Phase-Charge Duality in a Josephson Junction coupled to an - PowerPoint PPT Presentation

Phase-Charge Duality in a Josephson Junction coupled to an electromagnetic environment Silvia Corlevi, David B. Haviland Nanostructure Physics, Royal Institute of Technology Stockholm, Sweden Wiebke Guichard, Frank W. J. Hekking Universite´ Joseph Fourier – CNRS Grenoble, France

Outline Small-capacitance Josephson junctions: Phase and Charge dynamics Measurement of Bloch oscillations: Single SQUID in a tunable electromagnetic environment Thermal fluctuations in the overdamped quasicharge regime: Quasicharge diffusion Cooper pair transistor in the high impedance environment

Small-capacitance Josephson Junctions 2 [ ] Q ϕ = − ϕ = Q H E cos , 2 ei J 2 C 2 e E C = Charging energy 2 C ∆ R ( 0 ) = R Q = h/(2e) 2 ~ 6.45 k Ω Q Josephson energy E J 2 R N ω ω = 1 / 2 10 Z( ω ) ~ ( 8 E E ) / h ~ 10 Hz Electromagnetic environment p J C = ω I V / Z ( ) b b Current-biased junction Voltage-biased junction

Phase and Charge Dynamics of a Josephson Junction ∞ ϕ - tunneling Classical Josephson effect E J /E C q- tunneling Bloch oscillations Schmid ´83 ∞ 0 R Q /R Z ( ω )=R << R Q E J >>E C Z ( ω )=R >> R Q E C >>E J Phase dynamics Quasicharge dynamics

Classical Dynamics of the Josephson Phase ϕ ϕ ⎛ ⎞ ⎛ ⎞ & & & h h V dV 1 = ϕ + + = ϕ + + ⎜ ⎟ ⎜ ⎟ I I sin C I sin C b C C ⎝ ⎠ ⎝ ⎠ R dt R 2 e 2 e ⎛ ⎞ I = ϕ + ϕ + ϕ R E 2 & & & ⎜ ⎟ b Q sin = π 2 J Q ⎜ ⎟ I R 2 E ⎝ ⎠ C Q C = π ∆ I ( 0 ) / 2 eR C N << Overdamped phase dynamics Q 1 >> Underdamped phase dynamics ⎛ ⎞ Q 1 I ⎜ ⎟ ϕ = − ϕ + ϕ b U ( ) E ⎜ cos ⎟ J ⎝ ⎠ I C I < ϕ = & I I 0 b C “Load Line” I C slope=-1/R I = ϕ ≠ & I 0 b C << I R Q 1 I > I V = b − b C I I R >> Q 1 V

Thermal fluctuations in the case of overdamped-phase dynamics (theory) 2 k T δ δ + τ = δ τ B Overdamped phase dynamics: I ( t ) I ( t ) ( ) n n R E J <<E C Z( ω ) =R <<R Q Langevin-type eq. for the phase V dV I b +I n = ϕ + + I C sin C α = ∞ , 10, 3 R dt ⎛ ⎞ α I I ( ) ⎜ ⎟ = − α 1 i v Im ⎜ ⎟ α ⎝ ⎠ I I ( ) − α C i v v = α = V B RI / E / k T C J B Ivanchenko and Zil’berman ´68 Phase diffusion Supercurrent peak at finite voltage

Thermal fluctuations in the case of overdamped-phase dynamics (experiment) Steinbach, Joyez et al. 2001 Environment suppresses phase fluctuations at all frequencies − − = + ω ω 1 1 Z ( ) R iC B B

Quantum fluctuations of the phase: ”P(E) theory” ∞ ϕ - tunneling Classical Josephson effect E J /E C q- tunneling Bloch Limit of small E J oscillations Incoherent << CP tunneling E / E ( R / R) J C Q ∞ 0 R Q /R Z ( ω )=R >> R Q E C >>E J Z ( ω )=R << R Q E J >>E C Quasicharge dynamics Phase dynamics

Quantum fluctuations of the phase: ”P(E) theory” LC-harmonic oscillators Z( ω )=R R/R Q = 2, 20,100 2eV b =E C π 2 [ ] eE = − − J I P ( 2 eV ) P ( 2 eV ) S b b Kuzmin, Nazarov et al. ´91 h [ ] 1 ∫ = + h P ( E ) dt exp J ( t ) iEt / π 2 h − ω ω ω − = ∫ i t d Re[ Z ( )] e 1 t J ( t ) 2 − ω ω − h / k T R 1 e B Q Ingold and Nazarov ´90 Ingold, Grabert and Eberhardt ´94 Grabert and Ingold ´99

Quasicharge description of a Josephson junction ∞ ϕ - tunneling Classical Josephson effect E J /E C q- tunneling Bloch oscillations Schmid ´83 ∞ 0 R Q /R Z ( ω )=R << R Q E J >>E C Z ( ω )=R >> R Q E C >>E J Quasicharge dynamics Phase dynamics

Quasicharge description of a Josephson junction ∂ 2 E J /E C < 1 = − − ϕ H 4 E E cos ∂ ϕ C J 2 π ≤ ≤ ψ ϕ + π = ψ ϕ i 2 q / 2 e h ω p e q e ( 2 ) e ( ) E J E J /E C > 1 n , q n , q quasicharge V Langevin equation for the quasicharge Z( ω )=R >> R Q dq V dE = = = − 0 +I n V I I b dq dt R E C ~ E J ∆ >> [E C , k B T] Likharev, Zorin 1985

I b -V curve: Coulomb blockade region E J /E C <<1 V C q 0 I < I Stationary solution: b C C = V max[ dE / dq ] 0 C = I max[ dE / dq ] / R 0 = = = & Coulomb blockade of Cooper pairs q I 0 V I R b

I b -V curve: Bloch oscillations region E J /E C <<1 V C Bloch oscillations: I b > I C I q ≠ & B = Coherent tunneling of Cooper pairs b 0 f 2 e = → d V E / d q 0 0

I b -V curve: Bloch oscillations region V(t) V C = < V I R V Coulomb blockade V = b b C RI b b V(t) t V = RI b b V C = > <V> Cooper pair tunneling V I R V b b C V(t) t V = RI b b -V C V C → V ( t ) V saw ( t ) C <V> t = → d V E / d q 0 0 -V C

I b -V curve: Zener tunneling region E J /E C <<1 V C Zener tunneling: I b > I Z = & V dE / dq q 0 ⎡ ⎤ 2 ⎡− ⎤ π E e I = ⎢ − ⎥ = J ⎢ Z ⎥ Transition to higher energy bands P exp exp Z ⎢ ⎥ ⎣ ⎦ 8 E h I I ⎣ ⎦ C b b

Thermal fluctuations in the case of overdamped-quasicharge dynamics (theory) Increasing E J /E C Overdamped quasicharge dynamics: E J >>E C Z( ω ) =R >>R Q VC/e VC/e dq V = π = − V V sin( 2 q / 2 e ) I +I n C b R dt q/2e k B T/eV C =0, 0.05, 0.1 ⎡ ⎤ β π I ( eV / ) − β π = ⎢ ⎥ 1 i eI R / c b V V Im β π I/I C C ⎢ ⎥ ⎣ ⎦ I ( eV / ) − β π i eI R / c b β = 1 / k B T noise Beloborodov, Hekking, Pistolesi 2003 <V>/V C Quasicharge diffusion Thermal fluctuations suppress V C

Phase-Charge Duality Beloborodov,Hekking and Pistolesi 2003 Ivanchenko and Zil’berman 1968 I/I C <V>/V C Overdamped phase dynamics Overdamped quasicharge dynamics E C >>E J Z( ω ) = R <<R Q E J >>E C Z( ω ) = R >>R Q = π = ϕ V V sin( 2 q / 2 e ) I I sin C C π Φ ϕ 2 dq d = = I 0 V π 2 e dt 2 dt

High impedance environment Fine structure constant 2 ω = µ ε = Ω << Z e = = α Z ( ) ~ Z / 377 R 8 8 0 0 0 Q α = πε h 1 / 137 R 4 c Q 0 E J I Φ ext Φ 0 /2 Φ 5 k Ω < R 0 < 50 M Ω Φ= 0 9 10 8 10 7 10 R 0 ( Ω ) Φ= Φ 0 /2 6 10 5 10 4 10 0,00 0,01 0,02 0,03 B (T)

The single-junction samples Single SQUID junction: area ~ 0.04 µ m 2 SQUID arrays: 60 junctions area ~ 0.06 µ m 2 A SQUID /A array ~ 8-10 Single junction: area ~ 0.02 µ m 2 SQUID arrays: 60 junctions area ~ 0.06 µ m 2 non-SQUID arrays: 16 junctions area ~ 0.01 µ m 2

Measurement scheme R f T = 15 mK V I=(V out -V b )/R f +V b -V b Φ= 0 Φ= Φ 0 /2 50 k Ω < R 0 < 50 M Ω R 0 ~ 10 M Ω

I-V curve of a tunable single junction in the high impedance limit A) E J /E C = 4.5 ≤ B) E J /E C 0.2 B A V C =max[dE 0 /dq] C = 1.8 fF , R N ~ 2.8 k Ω R 0 ~ 10 M Ω

Thermal fluctuations in the case of overdamped-quasicharge dynamics T cryo = 250 mK T cryo = 300 mK T cryo = 50 mK E J /E C ~3 T noise = 160 mK T noise = 260 mK T noise = 400 mK C ~0.9 fF R N ~ 2 k Ω R 0 ~ 10 M Ω V C = 30 µ V R fit = 150 k Ω

Comparison between Tnoise and Tcryostat Tcryo=Tnoise Saturation of V C inadequate filtering V C drop quasiparticles tunneling

The Cooper Pair Transistor ϕ = ϕ − ϕ φ = ϕ + ϕ ( ) / 2 ( ) φ = ϕ = 2 1 2 1 [ , Q ] [ , Q ] 2 ie φ = − + = 1 + Q Q Q Q Q ( Q Q ) / 2 C ~ 2C φ 1 2 g ∑ 2 + 2 2 Q ( Q Q ) φ = + − φ ϕ g H 2 E cos( / 2 ) cos( ) J C 2 C ∑ High impedance environment Low impedance environment modulation of the threshold voltage modulation of the supercurrent Zorin et al. 1999

Cooper Pair Transistor: 2D energy band diagram − 2 2 2 ( Q Q ) Q Q ( ) ( ) Q Q = + + + − ϕ − ϕ 1 2 g g 1 2 H E cos E cos J 1 J 2 2 C 2 C 2 C C E J <<E C Σ Q g = (2n+1)e 2D Bloch-band picture dE ( q ) dE ( q ) = + n 1 n 2 V dq dq 1 2 V C =max [V(q 1 ,q 2 ,Q g )] Q g = 2ne

I-V curve of the CPT for different environments 1.5 R 0 ~ 55 k Ω 1.0 R 0 ~ 1 M Ω R 0 ~ 20 M Ω 0.5 SQUID I (nA) 0.0 arrays -0.5 -1.0 -1.5 -10 -5 0 5 10 V (mV) 1.5 E J /E C Σ ~ 0.6 1.0 0.5 I (nA) CPT 0.0 -0.5 -1.0 -1.5 -0.2 -0.1 0.0 0.1 0.2 V (mV)

I-V curves of CPTs with different E J /E C values R 0 ~ 20 M Ω E J /E C Σ ~ 8.7 E J /E C Σ ~ 1.9 E J /E C Σ ~ 0.75

Gate-induced modulation of the Coulomb blockade E J /E C Σ ~ 0.75 R 0 ~ 20 M Ω Q g =(2n+1)e Q g =2ne