We consider the dynamical behavior of a strongly hysteretic SQUID ring coupled to a radio frequency resonator. By experiment we show that this system can display novel multiple level structures in its rf voltage-current characteristics which are solutions of the nonlinear equations of motion describing the system

E. Ben-Jacob

H. Prance

J. E. Zimmerman

J. F. Ralph

K. K. Likharev

L. Gammaitoni

M. Everitt

M. P. Soerensen

O. V. Lounasmaa

R. J. Prance

R. Whiteman

S. Al-Khawaja

T. D. Clark

V. Schöllmann

W. C. Schieve

R.J. Prance (7170587)

R. Whiteman (7170590)

T.D. Clark (7170353)

H. Prance (7170593)

V. Schollmann (7170596)

J.F. Ralph (7170359)

S. Al-Khawaja (7170599)

Mark Everitt (1250502)

Loughborough University Institutional Repository

   This item was submitted to Loughborough’s Institutional Repository (https://dspace.lboro.ac.uk/) by the author and is made available under the following Creative Commons Licence conditions.      For the full text of this licence, please go to: http://creativecommons.org/licenses/by-nc-nd/2.5/  VOLUME 82, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 28 JUNE 1999Nonlinear Multilevel Dynamics of a Coupled SQUID Ring–Resonator Systemin the Hysteretic RegimeR. J. Prance, R. Whiteman, T. D. Clark,* H. Prance, V. Schöllmann, J. F. Ralph, S. Al-Khawaja, and M. EverittSchool of Engineering, University of Sussex, Brighton, Sussex BN1 9QT, United Kingdom(Received 16 October 1998)We consider the dynamical behavior of a strongly hysteretic SQUID ring coupled to a radio frequencyresonator. By experiment we show that this system can display novel multiple level structures in its rfvoltage-current characteristics which are solutions of the nonlinear equations of motion describing thesystem. [S0031-9007(99)09459-4]PACS numbers: 85.25.Dq, 74.50.+rThe usefulness of the SQUID magnetometer [here, aweak link enclosed by a thick superconducting ring in-ductively coupled to a parallel LC resonant (tank) circuit–Fig. 1(b)] is well known [1]. Its sensitivity relies on theF0  h2e periodic response to an applied magneticflux Fx . This response is manifestly nonlinear and cangive rise to chaotic behavior [1–4] which can adversely af-fect the performance of SQUIDs as magnetometers. How-ever, the nonlinear dynamics of SQUID circuit systemsmay also generate other very interesting and potentiallyuseful behavior. An example is stochastic resonance [5]which can occur where multiple stable states coexist. Inthis paper we show that a new class of multiply coexist-ing dynamical states can develop in a driven SQUID ring–radio frequency (rf) tank circuit system when the ring isoperating in the strongly hysteretic regime.SQUID ring regimes of behavior can be categorizedin terms of the parameter b  2pLIcritF0, where Icritis the weak link critical current and L is the ringinductance. Thus, the inductive b # 1 and hystereticb . 1 regimes correspond to the ring screening currentIs, or total included magnetic flux F, being either singleor multivalued in F1x . In the hysteretic regime, IsFxconsists of a set of approximately diamagnetic brancheswith extremal values on any branch set by Icrit. In theabsence of noise, jumps between adjacent branches occurat 6Icrit so that for excursions Fx . LIcrit hysteresisloops are traced out in IsFx. These loops form the basisof the hysteretic SQUID magnetometer [1,6] where the(time dependent) flux Fxt is supplied via a coupled tankcircuit, typically resonant at rf. Ordinarily, it is assumedthat the ring stays at, or extremely close to, the minimumin its potential [1]UF,Fx F 2 Fx22L2IcritF02pcosµ2pFF0∂,so following Fxt adiabatically. In this situation thering-tank circuit dynamics (rf tank circuit voltage Voutvs rf input current Iin) are of the standard hystereticSQUID magnetometer form, i.e., a set of steplike featuresin Vout vs Iin, periodic in Iin, with a current interval setby F0L. In general, as successive pairs of hysteresis0031-90079982(26)5401(4)$15.00loops in IsFx are swept out with increasing Fxrf,a new step is generated in Vout vs Iin. Each step ismodulated (modulo F0) by a static or quasistatic appliedflux Fxqstat. We will refer to the Vout vs Iin forFxqstat  nF0 or n 1 12F0, n integer, as the extremalcharacteristics. In the standard treatment a comparisonof the rf voltages at which the first step starts on theseextremal characteristics (the breakpoints) yields [6] Icritand hence b. It was recognized early on [7] that theadiabatic constraint on ring-tank circuit dynamics canbreak down, at least when the circuit parameters lead tounderdamping. With this constraint removed, the ringneed not necessarily follow the minimum in UF,Fxas Fxt varies, implying that the simple hysteresis loopmodel may no longer provide an accurate picture for Ist.As we shall see, the lifting of the constraint at large bvalues leads to quite different dynamical characteristics tothose usually observed.As we have recently demonstrated [8], improved rfreceivers have helped us uncover new inductive modering-tank circuit dynamics. We have now used thesereceivers to probe the hysteretic regime. In particular, wehave concentrated on exploring the phase space for whichb is large (i.e., b  30 50). The rf electronics, usedto make rf diode detected plots of Vout vs Iin, consistedof a low noise, 4.2 K GaAsFET preamplifier (gain closeto 20 dB) and a low noise, room temperature receiver[8] [see Fig. 1(b)]. The SQUID rings studied were ofthe Zimmerman two hole type [9], fabricated in niobium,in which the enclosed weak link was a niobium pointcontact, allowing us to access a wide range of b values.In general, large b dynamical characteristics do not dis-play the familiar (adiabatic) steps in Vout vs Iin. Instead,gross step structures develop on Vout and Iin scales largecompared with those observed using standard SQUIDmagnetometers [1,6]. Furthermore, for certain rangesof ring parameters, these large b characteristics containnovel, and reproducible, multilevel structures. An experi-mental, diode detected example is shown in Fig. 1(a)(taken at 4.2 K with Fxqstat  nF0, a measured systemquality factor Q  500, and a coupling constant squaredK2  M2LtL  0.008 for tank circuit inductance Lt© 1999 The American Physical Society 5401VOLUME 82, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 28 JUNE 1999and ring-tank circuit mutual inductance M). As is com-mon practice, this characteristic was recorded at the nF0bias frequency frf (here 23.02 MHz) equal to the resonantfrequency for the coupled system when Fxqstat  nF0.As can be seen, the gross structure in Fig. 1(a) containsfour distinct sublevels. However, as we shall see, thisnumber is dependent on the circuit parameters of the cou-pled system. At larger amplitudes of Iin (not shown inFig. 1) the gross structure is repeated periodically in Iinalthough after the first structure those which follow mayFIG. 1. Large b, diode detected, 4.2 K dynamics; ( frf 23.02 MHz, Q  500, K2  0.008) at Fxqstat  nF0 andn 1 12F0, n integer. Insets in (a) 4.2 K adiabatic dynamicsat Fxqstat  nF0 and n 1 12F0 with b just less than one( frf  23.02 MHz, Q  500, K2  0.008) and in (b) systemblock diagram.5402be very smeared out. With sufficient resolution these pe-riodically repeated gross structures are always found tocontain the same number of sublevels, albeit less distinctthan those plotted in Fig. 1(a). When Fxqstat is changedfrom nF0 to n 1 12F0 the sublevels move in Voutand the number is reduced by one [here, from 4 to 3—seeFig. 1(b)]. This was the case for all the large b, sublevelstructure characteristics we studied. We note that froman adiabatic viewpoint the characteristics in Figs. 1(a)and 1(b) would appear to display multiple critical cur-rents, one for each matching pair of extremal breakpoints.However, as we shall see, these sublevel structures canbe modeled on the basis of just one critical current Icrit.We also note that any estimate of b involving gross struc-ture characteristics must rely on a best fit choice of Icrit tocalculate the system dynamics.For comparison with the Fig. 1 dynamics, we showinset standard (adiabatic) 4.2 K SQUID magnetometercharacteristics taken using the ring-tank circuit sys-tem of Fig. 1. Here, with all other circuit parametersheld the same, the weak link was readjusted to makeb very close to one. Clearly, compared with thesecharacteristics the large b dynamics display sublevelswhich are very extended along the Iin axis. Each ofthese sublevels has the same separation in Vout as thestandard characteristics, implying a system responseF0 periodic in Fxqstat, as is observed experimentally.We also note that the length of the sublevels in thegross step structures is always an integer multiple ofthe step length seen on the adiabatic characteristics.We have recorded a whole range of these gross struc-tures with varying numbers of sublevels, from a mini-mum of 2 (at Fxqstat  nF0, T  4.2 K) to a maximumof 13 (Fxqstat  nF0, T  1.9 K). To add to Fig. 1 weshow in Fig. 2 another example of these structures, takenfor a different ring-tank circuit system where Q  500,K2  0.004, frf  20.019 MHz, and T  4.2 K. Here,with the two extremal characteristics overlapped [nF0—filled circles, n 1 12F0—open circles], the first twogross structures can be seen, both displaying three nF0or two n 1 12F0 sublevels.It is clear in Fig. 1 that stochastic jumps can occurbetween sublevels. These jumps (with submilliseconddwell times in any particular sublevel) presumably arisebecause of the classical noise in the system. Since thering-tank circuit coupling is weak, with the latter driven bya 4.2 K source, it is reasonable to assume that this noisecomes from the tank circuit itself. The jumps can be betterappreciated by examining the time domain record of thesublevels in Fig. 1, as shown in Fig. 3(a). Here, with thepeak Iin set as arrowed in Fig. 1(a), and Fxqstat  nF0,we see apparently random time jumping between a set ofequally spaced levels. The phase space portrait of theseprocesses is revealing. In Fig. 3(b) we show this (pseudo)phase portrait for the continuing data set of Fig. 3(a).As is customary, we plot Voutt against the time shiftedVOLUME 82, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 28 JUNE 1999FIG. 2. Large b, diode detected 4.2 K characteristics showingthree and two sublevels, respectively, at Fxqstat  nF0 andn 1 12F0 ( frf  20.019 MHz, Q  500, K2  0.004).Voutt 1 t at all times t, where t is a suitable time delay(1024 sec in this example). We note that the numberof return points (nodes), where the ellipses intercept onthe phase portrait, is equal to the number of levels, asit should be. We also note that if t ! 0 the ellipses inFIG. 3. (a) Diode detected, time domain plot of the levelvoltages [at the peak Iin arrowed in Fig. 1(a)] showingtransitions (b) V t vs V t 1 t for the time domain voltagelevels of (a) where t  1024 sec.Fig. 3(b) close down to a point, while if t ! ` theseellipses become completely washed out.In modeling the structure typified by Fig. 1, we haveused the resistively shunted junction plus capacitanceRSJ 1 C description of a weak link [1,2]. Given thatthe Q is large 500, the resonant frequency of the baretank circuit (capacitance Ct , parallel resonance resistanceRt) is given by fR  vR2p  12ppLtCt , with Rt QvRLT . The magnetic fluxes, F in the ring (ringcurrent Is) and w in the tank circuit inductor (inductorcurrent ILt ), are then F  IsL 1 MILt and w  ILtLt 1MIs. Current conservation in the two electrically isolatedparts of this coupled system implies that ICt 1 IRt 1ILt  Iint and IC 1 IR 1 IJ 1 IL  0, where IJ Icrit sin2pFF0 is the weak link Josephson current, thesubscripts Ct , Lt , Rt , C, L, and R refer, respectively,to the currents flowing in the various elements of thetank circuit and SQUID ring, and Iint may include bothcoherent and noise terms [4,8]. Expressing the circuitcurrent elements above in terms of magnetic fluxes leadsto the coupled equationsCtd2wdt211Rtdwdt1wLt1 2 K2 Iint1mFL1 2 K2,(1)Cd2Fdt211RdFdt1 Icrit sinµ2pFF0∂1FL1 2 K2mwL1 2 K2,(2)where m  MLt is the fraction of rf flux w coupledbetween the tank circuit and the ring. In general, a biasflux Fxqstat is also applied. For convenience we assumethat the current generating this is fed through the rf coil(i.e., m is the same for both the fluxes seen by the ring).In the RSJ 1 C model we can write down charac-teristic times tsqcr  CR and tsqlc  2ppLC for thering and tt  1fR for the tank circuit. For parametervalues typically invoked in this model (e.g., C  10212 to10213 F, L  a few 3 10210 H, and R  1 to 10 V),and fR  20 MHz, tsqcr and tsqlc can be 3 or moreorders of magnitude smaller than tt . This means that(1) and (2) must be handled carefully if accurate dy-namical solutions for the system are to be found. Forthis reason we have made no simplifying assumptions.In particular, we do not neglect the derivative terms in(2). We solve the coupled equations using fourth orderRunge-Kutta with an adaptive step size algorithm and asufficient number of steps per rf cycle to follow the os-cillations in the SQUID ring accurately. Typically, wefind that we can obtain reliable solutions when our inte-grations are taken over a few thousand steps per rf cycle.5403VOLUME 82, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 28 JUNE 1999To make comparison with experiment we need to com-pute Vout vs Iin at a particular value of rf frequency. It iscommon experimental practice in the hysteretic regime toadjust the rf so that jVoutj vs Iin. For relatively large ris-ers (where b ¿ 1) this means driving at, or very closeto, the system resonant frequency at which the ring ismost diamagnetic, i.e., at the Fxqstat  nF0 bias fre-quency frf where Is is almost linear in Fx . We referto this as the pseudoresonant frequency [4,8] given byfR1 2 K2xnF01 2 K2	12, with a ring magneticsusceptibility xnF0  ≠IsnF0≠Fx  21.As we have shown [10], we can estimate C if we knowIcrit and the Fermi velocity of the ring material (niobium).Thus, for Icrit  50 mA, Cestimated is close to 10213 F,a value considered typical for high critical current pointcontact SQUID rings. Within the RSJ 1 C model therange of R can be arrived at by measuring the dc voltage-current characteristics of singly connected point contactweak links, e.g., for Icrit  50 mA, R is typically 10 V.Accordingly, for a best fit to the data of Fig. 1, Eqs. (1)and (2) were solved for Icrit  43 mA,C  10213 F, withR taken as an adjustable parameter between 5 and 30 V.The other ring-tank circuit parameters were set as for theFIG. 4. Computed Vout vs Iin (C  10213 F, R  19 V 14 K noise) at (a) Fxqstat  nF0 and (b) n 1 12F0 for thecircuit parameters of Fig. 1.5404experimental dynamics of Fig. 1. For accuracy a sim-ulated 4 K noise source [4,8], derived from a randomnumber generator, was included in Iin. The computed bestfit extremal characteristics (for R  19 V) are shown inFig. 4. Here, it can be seen that (i) for these chosen valuesof C and R at least, the RSJ 1 C model predicts the exis-tence of sublevels, as indicated by earlier computations [7],(ii) the number of sublevels in each gross structure changeswithFxqstat in the manner observed experimentally, with agood match to the data of Fig. 1, and (iii) stochastic jumpsare generated between the sublevels which follow closelythose observed experimentally. Extensive computationshave demonstrated that sublevels are generated only whenIcrit is large b ¿ 1. Its value, in turn, dictates the rangeof C and R. We note that by computation the maximumnumber of levels generated so far is 15, strictly comparablewith the maximum of 13 detected experimentally.In this paper we have demonstrated the existenceof multiple levels in the dynamical response of anexperimental rf SQUID magnetometer in the stronglyhysteretic regime. These levels, F0 periodic in Fx , arein very good agreement with the theoretical predictions ofthe full nonlinear RSJ 1 C model for the coupled system.Our results represent a significant improvement in theexperimental sensitivity and, together with the predictionsof the model, clearly expose a new and potentially veryrich field of nonlinear behavior in an extremely interestingsystem.*Email address: t.d.clark@sussex.ac.uk[1] K. K. Likharev, Dynamics of Josephson Junctions andCircuits (Gordon and Breach, Sidney, 1986).[2] W. C. Schieve, A. R. Bulsara, and E. W. Jacobs, Phys.Rev. A 37, 3541 (1988).[3] M. P. Soerensen, M. Barchelli, P. L. Christiansen, andA. R. Bishop, Phys. Lett. 109A, 347 (1985).[4] T. D. Clark, J. F. Ralph, R. J. Prance, H. Prance,J. Diggins, and R. Whiteman, Phys. Rev. E 57, 4035(1998).[5] L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni,Rev. Mod. Phys. 70, 1 (1998).[6] O. V. Lounasmaa, Experimental Principles and Methodsbelow 1 K (Academic Press, London, 1974), pp. 156–159.[7] E. Ben-Jacob and D. Abraham, Appl. Phys. Lett. 39, 835(1981).[8] R. Whiteman, J. Diggins, V. Schöllmann, T. D. Clark, R. J.Prance, H. Prance, and J. F. Ralph, Phys. Lett. A 234, 205(1997).[9] J. E. Zimmerman, P. Thiene, and J. T. Harding, J. Appl.Phys. 41, 1572 (1970).[10] J. F. Ralph, T. D. Clark, J. Diggins, R. J. Prance,H. Prance, and A. Widom, J. Phys. Condens. Matter 9,8275 (1997).

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Nonlinear multilevel dynamics of a coupled SQUID ring-resonator system in the hysteretic regime

Robert Prance (4460476)

R Whiteman (16059449)

T D Clark (16059452)

H Prance (16059455)

V Schollmann (16059458)

J F Ralph (16059461)

S Al-Khawaja (16059464)

Mark Everitt (16059467)

Sussex Research Online

Non-linear Multi-level Dynamics of a SQUID Ring-resonator System in the Hysteretic Regime

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Physical Review Letters

Nonlinear Multilevel Dynamics of a Coupled SQUID Ring–Resonator System in the Hysteretic Regime

http://dx.doi.org/10.1103/PhysRevLett.82.5401

Non-linear Multi-level Dynamics of a SQUID Ring-resonator System in the Hysteretic Regime

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