Shock wave in series connected Josephson transmission line: Theoretical foundations and effects of resistive elements
aa r X i v : . [ c ond - m a t . s up r- c on ] M a r Shock wave in series connected Josephson transmission line
Eugene Kogan ∗ Jack and Pearl Resnick Institute, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel (Dated: March 2, 2021)We analytically calculate the velocity of propagation and structure of shock waves in the Josephsontransmission line (JTL), for which the legs are constructed from Josephson junctions (JJ) and linearinductors in series; the capacitors are between the legs of the line. In the absence of ohmic dissipationthe shocks are infinitely sharp. As such they remain when ohmic resistors are introduced in serieswith JJ and linear inductors. However, ohmic dissipation in this case leads to decrease of the shockamplitude with time. When ohmic resistor shunts JJ or is in series with the capacitor, the shocksobtain a finite width. We write down the shock profile when JJ is shunted simultaneously by anohmic resistor and a capacitor using Weierstrass elliptic function. In all the cases considered, ohmicresistors (and shunting capacitors) don’t influence the shock propagation velocity. We presenta simple picture of shock wave formation for dissipationless JTL and formulate the simple waveapproximation for the JTL with ohmic dissipation.
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I. INTRODUCTION
The concept that in a nonlinear wave propagation sys-tem the various parts of the wave travel with different ve-locities, and that wave fronts (or tails) can sharpen intoshock waves, is deeply imbedded in the classical theory offluid dynamics . The methods developed in that field canbe profitably used to study signal propagation in nonlin-ear transmission lines . In the early studies of shockwaves in transmission lines, the origin of the nonlinearitywas due to nonlinear capacitance in the circuit .An interesting and potentially important case of non-linear transmission lines is the circuits containing Joseph-son junctions (JJ) - Josephson transmission lines(JTL) . The unique nonlinear properties of JTL al-low to construct soliton propagators, microwave oscil-lators, mixers, detectors, parametric amplifiers, analogamplifiers .Transmission lines formed by in series connectedJJ were studied beginning from 1990s, though muchless than transmission lines formed by JJ connectedin parallel . However, the former began to attractquite a lot of attention recently , especially in con-nection with possible JTL traveling wave parametricamplification .Although numerical calculations have revealed the ex-istence of shock waves in series connected JTL , theanalytical solutions have not been successfully derivedso far. (Some recent analytical results were presentedrecently ). In the present paper we study propagationof shock wave in the JTL analytically, paying especialattention to the influence of ohmic resistance which in-evitably exists in the system.The rest of the paper is constructed as follows. InSection II we analyse shocks in the JTL in the absence ofany ohmic dissipation. Shocks in several possible cases ofintroduction of ohmic resistors into the JTL are studiedin Section III. Shock wave generation is illustrated inSection IV. Application of the results obtained in the paper and opportunities for their generalization are verybriefly discussed in Section V. We conclude in Section VI.JTL equations in the framework of Lagrange approachare derived in Appendix A. Some mathematical detailsare relegated to Appendix B. II. DISSIPATIONLESS JTL
Let us start from the dissipationless model of JTL con-structed from identical JJ, linear inductors and capaci-tors, and indicated in Fig. 1. Here ∆ x is the period ofthe line, ℓ is the linear inductance, c is the capacity. The FIG. 1: Dissipationless JTL. circuit equations are given by c ( q n +1 − q n ) = − ℓ dI n dt − ~ e dϕ n dt , (1a) dq n dt = I n − − I n , (1b) I n = I c sin ϕ n , (1c)where the index n enumerates JJ, I c is the critical cur-rent, q n and ϕ n are the charge of the n th capacitor andthe phase difference of the n th JJ, I n is the currentthrough the junction, E J = ~ / e . Equation (1a) is thecombination of Lenz and the second Josephson laws, Eq.(1b) is the Kirchhoff’s law, Eq. (1c) is the first Josephsonlaw.Assuming smooth variance of ϕ n and q n with n , wecan go to the continuous approximation ∂V∂x = − L ∂∂t ( I + I L ϕ ) , (2a) C ∂V∂t = − ∂I∂x , (2b) I = I c sin ϕ , (2c)where C = c/ ∆ x is the capacitance per unit length, L = ℓ/ ∆ x is the linear inductance per unit length, and I L = E J /ℓ . The reader should constantly keep Eq. (2c)in mind because we will switch freely between I and ϕ description.We can eliminate V from Eq. (2) to obtain ∂ I∂x − u ∂∂t ( I + I L ϕ ) = 0 , (3)where u = 1 / √ LC is the signal propagation velocity inthe case when I L = 0, and the line is linear. Remind thatsuch line has characteristic impedance Z = p L/C . Note that it is the second term in the r.h.s. of Eq. (3)which is responsible both for the dispersion and for thenonlinearity, which are hence of the same order.When we consider propagation of a small amplitudedisturbance of a uniform state we can present the currentas I ( x, t ) = I + δI ( x, t ) and linearize with respect to δI . In this approximation (3) is reduced to linear waveequation, describing propagation of the disturbance withthe velocity1 u ( I ) = 1 u (cid:18) I L ∂ϕ∂I (cid:19) = 1 u (cid:20) I L I c cos ϕ (cid:21) = 1 u (cid:20) I L ( I c − I ) / (cid:21) . (4)For the analysis of propagation of strong signals thesystem of first order differential equations (2) is moreconvenient than a single second order differential equa-tion (3). Assume the existence of the shock wave propa-gating with velocity U . Consider two points, x and x ;the point x is assumed to be just to the left of the steepportion of the wave front, while the point x is just to theright of it. Now let us integrate equations (2) betweenthe two points. This gives∆ V = − L ∂∂t Z x x ( I + I L ϕ ) dx , (5a) C ∂∂t Z x x V dx = − ∆ I . (5b)These equations can be assumed to be accurately valid,even in the region where V and I are changing veryrapidly. Now if x , and x are sufficiently close to the steepregion, then the integrals involved in Eqs. (5) have atime derivative only because of the motion of the steepportion, and the slower changes due to the motion of theparts of the wave with moderate slope can be neglected.The integral changes because in a time dt a section ofline U dt in length and within the range of integration hasvalues V ( x ) and I ( x ) replaced by V ( x ) and I ( x ). Eqs. (5) therefore become:∆ V = − U L (∆ I + I L ∆ ϕ ) , (6a) U C ∆ V = − ∆ I . (6b)Eliminating ∆ V from Eqs. (6) we find1 U ( I , I ) = 1 u (cid:18) I L ∆ ϕ ∆ I (cid:19) = 1 u (cid:18) I L ∆ ϕ ∆ sin ϕ (cid:19) . (7)The values I and I enter into Eq. (7) in a symmetricalway. However, in Section III A Section we’ll show thatdue to ohmic resistance, inevitably present in the system,the absolute value of the current before the shock frontis always higher than that behind the shock front III. JTL WITH OHMIC DISSIPATIONA. Resistor shunting the JJ
Let us first introduce ohmic resistor which shunts theJJ, thus considering the transmission line presented onFig. 2. In this case Eq. (2) changes to
FIG. 2: JTL with ohmic resistor shunting the JJ. ∂V∂x = − L ∂∂t ( I + I L ϕ ) , (8a) C ∂V∂t = − ∂∂x (cid:18) I + E J r ∂ϕ∂t (cid:19) . (8b)These equations have a set of particularly simple solu-tions, called traveling waves, in the form( V ( x, t ) , I ( x, t )) = ( V ( X ) , I ( X )) , X = x − U t. (9)These solutions satisfy ordinary differential equations dVdt = U L ddt ( I + I L ϕ ) , (10a) U C dVdt = ddt (cid:18) I + E J r dϕdt (cid:19) . (10b)Eliminating V and integrating once we obtain E J r dϕdt + I − U ( I + I L ϕ ) + A = 0 , (11)where A is an arbitrary constant, and U = U/u .We are looking for a solution which tends to constantsat infinity lim t →−∞ I = I , lim t → + ∞ I = I . (12)Note that time independence of I at large X means thatthe dissipating current flows only in the vicinity of theshock. For given I and I , the parameters U, A mustsatisfy I i − U ( I i + I L ϕ i ) + A = 0 , i = 1 , . (13)Eliminating A we recover Eq. (7). Thus the relationbetween the shock velocity and the values of current onboth sides of the shock is dissipation independent.Equation (11) with the boundary conditions (12) canbe presented as ℓr dϕdt + U M ( ϕ ) = 0 , (14)where M ( ϕ ; ϕ , ϕ ) = 1sin ϕ − sin ϕ (15)[(sin ϕ − sin ϕ )( ϕ − ϕ ) − ( ϕ − ϕ )(sin ϕ − sin ϕ )] . Further on we will frequently suppress arguments ϕ and ϕ of M , like we did in Eq. (14).Zero of M ( ϕ ) at ϕ = ϕ between ϕ and ϕ would cor-respond to splitting of the single shock considered aboveinto two shocks: one between the current I and I , andthe other between the currents I and I . Thus M ( ϕ )has constant sign between ϕ and ϕ (and is equal tozero at ϕ and ϕ ). Looking at Eq. (14) we realize that dM ( ϕ ) /dϕ | ϕ = ϕ is always positive, and dM ( ϕ ) /dϕ | ϕ = ϕ is always negative. On the other hand, dM ( ϕ ) dϕ (cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ − dM ( ϕ ) dϕ (cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ = (cos ϕ − cos ϕ ) ( ϕ − ϕ )sin ϕ − sin ϕ . (16)Thus we realize that cos ϕ > cos ϕ , that is | I | > | I | - the absolute value of the current before the shock isalways higher than that behind the shock.Let us return to the issue of possible zero of M at some ϕ = ϕ . Such ϕ should satisfy equationsin ϕ − sin ϕ ϕ − ϕ = sin ϕ − sin ϕ ϕ − ϕ . (17)Geometrically, Eq. (17) means that the secant intersect-ing sin ϕ curve at the points ϕ and ϕ , intersects the curve also at some other point ϕ . Since sin ϕ is con-cave downward for 0 < ϕ < π/
2, and concave upwardfor − π/ < ϕ <
0, the secant can’t intersect the curvebetween ϕ and ϕ having the same sign. In other words,a single shock can exist between any currents I and I having the same direction. On the other hand, if ϕ and ϕ have opposite signs and cos ϕ > cos ϕ , there is nozero of M ( ϕ ) between ϕ and ϕ , only ifcos ϕ > ∆ ϕ ∆ sin ϕ > cos ϕ . (18)(Inequalities (18) for any ϕ and ϕ which have the samesign and such that cos ϕ > cos ϕ , are satisfied auto-matically because of concavity of function sin ϕ .) Look-ing back at Eq. (7), we realize that inequalities (18) areequivalent to u ( I ) > U ( I , I ) > u ( I ) . (19)The latter inequalities reflect the well known fact: shockvelocity is lower than the sound velocity in the regionbehind the shock, but higher than the sound velocity inthe region before the shock .Equation (14) can be integrated as X = Z RU ( I , I ) Z dϕM ( ϕ ) , (20)where R = r/ ∆ x is ohmic resistance per unit length. No-tice that, apart from the scale factors, the shape of theshock depends only upon ϕ and ϕ and is independentupon the parameters of the transition line. The width ofthe shock is obviously inversely proportional to the shunt-ing resistance. Equation (20) is presented graphically onFig. 3 - - -
10 0 10 20 300.00.20.40.60.81.0 Rx / Z I / I c FIG. 3: Shock profile according to Eq. (20). I L is chosen tobe I c /
2. Blue solid line corresponds to I = . I c , I = . I c ,red dot-dashed line - to I = . I c , I = . I c , green dashed line- to I = . I c , I = . I c . For weak shock ( | I − I | ≪ | I | ), Eq. (20) is simplifiedto X = 2 Z cot ϕRu ( ϕ ) Z dϕ ( ϕ − ϕ )( ϕ − ϕ ) . (21)Calculating the integral we obtain I = I + ∆ I αX ) , (22)where ∆ I = I − I , I = ( I + I ) /
2, and α = Ru ( I )4 Z I ∆ II c − I . (23)Thus the difference between the current and its limitingvalue decreases exponentially when | X | → ∞ . (As canbe seen from Eq. (14), this is the property of any shock,and not only of weak one.) B. Resistor in series with the capacitor
Alternatively, we can introduce an ohmic resistor inseries with the capacitor, thus considering JTL presentedon Fig. 4. Equations (2) in this case change to
FIG. 4: JTL with ohmic resistor in series with the capacitor. ∂V∂x = − L ∂∂t ( I + I L ϕ ) , (24a) ∂Q∂t = − ∂I∂x , (24b) V = QC + R ∂Q∂t (24c)where Q is the charge per unit length, R = r ∆ x is theohmic resistance per unit width.Looking for the solution in the form( V ( x, t ) , I ( x, t ) , Q ( x, t )) = ( V ( X ) , I ( X ) , Q ( X )) , (25)we obtain, after elimination of V and Q and one integra-tion, an ordinary differential equation R C dIdt + I − U ( I + I L ϕ ) + A = 0 . (26)Presented in Section III A analysis of the shock wave canbe repeated almost verbatim, and we recover, in partic-ular, Eq. (7). Instead of Eq. (20) we now obtain X = R I c Z U ( I , I ) I L Z cos ϕdϕM ( ϕ ) , (27) Eq. (22) remains as it is, and analog of Eq. (23) is α = Z u ( I )4 R I L I ∆ I ( I c − I ) / . (28)The width of the shock is proportional to the ohmic re-sistance between the legs. C. Resistor in series with the JJ
Let us now introduce ohmic resistor in series with theJJ, thus considering transmission line presented on Fig.5. In this case Eq. (2) changes to
FIG. 5: JTL with ohmic resistor in series with the JJ. ∂V∂x = − L ∂∂t ( I + I L ϕ ) − RI , (29a)
C ∂V∂t = − ∂I∂x . (29b)Let us look for traveling waves solutions of Eq. (29)(see Eq. (9)). For these solutions Eq. (29) is reducedto a system of ordinary differential equations, and Eq.(29b) can be easily integrated. Substituting the integralinto Eq. (29a) we obtain ddX (cid:20) IU C − U L ( I + I L ϕ ) (cid:21) = RI . (30)Integrating Eq. (30) in the infinitesimal vicinity of dis-continuity we recover Eq. (7). Thus Eq. (29) allowsinfinitely sharp shocks, same as Eq. (2), with the samevelocity of shock propagation. An equivalent method ofanalysis of moving discontinuities is presented in the Ap-pendix B.Equation (30) can be presented as (cid:20) U − u ( I ) (cid:21) dIdt = RCI . (31)The quantity in the square brackets in the l.h.s. of Eq.(31) is positive behind the shock and negative before theshock (see (19)). Thus absolute value of the current be-hind the shock increases with time, and absolute value ofthe current before the shock decreases with time. Henceohmic dissipation in the case considered leads to decreaseof amplitude of the shock with time.
D. Resistor and capacitor shunting the JJ
A model for a Josephson junction, more realistic thanthe one consider above, would include a capacitor in par-allel with the JJ. Let us modify JTL considered in SectionIII A, as indicated in Fig. 6. In this case Eq. (8) changes
FIG. 6: JTL with ohmic resistor and capacitor shunting theJJ. to ∂V∂x = − L ∂∂t ( I + I L ϕ ) , (32a) C ∂V∂t = − ∂∂x (cid:18) I + E J r ∂ϕ∂t + c E J ∂ ϕ∂t (cid:19) . (32b)Looking for traveling wave solutions (9), eliminating V and integrating once we obtain closed equation for thephase c E J d ϕdt + E J r dϕdt + I c sin ϕ − U ( I c sin ϕ + I L ϕ ) + A = 0 , (33)where A is an arbitrary constant. The boundary condi-tions (12) alow us to recover Eq. (7). Thus the relationbetween the shock velocity and the values of current onboth sides of the shock does not change in this case either.Equation (33) with the boundary conditions (12) canbe presented as c ℓ d ϕdt + ℓr dϕdt + U M ( ϕ ) = 0 , (34)It will be convenient for us to present Eq. (34) in theform d ϕdτ + γ dϕdτ + d Π( ϕ ) dϕ = 0 , (35)where γ = s ℓ ∆ ϕr c U , τ = t/ ( γc r ) , Π( ϕ ) = 24∆ ϕ h ϕ (cid:16) ϕ − ϕ (cid:17) − ( ϕ − ϕ )(cos ϕ + ϕ sin ϕ )sin ϕ − sin ϕ (cid:21) . (36) Thus the JTL equations are equivalent to equation de-scribing motion of the fictitious particle under the influ-ence of the potential force and the friction force . Forshock problem the particle (asymptotically) starts in oneequlibrium position and finishes in the other equilibriumposition with lower potential energy. Though Eq. (35) isnon integrable, qualitatively the picture is clear (at leastin the regimes γ ≫ γ ≪ γ ≪ (cid:18) dφdτ (cid:19) + Π( φ ) = E . (37)For the time being, let us limit ourselves by the case ofweak shocks. Let us make a change of variables ψ = 2 ϕ − ( ϕ + ϕ ) ϕ − ϕ ; (38)the t = −∞ state corresponds to ψ = 1, and the t =+ ∞ state corresponds to ψ = −
1. The potential energywritten down in the new variables isΠ w ( ψ ) = − ψ + 6 ψ , (39)and Eq. (37) takes the form (cid:18) dψdτ (cid:19) = 4 ψ − ψ + 2 E . (40)Equation (40) is the equation defining Weierstrass ellipticfunction (with g = 12). Thus, the solution can beimmediately written down as ψ = P ( τ ; 12 , − E ) . (41)Now small damping can be taken into account usingthe method of energetic balance . Equation (35) forweak shock is simplified to d ψdτ + γ dψdτ + d Π w ( ψ ) dψ = 0 . (42)From Eq. (42) follows ddτ " (cid:18) dψdτ (cid:19) + Π w ( ψ ) = − γ (cid:18) dψdτ (cid:19) . (43)We average Eq. (43) during the period of oscillations dEdt = − γT Z T (cid:18) dψdτ (cid:19) dτ . (44)The method of energetic balance is based on two assump-tions. First, that in the integrand in Eq. (44) we can use ψ ( τ ) calculated for the undamped motion. Here proba-bly is necessary to look at Fig. 7. The potential energyhas absolute minimum Π ( min ) w = − ψ = − ψ = 1. We make now an ad hoc approxi- - - - ψ E Π w ( ψ ) FIG. 7: Potential energy of the fictitious particle according toEq. (39) with γ = .
001 (dashed blue curve) and the particletrajectory in phase space according to Eq. (46) (solid redcurve). mation which is not the part of the method. We take thekinetic energy in the integrand, to be equal to one halfof E − Π ( min ) w , and consider it to be constant during thewhole period. The approximation is certainly bad for thevery initial phase of the motion and not too good for thevery final phase either. But we believe that it correctlydescribes the rest of the motion. Recalling our problem,we claim to describe the body of the shock wave. Afterthat approximation is made, the rest is trivial. The in-tegral in Eq. (44) is calculated, and the solution of theresulting differential equation gives the time dependenceof the energy E ( τ ) = Π min + [ E (0) − Π min ] e − γτ . (45)Now comes the second canonical assumption of themethod: we take the solution obtained in the absenceof friction and substitute the calculated time dependentenergy for the previously constant one. Thus, finally, weobtain the solution ψ ( τ ) = P ( τ ; 12 , − E ( τ )) . (46)This solution is presented on Fig. 39. Returning toour shock problem we see that for large shunting capac-itance the phase (current) in the shock wave oscillatesas presented on Fig. 7, which is in strong contrast tomonotonous change in the case of small shunting capac-itance.The result obtained above for weak shock certainly re-mains semi quantitatively valid for shock of arbitrary am-plitude. On Figure 8 we plotted Π( ϕ ) for some specificvalues of ϕ and ϕ . The plot looks very much similar tothe potential on Fig. 7. φ Π ( φ ) FIG. 8: Effective potential energy for the fictitious particleaccording to Eq. (36); ϕ = . ϕ = 1. IV. FORMATION OF SHOCK WAVE
A relevant question is how the shock waves describedabove are formed. The aim of this Section is to presentqualitative explanation of the process.
A. Dissipationless JTL
Let us consider dissipationless JTL and write down Eq.(2) in a more explicit way ∂V∂x = − Cu ( I ) ∂I∂t , (47a) C ∂V∂t = − ∂I∂x , (47b)where u ( I ) is given by Eq. (4).The simple wave approximation consists in seeking asolution of Eq. (47) in the form of a simple wave, i.e, inassuming that V = V ( I ). The assumption being made,Eq. (47) takes the form dVdI ∂I∂x = − Cu ( I ) ∂I∂t , (48a) C dVdI ∂I∂t = − ∂I∂x . (48b)These are two equations for one variable and so the co-efficients of the derivatives must be the same, i.e, C (cid:18) dVdI (cid:19) = 1 u ( I ) . (49)Solving Eq. (49) for dV /dI and substituting into any oneof Eqs. (48) we obtain, for the wave going to the right,the first order partial differential equation ∂I∂t + u ( I ) ∂I∂x = 0 (50)(for the wave going to the left, the sign before the deriva-tive with respect to x should be minus.Solutions of Eq. (50) can have discontinuities. If I and I are the values of currents to the left and to theright of the discontinuity respectively, such discontinuitywill propagate with the velocity e U = 1∆ I Z I I dIu ( I ) (51)(and is called shock).Equation (51) doesn’t coincide exactly with Eq. (7).The point is that the simple wave assumption, thoughtraditionally made , is in general case an uncontrollableapproximation. However the results of Eqs. (51) and (7)are very much alike, so one can believe that the assump-tion is not bad. In particular, for weak shock, when wecan restrict ourselves by linear with respect to I − I approximation, the results of both equations coincide e U = u ( I ) + u ( I )2 = U ( I , I ) . (52)Equation (50), in distinction to Eq. (2) (or Eq. (3)),can be easily solved. Solution of Eq. (50), written in aparametric form , is I ( x, t ) = f ( ξ ) , (53a) x = ξ + c ( f ( ξ )) t , (53b)where the function f is determined by the initial condi-tions f ( ξ ) = I ( ξ, . (54)The solution has a very simple meaning . Initial valueof I at a given point ξ is propagating without changealong the characteristic given by Eq. (53b).The solution (53) describes, in particular, shock for-mation. If the initial conditions are such that | I ( ξ ) | in-creases with ξ , and hence u ( ξ ) decreases with ξ , initialvalues corresponding to larger | I | will propagate slowerthan those corresponding to smaller | I | . The character-istics will start to cross, and the shock will form .Notice that if initial conditions I ( ξ,
0) have disconti-nuity and | I | > | I | , where I and I are the values ofcurrents to the left and to the right of the discontinuityrespectively, the solution, instead of shock, contains anexpansion fan . Thus if we take initial conditions in theform I ( ξ,
0) = I H ( − ξ ) + I H ( ξ ) , (55)the solution would be I ( x, t ) = I , for x > u ( I ) t I ( x/t ) , for u ( I ) t ≤ x ≤ u ( I ) tI , for x < u ( I ) t , (56)where the function I ( u ) is obtained by inverting Eq. (4) I ( u ) = (cid:20) I c − I L u ( u − u ) (cid:21) / . (57) B. Transmission line with ohmic dissipation
Let us start from JTL containing an ohmic resistorin series with the capacitor, considered in Section III B.Equation (24) after elimination of V and Q takes theform ∂ I∂x − ∂∂t (cid:20) u ( I ) ∂I∂t (cid:21) + R C ∂ I∂t∂x = 0 . (58)We assume that the right going wave satisfies equation ∂I∂t + u ( I ) ∂I∂x = R C ∂ I∂t . (59)If we consider travelling wave solution I ( x, t ) = I ( x − U t ) , (60)Eq. (59) is reduced to ordinary differential equation (cid:2) u ( I ) − U (cid:3) dIdX = R CU d IdX . (61)If we integrate Eq. (61) with respect to x from −∞ to + ∞ and take into account boundary conditions (12),the r.h.s. of the equation integrates to zero. Thus wereproduce Eq. (51) - the velocity of shock propagation isdissipation independent.For weak shock, taking into account Eq. (52), we maypresent Eq. (61) as( I − I ) β ( I ) dIdX = R Cu ( I ) d IdX , (62)where β ( I ) = dudI = − u ( I )2 u I L I ( I c − I ) / . (63)One can easily check up that the weak shock profile ob-tained in Section III B (Eq. (22) with α given by Eq.(28)) is the solution of Eq. (62).Equation (59) can not be solved as easily, as Eq. (50),though by all means is simpler than Eq. (58). In addi-tion, when the signal can be considered as a small ampli-tude disturbance of a uniform state, we can present thecurrent as I ( x, t ) = I + δI ( x, t ) and (in the lowest orderwith respect to δI ) simplify Eq. (59), writing it down as ∂δI∂t + h u ( I ) + β ( I ) δI i ∂δI∂x = R C ∂ δI∂t . (64)It can be checked up by inspection that Eq. (64) does notdepend upon I , provided it satisfies inequality | I ( x, t ) − I | ≪ | I | . Equation (64) allows, in particular, to studyweak shocks formation.Let us turn now to JTL with ohmic resistor shuntingthe JJ, considered in Section III A. Equation (8) afterelimination of V takes the form ∂ I∂x − ∂∂t (cid:20) u ( I ) ∂I∂t (cid:21) + LR I L ∂ ϕ∂t∂x = 0 . (65)We assume that the right going wave satisfies equation ∂I∂t + u ( I ) ∂I∂x = LR I L ∂ ϕ∂t . (66)Hence Eq. (61) changes to (cid:2) u ( I ) − U (cid:3) dIdX = LR U I L d ϕdX . (67)The proof of the dissipation independence of the shockpropagation velocity presented right after Eq. (61) re-mains exactly the same. Equation (62) changes to( I − I ) β ( I ) dIdX = LR I L ( I c − I ) / u ( I ) d IdX . (68)Again, one can easily check up that the weak shock pro-file obtained in Section III A (Eq. (22) with α given byEq. (23)) is the solution of Eq. (68). Finally, Eq. (64)changes to ∂δI∂t + h u ( I ) + β ( I ) δI i ∂δI∂x = LR I L ( I c − I ) / ∂ δI∂t . (69) V. DISCUSSION
We hope that the results obtained in the paper are ap-plicable to kinetic inductance based traveling wave para-metric amplifiers based on a coplanar waveguide archi-tecture. Onset of shock-waves in such amplifiers is anundesirable phenomenon. Therefore, shock waves in var-ious JTL should be further studied, which was one ofmotivations of the present work.Recently, quantum mechanical description of JTL ingeneral and parametric amplification in such lines in par-ticular started to be developed, based on quantisationtechniques in terms of discrete mode operators , contin-uous mode operators , a Hamiltonian approach in theHeisenberg and interaction pictures , or the quantumLangevin method It would be interesting to understandin what way the results of the present paper are changedby quantum mechanics.
VI. CONCLUSIONS
We have analytically calculated the velocity of propa-gation and structure of shock waves in the transmissionline, for which the legs are constructed from JJ and lin-ear inductors in series; the capacitors are between thelegs of the line. In the absence of any ohmic dissipa-tion the shocks are infinitely sharp. As such they remainwhen ohmic resistors are introduced in series with JJand linear inductors. However, ohmic dissipation in thiscase leads to decrease of the shock amplitude with time. When ohmic resistors shunts JJ or is in series with thecapacitor, the shock obtains a finite width. The shockwidth is inversely proportional to the resistance shuntingJJ, or proportional to the resistance in series with the ca-pacitor. In all the cases considered, ohmic resistors don’tinfluence the shock velocity. We write down the shockprofile when JJ is shunted simultaneously by an ohmic re-sistor and a capacitor using Weierstrass elliptic function.In all the cases considered, ohmic resistors (and shuntingcapacitors) don’t influence the shock propagation veloc-ity. We present a simple picture of shock wave formationfor dissipationless JTL and formulate the simple waveapproximation for the JTL with ohmic dissipation.
Acknowledgments
Discussions with L. Friedland, M. Goldstein, H.Katayama, K. O’Brien, T. H. A. van der Reep, B. Ya.Shapiro, A. Sinner, F. Vasko, M. Yarmohammadi, R.Zarghami and A. B. Zorin are gratefully acknowledged.
Appendix A: Derivation of JTL equations inLagrange approach
Equations (2) simply follows from Kirchhoff’s andJosephson law. However, in this Appendix we would liketo rederive the equations in the framework of Lagrangeapproach. We take as dynamical variables the phase atthe n th JJ and the charge which has passed through the n th inductor e q n . The charge at the n th capacitor q n issimply connected with e q n : e q n = q n +1 − q n . (A1)Differentiating Eq. (A1) with respect to t and taking intoaccount the relation I n = d e q n /dt we recover Eq. (1b).The Lagrangian L is L = ℓ X n (cid:18) d e q n dt (cid:19) − c X n q n + E J X n cos ϕ n + ~ e X n d e q n dt ϕ n , (A2)where E J = ~ I c / e Differentiating the Lagrangian withrespect to ϕ n we obtain − E J sin ϕ n + ~ e d e q n dt = 0 , (A3)thus recovering Eq. (1c). Finally, Lagrange equationcorresponding to e q n is ℓ d e q n dt + E J ddt ϕ n + 12 c ∂∂ e q n X n q n = 0 . (A4)Elementary algebra gives ∂∂ e q n X n q n = 2( q n +1 − q n ) , (A5)and we recover Eq. (1a). Appendix B: Alternative analysis of discontinuities
We present here the method to study discontinuities alternative (though equivalent) to that used in Section II.Consider JTL with resistor in series with the JJ. Askingwhether Eq. (29) admits moving discontinuities as partof the solution, we try to look for a solution in the form I ( x, t ) = I ( x, t ) H ( U t − x ) + I ( x, t ) H ( x − U t ) , (B1a) V ( x, t ) = V ( x, t ) H ( U t − x ) + V ( x, t ) H ( x − U t ) . (B1b)where H ( x ) is the Heaviside step function and I , I , V , V are continuous functions. On substitution(B1) into Eq. (29) we obtain (keeping only the singular terms) n [ V ( x ) − V ( x )] + U L [ I ( x ) − I ( x )]+ U I L [ ϕ ( x ) − ϕ ( x )] o δ ( x − U t ) = 0 , (B2a) n U C [ V ( x ) − V ( x )] + [ I ( x ) − I ( x )] o δ ( x − U t ) = 0 . (B2b)We deduce that the multipliers before the δ functions areequal to zero, thus recovering Eq. (5).Notice, that if we make substitution (B1) into Eq. (8),the second derivative ∂ /∂x∂t will introduce the termwith δ ′ ( x − U t ) and there is no other that singular termto balance it. Thus there are no current or voltage dis-continuities in the case of ohmic resistor shunting JJ.Similar arguments explain the absence of discontinuitiesin the case of ohmic resistor in series with the capacitor. ∗ Electronic address: [email protected] G. B. Whitham,
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