SUPERCONDUCTING QUANTUM BITS - PDF Free Download

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

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

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

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

measurement quantum system environment

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

NMR 10 18 identical molecules in liquid each molecule has N identifiable spins 10 18 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

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

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

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

ions in an ion trap c. monroe et al. PRL 75 4714 1995 trapped ions 0.2 mm Courtesy D. Wineland, NIST

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

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 1998 200 nm

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

n=1 E 0.04 0.02 2 0.46 0.48 0.52 0.54-0.02 p ext -0.04 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

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

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

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

'magic point'

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 0-1 0.5 Φ/Φ o -I p Science 285, 1036 (1999)

Φ/Φo

SQUID: Ι s max (Φ) Ι s max /Ι c 2 1.5 maximum supercurrent 1 Ι bias 0.5 0.5 1 1.5 2 Φ/Φ o Ι circulating 1 circulating current at f=0.5 Ι circ /Ι c 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 Ι bias /Ι c

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 60 9.00 GHz I switch 55 50 45 40 0.5 Φ/Φ o measurement switching current: (RF 9 GHz applied) 5000-10000 ramps

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

superposition of states with clockwise and anti-clockwise-current switching current Ι SW (0.4 na per division) ~ 9.711 GHz 8.650 GHz 6.985 GHz 5.895 GHz 4.344 GHz 3.208 GHz 2.013 GHz 1.437 GHz 1.120 GHz 10 f GHz f (GHz) 5 2 0 0 0.8 0.498 0.500 0.502 Φ ext (Φ 0 ) Φ/Φ o 0.850 GHz 0 x10-3 Φ/Φ o -0.5 0 1 2 3 Φ res (10-3 Φ 0 ) Van der Wal et al., Science 290, 773 (2000) also SUNY (Friedman, Lukens et al.)

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

quantum system environment

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

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

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

h set by external flux t set by fabrication H = hσ z + tσ x 6 excitation, measurement σ z E 4 2-6 -4-2 2 4 6-2 -4 2t (here t=1) H hσ z -6 h H hσ z H = tσ x

DC, RF drives quantum system environment SQUID measurement

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

-42 E / E C -44-46 -48 shift -50 operation point read-out point 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 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

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

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 90 80 70 60 50 40 30 20 10 switching probability (%)100 Isw 0 2.3 2.4 2.5 2.6 2.7 pulse height @ AW generator (V)

16 14 12 F (GHz) 10 8 6 4 2 0-0.005 0.000 0.005 Φ ext / Φ 0

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

90 60 switching probability (%) 30 90 60 30 90 60 A = 9 db A = 3 db Rabi frequency (MHz) 600 500 400 300 200 100 A = -3 db 30 0 20 40 60 80 100 120 140 160 180 RF pulse length (ns) 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 RF amplitude, 10 A/20 (a.u.)

Switching probability (%) 100 90 80 70 60 50 40 30 20 Psw (%) Psw (%) 90 60 30 0 10 20 30 RF pulse length (ns) 62 60 58 RF pulse length (ns) 500 510 520 530 0 50 100 150 200 250 300 350 400 450 500 RF pulse length (ns)

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 (%) 100 90 80 70 60 50 40 30 0 1 2 3 4 5 6 7 8 9 10 delay time (µs) 8.3 ns, A=-12dBm 6 ns, A=-9dBm 4.5 ns, A=-6dBm 3.2 ns, A=-3dBm 2.445 ns, A=0 dbm exp fit of A=-12dBm T 1 = 870 ns

measurement T 2 time Ramsey 90 trigger π/2 π/2 I b pulse 80 Switching probability (%) 70 60 50 A = 9 db F 0 = 6.607 GHz f = 250 MHz 0 5 10 15 20 25 30 35 40 45 50 time between two π/2 pulses (ns)

spin echo switching probability (%) 70 60 50 π/2 π π/2-25 -20-15 -10-5 0 5 10 15 20 25 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 0 50 100 Techo (ns)

Rabi with low number of measurements, single shot contains some information 100 80 F=6.512 GHz A=-3 dbm switching probability (%) 60 40 20 0 N- number of averaged events N=1 N=10 N=10000 0 10 20 30 40 50 60 70 80 RF pulse length (ns)

switching probability [ % ] 100 80 60 40 20 B=4.55 Gs F=5.63GHz 1 µs pulse No RF π-pulse (3 ns) 0 1.6 1.7 1.8 1.9 pulse height @ AWG TE3362 [ V ] dp/di b [ %/V ] 2000 1500 1000 500 No RF 1 µs pulse π-pulse (3 ns) 0 1.6 1.7 1.8 1.9 pulse height @ AWG TE3362 [ V ] Kouichi Semba, NTT, Delft TE3362 AWG

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

resonance curve at B=0: F res = 762.2 MHz, Q = 33 V (mv) 1.25 1.00 0.75 0.50 0.25 0.00 650 700 750 800 850 F (MHz) 90 60 30 0-30 -60-90 Phase (deg)

Typical spectroscopy curve A, mv 0.3 0.2 0.1 0.0-0.1-0.2-0.3-0.05-0.10-0.15-0.20 3.5 3.6 3.7 3.8 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 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

relaxation 0.470 The measurement is done with F=855 MHz T p =1µs, t mw =9.9 µs, F mw =17 GHz, A=9 dbm 0.465 T r =86 +/- 12 us U, mv 0.460 0.455 0.450 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 td, m s (preliminary) conclusion: relaxation time about 60 µs

T 1 T 2 (ns) τ φ and τ r (ns) 90 7.5 80 70 7.0 60 50 6.5 40 T 1 30 6.0 20 T 10 2 5.5 0 4.400 4.405 4.410 4.415 4.420 4.425 4.430 field (Gs) Larmor frequency (GHz) E 1 -E o (GHz)

influence of bias current on T 1 and T 2 1 T 2 = 1 2T 1 + 1 T ϕ 500 400 300 T1 T2 Tphi T φ calculated from measured T 1 and T 2 time (ns) 200 100 0 black: T 1 red: T 2 0 1 2 3 Ibias (µa)

Ramsey in presence of transport current 90 Ib=2.841µA Ib=2.976µA 85 Ib=2.565µA 120 F=5.675GHz Pswitch (%) 80 75 70 Pswitch (%) 100 80 F=5.685GHz F=5.695GHz 65 0.00 0.04 0.08 0.12 0.16 delay (microseconds) T 1 =70ns T 2 =120 ns 0.0 0.1 0.2 0.3 delay btwn pulses (µs)

Patrice Bertet, Irinel Chiorescu

Larmor frequency (GHz) 12 10 8 6 4 2 = 5.866 GHz I q = 272 na spectroscopy peaks fit: E J /E C =30.834 E C =7.281 GHz α=0.76 0 0.500 0.502 0.504 0.506 Φ/Φ 0 gap 5.866 GHz persistent current 272 na switching probability (%) 50 45 40 35 30 25 20 (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 15 1 2 3 4 5 6 7 8 9 10 11 12 frequency (GHz)

60 55 ONE-PHOTON F mw =7.16GHz 60 55 TWO-PHOTON F mw =3.62 GHz 60 55 THREE-PHOTON F mw =2.40 GHz 50 50 50 45 45 45 40 40 40 35 35 35 Switching probability (%) 30 25 20 60 55 50 45 40 35 30 25 20 60 55 A = -14 dbm A = -18 dbm 30 25 20 60 55 50 45 40 35 30 25 20 60 55 A = -15 dbm A = -17 dbm 30 25 20 60 55 50 45 40 35 30 25 20 60 55 A = 0 dbm A = -1 dbm 50 50 50 45 45 45 40 40 40 35 35 35 30 25 A = -22 dbm 20 0 5 10 15 20 25 30 35 40 30 25 A = -19 dbm 20 0 5 10 15 20 25 30 35 40 MW pulse length (ns) 30 25 A = -2 dbm 20 0 5 10 15 20 25 30 35 40

0.30 One-photon Rabi frequency (GHz) 3.0 2.5 2.0 1.5 1.0 0.5 one-photon Rabi frequency J 1 (b10 A/20 ) with b=0.92, =5.344 GHz 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 10 A/20 (a.u.) Two-photon Rabi frequency (GHz) 0.25 0.20 0.15 0.10 0.05 two-photon Rabi frequency J 2 (b10 A/20 ) with =5.344 GHz, b=5.15 0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 10 A/20 (a.u.)

Switching probability (%) 35 30 25 35 30 25 35 30 25 20 A Rabi oscillations A = 9.5 dbm 3 6 9 12 A = -1 dbm 3 6 9 12 A = -17 dbm 0 3 6 9 12 15 18 21 24 MW pulse length (ns) Rabi frequency (GHz) 6 5 4 3 2 1 0 6 5 4 3 2 1 0 B ω p 0 1 2 3 10 A/20 (a.u.) C 0 5 10 15 ω 1 (GHz)

flux qubit coupled to harmonic oscillator harmonic oscillator: measurement SQUID shunted by large external capacitance; Q about 150 strong coupling accidental 1 0 1 1 1 2 ω p E 1 -E o 0 0 0 1 0 2

frequency [GHz] 16 14 12 10 8 6 blue sideband red sideband 10 00 11 01 4 squid resonance 0.000 0.002 0.004 0.006 Φ/Φ ο

coherent oscillations of the coupled system qubit Larmor frequency 7.16 GHz plasma frequency : 2.91 GHz coupled system at 10.15 GHz (blue sideband) switching probability (%) 40 38 36 34 32 30 28 Rabi oscillations F=10.15 GHz, A=-5dBm 26 qubit: F 24 L =7.16 GHz squid: ω pl =2.91 GHz 22 0 2 4 6 8 10 12 14 16 pulse length (ns) switching probability (%) 26 24 22 20 18 Rabi oscillations F=10.15 GHz, A=3dBm qubit: F L =7.16 GHz squid: ω pl =2.91 GHz 0 2 4 6 8 10 12 14 16 pulse length (ns)

1 0 switching probability (%) 40 35 30 25 20 50 40 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 3 4 5 6 7 8 9 10 MW frequency (GHz) Rabi oscillations at F L =6.48 GHz 30 00> 10> 01> 11> 20 0 10 20 30 40 pulse length (ns) 1 1 0 0 0 1

SQUID readout with qubit-dependent resonant pulse Rabi pulse bias current Switching probability (%) 80 60 40 1 readout pulse 2 time 20 0.00 0.02 0.04 0.06 0.08 0.10 Pulse 1 length (µs) improved visibility Switching probability(%) 250 200 0 100 50 0 2.3 2.4 2.5 2.6 Frequency (GHz) increasing length of Rabi pulse

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

Hannes Majer Floor Pauw Alexander ter Haar

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 12 8 20 Energy (GHz) 4 0-4 + - Energy (GHz) 10 0-10 -8-20 -6-4 -2 0 2 4 6 Φ-1/2Φ 0 (mφ 0 ) symmetric qubits -8-4 0 4 8 Φ-1/2Φ 0 (mφ 0 ) asymmetric qubits

SQUID signal (a.u.) 12 8 4 14 GHz 12 GHz 10 GHz 8 GHz 6 GHz 0-8 -4 0 4 8 Φ-1/2Φ 0 (mφ 0 )

20 Energy (GHz) 16 12 8 4 0-8 -4 0 4 8 Φ-1/2Φ 0 (mφ 0 )

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 = -0.027% ± 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 1.0 0.8 0.6 0.4 0.2 T 01 T 02 T 03 0.0-8 -4 0 4 8 Φ-1/2Φ 0 (mφ 0 )

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