Confocal Annular Josephson Tunnel Junctions with Large Eccentricity
CConfocal Annular Josephson Tunnel Junctionswith Large Eccentricity
Roberto Monaco
CNR-ISASI, Institute of Applied Sciences and Intelligent Systems ”E. Caianello”,Comprensorio Olivetti, 80078 Pozzuoli, Italy ∗ Jesper Mygind
DTU Physics, B309, Technical University of Denmark, DK-2800 Lyngby, Denmark † Lyudmila V. Filippenko
Kotel’nikov Institute of Radio Engineering and Electronics,Russian Academy of Science, Mokhovaya 11, Bldg 7, 125009 Moscow, Russia ‡ (Dated: May 21, 2018) Abstract
Confocal Annular Josephson Tunnel Junctions (CAJTJs) which are the natural generalization of the circular annular Joseph-son tunnel junctions, have a rich nonlinear phenomenology due to the intrinsic non-uniformity of their planar tunnel barrierdelimited by two closely spaced confocal ellipses. In the presence of a uniform magnetic field in the barrier plane, the periodi-cally changing width of the elliptical annulus generates a asymmetric double-well for a Josephson vortex trapped in a long andnarrow CAJTJ. The preparation and readout of the vortex pinned in one of the two potential minima, which are important forthe possible realization of a vortex qubit, have been numerically and experimentally investigated for CAJTJs with the moderateaspect ratio 2 : 1. In this work we focus on the impact of the annulus eccentricity on the properties of the vortex potentialprofile and study the depinning mechanism of a fluxon in more eccentric samples with aspect ratio 4 : 1. We also discuss theeffects of the temperature-dependent losses as well as the influence of the current and magnetic noise.
I. INTRODUCTION
The problem of a particle in a double-well potential(DWP), characterized by two adjacent - in general un-equal - potential minima, is almost as old as QuantumMechanics , and one of the first applications was thecalculation of an inversion frequency of the ammoniamolecule back in 1932 . During the last two decades, thephenomenon of tunneling in asymmetric DWPs was ac-tively considered across several branches of physics andfound application in the study of systems, such as theBose-Einstein condensates in trapped potentials andthe quantum superconducting circuits based on of low-capacitance Josephson Tunnel Junctions (JTJs) . Thelatter have attracted great attention due to their po-tential use as elementary bits of quantum information(qubits, i.e., two-state quantum-mechanical systems) ca-pable of implementing quantum computing operations.Long and narrow, annular JTJs are potential qubit can-didates due to their unique capability to trap a Joseph-son vortex (a supercurrent loop carrying one magneticflux quantum also called fluxon) whose center of massbecomes the macroscopic collective coordinate of a mas-sive particle: at sufficiently small temperatures, the par-ticle enters the quantum regime as it shows discrete en-ergy levels within a potential well and can escape froma potential well via macroscopic quantum tunneling .Different techniques have been adopted to implement atwo-minima fluxon potential in a long JTJ by the applica-tion of an external magnetic field and/or on some abruptchanges of the tunnel barrier properties, either the curva-ture radius or the Josephson current density . An alternative approach has been proposed which takes ad-vantage of the proportionality between the fluxon spatialpotential and the local width of the long JTJ . It followsthat a large variety of spatially dependent fluxon poten-tials can be engineered by means of JTLs having a non-uniform width . In particular, a magnetically tunabledouble-well potential was conjectured in a variable-widthannular JTL named Confocal Annular Josephson TunnelJunction (CAJTJ) where the tunneling area is delim-ited by two ellipses having the same focal length. Fig. 1shows the scanning electron microscope image of a CA-JTJ made of N b doubly-connected electrodes. The CA-JTJs represent a generalization of the well-known circular(i.e., zero-eccentricity) annular JTJs intensively studiedto experimentally test the perturbation models developedto take into account the dissipative effects in the prop-agation with no collisions of sine-Gordon kinks andto investigate both the static and the dynamic propertiesof a fluxon in the spatially periodic potential induced byan in-plane magnetic field . At variance with thering-shaped JTLs which have a constant width, it is seenthat for the CAJTJs the width of the planar tunnel bar-rier is smallest at the equatorial point and largest at thepoles; the width variation is smoothly distributed alongone fourth of the elliptical annulus (mean) perimeter. Itis this smooth periodic change of the width of the planartunnel barrier that makes the physics of CAJTJs veryrich and interesting especially since the modeling is veryaccurate . Recently , experiments have shown thata fluxon trapped in a long and narrow CAJTJ experi-ences a finely tunable DWP and both the preparationand readout of the vortex states in either the left or right1 a r X i v : . [ c ond - m a t . s up r- c on ] M a y IG. 1. Scanning electron microscope image of a confocal an-nular Josephson tunnel junction (CAJTJ) made of Nb doubly-connected electrodes. The ratio of the major axis and theminor axes is 2 : 1 that implies that the equatorial annuluswidth is one half of the polar width. state, that are important with respect to the possible re-alization of a vortex qubit, can be achieved by simpleand robust procedures. The previously presented find-ings concerned CAJTJs with the moderate aspect ratioof 2 : 1 in which, as in Fig. 1, the (mean) major diame-ter is twice larger than the minor one. In this paper weinvestigate the effect of the annulus eccentricity on theproperties of the intrinsic fluxon DWP and present bothnumerical and experimental findings on CAJTs with as-pect ratio of 4 : 1. In particular, we will focus on themechanisms influencing the depinning of a fluxon fromeach of the potential wells.The paper is organized into five sections. Section II in-troduces the theoretical framework for the study of acurrent-biased CAJTJ subjected to an external magneticfield in a modified and perturbed sine-Gordon equation;we then consider the two-minima periodic potential expe-rienced by a trapped fluxon and discuss how the potentialchanges with the system aspect ratio and how it can betuned by means of an external in-plane magnetic fieldand/or bias current. In Sec. III we present numericalsimulations of the depinning of a fluxon from each of thetwo stable states of the DWP in underdamped CAJTJsand describe a protocol to reliably prepare and deter-mine the vortex state. In Sec.IV we present the experi-mental data obtained with high-quality low-loss N b/Al - AlOx/N b window-type CAJTJs in the presence of in-plane magnetic fields and discuss the role of the temper-ature and noise on the fluxon depinning. The conclusionsof our work are presented in Section V.
II. THEORY OF ONE-DIMENSIONAL CAJTJS
The tunneling area of two CAJTJs with different aspectratios are sketched by the hatched area in the top panelsof Figs. 2(a) and (b) where the principal diameters ofthe confocal ellipses are made parallel to the X and Y axes of a Cartesian coordinate system. The common foci(small gray closed circles) lie on the X -axis at ( ± c, ν, τ ), a two-dimensionalorthogonal coordinate system in which the coordinatelines are confocal ellipses and hyperbolae. In this sys-tem, any point ( x, y ) in the X - Y plane is uniquely ex-pressed as ( c cosh ν sin τ, c sinh ν cos τ ) with ν ≥ τ ∈ [ − π, π ] for a given positive c value. According tothese notations, the origin of τ lies on the positive Y -axis and increases for a clockwise rotation. In the limit c →
0, the elliptic coordinates ( ν, τ ) reduce to polar coor-dinates ( r, θ ), where θ is the angle relative to the Y -axis;the correspondence is given by τ → θ and c cosh ν → r (note that ν itself becomes infinite as c → ± c,
0) is uniquelyidentified by a value of ν ; we will name ν i and ν o > ν i the characteristic values of, respectively, the inner andouter elliptic boundaries of a CAJTJ. Their mean value,¯ ν = ( ν o + ν i ) /
2, labels one more confocal ellipse in be-tween, called mean or master ellipse - see the dashedellipses in the top panels of Figs. 2(a) and (b). As theminor and major axes of the master ellipse are given by,respectively, 2 c sinh ¯ ν and 2 c cosh ¯ ν , we define the aspectratio of a CAJTJ as ρ ≡ tanh ¯ ν and its (mean) eccentric-ity as e ≡ − ρ = sech ¯ ν .For closely spaced inner and outer ellipses, ∆ ν ≡ ν o − ν i <<
1, the expression of the local annulus width is :∆ w ( τ ) = c Q ( τ ) ∆ ν, (1)where Q ( τ ) is the elliptic scale factor defined by Q ( τ ) ≡ sinh ¯ ν sin τ + cosh ¯ ν cos τ = sinh ¯ ν + cos τ =cosh ¯ ν − sin τ = (cosh 2¯ ν + cos 2 τ ) /
2. The width ofthe confocal annulus is smallest at the equatorial points,with ∆ w min = c ∆ ν sinh ¯ ν , and largest at the poles, with∆ w max = c ∆ ν cosh ¯ ν ; then ∆ w min = ρ ∆ w max , i.e., in-terestingly, as we make the confocal annulus more eccen-tric we enhance the width spread ∆ w max / ∆ w min . Thisis clearly seen in Fig. 2 where the annuli are made tohave the same equatorial widths, ∆ w min = 0 .
1: as wehalve the aspect ratio, ρ , passing from 1 / /
4, thepolar width, ∆ w max , doubles from 0 . .
4. The an-nuli width variations with the angular elliptic coordinate, τ , are shown in the bottom panels of Figs. 2(a) and (b)as given by (1).In the small width approximation, ∆ w max << λ J , where λ J , called the Josephson penetration length, gives a mea-sure of the distance over which significant spatial vari-2tions of the Josephson phase occur, the system be-comes one-dimensional. It has been derived that the ν -independent Josephson phase, φ ( τ, ˆ t ), of a narrow CA-JTJ in the presence of a spatially homogeneous in-planemagnetic field H of arbitrary orientation, ¯ θ , relative tothe Y -axis, obeys a modified and perturbed sine-Gordonequation with a space dependent effective Josephson pen-etration, λ J /Q ( τ ), length inversely proportional to thelocal junction width : (cid:20) λ J c Q ( τ ) (cid:21) (cid:18) β ∂∂ ˆ t (cid:19) φ ττ − φ ˆ t ˆ t − sin φ = αφ ˆ t − γ ( τ )+ F h ( τ ) , (2)where ˆ t is the time normalized to the inverse of the so-called (maximum) plasma frequency, ω p . The criticalcurrent density, J c , was assumed to be uniform. Thesubscripts on φ are a shorthand for derivative with re-spect to the corresponding variable. Furthermore, γ ( τ ) = J Z ( τ ) /J c is the local normalized density of the bias cur-rent and F h ( τ ) ≡ h ∆ cos ¯ θ cosh ¯ ν sin τ − sin ¯ θ sinh ¯ ν cos τ Q ( τ ) (3)is an additional forcing term proportional to the ap-plied magnetic field; h ≡ H/J c c is the normalized fieldstrength for treating long CAJTJs and ∆ is a geometri-cal factor which sometimes has been referred to as thecoupling between the external field and the flux densityof the annular junction . As usual, the α and β terms in(2) account for, respectively, the quasi-particle shunt lossand the surface losses in the superconducting electrodes.The perimeter of the master ellipse is L = 4 c cosh ¯ ν E ( e ),where E ( e ) ≡ E ( π/ , e ) is the complete elliptic integralsof the second kind of argument e . Then the normalizedor electric length, (cid:96) = L/λ J , of the CAJTJ of a givenaspect ratio grows linearly with the foci distance, 2 c .When cooling an annular JTL below its critical tempera-ture one or more flux quanta may be trapped in its dou-bly connected electrodes. The algebraic sum of the fluxquanta trapped in each electrode is an integer number n , called the winding number, counting the number ofJosephson vortices (fluxons) trapped in the junction bar-rier. To take into account the number of trapped fluxons,(2) is supplemented by periodic boundary conditions : φ ( τ + 2 π, ˆ t ) = φ ( τ, ˆ t ) + 2 πn, (4a) φ τ ( τ + 2 π, ˆ t ) = φ τ ( τ, ˆ t ) . (4b) A. The single fluxon potential
In the absence of dissipative and driving forces, thesimplest topologically stable dynamic solution of (2) onan infinite line, in a first approximation, is a 2 π -kink(single fluxon) centered at a time-dependent coordinate s (ˆ t ) and moving with instantaneous (tangential) veloc-ity ˆ u ≡ d ( s /λ J ) /d ˆ t = ( c/λ J ) Q ( τ ) dτ /d ˆ t :˜ φ ( τ, ˆ t ) = 4 arctan exp (cid:8) ℘ [ s ( τ ) − s (ˆ t )] /λ J (cid:9) , (5)where ℘ = ± and s ( τ ) the non-linear curvilinear coordinate s ( τ ) = c (cid:82) τ Q ( τ (cid:48) ) dτ (cid:48) . Inserting the phase profile in (5)into (2) it was derived that , in the absence of externalforces, the energy of a non-relativistic fluxon (ˆ u << E = ˆ K + ˆ U w , is conserved. The circumflex accents de-notes normalized quantities. ˆ E is normalized to the char-acteristic energy, E = Φ J c λ J c ∆ ν/ π . Both the kineticenergy, ˆ K ( τ ) ≈ Q ( τ )ˆ u , and the intrinsic potential en-ergy, ˆ U w ( τ ) ≈ Q ( τ ), are position dependent throughthe scale factor Q , that is, in force of (1), they are propor-tional to the annulus width. This is consistent with therelativistic expression ˆ E = ˆ m ( τ ) / (cid:112) − ˆ u ( τ ) reportedby Nappi and Pagano , provided that we introduce theposition dependent reduced rest mass ˆ m ( τ ) = 8 Q ( τ )of the fluxon. Note that the energy, E , of a CAJTJcontaining one static vortex is 8 E sinh ¯ ν . With exper-imentally accessible geometrical and electrical parame-ters the normalizing energy is much larger than k B T - E = O (10 K ) - and the fluxon rest mass, m ≡ E / ¯ c ,happens to be much smaller than the electron rest mass m e - m = O (10 − m e ) -, where the so-called Swihartvelocity , ¯ c , is the characteristic velocity of the electro-magnetic waves in JTJs.As can be discerned from the plots in the bottom pan-els of Figs. 2(a) and (b), ˆ U w ∝ ∆ w expresses a π -periodicpotential energy function independent on the fluxon po-larity, and uniquely determined by the CAJTJ aspectratio. The potential wells are located at τ = ± π/ | L (cid:105) and right | R (cid:105) wells of the potential constitute stable clas-sical states for the vortex with degenerate ground stateenergy. Considering that sinh ¯ ν ≤ Q ( τ ) ≤ cosh ¯ ν , thepotential wells are separated by a normalized energy bar-rier, ∆ ˆ U w ≡ ˆ U w,max − ˆ U w,min = 8 exp − ¯ ν , uniquely de-termined by the system aspect ratio. As an example, thechange of the aspect ratio from 2 : 1 to 4 : 1 in Figs. 2results in the triplication of the energy barrier (and so ofthe potential gradient).In the presence of small applied magnetic field and biascurrent, two more terms contribute to the total potentialenergy, ˆ U , experienced by the fluxon:ˆ U ( τ ) = ˆ U w ( τ ) + ˆ U h ( τ ) + ˆ U γ ( τ ) . (6)ˆ U h ( τ ) is the magnetic potential such that dU h /dτ =2 π℘ ( λ J /c ) Q ( τ ) F h ( τ ), i.e.,:ˆ U h ( τ ) = ˆ U h ⊥ ( τ ) + ˆ U h || ( τ ) ≈≈ π℘ ( λ J /c )∆ (cid:0) h ⊥ cosh ¯ ν cos τ + h || sinh ¯ ν sin τ (cid:1) , (7)3here h ⊥ ≡ h cos ¯ θ and h || ≡ h sin ¯ θ are the componentsof the in-plane magnetic field, respectively, perpendicularand parallel to the CAJTJ’s major diameter. ˆ U h ( τ ) is2 π -periodic and π -antiperiodic in τ , i.e., ˆ U h ( τ + π ) = − ˆ U h ( τ ), then it averages to zero over one period. It isimportant to note that ˆ U h ⊥ is even in τ as the intrinsicpotential ˆ U w , while ˆ U h || is odd and breaks the systemparity.Furthermore, ˆ U γ ( τ ) is the current-induced potentialsuch that d ˆ U γ /dτ = 2 π℘ ( λ J /c ) Q ( τ ) γ ( τ ); assuming auniform current distribution γ ( τ ) = γ , it is:ˆ U γ ( τ ) ≡ π℘ ( λ J /c ) γ (cid:18) τ cosh 2¯ ν + 12 sin 2 τ (cid:19) . (8)Fig. 3 qualitatively explains how the width-dependentfluxon potential can be tuned by means of an externallyapplied magnetic field and/or a bias current. The dottedcurve at the bottom of Fig. 3 shows the fluxon potentialin the presence of a (negative) perpendicular magneticfield h ⊥ ; the potential ˆ U w + ˆ U h ⊥ is still invariant undera parity transformation ( τ → − τ ) and develops into afield-controlled symmetric potential with finite walls andtwo spatially separated minima. Increasing the ampli-tude of the magnetic field, eventually the minima coalesceand the perturbed potential becomes single-welled for a(perpendicular) threshold field strength h ∗⊥ . The evolu-tion of the potential ˆ U with an increasing parallel field, h || , follows a quite different pattern (not shown in Fig. 3),but again a threshold value, h ∗|| , exists where the poten-tial ˆ U w + ˆ U h || becomes single-welled. It means that theinter-well barrier can be fine-tuned and made arbitrarilysmall by means of both a perpendicular or a parallel in-plane magnetic field. The dashed line in Fig. 3 shows thetotal potential when a bias current is feeding the CA-JTJ. The resulting potential, ˆ U w + ˆ U γ , is qualitativelysimilar to the well-studied tilted washboard potential forthe phase difference of a small JTJ biased below its crit-ical current; the only difference is that in our case thedegree of freedom is the spatial coordinate, rather thanthe Josephson phase difference. Indeed, the potentialprofile can be tilted either to left or to right dependingon the polarity of the bias current, γ . The inclination isproportional to the Lorentz force on the vortex which isinduced by the bias current applied to the junction. Atlast, the total fluxon potential, ˆ U w + ˆ U h ⊥ + ˆ U γ , in thepresence of both an applied magnetic field and bias cur-rent is depicted by the solid line at the top of Fig. 3. Wenote that the left well is very shallow and an increment ofthe bias current would further tilt the potential; a staticfluxon pinned in the left well would become unstable andgets trapped in the right well as, in this specific example,it does not have enough energy to move further to theright. The smallest tilting that allows the vortex to es-cape from a well defines the so-called depinning current, γ d . Clearly, the depinning current depends on the ap-plied magnetic field and, in general, for a given field, thedepinning currents, γ Ld and γ Rd , from the left and right wells are different. The deeper is the original potentialwell from which the fluxon has to escape and the largeris the corresponding depinning current. III. THE NUMERICAL SIMULATIONS
In the Figs. 4(a) and (b) we report the numerically com-puted field dependencies of the positive left and rightdepinning currents, γ Ld + (open circles) and γ Rd + n = 1.We set the damping coefficients α = 0 .
05 to simulate aweakly underdamped regime and β = 0 as we are con-sidering quasi-static phase solutions. We assumed a uni-form current distribution, i.e., γ ( τ ) = γ . In addition,the field coupling constant, ∆, was set equal to 1. Inorder to compare the numerical results with the experi-mental findings presented in the next section, we set theannulus aspect ratio to ρ = 1 / (cid:96) = L/λ J , was set to be 10 π ;then, the (smooth) variation of the annulus width oc-curs over a length, L/ . πλ J ≈ λ J , quite largecompared to the fluxon size. A static fluxon centeredeither in the left ( τ = − π/
2) or right well ( τ = π/ γ = 0in (2); then the normalized bias current was ramped-upin small adiabatic increments of 0 .
05 and the stationary,i.e., time-independent solutions recorded until the fluxonwas depinned from its initial state.We first note that the field dependencies of the depin-ning current shown in Figs. 4(a) and (b) are qualita-tively similar despite the quite different ways the par-allel and perpendicular fields affect the fluxon potential.The most evident discrepancy resides in the magneticscales and reflects the fact that the junction cross-sectionseen by a perpendicular field is larger than its paral-lel counterpart . In both cases the exchange of thefluxon initial position is equivalent to a field reversal, i.e., γ Ld + ( h || ) = γ Rd + ( − h || ) and γ Ld + ( h ⊥ ) = γ Rd + ( − h ⊥ ). As ex-pected, the zero-field depinning currents are degenerate, γ Ld + (0) = γ Rd + (0), and constiture an appreciable fraction,that is 41%, of the zero-field critical current; a smallervalue, that is 19%, has been reported for a CAJTJ withaspect ratio ρ = 1 /
2. These different values reflect thefact that the deeper is the potential well and the largestis the bias current needed to unpin the fluxon. As a par-allel magnetic field is turned on, it is seen from Fig. 4(a)that, the left and right depinning currents change quite4inearly and in opposite directions, as long as the ampli-tude of the applied field is smaller than a characteristicfield value h ∗|| where one of the depinning currents van-ishes. In other words, for | h || | ≤ h ∗|| the fluxon escapefrom the | L (cid:105) and | R (cid:105) states occurs at quite different biascurrents. The state with the higher depinning currentcorresponds to a deeper potential well when the CAJTJis unbiased; numerical simulations show that when thefluxon escapes from the state with higher depinning cur-rent it starts to travel along the annulus perimeter andswitches the junction into a dynamic state with a finitevoltage across the junction proportional to the fluxontime-averaged speed. Therefore, in this particular case,the depinning current identifies with the switching cur-rent that can be easily determined experimentally. Thisis not necessarily the case for the state with lower de-pinning current, as it might happen that, once depinned,the fluxon gets trapped in the opposite well which hasan higher depinning current , so that the junction re-mains in a time-independent, i.e. zero-voltage, state.Numerical analysis shows that this situation only occursfor magnetic fields whose absolute values are close to -but lower that - h ∗|| . Furthermore, the re-trapping dras-tically depends on the junction losses that may dissipatethe energy of the depinned fluxon well before it over-comes the opposite well. When the loss parameter α in (2) is decreased to 0 .
01 the re-trapping field rangeshrinks, but does not vanish; therefore, even lower lossesshould be used in the numerical analysis to investigatethe re-trapping conditions. However, great care must betaken to simulate low-damping nonlinear systems, since,besides the longer transients, the results are very sen-sitive to the numerical algorithm adopted to integratethe partial differential equation. Nevertheless, setting α = 0 .
05 the fluxon escaping from the state with thelower depinning range jumps over the opposite well andenters the running mode in a field range of approximately | h || | ≤ h ∗|| /
2. In this range we can talk again of a switch-ing current that coincides with the depinning current. Werecall that in the experiments with high-quality JTJs thelosses drastically decrease with the temperature andit is not difficult to reach damping parameters as smallas 0 . . Notably, the existence of a range of parallelfields in which also the lower depinning current turns theCAJTJ into a finite voltage state implies that a switchingcurrent measurements allows to localize the vortex inone of the two states (this technique has been successfullyused to prove the existence of a DWP in other Joseph-son vortex qubit prototypes ). As | h || | exceeds h ∗|| the unbiased fluxon potential becomes single welled, theinformation about the vortex initial state is lost and theleft and right depinning currents suddenly coincide. Atthis point an eventual reduction of the field amplitudebelow h ∗|| and even its full removal leaves the fluxon inthe left or in the right well depending on its original sign;in different words, the proper ramping up and down ofjust the parallel field represents a viable procedure to TABLE I. Some electrical parameters (at 4 . K ) of our CA-JTJs and the geometrical details of their tunneling area. J c λ J ρ ¯ ν ∆ ν c ∆ w min ∆ w max A LkA/cm µm µm µm µm µm µm prepare the vortex state. Similar considerations also ap-ply to the case of a perpendicular magnetic field; the onlysignificant difference is that the preparation of the fluxonstate, beside a magnetic field | h ⊥ | > h ∗⊥ , also requires asmall bias current that breaks the system symmetry . IV. THE MEASUREMENTS
In this section we report on the switching currents mea-sured on high-quality window-type
N b/Al - AlOx - Al/N b
CAJTJs with a single fluxon trapped during the zero-fieldcooling of the samples through their critical temperature, T c ≈ . K . This process is known to spontaneouslygenerated one or more fluxons on a statistical basis with a probability that increases with the speed of thenormal-to-superconducting transition; at the end of eachquench the number of trapped fluxons is determined bycarefully inspecting the junction current-voltage char-acteristic (IVC) and measuring the voltage of possiblecurrent branches, the so-called zero-field steps , associ-ated with the traveling of the Josephson vortices aroundthe annulus. The samples designed to test the theoryhad the so called Lyngby-type geometry that refers toa specularly symmetric configuration, as that shown inFig. 1, in which the width of the current carrying elec-trodes matches one of the axes of the outer ellipse andthe tunneling area is obtained by the superposition oftwo superconducting rings. More details on the samplesdescription end their fabrication were reported, respec-tively, in Ref. and Ref. . Some electrical parameters(measured at 4 . K ) and the geometrical details of thetunneling area for our CAJTJs are listed in Table I. Thesamples normalized length is L/λ J ≈ . ≈ π . Weobserve that the annulus polar width, ∆ w max , slightlyexceeds the fluxon size, λ J , meaning that the samplesare not strictly one-dimensional. In addition, the small-est curvature radius of the master ellipse occurring atthe equatorial points, cρ sinh ¯ ν ≈ . µm , is smaller thanthe fluxon size, a fact that may induce an interaction(repulsion) between the leading and trailing edges of thefluxon.Several nominally identical CAJTJs fabricated on differ-ent chips within the same run of a standard N b − AlO x − N b trilayer process were investigated. Their electrodeswere either parallel, as shown in Fig. 1, or perpendicularto the annulus major diameter and they all gave qual-itatively similar results regardless of the orientation ofthe bias. More important was found to be the orien-tation of the externally applied in-plane magnetic field;5n Figs. 5(a) and (b) we report the field dependence ofthe positive and negative switching currents measuredat T = 4 . K on two representative samples subject toin-plane magnetic fields having orthogonal orientations,respectively, H || and H ⊥ . The switching currents, I SW ,were obtained by continuously sweeping the bias cur-rent with a sufficiently large symmetric triangular wave-form with a frequency of few hertz and automaticallyrecording the largest (and smallest) zero-voltage currentten times for each value of the externally applied mag-netic field. In agreement with the numerical expectationspresented in the previous Section, in a field range cen-tered around zero, the switching current was found to bedouble-valued. This is a clear indication that the fluxonexperiences a potential profile with two stable states and,when the sweeping current crosses zero, the deceleratingfluxon stops on a statistical basis in either one of the twodifferent-depth potential wells. The pinning process ofa particle slowing down in a DWP drastically dependson the drag force experienced by the particle; in fact, byrepeating the measurements at different temperatures,the double-valued field range progressively shrank as thetemperature, and so the losses, was increased. Highertemperatures also means larger thermal fluctuations thatmight induce hopping between the two states. Othernoise sources, such as current and/or field noise, con-cur on limiting the field range of the fluxon state havingthe lower switching current; in fact, few more values ofthe magnetic field were found to show a double-valuedswitching current using a manual battery-operated ramp-ing of the bias current. The moderate skewness of theswitching current plots is ascribed to fact that for bothsamples the bias current flow occurs in the direction or-thogonal to the applied field ; in this configuration theself-field adds to the external field in the second andfourth quadrants, while in the first and third quadrantsit partially compensates the applied field. We found theswitching current field dependence to be quite linear andthe dashed lines in Figs. 5(a) and (b) are the linear ex-trapolations that help to locate the threshold field H ∗|| , ⊥ ,i.e., the largest theoretical absolute value of the field thatwould yield a double-valued switching current wheneverboth the dissipation and noises can be neglected. There-fore, according to our measurements, the determinationof the fluxon state can be reliably achieved by applyingto an unbiased CAJTJ an in-plane magnetic field that isknown to have quite different switching currents for thethe two states and then incrementing the bias currentwith a constant rate until a sudden jump of the voltagefrom zero to any finite value is detected.It is worthwhile to mention that the finite voltage ob-served after a switch does not necessarily have to be v Φ /L corresponding to the fluxon continuous motionalong the annulus perimeter, L , with a certain time-averaged speed v . In most cases the junction switchedto a state of free-running phase on the McCumber curve,i.e., at a voltage close to the junction gap voltage V g ≈ . mV , in particular when the switch occurred from the state with the higher switching current. It means thatthe fluxon potential is too steep to allow a steady motion;in fact, as the numerical analysis shows, the plasma wavesemitted by the leading (trailing) edge of the accelerating(decelerating) fluxon might grow in size as far as theybreak in a fluxon-antifluxon pair; the process of pair nu-cleation continues until the system becomes unstable .Beside the potential gradient, this radiative process heav-ily depends on several factors, such as the fluxon speedand the annulus perimeter, but again, above all it is de-termined by the system dissipation. Indeed, for the highaspect-ratio CAJTJs considered in this paper, tempera-tures larger than T = 4 . K were needed to observe thezero field steps. This fact indicates that, as the systemlosses are increased, for a given bias current, the fluxonaverage speed gets smaller; in turn, the emitted plasmawaves not only have a smaller amplitude but also dieaway faster and then the conditions are restored for aregular stable motion of the fluxon. V. CONCLUSION
A Josephson vortex trapped in a long and narrow CA-JTJ in the presence of a in-plane magnetic field is subjectto a periodic asymmetric and fine-tunable double well po-tential accurately modeled by a modified and perturbedone-dimensional sine-Gordon equation. The key ingredi-ent of this potential is the smoothly distributed modula-tion of the planar tunnel barrier width whose gradient isrelated one-to-one to the eccentricity of the elliptical an-nulus. Numerical simulations and experiments carriedon high-quality
N b/Al - AlOx - Al/N b samples with onetrapped fluxon demonstrate that a robust two-minimapotential can be tailored whose inter-well barrier growsas the annulus is made more eccentric. We have foundthat the depinning of the fluxon occurs at bias currentthat, among other things, depends on the potential wellfrom which the fluxon is escaping. We considered uni-form magnetic fields applied in the barrier plane eitherperpendicular or parallel to the major axis of the CA-JTJ. In both cases, although for different reasons, a widerange of magnetic field was observed characterized bytwo-valued depinning currents, each value correspondingto either one of the potential stable states. Experimentscarried out on CAJTJs with an aspect ratio of 4 : 1 indi-cate that these ranges grow as the temperature, and sothe fluxon losses, and/or as the noise sources are reduced.The eccentricity of a CAJTJ also has a drastic effect onthe motion of a fluxon whose total energy exceeds thepotential energy. In fact, as the aspect ratio is increased,the growing accelerations and decelerations the travelingfluxon experiences when approaches the region of, respec-tively, smallest and larger width are responsible of a peri-odic radiation of small amplitude waves that destabilizethe forward fluxon advancement. Both numerical simu-lations and experiments with slightly damped CAJTJshaving an aspect ratio of 4 : 1 showed no manifestation of6he fluxon propagation along the annulus perimeter.Despite all the above considerations, we observed thatthe eccentricity of the CAJTJs does not affect the ro-bustness and reliability of the operation of a CAJTJ as aJosephson vortex two-state system. Only small quantita-tive differences were observed in the depinning currentsof samples with different aspect ratios. In conclusion, wehave demonstrated the full reliability of the proceduresfor both the preparation and the determination of thevortex in either one of the two potential minima, that are important for the possible realization of a vortex qubit.
ACKNOWLEDGMENT
RM and JM acknowledge the support from the DanishCouncil for Strategic Research under the program theDanish National Research Foundation (bigQ). LVF ac-knowledges support from the Russian Foundation for Ba-sic Research, grant No. 17-52-12051. ∗ Corresponding author’s e-mail: [email protected] † Author’s e-mail: [email protected] ‡ Author’s e-mail: [email protected] F. Hund,
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0) -, but different aspectratios, ρ ,: (a) ρ = 1 / ν o ≈ . ν i ≈ . ρ = 1 / ν o ≈ . ν i ≈ . ν ≡ ( ν o + ν i ) /
2. The annuli are built to have the sameequatorial or minimum width, ∆ w min = c sinh ¯ ν ∆ ν = 0 . ν ≡ ν o − ν i . Bottom panels: The annuli width, ∆ w ,varies with the angular elliptic coordinate, τ , as given by (1). IG. 3. (Color online) Schematic representation of the fluxonpotential under different conditions. The dotted line atthe bottom refers to the symmetric double-well potentialˆ U w + ˆ U h ⊥ in the presence of a uniform in-plane magneticfield perpendicular to the long annulus diameter with twominima at τ ≈ ± π/ | R (cid:105) and | L (cid:105) ; the dashed line corresponds to ˆ U w + ˆ U γ and dis-plays the tilting of the potential due to a uniform bias current;the solid line shows the asymmetric potential ˆ U w + ˆ U h ⊥ + ˆ U γ in the more general case of applied (perpendicular) magneticfield and bias current. The three potentials are shifted byarbitrary vertical offsets. a)(b) FIG. 4. (Color online) Numerically computed field dependen-cies of the positive fluxon depinning currents of the | L (cid:105) (opencircles for γ Ld + ) and | R (cid:105) (stars for γ Rd + ) states for two valuesof the in-plane field orientation with respect to the annulusmajor axis: (a) parallel field, h || , and (b) perpendicular field, h ⊥ . The magnetic fields are normalized to J c c . a)(b) FIG. 5. (Color online) Positive and negative switching cur-rents, I SW , recorded at T = 4 . K , by continuously sweep-ing the bias current as an in-plane magnetic field is changed;the dashed lines are the extrapolations of the almost linearbranches that help to locate the threshold field H ∗|| , ⊥ ; (a)parallel field, H || , and (b) perpendicular field, H ⊥ ..