IFF Spring School Folie 25

DIQPC Vpulse Spin read-out procedure empty QD 18. March 2009 IFF Spring School inject DIQPC Vpulse Spin read-out procedure empty QD 18. March 2009 IFF Spring School inject & wait read-out spin empty QD Nature 430, 431(2004) Folie 26

Spin read-out results empty QD DIQPC (n. A) DIQPC Vpulse Elzerman et al. , Spin read-out results empty QD DIQPC (n. A) DIQPC Vpulse Elzerman et al. , Nature 430, 431, 2004 inject & wait 2 read-out spin “SPIN UP” empty QD “SPIN DOWN” 1 0 0 18. March 2009 IFF Spring School 0. 5 1. 0 Time (ms) 1. 5 0 0. 5 1. 0 Time (ms) 1. 5 Folie 27

More spin down traces DIQPC (n. A) twait tread 2 1 Threshold value thold More spin down traces DIQPC (n. A) twait tread 2 1 Threshold value thold 0 0 0. 5 1. 0 1. 5 Time (ms) thold : time the electron spends in the dot tdetect : 1/G 1 tunneling time 18. March 2009 IFF Spring School Folie 28

Spin down fraction Verification spin read-out Waiting time (ms) 18. March 2009 IFF Spring Spin down fraction Verification spin read-out Waiting time (ms) 18. March 2009 IFF Spring School Spin flip Folie 29

Measurement of T 1 B=8 T T 1 ~ 0. 85 ms B = Measurement of T 1 B=8 T T 1 ~ 0. 85 ms B = 10 T T 1 ~ 0. 55 ms • • Surprisingly long T 1 goes up at low B B = 14 T T 1 ~ 0. 12 ms Elzerman et al. , Nature 430, 431, 2004 18. March 2009 IFF Spring School Folie 30

Read-Out of Electron Spin Requirements • Charges are measured → Spin to charge conversion Read-Out of Electron Spin Requirements • Charges are measured → Spin to charge conversion • Read-out has to be fast enough → Shorter than T 1 (spin energy relaxation) • Back-action on qubit system should be small → decouple read-out from qubit system 18. March 2009 IFF Spring School Folie 31

„Di. Vincenzo“ Criteria Di. Vincenzo: Fortschr. Phys. 48 (2000) 9 -11, pp. 771 -783 „Di. Vincenzo“ Criteria Di. Vincenzo: Fortschr. Phys. 48 (2000) 9 -11, pp. 771 -783 A(scalable)system with well characterized qubits A qubit-specific measurement capability A („read-out“) The ability to initialize the state of the qubits to a simple fiducial state, e. g. |00. . . 0> A „universal“ set of quantum gates U Long relevant decoherence times, much longer than the gate operation time 18. March 2009 IFF Spring School Folie 32

quantum measurement Any more questions about this point? 18. March 2009 IFF Spring School quantum measurement Any more questions about this point? 18. March 2009 IFF Spring School Folie 33

Drawbacks of read-out So far: energy-selective read-out (E-RO) Drawbacks: (1) energy splitting must be Drawbacks of read-out So far: energy-selective read-out (E-RO) Drawbacks: (1) energy splitting must be larger than thermal energy (2) very sensitive to fluctuations in electrostatic potential (3) high-frequency noise can spoil E-RO (photo-assisted tunneling) 18. March 2009 IFF Spring School Folie 34

Alternative read-out scheme Now: tunnel-rate-selective read-out (TR-RO) >> (1) t = 0 : position Alternative read-out scheme Now: tunnel-rate-selective read-out (TR-RO) >> (1) t = 0 : position both levels above chemical potential (2) electron will tunnel regardless of spin state (3) t = t: with >> >> high PR that electron was in state ES low PR that electron was in state GS 18. March 2009 IFF Spring School Folie 35

Alternative read-out scheme Now: tunnel-rate-selective read-out (TR-RO) >> Advantage: (1) does NOT rely on Alternative read-out scheme Now: tunnel-rate-selective read-out (TR-RO) >> Advantage: (1) does NOT rely on large energy splitting (2) robust against background charge fluctuations (cause small variation of tunneling rate) (3) photon-assisted tunneling not important 18. March 2009 IFF Spring School Folie 36

Singlet-triplet read-out Experimental conditions: (1) can be achieved in Quantum Hall regime, where high Singlet-triplet read-out Experimental conditions: (1) can be achieved in Quantum Hall regime, where high spin-selectivity is induced by spatial separation of spin-resolved edge channels (2) can be used for read-out of two-electron dot with electrons in (a) spin singlet ground state (b) spin triplet state 18. March 2009 IFF Spring School Folie 37

Single-shot read-out 18. March 2009 IFF Spring School Folie 38 Single-shot read-out 18. March 2009 IFF Spring School Folie 38

Single-shot read-out 18. March 2009 IFF Spring School Folie 39 Single-shot read-out 18. March 2009 IFF Spring School Folie 39

On chip generation of oscillating magnetic fields On-chip design Minimum field Bac = 5 On chip generation of oscillating magnetic fields On-chip design Minimum field Bac = 5 m. T f. Rabi ~ 30 MHz Single Qubit gate operation 1/2 f. Rabi ~ 15 ns Compare to spin coherence time dissipation: 10 m. W at 1 m. T 250 m. W at 5 m. T thermal “budget” dilution fridge: 300 m. W at 100 m. K 18. March 2009 IFF Spring School Folie 51

Basics of electron spin resonance energy m. S = 1/2 B 0 magnetic field Basics of electron spin resonance energy m. S = 1/2 B 0 magnetic field modulation m. S = -1/2 DE = hn = giµBB 0 = 30 µe. V für n = 9 GHz 18. March 2009 IFF Spring School Folie 52

Detection of continuous wave ESR Ground state Engel & Loss, PRL 86, 4648 (01) Detection of continuous wave ESR Ground state Engel & Loss, PRL 86, 4648 (01) AC field lifts Coulomb blockade Simple concept: BUT hard to prove that signal in current is due to single spin rotation 18. March 2009 IFF Spring School Folie 53

Photon-assisted tunneling Electric field couples to charge for G< f: - Electron in dot Photon-assisted tunneling Electric field couples to charge for G< f: - Electron in dot absorbs photon (N+1) → N - Electron in lead absorbs photon N → (N+1) Two side-peaks arise e 0 - hf N electrons 18. March 2009 IFF Spring School e 0 + hf N+1 electrons Folie 54

Spin manipulation and detection 18. March 2009 IFF Spring School Pull dot levels far Spin manipulation and detection 18. March 2009 IFF Spring School Pull dot levels far below Fermi level to avoid PAT Pulse spin down level in bias window Switch on hf to change the spin state Initialization Single shot read-out Folie 56

Spin manipulation and detection Double quantum dot with one electron in the right dot Spin manipulation and detection Double quantum dot with one electron in the right dot T(0, 2) S(0, 2) by spin blockade 18. March 2009 IFF Spring School Pull dot levels far below Fermi level to avoid PAT Pulse spin down level in bias window Switch on hf to change the spin state Initialization Single shot read-out Read-out by lifted spin blockade Folie 57

Coherent Rabi oscillations 18. March 2009 IFF Spring School Folie 58 Coherent Rabi oscillations 18. March 2009 IFF Spring School Folie 58

Coherent Rabi oscillations Idot large Idot small 18. March 2009 IFF Spring School Folie Coherent Rabi oscillations Idot large Idot small 18. March 2009 IFF Spring School Folie 59

SWAP gate implementation in a Double Quantum Dot Few electron double quantum dot • SWAP gate implementation in a Double Quantum Dot Few electron double quantum dot • Fully tunable 2 Qubit system • Quantum point contact (QPC) as charge detector • Measure d. IQPC/d. VL : change of total electron number in double dot • VL controls number of electrons in left dot • VP controls number of electrons in right dot R 18. March 2009 IFF Spring School Folie 60

Current in a double quantum dot Vtgl (2, 0) (2, 1) Vtgm Vtgr (2, Current in a double quantum dot Vtgl (2, 0) (2, 1) Vtgm Vtgr (2, 2) Vleft source (1, 0) (1, 1) (0, 0) (0, 1) drain (1, 2) (0, 2) Vleft 18. March 2009 IFF Spring School Vright Folie 61

Current in a double quantum dot 4 12 103 8 (0, 2) 2 6 Current in a double quantum dot 4 12 103 8 (0, 2) 2 6 (2, 1) 41 2 0 0 Vtgl Vtgm Vtgr (2, 2) Vleft source (1, 0) (1, 1) drain (1, 2) h e (0, 0) (0, 1) (0, 2) Vleft 18. March 2009 IFF Spring School Vright Folie 62

Two electron double quantum dot e=0 VL VR • QPC can detect all charge Two electron double quantum dot e=0 VL VR • QPC can detect all charge transitions • 2 electron double quantum dot • Tuned between (1, 1) and (0, 2) state 18. March 2009 IFF Spring School Folie 63

Spin configurations in a DQD Spin-Singlet Spin-Triplet S=0 S = 1; ms = +1, Spin configurations in a DQD Spin-Singlet Spin-Triplet S=0 S = 1; ms = +1, 0, -1 antisymmetric 18. March 2009 IFF Spring School Folie 64

Hyperfine coupling in a DQD • Ga and Ar have a nuclear spin: about Hyperfine coupling in a DQD • Ga and Ar have a nuclear spin: about 106 nuclear spins in a quantum dot • Electrons feel a magnetic field due to hyperfine interaction with these nuclei “Overhauser field” • Nuclear spins are not fully polarized fluctuations lead to a field • Singlet and Triplet states become mixed • In an external magnetic field in , |S and |T 0 become mixed 18. March 2009 IFF Spring School Folie 65

Harvard scheme spin selection rules: Singlet ground state • (1, 1) S can tunnel Harvard scheme spin selection rules: Singlet ground state • (1, 1) S can tunnel to (0, 2) S • (1, 1) T to (0, 2) S transition is blocked Tilt potential: new charge ground state If charge does NOT return to (0, 2) state, spin (1, 1) dephasing during time ts B > 0: (1, 1) S and (1, 1) To mixing t = ts: transfer to (0, 2) ground state 18. March 2009 IFF Spring School Folie 66

Harvard scheme Interdot tunneling: • hybridization (0, 2) – (1, 1) • exchange splitting Harvard scheme Interdot tunneling: • hybridization (0, 2) – (1, 1) • exchange splitting J(e) Strength of J(e) controlled by gates B = 100 m. T perp. field 18. March 2009 IFF Spring School Folie 67

The logical Qubit T 2* ~ 8 ns How long can the electrons be The logical Qubit T 2* ~ 8 ns How long can the electrons be separated spatially before they loose phase coherence? 12 3 1. prepare singlet (0, 2) S 2. rapid pulse (1 ns) : slow compared to tunnel splitting separated singlet 3. separation time ts: rapid back projection into (0, 2) S state 18. March 2009 IFF Spring School Folie 68

Spin swap and Rabi oscillations Slow detuning: rotate for J into 0 18. March Spin swap and Rabi oscillations Slow detuning: rotate for J into 0 18. March 2009 IFF Spring School Folie 69

Spin swap and Rabi oscillations Read-out 18. March 2009 IFF Spring School Folie 70 Spin swap and Rabi oscillations Read-out 18. March 2009 IFF Spring School Folie 70

Spin swap and Rabi oscillations 18. March 2009 IFF Spring School turn on J(e) Spin swap and Rabi oscillations 18. March 2009 IFF Spring School turn on J(e) Folie 71

Spin SWAP and Rabi oscillations 18. March 2009 IFF Spring School p 3 p Spin SWAP and Rabi oscillations 18. March 2009 IFF Spring School p 3 p 5 p Folie 72

A universal set of quantum gates Single qubit rotations and the CNOT gate form A universal set of quantum gates Single qubit rotations and the CNOT gate form a universal set • Single qubit rotations Idot (f. A) 100 • CNOT can be composed from single qubit rotations and √SWAP Rotation of spin 2 18. March 2009 IFF Spring School Rotation of spin 1 Folie 73

„Di. Vincenzo“ Criteria Di. Vincenzo: Fortschr. Phys. 48 (2000) 9 -11, pp. 771 -783 „Di. Vincenzo“ Criteria Di. Vincenzo: Fortschr. Phys. 48 (2000) 9 -11, pp. 771 -783 A(scalable)system with well characterized qubits A qubit-specific measurement capability A („read-out“) The ability to initialize the state of the qubits to a simple fiducial state, e. g. |00. . . 0> A „universal“ set of quantum gates U Long relevant decoherence times, much longer than the gate operation time 18. March 2009 IFF Spring School Folie 74

Entanglement and decoherence 18. March 2009 IFF Spring School Folie 75 Entanglement and decoherence 18. March 2009 IFF Spring School Folie 75

Singlet-triplet spin echo • refocus separated singlet to undo inhomogeneous dephasing • apply p Singlet-triplet spin echo • refocus separated singlet to undo inhomogeneous dephasing • apply p pulse by pulsed J(e) 18. March 2009 IFF Spring School Folie 76

Singlet-triplet spin echo Singlet probability as a function of detuning and t. E. singlet Singlet-triplet spin echo Singlet probability as a function of detuning and t. E. singlet recovery 18. March 2009 IFF Spring School Folie 77

Singlet-triplet spin echo 18. March 2009 IFF Spring School Folie 78 Singlet-triplet spin echo 18. March 2009 IFF Spring School Folie 78

Spin-spin relaxation times Spin dephasing time: ~ 8 ns Spin coherence time: ~ 1. Spin-spin relaxation times Spin dephasing time: ~ 8 ns Spin coherence time: ~ 1. 2 ms Time for √SWAP: ~ 180 ps about 7000 √SWAP operations can be performed during T 2 However 18. March 2009 IFF Spring School Folie 79

„Di. Vincenzo“ Criteria Di. Vincenzo: Fortschr. already buy a quantum computer ? Why can’t „Di. Vincenzo“ Criteria Di. Vincenzo: Fortschr. already buy a quantum computer ? Why can’t we Phys. 48 (2000) 9 -11, pp. 771 -783 ( ) A(scalable)system with well characterized qubits A qubit-specific measurement capability A („read-out“) The ability to initialize the state of the qubits to a simple fiducial state, e. g. |00. . . 0> A „universal“ set of quantum gates U Long relevant decoherence times, much longer than the gate operation time 18. March 2009 IFF Spring School Folie 80

Spin energy relaxation spin system is in excited state |1 relaxation to ground state Spin energy relaxation spin system is in excited state |1 relaxation to ground state due to spin-phonon interaction read-out within T 1 nuclei: T 1 ~ hours – days electrons: T 1 ~ ms |0 d. Mz M - M 0 = g (Mx(t)By - My(t)Bx) - z dt T 1 18. March 2009 IFF Spring School Folie 81

Origin of spin-phonon coupling Spin-orbit interaction is the most important contribution HSO cannot couple Origin of spin-phonon coupling Spin-orbit interaction is the most important contribution HSO cannot couple different spin states of the same orbital New eigenstates can couple to the electric field Lattice vibrations lead to fluctuations of the electric field Spin relaxation 18. March 2009 IFF Spring School Folie 82

Different contributions new eigenstates Only acoustic phonons are relevant → linear dispersion relation Matrix Different contributions new eigenstates Only acoustic phonons are relevant → linear dispersion relation Matrix element: Piezoelectric phonons dominate Phonon wavelength much larger than dot size 18. March 2009 IFF Spring School Folie 83

Breaking time reversal symmetry All contributions would cancel out without magnetic field applied “van Breaking time reversal symmetry All contributions would cancel out without magnetic field applied “van Vleck” cancellation Follow one period of lattice vibration (harmonic oscillator) SO SO B 0 18. March 2009 IFF Spring School Folie 84

Magnetic field dependence All contributions add up to: 18. March 2009 IFF Spring School Magnetic field dependence All contributions add up to: 18. March 2009 IFF Spring School G DEZee 5 Folie 85

Decoherence due to dephasing spins magnetization in x, y-plane (superposition) |1 |1 superposition decays Decoherence due to dephasing spins magnetization in x, y-plane (superposition) |1 |1 superposition decays because of dephasing Slow fluctuations can be refocused However: Time ensemble is needed for presented Hahn-echo |0 |0 From one Hahn-Echo sequence to the next nuclear field takes a new, random and unknown value 18. March 2009 IFF Spring School Folie 86

Magnetic field fluctuations Unknown magnetic field electron spin evolves in an unknown way BN Magnetic field fluctuations Unknown magnetic field electron spin evolves in an unknown way BN Gaussian distribution with standard deviation In experiment: T 2* = 10 ns ^ = BN = 2. 3 m. T Reduce dephasing Find a way to decrease s of magnetic field 18. March 2009 IFF Spring School Folie 87

Summary Proposal for quantum computing with quantum dots electron spin as qubit exchange interaction Summary Proposal for quantum computing with quantum dots electron spin as qubit exchange interaction as qubit coupling Single spin read-out spin to charge conversion quantum point contact as charge detector spin-energy relaxation time (T 1) measurement Quantum gates single spin rotation SWAP operation between two qubits spin-phase relaxation time (T 2) measurement Origin of spin relaxation spin orbit coupling (T 1) nuclear hyperfine field (T 2) 18. March 2009 IFF Spring School Folie 88

Outlook Why can’t we already buy a quantum computer ? • All necessary components Outlook Why can’t we already buy a quantum computer ? • All necessary components not yet implemented in the same device • Gate implementation still too slow • Scaling to ~1000 qubits not straight forward Any solutions possible? • Improve T 2 : Polarize nuclei to >99% Find materials without nuclear spins and SO coupling → carbon based (graphene, carbon nanotubes) → silicon (2 DEG charge carrier mobility too low) 18. March 2009 IFF Spring School Folie 89

Dilbert 18. March 2009 IFF Spring School Folie 90 Dilbert 18. March 2009 IFF Spring School Folie 90