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Observation of breather-like states in a single Josephson cell

F. Pignatelli and A. V. UstinovPhysikalisches Institut III, Universitat Erlangen-Nurnberg,

Erwin-Rommel-str. 1, D-91058 Erlangen, Germany.

(Dated: February 1, 2008)

We present experimental observation of broken-symmetry states in a superconducting loop withthree Josephson junctions. These states are generic for discrete breathers in Josephson ladders. Theexistence region of the breather-like states is found to be in good accordance with the theoreticalexpectations. We observed three different resonant states in the current-voltage characteristics ofthe broken-symmetry state, as predicted by theory. The experimental dependence of the resonanceson the external magnetic field is studied in detail.

I. INTRODUCTION

Nonlinearity and discreteness in lattices lead to theexistence of spatially localized stable solutions [1, 2, 3].These solutions, called discrete breathers (DBs), arecharacterized by time periodicity and a strong spatial lo-calization of energy, on the scale of the lattice constant.The localized mode can be vibrational or rotational, de-pending on the lattice and type of excitation. In the sec-ond case the DB is called rotobreather. DB were foundto exist in a large variety of nonlinear discrete latticesincluding weakly coupled optical wave guides [4], crystallattice in solids [5], antiferromagnets [6] and Josephsonjunction arrays [7, 8, 9, 10, 11]. The latter system isa convenient experimental tool to investigate the exis-tence and dynamics of DBs. The discrete lattice here isconstituted by an array of coupled underdamped smallJosephson junctions (JJs).

A JJ is composed by two superconducting layers sep-arated by a thin tunnel barrier. Its dynamics can bedescribed in the resistively-capacitively-shunted-junctionmodel (RCSJ) [12]. The equation for the superconduct-ing phase difference between the two superconductor lay-ers constituting the junction, ϕ, is written in the normal-ized form as

γ = ϕ + αϕ + sin ϕ. (1)

Here the time unit is normalized by the inverse of theJosephson plasma frequency ωp, γ is the bias current Inormalized by its critical value and α is the damping.This equation is formally equivalent to the equation ofmotion of a weakly damped pendulum in presence of anexternal torque γ. This mechanical analog allows for aneasy description of the two possible states of the JJ. If weapply a torque γ > 1 the pendulum will start to rotatewith an average frequency < ϕ >. After the pendulum isdriven to the rotating state, it will stay in this state, evenfor γ < 1, as long as the external torque is large enoughto compensate the damping term. So that at some rangeof γ both static and rotating states are possible, depend-ing on the initial conditions. By the second Josephsonequation V = Φ0/2π < ϕ >, these two states are theequivalent, respectively, to the superconducting and re-sistive states of the JJ. In experiment by measuring the

voltage on the JJ it is possible to distinguish betweenthese two states.

Josephson ladders allow for the existence of both oscil-latory and rotational localized modes [13]. However onlythe latter ones are easy to detect experimentally. A roto-breather in the Josephson ladder is constituted by someof the JJs in the resistive (rotating) state, while the restof JJs stays in the superconducting (oscillating) state.The rotobreathers were experimentally detected by mea-suring the I − V characteristics of the JJs in differentpoints of the ladder [7, 10] and also directly visualizedby low temperature scanning laser microscopy [8, 9].

Josephson ladders can support linear cavity modesthat can interact with the localized modes [10, 11, 14, 15].The interaction of the linear modes of the ladder with thelocalized modes was found to lead to nonlinear featuresin the characteristic of DBs. Their complex dynamic inthe region of the linear cavity resonances was investi-gated experimentally by the measurements of the I − Vcharacteristics of various DBs [11].

Single Josephson cells have been proposed [16, 17] toprovide an insight into complex discrete breather statesof larger systems such as Josephson ladders and two di-mensional Josephson junction arrays. Broken-symmetrystates with characteristics similar to those of discretebreathers were proposed to exist [16, 17, 18]. We willcall in the following these broken-symmetry states asbreather-like states.

Superconducting loops with three Josephson junctionhave also been recently proposed for the realization of aqubit [19, 20]. Such a loop has an important property re-quired for the qubit, namely two closely positioned min-ima in the dependence of its energy on the external mag-netic field [21]. Note, however, that in this work we areinterested into phase-rotational states, which are clearlydifferent from oscillatory states discussed for qubits.

In this paper, we study a single Josephson cell withthree small underdamped Josephson junctions. Twojunctions of same area placed along the direction parallelto the uniform bias current, V1 V2, are called verticaljunctions. The third junction located in one of the twotransverse branches of the cell, is called horizontal H, seeFig. 1.

A parameter of cell anisotropy, η, is defined as the ra-tio between the critical current of the horizontal junction

2

FIG. 1: a) Sketch of the single Josephson cell, with three JJs(indicated by crosses), two vertical V1, V2 and one horizontalH. b) Optical image of our experimental cell with 4 × 4 µm2

hole and three underdamped Nb/Al-AlOx/Nb JJs.

and the one of the vertical junctions, η = ICH/ICV . Theparameter of anisotropy describes the coupling betweenthe two vertical junctions, so that in the limit η → 0the two vertical junctions are entirely decoupled, whilethe larger is η the stronger is the coupling. The cell iscalled isotropic if all three junctions have same parame-ters, anisotropic otherwise. The system is uniformly bi-ased by two equal currents I, applied through the verticalbranches, 1-2 and 3-4, Fig. 1. As long as the uniform biascurrent I is smaller than the critical current ICV , all thejunctions stay in the superconducting state, Fig. 2a. Asthe current I exceeds the critical value ICV , the systemswitches to the homogeneous whirling state, constitutedby both vertical junctions in the resistive state and thehorizontal one in the superconducting state, Fig. 2b.Both these cases are symmetric states and there is nocurrent on the horizontal junction, so that the two ver-tical junctions behave identically. We are interested inthe last two cases, Fig. 2c and d, which are the inhomo-geneous breather-like states. These two states have thesame behavior and are characterized by the horizontaljunction and one of the vertical junctions in the resistivestate, while the other vertical junction remains in the su-perconducting state. In these cases, obviously, some partof the bias current flows through the transverse branch.

FIG. 2: Possible states of the system, (a) superconductingstate, (b) homogeneous whirling state, (c) (d) the two sym-metric breather-like states

The procedure to generate breather-like states in ex-periments is similar to that used for Josephson ladders

[7, 8]. We bias the system by a local current, Iloc, throughthe transverse branch 4-2, Fig. 1, and rise this currentuntil the system switches to a breather-like state. After-wards, we increase the bias current I and simultaneouslydecrease Iloc to zero, keeping the system in the generatedstate. Alternative procedure is first apply a homogeneousbias current I < ICV at a value where we expect to findthe breather-like state stable, then rise the local currentIloc to generate it, and finally reduce Iloc back to zero.By using the two different directions of the initial localcurrent it is possible to generate both breather-like statesshown in Fig. 2.

Different states of the system can be experimentallydetected by measuring the dc voltage on Josephson junc-tions. As for Josephson ladders, the broken-symmetrystates are the consequence of the characteristic propertyof small underdamped Josephson junctions, that is thecoexistence, for some ranges of the bias current, of twodifferent stable states, superconducting and resistive.

II. DYNAMICS OF THE SYSTEM

The dynamics of the system are described by the phasedifferences across the three Josephson junctions, ϕV1

,ϕV2

, ϕH . Using the current conservation law in each nodeof the cell and the flux quantization for the cell, one canget the equations of motion for the system [17, 18]:

N (ϕV1) = γ −

1

βL

(ϕV1− ϕV2

+ ϕH + 2πf)

N (ϕV2) = γ +

1

βL

(ϕV1− ϕV2

+ ϕH + 2πf) (2)

N (ϕH) = −1

βLη(ϕV1

− ϕV2+ ϕH + 2πf),

where the time is normalized by the inverse of the Joseph-son plasma frequency ωp. The influence of the externalmagnetic field is introduced by the term 2πf/βL. Theparameters of the system are the uniform bias current

normalized by the critical current of the vertical junc-tions, γ = I/ICV , the normalized self-inductance of thecell, βL = 2πLICV /Φ0 (where L is its real inductance),

the damping, α =√

Φ0/(2πICR2C), and the frustrationparameter f = Φext/Φ0, that can be tuned by the ex-ternal magnetic field H . The operator N represents thecurrent through a single junction in the RCSJ model andis defined as: N (ϕ) = ϕ + αϕ + sin ϕ.

Using a simplified dc model [17], where the junctionsin the resistive state are considered as resistances andthe ones in the superconducting state just as shorts, onecan estimate the dimensionless breather frequency Ω =<ϕ >, that is in fact the voltage across a junction in theresistive state normalized by Φ0ωp/2π:

Ω =γ

α(1 + η). (3)

3

From that one can also evaluate the region of existenceof the breather-like state [17]. The minimum current atwhich a broken-symmetry state may exist is

γr =4α

π(1 + η). (4)

At γ < γr the system is retrapped to the superconductingstate. The maximum current for a breather-like state is

γc =1 + η

1 + 2η. (5)

Above this current the system switches to the homoge-neous rotating state.

The simplified dc model neglects the influence of thesmall oscillations of the Josephson phase that may causean instability and nonlinearity in the characteristic of thebreather-like state. Neglecting the damping and lineariz-ing the equation of motion, Eqs. (2), around the breather-like state, Eq. (3), it is possible to calculate the two char-acteristic frequencies of these oscillations [17, 18]:

ω± =

√

F ±√

F 2 − G, (6)

where: F =1

2cos(c1) +

1 + 2η

2ηβL

G =1 + η

ηβL

cos(c1).

Here c1 is the phase shift of the vertical junction inthe superconducting state evaluated by the dc model,c1 = arcsin[γ(1 + 2η)/(1 + η)]. The presence of theelectromagnetic oscillations leads then to primary res-onances in the breather-like state, when the frequencyof the breather Ω locks to the frequency of the electro-magnetic oscillations ω+ or ω−, and to parametric andcombination resonances, respectively, at Ω = nω±, wheren ≥ 2 is an integer, and Ω = n(ω+ ± ω−), where n is aninteger [17, 18].

Using the rotational approximation [22], two differentdependencies of the resonant step amplitude ∆γ on thefrustration have been obtained for primary and paramet-

ric resonances [18]:

∆γω± ∝

∣

∣

∣

∣

sin

(

c1

2+

βLγη

2(1 + η)+ πf

)∣

∣

∣

∣

(7)

∆γnω± ∝

√

(1 − η)2 + 4η cos2(

c1

2+

βLγη

2(1 + η)+ πf

)

.

(8)

Expressions (7) and (8) have the same periodicity in f asthe critical current in the whirling state, but, dependingon the parameters of the system, they may be asymmetric

with respect to f = 0. Moreover, the external magneticfield can considerably change the amplitude of the reso-nance and, in the case of primary resonance, even makeit to vanish for some values of the frustration.

III. EXPERIMENTAL RESULTS

Our experimental system is a single cell with the holearea 4 × 4 µm2, containing three small underdampedNb/Al-AlOx/Nb Josephson junctions [23], see Fig. 1.The critical current density, measured at 4.2 K, is JC =133 A/cm2. The area of each vertical junction is about7 × 7 µm2, while the horizontal one is 3 × 4 µm2. Theanisotropy of the system, evaluated from the design, isthen η = 0.24. The parameter of self-inductance of thecell at 4.2 K, evaluated by measurement of a SQUID ofsame geometry, is βL = 1.47. The two equal bias currentsare introduced in parallel using two large external resis-tances of 1.5 kΩ, so that the bias current remains uniformalso in the broken-symmetry states. The damping of thesystem at 4.2 K is evaluated from the retrapping currentof the homogeneous whirling state, α ≃ (π/4)(Ir/2ICV ),that yields α ≃ 0.03. The measurements are performedin a 4He cryostat [24] in the temperature range between4 K and 8 K, so that the parameters of the system JC ,α, βL and ωp can be varied. For each temperature thedamping α and the critical current density JC are evalu-ated from the characteristic of the homogeneous whirlingstate, while the parameter of self inductance βL is evalu-ated from the critical current density. Finally, the valueof the plasma frequency is obtained at every temperaturefrom fitting the voltage of the parametric resonant stepto the expected value 2ω+.

For convenience of comparing with theory, the experi-mental data are shown in the previously introduced nor-malized units of the bias current γ = I/ICV , breather fre-quency Ω = V/(ωpΦ0/2π) and frustration f = Φext/Φ0.

In the range of temperatures from 4.2 K to 7.75 K(0.02 ≤ α ≤ 0.35) it was possible to excite both breather-like states and they were found stable in a finite range ofthe bias current. No breather-like state was found stablefor temperatures above 7.75 K (α > 0.35).

The minimum bias current below which the breather isretrapped increases linearly with the damping, see Fig. 3,in good agreement with the theoretical expectation ofEq (4). The maximum current that allows for the ex-istence of the breather-like state slightly decreases withincreasing the damping, see Fig. 3, whereas from Eq. (5)it is expected to be constant (γc = 0.84).

In fact, the temperature variation leads to an alter-ation of several parameters, see Fig. 4. In particular thedecrease of the self inductance βL at high temperaturesleads to stronger interactions of the breather-like statewith the small oscillations. This effect may strongly af-fect (in the range 0.1 < βL < 1) the region of existence ofthe breather-like, especially its upper border [17]. In fact,the experimental data show deviation from the expected

4

FIG. 3: Maximum (solid squares) and minimum (solid dia-monds) currents for the breather-like state. The solid linesshow the theoretical expectation of the dc model, Eqs. (4)and (5).

FIG. 4: Dependence of the damping, evaluated by the retrap-ping current (solid squares) and by the sub-gap resistance(solid circles), and of the self-inductance (open triangles) onthe temperature of the system.

value for α ≥ 1.15, which is in the range of the self induc-tance between 0.9 and 0.3. No resonances were observedin the I − V characteristics at temperatures below 6 K.The small damping of the junctions at this temperaturesseems to be unfavorable for reaching the small frequenciesof the electromagnetic oscillations where the resonancesare expected (Ωmin ≃ 5.5). Increasing the temperatureof the system had two helpful effects for seeing the res-onances, namely the increase of damping and decreaseof self inductance, see Fig. 4. As the consequence ofthe larger damping, the breather-like states were stabledown to smaller frequencies, whereas a smaller induc-tance moved the frequencies of the resonances up.

At the temperature of T = 6 K it was possible toobserve the first resonant step. The parameters of thesystem at this temperature are α ≃ 0.11, Jc = 92 A/cm2,

βL = 1.02 and ωp/2π = 37 GHz. This resonance is a stepin the characteristic, shown in Fig. 5. Comparing thisresonance with that at higher temperatures and with thetheoretical expectation we deduce it to be the parametricresonance at 2ω+.

FIG. 5: Measured current-voltage characteristic of thebreather-like state at damping α ≃ 0.11 and frustrationf = −0.1, where the magnitude of the resonant step is maxi-mum. The dotted lines show the expected frequencies of theelectromagnetic oscillations, Eq. (6).

At the temperature of T = 6.65 K we detect two addi-tional resonances in the characteristic of the breather-likestate, see Fig. 6. At this temperature the parameters ofthe system are α ≃ 0.18, Jc = 70 A/cm2, βL = 0.77 andωp/2π = 30 GHz. The two larger resonances, A and B,are different by the integer factor of two, as expected for2ω+ and ω+.

The dependencies of the current amplitude on the ex-ternal magnetic field for steps A and B are strongly differ-ent. The resonance A has its maximum amplitude closeto zero frustration, see Fig. 6a, and disappears in a widerange around f = 0.5, see Fig. 7. An other resonance Bis always present but has nearly opposite behavior. Atthe top of the resonance B, around frustration f = 0.4or f = −0.6, the system shows an instability by switch-ing either to the homogeneous whirling state or to thesuperconducting state, as shows Fig. 7. In this rangeof frustration the measurements of the resonant step Bwere performed with less accuracy and faster because ofits poor stability. We argue the larger step B is due tothe primary resonance at Ω = ω+, whereas the step A isdue to the parametric resonance at Ω = 2ω+. The thirdsmaller nonlinearity, C, is at a frequency between theother two, close to the theoretically expected value forthe composed resonance at Ω = ω+ +ω−, see Fig 6. Thisresonance was only present in a small range of frustra-tion around zero, where from the theory, the composedresonance is expected to be maximum. Because of its re-duced amplitude this resonant step has no hysteresis andits dependence on the magnetic field was not measured.

5

FIG. 6: Measured current-voltage characteristic of thebreather-like state at the damping α ≃ 0.18 and for:(a)f = −0.1, (b) f = 0. The dotted lines show the expectedfrequencies of the electromagnetic oscillations, Eq. (6). Theinsets show a zoom into the region of interest.

In Fig. 8b, c we present measurements of the currentamplitude of the resonant steps A and B on the exter-nal magnetic field. The parametric step at T = 6 K,see Fig. 5, is higher than that at T = 6.65 K, Fig. 6.Moreover, because of the larger damping in the lattercase the step vanishes in a larger range of the frustration(−0.95 < f < −0.25 and 0.05 < f < 0.75). Therefore,in order to observe a more clear modulation of this reso-nance, we report in Fig. 8c its dependence on the externalmagnetic field at T = 6 K.

As expected from the theory, Eq. (7) and Eq. (8), themagnetic field dependence of the resonance amplitudesshow an asymmetry with respect to f = 0. Note, thatsuch asymmetry is not observed for IC(f), see Fig. 8(a).The asymmetry depends on the parameter of the systemand particularly on the bias current where the resonanceoccurs. This behavior can be qualitatively attributed tothe current flowing in the transverse branch of the cellin the broken-symmetry state. The experimental dataare in rather good agreement with the theoretical expec-

FIG. 7: Measured current-voltage characteristic of thebreather-like state in presence of instabilities at dampingα ≃ 0.18 (a) f = −0.6, (b) f = 0.4. The dotted lines showthe frequencies of the electromagnetic oscillations evaluatedfrom Eq. (6).

tations. We fitted the maximum and minimum of theresonant steps in order to find the proportionality coef-ficient in Eqs. (7) and (8). The small variations of theexperimental data around the maximum of the primaryresonant step could be due to the appearance of insta-bilities, as discussed in relation to Fig. 7. This behaviorwas also found in simulations [18]. In the case of para-metric resonance the presence of a range of frustration,that increases with the damping α, where resonant stepdisappears, is also expected [18]. Note, that the regionwhere the experimental data show the largest deviationfrom the expected behavior is nearly the same for thetwo cases (0.3 < f < 0.6).

IV. CONCLUSIONS

We observed breather-like states of broken symmetryin an anisotropic single-cell system. For temperatures

6

FIG. 8: (a) Dependence of the critical current of the homo-geneous state on the frustration. (b), (c) Dependence of themagnitude of the resonant step on the frustration for primaryresonance A (measured at T = 6.65 K) and parametric res-onance B (measured at T = 6 K) respectively. The solidlines are the theoretical curves given by Eq. (7) and Eq. (8),respectively.

lower than 6 K the current-voltage characteristics of thebreather-like states were stable and did not show anyresonances. Increasing the dissipation by rising the tem-perature of the system we decreased the frequencies ofrotation of the breather-like state in such a way that itmatched the frequencies of the resonances in the cell. Inthis region, the primary, composed and parametric reso-nances were observed. As expected, in all three cases, theinteraction of the electromagnetic oscillations with thebreather-like state leads to resonant steps in the current-voltage characteristic. The large steps due to the primaryand parametric resonances show hysteresis, whereas thecomposed one induces just a small nonlinearity in thecharacteristic. The measured dependence on the externalmagnetic field of the magnitude of the primary and para-metric resonant steps is in good agreement with theory.In the case of primary resonance, the external frustrationcan strongly increase the amplitude of the resonant stepand lead to instability of the breather-like state. Thecomposed resonance, for this values of the damping, waspresent just in a small range around zero frustration anddid not lead to instabilities. The region of existence of thebreather-like state was also measured and found in goodaccordance with the expectations. At smaller values ofthe self inductance parameter, the stronger interaction ofthe breather with the electromagnetic oscillations leadsto a contraction of the region of existence.

V. ACKNOWLEDGMENTS

We would like to thank A. Benabdallah, P. Binder, M.V. Fistul, S. Flach, A. E. Mirishnichenko and M. Schusterfor the fruitful discussions and for sending us their latestworks before publication. This work was supported bythe Deutsche Forschungsgemeinschaft (DFG) and by theEuropean Union under RTN project LOCNET HPRN-CT-1999-00163.

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