1. Nonlinear Effects in Superconducting Resonators
Eyal Buks
Technion, Israel Institute of Technology
Technion Eran Segev
Baleegh Abdo
Oren Suchoi
Gil Bahar
Oleg Shtempluck
Fei Xue
U Waterloo Adrian Lupascu
Jean-Luc F.X. Orgazzi
Chunqing Deng
Marty Otto
Dartmouth Miles Blencowe
Nano-Electronics Research Center-Technion

2. Outline superharmonic resonances in circuit CQED intermode dephasingZ0
superharmonic resonances in circuit CQED
intermode dephasing
thermal instability and self-excited oscillation

3. Superconducting Qubits and Microwave Cavities
Nakamura et al., Nature 398, 786 (1999)
demonstrations of a Josephson qubit having a coherence time approaching 0.1ms [C. Rigetti et al., PRB, 86(10), (2012)]
two-qubit gate implementation [J. Chow et al., PRL, 102(9), (2009)]
three-qubit entanglement [L. DiCarlo et al., Nature, 467(7315), 574–578 (2010), M. Neeley et al., Nature, 467(7315), 570–573 (2010)],
Currently, it is widely believed that the most promising route towards the realization of large scale quantum computation is based on superconducting devices [Josephson qubits and microwave (MW) resonators]. Some of the recent achievements in this field include:
universal set of quantum gates implementation [J. M. Chow et al., PRL, 109(6), (2012)]
violation of a Bell-type inequality [M. Ansmann et al., Nature, 461(7263), 504–506 (2009)].

4. Microwave Cavity Coupled to Flux Qubit
cavity quantum electrodynamics (QED)
Waterloo SQD group:
Adrian Lupascu
Jean-Luc F.X. Orgazzi
Chunqing Deng
Marty Otto
cavity – superconducting coplanar microwave resonator
atom – persistent current flux qubit with 4 Josephson junctions

5. The Flux Qubit

6. Jaynes-Cummings Model
qubit frequency
coupling between qubit and cavity
eigenenergies

7. Cavity Linear Response
experiment
Pin=− 126dBm
T=23mK
theory

8. Weak Nonlinear Response
Pin=− 111dBm
T=23mK

9. Weak Nonlinear Response
hardening

10. Weak Nonlinear Response
softening
hardening

11. Weak Nonlinear Response - Theory
complex and amplitude dependent resonance frequency
frequency shift due to coupling to the flux qubit
experiment
theory
Boissonneault et al., PRA, 77, (2008)

12. Superharmonic Resonances
Near the n’th superharmonic resonances
the effective coupling constant is given by
experiment
theory

13. Intermodulation Response
pump
isolator
combiner
Spectrum
Analyzer
signal
strong pump
weak signal
Idler
Pump
Signal
-1500
-1000
-500
500
1000
1500
-100
-80
-60
D f
[Hz] P
tra
[dBm]
signal and idler gains

14. IMD – Experiment vs. Theory
pump transmission
signal and idler gains

15. Z0
intermode dephasing

16. Nondemolition Photon Detection
‘detector’
The path taken by a photon in the MZ interferometer can be inferred from a probe of the Kerr medium.
Kerr nonlinearity in the optical band is typically far too weak…
Kerr medium

17. Superconducting Stripline Resonator
dielectric wafer Z0
SC stripline
side view
dielectric wafer
SC stripline
SC ground plane
top view

18. Superconducting Stripline Resonator
current waveform
Pin
Pout
|S11|2=Pout/Pin
|S11|2
feedline
1st mode Z0
2nd mode Z0
3rd mode Z0

19. Integrating a Weak Link
DC SQUID F
Pin

20. DC-SQUID with FIB made microbridges
Niobium Devices
AFM
SEM
DC-SQUID with FIB made microbridges

21. DC-SQUID with FIB made microbridges
Niobium Devices
I-V - no external magnetic field (with lockin amplifier)
DC-SQUID with FIB made microbridges

22. Magnetic Flux Dependence2 4 6 8
x 10 -4
0.5 1
1.5
2.5 3 -6
I=10μA
I=1μA
I=5μA
bias current [A]
voltage [V]

23. DC-SQUID Incorporated into Stripline Resonator
feedline F Z0

24. Network Analyzers Measurements - I
Pin=-96dBm Z0
|S11|2=Pout/Pin F
Pout
S11[dB]

25. Network Analyzers Measurements - II
Pin=-96dBm Z0
S11=Pout/Pin
Pout
S11[dB]
current waveform
S11[dB]

26. Network Analyzers Measurements - III
Pin=-96dBm
nonlinear inductance Z0
S11=Pout/Pin
Pout
S11[dB]
S11[dB]

27. Nonlinear Inductance
Effective Hamiltonian wp
amp. w1 Z0
feedline

28. Intermodulation Measurement - I
pump
strong pump
circulator
resonator
isolator
combiner Z0 Z0
weak signal F
isolator
Spectrum
signal
Analyzer
Idler
Pump
Signal
-1500
-1000
-500
500
1000
1500
-100
-80
-60
D f
[Hz] P
ref
[dBm]

29. Intermodulation Measurement - II
Idler
Pump
Signal
pump power: -62.1dBm, signal power -81dBm.

30. Intermode Coupling 2.5 GHz 7.5 GHz
The driven ‘detector’ mode measures the energy stored in the system mode.
detector mode
system mode
7.5 GHz
2.5 GHz
Sanders and Milburn, PRA 39, 694 (1989)
Santamore, Doherty and Cross, PRB 70, (2004)
Santamore, Goan, Milburn, and Roukes, PRA 70, (2004)

31. Intermode Dephasing
Consequently, dephasing of system mode photons occurs. Z0
feedline
~ l1,3 N1 N3
detector mode
(2.5 GHz)
system mode
(7.5 GHz) w3 w
dephasing rate

32. Intermodulation Gain and Dephasing - I
Pump power: -62.1dBm, Signal power -81dBm, Network Analyzer power -101dBm

33. Intermodulation Gain and Dephasing - II
detector mode
system mode
resonance
idler
signal
Pump power: -62.1dBm, Signal power -81dBm, Network Analyzer power -101dbm

34. Intermodulation Gain and Dephasing - V
theory
experiment -2 2
0.7
0.8
0.9 G S
0.005
0.01 I 1 3 1/ g t f x /
signal gain
idler gain
dephasing rate
J. Lightwave Tech. 24, 5054 (2006)
Phys. Rev. A 73, (2006)

35. thermal instability and self-excited oscillations

36. DC SQUID with Stripline ResonatorZ0
nano-bridge
~ 100 nm wide

37. Response to Monochromatic Excitation
Pin Z0
Spectrum
Analyzer
-100
-50 50
100
-95
-90
-85
-80
-75
-70
-65
-60
-55 f c
[MHz] P
ref
[dBm]
Pin < -66 dBm

38. Response to Monochromatic Excitation
Pin Z0
Spectrum
Analyzer
-100
-50 50
100
-90
-80
-70
-60
-40
-30 f c
[MHz] P
ref
[dBm]
Self-Modulation
Pin > -66 dBm

39. Self-Modulation Versus Power
Pin Z0
Spectrum
Analyzer

40. Self-Modulation Versus Flux
Pin Z0
Spectrum
Analyzer

41. Temperature Dependent Resonance Frequency
Resonance frequencies and damping rates depend on Z1 Z0 Z1
Pin T
The impedance Z1=R+iX changes abruptly at the critical temperature Tc.
-60
-40 20
-14
-12
-10 -4 -2 f c
[MHz] |S 11
| [dB] S
T<Tc
3.8 GHz N
T>Tc

42. Equations of Motion heating power cooling power
equation of motion for the mode amplitude
input white noise
thermal balance equation
heating power
cooling power
dependence of resonator’s parameters on temperature T

43. Stability Zones N S heating power cooling power
equation of motion for the mode amplitude
input white noise
thermal balance equation
heating power
cooling power
mono-stable (S)
mono-stable (N)
bistable
astable N S

44. Experiment vs. Theory
experiment
theory
D f [GHz]
D f [GHz]

45. Nonlinear Induction Detection of ESR
2,2-Diphenyl-1-picrylhydrazyl (DPPH) `
4.2K
monostable
normal
conductive
superconductive
astable
bistable
Synthesizer, P , PP
cavity
Lock-in Amp bz
Signal Generator Bz
300K
up to 100 times larger responsivity!
linear
response
non-linear
ωL=ΔE/ħ = γESRBZ
APL 101, (2012)

46. Summary Superharmonic resonances found in circuit CQED.Z0
Intermode coupling results in dephasing of microwave photons.
Microwave heating results in thermal instability and self-excited oscillation.

47. Intermodulation Measurement
pump
Strong Pump
circulator
isolator
combiner
resonator
Weak Signal
isolator
Spectrum
signal
Analyzer
-1500
-1000
-500
500
1000
1500
-100
-80
-60 f c
[Hz] P
ref
[dBm]
Pump
Signal
Idler D
IM Gain

48. Intermodulation Gain E15 5nW [dB] [MHz] G f P [dBm] 30.7dB 17.8dB -34
-33
-32
-28
-27
-26
-20
-10 10 20 30 40 P
pump
[dBm] G IM
[dB] f SM
[MHz]
5nW
30.7dB
17.8dB SA
E15

49. Intermodulation Gain E15 [dB] [MHz] G f P [dBm] -34 -33 -32 -28 -27
-26
-20
-10 10 20 30 40 P
pump
[dBm] G IM
[dB] f SM
[MHz] SA
E15

50. Self Modulation E15 [dB] [MHz] G f P [dBm] -34 -33 -32 -28 -27 -26 -20
-10 10 20 30 40 P
pump
[dBm] G IM
[dB] f SM
[MHz]
E15
signal off SA

51. Intermodulation Gain vs. Self Modulation
mono-stable (S)
mono-stable (N)
bistable
astable
Extremely high IMD gain along the edge of the astable region. 
-34
-33
-32
-28
-27
-26
-20
-10 10 20 30 40 P
pump
[dBm] G IM
[dB] f SM
[MHz]

52. Underlying Mechanism of IMD Gain
mono-stable (S)
mono-stable (N)
bistable
astable
The pump power is modulated at frequency wp-ws.
Consequently, the lifetime of the metastable state is modulated in time.
wp- ws
pump, wp
isolator
combiner
signal, ws

53. Direct 10 KHz Amplitude Modulation
[mSec]
0.2
0.4
0.6
0.8
0.05
0.1
0.15 
mono-stable (S)
mono-stable (N)
bistable
astable 
0.1
0.2
0.3
0.4
0.6
0.8 P
refl
[a.u.]

54. Experiment vs. Theory
0.2
0.4
0.6
0.8 1
0.05
0.1
0.15
0.05
0.1
0.15
0.2
0.4
0.6
0.8
-20 20
-100
-90
-80
-70
-60
experiment
theory 1
-60
0.8
0.8
[a.u.]
[a.u.]
[dBm]
-70
0.6
0.6
-80
0.4
0.4
refl
refl
-90 P
0.2 P
0.2 P
-100
0.1
0.2
0.3
0.1
0.2
0.3
-20 20
0.2
0.4
0.6
0.8 1
0.2
0.4
0.6
0.8
-20 20
-110
-100
-90
-80
-70
-60
0.6 t
[mSec]
t [mSec] f
[KHz] c

55. Gain – Theory vs. Experiment1
0.05
0.1
0.15
0.2
0.4
0.6
0.8
0.8
-40.25
0.6
0.4
-40.2
0.2
-40.15
0.05
0.1
0.15 1
0.8
0.8
[dBm]
[a.u.]
[a.u.]
[dBm]
0.6
0.6
0.4
0.4
refl
refl
pump P
0.2 P
0.2 P P
0.1
0.2
0.3
0.1
0.2
0.3 1
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
-39.95
-39.9
Sim
Exp
-39.85
-20 20 G
[dB] t
[mSec]
t [mSec]
sig

56. Typical recovery time ¼ 0.1-5 ms
Amplifier Bandwidth
-50 50
-80
-60
-40
Frequency [MHz]
Power [dBm]
Frequency domain 2 4 6 8 10
-20 20 40
Time [ 
Sec]
Volt [mV]
Time [dBm] Pump Power
Typical recovery time ¼ ms

57. Period Doubling Model Ex. data +1 +1/2 -1/2 -1 [n.u.] t [mSec] P4 6 8
0.5 1
t [ m
sec]
bin=
-1.8
-1.2
-0.6
0.6
1.2
-140
-120
-100
-80
-60
Freq [MHz] P
refl
[dBm]
[n.u.]
Ex. data
0.01
0.012
0.014
0.016
0.018
0.5 1 t
[mSec] P
refl
[n.u.]
MF:1200KHz, PP:-39.07dBm
-1800
-1200
-600
600
1200
1800
-90
-80
-70
-60
-50 f c
[KHz]
[dBm] +1
+1/2
-1/2 -1

58. Period Tripling Model Ex. data -2/3 +2/3 -1 -1/3 +1/3 +1
refl
[n.u.] 2 4 6 8 10
0.5
t [ m
sec]
bin=1109
-1.8
-1.2
-0.6
0.6
1.2
1.8
-120
-100
-80
-60
Freq [MHz]
[dBm]
0.01
0.012
0.014
0.016
0.018
0.5 1 t
[mSec]
MF:1200KHz, PP:-38.93dBm
-1800
-1200
-600
600
1200
1800
-80
-70
-60
-50 f c
[KHz] P
refl
[dBm]
[n.u.]
-2/3
+2/3 -1
-1/3
+1/3 +1
Higher orders (up to 6) of period doubling are found both experimentally and in numerical simulations.

59. Self-Modulation - Time Domain
Frequency Domain
mono-stable (S)
mono-stable (N)
bistable
astable
Spectrum
Analyzer ~
Oscilloscope

60. Self-Modulation Versus Flux