We investigate the roto-breathers recently observed in experiments on
Josephson ladders subjected to a uniform transverse bias current. We describe
the switching mechanism in which the number of rotating junctions increases. In
the region close to switching we find that frequency locking, period doubling,
quasi-periodic behaviour and symmetry breaking all occur. This suggests that a
chaotic dynamic occurs in the switching process. Close to switching the induced
flux increases sharply and clearly plays an important role in the switching
mechanism. We also find three critical frequencies which are independent of the
dissipation constant $\alpha$, provided that $\alpha$ is not too large

Giles, Richard T

Kusmartsev, Feodor V

English

arXiv

We investigate the roto-breathers recently observed in experiments on Josephson ladders subjected to a uniform transverse bias current. We describe the switching mechanism in which the number of rotating junctions increases. In the region close to switching we find that frequency locking, period doubling, quasi-periodic behaviour and symmetry breaking all occur. This suggests that a chaotic dynamic occurs in the switching process. Close to switching the induced flux increases sharply and clearly plays an important role in the switching mechanism. We also find three critical frequencies which are independent of the dissipation constant $\alpha$, provided that $\alpha$ is not too large

Richard Giles (1251693)

Feodor Kusmartsev (1251207)

Loughborough University Institutional Repository

arXiv:cond-mat/0004078 v1   5 Apr 2000Switching and symmetry breaking behaviour of discretebreathers in Josephson laddersRichard T Giles and Feodor V KusmartsevDepartment of Physics, Loughborough University, Loughborough, LE11 3TU, U.K.April 4, 2000We investigate the roto-breathers recently ob-served in experiments on Josephson ladders sub-jected to a uniform transverse bias current. We de-scribe the switching mechanism in which the num-ber of rotating junctions increases. In the regionclose to switching we find that frequency locking, pe-riod doubling, quasi-periodic behaviour and symme-try breaking all occur. This suggests that a chaoticdynamic occurs in the switching process. Close toswitching the induced flux increases sharply andclearly plays an important role in the switchingmechanism. We also find three critical frequencieswhich are independent of the dissipation constantα, provided that α is not too large.The recent discovery[1, 2] of discrete breathers inJosephson ladder arrays (Fig. 1) driven by a trans-verse dc bias current has shown not only that theselocalised excitations exist but that they exhibit re-markable behaviour, which is the subject of thispaper.The Josephson ladder has been studied theoret-ically for many years. In the absence of a driv-ing current the interaction between vortices hasbeen found to be exponential and this leads tothe vortex density exhibiting a devil’s staircaseas the magnetic field is increased[3, 4]. Quan-tum fluctuations[5], meta-stable states[6] and in-ductance effects[7] have been studied as has theinteraction of vortices with a transverse dc biascurrent[8, 9].In the case of a Josephson ladder, a roto-breatheris a stable group of vertical junctions rotating to-gether (θ′j − θj ≈ ωt, see Fig. 1). Until recently,the interest has focussed on the case of a sinusoidalbias current, where it has been shown that roto-breather solutions exist and that a discrete breathermay even include a chaotic trajectory[10].Roto-breathers should also be stable[11] in theexperimentally easier case of a ladder array sub-jected to a uniform dc transverse driving current.The case of rotation occurring at a single verticaljunction has been studied numerically[12, 13]. Twoexperimental groups[1, 2] have now independentlyconfirmed the existence of such solutions and dis-covered further unpredicted behaviour. In an an-nular ladder[1] (Fig. 1a) and in a linear ladder[2](Fig. 1b) breathers were observed with variousnumbers of “vertical” (i.e. radial) junctions rotat-ing. However the most interesting features of theobservations were the switching between breathersand their symmetry breaking behaviour. Once abreather has been initialised it is stable, but if thebias current is slowly decreased then, at some crit-ical value of the current, the number of rotatingjunctions switches to a larger number and a newbreather is formed. Furthermore, the new breathermay have a different centre of symmetry from theold breather despite the symmetry of the ladder it-self and the symmetry of the driving currents abouta single vertical junction; this is particularly re-markable in the case of the annular ladder wheretranslational invariance should be exact. The pro-duction of breathers with an even number of ro-tating junctions is, by itself, a demonstration ofsymmetry breaking. The symmetry breaking effectwas not mentioned by the experimenters, presum-ably because it could be due to experimental im-perfections. However we show that it should alsoarise in a perfect experiment, the apparent mech-anism being the occurrence of chaotic dynamics in1the switching region.First we construct a new model, similar tobut significantly different from those proposedpreviously[12, 13, 14, 10]. The current through ajunction is determined by the RCSJ modelIIc=d2ϕdt2+ αdϕdt+ sinϕ (1)where Ic is the critical current andϕ = ∆θ −2piΦ0∫A.dl (2)where ∆θ is the change in superconducting orderparameter θ across the junction and A is the vec-tor potential. The “vertical” (i.e. radial) junc-tions may differ in area from the “horizontal” junc-tions by an anisotropy parameter η = Ich/Icv =Ch/Cv = Rv/Rh where Ich (Icv), Ch (Cv) and Rh(Rv) are, respectively, the critical current, capaci-tance and resistance of a horizontal (vertical) junc-tion. From Fig. 1a we see that there are three un-knowns per plaquette: θj , θ′j and fj , where fj is thetotal flux threading the jth plaquette. To solve forthese unknowns we construct three equations perplaquette as follows. The first two equations areobtained from current conservation at the top (in-ner) and bottom (outer) rails. The third equationis obtained by making the approximation that theinduced flux fj−fa (where fa is the applied flux) isproduced solely by the currents flowing around theimmediate perimeter of the jth plaquette (Fig. 1c):fj − faΦ0=βL8pi(Ivj + Ihj − Ivj+1 − I′hj)(3)where βL is an inductance parameter. This lastequation makes the plausible assumption that theplaquettes are more or less square and that it is onlythe currents flowing around the plaquette that pro-duce the induced field. This differs from, and forsquare plaquettes is more accurate than, the com-mon assumption[13, 12] that the induced field isproportional to the loop (or “mesh”) current circu-lating the plaquette.We impose the initial conditions, mimicing theexperiments, that at t = 0, θj = θ′j = 0, fj = 0 andIj = 0 for all j. This means that θj + θ′j = 0 forall time, i.e. there is a symmetry between the innerand outer rails. Using Landau gauge we then havethe following pair of coupled differential equationsfor each plaquette:{ddt2+ αddt}(−θ−j−1 + 4θ−j − θ−j+1)= 2Ij + sin θ−j−1−4 sin θ−j + sin θ−j+1 +8piβL(fj−1 − fj) (4){ddt2+ αddt}(−2piηfj + (1− η)(θ−j+1 − θ−j ))= sin θ−j− sin θ−j+1 +8piβL(fj − fa) + 2η sinχ−j (5)where θ−j = θ′j − θj and χ−j =12(θ−j+1− θ−j +2pifj).We now focus on determining whether or not ourmodel exhibits the interesting switching and sym-metry breaking behaviour observed in the data ofBinder et al[1]. All parameter values are chosento mimic the experimental setup i.e. η = 0.44,βL = 2.7, α = 0.07 and fa = 0. LetIj ={IB + I∆ if j = 0IB otherwise(6)where IB is called the bias current. Again follow-ing the experiment (Fig. 1a), we slowly increaseI∆ while keeping IB = 0 until rotation starts atsite j = 0. I∆ is then slowly decreased while at thesame time increasing IB to keep I0 constant. WhenIB has reached the desired value it is then heldfixed while I∆ is slowly reduced to zero. Finally IBis slowly reduced while keeping I∆ = 0. Note thatthe dynamical equations, initial conditions and in-jected currents are exactly symmetrical about sitej = 0. One would expect only solutions which arealso symmetrical about j = 0. However, like the ex-periments, the simulations also produce breatherswhich are not symmetrical about j = 0. Fig. 2shows how the number NR of rotating junctionschanges as the bias current IB is slowly reducedfrom various starting values. In agreement withexperiment, breathers with even NR are commonlyproduced (these cannot be symmetrical about j =0), and NR switches to larger and larger values un-til eventually all junctions are rotating. While wehave found that similar behaviour occurs in a pre-viously published model[13, 12], our model givesconsiderably better agreement with the experimen-tally observed switching currents, thus indicating2the importance of inductance effects in the switch-ing mechanism. We believe the origin of the sym-metry breaking is the occurrence of chaotic dynam-ics in the switching region.The chaotic nature of the switching is furthersuggested by the fact that when the same computercode which produced Fig. 2 is performed on a differ-ent computer (different floating point processor) wesee significant changes in the switching behaviourof nearly all trajectories although the overall pat-tern remains identical. The minute differences inthe handling of floating point numbers are ampli-fied in the chaotic regime to produce significantlyaltered trajectories.Fig. 3 shows the voltage-current characteristicsobtained from the same simulations used in produc-ing Fig. 2. At high frequencies (V = 〈dθ−/dt〉 >4.3) most of the breathers show purely resistive be-haviour (i.e. V = 〈dθ−/dt〉 = IBR), the value ofthe resistance R increasing with NR according toan expression deduced from experiment[1]:αR ≈ 1/(1 + η/NR) (7)This is the relationship expected if the currentthrough each rotating vertical junction is αV andthe four horizontal junctions surrounding the ro-tating region each carry current 12ηαV (to satisfyKirchoff’s rule). For NR = 1 this expression wasderived in ref. [13]. We find that no such resistivebehaviour occurs for V < 4.3.A typical example of a resistive breather (faraway from any switching region) is shown in Fig. 4.Although the breather is not symmetric about j =0 it shows exact symmetry about the midpoint ofthe rotating region and also appears to be exactlyperiodic, the period being two revolutions of a ver-tical junction (we call this “period 2”). The non-rotating junctions oscillate, the amplitude decreas-ing exponentially with distance from the rotatingregion. Far away from switching the magnitude ofthe flux fj is everywhere small (< 0.1) and is lim-ited to the edges of the rotating region.Fig. 3 shows that there are at least two criticalfrequencies, 〈dθ−/dt〉 = 6.5 and 〈dθ−/dt〉 = 5.0, atwhich breathers become unstable and switching oc-curs. As the current is reduced the frequency fallsuntil it reaches the critical value, at which pointthe breather becomes unstable and switches to alarger breather with a larger resistance (Eq. (7))and therefore a higher frequency. The processthen repeats as the current continues to be rampeddown. We identify the larger of these two criticalfrequencies with that observed experimentally[1].Although, at fixed IB , the frequency of rotation de-pends on the dissipation constant α, we find thatthese two critical frequencies are more or less in-dependent of α, as is the critical frequency belowwhich breathers cease to show constant resistance.This of course breaks down when the bias currentrequired to achieve the upper critical frequency ex-ceeds unity, i.e. when α > 0.15.Note also that in the above simulations: (i) nobreathers are seen for IB > 0.85, (ii) only the sin-gle site breather is seen for 0.68 < IB < 0.85 and(iii) all rotating junctions normally stop togetherwhen the current IB is reduced below 0.17, al-though sometimes a new breather containing onlya few rotating junctions is produced which is thendestroyed when IB falls below 0.14.As well as resistive period 2 breathers we havefound frequency-locked breathers. In fact the usualNR = 1 resistive breather makes a transition toa frequency-locked breather as the bias current isreduced below IB = 0.699. At this point theperiod doubles, and the breather becomes asym-metric and jumps to a higher rotation frequency〈dθ−/dt〉 = 6.32. At IB = 0.6895 the period dou-bles again and at IB = 0.6892 it becomes quasi-periodic (Fig. 5). Switching to multi-site breathersoccurs at IB ≈ 0.6891. Such frequency lockedbreathers have not been reported experimentallybut have probably been overlooked as they are onlystable in narrow current intervals.While the maximum flux fmax is normally small(fmax < 0.1), it sharply increase at switching tofmax ∼ 1. It would appear that an important partof the switching mechanism is the instability causedby having a large flux concentrated in a small area.The importance of the flux in the switching processis further confirmed by the fact that when IB hasfinally been reduced all the way to zero we find thatthe ladder may contain one or more stable vortex-antivortex pairs.The occurrence of frequency locking, perioddoubling, quasi-periodic behaviour and symmetrybreaking as switching is approached strongly sug-gests that the origin of the symmetry breaking isthe occurrence of chaotic dynamics in the switch-ing region. From the theory of chaotic dynamics3it is known that two or more coupled Josephsonjunctions (or, equivalently, two or more coupledpendula) may exhibit three types of motion: os-cillatory, rotational and chaotic. Chaotic motionarises for particular initial conditions. An anal-ogous situation arises in non-linear coupled lat-tices which also display both breathers and chaoticdynamics[15]. In a large array of coupled Joseph-son junctions (or, analagously, an array of coupledpendula) we must also expect these three types ofmotion. While roto-breathers are the most charac-teristic stable solutions it appears that the switch-ing between breathers occurs only in the chaoticregime. Of course, the chaotic dynamics has aninfluence on the selection of breather type. Anysmall difference in the initial conditions leads to acompletely different breather. Thus, in this chaoticregime, any small experimental imperfections orsmall inaccuracies in a computer simulation maylead to different chaotic trajectories and hence dif-ferent roto-breathers. Symmetry breaking arises ifthe perturbation itself breaks symmetry.We conclude that our model exhibits most of themain features [16] of the roto-breathers recently ob-served in Josephson ladder arrays, sheds light onthe switching mechanism and predicts further ob-servable behaviour. The occurrence of frequencylocking, period doubling, quasi-periodic behaviourand symmetry breaking in the switching region sug-gests that switching occurs in the chaotic regime.We find that the maximum flux increases sharplyin the switching region and that the flux clearlyplays an important role in the switching mecha-nism. We also find two critical switching frequen-cies and a critical frequency below which no resis-tive behaviour is observed. All three frequenciesare approximately independent of the dissipationconstant α (for α < 0.15),We are grateful to A.V. Ustinov and P. Binder forkindly giving us their data and explaining it priorto publication. We are also grateful for discussionswith M.V. Fistul and S. Flach and for the hospi-tality of the Max-Planck-Institut, Dresden and theUniversita¨t Erlangen-Nu¨rnberg.References[1] P. Binder et al., Phys. Rev. Lett. 84, 745(2000).[2] E. Tr´ias, J. Mazo, and T. Orlando, Phys. Rev.Lett. 84, 741 (2000).[3] M. Kardar, Phys. Rev. B 30, 6368 (1984).[4] R. Giles and F. Kusmartsev, J. Low Temp.Phys. 117, 623 (1999).[5] M. Kardar, Phys. Rev. B 33, 3125 (1986).[6] J. Mazo, F. Falo, and L. Flor´ia, Phys. Rev. B52, 10433 (1995).[7] J. Mazo and J. Ciria, Phys. Rev. B 54, 16068(1996).[8] S. Kim, Phys. Lett. A 235, 408 (1997).[9] M. Barahona, S. Strogatz, and T. Orlando,Phys. Rev. B 57, 1181 (1998), and referencestherein.[10] P. Martinez, L. Floria, F. Falo, and J. Mazo,Europhys. Lett. 45, 444 (1999).[11] R. MacKay and J.-A. Sepulchre, Physica D119, 148 (1998).[12] J. Mazo, E. Tr´ias, and T. Orlando, Phys. Rev.B 59, 13604 (1999).[13] S. Flach and M. Spicci, J. Phys.: Condens.Matter 11, 321 (1999).[14] S. Kim, Phys. Lett. A 229, 190 (1997).[15] S. Flach and C. Willis, Phys. Rep. 295, 181(1998).[16] Ref. [1] finds that breathers with even NR aremost common in the annular ladder. In agree-ment with this we find that, when switching,breathers with NR = 2 or NR = 4 have astrong tendency to grow symmetrically thusmaintaining even NR. No such effect is seenfor breathers with NR = 1 or NR = 3. Thisis strongly suggestive of a parity effect, i.e. adifference in behaviour according to whetherNR is even or odd.4θjθ′j+1θ′j-1θ′j-2θ′j+2θ′jθj-1θj-2 θj+1θj+2fjfj+1fj+2fj-1fj-2fj-3IjIj-1Ij-2Ij+1Ij+2(a)IB(b)Ivj+1Ihj(c)I′hjIvjFigure 1: (a) An annular ladder subjected to a uni-form transverse bias current IB. Each circle repre-sents a superconducting island. Each link betweenislands represents a Josephson junction. θ is thephase of the superconducting order parameter andf is the flux threading a plaquette. (b) A linearladder. (c) Explanation of the notation used inEq. (3).2468101214160.2 0.4 0.6 0.8Bias current IBNumberofrotatingjunctions,NRFigure 2: Result of smoothly ramping down thebias current. Each small circle marks the start of anew ramp.0.60.70.80.914 5 6 7 8 9Voltage V =〈dθ−dt〉Resistance,αRNR = 1NR = 2NR = 366 Frequency locking?? Switching frequenciesFigure 3: Resistance αR (where R = V/IB) plottedagainst voltage V = 〈dθ−/dt〉 for the data shown inFig. 2. Note the critical switching frequencies andfrequency locked states.5161001610216104–8–6–4–2 0 2 4 602468tjdθ−jdtFigure 4: An example of a resistive breather farfrom switching. dθ−j /dt is plotted against t for all j.Note that six junctions (−2 ≤ j ≤ 3) are rotatingand that the breather is not symmetrical about theinjection site j = 0. The breather is periodic andexactly symmetrical about its centre. The peakflux (not shown) is small (∼ 0.06).198001980219804198061980819810–8–6–4–2 0 2 4 60246tjdθ−jdtFigure 5: The quasi-periodic breather (approxi-mately period 8) which occurs near IB = 0.6892,just above the bias current at which switching tomulti-site breathers occurs. It is frequency-lockedat 〈dθ−/dt〉 = 6.32. Note that dθ−j /dt is nearlyanti-symmetric about j = 0. The maximum flux(not shown) is large (0.5) and increases still furtherat switching.6

Switching and symmetry breaking behaviour of discrete breathers in Josephson ladders

https://repository.lboro.ac.uk/articles/Switching_and_symmetry_breaking_behaviour_of_discrete_breathers_in_Josephson_ladders/9411815/files/17029649.pdf

Switching and symmetry breaking behaviour of discrete breathers in Josephson ladders

Abstract

Similar works

Full text

Available Versions

Loughborough University Institutional Repository