Phase-charge duality in Josephson junction circuits: effect of microwave irradiation*

Transcription

1 Phase-charge duality in Josephson junction circuits: effect of microwave irradiation* Frank Hekking Université Joseph Fourier Laboratoire de Physique et Modélisation des Milieux Condensés Maison des Magistères Jean Perrin CNRS-Grenoble, France Collaboration with Wiebke Guichard Institut Néel, CNRS-Grenoble, France *arxiv: Financial support: European network MIDAS Frequency- and time-resolved electron transport in nano-circuits Mini Workshop December 7, 2009, CPT, Luminy, Marseille, France

2 I. Ultrasmall Josephson junctions Some basic notions

3 Small Josephson junction Josephson relations: Small Josephson junction: two energy scales Hamiltonian Commutator

4 Superconductor vs. insulator behavior: consequence of phase-charge duality Localized charge Localized phase Coulomb blockade Josephson effect Q φ I I I c Zero-current state Zero-voltage state V c V V

5 Effect of environment: RCSJ model I 0 displacement current current trhu resistor supercurrent Dynamics of fictitious phase particle

6 Effect of environment: RCSJ model + δ ither I 0 displacement current current trhu resistor supercurrent Overdamped limit: effect of thermal noise Steinbach et al., 01 Ivanchenko & Zilbermann 69

7 Effect of microwaves: Shapiro voltage steps Dynamics of phase Finite voltage: Josephson oscillations Phase-locking: voltage-shapiro steps at dφ V = = dt V n = nhf / 2e 1 f J σ > 1 σ < 1 Kautz 96

8 Duality: Josephson junction versus phase-slip junction Cooper pair tunneling I c Duality Phase-slip V c Hˆ E ˆ 2 Q = 2C h = 2e C E J cosθˆ [ θ, Q] = 2ie θˆ qˆ L 2 Hˆ θˆ = Δ 2L Δ0 = ev c 0 cos qˆ J I c Schmid (1983) Likharev, Averin, Zorin (1985) Schön, Zaikin (1990) Ingold and Nazarov (1992) - - Weiss (1999) Beloborodov et al. (2002) Zazunov et al. (2008)? Arutyonov, Golubev & Zaikin (2008) J.E. Mooij and Yu.V. Nazarov (2006)

9 Consequences of duality: IV characteristics & Shapiro steps V 0 R inductance resistor phase slip element Phase vs. charge particle

10 Consequences of duality: IV characteristics & Shapiro steps V 0 R + δvther vmw (t) inductance resistor phase slip element Expected I-V characteristics Effect of microwaves: current-shapiro steps ρ > 1 ρ < 1

11 II. Voltage-biased junction in an inductive environment Single Josephson junction as a phase-slip element

12 CJRL-model: Circuit & Hamiltonian Apenko (1989), Schön & Zaikin (1990), Zazunov, Didier & FH (2008), Guichard & FH (2009) H ˆ = Hˆ + Hˆ + J L Hˆ ' B Josephson junction Inductance Bath Hamiltonian Coupling Bias phase h ϕ& 2e = b V b

13 Single junction: Bloch Bands Likharev,Averin,Zorin (1985) Periodic Hamiltonian: E J >>E C Tight-Binding Model Sinusoïdal lowest band: 1> 0> 2π Phase slip!!! Phase-slip amplitude: Δ 1/ 4 ( E E ) exp( 8E / E ) 3 0 J C J C

14 Quasicharge Qˆ 2 H ˆ 1 = EJ cos ˆ θ + + 2C 2L ( ) 2 ϕb ˆ ϕr ˆ θ Hbath Dynamics in lowest Bloch band H ˆ 1 = ε 0( qˆ) + + 2L ( ) 2 ϕb ˆ ϕr ˆ θ Hbath Equation of motion for quasicharge dqˆ dt = i[ Hˆ, qˆ] = dhˆ / d ϕ b = Î Quasicharge = total momentum of JJ that is conserved without external current

15 Equation of motion for quasicharge Guichard & FH (2009) E J >> E C Z(ω) R Exactly dual to usual RCSJ model!

16 Equation of motion for quasicharge: quasiclassical approach Guichard & FH (2009) Quasiclassical dynamics Narrow wave packets, thermal noise L 2 d q 2 dt dq + R + Vc sin( πq / e) = Vb + v dt Ignore wave-packet spreading on scale 1/ω c, L/R ε 0 (q) q max Classical noise: ev c ω c, R/L < T < Δ 0 q

17 Overdamped quasicharge dynamics in a phase-slip junction: experiment Corlevi, Guichard, FH & Haviland (2006); Beloborodov, FH & Pistolesi (2002) 2 d q dq L + R + Vc sin( πq / e) = Vb + v 2 dt dt ther R 0 (Ω) T=50mK T=200mK T=300mK B(T) R = 150 kohm Dual to Steinbach et al.! V c

18 III. One-dimensional junction arrays Josephson junction chain as a phase-slip element (providing its own inductance)

19 Phase-biased Josephson junction array Matveev, Larkin, Glazman (2002) ϕ E C Ignore self-capacitance: E J E C Action: Phase bias leads to constraint: Strong Josephson coupling & long array: Classical energy of chain: minimize potential energy for large N, imposing the constraint (inductor)

20 Phase-slips in phase-biased arrays Matveev, Larkin, Glazman (2002) Phase slip: energy unchanged, but constraint violated Phase slip combined with small adjustment: constraint satisfied

21 Hamiltonian for quantum phase-slip array Matveev, Larkin, Glazman (2002) Classicalenergyfor m phase slips: Effect of capacitance: quantum fluctuations E(φ) Phase slip at junction k: 2 π φ Phase slip amplitude: Δ 0 exp ( 3 E E ) J C 1/ 4 ( 8E / E ) J C Hamiltonian: Phase-slip can occur on any junction

22 Phase-slip array in quasi-charge representation Guichard & FH (2009) Introduce «quasicharge representation» Rewrite Hamiltonian Quasicharge dynamics related to current V b R Tunable parameters! Again exactly dual to usual RCSJ model!

23 IV. Phase-slip junction under microwave irradiation Current Shapiro steps: towards a current standard?

24 Microwaves & Shapiro steps Guichard & FH (2009) v 0 = V b /V c Overdamped case ρ=2, V MW =5V c, ω MW =1.5ω c Underdamped case ρ=0.15 V MW =3V c, ω MW =2ω c

25 Phase-slip junctions, Shapiro steps & the metrological triangle Josephson junction V = n h f/ 2e f Phase-slip? junction I = m e f V I Quantum Hall effect I = p e 2 V/h

26 Conclusions I. Ultrasmall Josephson junctions Some basic notions: -Junction in electromagnetic environment -Voltage-Shapiro steps -Duality II. Voltage-biased junction in an inductive environment Single Josephson junction as a phase-slip element: - Hamiltonian - Duality with an external inductance III. One-dimensional junction arrays Josephson junction chain as a phase-slip element: - Quantum phase-slips in a phase-biased array - Duality with intrinsic inductance IV. Effect of microwaves - Current-Shapiro steps: closing the metrological triangle? E J E C