Stochastic Resonance with a Single Metastable State
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Stochastic Resonance with a Single Metastable State
Eran Segev, ∗ Baleegh Abdo, Oleg Shtempluck, and Eyal Buks
Department of Electrical Engineering, Technion, Haifa 32000, Israel (Dated: November 21, 2018)We study thermal instability in NbN superconducting stripline resonators. The system exhibitsextreme nonlinearity near a bifurcation, which separates a monostable zone and an astable one. Thelifetime of the metastable state, which is locally stable in the monostable zone, is measure near thebifurcation and the results are compared with a theory. Near bifurcation, where the lifetime becomesrelatively short, the system exhibits strong amplification of a weak input modulation signal. Wefind that the frequency bandwidth of this amplification mechanism is limited by the rate of thermalrelaxation. When the frequency of the input modulation signal becomes comparable or larger thanthis rate the response of the system exhibits sub-harmonics of various orders.
Stochastic resonance (SR) is a phenomenon in whichmetastability in nonlinear systems is exploited to achieveamplification of weak signals [1, 2, 3]. SR has been ex-perimentally demonstrated in electrical, optical, super-conducting, neuronal and mechanical systems [4, 5, 6,7, 8, 9, 10, 11, 12]. Usually, SR is achieved by oper-ating the system in a region in which it has more thanone locally stable (metastable) steady state and its re-sponse exhibits hysteresis. Under some appropriate con-ditions, a weak input signal, which modulates the transi-tion rates between these states, can lead to synchronizednoise-induced transitions, allowing thus strong amplifica-tion.In the present paper we investigate SR and amplifica-tion in a superconducting (SC) NbN stripline resonator.Contrary to previous studies, we operate the system neara bifurcation between a monostable zone, in which thesystem has a single metastable state, and an astable zone,in which this state ceases to exist and the system lacksany steady states. In our previous studies we have in-vestigated several effects, e.g. strong amplification [13],noise squeezing [13], and response to optical illumination[14, 15], which occur near this bifurcation, and limit cy-cle oscillations, which are observed in the astable zone[16, 17]. In the present work we investigate experimen-tally and theoretically the response of the system to am-plitude modulated input signal, and find an unusual SRmechanism that has both properties of strong responsiv-ity and non-hysteretic behavior. The frequency band-width of this mechanism is found to be limited by therate of thermal relaxation. We find that rather uniquesub-harmonics of various orders are generated when themodulation frequency becomes comparable or larger thanthe relaxation rate. Moreover, we measure the lifetimeof the metastable state in the monostable zone near thebifurcation and compare the results with a theory.Our experiments are performed using a novel devicethat integrates a narrow microbridge into a SC striplineelectromagnetic resonator (see Fig. 1 (A)). Design con-siderations, fabrication details as well as resonance modescalculation can be found elsewhere [14]. The dynamics ofour system can be captured by two coupled equations of motion, which are hereby briefly described (see Ref. [17]for a detailed derivation). Consider a resonator driven bya weakly coupled feed-line carrying an incident amplitudemodulated coherent tone b in = b in0 (1 + a cos( ω m t )) e − iω p t ,where b in0 is constant complex amplitude, ω p is the driv-ing angular frequency, a is the modulation depth, and ω m ≪ ω p is the modulation frequency. The mode am-plitude inside the resonator can be written as Be − iω p t ,where B ( t ) is a complex amplitude, which is assumed tovary slowly on a time scale of 1 /ω p . In this approxima-tion, the equation of motion of B reads [18]d B d t = [ i ( ω p − ω ) − γ ] B − i p γ b in + c in , (1)where ω is the angular resonance frequency and γ ( T ) = γ + γ ( T ), where γ is the coupling coefficient betweenthe resonator and the feed-line and γ ( T ) is the temper-ature dependant damping rate of the mode, and T is thetemperature of the microbridge. The term c in representsan input Gaussian noise. The microbridge heat balanceequation reads C d T d t = 2 ℏ ω γ | B | − H ( T − T ) , (2)where C is the thermal heat capacity, H is the heat trans-fer coefficient, and T = 4 . γ ( T ) of the drivenmode on the resistance of the microbridge [19], whichin turn depends on its temperature. We assume the sim-plest case, where this dependence is a step function thatoccurs at the critical temperature T c ≃
10 K of the su-perconductor, namely γ takes the value γ for the SC T < T c phase of the microbridge and γ for the normal-conducting (NC) T > T c phase.Solutions of steady state response to a monochromaticexcitation (no modulation a = 0) are found by seekingstationary solutions to Eqs. (1) and (2) for the noiselesscase c in = 0. Due to the coupling the system may have,in general, up to two locally-stable steady-states, corre-sponding to the SC and NC phases of the microbridge. FIG. 1: (A) Experimental setup. (B) System stability dia-gram.
The stability of each of these phases depends on boththe power, P pump ∝ (cid:12)(cid:12) b in (cid:12)(cid:12) , and frequency ω p parame-ters of the injected pump tone. Our system has fourstability zones (Fig. 1(b)) [17]. Two are mono-stablezones (MS(S) and MS(N)), where either the SC or theNC phases is locally stable, respectively. Another is abistable zone (BiS), where both phases are locally stable[20, 21]. The third is an astable zone (aS), where none ofthe phases are locally stable. Consequently, when the res-onator is biased to this zone, the microbridge oscillatesbetween the two phases. The onset of this instability,namely the bifurcation threshold (BT), is defined as theboundary of the astable zone (see Fig. 1(b)).The experimental setup is depicted in Fig. 1(a). Weinject an amplitude modulated pump tone into the res-onator and measure the reflected power in the frequencydomain using a spectrum analyzer and in the time do-main using an oscilloscope. The parameters used for thenumerical simulation were obtained as follows. The cou-pling coefficient γ = 2 MHz and the damping rates γ =2 . γ = 64 MHz were extracted from frequencyresponse measurement [14, 20], whereas the thermal heatcapacity C = 54 nJ cm − K − and the heat transfer coef-ficient H = 12 W cm − K − were calculated analyticallyaccording to Refs. [22, 23].Our system exhibits an extremely strong amplifica-tion when tuned to the BT. Figure 2 shows both ex-perimental (Blue curves) and numerical (Red curves)results for the case where the system is driven by amodulated pump tone having the following parameters: ω p = ω = 2 π × .
363 GHz, ω m = 2 π ×
10 kHz, a = 0 . T eff = 75 K.Panel (A) plots the signal gain G sig , defined as the ratiobetween the reflected power at frequency ω p + ω m and thesum of the injected powers at frequencies ω p ± ω m , as afunction of the mean injected pump power h P pump i . Thesystem exhibits large gain of approximately 20dB aroundthe BT. The experimental results exhibits excess gain be-low BT relative to the numerical results. This can beexplained by additional nonlinear mechanisms [24] thatmay induce small amplification, and are not theoreticallyincluded in our piecewise linear model.Figure 2 (B), shows time and frequency domain results −40.2 −40.15 −40.1 −40.05 −40 −39.95−30−20−1001020 < P pump > [dBm] G s i g [ d B ] (a) (b) (c) NumExp (A) (a1) −100−80−60 (a2) (b1) −100−80−60 (b2) (c1) −100−80−60 (c2) (a3) −100−80−60 (a4) (b3) −100−80−60 (b4) t [msec] P r e f l [ n . u . ] (c3) −30 −20 −10 0 10 20 30−100−80−60 ∆ f SA [KHz] P r e f l [ d B m ] (c4) (B)FIG. 2: (A) Experimental (dotted-blue) and numerical(crossed-red) results of the signal amplification G sig as a func-tion of the mean injected pump power < P pump > . (B) Ex-perimental (subplots ( x
1) and ( x x
3) and ( x P refl as a functionof time (subplots ( x
1) and ( x f SA (subplots ( x
2) and ( x f = 4 . f SA = f SA − f ), where x denotes a , b , and c , corresponding to the marked points in panel (A). Thedashed-green curve represents the modulation signal. Thetime domain measurements are normalized by their maximumpeak to peak value. of the reflected power, for three pairs of input power val-ues, corresponding to the marked points (a − c) in panel(A). In addition, the time domain measurements containa green-dashed curve showing the modulating signal. Theresults shown in subplots (a1 − a4) were obtained whilebiasing the system below the BT, namely, h P pump i wasset below the power threshold, P c . In general, the spikesin the time domain plots of Fig. 2 (B) indicate eventsin which the temperature T temporarily exceeds T c [17].Below threshold, the average time between such events,which are induced by input noise, is the lifetime Γ − ofthe metastable state of the resonator. As we will showin the last part of this paper, Γ strongly depends on thepump power near BT, thus power modulation results ina modulation of the rate of spikes, as can be seen bothin the experimental and simulation results.Subplots (b1 − b4) of Fig. 2(B) show experimentsin which h P pump i ≃ P c and thus, the modulation itselfdrives the resonator in and out the astable zone. As aresult, during approximately half of the modulation pe-riod nearly regular spikes in reflected power are observed,whereas during the other half only few noise-inducedspikes are triggered. This behavior leads to a very stronggain as well as to the creation of higher order frequencycomponents (subplots (b2 , b4)). Figure 2(B), Subplots(c1 − c4), show experiments in which h P pump i > P c , andthus the regular spikes occur throughout the modula-tion period. The rate of the spikes is strongly correlatedto the injected power [16], and it is higher for strongerpump powers. Therefore, as the injected pump power ismodulated, so is that rate. This behavior also creates arather strong amplification, though weaker than the oneachieved in the previous case.The amplification mechanism in our system is uniquein several aspects. First it is extremely strong. Toemphasize the strength of the amplification we notethat, usually, no amplification greater than unity (0 dB)is achieved in such measurements with SC resonators[25, 26], unless the resonator is driven near BT [27]. Inaddition, it does not exhibit a hysteretic behavior.Each spike in subplots (a1 − a2) of Fig. 2(B) lasts ap-proximately 1 µ s, after which the device is ready to detecta new event. This recovery time determines the detectionbandwidth. A measurement of the dependence of the am-plification mechanism on the modulation frequency ω m has reveled a mechanism in which sub-harmonics of themodulation frequency are generated by the device. Thegeneration occurs when the modulation period is compa-rable to the recovery time of the system. The results areshown in Fig. 3 which shows both experimental (Bluecurves) and numerical results (Red curves) for the caseof ω p = 2 π × .
363 GHz, a = 0 . T eff = 75 K, and ω m = 2 π × . ω m = 2 π × , a3)). Each quasi-period lasts three modulation cycles, where only duringthe first two a spike occurs, namely a spike is absentonce every three modulation cycles. This behavior origi-nates from the mismatch between the modulation periodand the recovery time of a spike, which induces a phasedifference, that is monotonically accumulated, betweenthe two. Once every n = 3 modulation cycles, in thiscase, the system fails to achieve critical conditions nearthe time where the peak in the modulation occurs, andtherefore a spike is not triggered. Similar behavior is alsoshown in subplots (a1 , a3) and (c1 , c3), where the quasi- (a1) −90−60 (a2) (b1) −90−60 (b2) (c1) −90−60 (c2) (a3) −90−60 (a4) (b3) −90−60 (b4) t [ µ sec] P r e f l [ n . u . ] (c3) −1.2 −0.6 0 0.6 1.2−90−60 ∆ f SA [MHz] P r e f l [ d B m ] (c4) (A) (a1) −90−60 (a2) t [ µ sec] P r e f l [ n . u . ] (a3) −2 −1 0 1 2−90−60 ∆ f SA [MHz] P r e f l [ d B m ] (a4) (B)FIG. 3: Sub-harmonics generation. period lasts two and four modulation cycles respectively.Another mechanism for SHG is observed when themodulation frequency is increased. Fig. 3, Panel (B),shows measurement results for ω m = 2 π × n = 3. Unlike the previouscase, this SHG is characterized by a single spike thatoccurs once every three modulation cycles.We further study our system by measuring thefluctuation-induced escape rate Γ of the metastable statein the MS(S) zone. In Ref. [28] we have found theoreti-cally that Γ = Γ exp( − γ ∆ P γ k b T eff P pump ) , (3)where Γ = p Hγ/C/ π , and the power difference isgiven by ∆ P pump ≡ P c − P pump . Note that the unusualscaling law in the present case log (Γ / Γ ) ∝ ∆ P ,which differs from the commonly obtained scaling lowof log (Γ / Γ ) ∝ ∆ P / [29, 30], is a signature of thepiecewise linear dynamics of our system.The escape rate was experimentally measured for sev-eral levels of T eff , which are given in the first row oftable I. The noise was generated by an external whitenoise source, and combined with the amplitude modu- ∆ P pump 2 [nW ] l og ( E sca p e R a t e [ H z ]) (1)(2)(3)(4)(5)(6) FIG. 4: Escape rate of metastable states for several levels of T eff , summarized in table I. The graphs are plotted in pairs,where the solid curves show the experimental data and thedashed curves show the corresponding theoretical fit. lated pump tone. The modulation frequency was setto 500 Hz, which is more than three orders of magni-tude lower than the relaxation rate of the system, andtherefore to a good approximation the system follows thismodulation adiabatically [29].The results are shown in Fig. 4, which plots the escaperate in logarithmic scale as a function of ∆ P . Sixpairs of solid and dashed curves are shown, correspond-ing to the six different levels of injected noise intensities.The solid curves were extracted from time domain mea-surements of the reflected power. The dashed curves wereobtained by numerically fitting the experimental data toEq. (3) and show good quantitative agreement betweenthe experimental results and Eq. (3). The fitting param-eters included the pre-factor Γ = 0 .
86 MHz that wasdetermined by a separate fitting process, and P c (seetable I) that slightly decreases with the thermal noise.This behavior can be explained by local heating of themicrobridge, induced by the noise that is injected intothe resonator through additional resonance modes. Notethat T eff was extracted from a direct measurement of theinjected noise intensity (see table I). Note also that thesystem recovery time at the threshold imposes a limit onthe measured escape rate. Thus the escape rate close tothe threshold might be higher than measured.In summary, a novel mechanism of SR with a singlemetastable state has been demonstrated. Near BT thesystem exhibits rich dynamical effects including bifurca-tion amplification and SHG. In spite of its simplicity, ourtheoretical model successfully accounts for most of theexperimental results.We thank Steve Shaw and Mark Dykman for valu-able discussions and helpful comments. This work TABLE I: Escape rate parameters1 2 3 4 5 6 Note T eff [10 K] 0 .
52 1 1 .
36 1 .
64 2 . .
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