The SQUID-tunable resonator as a microwave parametric oscillator Tim Duty Yarema Reshitnyk Charles Meaney Gerard Milburn University of Queensland Brisbane, Australia Chris Wilson Martin Sandberg Per Delsing Chalmers University of Technology Göteborg, Sweden
Parametric Oscillators self-oscillation instability---very well understood classical non-linear system with exact mathematical results parametric amplifier when operated below threshhold, a threshold detector latched amplifier above threshold optics example: generate squeezed radiation, e.g. will be used in gravity wave detectors to beat standard quantum limit (shot-noise limit), photon-pairs for QIP quantum phenomena with large non-linearities, quantum noise at a bifurcation, strange quantum tunneling...
mechanical parametric oscillators child on a swing (perhaps) skate-boarder in a half pipe a parametrically driven electrical oscillator L Parametric Oscillators C φ EJ[φ(t)] C Leff[φ(t)] key feature: modulate resonant frequency (center of mass, length of pendulum, L or C) at ~ 2 times bare oscillation frequency strong enough driving self-oscillation instability
A SQUID-tunable ~5 GHz resonator a method for coupling qubits Wallquist et al. PRB (2006) parametric amplifier: qubit readout and quantum-limited sensing of nano-mechanics, charge, flux. important since quantum-feedback requires quantum-limited readout a parametric oscillator: novel dynamical tunneling single SQUID SQUID array tuning line-widths
Classical parametric oscillator d 2 q dq +2γ dt2 dt +(ω2 0 + F cosω F t)q + χq 3 = ξ(t) damping driving nonlinearity noise Dykman et al. PRE (1998) near resonant driving ω F 2ω 0 γ, 2ω 0 ω F ω 0 rotating frame q(t) = ( 4ωF γ 3 χ ) 1/2 [ q 1 cos ω F t 2 q 2 sin ω ] F t 2, equations of motion quasi-energy g(q 1,q 2 )= 1 2 ( q 2 1 + q 2 2 dq 1 dτ = q 1 + g + ξ 1 (τ/γ), q 2 dq 2 dτ = q 2 g + ξ 2 (τ/γ) q 2 )[ Ω 1 2 ( q 2 1 + q 2 2 τ = γt ) sgnχ ] + 1 2 ζ ( q 2 1 q 2 2). with Ω = [(ω F /2) ω 0 ] /γ, ζ = F/2ω F γ
Classical parametric oscillator above threshold quasi-energy Dykman et al. PRE (1998) phase poitrait q2 q2 q1 q1 self-oscillation amplitude 2 π-shifted stable states tunneling activation energy - π states - 0 state detuning 2 unstable states detuning
Quantum noise in the parametric oscillator with quartic (Kerr) non-linearity Strength of quantum noise determined by χ/γ frequency shift per photon photon decay rate also need to consider temperature, i.e. number of thermal photons classical except at bifurcation for χ γ quantum activation for low enough temperature extreme quantum limit χ γ
A microwave parametric oscillator Threshold for photon generation "f f 0 Q >1 output signal around fpump/2 μ=0.5 We are well above this threshold ~250 μ=-2.0 signal (db) 50 55 60 65 70 75 80 5.1965 5.197 5.1975 5.198 0.5 Pump frequency (GHz)
A microwave parametric oscillator μ=0.5 μ=2.0
Quantum activation in the parametric oscillator with a quartic non-linearity Marthaler and Dykman (2006) quasi-energy spacing given by effective Planck constant λ = 3χ F ω F scaled tunneling exponent double well separation quantum activation when relaxation greater than interstate tunneling rate limit of large quasi-energy spacing somewhat analogous to MQT of phase
Some data on the tunneling rates... Chalmers data Marthaler and Dykman (2006) one can tune quasi-energy spacing via F Different scaling for small vs large F double well separation quasi-energy spacing given by effective Planck constant λ = 3χ F ω F
Quantum interference in the classically forbidden region Marthaler and Dykman (2007) tunneling matrix element double well separation Tunneling matrix element oscillates with well separation! An unresolved question: can we achieve such a large non-linearity? frequency shift per photon χ γ resonator photon lifetime If so, how is such an interferometer useful?
Phase insensitive amplifiers: QM does not allow noiseless amplification! Есть усилитель, есть проблема... Нет усилитель, нет проблем! linear amplifier input mode output mode a b [a, a ]=1 [b, b ]=1 we want b = Ga b = Ga extra mode c but then [b, b ]=G[a, a ] 1 extra mode is needed b = Ga+ G 1 c b = Ga + G 1 c [b, b ]=G[a, a ]+(G 1)[c,c] = 1
No noiseless (phase-insensitive) amplification! (cont d) linear amplifier input mode output mode a b [a, a ]=1 [b, b ]=1 b = Ga+ G 1 c b = Ga + G 1 c extra mode c ( x in ) 2 = 1 2 aa + a a = n a + 1 2 output noise G 1 ( x out ) 2 = 1 2 = G bb + b b = G 2 ( n a + 1 2 + n c + 1 2 {a + c,a + c} ) amplified input vacuum added noise
Quantum noise related to gain (Caves 1982) I. phase insensitive amplifier A 1 2 1 G 1 added noise number A --- number of noise quanta, referred to input II. phase sensitive amplifier A 1 A 2 1 16 1 (G 1 G 2 ) 1/2 2 reduction of noise added to X1 requires an increase in noise added to X2, except special case G 1 G 2 =1 squeezing need not add noise to either quadrature. de-amplification Degenerate Parametric Amplifier --- requires non-linearity Is the special case (noiseless amplification) really possible using a JPA based on the Josephson non-linearity? If not what are the limits of such devices? a challenging problem: quantum noise at a bifurcation
quantum noise in the parametric oscillator with a Kerr non-linearity mean-field exact gain mean-field breaks down for exact steady-state solution can be found using complex-p representation (FP-like equation, potential conditions) can calculate steady-state (zero-frequency) moments critical slowing down occurs only in amplified quadrature output modes are not a minimum uncertainty state added noise 1 2 1 µ χ/γ 1+SX thr(0) 1 1+Sy 2 thr (0) = (γ/χ) 1/4 1 4 best to operate above or below bifurcation (precise optimum difficult to find) For our JPA best to simultaneously minimize tunability and higher-order non-linearities, or use low Q
Degenerate parametric amplification Below threshold Above threshold Below threshold Above threshold experimentally does not satisfy gain relation G 1 G 2 =1 It must add quantum noise! see also Castellanos-Beltran et al. Nature Physics (2008), related device, squeezing of quantum noise Yamamoto et al. APL (2008) similar device
open questions noise/gain limits on parametric amplifier, is quantum-limed measurement, quantum feedback possible? are we seeing quantum activation; can we see dynamical tunneling? value of Kerr constant (theory needs work) generation and use of squeezed microwaves coherent quantum behavior?
UQ Quantum Devices Lab October 2009 April 2009 first cryogen-free DR in the Southern Hemisphere Postdoc positions available
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