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ThP29 QUANTUM JOSEPHSON D/A CONVERTER DRIVEN BY TRAINS OF SHORT 2~-PULSES

V.M.Buchstaber, O.V.Karpov*, S.I.Tcrtychniy ' E m a i l meraomi i f t r i .TU

National Scientific and Research Institute for Physical-Technical and Radiotechnical Measurements (VNIIFTRI) VNIIFTRI. Mendeleevo, Moscow region 141570, Russia

Abs t rac t The effects used for quantum Josephson digital-to- analogue (D/A) conversion are discussed. Dynamics of an overdamped Josephson junction (4 driven by a series of short 2?r-pulses of programmable intensity and repetition rate is described. The results are presented both analytically and in graphical form.

Summary Operation principles of Josephson D/A converters driven by a series of pulses of programmable repetition rate are described in a number of papers [l]. In this report we consider the operation principles of quantum JI D/A con- version driven by digitally programming series of short Z?r-pulses. Our novel analytical method [2,3] and WJ model based equation

+ ( t ) + S i n & ) = c(t) , (1) are used for analysis of dynamicj of overdamped JI ex- cited by periodic series of short pulses. Here ~ ( t ) is the phase of the order parameter function, t = wee, 8 is a current dimensional time, w. = I c R ~ K j , I , is the JI critical current, RN is .U normal resistance, K J is Joseph- son constant, ~ ( t ) is JI bias current. Such problems as stability of solutions, transient processes caused by the changing of exitation conditions and accuracy of DfA conversion are discussed in section 4. 1. Let us represent the periodic series of short pulses of a bias current as follows

L ( t ) = LO + 271q1 UT(t - kT) k

Eqs. (3a) and (3b) describe unipolar and bipolar pulses, respectively, T is the pulse repetition period, T is the lapse between pairs of pulses of opposite polarity, Np = 0 , f l is a PO hit parallel code of 2?r-pulses, LO = I / I , where I is external DC bias, q1 = ( % / I c ) Q1, where Q1 is a dimensional charge carried by a pulse, qo is dimen- sionless charge carried by a single 2n-pulse which numer- ically equals the unity.

Note that the mathematical representation of driving short pulses in the form of delta functions is correct if pulse duration is significantly less than w;'. The current LO, the dimensionless pulse charge 2nql and the pulse r e p etition period T determine the JJ operation mode. The current ca averaged over n = 2POm periods equals

271 271 T T

t+nT

L(t)dt = LO + C - q i + -qoA, (5)

where lAl = ICz, 2-PNpl < 1, m = 1 , 2 , 3 . . . , L = 1 for unipolar and c = 0 for bipolar pulses. 2. The phase locking condition for the operating mode ( N p = 0) in the form of inequality ID1 > 1 can be ob- tained [2,3], where for LO[ < 1

D = cosh ( + T f i ) cos"q1

in the case of unipolar pulses and

D = cosh (42 ' f i )

-- sin' nql [cosh (6.m) 1 - LE

-cosh ((:T - 7 ) m)] (7)

for bipolar pulses. (For 1 ~ 0 1 > 1 the corresponding for- mulas can be obtained from the above ones by means of formal analytic continuation.) 3. The solution to the problem (1)-(3) is

PO

~ 2 1 k

where 'po is the solution for (1) with RHS (2) without the last term (operation mode), h(t) is the Heaviside step function. Then, the average rate of a phase increment in an interval ( t , t + At) , At = n T determining the voltage across JI, can be represented as follows

p ( t ) = q o ( t ) + 2~ N,h(t - ZPkT),

0-7803-7242-WO231 0.00 02002 IEEE

SO3

In prcscncc of phase locking, (Apo/At) = 27r1, wherc I is the intcger Shapiro stcp number. Thereforc. thc voltage across JI avcraged over the time interval At is the hnear function of A. This is thc first effect used for quantum Joscphson D/A conversion. 4. It can bc shown that the substitution expiq = (z - iy)/(z + iy) converts (1) to the system of two linear equations of the following form

i ( t ) = fX(t) + $L(t)Y(t), j ( t ) = +(t)Z(t) - iy(t).

If i ( t ) is any periodic function, there exists Lapunov transformation 141 . . . .

where 2 x 2 matrix L is periodic and either

At x1 = eAIt, y1 = e*lt, or x1 = ( t + .)eAt, yI = e , A i , X p , A, a arc some real constants. For the system (1)-(3), when the phase locking effect takes place, the periodic matrix function L describes the system reaction to periodic short pulse exitation while the exponent exp(-xt/T) = exp(-/X1 - Azjt/T) describes a relaxation of transient processes caused by an exitation condition changing, where

Accordingly, a variation of 2sq0A being a function of PO and A’, leads to transient processes with the same re- laxation rate. Therefore for a sufiiciently large value of the parameter xt/T the above variation cannot violate the operation mode and necessary accuracy of D/A con- version. This is the second effect used for the quantum Josephson D/A conversion. To illustrate the above, the areas (non-shaded) of a phase locking on the plane of parameters ( average current ia

- effective short pulse charge 27rq ) are plotted in fig. 1, where q = q1 + qoA. The curves x = 0 determine range margins of variation of q and io in accordance with (4),(5). Fig. 2 displays similar areas and the x constant level curves for x = 1 , 2 l , . . . , 211 on the plane ( average current io - pulse repetition period T ) . Therefore, choosing appropriate values for the parame- ters m, PO, T , the desirable accuracy of D/A conversion can be ensured.

Acknowledgment The authors thanks Prof. J.Niemeyer (PTB. Germany)

for attention to the work.

References [l] B.Y. Shapiro, T.Dayan, M. Gitterman, et al. “Ex-

act calculation of Shapiro step sizes for pulse- driven Josephson junctions” PhysXev., Vol. B46, No. 13, p.8349, 1992; J.Kim, ASosso, and

dimennionlesa dc bias

FIG. 1: Nan-shaded areas of phase locking are shown for T = 5.3. Solid lines represent levels of constant values x = 1,2,. . . , 7 (from boundaries towards the centers).

2

.n 1 P

” r(

m 0 0

dimensionless pulse repetition period

FIG. 2 Non-shaded a r e s of phase locking are shown for 2sql = 0.333.

junctions driven by a squarewave pulse” J.Appl.Phys., Vol. 83, No. 6. pp. 3225-3332, 1998; S.P.Benz, C.A.Hamilton, C.J.,Burroughs, and L.A.Christian, “JoseDhson standard for AC voltage metrolom”, in _. NCSi Workshop & Symposium, 1997.

[2] V.M.Buchstaber, O.V.Karpov, and S.LTertychniy, “Electrodynamical properties of the Josephson junc- tion driven by a delta pulse series” JETp. Vol. 120, N. 6(12), pp.14781485, 2001.

131 SLTertychniy, “On asymptotic properties of solu- tions to equation @ + siny = f for periodical f” Ru. Math. Survey, Vol. 55, N1, p.186187, 2000.

[4] F.R.Gantmaher, The theory of matrices, Vol.11, New York, AMS Chelsea Publishing Company, p.121, 1960.