1. Metastability and self-oscillations in superconducting microwave Eran Segev Quantum Engineering Laboratory, Technion, Israel resonators integrated with a dc-SQUID

2. Quantum Measurements of Solid-State Devices Indirect measurements approach: –Resonance Readout - The quantum device is coupled to a superconducting resonator. Direct measurements of solid-state quantum devices has many drawbacks. V Input Probe Output Signal Freq Resonance Curve S12 Input Probe Output Signal Quantum Device –The state of the device modifies the resonance frequencies. –Readout is done by probing these resonance frequencies.

3. Resonance readout and Thermal Instability 1µm Feed line Resonator Weak link : Micro-Bridge Nonlinear thermal instability is expected under dc current bias. A. VI. Gurevich and R. G. Mints, Rev. Mod. Phys. 59, 941 (1987) Heat production: Heat balance condition: Heat transfer to a coolant hot spot Heat productionCooling power unstable Test bed for resonance readout – Superconducting micro-bridge as artificial weak link.

4. Self-Oscillations SC Threshold NC Threshold S.C Phase N.C Phase P res T Oscillation Cycle 1.Energy Buildup + Temperature increase 2.Switching the NC phase at T >= Tc 3.Energy relaxation + Temperature cool down 4.Switching back to the SC phase at T <= Tc Power When embedded in a resonator, the resonator applies negative feedback to the thermal instability mechanism, leading to self-oscillations.

5. Measurement Setup Spectrum Analyzer Synthesizer ~ Oscilloscope E. Segev et al., Euro. Phys. Lett. 78, (2007) E. Segev et al., J. Phys.: Condense. Matter 19, (2007) Feed Line

6. Self-Modulation - Time Domain I Spectrum Analyzer ~ Oscillo- scope Time Domain Frequency Domain

7. Self-Modulation - Time Domain II Time Domain Frequency Domain Spectrum Analyzer ~ Oscillo- scope

8. Self-Modulation - Time Domain III Time Domain Frequency Domain -50050 -70 -60 -50 -40 -30 Frequency [MHz] Power [dBm] Frequency domain 02004006008001000 -20 0 20 40 Time [ n Sec] |B| 2 Time Domain @ -27.72[dBm] Pump Power Spectrum Analyzer ~ Oscillo- scope

9. Self-Modulation - Time Domain IV Time Domain Frequency Domain -50050 -80 -60 -40 Frequency [MHz] Power [dBm] Frequency domain 0100200300400500 -20 0 20 40 Time [  n Sec] |B| 2 Time Domain @ -21.81[dBm] Pump Power Spectrum Analyzer ~ Oscillo- scope

10. Self-Modulation - Time Domain V Time Domain Frequency Domain Spectrum Analyzer ~ Oscillo- scope

11. Self-Modulation – Power Dependence Spectrum Analyzer ~ Oscillo- scope

12. System Model Control parameters Input signal amplitude Input signal frequency Internal variables Mode Amplitude Micro-Bridge Temperature Parameters Coupling rate to environment Coupling rate to losses Resonance frequency Heat capacity of micro-bridge Heat Transfer rate Resonance mode amplitude EOM Force Stored amplitude (energy) Thermal balance EOM heating powercooling power Equations of motion

13. Stability diagram mono-stable (S) mono-stable (N) bi-stable unstable bi-stable MB is superconducting MB is normal-conducting MB is either super or normal-conducting. MB oscillates between super and normal- conducting states. The shape of the stability diagram may vary depending on the tunability strength of the resonance frequency.

14. Self-Modulation Frequency ms (S) ms (N) bs us E15 E16 E15 m-s (S) m-s (N) bs us Self-Oscillation Frequency 1.Resonance frequency is negligibly tuned. 2.Resonance frequency is substantially tuned.

15. Theory vs. Experiment – Time Domain mono- stable (S) mono-stable (N) bistable Un-s working point   Theoretical Results Experimental Results

16. Theory vs. Experiment – Threshold phenomenon mono- stable (S) mono-stable (N) bistable un- stable working point   Theoretical Results Experimental Results Noise is added to simulation

17. Thermal instability as sensitive detection mechanism Spectrum Analyzer ~ Oscilloscope Weak AM modulation The AM creates small oscillations around the working point. mono- stable (S) mono- stable (N) bistable un- stable x x The thermal non-linearity in our device has two advantages in terms of detection. 1.The response of the system to a detectable stimulation is fast and strong. 2.The system has a natural feedback mechanism that drives it back to its original state once the response to the stimulation is ended.

18. Amplification mechanism P ref [a.u.] x un-s ms P ref [dBm] E. Segev et al., Phys Rev B. 77 (2008) Experiments Simulation Strongest amplification at the threshold of self-oscillations

19. Non-Linear Optical detection Spectrum Analyzer Synthesizer ~ Modulated IR Illumination E. Segev et al., IEEE Trans. Appl. Supercond., 16 (2006). E. Segev et al., IEEE Trans. Appl. Supercond., 17 (2007). The amplitude modulation is replaced by modulated IR laser illumination Threshold of Self- Oscillations

20. Fresnel Zone Plate Superconducting detectors must be kept small. Therefore: 1.Signal degraded due to light beam expansion between fiber tip and detector. 2.Cryogenic alignment between fiber and detector is needed. Problem is reduced by an order of magnitude using Fresnel zone plate. Alignment between FZP and detector is done in lithography. NbN Optical fiber 1550 nm Fresnel zone plate NbN meander

21. Additional thermal driven non-linear phenomena Noise SqueezingMode coupling Period doubling and Stochastic resonance Sub-Harmonics Unusual escape rate

22. Low noise non-linearity Input Probe Output Signal Strong nonlinearity. Self-Oscillations. Strong non-linear amplification and detection But – Thermal noise creates a major drawback. Solution – Inductive nonlinearity Input Probe Output Signal Thermal (resistive) driven nonlinearity. SQUIDs - Superconducting Quantum Interference Devices may behave as ideal non-dissipative inductors. In practice – SQUID dynamics might be hysteretic and dissipative.

23. SQUID: The ideal nonlinear Inductor JJ Character DC SQUID Model Nano-Bridge based JJ 80nm 60nm

24. Resonance Frequency Tuning E. Segev et al., Appl. Phys. Lett. 95, (2009) S11 vs. magnetic field Input Probe Output Signal Magnetic Flux [a.u.]

25. Self-Oscillations in superconducting resonator integrated with a DC-SQUID Flux dependant self-oscillations Flux triggering of self-oscillations Self-Oscillation without magnetic flux

26. Simulation of flux dependant self-oscillations Spectrum Analyzer ~ Resonance mode amplitude EOMThermal balance EOM ExperimentSimulation

27. Physical model of DC-SQUID Control parameters Internal variable Sine TermQuadric TermSource Squid Potential: Parameters_______________________

28. DC-SQUID Potential – Roll of Hysteresis Parameter Sine TermQuadric TermSource Hysteretic parameter that control the degree of metastability.

29. DC-SQUID Potential – Roll of control parameters Tilt by Current Tilt by magnetic Flux Control parameters

30. DC-SQUID Equations of Motions DC-SQUID EOM DC-SQUID Circuit Model JJ Current Coupling Circuit model includes: R J – Shunting resistor. C J – JJ capacitance. L – Self-inductance. Control parameters Internal variable Parameters Kirchhoff Equations

31. Stability boundaries – Phase space HessianLocal Stable ZonesLocal Extremum Points Stability Diagram in the plane of Local stability zones

32. Stability boundaries – Alternating excitation Stability Diagram in the plane of $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ Periodic dissipative zone – Static stability zones were dissipation of energy occurs under periodic excitation.

33. Numerical results Periodic dissipative static zones

34. 0.960.970.980.991 -2 0 1 2 I x /I C  x /  0 Periodic dissipative static zone Periodic non- dissipative static zone Periodic dissipative static zone E38 Parameters: Free running zone Periodic dissipative static zone

35. Experimental data Vs. Simulation Simulation Experiment E. Segev et al., arxiv:1007.5225v1 (2010)

36. Double Threshold to Oscillatory Zone Periodic non-Dissipative Static zone Periodic Dissipative Static zone Oscillatory zone Oscillatory zone only for negative excitation values Oscillatory zone only for positive excitation values

37. Double Threshold to Oscillatory Zone Experimental Results Simulation Results Split Threshold

38. Hybrid zones Experimental Results SQUID Voltage Noise LevelTD Statistics

39. Parametric Excitation Of Superconducting Resonator The reflected tone is measured with a spectrum analyzer. The reflected power has many sidebands originated by the nonlinear mixing Magnetic flux can be used to create parametric excitation of superconducting resonators Spectrum Analyzer Synthesizer ~ Current Sources ~

40. Stability Diagram for Parametric Excitation Only flux excitation: 0 Stability diagram in the plane of________ No Free-Running Zone The current through the SQUID is negligible. Periodic non-dissipative static zone Periodic dissipative static zone

41. Parametric excitation – Numerical results Simulation results Stability diagram in the plane of The effect of SQUID inductivity emerges at high frequencies. Boundaries between local stable zones are observed in the periodic non-dissipative zone. The variance of the SQUID inductance within a local stable state is observed. PNDSZ PDSZ PNDSZ PDSZ

42. Parametric excitation – Experimental results Location and shape of threshold is different Simulation results Exc. Heat Production Experimental results Many features agree between simulation and experimental results, but: Location and shape of PDSZ threshold is different. Different β L fits the PNDSZ and the PDSZ.

43. Threshold point to PDSZ Heat relaxation rates are comparable to the excitation frequency! Only heat degree of Freedom Can explain this change

44. DC-SQUID Model inc. heat balance equation EOM for the Josephson junction phases JJ Current Coupling represents the dependence of the k th JJ critical current of the temperature. Heat balance EOMs Heat Production Heat transfer to coolant Parameters

45. Numerical results inc. Heat production PNDSZ PDSZ + Stability diagram in the plane of Time domain simulation Stability Diagram in the plane of First Cycle Additional Cycles Legend

46. Heat dependant Hysteresis 6.577.588.5 |  x ac |/  0 6 -2 0 1 2 3  x dc /  0 + The hysteresis parameter depends on temperature. When β L decreases the stability diagram shifts to the left. The effective working point corresponds to enhanced number of transitions between LSZs. Stability diagram in the plane of Transitions between local stable states produces heat. The heat induces transient and average changes in the local temperature of the SQUID.

47. Future Research – Quantum Nano-Mechanics Quantum Nano-Mechanics – emerging research field in which quantum phenomena are measured in nano-mechanical beams. Question – Does stress or strain in Nano-beams affects material coherency ? Method – Study the effect of a mechanical degree of freedom on the Aharonov-Bohm effect. 100nm 1um 30nm-thick Aluminum I V V I Side electrode 1um 2 AB rings, 30x90nm 2 cross section.

48. Future Research – Quantum Nano-Mechanics Question – Can nano-mechanical beam behave like two level system, showing superposition of states? Method – Suspend one side of a DC-SQUID embedded in a resonator.

49. Summary Thermal (resistive) nonlinearity. Metastable and Hysteretic SQUID Self-Oscillations Detection and amplification Periodic dissipative stability zone Tunable resonators and self-oscillations Parametric Excitation