Introduction to Quantum Mechanics of Superconducting Electrical Circuits

Transcription

1 Introduction to Quantum Mechanics of Superconducting lectrical Circuits What is superconductivity? What is a osephson junction? What is a Cooper Pair Box Qubit? Quantum Modes of Superconducting Transmission Lines See R.-S. Huang PhD thesis on Boulder 004 web page

2 What is superconductivity? See: Boulder Lectures 000 Cooper pairs are like bosons in a BC -- except -- size of Cooper pair spacing between electrons Complex order parameter like BC ψ ( r) c c k k kfξ >> 1 k ψ ( r) ψ e i ϕ ( r) k

3 xcitation Gap for Fermions broken pair In a semiconductor the gap is tied to the spatial lattice (only occurs at one special density commensurate with lattice) In a superconductor, gap is tied to fermi level. Compressible. k

4 WHY SUPRCONDUCTIVITY? few electrons N ~ 10 9 forest of states total number of electrons N even ~ 1 ev ~ 1meV superconducting gap ATOM SUPRCONDUCTING NANOLCTROD

5 nergy vs. Particle Number on Grain. N-1 N N+1 N+ N+3 N+4... osephson tunneling between two grains N1, N N1 N 1, + 1 N1, N +

6 osephson Tunneling I Coherent tunneling of Cooper pairs Restrict Hilbert space to quantum ground states of the form N n, N + n, n =... 3,, 1,0, + 1, +, n

7 osephson Tunneling II Restrict Hilbert space to quantum ground states of the form N n, N + n n, n =... 3,, 1,0, + 1, +, n=+ T = n+ 1 n + n n+ 1 n= = G t 8 / e h normal state conductance sc gap Tight binding model: single particle hopping on a 1D lattice N.B. not 1/

8 osephson Tunneling III Tight binding model n=+ T = n+ 1 n + n n+ 1 n= position n wave vector ϕ (compact!) plane wave eigenstate n' =+ n' = ϕ n=+ = e iϕ n n = " e ikx " n= n=+ iϕ n T ϕ = n' + 1 n' + n' n' + 1 e n n=

9 osephson Tunneling IV n' =+ iϕ n T ϕ = n' + 1 n' + n' n' + 1 e n n' = n=+ n= n=+ iϕ n T ϕ = e n+ 1 + n 1 n= T ϕ = cos( ϕ) ϕ ϕ

10 Supercurrent through a position n momentum ϕ T ϕ = cos( ) ϕ ϕ Wave packet group velocity ϕ dn 1 dt = = dt dϕ current: sinϕ dn e I = ( e) = sinϕ dt e I = Icsin ϕ Ic

11 Charging nergy N n, N + n n, n=... 3,, 1,0, + 1, +, e c C U = n = 4 n ( e) C c U n

12 Second Quantization ϕ n=+ = e iϕ n n = " e ikx " n= Number Operator: ˆn i H = nˆ ϕ 4 c cos( ) H = 4c cos( ϕ) ϕ HΨ ( ϕ) = ( ϕ) Ψ ( ϕ) j j j ϕ ϕ Now viewed as coordinate Ψ j ( ϕ) Describes quantum amplitude for (quantum) phase

13 Weak Charging Limit: osephson Plasma Oscillations c Simple harmonic oscillator H = nˆ ϕ 4 c cos( ) H = 4c cos( ϕ) ϕ ϕ H 4c + 1+ ϕ spring constant K = mass M = 1 8 c ω = 8 c

14 Weak Charging Limit: osephson Plasma Oscillations c (heavy mass, small quantum fluctuations) spring constant K = mass M = 1 8 c ϕ Weak anharmonicity can be used to form qubit and readout levels ω = 8 c

15 Strong Charging Limit: Cooper Pair Box 4 c ϕ Fluctuates too wildly; better to use number (position) representation C 500aF C K U n Random offset charge due to built in asymmetry

16 TH BAR OSPHSON UNCTION: SIMPLST SOLID STAT ATOM osephson junction with no wires to rest of world: superconductor tunnel oxide layer superconductor Hamiltonian: ( ˆ ) H = 4 n Q / e cos ϕ +... C r 3 parameters: Q r = G t 8 e / h residual offset charge osephson energy negligible terms at small T C = e C Coulomb energy

17 IMPRFCTIONS OF UNCTION PARAMTRS CHARGD IMPURITY TUNNL CHANNL LCTRIC DIPOLS _ + - 1nm CAN IN PRINCIPL MAK UNCTION PRFCT, BUT MANWHIL...

18 UNCTION PARAMTR FLUCTUATIONS stat r = r + r Q Q Q t stat = + stat C = C + C ( ) t () t () param. dispers. Q r stat random! stat 10% stat C 10% noise sd@1hz Q r /e ~10-3 Hz -1/ / Hz -1/ C / C <10-6 Hz -1/? O th order problem: get rid of randomness of static offset charge! solutions: - control offset charge with a gate Cooper pair box (charge qubit) or - make / very large RF-SQUID C (flux qubit) Cur. Biased. (phase qubit)

19 SNSITIVITY TO NOIS OF VARIOUS QUBITS junction noise port noise qubits Q r C technical quantum backaction charge flux phase charge qubit flux qubit phase qubit

20 COOPR BOX STRATGY U charge deg. point U U n n compensate static offset charge with gate ( ˆ ) H = 4 n N cos ϕ +... gate charge can be controlled N g =C g U/e C n=+ T = cosϕ = n+ 1 n + n n+ 1 n= g

21 Mapping to two level system U 0 1 n z 1 σ nˆ = n= ( ˆ ) H = 4 n N cos ϕ +... n=+ T = cosϕ = n+ 1 n + n n+ 1 C g x σ

22 Mapping to two level system U 0 1 n nˆ = z 1 σ ˆ = ˆ [ n n] ( ) z x H = C 1 Ng σ σ + constant... x z Rotate: z z + x x ( ) x z H = C 1 Ng σ + σ + constant...

23 Mapping to two level system ( ) x z H = C 1 Ng σ + σ + constant... B H = Biσ ε ± =± B + B x z 0 ( Ng 1) U U n n

24 SPLIT COOPR PAIR BOX U gate N g I is actually a SQUID Q φ I coil φ cos π φ max = 0

25 Split Cooper Pair Box ( ) x z H = C 1 Ng σ + σ + constant... is actually a SQUID V g φ max = cos π φ 0 (Buttiker 87; Bouchiat et al., 98)

26 Superconducting Tunnel unction as a Covalently Bonded Diatomic Molecule (simplified view at charge deg. point) 8 N 10 N +1 pairs N pairs aluminum island tunnel barrier aluminum island N pairs N +1 pairs 1µ m Cooper Pair osephson Tunneling Splits the Bonding and Anti-bonding Molecular Orbitals anti-bonding bonding

27 Bonding Anti-bonding Splitting 1 ψ = ± ± ( ) = 7 GHz 0.3 K anti-bonding bonding = = bonding anti-bonding osephson coupling H = σ z

28 Dipole Moment of the Cooper-Pair Box (determines polarizability) V g L = 10 µm 1 nm = V g / L C 1 C V g + + C 3 0 = = bonding anti-bonding H = z σ d Vg L x σ 1/ C d = ( e ) L 1/ C 1+ 1/ C + 1/ C 3 d ~e -µ m

29 nergy nergy, Charge, and Capacitance of the CPB H z = σ d V L g x σ Charge no charge signal Q = d dv charge Capacitance C = dq dv polarizability is state dependent 0 1 CV g g / e deg. pt. = coherence sweet spot

30 RSPONSS OF BOX IN ACH LVL sweet spot nergy ( k B K) hν loop current box inductance L φ/φ 0 =δ/π I k k φ k = φ k = island charge 1 box capacitance N g =C g U/e Q C k k k = U = U k

31 Dephasing, decoherence, decay ev-s 15 1 nv acting on charge e for 1 µ s = 10 ev-s Spins very coherent but hard to couple and very hard to measure.

32 Avoiding Dephasing box / C V g e g ν 'sweet spot' for T ϕ() t = i dt ν( t ) t ν V 0 ψ() t = cos θ / 1+ e ϕ sin θ / 0 ( ) i () t ( ) g Vion et al. Science 00 = Q Q = 0

33 Atomic spectral line 38 switching probability (%) N g = 0.5 φ = 0 τ = 4 µs Lorentzian fit: ν 01 = MHz Q ~ ν (MHz) Courtesy M. Devoret

34 RLAXATION TIM AT OPTIMAL POINT 50 t w switching probability (%) xponential fit: T 1 = 1.84 µs Q 1 ~ time t w (µs)

35 PRINCIPL OF RAMSY XPRIMNT 90º pulse free evolution 90º pulse preparation measurement time 90º pulse free evolution 90º pulse preparation measurement

36 switching probability (%) Courtesy M. Devoret RAMSY FRINGS ν RF = MHz Q ϕ ~ 5000 (8000 fringes) time between pulses t (µs) t