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Thermal depinning of fluxons in discrete Josephson rings

J. J. Mazo,1, 2 F. Naranjo,1, 2, 3 and K. Segall4

1Dpto. de Fısica de la Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza, Spain2Instituto de Ciencia de Materiales de Aragon, C.S.I.C.-Universidad de Zaragoza, 50009 Zaragoza, Spain

3Universidad Pedagogica y Tecnologica de Colombia, Tunja, Colombia4Department of Physics and Astronomy, Colgate University, Hamilton NY 13346

(Dated: December 8, 2013)

We study the thermal depinning of single fluxons in rings made of Josephson junctions. Due tothermal fluctuations a fluxon can be excited from its energy minima and move through the array,causing a voltage across each junction. We find that for the initial depinning, the fluxon behaves asa single particle and follows a Kramers-type escape law. However, under some conditions this singleparticle description breaks down. At low values of the discreteness parameter and low values of thedamping, the depinning rate is larger than the single particle result would suggest. In addition, forsome values of the parameters the fluxon can undergo low-voltage diffusion before switching to thehigh-voltage whirling mode. This type of diffusion is similar to phase diffusion in a single junction,but occurs without frequency-dependent damping. We study the switching to the whirling state aswell.

I. INTRODUCTION

In past years many works have been devoted to thestudy of extended discrete nonlinear systems. On theone hand, it is important to deepen our knowledge ofgeneral properties of such systems since they often haveapplication to many different physical situations. On theother hand, many physical systems are well described bynonlinear discrete models. In this field, the emergenceof the concept of soliton for instance (in the continu-ous and its discrete counterparts) was paradigmatic.1,2

A well known example of model system supporting thistype of nonlinear excitations is the discrete Sine-Gordonequation (also called Frenkel-Kontorova model).3,4,5

A Josephson-junction (JJ) array is by construction adiscrete system made of interacting nonlinear solid statedevices. JJ arrays are an example of physical systemswith great fundamental and technological interest whichare well described by discrete nonlinear models.6,7 Fromthe experimental point of view, Josephson junctions area privileged place to study solitons and to explore theirpossible applications.8

Of the many different geometries for a JJ array the so-called Josephson ring (a set of JJ connected in paralleland closed forming a ring, see Fig. 1) is well described bya discrete sine-Gordon equation and supports nonlineardiscrete solitons or kinks, usually called fluxons in thiscontext.9 Thus, many studies of the discrete sine-Gordonequation or the role of kinks in nonlinear arrays havedirect application when studying JJ rings. Converselythe study and modelization of JJ rings allows explorationof new issues concerning the behavior of these nonlinearphenomena.

Our model system is the so-called Josephson ring, acollection of JJ coupled in parallel and forming a ring(see figure 1). When cooled below the superconductingcritical temperature an integer number of magnetic fluxquanta, fluxons, can be trapped in the ring. Then, thephysical properties of the array are dominated by the

presence of the fluxons in the system. The I-V curve ofthe array shows the mean voltage across the array as afunction of a constant external current applied to everyjunction of the array. When current is applied, the arrayremains superconducting up to a certain critical valuewhich defines the critical current of the array. In thepresence of fluxons this current corresponds to the fluxondepinning current. At this current, the fluxon starts tomove around the array and finite voltage is measured.

In many cases the energy exchange between the systemand the environment is relevant, so thermal fluctuationshave to be considered.7 Because of this, at finite temper-ature the value of the depinning current and the shape ofthe I-V curve can be strongly affected by thermal fluctu-ations. Motivated by recent experiments on the thermaldepinning and dynamics of fluxons (kinks) in small ringsmade of 9 junctions,10 in this work we numerically ex-plore some of these issues. In addition, in some casesthe fluxons can be understood as particles on a substrateperiodic potential.

The main object of this paper is to study numericallythe thermal depinning of a single fluxon in a JJ ringand compare it with numerical simulations and analyti-cal predictions for the case of a single particle. We have

FIG. 1: Left: Scheme of the JJ ring. Lines are for super-conducting wires and crosses for Josephson junctions. Right:Phase configuration profile (top) and magnetic flux (bottom)for a fluxon in JJ ring.

2

found excellent agreement in many cases. However, un-der some conditions the single particle description fails.In addition, for some values of the parameters the fluxoncan undergo low-voltage diffusion before switching to thehigh-voltage whirling mode. This type of diffusion is sim-ilar to phase diffusion in a single underdamped junction,but occurs without frequency-dependent damping.

II. EQUATIONS

Josephson junctions are made of two superconduct-ing materials separated by a thin insulating barrier.Driven by an external current, this system behaves asa solid-state nonlinear oscillator and is modeled by thesame dynamical equations that describe a driven pen-dulum:7 i = ϕ + Γϕ + sinϕ + ξ(τ). Here ϕ, the vari-able that describes the behavior of the junction, is thegauge-invariant phase difference of the superconductingorder parameter at both sides of the junction. In thisequation current is normalized by the junction criticalcurrent Ic and time by the junction plasma frequencyωp =

√

2πIc/Φ0C (Φ0 = h/2e is the magnetic flux quan-tum and C the junction capacitance). Γ is an impor-tant parameter which measures the dissipation in thesystem (Γ =

√

Φ0/2πIcCR2, with R the effective re-sistance of the junction). The last term, ξ(τ), describesthe effect of thermal noise in the dynamics (Johnson cur-rent noise) and satisfies 〈ξ(τ)〉 = 0 and 〈ξ(τ)ξ(τ ′)〉 =2ΓTδ(τ − τ ′) where we use T for a normalized temper-ature T = kBTexp/EJ (with EJ the Josephson energyEJ = Φ0IC/2π). The normalized dc voltage v whichgives the response of the system to the external cur-rent is defined by v = Vdc/IcR = (Φ0/2πIcR)〈dϕ/dt〉 =Γ〈dϕ/dτ〉.

As previously stated, the JJ ring consists of a series ofindividual junctions connected in parallel. Such systemcan be though of as a series of coupled pendula. Followingthe usual model for the system, the equations for an arraymade of N coupled junctions driven the same externalcurrent are given by:9

ϕj +Γϕj +sinϕj +ξj(τ) = λ(ϕj+1−2ϕj +ϕj−1)+ i (1)

Index j denotes different junctions and run from 1 to N .The new parameter λ accounts for the coupling betweenthe junctions which, in the framework of this model, oc-curs between neighbors and has an inductive characterλ = Φ0/2πIcL, with L the self inductance of every cellin the array. Boundary conditions are defined by thetopology of the array (here we consider circular arrays)and the number M of trapped fluxons in the system:ϕj+N = ϕj + 2πM .

In this article we will consider the case of a singlefluxon. We have studied different sizes for the array, buthere we will present results for an array made of 9 junc-tions, similar to those being experimentally studied. Wewill also study different values of λ and restrict our in-

terest to underdamped arrays biased by a dc current ina broad range of temperatures.

III. RESULTS

In this section we are going to present numerical simu-lations of the dynamics of one fluxon in a Josephson ringand one particle in a periodic potential. We will alsoshow numerical calculations from single particle thermalescape theory.

A. I-V curves: damping regimes

Figure 2 shows single I-V curves for one fluxon in a 9junctions Josephson ring with λ = 0.4. In this case thewidth of the fluxon is close to 2, so it is well localized inthe array and discreteness effects are important.11 Curveswere simulated at three different values of damping andfive temperatures. Current was increased from zero tosome maximum at an average ramp equal to 8

3× 10−7 in

normalized units.Let us look first at the T = 0 curves. If we start at

zero bias, in all the cases the fluxon is pinned to the arrayup to the so-called depinning current i0dep is reached (for

λ = 0.4, i0dep ≃ 0.155). Above this current very differentI-V characteristics are observed depending on the valueof the damping.

At small damping the system switches from the v = 0state to a high-voltage ohmic state (v = i) where all thejunctions rotate uniformly (whirling branch). The damp-ing is so small that when the fluxon moves through thearray it excites all the junctions to the high-voltage state.In this voltage state the fluxon is totally delocalized inthe array. For clarity, we have shown only curves forincreasing current. If current is decreased from the high-voltage state the junctions retrap at a small value of thecurrent (mostly defined by Γ). The curve is hystereticand shows bistability for a wide range of currents.

At intermediate values of damping the dynamics ismuch more complex. Now the I-V curve shows a low-voltage region dominated by a series of steps which cor-respond to resonances between the fluxon velocity andthe linear modes of the array. These resonances has beenthe object of great attention in the past.9,12,13,14,15,16,17

For these currents the fluxon moves around the ring ina localized manner. This regime persists up to a givenvalue of the current for which the fluxon reaches a highvelocity and all the junctions switch to the high-voltagepart of the curve. At moderate damping the IV curve canalso be multistable with hysteresis loops on every step.

At high values of the damping dissipation governs thedynamics, multistability disappears and the voltage in-creases from zero without discontinuities and jumps assoon as current reaches the depinning value. In this partof the curve a localized fluxon is moving around the ring

3

FIG. 2: (Color online) I − V curves for one fluxon in a 9 junctions ring with λ = 0.4 at Γ = 0.01 (left), Γ = 0.1 (middle) andΓ = 1.0 (right). Each figure shows 5 different temperatures (T = 0, 0.002, 0.01, 0.02, 0.1).

and voltage is related with the fluxon velocity. For cur-rents close to 1 (in normalized units) it starts the tran-sient to another regime where the fluxon delocalizes andall the junctions rotate and contribute to the overall volt-age in the array.5

Figure 2 also shows the dynamics of the array in thepresence of thermal noise. As can be seen in the figure,the first thermal effect is that the fluxon depins at smallercurrents. For small damping we find that if tempera-ture is high enough, noise also induces a fluxon diffusionbranch, which we discuss later in the paper. At moder-ate damping, the low-voltage resonances are rounded andat high enough temperature voltage increases smoothlyfrom zero to some value on the fluxon diffusion branch,and then switches to the high-voltage region of the I-Vcurve. At high damping temperature causes a roundingof the curve.

We will study how temperature affects the I-V curveat low damping since this is the case for the experimen-tal system we are trying to model. Usually the depinningcurrent is experimentally defined as the current for whichmeasured voltage is above a certain threshold. We havefollowed the same definition in our simulations. This is agood definition in the low damping and low temperatureregime where the system switches between two very dif-ferent voltage values so a threshold independent currentis expected. However, the election of the threshold volt-age is not trivial. A small threshold can give problemsat large temperatures where voltage fluctuations are alsolarge. A large threshold is not a good choice since ignorepossible low-voltage states like the fluxon diffusion one.If low-voltage states are present we distinguish between

the depinning current idep and the switching current isw,where depinning marks the end of the superconductingstate and switching the transition to the high-voltagebranch. At very high temperatures noise can reach thethreshold level and first switching is suppressed. To studythis issue we have used in our simulation three or five dif-ferent thresholds and compare results for all of them.

B. 〈idep(T )〉

The depinnig current is a stochastic variable with agiven probability distribution. We present results for themean value of the depinning current 〈idep〉 and its stan-dard deviation σ. Results were obtained after the numer-ical simulation of the dynamical equations of the systemfor 1000 samples. We used different values of dampingΓ (typically from 0.001 to 0.1) and coupling λ (usuallyfrom 0.2 to 1.0). The number of junctions in the array isN = 9. Another important parameter of the simulationsis the current ramp; in our case this ramp change fromone simulation to other but it is of the order of 10−7.

Figure 3 shows results for Γ = 0.01 and five differentvalues of the coupling (λ = 0.2, 0.4, 0.6, 0.8 and 1.0). Themain physical properties of the fluxon in the array changeimportantly with the value of λ. Thus, the zero tempera-ture depinning current of the array i0dep decreases a factor

of 30 from λ = 0.2 to λ = 1.0 (see table below). In or-der to compare the five curves, in figure 3 both axis havebeen scaled by the value of the zero temperature depin-ning current for every case.18 We see that once scaled allcurves are similar showing that in this range of parame-

4

FIG. 3: (Color online) Numerical calculation of normalized〈idep〉 and σ versus normalized T at Γ = 0.01 and 5 differentvalues of λ (λ = 0.2, 0.4, 0.6, 0.8, 1.0).

ters these results can be understood in a unified manner.

In the figure we see that 〈idep〉 decreases to zero aseffect of temperature. All the curves follow the samebehavior but at high temperatures the small λ curvesslightly deviates from the others. For temperatures ofthe order of the barrier thermal fluctuations dominatethe dynamics and the depinning current goes to zero. Infact, at high temperatures there is no a good definitionof depinning current since different thresholds may givedifferent results. With respect to the standard devia-tion, we can see that it grows with T 2/3 as predicted bystandard thermal activation theory and reaches a maxi-mum at high temperatures, when 〈idep〉 has an inflectionpoint, close to 〈idep〉 → 0. Then the standard deviationdecreases since all escape events happens in a narrowrange of current values.

C. The fluxon as a single particle.

When studying the dynamics of one fluxon in the ar-ray it is very common to use the picture of this extendedand collective object as a single particle.3,4,19,20 This ap-proach has been extensively used in the past and, as wewill see, it is very useful although not exact.

TABLE I: Fluxon parameters at different values of λ

λ EPN ω2PN m i0dep

0.2 0.77842 0.77405 0.5028 0.384820.4 0.30974 0.44775 0.3459 0.154350.6 0.12744 0.23151 0.2757 0.063670.8 0.05539 0.11721 0.2363 0.027671.0 0.02550 0.06041 0.2110 0.012725

Let us consider a new variable representing the centerof masses of the fluxon or the position of the fluxon inthe array. Then, in the simplest approach, the dynamicsof a fluxon in a ring can be approach by the dynamicsof a driven, damped, massive particle experiencing a si-nusoidal substrate potential (Peierls-Nabarro potential)and subjected to thermal fluctuations:

mX + ΓmX + i0dep sinX = i+ ξ(τ) (2)

where

〈ξ(τ)〉 = 0 and 〈ξ(τ)ξ(τ ′)〉 = 2mΓTδ(τ − τ ′) (3)

In this simple approach we are neglecting for instancethe spatial dependence of the mass, effective damping dueto the other degrees of freedom of the system and higherorder terms in the expansion of the substrate potentialfor the fluxon.

Table I gives a relation of numerically computed valuesof some of the parameters of the fluxon and its effectivepotential in the single particle picture: EPN is the zerocurrent potential barrier; ω2

PN is the squared frequencyfor small amplitude oscillations of the fluxon around equi-librium; m is the fluxon effective mass at rest (computedasm = EPN/2ω

2PN) and i0dep the depinning current. For a

perfect sinusoidal potential we should get i0dep = EPN/2.The exact results are close to it.

Figures 4 and 5 show for λ = 0.4 and λ = 0.8 respec-tively (in both cases Γ = 0.01) the comparison betweenthe results for the fluxon in the array, numerical simula-tions of the depinning of a single particle in a sinusoidalpotential (Eq. 2) and a theoretical calculation based onanalytical results for the thermal activation rate of par-ticles in sinusoidal potentials in the low damping regi-men.21,22,23,24,25,26 In the figure we plot the result com-puted from the Buttiker, Harris and Landauer equationfor escape rate at α = 1 [Eq. (3.11) in reference23]. Wehave checked that a very close result (indistinguishableat the scale of the figure) is got when using α = 1.45 27

or the Melnikov and Meshov theory.24

We can see that the single particle simulations repro-duce the results for the fluxon in an excellent way atthis value of the damping in all the temperature rangealthough a small deviation in a range of temperatures isobserved for λ = 0.4. Theoretical estimations disagreeat high values of T since escape rate equations were ob-tained in the infinite barrier limit (Eb ≫ kBT ).

5

FIG. 4: 〈idep〉 and σ versus T at Γ = 0.01 and λ = 0.4 forthe fluxon and a single particle in a periodic potential andcomparison with the theoretical prediction.

The agreement shown in Figures 4 and 5 does not oc-cur for other values of coupling and damping. For in-stance, for λ = 0.4 and Γ = 0.001 (figure 9 below) adeviation of the simulated curve with respect to the the-oretical prediction is found. To study further such resultwe have done numerical simulations at fixed T for differ-ent values of Γ and for λ = 0.4 (T = 0.01) and λ = 0.8(T = 0.0018).28 Results are shown in figures 6 and 7.There we can see that for λ = 0.4 the fluxon results devi-ate importantly from the expected for the single particle(or the theory) when damping is decreased. However thisis not the case for λ = 0.8. In this respect it seems to beimportant the degree of discreteness of the system. Suchdegree is measured by the coupling parameter λ (high λapproach the continuum limit of the system, and smallλ increases the discreteness effects). At small values ofλ effects due to other degrees of freedom are more im-portant and if damping is small such excitations persistlonger in the system.

D. Fluxon diffusion

In figure 2 we have seen that at low damping and highenough temperature a low-voltage branch appears in theI − V curves before the escape to the full running state.

FIG. 5: 〈idep〉 and σ versus T at Γ = 0.01 and λ = 0.8 forthe fluxon and a single particle in a periodic potential andcomparison with the theoretical prediction.

In this low-voltage state, the transport of the fluxon oc-curs through a series of noise-induce 2π phase slips (or2π/N depending on the definition of the fluxon center ofmasses) where every jump corresponds to a fluxon whichadvances one cell in the array. Such state can not beunderstood in terms of the single particle picture. Inanalogy with the phase diffusion that occurs in a singlejunction, we label this mode of transport fluxon diffusion.In spite of the fact of we are in the low damping regime,the fluxon is able to travel along the ring without excit-ing the whirling branch. Remarkably this is a thermallyexcited state and it is not seen at low temperatures.

In figure 8 we show the time evolution of the phaseof one junction (junction 1) and a phase associated withthe center of mass of the fluxon defined as ψ = 1

N

∑

j ϕj

to allow a better comparison. Current values have beenchosen in the low-voltage branch The random jumps of2π for the junction or 2π/N for the fluxon phase ψ areeasily observed.

We have also done numerical experiments to simulatethe jump from the diffusion branch to the full whirlingstate. These simulations are done with a higher voltagethreshold, but are otherwise similar to the previous ex-periments. Results for Γ = 0.01 and 0.001 are shownin figure 9, here λ = 0.4. We see that theory and sin-gle particle results agree quite well in all the range. At

6

FIG. 6: 〈idep〉 as a function of Γ for λ = 0.4 at T = 0.01.

FIG. 7: 〈idep〉 as a function of Γ for λ = 0.8 at T = 0.0018.

high temperature thermal fluctuations dominate, are ofthe order of the barrier, and theoretical results do notapply. We also see that the standard deviation increaseswith temperature following the expected law up to a cer-tain value where reaches a maximum and then decreaseswhen 〈idep〉 is close to zero.

However, in figure 9 we can also see that for the fluxoncurve there exists a value of T such that at higher tem-peratures, the value of the current for which the arrayswitches to the whirling branch is temperature indepen-dent. This temperature shows the emergence of a fluxondiffusion branch in the I-V curve. Looking at I−V curvesit appears that the switching current value is defined bysome value of the fluxon velocity. Comparing the stan-dard deviation curve in figure 9 to the similar one withthe lower threshold in figure 4, as expected, we can seethat a peak occurs much earlier in temperature for thejump from the diffusion state.

FIG. 8: λ = 0.4, Γ = 0.01 and T = 0.1. Top: I-V curveshowing the small voltage fluxon diffusion branch. Mediumand bottom: time evolution of the phase of junction 1 (blackline) and the phase of the center of masses of the fluxon (greyline) divided by 2π at i = 0.005, 0.01, 0.02, 0.05 and 0.07.

IV. DISCUSSION AND CONCLUSIONS

We have studied the thermal depinning of fluxons insmall Josephson rings at small values of damping. At zerotemperature as current is increased the system switches

7

FIG. 9: (Color online) 〈isw〉 and σ as a function of T atλ = 0.4, for two different values of the damping Γ = 0.01and Γ = 0.001. We show plots for the theoretical prediction(solid line), the particle (open symbols) and the fluxon (solidsymbols).

from a superconducting zero voltage state to a resistivestate where v = i. This happens at the so called de-pinning current. Beyond this current there is not staticconfiguration for one fluxon in the array, and the fluxonstarts to move. Due to the low value of the damping whenfluxon goes through the array causes all the junctions toswitch to a high-voltage state. Then all the junctionsdo the same but with a phase difference that accountsfor the presence of one quantum of flux homogeneouslydistributed along the whole array.

Due to thermal fluctuations, in an experiment thevalue of the measured depinning current changes fromone I-V to another and only a probability distributionfunction has sense. This function is usually character-ized by its mean value and its standard deviation. Themain object of this paper has been to numerically studyhow these observables behave for different system param-eters (coupling λ, damping Γ and temperature T ).29 Wealso have compared these results with numerical simula-tions and theoretical estimations for the depinning of asingle particle in a sinusoidal potential.

As expected, the mean value of the depinning or theswitching current decreases as temperature is raised.At low temperatures the standard deviation follows the

usual T 2/3 law. At higher temperatures the σ(T ) func-tion reaches a peak, that we can identify with the pointson the 〈idep(T )〉 or 〈isw(T )〉 curves at which the curvaturechanges sign (inflection point).

Roughly speaking our results show that the depinningof the fluxon can be understood in terms of the stochasticdynamics of a single particle in a tilted sinusoidal poten-tial. However, we have seen some unexpected effects thatwe attribute to discreteness. Then for the case of smallcoupling (λ = 0.4 and smaller) we have seen an increasingdeviation of the fluxon depinning behavior from our ex-pectations from the single particle picture. Arrays witha larger coupling are closer to the continuous limit anddiscreteness effects are smaller; here the single particlepicture works much better.

The other effect we have observed is the emergence atsmall damping of a low-voltage thermally excited statethat we call fluxon diffusion. In such cases the zero tem-perature I-V curve does not show any resonance or low-voltage state and the system switches from zero voltageto the high-voltage branch. However, after some tem-perature a low-voltage branch is observed. The systemfirst switches from zero to this branch and then to thehigh-voltage state. At higher temperatures the systemfirst continuously increases voltage from zero (then is notclear how to define idep) and at larger currents switchesfrom the low-voltage state to the high-voltage one. Wehave also seen that the values of damping and current forwhich this behavior is observed depends importantly inλ and also in the number of junctions in the array, N .30

An important point of our work has been to compareour numerical results with results based on the singleparticle picture. The main conclusion is that for mostof the cases this picture gives a good estimation of thefluxon depinning current. In fact, in an experimentalcase, where the different parameters (mainly λ, Γ andIc) are known with some imprecision will be difficult toidentify deviations from the expected behavior. In ad-dition, there are some points difficult to address: theeffective one-dimensional potential for the fluxon in thearray (Peierls-Nabarro potential) is not purely sinusoidal,and the value of the fluxon mass is not constant since itdepends on the fluxon position and the current value.We are also neglecting all the system degrees of freedomexcept one and we know that in some cases this is notvalid: for instance, to understand resonant steps whichare due to coupling between the fluxon velocity and thelinear waves of the discrete array or to understand thefluxon diffusion branch. We have also considered the ex-pression for the escape rate in a multidimensional case.In this expression usually the attempt frequency dependson the frequency of all the stable modes in the minimumand the saddle.25 We have computed such numbers andcheck that the maximum error is smaller of 7% (and oc-curs for λ = 0.125).

Our numerical results for the single particle agreepretty well the predictions from Kramers theory for es-cape rate except for some limits. Disagreement at high

8

temperatures is expected since theoretical expressionsare computed in the infinite barrier limit of the system(Eb ≫ kT ). This limit is not fulfilled at high temper-atures. This is also true at small temperatures, wheremost of the escape events occur at currents very close toidep where the barrier is also very small. We have checkedthat the Eb/kT ratio in this case is also small. To fin-ish we have to mention the unexpected disagreement atsmall values of Γ. We are currently study further suchresults.

In the single-particle picture, or the RCSJ model fora single junction, it has become generally accepted thatdiffusion cannot coexist with hysteresis. This was elu-cidated nicely in Kautz and Martinis31 through phasespace arguments. Simply put, if the value of applied cur-rent is sufficient to allow a stable running state to coexistwith the fixed points of zero voltage, the basins of at-traction for that running state necessarily separates thebasins of attraction for any two neighboring fixed points.Phase jumps between two fixed points are thus forbid-den, as the system must first pass through the basin ofattraction for the running state. While we have shownin previous sections that the initial escape of the fluxonfrom its minima can be explained by thermal activationof a single particle, the fluxon diffusion state in the I-Vcurves of figure 2 cannot be explained in a similar way.

In the original single-junction experiments on phasediffusion, the coexistence of phase diffusion and hystere-sis was explained by the presence of frequency-dependentdamping from the junction leads. A simple model offrequency-dependent damping is a series-RC circuit inparallel with the junction, which adds an extra degree offreedom to the phase space for the junction dynamics.This extra dimension resolves the above-mentioned issueregarding the non-overlapping basins of attraction. Amain result of the paper is the observation of fluxon dif-fusion in our simulations without frequency dependent-

damping, which has not been included in our equations.Instead of the extra dimension introduced by frequency-dependent damping, fluxon diffusion must occur becauseof the additional degrees of freedom from the multiplejunctions in the array. We plan to explore the physics ofthis new diffusion mechanism in future experiments andsimulations.

When studying the behavior of the system at differ-ent temperatures we have seen that the mean value ofthe distribution of the switching current from the fluxondiffusion branch is almost constant and the standard de-viation is very small. Comparing the standard deviationcurve in figure 9 to the similar one with the lower thresh-old in figure 4, we also see that a peak occurs much ear-lier in temperature for the jump from the diffusion state.This peak in the standard deviation is also reminiscent ofexperiments on single junctions,32,33,34 where a peak inthe standard deviation indicated the collapse of thermalactivation and the onset of diffusion.

To finish we want to mention that to our knowledgethere are not experimental or numerical results studyingsystematically the thermal escape of fluxons or solitonsin discrete arrays. However we want here to mentionthe work by A. Wallraff et al on vortices in long JJ.35

With respect with theoretical advances we want also tocite recent work36 where major differences between themacroscopic quantum tunnelling in JJ from tunnelling ofa quantum particle are reported.

Acknowledgments

We acknowledge F. Falo for a carefully reading of themanuscript. Work is partially supported by DGICYTproject FIS2005-00337 and NSF Grant DMR 0509450.

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9

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28 Note that the T/i0dep ratio is the same for both cases.29 This value also depend on the current ramp which enters

into the equations in a way well established by theory.30 This paper only shows results for 9 junctions arrays. How-

ever, we have done also numerical simulations for largerarrays.

31 R. L. Kautz, and J. M. Martinis, Phys. Rev. B 42, 9903(1990).

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