Metastable states in an RF driven Josephson oscillator R. VIJAYARAGHAVAN Daniel Prober Robert Schoelkopf Steve Girvin Department of Applied Physics Yale University 3-16-2006 APS March Meeting I. Siddiqi M. Metcalfe E. Boaknin V. Manucharyan P. Hyafil C. Rigetti L. Frunzio M.H. Devoret Acknowledgements: M. Dykman
INTRODUCTION DOUBLY CLAMPED MECHANICAL RESONATORS MICROMECHANICAL TORSIONAL OSCILLATOR RESONANT FREQ. ~ 90 MHz ~3 KHz QUALITY FACTOR ~7000 ~2000-8000 TEMPERATURE 4K 4K - 77K TRANSITIONS ADDED NOISE ADDED NOISE
INTRODUCTION ELECTRICAL OSCILLATOR RESONANT FREQUENCY ~ 1.5 GHz QUALITY FACTOR ~ 20 FAST DYNAMICS ~ nanoseconds THERMALLY INDUCED TRANSITIONS TEMPERATURE: 10-200 mk CLASSICAL TO QUANTUM DYNAMICS
Josephson Junction OUTLINE Non-dissipative, non-linear circuit element RF biased Josephson Junction Driven, non-linear oscillator Metastable states; transitions Quantum regime Applications
JOSEPHSON TUNNEL JUNCTION S I, 0 CJ Josephson relations I S I = I sin( δ ) 0 I0 critical current δ Superconducting phase Non-linear inductor h dδ V = = 2edt dδ dt I 0 V = ϕ0 = LJ = L C J J ϕ0 = I 0 1 cos( δ ) ϕ di dt 0 dδ dt
DC CURRENT BIAS I < I : V = 0 DC 0 SUPERCONDUCTING I > I : V 0 DC 0 DISSIPATIVE
RF DRIVEN NON-LINEAR OSCILLATOR 2 2 2 0 { C d δ ϕ dδ ϕ0 J + + ϕ 2 0 0sin 0 RF sin t dt { R dt I δ = ϕ I ω 14243 1 42443 mass drag force drive DRIVEN STATES IN THE SAME WELL NO TRANSITIONS OUT OF THE WELL
TWO DYNAMICAL STATES I RF /I 0 δ() t = δ sin( ωt+ γ ) ω p = max I0 ϕ C 0 J PLASMA FREQUENCY Q QUALITY = ω prcj FACTOR ω p 3 If ωp ω > bistability Q 2 Dynamical states differ in oscillation amplitude & phase
THE REFLECTION EXPERIMENT OSCILLATOR ENVIRONMENT 50 OHM CHARACTERISTIC IMPEDANCE
MINIMIZING NOISE USE CIRCULATOR TO PROTECT SAMPLE FROM IN-BAND NOISE
MINIMIZING NOISE USE FILTERS TO PROTECT SAMPLE FROM OUT OF BAND NOISE
JUNCTION + MICROWAVE CAPACITOR Si 3 N 4 Cu Al = Cu Al ε r =7.5 30 pf 1 mm 200 nm Si 3 N 4 15 nm Ti 35 nm Cr Al 1000 nm Cu 20 nm Cr Silicon Substrate METALLIC UNDERLAYER Al/Al 2 0 3 /Al Junction (1 µa, 0.330 nh)
NON-LINEAR RESONANCE k B T/E J 1/Q Q=ω p RC Siddiqi. et al, PRL. 94, 027005 (2005)
HYSTERESIS AND BISTABILITY EXPLOIT HYSTERESIS TO IMPROVE SIGNAL TO NOISE RATIO Siddiqi. et al, PRL 93, 207002 (2004).
TRANSITION RATES ω a Γ= exp 2π U kt B ( kt >> hω) U dyn 0 1 RF U = IB I 2 3/2 Dykman et. al. Physica 104 A, pp. 480-94 (1980). β 2/3 2 2/3 U0 I RF = 1 2 kt B esc I B β ωa = ln 2 π Γ Extract I B and escape temperature T esc
MEASURING ESCAPE TEMPERATURE I RF β 2/3 2 2/3 U0 I RF = 1 2 kt B esc I B State 0 Probability 2/3 β T=200 mk I / I 2 2 RF B Exponential decay of population Escape rate vs drive amplitude
MEASURING ESCAPE TEMPERATURE 2/3 β 200 mk 12 mk I / I 2 2 RF B
QUANTUM REGIME kt B Q esc = hω n+ 1 2 n = exp 1 / 1 ( hω k T ) B ω 2π = 1.525 GHz Good agreement with quantum activation theory Need higher oscillator frequencies Marthaler et. al. arxiv:cond-mat/0602288
JOSEPHSON BIFURCATION AMPLIFIER JBA: INPUT COUPLES TO I 0 - φ (i rf,i 0 ) -P switch (i rf,i 0 )
QUBIT READOUT QUBIT CONTROL PULSE SEQUENCE (~ 20 GHz) QUBIT STATE ENCODED IN REFL. PULSE PHASE φ A 1 0 A I RF / I 0 READOUT PROBING PULSE (~ 1 GHz)
1 Siddiqi et. al. Phys. Rev. B 73, 054510 (2006) 0 1 0
NON-LINEAR CAVITY RESONATORS Superconducting Nb 1D cavity (1-10GHz) Al-AlO-Al junction (I 0 ~0.5-5 µa) 10mm Power in Power out 100µm 10µm See the following talks later today for more details: W39.0002 (Vladimir Manucharyan, 2.42 pm, Room 342) W39.0003 (Etienne Boaknin, 2.54 pm, Room 342) W39.0004 (Michael Metcalfe, 5.06 pm, Room 342)
CONCLUSIONS WELL CONTROLLED NON-LINEAR OSCILLATOR AT GHz FREQUENCIES INTERESTING ESCAPE PHYSICS IN THE QUANTUM REGIME HIGH FIDELITY QUBIT READOUT TOOLBOX FOR SENSITIVE DETECTORS AND AMPLIFIERS
SLIDES AFTER THIS ARE ADDITIONAL
DC CURRENT BIAS II: Metastability & Switching (,, ω p 0 1 I0 I ) 2 exp Γ DC T = π U kt 2 2 h I DC U( I0, IDC ) = I0 1 3 e I0 3/2 2/3 2/3 ω 2 2 h I I p 0 DC ln 1 2π ( IDC ) = Γ 3 e kbt I0 Extract I 0 and escape temperature T esc
DC CURRENT BIAS III: Macroscopic Quantum Tunneling (MQT) 300 250 I 0 = 1.14 µa Martinis et al, PRB 35 (1987) T esc (K) 200 150 100 50 T * = 5mK 0 0 50 100 150 200 250 300 T bath (K) hω U * T > T = p kt : Γ e T 7.2k hω * p < T = : Γ= constant 7.2k thermal activation MQT
SOFTENING POTENTIAL Frequency decreases with drive amplitude For ω<ω p, weak drive off resonance strong drive on resonance
ATTRACTORS 0 1 THY δ t = δ sin ωt + δ cos ωt ( ) ( ) ( )
PHASE DIAGRAM: EXP & THY IN GOOD AGREEMENT All parameters in prediction measured experimentally! Siddiqi. et al, PRL. 94, 027005 (2005)
QUBIT + JBA CHIP WRITE PORT C C READ PORT