Driven Duffing Oscillator

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1 Driven Duffing Oscillator Consider an oscillator with cubic nonlinearity driven by an harmonic force The classical response is calculated employing the slow envelope approximation The response in the absence of noise f < f C f = f C f > f C A 2 A 2 critical point A 2 ω - ω ω - ω ω - ω

Ronen Almog, Stav Zaitsev, Baleegh Abdo, Eran Segev, Oleg Shtempluck Bernard Yurke 1

2 Nonlinear Resonators - From Nanomechanical to Superconducting Stripline Eyal Buks Department of Electrical Engineering, Technion Collaborators: Technion Bell Labs Ronen Almog, Stav Zaitsev, Baleegh Abdo, Eran Segev, Oleg Shtempluck Bernard Yurke 1 µm NbN Intermodulation amplifiers Gain Dissipation (linear and nonlinear) Noise squeezing

3 Nanomechanical Resonators - Fabrication and Characterization Au Si 3 N 4 Si Si 3 N 4 secondary electron detector Spectrum Analyzer Signal [a.u.] Frequency [Hz] V dc e-beam in ~ spectrum analyzer 1 µm

4 Damping in Nanomechanical Resonators Nanomechanical systems suffer from low quality factor Q relative to their macroscopic counterparts. Damping mechanisms: bulk and surface defects, thermoelastic damping, nonlinear coupling to other modes, phonon-electron coupling, clamping loss, etc. Surface properties are important in NEMS. For intermodulation amplifiers Nonlinear Damping may contribute to the total noise. M.L. Roukes, 2 Solid State Sensor and Actuator Workshop

5 Nonlinear Damping A nonlinear damping term is added to the equation of motion Bistable regime is accessible only when p<1, where f f C < f = fc f > fc A 2 A 2 critical point A 2 ω - ω ω - ω ω - ω Bernard Yurke and EB, unpublished Stav Zaitsev and EB, cond-mat/5313 (25)

6 Extracting the Parameter p -I secondary electron detector e-beam in ~ lockin amp. V dc Stav Zaitsev and EB, cond-mat/5313 (25)

7 Extracting the Parameter p -II Stav Zaitsev and EB, cond-mat/5313 (25)

8 Extracting the Parameter p - III Nonlinear damping plays an important role! What are the underlying mechanisms? Stav Zaitsev and EB, cond-mat/5313 (25)

9 Intermodulation Characterization - I secondary electron detector Pump power combiner e-beam in spectrum analyzer Signal Power spectrum Pump The frequencies of the pump, signal and idler are all within the bandwidth of the fundamental mode. Idler Signal offset Frequency

10 Intermodulation Characterization - II -4 Pump [dbm] frequency [Hz] x Freq. sweep up Freq. sweep down Signal [dbm] frequency [Hz] -7 x 1 5 Pump Idler [dbm] Idler Signal frequency [Hz] x 1 5

11 Intermodulation Characterization - III -63dbm -65dbm -65dbm The signal gain and intermodulation gain are both limited by pump depletion!

12 Hysteresis and Bistability Amplitude sweep up Amplitude sweep down 3 4 Bistability

13 Superconducting Stripline Resonator Nb / NbN Sapphire Nb / NbN Sapphire Nb / NbN

14 Nonlinear Response of Nb Resonator Meissner effect leads to non-uniform current profile with very high current density near the edges of the stripline. Kerr Nonlinearity (kinetic inductance) Nonlinear dissipation: S11 [db] T=4.2K -15 dbm P in vector network analyzer Pin S 11 Nb Pin frequency [GHz] E. Buks unpublished results

15 1 NbN Resonator - I S 11 [db] dbm -9 dbm -8 dbm -7 dbm -6 dbm -5 dbm -4 dbm -3 dbm -2 dbm -1 dbm dbm 1 dbm 2 dbm 3 dbm 4 dbm 5 dbm 6 dbm 7 dbm 8 dbm 9 dbm 1 dbm Network Analyzer Lets look closer Frequency [GHz] at this resonance!

NbN Resonator - II 2 S 11 [db] -2-4 -6-8 -1-12 -23.5 dbm -23.25 dbm -23 dbm -22.75 dbm -22.

75 dbm -19.5 dbm -19.25 dbm -19 dbm -18.75 dbm -18.

16 NbN Resonator - II 2 S 11 [db] dbm dbm -23 dbm dbm dbm dbm -22 dbm dbm dbm dbm -21 dbm dbm -2.5 dbm dbm -2 dbm dbm dbm dbm -19 dbm dbm dbm dbm -18 dbm Onset of bistability - 3 orders of magnitude lower than Nb! Frequency [Ghz] Critical coupling

17 2 NbN Resonator - III S 11 [db] dbm -9 dbm -8 dbm -7 dbm -6 dbm -5 dbm -4 dbm -3 dbm -2 dbm -1 dbm dbm 1 dbm 2 dbm 3 dbm 4 dbm 5 dbm 6 dbm 7 dbm 8 dbm 9 dbm 1dbm Network Analyzer Lets look 4 closer at this resonance! Frequency [GHz]

18 1 NbN Resonator - IV S 11 [db] dbm -9 dbm -8.5 dbm -8 dbm -7.5 dbm -7 dbm -6.5 dbm -6 dbm -5.5 dbm -5 dbm -4.5 dbm -4 dbm -3.5 dbm -3 dbm -2.5 dbm -2 dbm -1.5 dbm -5-6 critical coupling Frequency [GHz] Critical coupling Baleegh Abdo, Eran Segev, Oleg Shtempluck, EB, cond-mat/51236 (25)

19 Hysteresis Loop Changes Direction Counter clockwise hysteresis loop No hysteresis loop Clockwise hysteresis loop S 11 db CW frequency scan in both directions -8.5 dbm -8.4 dbm -8.3 dbm -8.2 dbm -8.1 dbm -8 dbm dbm dbm dbm dbm dbm dbm dbm dbm dbm -7.9 dbm dbm dbm dbm dbm dbm dbm dbm dbm dbm -7.8 dbm CW Forward CW Backward Frequency [GHz]

20 Multiple Jumps dbm -1 S 11 [db] CW scan forw ard CW scan backw ard Frequency [Ghz] Baleegh Abdo, Eran Segev, Oleg Shtempluck, EB, cond-mat/51114 (25)

21 NbN vs. Nb Contrary to the case of Nb 1. A simple model of a one-dimensional Duffing resonator cannot account for the experimental results of NbN resonators. 2. Besides kinetic inductance, another mechanism contributes to nonlinearity. What is the underlying mechanism?

22 Is it Heating? signal generator RF output circulator Dewar 4.2K Dynamic state 1 FM power signal diode generator Scope sync 12 Load stripline resonator 1*log V out Dynamic state dbm Global heating effect? - unlikely However, local heating of hot spots in the NbN film (out of equilibrium) are not rolled out τ = Cd / α 5 ns Frequency [Ghz] 2µ s Baleegh Abdo, Eran Segev, Oleg Shtempluck, EB, cond-mat/54582 (25)

23 Strong Dependence on Temperature 5 Constant P in =-1dbm S 11 [db] k 5.35k 5.4k 5.449k 5.499k 5.55k 5.599k 5.65k 5.699k 5.749k 5.8k 5.849k 5.899k Frequency [Ghz] Temp. [k] Baleegh Abdo, Eran Segev, Oleg Shtempluck, EB, cond-mat/54582 (25)

24 Strong Dependence on Magnetic Field 1 Constant power of -5dbm S 11 [db] Jump disappears -6 mt.99 m T 1.82 mt 2.73 mt 3.64 mt 4.55 mt 5.45 mt 6.36 mt 7.27 mt 8.18 mt 9.9 mt 1 mt 1.9 mt 11.8 mt 12.7 mt 13.6 mt 11.8 mt Frequency [Ghz]

Weak Link Hypothesis Microscopic Josephson junctions forming at the grain boundaries of the columnar structure of the NbN film are suspected to be responsible for the observed behavior.

25 Weak Link Hypothesis Microscopic Josephson junctions forming at the grain boundaries of the columnar structure of the NbN film are suspected to be responsible for the observed behavior. cross section NbN film I b I N () t R I c C V C Mass R friction I b driving force Potential Energy Quadratic approximation U [I c φ /π] RCSJ with AC bias current driven Duffing oscillator θ

26 IMD Measurement Setup Pump Isolator Power combiner Circulator Dewar 4.2K Signal Isolator Spectrum analyzer Superconducting resonator Offset idler pump signal

27 IMD Gain - I [dbm] Frequency [GHz] idler gain [db] Frequency [GHz] Frequency [GHz] reflected pump A A Pump power [dbm] signal gain A A A A [db] Pump power [dbm] Pump power [dbm]

28 IMD Gain - II idler gain signal gain 2 A-A" : Frequency GHz -3 2 A-A" : Frequency GHz db db 1-35 Intermodulation gain [db] Reflected pump power [dbm] Signal gain [db] Reflected pump power [dbm] Pump power [dbm] Pump power [dbm]

29 Frequency [GHz] Hysteresis Reflected pump power [dbm] Decreasing pump Increasing pump Frequency [GHz] Pump power [dbm] Pump power [dbm] Reflected pump power [dbm]

30 Bistability of a Duffing Resonator In the bistable regime the phase space contains two basins of attaraction. f > f C A 2 Can noise induce transitions? ω - ω

31 Adding White Noise Network analyzer Dewar 4.2K Power combiner 5 ohm Circulator Superconducting resonator Uniphase amplifier Miteq amplifier Baleegh Abdo, Eran Segev, Oleg Shtempluck, EB, cond-mat/54582 (25)

32 Noise Induced Transitions S 21 [db] frequent transitions With -58 dbm white noise T = Frequency [Ghz] eff dbm dbm dbm dbm dbm dbm dbm dbm -2.9 dbm -2.6 dbm -2.3 dbm -2 dbm 14 1 K dbm hysteresis -23 dbm eliminated S 21 [db] With -8 dbm white noise dbm dbm dbm -23 dbm dbm dbm dbm dbm dbm dbm -2.9 dbm -2.6 dbm -2.3 dbm -2 dbm cw forward cw backward Frequency [Ghz] Baleegh Abdo, Eran Segev, Oleg Shtempluck, EB, cond-mat/54582 (25)

33 Nonlinear Resonator Model Test port a 1 Nonlinear resonator γ 2 a γ H 2 1 = ~ ω A A Nonlinear dissipation port + (~/2) KA A AA γ 3 Linear dissipation port a 3 linear dissipation Kerr Nonlinearity nonlinear dissipation Heisenberg equation of motion: B. Yurke and EB, unpublished

34 Classical Response in the Absence of Noise A.1.5 cavity mode amplitude a out /a in reflection A critical point A = ω a out /a in A a out /a in ω/ω ω/ω B. Yurke and EB, unpublished

35 The Correlation Function - I Test port a 1 Nonlinear resonator γ 2 a γ H 2 1 = ~ ω A A Nonlinear dissipation port + (~/2) KA A AA γ 3 Linear dissipation port a 3 Define where The correlation function The low frequency limit of the power spectrum of

36 The Correlation Function - II in b =.5 in b1c 1 in in b =b1c 1 in in b =2b1c 1 4 (a) 4 (c) 4 (e) B 3 2 B 3 critical 2 point B ω/ ω ω/ ω ω/ ω (b) 15 (d) 15 (f) Log[S ()] 1 5 Log[S ()] 1 5 Log[ S ()] ω/ ω ω/ ω ω/ ω S () diverges at the bifurcation points!

Intermodulation Gain In the limit where the offset frequency ω the intermodulation gain close to the critical point is Thus G I diverges at the critical point.

37 Intermodulation Gain In the limit where the offset frequency ω the intermodulation gain close to the critical point is Thus G I diverges at the critical point. However, the assumption that the idler amplitude is small is violated, and the model thus breaks down close to the critical point critical 3 point B. Yurke and EB, unpublished SA

38 Noise Squeezing The noise properties can be characterized by homodyning with coherent radiation. Assume for simplicity In the limit where the offset frequency ω close to a bifurcation point coherent state t squeezed state with reduced phase uncertainty t Thus P min /P max at the bifurcation point. squeezed state with reduced amplitude uncertainty t B. Yurke and EB, unpublished

39 3-15 Inter-mode Coupling - I S 11 [db] pump mode GHz signal mode GHz Lets look closer at the signal mode S 11 [db] signal power 2.27 dbm pump power dbm pump on pump off Freq. [GHz] signal network analyzer Freq [GHz] Dewar 4.2K power combiner pump circulator Superconducting resonator

Inter-mode Coupling - II In the rotating wave approximation the nonlinear coupling

measuring the number of photons in the signal mode N s.

quantum non-demolition one [Sanders and Milburn, PRA 39, 694 ( 89)].

40 Inter-mode Coupling - II In the rotating wave approximation the nonlinear coupling between the modes is given by The pump mode can be considered as a detector measuring the number of photons in the signal mode N s. Since V commutes with the Hamiltonian of the system, such a measurement is a quantum non-demolition one [Sanders and Milburn, PRA 39, 694 ( 89)]. The coupling also leads to dephasing induced on the signal mode, with a rate bridge QD QPC The dephasing rate diverges at bifurcation points.

41 Summary Intermodulation amplification is demonstrated for both nano-mechanical resonators and superconducting stripline resonators. High gain is observed near the bifurcation points of both nanomechanical and superconducting stripline resonators. Nonlinear damping in nanomechanical resonators plays an important role. Injected noise induces transitions between basins of attraction. Microscopic Josephson junctions forming at the grain boundaries of the columnar structure of the NbN films are suspected to be responsible for the nonlinear response. The noise at the output of the intermodulation amplifiers is strongly squeezed. Inter-mode coupling may induce strong dephasing when driving the pump mode close to a bifurcation.

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