SUPERCONDUCTING QUANTUM BITS

Transcription

1 I0> SUPERCONDUCTING QUANTUM BITS I1> Hans Mooij Summer School on Condensed Matter Theory Windsor, August 18, 2004

2 quantum computer U quantum bits states l0>, l1> Ψ = αl0> + βl1> input - unitary transformations - output N qubits, 2 N states quantum algorithms factorization (Shor) searching (Grover) initialization controlled operations interaction large number individual readout non-trivial two-bit gates controlled-not, swap error correction

3 quantum bit: two level quantum system I1> Ψ = α I0> + β I1> E = ħω o I0> IαI 2 + IβI 2 = 1 α = cos θ β = e iφ sin θ latitude: angle θ content l0> and l1> measured in readout θ I0> I1> longitude: phase φ irrelevant in readout dφ/dt = E/ħ φ

4 quantum manipulations resonant electromagnetic wave ν= E/h Rabi oscillation π I1> Ψ=αI0>+βI1> I0> π/2 2π α(t)=cos(θ o +ω R t) β(t)=sin(θ o +ω R t) length of microwave pulse determines final state

5 measurement quantum system environment

6 well-established quantum technologies - nuclear magnetic resonance - photons - ions / atoms basic manipulations well-controlled in present form not scalable

7 NMR identical molecules in liquid each molecule has N identifiable spins parallel N-qubit computers ensemble-properties represent qubits readout: measurement of spin magnetic moments E 2 nuclear spins E 1 E 2 -E 1 proportional to magnetic field strength weak interaction between the spins in the same molecule no interaction between molecules

8 Deutsch-Jozsa 7-spin coherence D H C N O O D C O F H 3 C H C H C C O OH Order-finding F F C C F C Fe (CH) 5 (CO) 2 C F F

9 factoring 15 with nuclear spins 2 F 12 C 6 13 C F 1 7 F 3 Vandersypen et al., Nature 414, 883 (2001) 15 3 x 5 4 F Fe 13 C 12 C F 5 C 5 H 5 CO CO

10 photons 'free' single photons beam splitters polarizers non-linear Kerr cells single photon detectors photons in a cavity interaction through an atom

11 ions in an ion trap c. monroe et al. PRL trapped ions 0.2 mm Courtesy D. Wineland, NIST

12 new scalable quantum technologies - ion traps with micro-control electrodes 'atom chip' - superconducting systems - single spins in semiconductors - x x x x

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14 single spins in a semiconductor h X X - e X S L S R Gammon et al Imamoglu et al, PRL 1999 Loss & DiVincenzo, PRA nm

15 mesoscopic Josephson junction circuits Josephson coupling energy U J = E J ( 1 - cosφ ) E J = 2 (h/e 2 ) / (8R n) Coulomb charging energy U C = E C 4n 2 E C = e 2 / 2C E J /E C >>1 phase excitations fluxons n, φ δnδφ>1 E C /E J >>1 charge excitations V g Φ m

16 n=1 E p ext n=0 charge qubit p ext = V g C g /2e = E J n : Cooper pairs flux qubit p ext = Φ/Φ o : exp{ -(E J /E c ) 1/2 } n : fluxons

17 Nakamura, Pashkin, Tsai Nature 398, 786 (1999)

18 Yamamoto, Pashkin, Astaflev, Nakamura, Tsai Nature 425, 941 (2003) CNOT

19 charge qubit with phase read-out D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, M.H. Devoret, Science 296, 886 (2002) U circulating current in squid loop depends on charge state Ι b current pulse switch or no switch

20 'magic point'

21 persistent-current quantum bit flux qubit with three junctions, small geometric loop inductance E Φ 0 2t H = hσ z + tσ x with h=(φ/φ o -0.5) Φ o I p I circ 1 0 +I p Φ/Φ o -I p Science 285, 1036 (1999)

22 Φ/Φo

23 SQUID: Ι s max (Φ) Ι s max /Ι c maximum supercurrent 1 Ι bias Φ/Φ o Ι circulating 1 circulating current at f=0.5 Ι circ /Ι c Ι bias /Ι c

24 SQUID measurement I bias qubit generates flux ±LI p 10-3 Φ o measured with hysteretic (unshunted) SQUID maximum supercurrent depends on flux in the SQUID loop GHz I switch Φ/Φ o measurement switching current: (RF 9 GHz applied) ramps

25 SQUID I c 200 na underdamped, no ohmic shunt measurement of I switch by repeated ramping histogram of 5,000-10,000 points average value recorded

26 superposition of states with clockwise and anti-clockwise-current switching current Ι SW (0.4 na per division) ~ GHz GHz GHz GHz GHz GHz GHz GHz GHz 10 f GHz f (GHz) Φ ext (Φ 0 ) Φ/Φ o GHz 0 x10-3 Φ/Φ o Φ res (10-3 Φ 0 ) Van der Wal et al., Science 290, 773 (2000) also SUNY (Friedman, Lukens et al.)

27 Friedman, Patel, Chen, Tolpygo, Lukens: Nature 406, 43 (2000) E and E B manipulated with flux E J =76 K (cosφ xdc /Φ o ), E C = 9 mk U o = 9 K

28 quantum system environment

29 Decoherence relaxation T 1 τ relax dephasing T 2 τ φ - pseudospin - boson bath (harmonic oscillators) Milena Grifoni, Frank Wilhelm, Gerd Schon et al τ relax determined by spectral density at E τ φ determined by spectral density at low ω measurement circuit designed to optimize - pseudospin - spin bath? Phil Stamp - 1/f type noise: flux, charge, critical current

30 1/f noise charge noise: - charge noise: charged defects in barrier, substrate or surface lead to non-integer induced charge. Static offset, 1/f noise. - critical current noise: neutral defects in barrier. - flux noise: trapped vortices, magnetic domains, magnetic impurities, nuclear spins

31 Delft flux qubits Kees Harmans Alexander ter Haar Irinel Chiorescu Adrian Lupascu Floor Pauw Patrice Bertet Jelle Plantenga Jonathan Eroms Yasunobu Nakamura (NEC) Kouichi Semba (NTT) funding: FOM, EU ARO, Kavli

32 h set by external flux t set by fabrication H = hσ z + tσ x 6 excitation, measurement σ z E t (here t=1) H hσ z -6 h H hσ z H = tσ x

33 DC, RF drives quantum system environment SQUID measurement

34 300K DC, RF drives qubit SQUID circulating current SQUID bias current 300K variations of q, E J, Φ SQUID voltage 300K

35 -42 E / E C shift -50 operation point read-out point total flux (Φ intended principle: 0 ) - quantum operations at symmetry point where de/dφ=0 (but also qubit circulating currents are zero) - use changing SQUID circulating current for automatic shift to a read-out bias where there is qubit current to measure

36 well-designed electromagnetic environment calculated T 1 = 100 µs T 2 = 20 µs operation time 1 ns SQUID shunted by large capacitance on chip

37 SQUID readout: reference trigger quantum operations pulsed bias current ~ 30 ns rise/fall time time only two possible outputs: SQUID switched to gap voltage SQUID still at V=0 pulse height adjusted to give ~50% switching switching probability (%)100 Isw pulse AW generator (V)

38 F (GHz) Φ ext / Φ 0

39 Rabi: microwave pulse with varying length trigger pulse length RF I b pulse ~50ns rise/fall time time

40 90 60 switching probability (%) A = 9 db A = 3 db Rabi frequency (MHz) A = -3 db RF pulse length (ns) RF amplitude, 10 A/20 (a.u.)

41 Switching probability (%) Psw (%) Psw (%) RF pulse length (ns) RF pulse length (ns) RF pulse length (ns)

42 measurement of relaxation time observed values: 40 ns to 300 µs one π pulse and read-out pulse delayed trigger π delay time Ib pulse time operation read-out switching probability (%) delay time (µs) 8.3 ns, A=-12dBm 6 ns, A=-9dBm 4.5 ns, A=-6dBm 3.2 ns, A=-3dBm ns, A=0 dbm exp fit of A=-12dBm T 1 = 870 ns

43 measurement T 2 time Ramsey 90 trigger π/2 π/2 I b pulse 80 Switching probability (%) A = 9 db F 0 = GHz f = 250 MHz time between two π/2 pulses (ns)

44 spin echo switching probability (%) π/2 π π/ position of π pulse (ns) E 10 =5.7 GHz δf=220 MHz T π =4.5 ns fixed time between π/2 pulses, π pulse shifting 30 ns decay of maximum spin echo signal versus time between π/2 pulses T 2 (higher frequencies) 30 ns Techo (ns)

45 Rabi with low number of measurements, single shot contains some information F=6.512 GHz A=-3 dbm switching probability (%) N- number of averaged events N=1 N=10 N= RF pulse length (ns)

46 switching probability [ % ] B=4.55 Gs F=5.63GHz 1 µs pulse No RF π-pulse (3 ns) pulse AWG TE3362 [ V ] dp/di b [ %/V ] No RF 1 µs pulse π-pulse (3 ns) pulse AWG TE3362 [ V ] Kouichi Semba, NTT, Delft TE3362 AWG

47 inductive readout, circuit MW in I MW M MW F M Φ q V DC 30 mk 4.2 K 300 K DC out R L m R b s M sq F squid R Rl l V out G 1 G 2 RF out L J C L J (Φ sq ) V in (RF+ +DC) ) in G 1 G 2 RF out in circuit parameters C = 12 pf L s ~ 2.5 nh R L = 820 W R b = 5.6 kw R m = 11 kw Q~40 Adrian Lupascu, Kees Harmans, Raymond Schouten

48 resonance curve at B=0: F res = MHz, Q = 33 V (mv) F (MHz) Phase (deg)

49 Typical spectroscopy curve A, mv B, mgs Measurements at step s1 Measurement scheme: 1µs MW followed by 2 µs readout pulse F MW =17 GHz, F IM =855 MHz, F res =838 MHz SQUID excitation: current amplitude less that 1/2I c Estimated sub-peaks period:750 MHz

50 relaxation The measurement is done with F=855 MHz T p =1µs, t mw =9.9 µs, F mw =17 GHz, A=9 dbm T r =86 +/- 12 us U, mv td, m s (preliminary) conclusion: relaxation time about 60 µs

51 T 1 T 2 (ns) τ φ and τ r (ns) T T field (Gs) Larmor frequency (GHz) E 1 -E o (GHz)

52 influence of bias current on T 1 and T 2 1 T 2 = 1 2T T ϕ T1 T2 Tphi T φ calculated from measured T 1 and T 2 time (ns) black: T 1 red: T Ibias (µa)

53 Ramsey in presence of transport current 90 Ib=2.841µA Ib=2.976µA 85 Ib=2.565µA 120 F=5.675GHz Pswitch (%) Pswitch (%) F=5.685GHz F=5.695GHz delay (microseconds) T 1 =70ns T 2 =120 ns delay btwn pulses (µs)

54 Patrice Bertet, Irinel Chiorescu

55 Larmor frequency (GHz) = GHz I q = 272 na spectroscopy peaks fit: E J /E C = E C =7.281 GHz α= Φ/Φ 0 gap GHz persistent current 272 na switching probability (%) (Q+3) /2 Q/2 Q/3 Q-3 3 Q Q+3 spectroscopy peaks Q: qubit 1,2,3 fotons 3: ω p squid 2.91 GHz Q±3: sidebands frequency (GHz)

56 60 55 ONE-PHOTON F mw =7.16GHz TWO-PHOTON F mw =3.62 GHz THREE-PHOTON F mw =2.40 GHz Switching probability (%) A = -14 dbm A = -18 dbm A = -15 dbm A = -17 dbm A = 0 dbm A = -1 dbm A = -22 dbm A = -19 dbm MW pulse length (ns) A = -2 dbm

57 0.30 One-photon Rabi frequency (GHz) one-photon Rabi frequency J 1 (b10 A/20 ) with b=0.92, =5.344 GHz A/20 (a.u.) Two-photon Rabi frequency (GHz) two-photon Rabi frequency J 2 (b10 A/20 ) with =5.344 GHz, b= A/20 (a.u.)

58 Switching probability (%) A Rabi oscillations A = 9.5 dbm A = -1 dbm A = -17 dbm MW pulse length (ns) Rabi frequency (GHz) B ω p A/20 (a.u.) C ω 1 (GHz)

59 flux qubit coupled to harmonic oscillator harmonic oscillator: measurement SQUID shunted by large external capacitance; Q about 150 strong coupling accidental ω p E 1 -E o

60 frequency [GHz] blue sideband red sideband squid resonance Φ/Φ ο

61 coherent oscillations of the coupled system qubit Larmor frequency 7.16 GHz plasma frequency : 2.91 GHz coupled system at GHz (blue sideband) switching probability (%) Rabi oscillations F=10.15 GHz, A=-5dBm 26 qubit: F 24 L =7.16 GHz squid: ω pl =2.91 GHz pulse length (ns) switching probability (%) Rabi oscillations F=10.15 GHz, A=3dBm qubit: F L =7.16 GHz squid: ω pl =2.91 GHz pulse length (ns)

62 1 0 switching probability (%) after a π pulse 00> 10> A after a 2π pulse 00> 00> B red-side band 3.58 GHz F L = 6.48 GHz blue-side band 9.48 GHz +13 dbm MW frequency (GHz) Rabi oscillations at F L =6.48 GHz 30 00> 10> 01> 11> pulse length (ns)

63 SQUID readout with qubit-dependent resonant pulse Rabi pulse bias current Switching probability (%) readout pulse 2 time Pulse 1 length (µs) improved visibility Switching probability(%) Frequency (GHz) increasing length of Rabi pulse

64 coupling of qubits 0.01 ph 4 MHz 2 ph 0.8 GHz 0.3 ph 0.1 GHz 10 ph 4 GHz 1D qubit chain

65 Hannes Majer Floor Pauw Alexander ter Haar

66 H = h 1 σ z1 + t 1 σ x1 + h 2 σ z2 + t 2 σ x2 +jσ z1 σ z 1 asymmetry in h: E J and area (shift of f=0.5) asymmetry t: E J, junction ratio α Energy (GHz) Energy (GHz) Φ-1/2Φ 0 (mφ 0 ) symmetric qubits Φ-1/2Φ 0 (mφ 0 ) asymmetric qubits

67 SQUID signal (a.u.) GHz 12 GHz 10 GHz 8 GHz 6 GHz Φ-1/2Φ 0 (mφ 0 )

68 20 Energy (GHz) Φ-1/2Φ 0 (mφ 0 )

69 results of fit I p1 = 512 na ± 6nA t 1 = 0.45 GHz ± 0.2 GHz I p2 = 392nA ± 5nA t 2 = 1.9 GHz ± 0.1 GHz sigma = % ± 0.004% (difference areas) j = 0.50 GHz ± 0.03 GHz (coupling strength) no transitions to 3rd excited state: low transition probability Matrix element T if T 01 T 02 T Φ-1/2Φ 0 (mφ 0 )

70 conclusions - flux qubit behaves quantum mechanically - coherent driving possible (to a certain extent) - decoherence partly due to circuit, partly to defects - coherent driving observed of 2-level system +harmonic oscillator

71 Delft flux qubits Kees Harmans Alexander ter Haar Irinel Chiorescu Adrian Lupascu Floor Pauw Patrice Bertet Jelle Plantenga Jonathan Eroms Yasunobu Nakamura (NEC) Kouichi Semba (NTT) funding: FOM, EU ARO, Kavli