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

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

18. March 2009 IFF Spring SchoolFolie 3 Quantum computing calculationpreparationread-out time classical bit 1  ON  3.2 – 5.5 V 0  OFF  -0.5 – 0.8 V exponentially faster for Fourier transformation (Shor algorithm) quantum-bit (qubit)  0     1    a 1  0  + a 2  1  = a1a1 a2a2 HH -1 U    |A|    time   decoherence

18. March 2009 IFF Spring SchoolFolie 4 „DiVincenzo“ Criteria DiVincenzo: Fortschr. Phys. 48 (2000) 9-11, pp 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 SchoolFolie 5 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 SchoolFolie 6 Quantum dots single molecule metallic (superconducting) nanoparticle self-assembled QDnanotube nanowire 1 nm10 nm 1m1m vertical QD lateral QD 100 nm

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

18. March 2009 IFF Spring SchoolFolie 8 From 3D to 0D E D(E) EFEF EFEF E 3D 2D 1D 0D

18. March 2009 IFF Spring SchoolFolie 9 Gate fabrication

18. March 2009 IFF Spring SchoolFolie 10 Real Quantum Dot structures Ohmic contacts by RTA of Ni/Au/Ge (diffusion from surface to 2DEG) Electrical control of dot potential and tunnel barriers Electron spins can be polarized with large B and low T T el = 150 mK, B = 7 T, g = 0.44 |  P = 99.9%

18. March 2009 IFF Spring SchoolFolie 11 transport measurements sourcedrain VgVg sourcedrain VgVg sourcedrain VgVg Kouwenhoven et al., Science 278, 1788 (`97)

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

18. March 2009 IFF Spring SchoolFolie 13 Loss & DiVincenzo: Qubit Manipulation 2DEG gates eeee high-g layer B Addressing of single qubits by manipulation of g-factor Loss & DiVincenzo, Phys. Rev. A 57, 120 (1998) back gates Qubit manipulation using spin resonance B ac 2 Qubit operations using J coupling J-gates A-gates

18. March 2009 IFF Spring SchoolFolie 14 „DiVincenzo“ Criteria DiVincenzo: Fortschr. Phys. 48 (2000) 9-11, pp 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 SchoolFolie 15 Read-Out of Electron Spin Requirements Read-out has to be fast enough → Shorter than T 1 (spin energy relaxation) Charges are measured → Spin to charge conversion Back-action on qubit system should be small → decouple read-out from qubit system

18. March 2009 IFF Spring SchoolFolie 16 QPC as charge detector I QPC DRAIN SOURCE RESERVOIR 200 nm M R Q T P 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 V M more negative Working point: max. sensitivity to electrostatic environment

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

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

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

18. March 2009 IFF Spring SchoolFolie 20 How fast is the charge detection? I QPC DRAIN SOURCE RESERVOIR  200 nm MP R Q T V SD = 1 mV I QPC ~ 30 nA ∆I QPC ~ 0.3 nA Observation of singel tunneling events Spontaneous back and forth tunneling between dot and reservoir (a) electron predominantly in reservoir (b) electron predominantly in dot (a) (b) shortest steps ~ 8 µs

18. March 2009 IFF Spring SchoolFolie 21 Pulsed-induced tunneling response to pulse  I QPC (nA) Time  (ms) response to electron tunneling Real time single electron tunneling

18. March 2009 IFF Spring SchoolFolie 22 Histograms tunnel time  ~ (60  s) -1  ~ (230  s) -1 Increase tunnel barrier

18. March 2009 IFF Spring SchoolFolie 23 Spin read out principle: N = 1 SPIN UP time charge 0 -e-e N = 1 N = 0 SPIN DOWN time charge 0 -e-e  -1 convert spin to charge

18. March 2009 IFF Spring SchoolFolie 24 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 SchoolFolie 25 Read-Out of Electron Spin Requirements Read-out has to be fast enough → Shorter than T 1 (spin energy relaxation) Charges are measured → Spin to charge conversion Back-action on qubit system should be small → decouple read-out from qubit system

18. March 2009 IFF Spring SchoolFolie 26 Spin read-out procedure inject & wait empty QD V pulse read-out spin empty QD  I QPC Nature 430, 431(2004)

18. March 2009 IFF Spring SchoolFolie 27 Spin read-out results inject & wait empty QD V pulse read-out spin empty QD  I QPC “SPIN DOWN” Time (ms) “SPIN UP” Time (ms)  I QPC (nA) Elzerman et al., Nature 430, 431, 2004

18. March 2009 IFF Spring SchoolFolie 28 More spin down traces Time (ms)  I QPC (nA) t read t wait t hold Threshold value t hold : time the electron spends in the dot t detect : 1/  1 tunneling time

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

18. March 2009 IFF Spring SchoolFolie 30 Measurement of T 1 B = 8 T T 1 ~ 0.85 ms B = 10 T T 1 ~ 0.55 ms B = 14 T T 1 ~ 0.12 ms Surprisingly long T 1 T 1 goes up at low B Elzerman et al., Nature 430, 431, 2004

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

18. March 2009 IFF Spring SchoolFolie 32 „DiVincenzo“ Criteria DiVincenzo: Fortschr. Phys. 48 (2000) 9-11, pp 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 SchoolFolie 33 quantum measurement Any more questions about this point?

18. March 2009 IFF Spring SchoolFolie 34 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 SchoolFolie 35 (3) t =  : with high PR that electron was in state ES low PR that electron was in state GS 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 >>

18. March 2009 IFF Spring SchoolFolie 36 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 SchoolFolie 37 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 SchoolFolie 38 Single-shot read-out

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

18. March 2009 IFF Spring SchoolFolie 40 Read-out characteristics spin: “down”  “up” outcome:   Measurement fidelity: “SPIN UP” Time (ms)  I QPC (nA)  : probability that QPC-current exceeds threshold (spin-down), even if electron was spin-up Time (ms)  I QPC (nA) Threshold value spin-down Threshold (nA)  Up detected as up Up detected as down thermally activated tunneling electrical noise

18. March 2009 IFF Spring SchoolFolie 41 Read-out characteristics spin: “down”  “up” outcome:    Measurement fidelity: “SPIN UP” Time (ms)  I QPC (nA)  : probability that QPC-current stays below threshold (spin-up), even if electron was spin-down spin can relax into spin prior to charge detection spin can relax into spin

18. March 2009 IFF Spring SchoolFolie 42 More spin down traces Time (ms)  I QPC (nA) t read t wait t hold Threshold value t wait : T 1 time t detect : 1/  1 tunneling time

18. March 2009 IFF Spring SchoolFolie 43 Read-out characteristics spin: “down”  “up” outcome:    Measurement fidelity: “SPIN UP” Time (ms)  I QPC (nA)  : probability that QPC-current stays below threshold (spin-up), even if electron was spin-down spin can relax into spin prior to charge detection spin can relax into spin

18. March 2009 IFF Spring SchoolFolie 44 Read-out characteristics spin: “down”  “up” outcome:    Measurement fidelity:  : probability that QPC-current stays below threshold (spin-up), even if electron was spin-down spin tunnels out, but is replaced by spin within 8  s tunnels out replaces by spin Test: apply a reverse pulse Time (ms) 0 1  I QPC (nA)  2 = Pr [ miss step ] Electron is injected into dot each time: step at start pulse

18. March 2009 IFF Spring SchoolFolie 45 Measurement fidelity Threshold (nA)  1  2      Maximal visibility: 65% in single shot experiments Fidelity spin ~ 0.93 Fidelity spin ~ 0.72 Probability

18. March 2009 IFF Spring SchoolFolie 46 Read-out characteristics spin: “down”  “up” outcome:   Measurement fidelity: “SPIN UP” Time (ms)  I QPC (nA)  : probability that QPC-current exceeds threshold (spin-down), even if electron was spin-up Time (ms)  I QPC (nA) Threshold value spin-down Threshold (nA)  Up detected as up Up detected as down thermally activated tunneling electrical noise

18. March 2009 IFF Spring SchoolFolie 47 Read-out characteristics spin: “down”  “up” outcome:    Measurement fidelity: “SPIN UP” Time (ms)  I QPC (nA)  : probability that QPC-current stays below threshold (spin-up), even if electron was spin-down spin can relax into spin prior to charge detection spin can relax into spin

18. March 2009 IFF Spring SchoolFolie 48 Read-out characteristics spin: “down”  “up” outcome:    Measurement fidelity: “SPIN UP” Time (ms)  I QPC (nA)  : probability that QPC-current stays below threshold (spin-up), even if electron was spin-down spin can relax into spin prior to charge detection spin can relax into spin

18. March 2009 IFF Spring SchoolFolie 49 Read-out characteristics spin: “down”  “up” outcome:    Measurement fidelity:  : probability that QPC-current stays below threshold (spin-up), even if electron was spin-down spin tunnels out, but is replaced by spin within 8  s tunnels out replaces by spin Test: apply a reverse pulse Time (ms) 0 1  I QPC (nA)  2 = Pr [ miss step ] Electron is injected into dot each time: step at start pulse

18. March 2009 IFF Spring SchoolFolie 50 Measurement fidelity Threshold (nA)  1  2      Maximal visibility: 65% in single shot experiments Fidelity spin ~ 0.93 Fidelity spin ~ 0.72 Probability

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

18. March 2009 IFF Spring SchoolFolie 52 Basics of electron spin resonance field modulation  E = h = g i µ B B 0 = 30 µeV für  GHz m S = 1/2 m S = -1/2 B0B0 energy magnetic field

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

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

18. March 2009 IFF Spring SchoolFolie 55 Problems ESR resonance frequency in leads same as for electrons in dots Current flows independent of whether spin is rotated heat dissipation Current flows independent of whether spin is rotated

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

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

18. March 2009 IFF Spring SchoolFolie 58 Coherent Rabi oscillations

18. March 2009 IFF Spring SchoolFolie 59 Coherent Rabi oscillations I dot large I dot small

18. March 2009 IFF Spring SchoolFolie 60 SWAP gate implementation in a Double Quantum Dot Few electron double quantum dot Fully tunable 2Qubit system Quantum point contact (QPC) as charge detector Measure dI QPC /dV L : change of total electron number in double dot V L controls number of electrons in left dot V P R controls number of electrons in right dot

18. March 2009 IFF Spring SchoolFolie 61 source V left drain V right V tgl V tgm V tgr Current in a double quantum dot (0,0) (1,0) (0,1) (0,2) (1,1) (2,0) (1,2) V right (2,1)(2,2) V left

18. March 2009 IFF Spring SchoolFolie 62 source V left drain V right V tgl V tgm V tgr Current in a double quantum dot (0,0) (1,0) (0,1) (0,2) (1,1) (0,2) (1,2) (2,1)(2,2) e h V right V left

18. March 2009 IFF Spring SchoolFolie 63 VLVL VRVR Two electron double quantum dot QPC can detect all charge transitions 2 electron double quantum dot Tuned between (1,1) and (0,2) state  

18. March 2009 IFF Spring SchoolFolie 64 Spin configurations in a DQD Spin-Singlet S = 0 antisymmetric Spin-Triplet S = 1; m s = +1, 0, -1 symmetric

18. March 2009 IFF Spring SchoolFolie 65 Hyperfine coupling in a DQD Ga and Ar have a nuclear spin:  about 10 6 nuclear spins in a quantum dot Electrons feel a magnetic field due to hyperfine interaction with these nuclei Nuclear spins are not fully polarized  fluctuations lead to a field Singlet and Triplet states become mixed “Overhauser field” In an external magnetic field in, |S  and |T 0  become mixed

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

18. March 2009 IFF Spring SchoolFolie 67 Harvard scheme Interdot tunneling: hybridization (0,2) – (1,1) exchange splitting J(  ) B = 100 mT perp. field Strength of J(  ) controlled by gates

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

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

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

18. March 2009 IFF Spring SchoolFolie 71 Spin swap and Rabi oscillations turn on J(  )

18. March 2009 IFF Spring SchoolFolie 72 Spin SWAP and Rabi oscillations  

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

18. March 2009 IFF Spring SchoolFolie 74 „DiVincenzo“ Criteria DiVincenzo: Fortschr. Phys. 48 (2000) 9-11, pp 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 SchoolFolie 75 Entanglement and decoherence

18. March 2009 IFF Spring SchoolFolie 76 Singlet-triplet spin echo refocus separated singlet to undo inhomogeneous dephasing apply  pulse by pulsed J(  )

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

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

18. March 2009 IFF Spring SchoolFolie 79 Spin-spin relaxation times Spin dephasing time: ~ 8 ns Spin coherence time: ~ 1.2  s Time for √SWAP: ~ 180 ps  about 7000 √SWAP operations can be performed during T 2 However

18. March 2009 IFF Spring SchoolFolie 80 „DiVincenzo“ Criteria DiVincenzo: Fortschr. Phys. 48 (2000) 9-11, pp 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 ( ) Why can’t we already buy a quantum computer ? ( )

18. March 2009 IFF Spring SchoolFolie 81 Spin energy relaxation nuclei: T 1 ~ hours – days electrons: T 1 ~ ms   spin system is in excited state relaxation to ground state due to spin-phonon interaction read-out within T 1 dM z dt =  (M x (t)B y  M y (t)B x ) M z  M 0 T1T1 

18. March 2009 IFF Spring SchoolFolie 82 Origin of spin-phonon coupling Spin-orbit interaction is the most important contribution H SO 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 SchoolFolie 83 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 SchoolFolie 84 Breaking time reversal symmetry All contributions would cancel out without magnetic field applied Follow one period of lattice vibration (harmonic oscillator)  “van Vleck” cancellation SO B0B0 

18. March 2009 IFF Spring SchoolFolie 85 Magnetic field dependence All contributions add up to:    E Zee 5

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

18. March 2009 IFF Spring SchoolFolie 87 Magnetic field fluctuations Unknown magnetic field  electron spin evolves in an unknown way Gaussian distribution with standard deviation T 2 * = 10 ns In experiment: ^ = B N = 2.3 mT Reduce dephasing  Find a way to decrease  of magnetic field BNBN

18. March 2009 IFF Spring SchoolFolie 88 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 SchoolFolie 89 Outlook All necessary components not yet implemented in the same device Gate implementation still too slow Scaling to ~1000 qubits not straight forward Improve T 2 : Polarize nuclei to >99% Find materials without nuclear spins and SO coupling → carbon based (graphene, carbon nanotubes) → silicon (2DEG charge carrier mobility too low) Any solutions possible? Why can’t we already buy a quantum computer ?

18. March 2009 IFF Spring SchoolFolie 90 Dilbert