JOSEPHSON QUBIT CIRCUITS AND THEIR READOUT

Transcription

1 JOSEPHSON QUBIT CIRCUITS AND THEIR READOUT Michel DEVORET, Applied Physics and Physics, Yale University Sponsors: W.M. KECK Final version of this presentation available at APS March Meeting Pittsburgh, PA

2 QUANTUM INFORMATION PROCESSING # of atoms micro MACRO OPTICAL PHOTONS NUCLEAR SPINS IONS ATOMS MOLECULES QUANTUM DOTS SUPERCONDUCTING JOSEPHSON TUNNEL CIRCUITS coupling with env t. 10µm 50µm from A. O. Niskanen et al. Science 316, 723 (2007) courtesy of J. Martinis, 2009 from Metcalfe et al., 2007

3 ELECTROMAGNETIC SPECTRUM f (Hz) λ (m) AUDIO VIDEO RF / μw IR UV RX γ ELECTRICITY OPTICS 1 bit = RF signal with 0/1 photon? 10 GHz ~ 0.5K

4 HOW CAN A SUPERCONDUCTING CIRCUIT BEHAVE LIKE AN ATOM? SIMPLEST EXAMPLE: SUPERCONDUCTING LC OSCILLATOR CIRCUIT MICROFABRICATION L ~ 3nH, C ~ 10pF, ω r /2π ~ 2GHz ELECTRONIC FLUID SLOSHES BACK AND FORTH FROM ONE PLATE TO THE OTHER, INTERNAL MODES FROZEN, BEHAVES AS A SINGLE CHARGE CARRIER

5 DEGREE OF FREEDOM IN ATOM vs CIRCUIT Example of H atom with large principal quantum number Superconducting LC oscillator L C velocity of electron voltage across capacitor force on electron current through inductor

6 FLUX AND CHARGE DO NOT COMMUTE φ I +Q -Q V φˆ, Qˆ = i φ = LI Q= CV

7 LC CIRCUIT AS QUANTUM HARMONIC OSCILLATOR E φ +Q -Q hω r ( aˆ aˆ ) Hˆ = ω 1 r + 2 ˆ φ Qˆ ˆ φ Qˆ ˆ ; ˆ φ Q φ Q a = + i a = i φ = r Q r = r r r r 2 ω L r 2 ω C r φ annihilation and creation operators for mesoscopic excitation of circuit

8 φ WAVEFUNCTIONS OF LC CIRCUIT E I hω r In every energy eigenstate, (photon state) current flows in opposite directions simultaneously! Ψ(φ) Ψ 1 2φ r 0 Ψ 0 φ φ

9 EFFECT OF DAMPING E φ important: as little dissipation as possible φ dissipation broadens energy levels i 1 En = ωr n Q Q = RCω r

10 CAN PLACE CIRCUIT IN ITS GROUND STATE E φ hω r residual dissipation provides reset of circuit ω r kt 10 GHz 25mK B

11 PB: ALL TRANSITIONS ARE DEGENERATE! φ E hω r φ CANNOT STEER THE SYSTEM TO AN ARBITRARY STATE IF PERFECTLY LINEAR

12 NEED NON-LINEARITY TO FULLY REVEAL QUANTUM MECHANICS Potential energy Position coordinate

13 JOSEPHSON TUNNEL JUNCTION PROVIDES A NON-LINEAR INDUCTOR WITH NO DISSIPATION 1nm S I S superconductorinsulatorsuperconductor tunnel junction L J C J Ι Ι Ι = φ / L J L J 2 φ0 φ0 = = E I J 0 φ t ( ') = V t dt ' φ ( ) I = I sin φ / φ 0 0 φ 0 = 2e

14 JOSEPHSON TUNNEL JUNCTION PROVIDES A NON-LINEAR INDUCTOR WITH NO DISSIPATION 1nm S I S superconductorinsulatorsuperconductor tunnel junction L J C J Ι ( ) U = E cos φ / φ J 0 L J 2 φ0 φ0 = = E I J 0 φ t ( ') = V t dt ' φ bare Josephson potential φ 0 = 2e

15 COUPLING PARAMETERS OF THE JOSEPHSON "ATOM" REST OF CIRCUIT (CONT ROL & READ OUT ) THE Hamiltonian: (we mean it!) qˆext H J 1 ( ˆ ) 2 cos 2 eφ = Q qext EJ 2C J Josephson tunnel junction contribution to total circuit Hamiltonian Comparable with lowest order model for hydrogen atom H 1 eaˆ = pˆ 2m e 2 2 e 4πε 0 1 rˆ

16 Two dimensionless variables: Hamiltonian becomes : TWO ENERGY SCALES H 2e ˆ φ ˆ ϕ = ˆ ˆ Q N = 2e J ˆ, ϕ Nˆ = i ( N N ) 2 ext = 8E cos ˆ C EJ ϕ 2 Coulomb charging energy for 1e Josephson energy E C = 2 e 2C J reduced offset charge N ext = q et x 2e E J = valid for opaque barrier barrier transp cy 1 NT Δ 8 gap # cond ion channels

17 HARMONIC APPROXIMATION H J ( N N ) 2 ext = 8E cos ˆ C EJ ϕ 2 H Jh, ( N N ) 2 2 ext ˆ ϕ = 8EC + EJ 2 2 Josephson "plasma" frequency: ω = P 8 C E E J Josephson RF impedance: ( 2e) 2 Spectrum independent of DC value of N ext Z J = 8E E J C

18 credit I. Siddiqi and F.Pierre SOME JOSEPHSON TUNNEL JUNCTIONS IN REAL LIFE credit L. Frunzio and D. Schuster E J ~ 50K ω p ~ 30-40GHz E J ~ 0.5K 100nm

19 RF CONTROL & BIAS vs SENSITIVITY TO NOISE Devoret and Martinis, Quant. Inf. Proc see R. McDermot's tutorial "charge" qubit island "flux" qubit Φ = off Φ 2 0 "phase" qubit Φ > off Φ _ a.k.a. Cooper Pair Box Φ <Φ /2 off 0 THE MONSTERS WE FEAR: charged impurity unstable tunnel channel flavors ΔQ off noises ΔΦ off ΔE J "charge" "flux" freak electron spin loose elec. dipoles "phase"

20 EFFECTIVE POTENTIAL OF 3 MAIN BIAS SCHEMES see also proposals for topologically protected qubits, for example Feigelman et al. PRL 92, (2004) "charge" bias "flux" bias 1 2 ϕ = 2π 2eφ h "phase" bias in e g CEA Saclay, NEC, Yale Chalmers, JPL,... TU Delft, NEC, NTT, IBM, MIT, UC Berkeley, SUNY, IPHT Jena... 2eφ h 2eφ h UC Berkeley, NIST, UCSB, U. Maryland, I. Neel Grenoble...

21 HOW DO WE FIND THE HAMILTONIAN OF AN ARBITRARY CIRCUIT? Yurke B. and Denker J.S., Phys. Rev. A 29, 1419 (1984) Devoret M. H. in "Quantum Fluctuations", S. Reynaud, E. Giacobino, J. Zinn-Justin, Eds. (Elsevier, Amsterdam, 1997) p G. Burkard, R. H. Koch, and D. P. DiVincenzo, Phys. Rev. B 69, (2004) Example: 2 Josephson junctions capacitively coupled to 1 resonator mode C c C c L J C J L C L' J C' J

22 COOPER PAIR "BOX" U Φ U Φ island C g Φ controls E J, C g U/2e controls N g qbit space = Hilbert space of 0 or 1 quanta in this non-linear oscillator Bouchiat PhD Thesis 97, et al. Physica Scripta '98 Nakamura, Pashkin & Tsai, Nature '99

23 SCHEMATIC OF COOPER PAIR BOXES IN A MICROWAVE CAVITY Review by Blais et al., Phys. Rev. A 75, (2007)

24 TWO-QUBIT QUANTUM PROCESSOR slide courtesy of L. DiCarlo & Rob Schoelkopf V17.6 Thu. March 19 see also 1 qubit and 2 cavities: B. Johnson et al. V17.4

25 "FLUXONIUM QUBIT" V. Manucharyan et al. Q17.5 Wednesday resonator 50 Ω port 1 50 Ω port 2 small junction array junctions

26 A FEW USEFUL IDEAS FOR CIRCUIT HAMILTONIANS...

27 BRANCH VARIABLES I β banch β Introduce branch flux and charge rest of circuit V β φ β Q β t () t = V ( ') β t dt' t () t = I ( ') β t dt ' position variable: φ X momentum variable: Q P gener alized force : Ι f gener alized velocity: V V

28 BRANCH VARIABLES I β banch β Introduce branch flux and charge rest of circuit V β φ β Q β t () t = V ( ') β t dt' t () t = I ( ') β t dt ' For every branch β in the circuit: φˆ, Qˆ = i β β

29 PROBLEM: NOT ALL BRANCH VARIABLES ARE INDEPENDENT a KIRCHHOFF'S LAWS: b db b a c c bd branches λ around loop V λ = 0 branchesν tied to node I ν = 0 IMPOSE CONSTRAINTS ON BRANCH VARIABLES

30 TWO METHODS FOR DEFINING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes Defines node fluxes Method of loops Defines loop charges 08-II-11a

31 EXAMPLE OF A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes Φ 2 Φ 3 Φ 7 Φ 1 φ f φ h Φ 6 φ g 1) Choose a reference electrode (ground) 2) Choose a spanning tree (accesses every node, no loop) 3) Select tree branch fluxes (closure branches left out) 4) Node flux Φ n is sum of branch fluxes to ground (closure branch fluxes are expressed as differences between node fluxes) φ d Φ 4 example: Φ = φ e Φ 5 Φ 7 = φ d + φ e + φ g φ h = Φ 7 - Φ 6 + cst n tree branches leading to n φ γ Φn γ = + ( ) Φ ( ) cst γ n + φ β β

32 INDUCTIVE vs CAPACITIVE ELEMENTS I φ Inductance : current I function only of flux φ t E I Vdt ' I φ ' dφ ' = = φ 0 ( ) Electrical equivalent of spring: φ X ; I f Q V Capacitance : voltage V function only of charge Q t E V Idt' V Q' dq' = = Q 0 ( ) Electrical equivalent of mass: Q P ; V V

33 NODE CHARGES The conjugate coordinates of node fluxes are node charges: they are the sum of all the charges going into capacitances linked to this node. Q n = capacitive branches exiting from n Q β n for each node: [ Φ Q ] n, n This can be demonstrated by writing the Lagrangian of the circuit from its dynamical equations and performing a Legendre transform to obtain the Hamiltonian = i Dynamical equations Lagrangian conjugate coordinates Hamiltonian (,, ) H( x, p, t) L x x t p = L x

34 HAMILTONIAN OF TWO CAPACITIVELY COUPLED RESONATOR MODES Φ 1 Φ 2 C c C b L a C a L b Reduced capacitance matrix: C C C + C C a c c = Cc Cb + C c Inverse of reduced capacitance matrix: 1 1 Cb Cc C + c = CC a b + CC a c + CC b c Cc Ca + C c ˆ ˆ ˆ ( ˆ ˆ ˆ ˆ ) Φ Φ H Φ1, Q1; Φ 2, Q2 = + 2L 2L Qˆ Qˆ QQ ˆ ˆ C 2C C a b

35 ANHARMONICITY vs CHARGE SENSITIVITY Cooper pair box levels are "exactly soluble" (A. Cottet, PhD thesis, Orsay, 2002)

36 ANHARMONICITY vs CHARGE SENSITIVITY IN THE LIMIT E J /E C >> 1 anharmonicity: ω12 ω01 E /2 8E ( ω + ω ) C J J. Koch et al. Phys. Rev. A '07 peak-to-peak charge modulation amplitude of level m: TRANSMON: SHUNT CPB JUNCTION WITH CAPACITANCE 290 μm Courtesy of J. Schreier and R. Schoelkopf

37 SPECTROSCOPY OF A JOSEPHSON ATOM J. Schreier et al., Phys. Rev. B '08 ω 12 ω 02 2 ω 12 ω 02 2 ω 01 ω 01 Anharmonicity: ω01 ω12 = 455MHz EC Sufficient to control the artificial atom as a two level system: Qubit Slide courtesy of J. Schreier and R. Schoelkopf

38 THE MEMORY READOUT PROBLEM QUBIT OFF READOUT 0 1 QUBIT OFF READOUT 0 1 or ON ON QUBIT OFF READOUT 0 1 ON 1 0 ε 1 ε FIDELITY: F = 1 ε ε 0 1 WANT: 1) SWITCH WITH ON/OFF RATIO AS LARGE AS POSSIBLE 2) READOUT WITH F AS CLOSE TO 1 AS POSSIBLE

39 STATE DECAY STRATEGY Martinis, Devoret and Clarke, PRL 55 (1985) Martinis, Nam, Aumentado and Urbina, PRL 89 (2002)

40 DISPERSIVE READOUT STRATEGY Blais et al. PRA 2004, Walraff et al., Nature 2004 rf signal in ω ω 01 or rf signal out QUBIT CIRCUIT 0 or 1 QUBIT STATE ENCODED IN PHASE OF OUTGOING SIGNAL, NO ENERGY DISSIPATED ON-CHIP A) FILTER OUT EVERYTHING ELSE THAN READOUT RF B) REPEAT WITH ENOUGH PHOTONS TO BEAT NOISE : USE THE BEST AMPLIFIER AS POSSIBLE (see whole sessions J3 & L17 + Q17.6& Y34.4)

41 SUPERCONDUCTING MICROWAVE RESONATOR: ANALOG OF FABRY-PEROT CAVITY FILTER ν ~ 10GHz w~20μm Al or Nb Si or Al 2 O 3 can get Q up to 10 6 ( in bulk: Q ~ ) length ~ 1cm, but photon decay length ~ 10km! CHIP ~ IN OUT ADC VERY SMALL MODE VOLUME 1/2 photon: ~1nA ~100nV

42 SUPERCONDUCTING CAVITY = FABRY-PEROT 10mm 3mm Nb 100μm qubit goes here (see slide 2, lower right, in this talk)

43 EXAMPLE OF QUANTRONIUM IN MICROWAVE CAVITY OFF-RESONANT NMR-TYPE PULSE SEQUENCE FOR QUBIT MANIPULATION

44 QUANTRONIUM IN MICROWAVE CAVITY READOUT PROBING PULSE or 0> 1> QUBIT STATE ENCODED IN PHASE OF TRANSMITTED PULSE

45 QUANTRONIUM IN MICROWAVE CAVITY Metcalfe et al. Phys. Rev. B (2007) 0> 200ns 40ns 40ns t 1> t AWG ~ ADC IN OUT latching effect due to bifurcation of cavity mode

46 IN-LINE TRANSMON WITH BIFURCATING MICROWAVE CAVITY READOUT See V17.6 M. Brink et al. Thursday morning, and also recent Saclay group results (in preparation) C in ~200aF qubit C couple ~6fF readout junction C out ~20fF 50Ω qubit cavity readout cavity 50Ω

47 P.I.'s Grads Post-Docs Res. Sc. Undrgrds Collab. Acknowledgements: Circuit Quantum Electrodynamics Groups Depts. Applied Physics and Physics, Yale M. D. R. VIJAY (UCB) C. RIGETTI M. METCALFE (NIST) V. MANUCHARIAN F. SCHAKERT N. MASLUK A. KAMAL I. SIDDIQI (UCB) C. WILSON (Chalmers) E. BOAKNIN (McK) N. BERGEAL (ESPCI) M. BRINK A. MARBLESTONE D. ESTEVE et coll. (Saclay) B. HUARD (LPA/ENS) R. SCHOELKOPF J. CHOW B. TUREK (MIT) J. SCHREIER B. JOHNSON A. SEARS M. READ A. WALRAFF (ETH) H. MAJER (Vienna) A. HOUCK (Princeton) D. SCHUSTER L. DiCARLO L. FRUNZIO J. SCHWEDE P. ZOLLER (Innsbruck) S. GIRVIN T. YU L. BISHOP J. KOCH J. GAMBETTA (U. Waterloo) E. GINOSSAR A. NUNNENKAMP F. MARQUARDT (Munich) A. BLAIS (Sherbrooke) A. CLERK (McGill) W.M. KECK

48 P.I.'s Grads Post-Docs Res. Sc. Undrgrds Collab. Acknowledgements: Circuit Quantum Electrodynamics Groups Depts. Applied Physics and Physics, Yale M. D. R. VIJAY (UCB) C. RIGETTI M. METCALFE (NIST) V. MANUCHARIAN F. SCHAKERT N. MASLUK A. KAMAL I. SIDDIQI (UCB) C. WILSON (Chalmers) E. BOAKNIN (McK) N. BERGEAL (ESPCI) M. BRINK A. MARBLESTONE D. ESTEVE et coll. (Saclay) B. HUARD (LPA/ENS) R. SCHOELKOPF (Keithley Award, T3) J. CHOW B. TUREK (MIT) J. SCHREIER B. JOHNSON A. SEARS M. READ A. WALRAFF (ETH) H. MAJER (Vienna) A. HOUCK (Princeton) D. SCHUSTER L. DiCARLO L. FRUNZIO J. SCHWEDE P. ZOLLER (Innsbruck) S. GIRVIN T. YU L. BISHOP J. KOCH J. GAMBETTA (U. Waterloo) E. GINOSSAR A. NUNNENKAMP F. MARQUARDT (Munich) A. BLAIS (Sherbrooke) A. CLERK (McGill) W.M. KECK