This is the published version, also available here: http://dx.doi.org/10.1063/1.114161.The flux change δ Φ through a bistable superconducting quantum interference device has been measured in the presence of thermally induced switching (with rate Γ) versus δ Φ x , the change in the applied flux. For small δ Φ x , δ Φ is proportional to δ Φ x with a measured flux gain g, depending on the temperature, barrier height, and frequency Ω, with a maximum of about 16. In agreement with theories of periodically driven stochastic bistable systems,g(Ω) is nearly frequency independent up to Γ and is proportional to Ω−1 for Ω≫Γ. For larger amplitude signals, harmonic generation has been measured in the adiabatic limit (Ω≪Γ) and found to be in good agreement with theory. Possible applications of this system for flux measurement are discussed

Han,

J. E. Lukens

Jung

McNamara

R. Rouse

Siyuan Han

English

Rouse, R.

Han, Siyuan

Lukens, J. E.

KU ScholarWorks

 ThisFlux amplification using stochastic superconducting quantum interferencedevicesR. Rouse, Siyuan Han, and J. E. Lukensa)Department of Physics, SUNY at Stony Brook, Stony Brook, New York 11794-3800~Received 25 August 1994; accepted for publication 28 October 1994!The flux changed F through a bistable superconducting quantum interference device has beenmeasured in the presence of thermally induced switching~with rateG! versusd Fx , the change inthe applied flux. For smalld Fx , d F is proportional tod Fx with a measured flux gaing, dependingon the temperature, barrier height, and frequencyV, with a maximum of about 16. In agreementwith theories of periodically driven stochastic bistable systems,g~V! is nearly frequencyindependent up toG and is proportional toV21 for V@G. For larger amplitude signals, harmonicgeneration has been measured in the adiabatic limit~V!G! and found to be in good agreement withtheory. Possible applications of this system for flux measurement are discussed. ©1995 AmericanInstitute of Physics.-p-lultawsThe superconducting quantum interference devic~SQUID!, a superconducting loop interrupted by one or twJosephson junctions, has proven to be an extraordinarily ssitive magnetometer~see, e.g., Ref. 1 for a review!. Essen-tially, the operation of these devices involves switching thloop between fluxoid states using either a rf flux~single junc-tion rf SQUID! or the Josephson oscillations resulting fromdc voltage across the junctions~two junction dc SQUID!. Ingeneral, the noise per unit bandwidth is reduced as thswitching frequency is increased. Recent theories of periocally driven stochastic bistable systems2–4 show that, in prin-ciple, high sensitivity can also be achieved if the intrinsithermal fluctuations of the device are the source of thswitching. In this letter, we apply these results to a SQUIsystem and demonstrate that substantial flux gain canachieved in both passive and active versions of the systemThe system being studied here, which we refer to asstochastic~or S-! SQUID, uses the rf SQUID configuration,i.e., a superconducting loop of inductanceL interrupted by aJosephson junction of critical currentI c .5 The equation ofmotion for the fluxF through this S-SQUID is homologousto that of a particle of massC ~junction’s shunt capacitance!,moving in a potentialU~F! and having a friction coefficientg51/R, whereR is the junction’s shunt resistance. This potential, shown in Fig. 1~insert!, is given by:U~F!5~F2Fx!22L1EJ cosS 2p FF0 D . ~1!Here,Fx is the externally applied magnetic flux through theSQUID. F0 is the flux quantum, andEJ5F0 I c/2p is themaximum Josephson coupling energy of the junction. In E~1! the origins ofF andFx have been chosen atF0 /2 so thatatFx50 the potential is symmetric inF with two minima at6Fm separated by a barrierDU @cf. Fig. 1 ~insert!#. ForFxÞ0, the symmetry of the potential is broken and an energdifference 2e between the left and right wells appears withe5FmFx/L for smallFx .a!Electronic mail: jlukens@ccmail.sunysb.edu108 Appl. Phys. Lett. 66 (1), 2 January 1995 0003-6951 article is copyrighted as indicated in the article. Reuse of AIP content is sub129.237.46.100 On: Thu,eoen-eaisdi-cisDbe.a-q.yThermally induced switching between the two wells occurs with a rate given~for DU/kBT@1! by the Kramers re-lation asG5G0 expS 2DUkBT D . ~2!G0 depends on the S-SQUID parameters with values aproaching 1012 s21 possible. This switching will produce ran-dom telegraph noise inF with a spectral density given bySF~N!~v!5Fm2 t2p@11~vt /2!2#, ~3!wheret51/G. Theory2,4 shows that the addition of a smallsinusoidal energy modulation of the forme5e0 sin ~Vt!@i.e., Fx5F̃x sin~Vt!# to such a system results in a totaspectral densitySF~v! given by Eq.~3! plusd peaks atV andits harmonics. The spectral weightsSF(S)~V! of thesed peakshave been calculated for a two level system with the resfor the fundamental thatFIG. 1. Spectral weight of the first harmonic of the S-SQUID response tosinusoidal applied flux at frequencyV. The two curves are for differentbarrier heights~circles—higher barrier, triangles—lower barrier! with result-ing differences int. The lines are the fit of the data to Eq.~4! using thevalues oft obtained from the corresponding noise spectra. The insert shothe S-SQUID potential forFx50./95/66(1)/108/3/$6.00 © 1995 American Institute of Physicsject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 11 Sep 2014 16:05:57osenlealar-n ThiSF~S!~V!5~Fmbe0!211~Vt /2!2, ~4!whereb[1/kBT. Thus, the signal response is also Lorentziawith the same cutoff frequency, determined byt, as thenoise. As can be seen from Eq.~4!, at low frequencies thesignal gain,ASF(S)(V)/F̃x2, is given byg~0!5Fm2LkBT. ~5!Again, this is valid only in the small signal limit whereF!Fm .Measurements were made in a4He cryostat, using asetup similar to that described in detail elsewhere.5,6 In theS-SQUID ~L5200 pH!, the single junction is replaced bytwo junctions in parallel in a low inductance~L8510 pH!superconducting loop.7,8 This sample is well described byEq. ~1! with a coupling energyEJ , which can be variedinsitu by changing the flux through the small loop. This pemits DU and thusG to be varied at fixed temperature. Inorder to ensure that intrinsic fluctuations are the source oftransition, the sample is carefully shielded and is weakcoupled to the field coils used to modulateFx andEJ and toa dc SQUID magnetometer used to measureF. Figure 1shows the measured square of the small signal gainversusV for two barrier heights atT53.0 K. The lines show the fitto Eq. ~4! using the values oft obtained by fitting the corre-sponding noise spectra to Eq.~3!. As can be seen, the agreement is excellent giving values ofFm50.178 F0 andFm50.183F0 for the low and high barrier cases, respetively.It is clear from Eq.~4! that for a given signal frequencyV the maximum gain is achieved forG@V. IncreasingG byreducingDU or increasingT also reduces the noiseSF(N)~V!thus increasing the signal-to-noise ratio, at least as longEq. ~2! is valid. In this so-called adiabatic limit~G@V!, theanalysis of the system is straightforward even for larger snal amplitudes. Here, one can assume that thermal equrium between the wells is maintained at all times during tvariation ofFx giving^F~ t !&5Fm tanhS FmFx~ t !LkBT D . ~6!While Eq. ~6! is rigorously true only for a discrete two levesystem, it is an excellent approximation for the S-SQUIstudied here as can be seen by the agreement of the measstaticF2Fx curve with Eq.~6! shown in Fig. 2. The gain,dF/dFx , obtained from Eq.~6! also agrees with that givenin Eq. ~5!. As an example, a SQUID withL564 pH andI c56.9 mA would have a small signal gain of 48 at 4.2 Kwith a sensitivity of 0.1mF0/AHz, assuming the noise isgiven by Eq.~3!. Of course, another device such as a dSQUID would be required for reading out the flux so ampfied. A detailed analysis shows that if the S-SQUIDcoupled to the dc SQUID through a superconducting tranformer, the dc SQUID noise is not amplified by thS-SQUID. Thus the signal-to-noise ratio of the system ceasily be that due to the S-SQUID itself. This avoids thproblems of shunt resistors and heating associated withAppl. Phys. Lett., Vol. 66, No. 1, 2 January 1995s article is copyrighted as indicated in the article. Reuse of AIP content is su129.237.46.100 On: Thunr-thely-c-asig-ilib-helDuredcli-iss-eanedcSQUIDs. The signal gain provided by the S-SQUID alsconsiderably relaxes the stringent requirements for low noiamplifiers usually associated with dc SQUID systems.Another mode of operation for the S-SQUID is based othe selection rules of periodically driven stochastic bistabsystems first introduced by Jung and Ha¨nggi.3 These rulesstate that when an external additive periodic driving force@inour caseF̃x sin~Vt!# of arbitrary amplitude is applied to astochastic bistable system of even parity~e.g., the S-SQUIDwith dc flux F̄x50! the power spectral density of the systemcontainsd peaks only atV and its odd harmonics. For abistable system having no definitive parity the power spectrdensity in general contains peaks at both odd and even hmonics. This effect can be clearly seen in the spectra showin Figs. 3~a! and 3~b!, where a flux modulation of amplitudeF̃x542.5 mF0 and frequencyV/2p553 Hz was applied tothe S-SQUID. The spectrum in Fig. 3~a!, which shows noFIG. 2. Measured best fit ofF vs Fx for largeG to Eq. ~6! along with theresulting residual.FIG. 3. Spectra ofF resulting from large amplitude modulation of theS-SQUID potential byF̃x for ~a! F̄x50 and ~b! F̄x58.5 mF0. Note theabsence of even harmonics for the symmetric potential withF̄x50.109Rouse, Han, and Lukensbject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:, 11 Sep 2014 16:05:57asenem:umtalnduticuxen-dia-n-behsyyttoA.s, J. Thieven harmonics, was obtained withF̄x50, while F̄x58.5mF0 for Fig. 3~b!, which contains all harmonics.In the adiabatic limit it is straightforward to calculate thamplitude of the harmonics. We obtain for thenth Fouriercoefficient of the response,MnMn5Vp E2p / Vp / VFm tanhS FmFx~ t !LkBT Dexp~ inVt !dt, ~7!whereFx(t)5F̄x1F̃x sin~Vt!. SF(S)(nV)5uMn~V!u2. Figure4 shows the measured harmonic response of the S-SQUIDa function ofF̄x for the first three harmonics,9 compared tothat predicted from Eq.~7!, with no adjustable parametersCare was taken to ensure that all power from a given pewas registered in a single bin of the spectrum analyzer. Tnormalized spectral weights of the peaks plotted in Fig. 4 acalculated from the recorded spectral density times thesize in hertz.SinceM2 is a linear function ofF̄x for small F̄x , theamplitude of the second harmonic can serve as a measurFIG. 4. The absolute value of the amplitude of the first three Fourier coponents~normalized toFm! of the S-SQUID response to a large amplitudsignalF̃x as a function of the mean applied fluxF̄x . The solid lines are thepredictions of Eq.~7! with no adjustable parameters. The insert shows tslope of the second harmonic nearF̄x50 as a function of the modulationamplitudeF̃x compared to that calculated from Eq.~7! again, with no ad-justable parameters.110 Appl. Phys. Lett., Vol. 66, No. 1, 2 January 1995s article is copyrighted as indicated in the article. Reuse of AIP content is su129.237.46.100 On: Thueas.akherebine ofthe low frequency~!V! flux applied to the S-SQUID. Thephase of this signal shifts byp as F̄x changes sign, and socan serve as the basis for a feedback scheme using phsensitive detection for the measurement of large signals. Ocan define a second gain parameter for the systeg2(F̃x)5dM2(F̄x)/dF̄x . Figure 4 ~insert! shows the mea-suredg2(F̃x! for F̄x54.32 mF0 andV/2p5250 Hz!G. Thesolid line is the prediction from Eq.~7! with all parametersindependently measured. As can be seen, the maximvalue ofg2, obtained by optimizingF̃x is of the same orderasg, so the sensitivity calculated above for the fundamenmode of operation would serve as a guide to the secoharmonic mode as well.In summary, we have demonstrated a flux gain of abo16 in a stochastic SQUID~using thermally activated fluxswitching!. Theories of the signal to noise ratio in stochastbistable systems, applied to this S-SQUID, indicate that flgains much greater than 100 should be possible with a ssitivity greater than 1027 F0/AHz. The large signal responseof the S-SQUID has been analyzed and measured in the abatic limit with excellent agreement obtained. The substatial second harmonic gain shows that the S-SQUID canused in a flux-locked loop system.The authors are grateful for helpful discussions witPeter Ha¨nggi and thank Sue Coppersmith for bringing thisubject to our attention.10 This work is supported by the U.S.Office of Naval Research.1J. Clarke, in The New Superconducting Electronics, edited by H.Weinstock and R. W. Ralston~Kluwer, Boston, 1993!, Chap. 5, pp. 123–180.2B. McNamara and K. Wiesenfeld, Phys. Rev. A39, 4854~1989!.3P. Jung and P. Ha¨nggi, Phys. Rev. A44, 8032~1991!.4P. Jung, Phys. Rep.234, 175 ~1993!, and references therein.5S. Han, J. Lapointe, and J. Lukens, inActivated Barrier Crossing, 1st ed.,edited by G. Fleming and P. Ha¨nggi ~World Scientific, Singapore, 1993!,Chap. 9, pp. 241–267.6J. Lapointe, Ph.D. dissertation, State University of New York at StonBrook, Department of Physics, 1993.7S. Han, J. Lapointe, and J. E. Lukens, Phys. Rev. Lett.63, 1712~1989!.8B. K. Sen, Ph.D. dissertation, State University of New York at StonBrook, Department of Physics, 1986.9The null in then53 response~Fig. 4! has recently been predicted as parof a more detailed analysis: R. Bartussek, P. Jung, and P. Ha¨nggi, Phys.Rev. E49, 3930~1994!.10Note added in proof. We thank A. R. Bulsara for sending us, priorpublication, his paper on stochastic resonance in SQUIDs: A. D. Hibbs,L. Singsaas, E. W. Jacobs, A. R. Bulsara, J. J. Bekkedahl, and F. MosAppl. Phys.~in press!.m-eheRouse, Han, and Lukensbject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:, 11 Sep 2014 16:05:57

Flux amplification using stochastic superconducting quantum interference devices

Crossref

https://kuscholarworks.ku.edu/bitstream/1808/17491/1/HanSiyuan_APL_1995%2866%29108.pdf

Flux amplification using stochastic superconducting quantum interference devices

Abstract

Similar works

Full text

Available Versions

KU ScholarWorks

Crossref