Detection of small single-cycle signals by stochastic resonance using a bistable superconducting quantum interference device
Guozhu Sun, Jiquan Zhai, Xueda Wen, Yang Yu, Lin Kang, Weiwei Xu, Jian Chen, Peiheng Wu, Siyuan Han
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Detection of small single-cycle signals by stochastic resonance using a bistablesuperconducting quantum interference device
Guozhu Sun,
1, 2, ∗ Jiquan Zhai,
1, 2
Xueda Wen, Yang Yu,
4, 2
LinKang,
1, 2
Weiwei Xu,
1, 2
Jian Chen, Peiheng Wu,
1, 2 and Siyuan Han National Laboratory of Solid State Microstructures and Research Institute of Superconductor Electronics,School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA School of Physics, Nanjing University, Nanjing 210093, China Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA
We propose and experimentally demonstrate detecting small single-cycle and few-cycle signals byusing the symmetric double-well potential of a radio frequency superconducting quantum interfer-ence device (rf-SQUID). We show that the response of this bistable system to single- and few-cyclesignals has a non-monotonic dependence on the noise strength. The response, measured by theprobability of transition from initial potential well to the opposite one, becomes maximum when thenoise-induced transition rate between the two stable states of the rf-SQUID is comparable to the sig-nal frequency. Comparison to numerical simulations shows that the phenomenon is a manifestationof stochastic resonance.
It is a long-held belief that noise is detrimental or evendestructive to detecting signals which often appear asweak periodic modulations. However, during the last35 years theoretical and experimental investigations haveshown that in nonlinear systems a proper amount of noisecan actually increase the signal-to-noise ratio (SNR) andthus become beneficial for signal detection. This interest-ing phenomenon is named as stochastic resonance (SR)[1–4]. For example, suppose that a particle is moving ina periodically perturbed symmetric double-well poten-tial under the influence of a Gaussian-white noise suchas thermal fluctuation. Then SNR of the power spec-tral density of the particle’s trajectory displays a broadmaximum when the rate of inter-well transitions, whichdepends on noise strength exponentially, is comparableto the frequency of periodic signal. This is the essence ofSR.Due to its simplicity and ubiquity of the underlyingmechanism, SR has attracted much interest from physi-cists, chemists, biologists, and electronic engineers [1–11].It has also been observed in Josephson junction-basedsystems [12–16], which have recently attracted much in-terest and been applied in many fields such as quantuminformation [17–19]. However SR has been only investi-gated for periodic signals that last many cycles. Namely,only the steady-state properties of the noisy periodicallydriven systems have been studied. On the other hand,in a variety of science and engineering disciplines, it isa significant challenge to detect small signals which notonly last a few cycles but also are buried in noise. Upto this point, whether SR can also enhance single-cyclesignal detection remains an open question.In this Letter, we report on the observation of SR in ∗ Electronic address: [email protected] a radio frequency superconducting quantum interferencedevice’s (rf-SQUID’s) [20, 21] response to weak single-cycle and few-cycle signals by measuring the inter-welltransition probability as a function of the noise strength D and the signal frequency f s systematically. Our ex-perimental and numerical results show that one can dis-till small single-cycle and few-cycle signals from noisyenvironment by using bistable systems configured as bi-nary threshold detectors. The maximum sensitivity isachieved at the value of D that matches well with theposition of SR. We also show that the sensitivity of de-tecting single-cycle signals is similar to that of detectingmany-cycle signals.In our experiment we use an rf-SQUID, which is asuperconducting loop of inductance L interrupted by aJosephson junction of critical current I c , as our bistabledetector. An optical micrograph of the sample is shownin the inset of Fig. 1. The Josephson junction is made ofNb/AlOx/Nb on a silicon substrate. The critical current I c and the capacitance C of the junction are approxi-mately 0 . µ A and 90 fF, respectively. The inductance L of the Nb superconducting loop is approximately 1053pH. The potential energy of the rf-SQUID is given by U (Φ) = (Φ − Φ e )2 L − E J cos (cid:18) π ΦΦ (cid:19) , (1)where Φ is the flux quantum and E J = I c Φ / π is theJosephson coupling energy of the junction. The shape ofthe double well potential can be controlled in situ by aflux bias Φ e applied via a flux bias line coupled induc-tively to the rf-SQUID. In particular, at Φ e = Φ / U as shown in Fig. 1. For the SQUIDstudied here, ∆ U /k B ≃ . k B is the Boltz-mann’s constant. The dynamics of the rf-SQUID, identi-cal to that of a fictitious flux particle of mass C movingin the potential U (Φ) with a friction coefficient R − , isgoverned by the corresponding Langevin equation: C d Φ dt + 1 R d Φ dt = − dUd Φ + I n ( t ) . (2)Here, I n is the noise current and R is the damping re-sistance of the Josephson junction. Without externallyinjected noise, I n and R are related by the fluctuation-dissipation theorem h I n ( t ) I n ( t ′ ) i = k B TR δ ( t − t ′ ) in ther-mal equilibrium, where T is temperature. The small os-cillation frequency of the system around the bottom ofthe potential wells is denoted as ω . At T ≫ ~ ω /k B , where ~ is the Planck constant, thermal activation causesinter-well hopping with the characteristic transition rategiven by [22] Γ = ω π a t exp (cid:18) − ∆ U k B T (cid:19) , (3)where a t is a damping dependent constant of order ofunity. When transitions are dominated by an externalnoise source of strength D ≫ k B T , the denominator inthe exponent of Eq. (3) is replaced by D which is pro-portional to the square of the rms noise current, I n, rms , applied to the system. For the sake of simplicity, here-after we set k B = 1 so that D is measured in units ofkelvin. Note that because the potential is symmetric, Γ is identical for left-to-right and right-to-left transitions.Because all key parameters of the rf-SQUID potentialand its control circuit can be accurately determined, thedouble-well potential of the rf-SQUID is an ideal systemfor investigating SR [12–14] and noise-enhanced detectionof single-cycle and few-cycle signals. In our experiment,the Gaussian-white noise has a bandwidth of about 9MHz, which is generated by an arbitrary waveform gen-erator. The signal and noise are applied to the rf-SQUIDthrough a second flux bias line with higher bandwidth(up to 18 GHz). The relationship between D and I n, rms is calibrated by measuring Γ versus I n, rms and compar-ing the result to Eq. (3).As shown schematically in Fig. 1, each measurementcycle begins by ramping up the quasi-static flux bias from0 to Φ / U + ε sin(2 πf s t ),where ε is proportional to the amplitude of the fluxmodulation which is kept at ε = 0 . U ≃ .
86 Kin the experiment. The position of the flux particle ismeasured by using a dc-SQUID switching magnetome-ter inductively coupled to the rf-SQUID, either after asingle signal cycle or a fixed duration of signal time asdiscussed later, as a function of 0 . ≤ D ≤ . ≤ f s ≤
200 kHz. The quasi-static flux bias isthen ramped down to zero to complete the measurementcycle. To obtain the fractional population in the rightwell ρ R , the procedure is repeated 2000 times at eachvalue of D and f s . All data are measured at T ≈
20 mK ≪ D in a cryogen-free dilution fridge carefully shieldedfrom the environmental electromagnetic interference sothat the effects of thermal fluctuation and extra noise onthe experiment are negligible.We first measure ρ R as a function of the noise strength D by using single-cycle signals ε ± ( t ) = ± ε sin(2 πf s t ) asdepicted in Fig. 1, where the signal frequency f s = 10kHz. The result is shown in Fig. 2(a). The noise strength D is varied between 0 . . ε = 0 (no sig-nal) are noise activated. Note that without the noise, ε ( t ) alone would be too small to cause transitions be-tween the potential wells because (∆ U − ε ) /T > D > . ε = 0 the transi-tion rate Γ grows exponentially from approximately 1 / sat D = 0 . / s at D = 2 . D . . ρ R is negli-gible at t = τ = 1 /f s . The data indicated by the bluesquares in Fig. 2(a) are taken with ε + ( t ) which showthat as D is increased from 0 . ρ R rises rapidly toreach maximum when Γ ( D m ) ∼ f s , where D m denotesthe noise strength corresponding to the maximum ρ R .When D > D m , the probability of hopping back fromthe right well to the left well increases rapidly, causing ρ R ( D ) to decrease. Finally, when D ≫ ε the populationof each potential well is equalized to 50%.SR has two most prominent signatures: One is thepeak in the system’s response versus noise strength D .The other is the position of the peak and the sig-nal frequency satisfying Γ ( D m ) ∼ f s , or equivalently,1 /D m ∼ − ln( f s ) according to Eq. (3). In Fig. 2(b),where ε + ( t ) is applied, we plot ρ R versus 1 /D and f s which shows clearly both signatures of SR. In particu-lar, the nearly linear relationship between Γ ( D m ) and f s is demonstrated as shown in the inset of Fig. 2(b).The slope of Γ ( D m ) versus f s obtained from the best-fitto a line is 2.7, which is consistent with the numericalresult previously obtained for SR under continuous mod-ulation [23]. In addition, we numerically calculate thepower spectral density S ( f s ) of the flux particle’s trajec-tories Φ( t ) generated by Monte Carlo simulation of Eq.(2). It is found that S ( f s ) reaches its maximum at thesame value of D as ρ R does. We thus conclude that SRplays a central role in the bistable system’s response tosingle-cycle signals.In order to compare the result of our measurementswith that of numerical study over the entire parameterspace covered by the experiment, we adopt the two-statemodel [4, 24] and introduce the rate equation: dρ R ( t ) dt = − Γ − ρ R ( t ) + Γ + [1 − ρ R ( t )] (4)with the initial condition ρ L (0) = 1 , ρ R (0) = 0 . Here, ρ R ( ρ L = 1 − ρ R ) is the fractional population of the right(left) potential well. When ε = 0, the barrier height isoscillating between ∆ U ± ε and the transition rates aretime-dependentΓ ± ( t ) = Γ exp (cid:20) − ± ε sin(2 πf t ) D (cid:21) , (5)where Γ + and Γ − denote the rates of left-to-right andright-to-left transitions, respectively. Γ is given by Eq.(3). Using the system parameters given above, we nu-merically integrate Eq.(4) to obtain ρ R ( t ) as a functionof f s and D. The result is shown in Fig. 2(c). It can beseen that the key features of the experimental data arewell reproduced.Next, we show that the sensitivity of detecting single-cycle signals is comparable to that of detecting many-cycle signals and that one can predict the population dis-tribution of the bistable systems at the end of N -cyclemodulations ρ R,N ≡ ρ R ( N τ ) = 1 − ρ L,N from that ofsingle-cycle modulation ρ R, . It is straightforward to ob-tain the recursion relation ρ R,n +1 = ρ L,n P + + ρ R,n (1 − P − ) (6)= (1 − ρ R,n ) P + + ρ R,n (1 − P − ) . The first (second) term of the r.h.s. of Eq. (6) is thefractional population of the left (right) well at t = nτ that ends (remains) in the right well at t = ( n + 1) τ . P +( − ) is the probability of switching from the left (right)to the right (left) well during the time interval nτ ≤ t ≤ ( n + 1) τ . Notice that with the single-cycle perturbation ε ± (0 ≤ t ≤ τ ) and the initial condition ρ R, = 0, onehas P ± = ρ R, by taking into consideration the spatialand temporal symmetry properties of the rf-SQUID po-tential and ε ± . Thus, we can obtain P ± directly from thedata presented in Fig. 2(a). Because Eq. (6) is valid forarbitrary noise strength D and signal frequency f s , wecan compute ρ R,N from P ± for any integer N >
1. Wefind that as N increases ρ R,N converges rapidly. In orderto investigate the dependence of SR on the number ofsignal cycles N , we modify the experimental procedureby changing the duration of the applied signal and noisefrom τ to 0 . N = 3 for f s = 10 kHz, which increases ultimately to N = 60 for f s = 200kHz. In Fig. 3(a), the measured ρ R,N is plotted against1 /D and f s , which compares well with ρ R,N computedfrom Eq. (6) by using the measured P ± as inputs [see Fig.3(b)] and that obtained by solving the corresponding rateequation (4) [see Fig. 3(c)]. The results presented in Fig.3 all have two distinctive features: (i) The threshold noisestrength D which demarcates the blue region ( ρ R,N ≈ N, and (ii) 1 /D m ∝ − ln( f s ) remains validfor the entire range of 3 ≤ N ≤
60. As shown in theinset of Fig. 3(a), the dependance of Γ ( D m ) on f s is ap-proximately linear with a slope of about 2.6. These twofeatures strongly indicate that the sensitivity of detect-ing single-cycle signals is similar to that of many-cycleand continuous wave signals and that SR does exist inthe systems driven by small single-cycle signals.In summary, using an rf-SQUID as a prototypicalbistable system, we have demonstrated the existenceof SR with single-cycle perturbation to the symmetricdouble-well potential of the system. Furthermore, wehave investigated the possibility of exploiting SR for de-tecting small single-cycle and few-cycle signals in noisyenvironment. We have found that a proper amount ofnoise can lead to SR which enhances the sensitivity ofdetection. Our work provides insights into the behav-ior of bistable systems under the combined influence ofweak single-cycle (or few-cycle) periodic modulation andnoise. Because conventional techniques, such as phasesensitive lock-in and heterodyne detection schemes, arenot applicable to detecting single-cycle and few-cycle sig-nals buried in noise, the method demonstrated here ispromising for applications where signals are unavoidablymixed up with noise and only last a very small numberof cycles.We thank Dan-Wei Zhang and Shi-Liang Zhu forthe valuable discussions. This work was partially sup-ported by MOST (Grant Nos. 2011CB922104 and2011CBA00200), NSFC (11474154, BK2012013), PAPD,a doctoral program (20120091110030) and DengfengProject B of Nanjing University. S.H. was supported inpart by NSF (PHY-1314861). [1] K.Wiesenfeld and F.Moss, Nature ,33 (1995).[2] L.Gammaitoni,P.H¨anggi,P.Jung,and F.Marchesoni,
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Measured ρ R as a function of 1 /D and f s witha constant duration (0 . (b) ρ R as a function of 1 /D and f s with a constantduration (0 . P ± obtained from the experimental results when single-cycle signals are used. (c) Numerical calculation of ρ R as afunction of 1 /D and f s with a constant duration (0 ..