Nonlinear kinetic inductance in TiN/NbTiN microresonators and Transmission lines Peter Day Byeong-Ho Eom Rick Leduc Jonas Zmuidzinas
Phase shift at 2GHz (rad) Nonlinear kinetic inductance 0.1 meter long CPW line 5um mwave Vector Network Analyzer 0 5um 5um 50nm -0.1 TiN CPW TRL IDC -0.2-0.3-0.4-0.5 0 1 2 3 4 5 6 7 DC current (A) x 10-3
Phase shift at 2GHz (rad) Nonlinear kinetic inductance 5um Dq ~ I 2 L = L 0 (1 + I 2 /I *2 ) dv ph ~ I 2 ~P Kerr medium Line length =0.1m -> 21 radians Dl/l ~ 2.5% 0-0.1-0.2-0.3-0.4-0.5 5um 5um 50nm 0 1 2 3 4 5 6 7 DC current (A) x 10-3
Superconducting Nitrides large kinetic inductance l 105 nm x (r n [mw.cm] / T c [K]) 1/2 Nitrides: r s 100 mw.cm l (TiN, NbTiN) 500 nm Example: TiN CPW line L s / L tot 0.94 1um 1um 1um 50nm v ph 0.1 c Z0 220 W
Kinetic Inductance Non-linearity Ginsberg-Landau theory L s (I), l(i) dv ph v ph (I = 0) = -L kin m 0 l2 J 2 s 4 L tot m 0 H 2 c Supercurrent kinetic energy Condensation energy
TiN Resonator measurements meander Inductor (absorber) LEKID style FIR MKID pixel capacitor 1mm
Nonlinear Duffing oscillator Resonance frequency depends on resonator current Hysteretic resonance curves: Increasing drive power
Complex transmission Blue: upward Green: downward
Non-linear resonator model See Yurke and Buks (2008); Dahm and Scalapino (1997) t
Fits using NL resonator model K Q = 0!
Model results for different ratios of inductive to dissipative response d resistance/ d reactance < 2 x 10-4
Measurements at elevated T
Transition to dissipative regime
Josephson parametric amplifiers Nonlinearity Josephson inductance Transfer of energy from strong pump to signal Idler tone Purely reactive nonlinearity May reach QL Resonant, narrow band Yurke et al. (1988)
Kinetic inductance cavity para-amp Tholen et al. (2007) Nb CPW resonator
Traveling wave Josephson paramp Proposed by Sweeny and Mahler (1985) Experimentally realized by Yurke et al. (1996) Gain x bandwidth ~ 500MHz
Optical fiber paramp Intensity dependent index: n ~ E 2 Kerr medium from Hansryd et al. (2002) Yurke s device an analogue of this
DTWKI paramp ver 1.0 DTWKI = Dispersionengineered Traveling Wave Kinetic Inductance Single layer TiN or NbTiN 0.8 m CPW line Tapers at input, output match 50 ohms 1um 1um 50nm
Coupled mode equations From the optical fiber paramp: (Stohlen et al.) SPM, XPM Energy transfer A p,s,i : slowly varying (complex) amplitudes g: non-linearity parameter Db= b(w s ) + b(w i ) 2b(w p )
Coupled mode equations From the optical fiber paramp: (Stohlen et al.) SPM, XPM Energy transfer Without dispersion (Db = 0): G = 1 + (gp p L) 2 = 1 + (Dq) 2 Total phase shift in radians in response to pump
Shock wave formation Superconducting TRLs are virtually dispersionless Harmonic generation is efficient Landauer (1960)
Adding dispersion from Hansryd et al. (2002) Phase mismatch Limited bandwith Optimal dispersion compensates nonlinear phase missmatch Exponential gain regime
b - b 0 (m -1 ) Dispersion-engineered traveling wave kinetic inductance (DTWKI) amplifier 1 0.9 0.8 0.7 0.6 Periodically load TRL Produce stop band at 3 x f pump 0.2 0.1 Dispersion around f pump 0 0 2 4 6 8 10 12 14 16 S 21 2 0.5 0.4 0.3 Frequency (GHz) v ph 2D 6 4 f pump 2 D 0-2 -4 Db = b s + b i 2b pump ~ const -6 0 2 4 6 8 10 Frequency (GHz)
Idler generation pump idler signal
output power (dbm) 3rd Harmonic/Idler Generation 3rd harmonic outside stop band -30-40 -50 idler 3rd harmonic ~P p 3 w p w p w p 3w p -60-70 w p w s -80-90 ~P p 2 w p w i -100-25 -20-15 -10-5 pump power (dbm) (signal power = -60dBm)
output power (dbm) 3rd Harmonic/Idler Generation -30-40 -50-60 -70-80 -90-100 3rd harmonic outside stop band idler 3rd harmonic ~P p 3 ~P p 2-25 -20-15 -10-5 pump power (dbm) (signal power = -60dBm) P out (3w p ) P out (w i ) P p = Ps eg. @ P p = -20 dbm, P s = -60 dbm P out (3w p ) P out (w i ) = 40 db
Gain (db) Gain (db) Coupled-mode prediction Assume harmonic generation blocked Dispersionless around pump (Db = 0) DQ = 3, Db = Constant 20 DQ = 1 radian 3 10 20 Db / gp p = 0 0.5 2 15 15 10 10 5 5 0 0 0 0.5 1 1.5 2 f signal / f pump 0 0.5 1 1.5 2 f signal / f pump
Paramp gain NbTiN device, pump at 11.56 GHz Near (2 nd ) engineered dispersion feature
Gain (db) Paramp gain NbTiN device, pump at 11.56 GHz Near (2 nd ) engineered dispersion feature Ripple due to reflections, amplifying medium 14 12 10 8 6 4 2 10 10.05 10.1 10.15 10.2 10.25 10.3 10.35 10.4 Frequency (GHz)
Conclusion Nonlinearity, low dissipation of superconducting nitride films enables a new broad-band paramp Measured gain roughly consistent with expectations Noise measurements are underway Applications Microwave amp for detector array readout Potential of QL performance Radio astronomy (mm submm) NbTiN gap frequency > 1 THz Quantum information Axion search