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INTRODUCTION TO QUANTUM SUPERCONDUCTING CIRCUITS: PART 1 QUAN ELEC UM – MECHANICAL RONICS LAB Applied Physics and Physics, Yale University R. VIJAY (U.C.Berkeley) I. SIDDIQI (U.C.Berkeley) C. RIGETTI M. METCALFE (NIST) V. MANUCHARIAN N. BERGEAL (ESPCI Paris) F. SCHACKERT A. KAMAL B. HUARD (ENS Paris) A. MARBLESTONE (MIT) N. MASLUK M. BRINK K. GEERLINGS L. FRUNZIO M. DEVORET Acknowledgements: R. Schoelkopf, S. Girvin, D. Prober & D. Esteve W.M. KECK Rutgers Dec. 09 MESOSCOPIC ELECTRONS vs PHOTONS INDEPENDENT ELECTRON INTERFERENCES DIRECTLY REVEAL QUANTUM MECHANICS Schrödinger's equation Example: Sharvin-Sharvin & A.B. effects Φ 2 ⎡ 2⎛ ⎤ eA ⎞ ∂ ∇ − + Ψ = Ψ i eU i ⎢ ⎜ ⎥ ⎟ ∂t ⎠ ⎢⎣ m ⎝ ⎥⎦ Interactions provide further quantum effects INDEPENDENT PHOTON INTERFERENCES DO NOT DIRECTLY REVEAL QUANTUM MECHANICS Maxwell's equations ( ) ( ∂ c∇ × E + icB = i E + icB ∂t Planck's constant drops out! Need ph-ph interactions.... ) QUANTUM OPTICS QUANTUM RF CIRCUITS FIBERS, BEAMS TRANSM. LINES, WIRES BEAM-SPLITTERS COUPLERS MIRRORS CAPACITORS LASERS ~ GENERATORS PHOTODETECTORS AMPLIFIERS ATOMS JOSEPHSON JUNCTIONS ADVANTAGES OF CIRCUITS: - PARALLEL FABRICATION METHODS - LEGO BLOCK CONSTRUCTION OF HAMILTONIAN - ARBITRARILY LARGE ATOM-FIELD COUPLING DRAWBACKS OF CIRCUITS: ARTIFICIAL ATOMS PRONE TO VARIATIONS SELECTED BIBLIOGRAPHY ON QUANTUM CIRCUITS Michel Devoret, in "Quantum Fluctuations", S. Reynaud, E. Giacobino, J. Zinn-Justin, Eds. (Elsevier, Amsterdam, 1997) p. 351-385 Michel Devoret & John Martinis, Quant. Inf. Proc., 3, 351-380 (2004) Robert J. Schoelkopf & Steve M. Girvin, Nature 451, 664-669 (2008) John Clarke & Frank K. Wilhelm, Nature 453, 1031-1042 (2008) OUTLINE TODAY: TOMORROW: PRESENT BASIC CIRCUIT ELEMENTS COMPARE QUBIT CIRCUITS SUPERCONDUCTING CIRCUIT ELEMENTS CAPACITANCE INDUCTANCE RESISTANCE SUPERCONDUCTING CIRCUIT ELEMENTS e2 EC = 2C CAPACITANCE EL = 2 2 4e L EJ = INDUCTANCE RESISTANCE 2 4e 2 L J TRANSMISSION LINES ARE JUST L's AND C's L L C L C L C L C LIKEWISE, EVERY LINEAR PASSIVE RECIPROCAL RF COMPONENT IS BUILT FROM INDUCTANCES AND CAPACITANCES RESISTORS CAN BE THOUGHT OF AS INFINITE TRANSMISSION LINES (NYQUIST, CALDEIRA & LEGGETT) DYNAMICAL VARIABLES OF CIRCUIT CIRCUIT : ARBITRARY NETWORK OF ELECTRICAL ELEMENTS node n element Vnp branch np loop Inp node p TWO DYNAMICAL VARIABLES CHARACTERIZE THE STATE OF EACH DIPOLE ELEMENT AT EVERY INSTANT: Voltage across the element: Vnp ( t ) = ∫ E ⋅ d p n Current through the element: I np ( t ) = ∫∫ j ⋅ dσ np Signals: any linear combination of these variables CONSTITUTIVE RELATIONS OF CIRCUIT ELEMENTS linear inductance: V = L dI/dt non-linear inductance: linear capacitance: I = C dV/dt E = LI2/2= Φ2/2L E = f(Φ) f non-quadratic E = CV2/2= Q2/2L Inductances and capacitances are "bottles" for electric and magnetic fields KIRCHHOFF’S LAWS b c a b b d ∑ λ Vλ = 0 branches around loop c d a b ∑ν branches tied to node Iν = 0 CONSTITUTIVE RELATIONS + KIRCHHOFF'S LAWS MAXWELL'S EQUATIONS ON A NETWORK QUANTUM TREATMENT OF CIRCUITS Iβ rest of circuit branch β Vβ Need to take branch flux and branch charge as basic variables: φβ ( t ) = ∫ Vβ ( t ')dt ' t −∞ Qβ ( t ) = ∫ I β ( t ')dt ' t −∞ For every branch β in the circuit: ⎡φˆβ , Qˆ β ⎤ = i ⎣ ⎦ FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes 1) Choose a reference electrode (ground) FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes 1) Choose a reference electrode (ground) 2) Choose a spanning tree (accesses every node, no loop) FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes 1) Choose a reference electrode (ground) 2) Choose a spanning tree (accesses every node, no loop) FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes 1) Choose a reference electrode (ground) 2) Choose a spanning tree (accesses every node, no loop) FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes φb φc φa 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) φd φg φf φe FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Φ2 Method of nodes Φ3 Φ7 Φ1 Φ6 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 is sum of branch fluxes to ground (closure branch fluxes are expressed as differences between node fluxes) Φ4 Φ5 FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Φ2 Method of nodes Φ3 φh Φ1 φf 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 is sum of branch fluxes to ground (closure branch fluxes are expressed as differences between node fluxes) φd Φ4 example: Φ7 φg Φ6 φe Φ5 Φ 7 = φd + φe + φg φh = Φ 7 - Φ 6 + cst FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Φ2 Method of nodes Φ3 φh Φ1 φf 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 is sum of branch fluxes to ground (closure branch fluxes are expressed as differences between node fluxes) φd φg Φ6 φe Φ4 example: Φ7 Φ5 Φ 7 = φd + φe + φg φh = Φ 7 - Φ 6 + cst Φn = ∑ tree branches β leading to n φβ φγ = Φ n (γ ) − Φ n (γ ) + cst + − TWO METHODS FOR DEFINING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes Method of loops Defines loop charges HAMILTONIAN OF CIRCUIT mechanical analog world electrical world φ I X +Q -Q V M Q (φ − φ0 ) H= + 2C 2L 2 φ= ∂H Q = ∂Q C ∂H − (φ − φ0 ) Q=− = L ∂φ ωr = f 2 1 LC k k ( X − X0 ) P H= + 2M 2 2 ∂H P = X= ∂P M ∂H P=− = −k ( X − X 0 ) ∂X 2 ωr = k M A SUPERCONDUCTING CIRCUIT BEHAVING LIKE AN ATOM? SIMPLEST EXAMPLE: SUPERCONDUCTING MICROFABRICATION LC OSCILLATOR CIRCUIT L ~ 3nH, C ~ 10pF, ωr /2π ~ 2GHz ELECTRONIC FLUID SLOSHES BACK AND FORTH FROM ONE PLATE TO THE OTHER, INTERNAL MODES FROZEN BEHAVES AS A SINGLE CHARGE CARRIER DEGREE OF FREEDOM IN ATOM vs CIRCUIT Superconducting LC oscillator Example of Rydberg atom L velocity of electron → force on electron → voltage across capacitor current through inductor C LC CIRCUIT AS A QUANTUM HARMONIC OSCILLATOR E φ +Q -Q hω r φ φˆ Qˆ φˆ Qˆ † aˆ = ; aˆ = +i −i φZPF QZPF φZPF QZPF φZPF = 2 ωr L ( ⎡⎣ aˆ , aˆ † ⎤⎦ = 1 trapped photons! QZPF = 2 ωr C Hˆ = ωr aˆ † aˆ + 1 annihilation and creation operators 2 ) ALL TRANSITIONS BETWEEN QUANTUM LEVELS ARE DEGENERATE IN PURELY LINEAR CIRCUITS! E φ hω r φ CANNOT STEER THE SYSTEM TO AN ARBITRARY STATE IF PERFECTLY LINEAR NEED NON-LINEARITY TO FULLY REVEAL QUANTUM MECHANICS Potential energy Position coordinate JOSEPHSON TUNNEL JUNCTION PROVIDES A NON-LINEAR INDUCTOR WITH NO DISSIPATION S 1nm I S Ι Ι LJ superconductorinsulatorsuperconductor tunnel junction Ι = φ / LJ CJ LJ = φ0 I0 I0 φ φ = ∫−∞ V ( t ')dt ' t I = I 0 sin (φ / φ0 ) φ0 = 2e JOSEPHSON TUNNEL JUNCTION PROVIDES A NON-LINEAR INDUCTOR WITH NO DISSIPATION S 1nm I S superconductorinsulatorsuperconductor tunnel junction LJ CJ LJ = U = − EJ cos (φ / φ0 ) Ι φ = ∫−∞ V ( t ')dt ' t φ02 EJ φ 2 EJ bare Josephson potential φ0 = 2e TRANSMISSION LINE AS 1D BOSON FIELD L n-1 C L Vn −1 I n −1 C n L L n+1 Vn I n C n+2 L Vn +1 I n +1 C a Dynamical equations: d Vn − Vn +1 = L I n dt d I n −1 − I n = C Vn dt Continuum limit: Field equations: Vn +1 − Vn ∂V → ∂x a I n +1 − I n ∂I → ∂x a C L →C ; → L a a ∂V = −L ∂x ∂I = −C ∂x ∂I ∂t ∂V ∂t CHARGE AND FLUX IN CONTINUUM LIMIT Φ ( x) Lδx Π ( x)δ x Φ ( x + δ x) Lδx C δx Lδx C δx Π ( x + δ x)δ x δx ⎡⎣Φ ( x ) , Π ( x ) δ x ⎤⎦ = i ⎡⎣Φ ( x ) , Π ( x + δ x ) δ x ⎤⎦ = 0 δx→0 ˆ ( x ),Π ˆ ( x )⎤ = i δ ( x − x ) ⎡Φ 1 2 ⎦ 1 2 ⎣ Hamiltonian : energy density as a function of field and conjugate momentum: 2 2⎫ +∞ ⎧ 1 ˆ 1 ˆ ˆ ⎡ ⎤ ⎡ ⎤ H = ∫ dx ⎨ Π ( x )⎦ + ∇Φ ( x ) ⎦ ⎬ ⎣ ⎣ −∞ 2L ⎩ 2C ⎭ COMPATIBILITY WITH STANDARD QED For simplest geometry, consider stripline waveguide: w y h x z B ( x1 , y1 , z1 ) E ( x2 , y2 , z2 ) Flux between strips: x yb + h −∞ yb ˆ ( x ) = 1 dx Φ 1 ∫ ∫ Strip charge per unit length: dy Bˆ z ( x, y, z1 ) ˆ (x ) = ε Π 2 0∫ z f +w zf dz Eˆ y ( x2 , y2 , z ) Commutation relations between field operators in standard QED: i ∂ ⎡ Bˆ z ( x1 , y1 , z1 ) , Eˆ y ( x2 , y2 , z2 ) ⎤ = ⎣ ⎦ ε ∂x δ ( x1 − x2 ) δ ( y1 − y2 ) δ ( z1 − z2 ) 0 1 ˆ ( x ),Π ˆ ( x )⎤ = i δ ( x − x ) ⎡Φ 1 2 ⎦ 1 2 ⎣ OK! END OF 1ST LECTURE INTRODUCTION TO QUANTUM SUPERCONDUCTING CIRCUITS: PART 2 QUAN ELEC UM – MECHANICAL RONICS LAB Applied Physics and Physics, Yale University R. VIJAY (U.C.Berkeley) I. SIDDIQI (U.C.Berkeley) C. RIGETTI M. METCALFE (NIST) V. MANUCHARIAN N. BERGEAL (ESPCI Paris) F. SCHACKERT A. KAMAL B. HUARD (ENS Paris) A. MARBLESTONE (MIT) N. MASLUK M. BRINK K. GEERLINGS L. FRUNZIO M. DEVORET Acknowledgements: R. Schoelkopf, S. Girvin, D. Prober & D. Esteve W.M. KECK Rutgers Dec. 09 OUTLINE YESTERDAY: TODAY: PRESENT BASIC CIRCUIT ELEMENTS COMPARE QUBIT CIRCUITS QUANTUM TREATMENT OF CIRCUITS Iβ rest of circuit branch β Branch flux and branch charge are primary variables: φβ ( t ) = ∫ Vβ ( t ')dt ' t −∞ Vβ Qβ ( t ) = ∫ I β ( t ')dt ' t −∞ For every branch β in the circuit: ⎡φˆβ , Qˆ β ⎤ = i ⎣ ⎦ IRREVERSIBLE/REVERSIBLE CHARGE TRANSFER e tunneling quasiparticle energy 2Δ rate before SIS TUNNEL JUNCTION after IRREVERSIBLE/REVERSIBLE CHARGE TRANSFER e tunneling quasiparticle energy 2Δ rate before SIS TUNNEL JUNCTION after 2e CP tunneling quasiparticle energy matrix element DYNAMICS OF JOSEPHSON ELEMENT FROM CHARGE POINT OF VIEW (IN ABSENCE OF Q.P. TUNNELING) integer Hopping N0 − 1 N0 N0 + 1 Q =N 2e 1 h / e2 M = EJ = Gt Δ 2 16 Charge states: N̂ N = N N Josephson tunneling hamiltonian in charge basis: EJ ˆ HJ = − 2 ∑( N N N +1 + N +1 N ) FROM NUMBER TO "PHASE" REPRESENTATION ϕˆ = 2eφˆ / ⎡ϕˆ , Nˆ ⎤ = i ⎣ ⎦ eiϕˆ Nˆ e − iϕˆ = Nˆ − 1 EJ ˆ HJ = − N N +1 + N +1 N ( ∑ 2 N EJ iϕˆ (from expression of − iϕˆ = − ( e + e ) translation operator) 2 = − EJ cos ϕˆ ( = − EJ cos φˆ / φ0 Josephson's ϕ ) ) ϕ Φ0 φ0 = = 2e 2π runs on a line but tunnel hamiltonian is periodic ELECTRODYNAMICS OF JUNCTION IN ITS ENVRONMENT REST OF CIRCUIT Z (ω ) CJ EJ U(t) 2 φ ( t ) = ∫−∞ ∫ E ( x,τ ) dxdτ t 1 N.B. The electric field E encompasses here all contributions of the force on the electrons doing work, including those usually called chemical potential effects. ELECTRODYNAMICS OF JUNCTION IN ITS ENVRONMENT t Qext = ∫ Idτ −∞ Q = 2eN REST OF CIRCUIT EJ QC Z (ω ) CJ U(t) 2 φ ( t ) = ∫−∞ ∫ E ( x,τ ) dxdτ t 1 Equation of motion: ∂ CJ φ + ∂φ ⎡ ⎛ φ ⎞⎤ ⎢ − EJ cos ⎜ ⎟ ⎥ = I φ , φ ,.... ⎝ φ0 ⎠ ⎦ ⎣ ( ) Can be obtained in general from a Lagrangian: d ∂L ∂L − =0 dt ∂φ ∂φ CJ 2 φ L = L J +L ext = φ + EJ cos + L ext φ , φ ,... 2 φ0 ( ) TWO CHARACTERISTIC ENERGIES OF ENVIRONMENT Total environment admittance: Effective shunt capacitance of junction: Effective shunt inductance of junction Ytot (ω ) = iCJ ω + Z −1 (ω ) Im ⎡⎣Ytot (ω ) ⎤⎦ CΣ = lim ω ω →0 Leff ⎧⎪ ⎡ ⎤ ⎫⎪ 1 = lim ⎨Im ⎢ ⎥⎬ ω →0 ⎩⎪ ⎣ ωYtot (ω ) ⎦ ⎭⎪ (electron charge appears here instead of Cooper pair charge for convenience in some formulas) 2 Coulomb charging energy Inductive energy e EC = 2CΣ EL h / 2e ) ( = Leff 2 (form chosen for easy comparison with Josephson energy) TWO BASIC SUPERCONDUCTING "ATOMS" flux charge superconducting island LJ superconducting loop ϕ∈ LJ N∈ CJ Φext L > LJ Cg CJ U I "HYSTERETIC RF SQUID" E J > EL EC Δϕ 2π Friedman et al. (2000) "COOPER PAIR BOX" 1 EL = 0; EJ EC ΔN < 1 Bouchiat et al. (1998), Nakamura, Pashkin, Tsai (1999) PHASE POTENTIALS OF SQUID AND BOX E J > EL EL = 0; EJ EC EC E SQUID BOX max. curvature = EL − E J 2EJ −1 2eΦ ext / 0 ϕ 2π +1 +2 − ϕ 2π 1 2 e + 1 2 iπ C gU / e RICH LEVEL STRUCTURE, BUT EXTREME SENSITIVITY TO NOISE 09-III-10 PHASE-CHARGE DUALITY E J > EL EL = 0; EJ EC EC E SQUID BOX 8 EC N g − 2π EL ϕext − π −1 0 ϕ 2π +1 +2 1 2 −1 +1 0 N +2 CHARGE QUBIT FAMILY THE SINGLE COOPER PAIR BOX ARTIFICIAL ATOM EJ Bouchiat et al. (1998) Nakamura, Tsai and Pashkin (1999) 0.5EC U Φ U Q=2Ne Φ Cg EJ 4 EC QUANTRONIUM Vion et al. (2001) U Φ Q=2Ne Φ U Cg TRANSMON COOPER PAIR BOX J. Koch et al. (2007) A. Houck et al. (2007) J. Schreier et al. (2007) U Φ EJ 50 EC U Q=2Ne Φ Cg Cottet et al. (2002) Koch et al. (2007) ANHARMONICITY vs OFFSET CHARGE INSENSITIVITY IN COOPER PAIR BOX gap EJ "C.P. BOX" "QUANTRONIUM" "TRANSMON" FLUX QUBIT FAMILY EXAMPLES OF QUANTUM CIRCUITS BELONGING TO THE RF-SQUID TYPE Readout Readout Friedman, Patel, Chen, Tolpygo and J. E. Lukens, Nature 406, 43 (2000). Chiorescu, Nakamura, Harmans & Mooij, Science 299, 1869 (2003). "flux" qubit "phase" qubit Readout Steffen et al., Phys. Rev. Lett. 97, 050502 (2006) FLUXONS PLAY WITH FLUX QUBITS A ROLE SIMILAR TO WHAT COOPER PAIRS PLAY WITH CHARGE QUBITS Inductive energy -1 0 1 Φ Φ0 ⎛ EJ ⎞ ~ exp ⎜⎜ −α ⎟⎟ E C ⎠ ⎝ OUR RECENT CHILD IN FLUX QUBIT FAMILY NOISE IN THE TWO BASIC QUBITS superconducting loop flux charge LJ LJ ϕ ΦN Φext flux superconducting island CJ L ∼ LJ noise! N Cg CJ UN U charge noise! I "RF SQUID" Δϕ 2π 1 "COOPER PAIR BOX" ΔN < 1 FLUXONIUM IDEA: SHUNT A COOPER PAIR BOX AT DC, LEAVE IT UNSHUNTED AT RF L LJ LJ Cg CJ UN U Manucharyan. et al., Science, 326, 113; arxiv 0906.0831 Koch. et al., Phys. Rev. Lett, to appear; arxiv 0902.2980 Manucharyan. et al., submitted to Nature, arxiv 0910.3039 PRACTICAL IMPLEMENTATION write / d a e r Small junction: EJ/EC = 3.6 Array junctions (N =43): EJA/ECA = 28 Readout f0 = 8 GHz; Q=400 Every island is shunted by at least one large junction: array performs as inductor CHIP AND MODEL “Atom” EC = ½ e2/(CJ+Cc)=2.5 GHz EJ = ½ (Φ0/2π)2/LJ=8.9GHz EL = ½ (Φ0/2π)2/L=0.52GHz “Cavity” ωR=(LRCR)-1/2/2π = 8.2GHz ZR=(LR/CR)1/2 = 82Ω Coupling constant g = ωR(ZR/2RQ)1/2 /(1+CJ/Cc) = 135MHz (RQ=1 kΩ) CJ/Cc=11 PARAMETERS OF THE ARRAY EJA=22.5GHz ECA=0.8GHz Cg~CJ/2000 N =43 <(CJ/Cg)1/2 TWO-TONE SPECTROSCOPY vs EXTERNAL FLUX Φ0 Φ≈ 2 0-3 transition Parasitic resonance in the array (accounted by a minor model correction) 0-2 transition Inset: 0-2 transition symmetry-forbidden at zero flux resonator & vacuum Rabi 0-1 transition More than 106 points, taken over >72 hours w/o jumps or drifts! NO OFFSET CHARGE THY vs EXP CONFIRMS SINGLE COOPER PAIR REGIME V.Manucharyan, J.Koch, L.Glazman, M.Devoret, Science, 326, 113; arxiv 0906.0831 LOCATING FLUXONIUM ON THE SUPERCONDUCTING QUBIT MAP A QUBIT OPERATING FOR ALL VALUES OF FLUX ~350MHz Inductive energy 9GHz Potential energy -1 1 0 Φext =1/4 Φ0 Φext =1/2 Φ0 Φ Φ0 Φext = 0 ~2EJ 2π 2π2EL 2π 4π ϕ Φ0 TWO-TONE SPECTROSCOPY vs EXTERNAL FLUX Φ ≈ 2 A 370MHz ATOM MEASURED THROUGH A 8.2GHz CAVITY? atom transition 0-1: store/manipulate atom transition 1-2: couple to readout g12 = g <1|N|2> ~ 100MHz ω12 - ωR ~ 1GHz χ= g122/(ω12 - ωR) ~10MHz At the same time ω01 << ωR reflected phase (deg) TIME DOMAIN COHERENCE time (μs) Time, ns Frequency, GHz COHERENCE AS A FUNCTION OF FLUX BIAS ∂Φ T21/ f = Q1=130,000 quasiparticles, quantum phase slips? External flux Φ ext / Φ 0 ∂ω01 A ⎡⎣ = 10−6 Φ 0 ⎤⎦ Time, ns Frequency, GHz COHERENCE AS A FUNCTION OF FLUX BIAS External flux Φ ext / Φ 0 CONCLUSIONS AND PERSPECTIVES QUANTUM CIRCUITS OFFER A RICH PLAYGROUND FOR EMULATING EXISTING MANY-BODY QUANTUM SYSTEMS AND CREATING NEW ONES STRONG COUPLING, NON-PERTURBATIVE REGIME IS EASILY ACCESSIBLE PRESENT CHALLENGE IS QUBIT WITH BUILT-IN ERROR-CORRECTION, AND MORE GENERALLY QUANTUM FEEDBACK MECHANICAL ANALOG OF RF SQUID angle of pendulum : gauge invariant phase difference moment of inertia of pendulum : junction capacitance spring : loop inductance torque due gravity : Josephson current potential energy of pendulum : Josephson energy U(ϕ ) ϕ ϕ MECHANICAL ANALOG OF COOPER PAIR BOX angular velocity: Cg U Ω= CΣ φ0 rotating magnet ϕ̂ compass needle needle angular momentum: Nh torque on needle due to magnet current through junction velocity of needle in magnet frame voltage across junction Introduction to quantum superconducting circuits 1) Usual 2deg physics: circuit with normal electrons. quantum circuits means that the electron obeys S. equation over the whole scal of circuit. Electron propagate as Fermi waves 2) What about electromagnetic Bose wave propagation? ⎡− ⎤ ∂ Δ + U ( x, t ) ⎥ Ψ ( x, t ) = i Ψ ( x, t ) ⎢ ∂t ⎣ m ⎦ 2 ( ) c∇ × E + icB = i ( ∂ E + icB ∂t ) 3) Need non-linearity. But usually, non-linear element are also dissipative. Dissipation not good for QM 4) Josephson element: non-linear, non-dissipative, point-like 5) Lumped element electromagnetism: L and C's 6) L bottle for B field. C bottle for E field. Constitutive equations + Kirchhoff equations for circuits are equivalent to Maxwell's equation Define flux and charge for one element Flux and charge are conjugate variables How do we understand it? 2 choices flux is position, charge is position Connections between elements Example of harmonic oscillator Always in the correspondence limit The Josephson junction non linear inductance Two superconducting electrodes: islands 1sole degree of freedom in an island : its number of Cooper pair Charge hopping between two islands (zero external field) Gives back Josephson formula Arrays of Josephson junctions tends toward inductances Quantum phase slips Conclusions: finite networks are caricature of distributed systems infinite networks are more powerful than distributed systems single junction is non-linear inductance. Infinite junction array is linear inductance Artificial Josephson junction atoms 3 types of energies Capacitive environem: easy Inductive environment: more sutble