Advances in Josephson Quantum Circuits

Transcription

1 APS 00 March Meeting, Tutorial #3 Advances in Josephson Quantum Circuits Instructors: Michel Devoret, Yale University "Introduction to superconducting quantum circuits" Yasunobu Nakamura, NEC Japan "Superconducting qubits coupled to a transmission line " John Martinis, University of California, Santa Barbara "Precision Control of Josephson Qubits" Leo DiCarlo, Yale University "Production and detection of entanglement in cqed processors" Portland Convention Center Sunday, March 4 8:30 a.m. - :30 p.m.

2 Introduction to superconducting quantum circuits Outline Motivation: quantum information Why Josephson junctions? Main flavors of Josephson qubits Readout of qubits -qubit qnd -qubit gates

3 RECENT REVIEWS ON JOSEPHSON QUANTUM CIRCUITS M.H. Devoret and J.M. Martinis, Quantum Information Processing 3,63 (004) A. Blais et al., Phys. Rev. A 75, 0339 (007) J. Clarke and F. Wilhelm, Nature 453, (008) R. Schoelkopf and S. M. Girvin, Nature 45, (008) J.M. Martinis, Quantum Information Processing 8, 8 (009) Final version of this presentation available at (talks)

4 CLASSICAL BIT = SWITCH Mechanical switch Electrical switch CMOS Transistor Energy B kt noise Coordinate Bit state is either 0 or : ) strong dissipation and ) kt noise << B

5 QUANTUM BIT: LEVELS FORMING EFFECTIVE SPIN / MOLECULE, ATOM, PARTICLE... ENERGY spin up z θ Bloch sphere representation α 0 + β 0 } x φ spin down y α = cos β = sin θ θ e e + iφ iφ Qubit state can be 0 and : ) no dissipation and ) kt noise << hω 0

6 RELAXATION TIMES OF QUANTUM MEMORY T PROCESS random fields in x,y plane T φ PROCESS random field along z T = + T T φ DECOHERENCE TIME ω0t DECOHERENCE QUALITY FACTOR

7 THE POWER OF QUANTUM SUPERPOSITION REGISTER WITH N=0 BITS: N = 04 POSSIBLE CONFIGURATIONS 0 classically, can store and work only on one number between 0 et 03

8 THE POWER OF QUANTUM SUPERPOSITION REGISTER WITH N=0 BITS: N = 04 POSSIBLE CONFIGURATIONS 0 classically, can store and work only on one number between 0 et 03 quantally, can store and work on an arbitrary superposition of these numbers! N Ψ = α0 0 + α + α α N

9 QUANTUM PARALLELISM suppose a function f j { 0,03} n= f ( j) { 0,03} Classically, need bit registers (0,000 bits) to store information about this function and to work on it. Quantum-mechanically, a 0-qubit register can suffice! Ψ = N / N j= 0 j f j ( ) Function encoded in a superposition of states of register

10 HOW CAN A SUPERCONDUCTING CIRCUIT BEHAVE LIKE AN ATOM? SIMPLEST EXAMPLE: SUPERCONDUCTING LC OSCILLATOR CIRCUIT MICROFABRICATION L ~ 3nH, C ~ pf, ω r /π ~ 4GHz ELECTRONIC FLUID FLOWS BACK AND FORTH BETWEEN PLATES: ALL ELECTRONS BEHAVE AS A SINGLE CHARGED ENTITY see practical LC superconducting resonators: Lindström et al., PRB 80, 350 (009) Paik & Osborn, APL 96, (00)

11 QUANTUM CIRCUITS IN A NUTSHELL: FLUX AND CHARGE DO NOT COMMUTE φ I +Q -Q V φˆ, Qˆ = i φ = LI Q= CV

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

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

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

15 CAN PLACE CIRCUIT IN ITS GROUND STATE E φ hω r residual dissipation provides reset of circuit ω r kt 5 GHz 5 mk B

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

17 NEED NON-LINEARITY TO FULLY REVEAL QUANTUM MECHANICS Potential energy Position coordinate Emission spectrum frequency ω 34 ω 3 ω ω 0

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

19 JOSEPHSON TUNNEL JUNCTION PROVIDES A NON-LINEAR INDUCTOR WITH NO DISSIPATION nm S I L J C J S Ι ( ) U = E cos φ / φ J 0 L J φ0 φ0 = = E I J 0 φ t ( ') = V t dt ' φ ordinary inductance Bare Josephson potential φ 0 = e

20 ENERGY SCALES OF THE JOSEPHSON JUNCTION "ATOM" REST OF CIRCUIT q ext t ( ) Q= ˆ I t' dt' φ e ˆ φ ˆ ϕ = ˆ ˆ Q N = e ˆ, ϕ Nˆ = i Hamiltonian: Hˆ J ( Nˆ N ) ext = 8E cos ˆ C E J ϕ Coulomb charging energy for e Josephson energy E C = e C j reduced offset charge N ext = q ext e E J = valid for opaque barrier barrier transp cy NT Δ 8 gap # cond ion channels

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

22 3 TYPES OF BIASES charge L J C g C J U C g ( Nˆ CU/) g 8E cos ˆ C EJ ϕ "Cooper pair box" ˆϕ lives on circle ˆN integer

23 charge LJ 3 TYPES OF BIASES flux L J C g CJ U C g L CJ Φ b ( Nˆ CU/) g 8E cos ˆ C EJ ϕ "Cooper pair box" ˆϕ lives on circle ˆN integer eφ b ˆ ˆ N ϕ 8E cos ˆ C E + L EJ ϕ "RF-Squid", ˆϕ lives on line, ˆN real number

24 3 TYPES OF BIASES charge flux current L L J J LJ C g CJ U ( Nˆ CU/) g C g 8E cos ˆ C EJ ϕ "Cooper pair box" ˆϕ lives on circle ˆN integer L CJ Φ b eφ b ˆ ˆ N ϕ 8E cos ˆ C E + L EJ ϕ C J I b 8 Nˆ cos "RF-Squid", ˆϕ lives on line, ˆN real number EC b E ˆ ˆ J ϕ ϕ I0 Φ b L I in the limit Φ L b I b

25 EFFECTIVE POTENTIAL OF 3 MAIN BIAS SCHEMES see also proposals for topologically protected qubits, for example Feigelman et al. PRL 9, (004) "charge" bias "phase" bias "flux" bias a few levels here. quasi-continuum there. ϕ = π eφ h + Φ b Φ = 0 in e g CEA Saclay, NEC, Yale Chalmers, JPL,... TU Delft, NEC, NTT, IBM, MIT, UC Berkeley, SUNY, IPHT Jena... eφ h eφ h NIST, UCSB, U. Maryland, I. Neel Grenoble...

26 SUPERCONDUCTING ARTIFICIAL ATOMS "MENDELEEV" TABLE Cooper Pair Box Quantronium Transmon Fluxonium E L / E J inverse of number of wells in potential Flux Qubit Phase qubit E J / E C charge fluctuations relative to phase fluctuations

27 THE MEMORY READOUT PROBLEM 0 or QUBIT OFF READOUT 0 or QUBIT OFF ON READOUT 0 ON pointer variable QUBIT OFF ON READOUT 0 0 ε ε FIDELITY: F = ε ε 0 WANT: ) SWITCH WITH ON/OFF RATIO AS LARGE AS POSSIBLE ) READOUT WITH F AS CLOSE TO AS POSSIBLE 3) FAST, 4) PRESERVE STATE (QND)

28 STATE DECAY STRATEGY 0 0 Martinis, Devoret and Clarke, PRL 55 (985) Martinis, Nam, Aumentado and Urbina, PRL 89 (00)

29 DISPERSIVE READOUT STRATEGY Blais et al. PRA 004, Walraff et al., Nature 004 rf signal in ω ω 0 or rf signal out QUBIT CIRCUIT 0 or 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 session V6 )

30 SCHEMATIC OF COOPER PAIR BOXES IN A MICROWAVE RESONATOR (CAVITY) OUT IN Roles of cavity: ) Filter, ) Dispersive measurement, 3) Quantum bus "Circuit QED": Review by Blais et al., Phys. Rev. A 75, 0339 (007)

31 σ σ σ σ z x y w PAULI SPIN MATRICES AND ROTATIONS 0 = = Z 0 0 = = X 0 0 i = = Y i 0 0 = = I 0 useful notation of Pauli spin matrices HERMITIAN (MEASUREMENT) Z i 0 = iσz = Rz 0 i π X 0 i = iσx = Rx i 0 π Y 0 = iσy = Ry 0 π 0 I = σw = Identity 0 [ ] ( ) [ ] ( ) [ ] ( ) UNITARY (GATE)

32 ELEMENTARY GATES ARE π/ ROTATIONS In lab frame: In rotating frame at Larmor freq.: Hˆ Hˆ Do rotating wave approximation ( t) ω area π/ x z = ω σ + ω () t cos ω t+ φ () t σ z x z x transverse osc. field amplitude σ σ x = ωx() t cos φ() t ωx() t sin φ() t + [ Z ] / : shift Zeeman field [ X ] / [ Y ] / x ( t) ω area π/ y Pulse 90 around x φ = 0 t Pulse 90 around y φ = π/ t

33 NATURAL ENTANGLING OPERATIONS Uˆ ( τ) = exp ( ih ) intτ / ˆ Secular interaction: Flip-flop interaction: Hˆ int = g σ σ z int + z adjustment of gate duration time: Hˆ = g σ σ + hc.. ( σ σ σ σ ) = g + x x y y τ = [ ZZ ] / π 4g [ ] / [ ] / XX YY

34 PAIRWISE COUPLING v.s. BUS COUPLING qubit coupling element: capacitor, inductor, auxiliary qubit microwave transmission line resonator qubit coupling element

35 TWO-QUBIT QUANTUM PROCESSOR slide courtesy of Leo DiCarlo & Rob Schoelkopf f V f V Q Q T9, V6, Y6 see qubit and cavities: B. Johnson et al., 3 qubits and cavity L. DiCarlo et al.: T6, W6

36 P.I.'s Grads Post-Docs Res. Sc. Collab. Acknowledgements: Circuit Quantum Electrodynamics Groups Depts. Applied Physics and Physics, Yale M. D. R. VIJAY (UCB) M. METCALFE (NIST) V. MANUCHARIAN F. SCHAKERT N. MASLUK A. KAMAL I. SIDDIQI (UCB) C. WILSON (Chalmers) E. BOAKNIN (McK) N. BERGEAL (ESPCI) C. RIGETTI M. BRINK D. ESTEVE et coll. (Saclay) B. HUARD (LPA/ENS) R. SCHOELKOPF B. TUREK (MIT) J. CHOW B. JOHNSON A. SEARS M. READ A. WALRAFF (ETH) H. MAJER (Vienna) A. HOUCK (Princeton) D. SCHUSTER L. DiCARLO L. SUN H. PAIK L. FRUNZIO 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