the josephson junction circuit family: network theory

INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONSInt. J. Circ. ¹heor. Appl. 2000; 28:371}420

The Josephson junction circuit family: network theory

Lutz Finger*s

School of Electrical, Electronic and Information Engineering, South Bank University, London, SE1 0AA, U.K.

SUMMARY

In this paper a qualitative theory for the Josephson junction circuit family is developed using dissipativeHamiltonian systems and the theory of ordinary and stochastic di!erential equations. Circuit models for thesingle junction and the #ux quantization are presented. Moreover, a novel classi"cation for the Josephsonjunction circuit family is introduced. Network models for Josephson junction circuits are developed and thephysical mechanism for oscillations is explained which is di!erent to the mechanism in semiconductorcircuits. It is shown that a general description of periodic solutions in the cylindrical phase space (periodicsolution of the second kind) is evident. Several types of hystereses in Josephson junction circuits areexplained. For the in#uence of thermal noise two concepts are presented, the Langevin approach and theapplication of the Fokker}Planck equation to the Josephson junction. Both are used for the Josephsonjunction circuit family. Two special kinds of stochastic resonance with practical importance are explained.The developed theory is applied to several circuits with practical applications and the software packageDONANS for the analysis of general non-linear and noisy systems is presented. Copyright ( 2000 JohnWiley & Sons, Ltd.

KEY WORDS: Josephson junction; dissipative Hamiltonian systems

1. INTRODUCTION

In this paper a qualitative theory for autonomous Josephson junction circuits is developed fromthe circuit theory point of view. Unlike in References [1}4] where a single Josephson junction isconsidered, and in [5,6], where coupled Josephson junctions are investigated, it is shown thata generalization to general Josephson junction networks with di!erent applications includingseveral array structures is possible. Moreover, a new classi"cation of Josephson junction net-works is introduced. Based on the theory of Hamiltonian systems applied to some Josephsonjunction circuits and qualitative descriptions by means of the theory of ordinary and stochasticdi!erential equations we unify several seemingly di!erent circuits under a single description andshow that previous investigations concerning the single Josephson junction and coupled junc-tions, the d.c.-SQUID, are special cases of this theory. Moreover, methods of the non-linear

*Correspondence to: Lutz Finger, Gartenstr. 6, D-98590 Wernshausen, GermanysE-mail: [email protected]

Contrant/grant sponsor: Alexander von Humboldt Foundation (AvH)

Received 21 December 1998Copyright ( 2000 John Wiley & Sons, Ltd. Revised 12 January 2000

circuit theory and stochastic dynamics are introduced and applied to Josephson junctionnetworks.

Josephson junction circuits have a very complex dynamical behaviour which has not yet beenfully understood. Especially thermal noise and the di$cult situation in the measurement tech-nique (only mean values of the voltages can be measured due to the high frequencies in the GHzregion) make it hard to understand and explain unwanted phenomena in applications by recentlyused experimental methods.

The paper is organized as follows. In Section 2 the dynamical description of Josephson junctioncircuits without in#uence of thermal noise is developed. Starting with a single Josephson junctionand its circuit model, the #ux quantization and the corresponding circuit model is introduced.Then we classify Josephson junction networks. After that, we derive energy formulations forpossible network elements and sources in Josephson junction networks. Using them a formula-tion for Josephson junction circuits as dissipative Hamiltonian systems is introduced. ThisHamiltonian formulation extends former considerations of Hamiltonian systems by includingdissipative elements. Moreover, we compare it with a$ne Hamiltonian systems which includesources but no dissipative elements. Then we present the physical mechanism for oscillations inJosephson junction circuits which is di!erent to the well-known mechanism with a negativeresistor in electronic circuits. It is shown, that every stable solution is either a equilibrium solutionor a running periodic solution winding around this cylinder. Using the Hamiltonian formulationand the mechanism for oscillations the motion of periodic solutions of Josephson junction circuitsin the cylindrical phase space is derived. Moreover, it can be shown, that this considerationconcerning periodic solutions of the second kind is independent on the order of the Josephsonjunction network. This makes it possible to unify autonomous Josephson junction circuits ina conception which will be called Josephson junction circuit family for further reference. Afterthat, the stability of periodic solutions by means of Floquet multipliers is investigated. Moreover,a formulation for a hyperplane for the calculation of the PoincareH map can be found for allmembers of the Josephson junction circuit family. Then several types of hystereses of periodicsolutions are presented.

Section 3 summarizes two possible descriptions if the inevitable thermal noise is included in theinvestigation. Firstly, the usual modelling with stochastic di!erential equations is shown. ThisLangevin description has the advantage that a numerical solution by means of stochasticintegration methods is straightforward. We show how the Langevin equation can be solvednumerically. Finally, an alternative formulation of the noise problem by means of the Fokker}Planck equation is demonstrated [7]. We show, how the coe$cients of the drift vector and thedi!usion matrix of the Fokker}Planck equation can be derived from the Langevin equations andapply it to the single junction. On the other hand, the numerical solution of this partial di!erentialequation is not straightforward which makes the application of this method rather di$cult.Nevertheless, we derive a numerical algorithm for the single Josephson junction. Then we presenttwo special cases of stochastic resonance. We show how these phenomena can cause erraticbehaviour which has an important in#uence on practical Josephson junction circuits.

In Section 4 the developed theory is applied to some practical circuits in order to show its usefor studying the dynamics of such circuits. We start with one basic element for a superconductingelectronics, the quantum #ux parametron (QFP). After that, several arrays of Josephson junctionsare considered and the complex dynamical behaviour is explained using the developed theory.Finally, a model of a complete SQUID with a coupling circuitry is investigated with experi-mentally obtained parameters.

372 L. FINGER

Copyright ( 2000 John Wiley & Sons, Ltd. Int. J. Circ. ¹heor. Appl. 2000; 28:371}420

In the last section the obtained results are summarized and an outview is given. Then thesoftware package dynamics of non-linear and noisy systems (DONANS) for the analysis ofgeneral non-linear and noisy systems is presented as in appendix.

2. THE DYNAMICS OF JOSEPHSON JUNCTION CIRCUITS

2.1. The model of a Josephson junction

In 1962 Josephson [8] predicted that:

(1) When two superconductors are separated by a very thin oxide layer ((20 As thick) a d.c.tunnel current of limited magnitude can #ow through the junction even when there is novoltage between the superconductors (the d.c. Josephson e!ect).

(2) When a voltage < is established across the junction the current oscillates at a frequency2e</h (the a.c. Josephson e!ect).

When two superconductors S1

and S2

are separated by the thin layer with a weak coupling A thewave functions t

i, i"1, 2, of the cooper pairs in every superconductor can be described by

ti"Jp

ie+ai , i"1, 2 (1)

with the cooper pair density pi, the superconducting phase a

iin the superconductor S

iand the imaginary unit j. For the Josephson junction the SchroK dinger equation can bewritten as

jh

2ntQ

1"!

q<

2t

1#Kt

2

jh

2ntQ2"

q<

2t2#Kt

1(2)

with the voltage < between the superconductors, the charge q"2e of a cooper pair and theconstant K which describes the coupling between the wave functions of the superconductors.Inserting Equations (1) into (2) and after sorting real and imaginary parts we obtain

p51"

4nK

hJp

1p2

sin a

a51"!

2nK

h Sp2

p1

cos a#2nq<

2h

pR2"!

4nK

hJp

1p2

sin a

a52"!

2nK

h Sp1

p2

cos a!2nq<

2h (3)

NETWORK THEORY 373


tAs a convention, capital letters are used for the circuit equations whereas small letters are used for the dimensionlessform. This is common in the superconducting literature.A In this section we still neglect the thermal noise.B In the following at most this formulation will be used. This is the usual convention.

with the phase di!erence a"a1!a

2. The current in the junction can be determined using

I"dp1/dt"!dp

2/dt. With

I0"

4nK

hJp

1p2

(4)

we obtain the equation of the Josephson direct current

Is"I

0sin a (5)

where a denotes the quantum phase di!erence between the two superconductors which made upthe junction and the critical current I

0. Subtracting the fourth from the second equation in

Equation (3) gives the equation of the Josephson alternating current:

<"'

02n

a5 (6)

Equation (5) gives only the ideal behaviour of a junction. From the circuit theory point of viewseveral circuit models are known which model this behaviour (see Reference [9] for a review). Forthe considerations in this paper the Stewart/McCumber (RCSJ) model is used for the two-terminal device of the Josephson junction [10,11] even though other more complex models existfor this junction. On the other hand, the qualitative behaviour remains the same with other circuitmodels. The RCSJ model is given in Figure 1 as a circuit model.

Beside the resistor R and the capacitor C a non-linear element representing the supercurrent isgiven by Equation (5).t The superconducting phase di!erence a is used as the state-space variable.

Using KCL the equation governing the second-order circuit in Figure 1 is given in thecurrent}#ux formulation by A

I"C'G#1

R'0 #I

0sin

2n'

0

', '0"

h

2e(7)

Here e denotes the electron charge and h denotes Planck's constant, and '0

is the #ux quantum.In the current-phase formulationB the equation is given by (aR "da/dt)

I"'

0C

2na(#

'0

2nRaR #I

0sin a (8)

Phase and #ux are related by

a"2n''

0

(9)

Therefore, this circuit element can be interpreted as a non-linear inductance whose non-linearityis given by the sine function.

374 L. FINGER


Figure 1. RCSJ model of a Josephson junction.

The output of the junction, the voltage, is given by the Josephson equation (6) (the a.c.Josephson e!ect) [8].

Equation (8) can be transformed into dimensionless form in two ways. Both normalizations usethe transformation i"I/I

0(external current). The "rst way uses the Josephson circle frequency

)C"2nI

0R/'

0and q")

Ct (normalized time), u")/)

C(normalized frequency), v"</I

0R"

aR "da/dq (normalized voltage), bC")

CRC (hysteresis parameter, McCumber parameter):

bCa(#aR #sin a"i (10)

The second way uses the plasma frequency )1"J2nI

0/C'

0and 0")

1t (normalized time),

u1")/)

1(normalized frequency), v"</I

0R"aR "da/d0 (normalized voltage), c"1/)

1RC

(damping parameter):

a(#caR #sin a"i (11)

Between these normalizations the following transformations apply: c")1/)

C"1/Jb

C,

u"u1c,0"qc. The "rst normalization has the advantage that the voltage corresponds to the

frequency in dimensionless units v"u and will at most be used in this paper. As can be seen, afternormalization the output of the Josephson junction, the voltage, is represented by the statevariable aR .

2.2. The Josephson junction in a superconducting loop

To obtain the set of equations for a Josephson junction in a superconducting loop, one shouldnote that the voltage < across the junction in this circuit can arise only as a result of a change ofthe magnetic "eld in the loop by the relation<"'0 . Substitution of< from this equation into theJosephson equation (6) and its integration over time yields the formula

a"2n''

0

(12)

In order to obtain Equation (12), the constant of integration is set to zero. Thus, the Josephsonphase di!erence is directly related to the magnetic #ux in the loop. Moreover, the supercurrentI0

sin a becomes now a periodic function of ' with the period '0. This phenomenon is called the

quantum interference or #ux quantization [9,12].

NETWORK THEORY 375


Figure 2. Equivalent circuit for a Josephson junction in a superconducting loop.

To complete the set of equations in a superconducting loop, one should take into account thatnot only the external magnetic "eld, but also the loop current I, which is equal to the junctioncurrent, can contribute to the #ux. This yields an equivalent circuit containing three elements, theJosephson junction (phase di!erence a), the inductance (phase drop a

L), and the external magnetic

"eld (represented here by a &phase generator' and interpreted as a source). The equivalent circuitdescribing the loop is given in Figure 2.

Due to the Josephson equation (6) the circuit equation describing the behaviour in a supercon-ducting loop can be written using the phases just as the voltages in the usual Kirchho! rule by thephase law with

N+i/1

vi"

'0

2nN+i/1

a5i"0 (13)

and

'0

2nai"c, c"const., c"2nn, n"0,$1,2 (14)

These formulas generalize Kirchho!'s voltage law for Josephson junction circuits.Evidently, the sum of the phase drops along any closed contour should be equal to zero [9,13].

Using this generalized rule the equation describing the loop (Figure 2) is given by

'"'a!¸I (15)

It is convenient to unite Equation (15) in the form

a"2n/!bLi (16)

Here i"I/I0

is the normalized current, /"'/'0

is the normalized &external' phase, andbL"2n¸I

0/'

0is the loop parameter which can be interpreted as a normalized inductance.

2.3. Classixcation of the Josephson junction circuit family

For further reference we introduce a novel classi"cation of the Josephson junction circuit familybased on circuit theory approaches and the structure of the state-space variables. This classi"ca-tion has an important in#uence on analytic treatment of the Josephson junction circuit family.Some examples for the di!erent types in the classi"cation follow.

I. The Josephson junction network is described by the state-space vector

(a1,2, a

N, q

1,2, q

M, aR

1,2, aR

N, qR

1,2, q(

1,2, q(3)

1,2)T (17)

376 L. FINGER


This is the most general description for Josephson junction networks which will not be consideredin this paper. For q

1, qR

1,2 several formulations are possible, e.g. all q

ican be a current (the

description of the network is given by the phase-current formulation), or all qiare charges (the

Josephson junction network is described by the phase-charge formulation), etc.From this general form of the state-space vector several special cases are possible which play an

important part in this paper. The state-space vector can be given by

(Ia) (a1,2, a

N, q

1,2, q

M, aR

1,2, aR

N, qR

1,2, qR

L)T, ¸(M.

(Ib) (a1,2, a

N, q

1,2, q

M, aR

1,2, aR

N, qR

1,2, qR

M)T.

(Ic) (a1,2, a

N, q

1,2, q

M, aR

1,2, aR

N)T.

These three cases occur if the system described by the state-space vector is only determined by thephases a

i, i"1(1)N, and their derivatives and q

iand only their "rst derivatives.

Further simpli"cations can be found when the Josephson junction network is only determinedby the superconducting phases:

(II) (a1,2, a

N, aR

1,2, aR

N)T

This is the situation where the Josephson junction network has besides the junctions onlyresistors and inductances or capacitances which do not contribute to the order of the system.Therefore, we distinguish further the following:

(IIa) The Josephson junction network is only determined by the Josephson junctions andadditional resistors.

(IIb) The Josephson junction network is only determined by the junctions and inductances orcapacitances which do not contribute to the order of the network. This is a typicalsituation in Josephson junction circuits due to the fact that superconducting strips can bemodelled by inductances.

Cases (Ib) and (II) are special systems which make the Hamiltonian theory applicable to theJosephson junction circuits. On the other hand, for systems (Ia) and (Ic) this is, in general, notpossible. Examples will follow below.

2.4. Dissipative Hamiltonian character of Josephson junction circuits

Recently, there is a great interest in a general description of electrical networks. One of thesedescriptions is the Hamiltonian formulation. However, due to the dissipation in resistive elementsthose cannot be included into the standard Hamiltonian description of electrical networks.

Here an extension of Hamiltonian systems, dissipative Hamiltonian systems, is introduced andapplied for a description of some Josephson junction circuits. It will be shown later, that this formis extremely convenient for a general description of those networks.

As can be seen in the previous section a Hamiltonian formulation is only possible for types (Ib),(Ic) (M even), (IIa), and (IIb).

De,nition 1A system which can be described in the form

x5 "JDxH!Sx (18)

NETWORK THEORY 377


where Dx

is the gradient and J and S are matrices of dimension N of the form

J"A0 I

!I 0B and S"A0 0

0 KB (19)

with H being the Hamiltonian function, J the simplectic, and I the identity matrix of dimensionN, K, 0 are of dimension N, will be called dissipative Hamiltonian system.

For type (Ib) x is given by x"(a1,2, a

N, q

1,2, q

M, aR

1,2, aR

N, qR

1,2, qR

M)T and for types (IIa)

and (IIb) by x"(a1,2, a

N, aR

1,2, aR

N)T. For type (IIb) of the classi"cation of Josephson junction

networks S has the special form

S"A0 0

0 IB (20)

As mentioned above, this case occurs if there are no additional resistors in Josephson junctionnetworks except those in the Stewart/McCumber model of the single junction.

For Josephson junction circuits we use the following energy formulations for the Josephsoncircuit elements in the phase formulation:

The energy stored in a capacitance is given by (aR "da/dt)

E"P <I dt"'

02n

C P aR d<"'2

04n2

C P aR daR "'2

04n2

CaR 22"

CR2I20

2 AdadqB

2(21)

An inductance delivers the following part to the energy formulation:

E"P <I dt"'

02n P I da"

'20

4n2¸ P a da"'2

04n2¸

a2

2"CR2I2

0

1

bLbC

a22

(22)

The part delivered by the supercurrent which can be interpreted as a non-linear inductance hasthe following energy stored:

E"P <I dt"'

02n P I da"

'0

2n P I0

sin a da"'

02n

I0(1!cos a)"CR2I2

0

1

bC

(1!cos a) (23)

In Equations (8) and (15) two sources are included. They are included in the energy formulation inthe following way. The current source delivers the part

E"P <I dt"'

02n P I da"

'0

2nIa"CR2I2

0

1

bC

ia (24)

The source delivering through the #ux quantization takes part with

E"P <I dt"'

02n P I da"

'0

2n P'¸

da"'

02n

'¸

a"CR2I20

1

bLbC

2n/a (25)

In all Equations (21)} (25) the initial value of the phase for the integration is set to zero whichrepresents the zero state of the energy.

378 L. FINGER


The last terms in these equations show the formulas after introducing the normalizationparameters. We observe a constant factor CR2I2

0in all equations. If we use the normalization for

the energy e"E/CR2I20

and write down the Hamiltonian function for normalized equations thisfactor disappears and we obtain for the energy in a capacitance

e"1

2 AdadqB

2(26)

in an inductance,

e"1

bLbC

a22

(27)

for the non-linear element in the junction through the supercurrent,

e"1

bC

(1!cos a) (28)

for the current source,

e"1

bC

ia (29)

and for the #ux quantization,

e"1

bLbC

2n/a (30)

For the second term on the right-hand side of Equation (18) the constant factor 1/bC

appears inall entries of the matrix S. Therefore, the following form of the dissipative Hamiltonian functionwill be used in Josephson junction circuits:

x5 "JDxH!

1

bC

Sx (31)

Due to the special structure of equations describing Josephson junction circuits the matrix S isa sparce matrix. For the special case (IIb) where no resistors except those of the junctions occurS has the form (20). If only linear resistors are assumed in type (IIa) K is of the form

K"K@#I (32)

Di!erentiation of time for the Hamiltonian function yields

dH

dq"(D

xH)T xR "(D

xH )T JD

xH!

1

bC

(DxH)T Sx"!

1

bC

N+j/1

N+i/1

aRjkjiaRi

(33)

where (DxH)TJD

xH"0 and k

jiare the entries of the linear matrix K. For type (IIb) we have

dH

dq"(D

xH)TxR "(D

xH)TJD

xH!

1

bC

(DxH)T Sx"!

1

bC

N+j/1

aR 2j

(34)

which shows that H is a monotonically decreasing function of time.

NETWORK THEORY 379


E Here we restrict the state space to the state variables of the Josephson junctions. This is, however, no restriction forgeneral Josephson junction circuits of type (I) where the following transformations can be applied as well.

**The 2n translation is in general not valid for individual junctions in the Josephson junction circuit family (see Section4.3.6.).

ss It is only important to use the transformation yM"1/M+M

j/1aj. The choice of the state space variables y

1,2, y

M~1can be made di!erent as in this paper, e.g. y

j"a

j!a

j`1, j"1(1)M!1.

In this Hamiltonian formulation the sources are included directly in the Hamiltonian function.Other possible formulations without resistive elements are given by a$ne Hamiltonian functions[14]. Here electrical networks are described by

d

dt Cq

pD"JDxH (q, p)!

m+i/1

JDxH

i(q, p) u

i(35)

with m inputs uiand m outputs y

iwhich are given by

y"(H1, H

2,2,H

m)T (36)

By comparing Equation (18) without the dissipative part with Equation (35) we observe that bothfunctions are related by H

ui"!H

iui. Here H

uirepresents the part of Equation (18) which is

delivered by the sources ui. Therefore, dissipative Hamiltonian systems include directly the

sources in the Hamiltonian function whereas in a$ne Hamiltonian systems they are described byadditive terms. This relation is reasonable if the sources in electrical networks are linear.

2.5. The behaviour of the Josephson junction circuit family in the cylindrical phase space

As can be easily seen Equation (10) is invariant under the operation aPa#2n. Furthermore,a solution which satis"es

a (q#k¹)"a (q)#2kn, k"0, 1,2 (37)

will be called running periodic solution or periodic solution of the second kind [15}17].Moreover, considerations concerning the cylindrical phase space and running periodic solu-

tions can be extended to the Josephson junction circuit family. Here we use the followingsymmetries for a system of N junctions where M,M)N, junctions follow the 2n translation:E,**

(aj)P(a

j#2n), j"1(1)M (38)

Using the transformation

ai"y

i#y

M, a

M"y

M!

M~1+j/1

yj, i"1(1)M!1 (39)

i.e.

yi"a

i!y

M, y

M"

1

M

M+j/1

aj, i"1(1)M!1 (40)

we can state the following qualitative propositions for running periodic solutions of allai, i"1(1)M:ss

380 L. FINGER


ttFor circuits of type (II) with M"N the subsystem is the complete system.AAThis preposition can only be proofed yet for systems of type (II) (see Theorem 2, Section 4.2).

1. The transformed subsystemtt is with Equations (38) and (40) invariant under the operation

yMPy

M#2n (41)

2. By identi"cation of the state variables of the Josephson junctions (y1,2, y

M~1, y

M,

yR1,2, yR

M) modulo this translation we obtain a phase #ow de"ned on the cylinder

R2M~1]S1"My1,2, y

M~1, y

Mmod 2n, yR

1,2, yR

MN (42)

3. For this subsystem we have

yM

(¹ )!yM

(0)"2nm, mO0 (43)

with some integer m.4. Any periodic solution in the Josephson junction circuit family is a running periodic

solution.AA

5. For numerical computations we consider a (2M!1)-dimensional hyperplane &3R2M~1]S1 de"ned by

& :"M(y1,2, y

M, yR

1,2, yR

M)3R2M~1]S1 : y

M"y

M0N (44)

Every 2n of yM

, the trajectory intersects &.6. The resulting PoincareH map

P :&P& (R2M~1PR2M~1) (45)

replaces the #ow of the 2M-order continuous-time subsystem formed of the junctions witha 2M!1 order discrete-time subsystem. By using the modulo function the state variableyM

is limited to [0, 2n) and the surface of the section of the PoincareH map is chosen asyM0

"n. Note that the choice of the hyperplane is independent of the number of Josephsonjunctions in the subsystem and therefore it is also independent of the order of the investi-gated Josephson junction network.

These propositions have an important in#uence on the analysis of the non-linear behaviour ofthe Josephson junction circuit family. Due to Equation (42) any member with M junctionsfollowing the 2n translation in the Josephson junction circuit family can be reduced to thecylindrical state space RM~1]S1 independent of the internal structure of the circuit.

RemarkThe most usual situation in the Josephson junction circuit family is the case where M"N, i.e.

all Josephson junctions follow the 2n translation property Equation (38) for M"N. An examplefor the exceptional case M(N follows in Section 4.3.6.

2.6. Physical mechanism of oscillations in Josephson junction circuits

In electronic circuits the basic structure in an oscillator consists of an inductor, capacitor, anda resistor. Since both inductor and capacitor are mostly linear and passive in semiconductor

NETWORK THEORY 381


devices, the resistive one port must be active for at least some points in the second andfourth quadrants of the v}i plane in order for oscillations to be possible. An additionalassumption for this mechanism is given by an unstable equilibrium which repels trajectoriesfrom it [18].

However, in the circuit theory community there has been a long discussion on activity andpassivity [19].

But the mechanism for oscillations in Josephson junction circuits is di!erent. Here both resistorand capacitor are linear in the RCSJ model. The only non-linear two-terminal device and hencepossible active element is described by I

0sin a which can be interpreted as a non-linear

inductance.For Josephson junction circuits this non-linear inductance must be active in the second and

fourth quadrants of the a}i plane to make oscillations possible. This can be easily seen inEquation (5). Here the non-linear part is passive for !n/2#2kn(a(n/2#2kn, k"0, 1,2and active for n/2#2kn(a(3n/2#2kn, k"0, 1,2.

On the other hand, equilibrium points occur only for Equation (10) if sin a"i andtherefore the condition i)1 holds for equilibrium solutions. There are no equilibrium pointsfor i'1 whose appearance is therefore not necessary for oscillations in Josephson junctioncircuits.

As long as the current I is smaller than the maximum supercurrent I0, i.e. i(1 in Equation

(10), there exists a constant phase

an"arcsin

I

I0

#2nn, n"0, 1,2 (46)

across the junction. Hence aRn"0, i.e. the voltage drop is zero and the junction functions as

a superconductor. This corresponds to the d.c. Josephson e!ect. Moreover, no energy is dissi-pated inside the Josephson junction in this case. Some energy is, however, stored in junctionwhich can be calculated by means of the energy of supercurrent (23). There exists a critical inputcurrent such that for I'I

0the phase (di!erence) across the junction assumes the time-varying

form

a(q)"2n¹

q#p (q) (47)

where ¹ is the period. The associated terminal voltage is, therefore, time varying and assumes theform

v"<

RI0

"

'0

2ndadt

"(A2n¹

q#p (q)B (48)

Due to the high frequencies in the GHz region it is only reasonable to experimentally specify thejunction characteristics by plotting the average voltage vN

i"SaR

iT as a function of the current

i (current}voltage characteristic) or as a function of the magnetic #ux / with a constant currenti for more than one Josephson junction (#ux}voltage characteristic). Consequently, this averageor d.c. voltage is proportional to the oscillation frequency )

C/¹:

vN"v$.#.

"

<$.#.

RI0

"

)C

RI0¹ P

T@)C

0

v(t) dt")

CRI

0¹

'0

2n PT@)C

0

da (q)"2n¹

"

)C'

0RI

0¹

(49)

382 L. FINGER


Figure 3. Current}voltage characteristic of a Josephson junction.

It follows that the d.c. voltage<$.#.

will increase with i and C when R)C

is held constant and withR when b

Cis held constant [3]. A typical characteristic is shown in Figure 3.

For I(IC

there is no output voltage. This corresponds to equilibrium points where aR "0. ForIC(I(I

0equilibrium points and oscillations are coexisting which is represented by zero

voltage (equilibrium points) and a voltage larger than zero (oscillations) where the mean value ofa5 is di!erent of zero. For I'I

0only oscillations appear.

It can happen that equilibrium solutions and periodic solutions coexist for some parameterregions and it has been shown that this depends strongly on the parameter b

C, e.g. [1,3]. If b

Cis

&large' (greater than 1) then there exists a iC(b

C) with the properties that for i(i

C(b

C) there are

only equilibrium solutions. For iC(b

C))i(1 equilibrium solutions and periodic solutions are

coexisting, whereas for i'1 only oscillations exist which excludes the necessity of equilibriumpoints for the oscillation mechanism.

If the current I is increased the junction remains longer in the basin of attraction of theequilibrium state and then switches into the voltage state (oscillations), whereas with decreasingof I a longer appearance of the trajectory in the basin of attractions of the oscillation solution canbe observed which causes the hysteresis.

2.7. Periodic solutions in the Josephson junction circuit family

Now, we consider the stability of periodic solutions. Here we deal only with the Hamiltonianstructure of type (IIb) where the matrix S has the form (20). The general case (19) is still an openproblem.

Stability is usually investigated by means of Floquet theory. Therefore, we formulate thefollowing theorem for the Floquet multipliers.

Theorem 1x(q) is a trajectory of the damped Hamiltonian system (18) with period ¹. Then the Floquet

multipliers of x (q) appear as

A1, e~T@bC, ae~T@2bC,1

ae~T@2bC, k

ie~T@2bC,

1

ki

e~T@2bC, kNie~T@2bC,

1

kNi

e~T@2bCB , a3R, ki, kN

i3C

NETWORK THEORY 383


i"1(1)N

2!1, N even (50)

A1, e~T@bC, kie~T@2bC,

1

ki

e~T@2bC, kNie~T@2bC,

1

kNi

e~T@2bCB , ki, kN

i3C, i"1(1)

N!1

2, N odd

Proof. The linearization of Equation (18) yields

y5 "AJD2xH!

1

bC

SB y (51)

where D2xH is the Hessian matrix of H:

D2xH

ij"

L2H

LxiLx

jKx(q) (52)

The solution of the fundamental matrix Y is de"ned by

Y0 "AJD2xH!

1

bC

SBY, Y(0)"I (53)

The product YTJY with A"JD2xH!S/b

Csatis"es

d

dqYTJY"YT(ATJ#JA)Y"!

1

bC

YT (STJ#JS)Y"!

1

bC

YTJY (54)

The solution YTJY is then

YTJY"e~1@bC, YT(0)JY(0)"e~1@bC J (55)

which implies the &dissipative simplectic' character of Y [20]. Thus eq@2bCY is also simplectic:

(e(1@2bC)qY)TJ (e(1@2bC)q Y)"J (56)

The eigenvalue theorem of simplectic matrices yields for the characteristic polynomialp(k)"k2Np(1/k) [21]. Therefore, the eigenvalues appear as

Aki, kN

i,

1

ki

,1

kNiB , i"1(1)

N

2, N even (57)

Al,1

l, k

i, kN

i,

1

ki

,1

kNiB , i"1(1)

N!1

2, N odd

where ki, kN

i3C, l3R.

For Y(¹ ) the Floquet multipliers are

(e~(1@2bC)T, ki, e~(1@2bC)T/k

i, e~(1@2bC)T, kN

i, e~(1@2bC)T/k6

i), i"1(1)

N

2, N even

384 L. FINGER


BB It should be mentioned that these di!erent types of hystereses are practically very important in the Josephsonjunction circuit family. This holds in particular if noise is taken into consideration (see Section 3.3).

(e~(1@2bC)T l, e~(1@2bC)T/l, e~(1@2bC)Tki, e~(1@2bC)T/k

i, e~(1@2bC)T, kN

i, e~(1@2bC)T/k6

i),

i"1(1)N!1

2, N odd

with period ¹. For this autonomous system one Floquet multiplier is 1. Thus kN@2

"eT@2bC

(N even) respectively l"eT@2bC (N odd) proves the theorem. K

There are several special cases and results which can be derived from this theorem.For the single Josephson junction we have the Floquet multipliers (1, e~T@bC ). This is analogous

to the single pendulum and has been investigated e.g. in Reference [22].The Floquet multipliers are (1, e~T@bC , ae~T@2bC , e~T@2bC/a), a3R, for a system of two Josephson

junctions of type (IIb). This is the situation, e.g. in the ideal d.c.-SQUID and it has beendemonstrated that only bifurcations by $1 are possible. Hopf bifurcations are not possible inthis system [23,24].

2.8. Hystereses of periodic solutions

In the single junction there is a unique running periodic solution for i'1, which is, moreover,globally exponentially stable, i.e. each trajectory in the phase plane tends to it exponentially astPR [15]. In systems with at least two Josephson junction this unambiguity of periodicsolutions is no longer valid and hysteresis phenomena of periodic solutions can occur.

There are several types of hystereses possible in the Josephson junction circuit family which willbe classi"ed below.BB

From the circuit theory point of view these hystereses are hystereses of transfer characteristicsof periodic solutions and they can occur both in the current}voltage characteristic and#ux}voltage characteristic.

To avoid clutter, we classify the several types of hystereses in the most simple form, e.g. with 1¹and 2¹ periodic solutions, even though hystereses with higher periodic solutions can occur andhave been observed [23,25] but they are qualitatively of same types as in the following classi"ca-tions.

2.8.1. One-dimensional consideration. In Figure 4 one possible hysteresis mechanism is shownwhich will be called hysteresis of type (I) for further reference.

For growing parameter p a 2¹ periodic solution is present. The system undergoes a saddle-node bifurcation at SN

2and a jump to a 1¹ periodic solution occurs. In the reverse direction

(falling p) the 1¹ periodic solution is pursued until PD1. Then a subcritical period-doubling

bifurcation occurs and a jump to the 1¹ periodic solution is visible which causes another jump inthe characteristic.

The mechanism in Figure 4 is not the only hysteresis phenomenon which occurs in practicalJosephson junction circuits. A second mechanism is shown in Figure 5 which will be classi"ed astype (II).

For growing parameter p a 1¹ periodic solution is present. The system undergoes a saddle-node bifurcation at SN

2and a jump to another 1¹ periodic solution occurs. In the reverse

NETWORK THEORY 385


Figure 4. Hysteresis type (I).

Figure 5. Hysteresis type (II).

direction (falling p) the 1¹ periodic solution is pursued until SN1. Then another saddle-node

bifurcation occurs and a jump to the "rst 1¹ periodic solution is visible which causes anotherjump in the characteristic.

These two mechanisms have been observed in practical Josephson junction circuits. Herea third possible hysteresis mechanism is presented (type (III)) in Figure 6 which has to the author'sbest knowledge not yet been observed in a practical Josephson junction circuit and is, therefore,at the moment a theoretical prediction.

For growing parameter p a 1¹ periodic solution is present. The system undergoes a "rstsubcritical period-doubling bifurcation at PD

2and a jump to a second 1¹ periodic solution

occurs which is coexisting with a unstable 1¹ periodic solution. In the reverse direction (falling p)the 1¹ periodic solution is pursued until PD

1. Then another subcritical period-doubling bifurca-

tion occurs and a jump to the "rst 1¹ periodic solution is visible (coexisting with another unstable1¹ periodic solution) which causes another jump in the characteristic.

The hysteretic behaviour distracts possible applications of d.c.-SQUIDs as magnetometers.Di!erent mean values of the voltage make it impossible to use the SQUID in this parameterregion for sensitive measurements. Unfortunately, this behaviour is often seen in parameterregions where the gain of the SQUID can be very high. This holds for currents i straight above the

386 L. FINGER


Figure 6. Hysteresis type (III).

Figure 7. Two-dimensional bifurcation diagram of hysteresis type (II) for system (129) with theparameters given in Section 4.4.

equilibrium solutions in the current}voltage characteristic. Therefore, it is often necessary towork with higher currents i, even though only a lower gain in the SQUID can be achieved due tounwanted non-linear phenomena.

2.8.2. Two-dimensional consideration. The explanation of hysteretic behaviour given in the lastsection clari"es only the mechanism of hystereses for the change of one system parameter. Inorder to understand the hystereses in Josephson junction circuits more deeply it is necessary toconsider them with two variable parameters.

We start with type (II) hysteresis. Figure 7 shows this hysteresis in the i!/ plane for the laterconsidered d.c.-SQUID with input coil (Equation (129)).

As can be seen, the two bifurcation branches merge into a cusp point by (/"0.173, i"2.4).On the left-hand side of the cusp point we observe resonances as a pre-state of hystereticbehaviour which will be discussed by means of Figure 8.

Resonances are a very common problem in Josephson junction circuits, e.g. Ref. [26] whosebehaviour can be discussed with methods of the linear circuit theory as an approximation overlong parameter regions. Hereby, the non-linear element in the Josephson junction, the non-linear

NETWORK THEORY 387


Figure 8. Resonances in the d.c.-SQUID with input coil (a) i}vN characteristic and (b) secondFloquet multiplier k

2, /"0.15.

Figure 9. Two-dimensional bifurcation diagram of hysteresis type (I) in the ideal d.c.-SQUID [23,24].

inductance, is approximated by a Fourier series truncated at the fundamental frequency. Theresulting equations are solved analytically [26]. Then the resonance frequency (and therefore theresonance voltage) can be calculated. In other methods the non-linear inductance is replaced byan ideal linear current source [27]. Although this method is very limited it gives su$cientapproximations in some regions of the parameter space. From the non-linear circuit theory pointof view the resonance appears as a structure in the characteristics and one Floquet multipliermoves to #1, but before crossing a reversion in the behaviour can be observed [24]. For highervalues of / this Floquet multiplier changes his direction more and more close to #1 up to thecusp point by /"0.173. Above this point the resonance has merged into a hysteresis.

For type (I) hysteresis the behaviour is more complicated. Figure 9 shows the two-parameterbifurcation diagram of the ideal d.c.-SQUID in the i}b

Cplane [24]. Again the two branches of the

hystereses and the resonance merges into a cusp point by bC"1.4462, i"1.993. But besides this

fact, chaos exists with a constant distance to the hysteresis type I. Both ends of the chaotic

388 L. FINGER


behavior move together in a &chaotic cusp point' by bC"1.71, i"1.89. This strong connection

between chaos and the hysteresis of type (I) has already been observed in other Josephsonjunction circuits, e.g. in the double-loop SQUID [25].

As mentioned before, type (III) hysteresis has not yet been observed. Therefore, no investiga-tion in the two-dimensional case can be given.

3. THE JOSEPHSON JUNCTION CIRCUIT FAMILY WITH THERMAL NOISE

We distinguish between two cases of #uctuations in physical systems. When the noise is due to anexternal source coupled to the system we call it external noise. The noise is proportional toa physical coupling parameter and can be switched o!. In this case the noise is not exactly white.Internal noise, which is present in Ohmic resistors, cannot be switched o! and is inevitable. Inevery dissipative electrical circuit thermal noise is inevitable due to the #uctuation}dissipationtheorem, e.g. Ref. [28]. Depending on the dissipative elements, the resistors in electrical networks,we distinguish linear and non-linear #uctuation}dissipation theorems [29].

Thermal noise is very important in the Josephson junction circuit family and cannot beneglected in the circuit description. This can be easily seen by locking at the size of the noiseparameter D. For the low-temperature superconductivity by 4.2 K we have a parameter ofD"0.0044. For the high-temperature superconductivity at 77 K this parameter is more than 18times as high for the same system parameters. However, a general theory for noisy networks isstill lacking. This holds in particular for non-linear resistors. Therefore, a discussion like in thispaper starting with the noise-free case and expanding it to the noisy case is only possible forcircuits containing linear resistors. A second important point lies in the fact that numericalalgorithms for the analysis of such networks are not very well known. Therefore, new methodsand descriptions for noisy non-linear networks have to be developed.

3.1. Langevin concept

3.1.1. Basics. Josephson junction circuits modelled with the Stewart/McCumber (RCSJ) modelinclude linear resistors as dissipative network elements. Therefore, considerations using the linear#uctuation}dissipation theorem are su$cient here. For more enhanced models with non-linearresistors these considerations are no longer possible.

For the investigation of noisy non-linear networks two methods are known, the Langevinapproach and the Fokker}Planck equation (or forward Kolmogorov equation). It has beenshown that both lead to analogic results [28]. We apply them to Josephson junction circuits inthis paper.

Due to the linear resistor in the RCSJ model the Langevin approach leads to linear stochasticdi!erential equations. Moreover, it is possible to add a noise term with given stochastic propertyto the deterministic model and, respectively, di!erential equation. As mentioned before, thisapproach is limited to linear dissipative network elements. For non-linear resistors a non-linearLangevin concept has to be applied which leads to di$culties. In particular, this approach haslimits in the application [29}31]. On the other hand, for non-linear resistors one has to deal withgeneral stochastic di!erential equations in the Stratonovich or Ito( interpretation [32,33] whichmakes especially the numerical solution of these stochastic di!erential equations more di$cultthan the numerical treatment of networks with pure additive noise [34].

NETWORK THEORY 389


Figure 10. RCSJ model of a Josephson junction with thermal noise.

EE D is used as noise parameter in the literature about stochastic systems, whereas ! is used in the superconductingliterature.

If we consider again the Stewart/McCumber model with the additive white noise term weobtain the circuit model for a single Josephson junction in Figure 10.

KCL yields the following stochastic di!erential equation for the model in Figure 10:

I"'

0C

2na(#

'0

2nRaR #I

0sin a!I

n(58)

Here In

is the (Johnson) noise current [35] with SInT"0 and the autocorrelation function

SIn(t)I

n(t@)T"

2k¹

Rd (t!t@) (59)

given by Nyquist's theorem [36] where k denotes Boltzmann's constant, ¹ denotes the temper-ature and d denotes Dirac's delta function. Using the normalization i

n"I

n/I

0(noise current)

with the Josephson frequency )C

the dimensionless linear stochastic di!erential equation for theJosephson junction reads

bCa(#aR #sin a"i#i

n(60)

with the noise parameter D"!"2nk¹/I0'

0and the properties of the noise process Si

nT"0

andEE

Sin(q)i

n(q@)T"2Dd(q!q@) (61)

Using the plasma frequency )1

the dimensionless stochastic di!erential equation is

a(#caR #sin a"i#in

(62)

with the properties of the noise process SinT"0 and

Sin(0 )i

n(0 @)T"2cDd (0!0 @ ) (63)

The spectral density S(u) of the Langevin force, which is the Fourier transform of the correlationfunction by the Wiener}Khintchine theorem, is independent of the frequency u for white noise:

S (u)"2 P=

~=

2De~+uqd(q) dq"4D (64)

390 L. FINGER


Due to the linear resistor in the RCSJ model the white noise process in a Josephson junction isuniquely determined through the Langevin equation.

The Langevin equation can be solved by numerical simulation by means of stochasticintegration methods [23,34,37].

3.1.2. Numerical integration of the Langevin equation. Several stochastic integration methods arepossible for numerical approximations of the Langevin equation. Besides predictor}correctorintegration methods and one-step-collocation methods, stochastic Runge}Kutta methods arepossible for the simulation of systems with additive noise sources. An overview over the numericalsolution of general stochastic di!erential equations is given in Ref. [34]. Because of the additivityof the noise the following general stochastic Runge}Kutta method can be used for the followingstate-di!erential equation:

xR "f (x, t)#g(t), x (t0)"x

0(65)

and the correlation properties Sg (t)T"0 and Sg(t)g (t@)T"2Dd(t!t@). ST is the mean value,D the noise parameter and d is the Dirac delta function. The general stochastic Runge}Kutta(SRK) method for additive white noise has the following form for one integration step witht3[t

m, t

m`1] [38,39]:

xm`1

"xm#h

m

s+i/1

piki#J2Dh

m

s+i/1

qiti

(66)

ki"f Atm#c

ihm, x

m#h

m

i~1+j/1

aijkj#J2Dh

m

i~1+j/1

bijtjB , i"1(1)s

Here hm

is the step size of the integration and s means the order of the Runge}Kutta method. aij,

bij, c

j, p

i, q

iare coe$cients with the conditions +s

i/1pi"+s

i/1qi"1. The deterministic part of

the integration algorithm (66) depends on the step size hm

whereas the stochastic part depends on

Jhm

which makes the developing of stochastic integration algorithms much more di$cult thandeterministic integration routines. t

ijare independent Gaussian random numbers with the

properties Sti,j

T"0 and St2i,j

T"1. For the simulation these numbers are produced witha pseudo-random generator. In the software package DONANS [23] the random generator ran2[40] is used. Then such two uniformly distributed random numbers r

1, r

2are transformed into

two independent normally distributed random numbers t1, t

2with the Box}Muller algorithm

or the Polar}Marsaglia method:

w"(2r1!1)2#(2r

2!1)2)1

t1"(2r

1!1) S

!2 lnw

w(67)

t2"(2r

2!1) S

!2 lnw

w

The algorithm for solving Equation (65) is then the following:

1. Generate uniformly distributed random numbers r1, r

2.

2. Transform them into normally distributed random numbers t1, t

2by Equation (67).

NETWORK THEORY 391


3. Solve Equation (65) by Equation (66) for one step.4. Repeat these steps until the desired simulation time is reached for the trajectory.

It has been shown that for the simulation of systems with additive noise the best numericalscheme has a local truncation error O(h3

m) [37,41]. Therefore, it is su$cient to use a stochastic

Heun method. Every numerical approximation by means of stochastic integration methodsconverges in the quadratic mean sense [34]. For our investigations we use mostly stochasticRunge}Kutta methods second [42] and third order [38] which minimizes the error O(h3

m). Some

important properties of stochastic Runge}Kutta methods, like convergence, consistence andnumerical stability, are given in Refs. [23,34,37].

3.2. Fokker}Planck equation

3.2.1. Basics. As mentioned before, a noise process can be equivalently described by the Fokker}Planck equation [7]. Prerequisite to the application is the ful"llment of the Markov property.A Markov process can be uniquely described by a transition (or conditional) probability densityP (x, t Dx

0, t

0). Here P (x, t Dx

0, t

0) is the probability for a transition of the system from x

0at the time

t0

into the interval [x, x#dx] at the time t.

De,nition 2A stochastic process is a Markov process if the conditional probability density at time

tn

depends only on the value at xn~1

at the next earlier time.

This follows from the fact that a system of stochastic di!erential equations is uniquelydetermined by the initial values and that a d-correlated Langevin force g(t) at a former timet(t

n~1cannot change the conditional probability at a later time t't

n~1[7].

This Markov property is destroyed if g (t) is no longer d-correlated which holds, e.g. forcoloured noise with an exponential correlated function. On the other hand, the Markov propertyis ful"lled for d-correlated additive white noise for a system of stochastic di!erential equationswith (x"(x

1,2,x

N)T)

x5 "f (x, t)#g (t) (68)

and the correlation properties (dij

Kronecker's symbol)

Sgi(t)T"0, Sg

i(t)g

j(t@)T"2Dd

ijd(t!t@), i, j"1(1)N (69)

The temporal behaviour of the transition probability density P(x, t Dx@, t@) is governed by a partialdi!erential equation which is denoted by Fokker and Planck [7,29]

LP

Lt"A!

LLx

i

Di(x, t)#

L2Lx

iLx

j

Dij(x, t)B P"¸

FP(x, t)P (70)

Here Einstein's summation convention is used. ¸FP

is the Fokker}Planck operator

¸FP

(x, t)"!

LLx

i

Di(x, t)#

L2Lx

iLx

j

Dij(x, t) (71)

392 L. FINGER


The initial condition is

P (x, t@ Dx@, t@)"d (x!x@) (72)

Here Di, i"1(1)N, is the drift vector and D

ij, i, j"1(1)N is the di!usion matrix. We obtain them

for Equation (70) from the Langevin equations (68) and (69) by the relations Di"f

i(x, t) and

Dij"Dd

ij. Here D"! is the noise parameter.

Although at most only a numerical solution is possible, there are some algorithms to solve theFokker}Planck equation in special cases [7].

For the single Josephson junction given by Equation (60) the drift vector is given by

Da"aR , DaR"1

bC

(!aR !sin a#i ) (73)

and the di!usion matrix reads

DaR aR "D

bC

, Daa"DaaR "DaR a"0 (74)

Thus the Fokker}Planck equation for the single Josephson junction normalized with theJosephson frequency )

Ctakes the form

LP

Lq"A!

LLa

aR !1

bC

LLaR

(i!sin a!aR )#D

bC

L2

LaR 2B P (75)

and using the plasma frequency )1

the Fokker}Planck equation is

LP

Lq"A!

LLa

aR !LLaR

(i!sin a!caR )#cDL2LaR 2B P (76)

Due to the behaviour of the Josephson junction circuit family in the cylindrical state space(Equation (41)) the Fokker}Planck operator for the single junction commutes with the transla-tion operator ¹ de"ned by

¹P (a, aR )"P (a#2n, aR ) (77)

A closed analytic investigation is not possible. Therefore, approximations or numerical methodshave to be used for this partial di!erential equation.

The 2n translation property of the Fokker}Planck equation applies as well to members of theJosephson junction circuit family with M junctions following the 2n translation (see Equation(42)) by the relation

¹P (y1,2, y

M, yR

1,2, yR

M)"P(y

1,2, y

M~1, y

M#2n, yR

1,2, yR

M) (78)

of the transformed subsystem by Equations (39) and (40).

3.2.2. Numerical calculation for the Josephson junction. For the single Josephson junction theFokker}Planck equation can be numerically solved using methods applied to the Kramersequation [7].

NETWORK THEORY 393


The Langevin equation (62) is a special case of the Klein}Kramers equation

x(#cxR #f @(x)"g(t) (79)

with Sg (t)"0T and Sg(t) g (t@)T"2cDd(t!t@) and the corresponding Fokker}Planck equation

LP

Lt"A!

LLx

xR #LLaR

(cxR #f @(x))#cDL2LxR 2BP (80)

with x"a, t"q, f (x)"!cos a!ia. The Klein}Kramers equation describes the Brownianmotion of a particle in a potential.

In order to solve Equation (76) the distribution function is expanded according to

P (a, aR , q)"t0(aR )

=+n/0

cn(a, q)t

n(aR ) (81)

Here tn(aR ) are eigenfunctions of the Fokker}Planck operator

¸aR t0(aR )t

n(aR )"!nct

0(aR )t

n(aR ) (82)

The functions tn(aR ) are the Hermite functions

tn(aR )"

hn(aR /J2D)

4JD

hn(m)"

Hn(m) e~m2@2

J2nn!J2n(83)

known also from the oscillator eigenfunctions of quantum mechanics (Hn(m) is the Hermite

polynomial). The eigenfunctions obey the orthonormality relation

P=

~=

tn(aR )t

m(aR ) daR "d

nm(84)

Inserting Equation (81) into Equation (76), multiplying it by tm(aR )/t

0(aR ) and using Equations (83)

and (84) we obtain the following in"nite system of coupled di!erential equations (n*0, c~1

"0):

Lcn

Lq"!JnD< c

n~1!ncc

n!Jn#1Dc

n`1(85)

with

D"JDLLa

, D< "JDLLa

#

sin a!i

JD(86)

System (85) has a tridiagonal structure and is called Brinkman's hierarchy. Its solution can beobtained in terms of matrix-continued fractions. Expanding the coe$cients c

n(a, q) of Equation

(85) into a complete set with respect to a

cn(a, q)"

Q+

q/~Q

cqn/q (a) (87)

394 L. FINGER


inserting into Equation (85), and using the orthonormality relation

P /p(a)*/q(a) da"dpq

(88)

we obtain the tridiagonal recurrence relation

c5n"!JnD< c

n~1!ncc

n!Jn#1Dc

n`1(89)

with cn"(c~Q

n,2, c0

n,2, cQ

n)T. For numerical computations this in"nite system has to be trun-

cated at an upper limit Q. For the cosine potential we use the truncated complex Fourier series asthe complete set in Equation (87)

/q(a)"1

J2ne+qa, q"0,$1,$2,2 (90)

The matrix elements of the operators D and D< in Equation (89) are given by

Dpq"

JD

2n P2n

0

e~+paLLa

e+pa da"j JDqdpq

(91)

DKpq"

JD

2n P2n

0

e~+paALLa

#

sin a!i

D B e+pa da"JD AAjq!i

DB dpq!

j

2D(d

p,q`1!d

p,q~1)B

For the stationary case, i.e. c5n"0, n"0, 1,2, we solve Equation (89) by introducing the

matrices Mn

which connect cn

and cn~1

JnDcn"M

ncn~1

(92)

By repeated insertion into Equation (89) this leads to

Mn"!

1

cDAI#

1

ncM

n`1B~1

D< , n*2 (93)

Equation (93) is a matrix-continued fraction. Using Equation (92) for n"1 and c0"Hc

1we

arrive at

H"!cD< ~1 AI#1

cM

2B (94)

By iterating down we obtain the in"nite matrix-continued fraction [7]

H"!cD< ~1CI!1

c2DAI!

1

2c2D (I!2)~1D< B

~1D< D (95)

Thus, the main task in determining the distribution function P is calculating the matrix-continuedfraction (95) which is approximated by its Nth approximant. The number N and the truncationnumber Q of the Fourier series (90) are to be determined that a further increase of N and Q does

NETWORK THEORY 395


Figure 11. Distribution function for the single junction (bC"1, D"1, i"0.01).

not alter the results [7]. For the numerical computation the downward iteration (starting withN and iterating downward) is preferred against upward iteration.

After carrying out the iteration the Fourier coe$cients of Equation (90) can be calculated by

cp0"

Hp0

J2nH00

, cp1"

dp0

J2nH00

(96)

By up-iteration of the stationary Equation (89) we obtain the higher expansion exponents

c0n"!

1

cnQ+

q/~Q

DK0q

cqn~1

, n*2, p"0 (97)

cpn"

j

pJnD A(n!1)ccpn~1

#Jn#1Q+

q/~Q

DKpq

cqn~2B , n*2, pO0

Inserting the Fourier coe$cients into Equations (90), (87) and (81) we "nally obtain the distribu-tion function P of the single Josephson junction.

In Figure 11 the calculated distribution function is shown for the Josephson junction with theparameters b

C"1, D"1, i"0.01. Due to the 2n translation (77) of the Fokker}Planck equation

for the single Josephson junction the distribution function is only shown for a3(!n, n).

3.3. Coexisting solutions in Josephson junction circuits under inyuence of thermal noise

3.3.1. Equilibrium points and periodic solutions. Stability for stochastic systems is usually givenby stochastic stability. One widely used de"nition in the stochastic literature goes back toHasminski [32,43].

De,nition 3The steady solution of Equation (68) is stochastically stable if for any e'0, t

0'0, and the

probability distribution P

limx0?0

PA supt*t0

Dx(t)D*eB"0 (98)

396 L. FINGER


Figure 12. Switching between equilibrium points and oscillations caused by noise.

and asymptotically stable if, in addition,

limx0?0

PA limt?=

Dx (t)DP0B"1 (99)

or stochastically asymptotically stable in the large when

PA limt?=

Dx (t)DP0B"1, ∀x03R (100)

Under the in#uence of thermal noise the behaviour of stable solutions is more complex, inparticular for coexisting solutions. Then a strong distinction between zero-voltage state (equili-brium points) and oscillations (periodic solutions) is no longer possible. Here a switching betweenthese solutions caused by noise can be observed [24,37]. This is shown as a numerical simulationin Figure 12 for Equation (60) b

C"1, i"0.975, and a noise parameter D"0.01 for one

realization of the noise process. Here the stable equilibrium point is a"1.429, aR "0. The noisecauses a switching between this stable equilibrium point and the oscillations depending onthe noise parameter D which is represented by a moving of the state-space variable a above 2nwhere a is taken modulo 2n corresponding to Equation (37). If i is increased and moves closer toi"1 the behaviour is more and more dominated by the periodic solutions with higher probabi-lity which causes a greater mean value of the voltage aR .

This behaviour has important in#uence on practical Josephson junction circuits. The switchingmechanism causes a voltage whose mean value is di!erent from zero in contrast to the noise-freecase where a clear distinction between static and periodic solutions is evident. Therefore, thetransition from the static to the periodic solutions is no longer abrupt and the current}voltagecharacteristic (Figure 3) is rounded which has been observed in experiments. Hence the boundarybetween the static solutions (equilibrium points) and oscillations is now given by the currentIn"I

n(D) which is schematically shown in Figure 3. Moreover, for a large parameter D the

coexisting equilibrium and periodic solutions are no longer present.

NETWORK THEORY 397


Figure 13. i}vN characteristic of the double-loop SQUID: (a) without and (b) with noise (D"0.01).

If we compare this behaviour with the de"nition of stochastic stability it can be observed thatnone of the de"nitions match. Therefore, the system behaviour is unstable in stochastic sense(according to Hasminski's de"nition) even though there are two stable coexisting solutions, anequilibrium point and a periodic solution, for the deterministic system. Hence, a new quality isintroduced through thermal noise.

This switching behaviour is a special case of stochastic resonance [44]. Normally, stochasticresonance has been observed for mostly bistable systems with two equilibrium points. In theabsence of noise a switching between the bistable states is not possible. By increasing the noiseintensity transitions between the states are possible. In Josephson junction circuits the behaviouris similar between the zero-voltage state and the voltage state except the fact that there exists anequilibrium point and a periodic solution as bistable states without an external stimulus.

3.3.2. Coexisting periodic solutions and hysteresis. We have seen in Section 2.8 the possibleoccurrence and practical importance of hysteresis phenomena of periodic solutions for circuitswith at least two Josephson junctions. Again these explanations are only valid without noise and,therefore, in a very limited temperature region close to 0 K. It is, on the other hand, described inthe literature that switching between equilibrium points and periodic solutions is not the onlyerratic phenomenon caused by noise.

It is well known that thermal noise can cause a smoothing of non-linear behaviour. Moreover,switching between periodic solutions caused by noise can be observed [45]. In addition to this,thermal noise can cause a shift in bifurcation behaviour. It happens in Josephson junction circuitsas well.

This is shown by means of the current}voltage characteristic in Figure 13 of the double-loopSQUID [23,25,37] without and with noise.

The large hysteresis in Figure 13, which will only be considered in this paper, is of type (I) withreverse direction compared to Figure 4 for the noise-free case. If we consider the same character-istic of the double-loop SQUID in Figure 13(b) under in#uence of noise we observe that onlya small region of this hysteresis remains. The reason of this behaviour will be discussed by meansof the probability density function of the PoincareH map [23,37]. As can be seen in Figure 14 forgrowing noise parameter D a switching from a 2¹ periodic solution to a 1¹ periodic solutionhappens caused by a shift in the basin of attraction for i"2.04. This is the coexisting 1¹ periodic

398 L. FINGER


Figure 14. Probability density function of the PoincareH map (i"2.04) for (a) D"10~4 and (b)D"0.01, 10 000 values.

Figure 15. Probability density function of the PoincareH map for i"2.055, D"0.01, 10 000 values.

solution of the deterministic model. But for i"2.055 the 2¹ periodic solution remains stableagainst noise for this higher value of D as can be seen in Figure 15. Therefore, the bifurcationpoint shifts under in#uence of noise causing the reducing of the largest hysteresis in thecurrent}voltage characteristic in Figure 13.

Figure 16 shows the /}vN characteristics of the SQUID model in Section 4.4. For the noise-freecase a hysteresis of type (II) exists for i"2.4. The mechanism of the hysteresis has already beenschematically shown in Figure 5. The situation is completely di!erent if noise is taken intoaccount. For the noise parameter D"0.006 for ¹"4.2 K the characteristics are shown in Figure16(b). The hysteresis in the noise-free case has vanished. The reason for this behaviour can bediscussed by means of the probability density function of the PoincareH map [37] which has beencalculated for 20000 points of the PoincareH map for /"0.175 and the state-space variable xR . Theresults can be seen in Figure 17.

NETWORK THEORY 399


Figure 16. /}vN characteristics of the SQUID model (129): (a) without noise and (b) with noise(i"2.4, D"0.006) and the parameters given in Section 4.4.

Figure 17. Probability density function of the PoincareH map for /"0.175 and 20 000 points(i"2.4, D"0.006).

In the noise-free case there exists two stable periodic solutions for /"0.175 with xR "0.1 and0.48, respectively. With noise a switching and jumping between these stable 1¹ periodic solutionscan be observed for hystereses of type (II) which causes a mean value between the mean values ofthe stable periodic solutions in the noise-free case.

This is another special case of stochastic resonance. Here the two bistable states are twoperiodic solutions and a su$cient intensity of noise even for the low-temperature superconducti-vity (4.2 K) causes the switching between the two periodic solutions without any further externalstimulus.

Table I summarizes the hystereses phenomena in Josephson junction circuits without noise andwith the in#uence of thermal #uctuations.

400 L. FINGER


Table I. Hystereses phenomena in Josephson junction circuits.

Type Floquet Bifurcation Types of solutions In#uence of noisemultipl.

(I) !1, #1 One subcriticalPD, SN

Two stable, two unstablestrong connection to chaos

Shift of the SN bifurcation,hysteresis vanishes for strongnoise

(II) #1, #1 Two SN Two stable, one coexistingunstable

Switching and jumping betweenthe stable periodic solutions,rounding of characteristics

(III) !1, !1 Two subcriticalPD

Two stable, three unstable Not yet observed

The smoothing and rounding property as well as the shift of bifurcations and switchingphenomena caused by noise has a strong in#uence on experimental situations. As mentionedbefore, only mean values of the voltage can be measured because of the high frequencies. Due tothe fact that thermal noise is ever present and unavoidable in experimental situations andapplications (already 4.2 K delivers a noise parameter D"0.0044 for e.g. I

0"40 lA), an

experimental observation and identi"cation of many unwanted non-linear phenomena is current-ly not possible in Josephson junction circuits.

4. EXAMPLES

In this section the developed theory is applied to several circuits with practical applications.We show how the developed network theory can be used to study the complex dynamicsin the Josephson junction circuit family. Necessary transformations and derivativeswill be carried out to demonstrate the novel classi"cation of the Josephson junction circuitfamily.

4.1. The quantum yux parametron

The quantum #ux parametron (QFP) is a basic element for a superconducting electronics [46]. Itconsists of a superconducting loop having two identical Josephson junctions, two-loop inductorsand a load inductor ¸

d. Here we assume identical inductors and junctions. An excitation line

coupled magnetically with the loop inductors is set close to the superconducting loop. Assumingideal coupling the circuit model for the QFP is given in Figure 18.

To derive a system of di!erential equations we use again Equation (58) for the Josephsonjunctions, Equation (15) and

'd"'

e#

'0

2na1

(101)

'e

is the #ux coupled into ¸ (through Ie), '

dis the #ux which corresponds to ¸

d(through I

d).

With the parameters of the single junction and b"2n¸I0/'

0, b

d"2n¸

dI0/'

0, p"'

e/'

0,

NETWORK THEORY 401


Figure 18. Circuit model of the quantum #ux parametron.

inj"I

nj/I

0, j"1, 2 we obtain the following set of normalized equations:

bCa(1#aR

1#sin a

1"

1

b/

d!

1

ba1#

2nb

p#in1

bCa(2#aR

2#sin a

2"

1

b/

d!

1

ba2!

2nb

p#in2

(102)

/d"

1

2#b/bd

(bi#a1#a

2)

with the correlation properties Sinj

T"0, j"1, 2 and Sini(q) i

nj(q@)T"2Dd

ijd(q!q@), i, j"1, 2.

Here /d

is the output of the QFP. System (102) is of type IIb and a dissipative Hamiltoniansystem (31) where S has the form (20). The Hamiltonian function H for Equation (102) withoutnoise is

H"

1

2(aR 2

1#aR 2

2)!

1

bCA

1

2#b/bdAbi(a

1#a

2)#

1

2(a2

1#a2

2)B

!(1!cos a1)!(1!cos a

2)!

1

2b(a2

1#a2

2)#

2np

b(a

1!a

2)B (103)

Using the Langevin equations (102) and relations (73) and (74) the Fokker}Planck equation (70)for the QFP yields

LP

Lq"A!

LLa

1

aR1!

LLa

2

aR2!

1

bCA

LLaR

1A!sin a

1!aR

1#

1

2#b/bd

(bi#a1#a

2)!

1

ba1#

2np

b B#

LLaR

2A!sin a

2!aR

2#

1

2#b/bd

(bi#a1#a

2)!

1

ba2!

2np

b BB#D

bCA

L2

LaR 21

#

L2LaR 2

2BBP

(104)

with P"P (a1, a

2, aR

1, aR

2, q).

402 L. FINGER


Figure 19. Characteristic of the QFP.

Depending on the parameters coexisting equilibrium solutions can occur. In Figure 19 resultsof a simulation for I

01"I

02"50 lA,¸"2pH,¸

d"5pH, I"0A are shown.

In the "gure, the output /dis represented as a function of the exciting magnetic #ux (depending

on Ie

respectively 'e). Depending on p the output is divided into several curves. Starting with

p"0 there is a single equilibrium solution with /d"0. For p+0.385 and 0.615 bifurcations

occur and now there are three equilibrium points, one is unstable (/d"0) and two are stable

(/d"$1.955 for p"0.5). These two stable equilibria can be used as logical states. They appear

as a current whose absolute value is equal for both stable equilibria. Only its sign di!ers whichcorresponds to di!erent directions of the current through the inductor /

dwhich can be measured

in experiments.This circuit operates only in the superconducting state (equilibrium points). Therefore, a trans-

formation of Equation (102) by means of Equation (39) will not be discussed here, even though itcan be easily carried out.

4.2. A one-dimensional parallel array of the Josephson junctions

The developed theory can be applied to arrays of the Josephson junctions too. Especially, severalarray structures show very complex behaviour with many unwanted phenomena which distractpossible applications.

In Figure 20 a circuit model of a one-dimensional parallel array of Josephson junction isshown. Such arrays have applications, e.g. as shift register, transmission line and a superconduct-ing transistor using vortex solutions [47,48].

This array consists of N Josephson junctions which are connected by superconducting stripsrepresented by an inductance ¸. A current I

jcan be injected onto the array at each node. For the

individual junction the Stewart/McCumber model is used (58).Using KCL we can write for the current I

Ljthrough ¸ connecting node j and j#1:

ILj"I

Lj~1!I

Jj#I

j(105)

NETWORK THEORY 403


Figure 20. A parallel array of Josephson junctions.

Using the #ux quantization (12), the normalizations for the single junction, and the parameterb"1/"2

j"2n¸I

0/'

0(discreteness parameter) the set of normalized stochastic di!erential equa-

tions describing the 1D parallel array of Josephson junctions becomes [49]

bCa(1#aR

1#sin a

1"i

1#

1

b(a

2!a

1)#

2n/

b#i

n1

bCa(j#aR

j#sin a

j"i

j#

1

b(a

j~1!2a

j#a

j`1)#i

nj(106)

bCa(N#aR

N#sin a

N"i

N#

1

b(a

N~1!a

N)!

2n/

b#i

nN

with j"2(1)N!1 and the correlation properties SinjT"0, j"1(1)N and Si

ni(q)i

nj(q@)T"

2Ddijd(q!q@), i, j"1(1)N.

In Ref. [50] it has been shown that these arrays can be modelled by the paradigm of acellular neural network (CNN) with a non-linear template and a non-memoryless outputfunction by neglecting thermal noise. For N"2 this array reduces to an ideal d.c.-SQUID[24].

System (106) without noise has only additional inductances to the junctions and no additionalresistors and is, therefore, of type (IIa) and a dissipative Hamiltonian system (31) where S has theform (20). The Hamiltonian function H for equation (106) without noise is

H"

1

2

N+j/1

aR 2j!

1

bCA

N+j/1

(1!cos aj#i

jaj)#

2n/b

(a1!a

N)!

1

2bN~1+j/1

(aj`1

!aj)2B (107)

With Equation (106) and relations (73) and (74) the Fokker}Planck Equation (70) for the parallelarray holds

LP

Lq"A!

N+j/1

LLa

j

aRj!

1

bCA

LLaR

1Ai1!aR

1!sin a

1#

1

b(a

2!a

1)#

2n/

b B

404 L. FINGER


#

N~1+j/2

LLa5

jAij!aR

j!sin a

j#

1

b(a

j~1!2a

j#a

j`1)BB

#

LLa5

NAiN!aR

N!sin a

N#

1

b(a

N~1!a

N)!

2n/

b B#D

bC

N+j/1

L2

LaR 2jBP (108)

with P"P (a1,2, a

N, aR

1,2, aR

N, q).

By means of Equation (39) (respectively Equation (40)) with M"N we obtain the followingsystem of stochastic di!erential equations:

bCy(1#yR

1#sin(y

1#y

N)"f (y, i)#i

1#

1

b(y

2!y

1)#

2n/

b#i

n1

bCy(j#yR

j#sin(y

j#y

N)"f (y, i)#i

j#

1

b(y

j~1!2y

j#y

j`1)#i

nj, j"2(1)N!2

bCy(N~1

#yRN~1

#sin(yN~1

#yN)"f (y, i)#i

N~1#

1

b AyN~2!2y

N~1!

N~1+j/1

yjB#i

nN~2(109)

bCy(N#yR

N"!f (y, i)

f (y, i)"1

N AN~1+j/1

sin(yj#y

N)#sin AyN!

N~1+j/1

yjB!

N+j/1

(ij#i

nj)B

with y"(y1,2, y

N)T. Here the state variable y

Nappears only within trigonometric functions and

Equation (41) holds. Moreover Equations (42)} (45) are valid for this parallel array of Josephsonjunctions. Therefore, it is possible to choose the hyperplane for the PoincareH map independent ofthe number of Josephson junctions and of the order of system (109). Furthermore, the followingtheorem can be formulated for this parallel array of Josephson junctions [50].

¹heorem 2Any periodic solution with period ¹ in the one-dimensional parallel array of Josephson

junctions is a running periodic solution.

Proof. Using Equations (43) and (39) we obtain with the translation property of periodicsolutions a

jPa

j#2nm, j"1(1)N. Equation (107) for the Hamiltonian system combined with

the translation property (37) of periodic solutions yields

H(¹)!H(0)"!

2nm

bC

N+j/1

ij

(110)

On the other hand, the dissipation relation (34) yields

H(¹)!H(0)"!

1

bCP

T

0

N+j/1

aR 2jdq (111)

If H(¹)"H(0) then all aRj"0 and the solution is an equilibrium point. Otherwise, m must be

di!erent from 0 and thus it is a running periodic solution. Therefore, for any periodic solutionwith Equations (43), Equation (111) proves the theorem for mO0. K

NETWORK THEORY 405


Figure 21. Energy values of vortex solutions in di!erent positions of a parallel array of Josephson junctions.

In this one-dimensional parallel array some special equilibrium solutions known as vorticescan occur which have recently found some applications [48]. If we set i

1"2"i

N"0, /"0,

and choose the initial conditions ai"2kn, i)z and a

i"2(k#1)n, i'z, k"0, 1,2;

1(z(N we obtain stable vortices between two Josephson junctions of the array. By settingai"2kn, i(z, a

i"(2k#1)n, i"z and a

i"2(k#1)n, i'z, k"0, 1,2; 1)z)N and using

Newton's method we obtain unstable vortex solutions. Without loss of generality we setk"0 [50].

In Ref. [50] it has been demonstrated that this parallel array is a discrete approximation ofa long Josephson junction which can be described by a partial di!erential equation, thesine-Gordon equation [9]. Moreover, it has been shown that a particular solution, the vortexsolution

a"$4 arctan eJb(x~x0 ) (112)

of the (stationary) sine-Gordon equation

1

baxx!sin a"0 (113)

can be approximated by this parallel array of Josephson junctions for N large.In Figure 21 the vortex solutions for an array of 9 junctions (N"9) with the parameters

bC"1, b"5.263158 are shown. Here the ordinate shows the size of the energy function H, the

Hamiltonian function (107), depending on the position of the vortex in the array. As can be seenstable as well as unstable vortex solution occur. Global minima occur if all phases are of the formaj"2kn, k"0, 1,2, j"1(1)N where H"0. The stable vortex solutions are represented in

Figure 21 as local minima whereas unstable vortices are given by local maxima.But this behaviour is only valid without noise. Under the in#uence of noise some stable vortex

solutions are no longer present. At "rst, the vortices at the boundaries of the array close to theglobal minima (i"1, 8) vanish by jumping over the barrier due to noise. This is shown in Figure22 for D"0.004. In this "gure the state-space variable a

9is shown depending on the time q.

Starting with the stable vortex caused by thermal noise the trajectory jumps after some time over

406 L. FINGER


Figure 22. In#uence of noise on stable vortex solutions.

the lower energy barrier (Figure 21) to the equilibrium point with lower energy where all phasesare in the zero state. For this noise parameter D only the stable vortices at the boundaries vanishwhereas those in the middle of the array remain stable against noise. For higher parameters D allvortices disappear which has an important in#uence on possible use of vortices for high-temperature superconductivity applications [48].

4.3. A series Josephson junction array

Figure 23 shows the circuit model of a series Josephson junction array. These arrays haveapplications as tuneable oscillators, e.g. References [51,52]. Here the mechanism of phase lockingis used in order to deliver high-frequency oscillations.

Unfortunately, these series Josephson junction arrays have like other Josephson junctioncircuits a very complex dynamics too with lots of unwanted phenomena like chaos, hyperchaos,semirotor solutions [53] and neutral stability [54] which can destroy the mechanism of phaselocking. With the developed theory in this paper it is possible to understand the behaviour andthe non-linear and stochastic dynamics of these arrays.

The series array consists of N Josephson junctions which are biased and shunted by a load[51]. Hadley et al. used several loads (resistive, inductive, capacitive) and combinations of thesecircuit elements which we discuss in this paper in connection with the novel classi"cation of theJosephson junction circuit family. Like in the examples above we use again Equation (58) for theJosephson junctions and equations for the currents

I"I1#I

L(114)

4.3.1. Resistive, capacitive, and inductive load. For a series resistive, capacitive, and inductiveload the voltage across the array which equals the voltage across the load is

<"¸L

dIL

dt#I

LR

L#

1

CLP I

Ldt"

'0

2nN+j/1

aRj

(115)

NETWORK THEORY 407


Figure 23. A series Josephson junction array.

with IL"QQ

L"I

RL#I

n0where Q

Lis the electric charge and I

n0the noise current of the external

resistor.Using the normalizations of the single junction and q

L"Q

L/I

0RC (normalized charge),

pR"R

L/R, p

C"C/C

L, i

L"I

L/I

0, i

n0"I

n0/I

0we obtain the following set of normalized

stochastic di!erential equations:

bCa(j#aR

j#sin a

j"i!qR

L#i

nj, j"1(1)N (116)

bbCq(L#b

CpRqRL#p

CqL"

N+i/1

aRi

with Sinj

T"0 and ini(q)i

nj(q@)"2Dd

ij(q!q@), i, j"1(1)N.

System (116) is of order 2N#2 with x"(a1,2, a

N, q

L, aR

1,2, aR

N, qR

L)T. It is of type (Ib) and

a dissipative Hamiltonian system where the Hamiltonian function without noise is

H"

1

2

N+j/1

aR 2j#

1

2qR 2L!

1

bCA

N+j/1

(1!cos aj)#i

N+j/1

aj!

pC

2bq2LB (117)

The matrix K is given by

K"A1 1

} F

1 1

~1b 2 ~1b bC

b pRB (118)

where all other entries of the matrix are 0.

408 L. FINGER


4.3.2. Resistive and capacitive load (¸"0). Here the system of normalized stochastic di!erentialequations yields

bCa(j#aR

j#sin a

j"i!qR

L#i

nj, j"1(1)N (119)

bCpRqRL#p

CqL"

N+i/1

aRi

System (119) is of order 2N#1 and of type (Ic).

4.3.3. Resistive and inductive load (CPR). Here the system of normalized stochastic di!erentialequations yields

bCa(j#aR

j#sin a

j"i!i

L#i

nj, j"1(1)N (120)

biQL#p

RiL"

N+i/1

aRi

System (120) is again of order 2N#1 and of type (Ic) where the state variable q1

is given by iL.

4.3.4. Resistive load (¸"0,CPR). This circuit is used for further investigations. Here weobtain the following system of stochastic di!erential equations:

bCa(j#aR

j#sin a

j"i!

1

pR

N+i/1

aRi#i

nj, j"1(1)N (121)

with the correlation properties Sinj

T"0 and Sini(q)i

nj(q@)T"2Dd

ijd (q!q@), i, j"1(1)N.

System (121) is of type (IIa). Again without noise it is a dissipative Hamiltonian system (31)where S is of the form (19). Here K and K@ are given by

kij"G

1#1

pR

, i"j

1

pR

otherwise, k@

ij"

1

pR

(122)

The Hamiltonian function H for Equation (121) without noise is

H"

1

2

N+j/1

aR 2j!

1

bCA

N+j/1

(1!cos aj)#i

N+j/1

ajB (123)

With Equation (121) and relations (73) and (74) the Fokker}Planck Equation (70) for the seriesarray holds

LP

Lq"A!

N+j/1

LLa

j

aRj!

1

bC

N+j/1

LLaR

jAi!aR

j!sin a

j!

1

pR

N+i/1

aRiB#

D

bC

N+j/1

L2LaR 2

jB P (124)

with P"P (a1,2, a

N, aR

1,2, aR

N, q).

NETWORK THEORY 409


***Due to the central limit theorem [32] it is possible to join the sums and di!erences of noise currents inj

, j"1(1)Ntogether into a new noise variable.

By means of Equation (39) respectively Equation (40) with M"N we obtain the followingsystem of stochastic di!erential equations:***

bCy(j#yR

j#sin(y

j#y

N)"f (y, i)#i

nj, j"1(1)N!1

bCy(N#yR

N"!f (y, i)#i!

N

pR

yRN

(125)

f (y, i)"1

N AN~1+j/1

sin(yj#y

N)#sin AyN!

N~1+j/1

yjB!

N+j/1

injB

with y"(y1,2, y

N)T. Again the state variable y

Nappears only within trigonometric functions

and Equation (41) holds. Moreover Equations (42)}(45) are valid for this series array ofJosephson junctions. Therefore, it is possible to choose the hyperplane for the PoincareH mapindependent on the number of Josephson junctions and on the order of system (125). Note thatbeside Equation (41) the translation properties y

jPy

j#2n, j"1(1)N!1 hold for this series

array. But unlike to yN

Equation (38) does not deliver a motion in the cylindrical phase space foryj, j"1(1)N!1.

4.3.5. Chaos and Hyperchaos. The presented theory shall be demonstrated on two examples.Firstly, we use N"2 for the series Equation (121) (respectively Equation (125)) array whichcorresponds to a system fourth order. Using the parameters b

C"1, p

R"2 [51] we observe

chaotic behaviour (without noise) which coexists to periodic solutions if i is varied which can beseen in Figure 24(a) by means of the largest Ljapunov exponent and in Figure 24(b) with theone-dimensional brute-force bifurcation diagram based on the PoincareH map.

At i"1.399 a saddle-node bifurcation to another periodic solution occurs. Later chaos arisesthrough a saddle-node bifurcation at i"1.515. From the right-hand side chaos arises throughFeigenbaum scenario. In Figure 25(a) the three-dimensional chaotic attractor is shownfor i"1.55. For the calculation of the PoincareH map a method based on Runge}Kutta triples isused [55].

As mentioned before, this chaotic attractor is coexisting with a stable periodic solution. Thequestion arises what happens with noise which delivers a more realistic model of this circuit? It isknown that new phenomena like noise-induced chaos and noise-induced order can occur throughthermal noise [24].

For the parameter of the series array of Josephson junctions the situation is di!erent. Here oneof the coexisting solutions can no longer be observed. In this case only the chaotic solution ispresent if the in#uence of thermal noise is considered. Moreover, this happens for very smallvalues of the noise parameter D far below of the technical relevant size which is demonstrated inFigure 25(b) by means of the PoincareH map using the programme DONANS [23]. Here thelargest Ljapunov exponent is j

1"0.022.

Without noise a stable 1¹ periodic solution is present which is coexisting to the chaoticsolution in Figure 25. With noise the stochastic trajectory jumps into the basin of attraction of thechaotic attractor. Unlike to noise-induced chaos here a jump mechanism is present which causesa qualitative change in the dynamical behaviour. Moreover, unlike to other mechanisms where

410 L. FINGER


Figure 24. Largest Ljapunov exponent and bifurcation diagram for the series array (N"2,bC"1, p

R"0.5).

Figure 25. PoincareH map (i"1.55, N"2): (a) of the chaotic attractor and (b) under in#uence of thermalnoise (D"0 and D"10~5) with initial conditions of the periodic solution.

the periodic solutions remain more stable rather than the chaotic solutions against noise here thedynamic behaviour is inverted.

The non-linear behaviour is more complex for more than 2 junctions. In the following we use3 junctions for the simulations. The parameters are similar as before. Here hyperplane (44) is "vedimensional. We obtain the Ljapunov exponents and the bifurcation diagram which are given inFigure 26.

As can be seen for some parameter regions chaotic behaviour occurs. Hyperchaos is present aswell which can be observed by means of the spectrum of the Ljapunov exponents. Moreover, forsome parameter regions periodic solutions, chaotic, and hyperchaotic attractors are coexisting.This is demonstrated in Figure 27 for i"1.7 by means of the PoincareH map. Due to the fact thatthe attractor is "ve dimensional only a three-dimensional projection can be shown.

It is visible that the hyperchaotic attractor is much more structured than the chaotic attractor.To "nd out the reason for this hyperchaotic behaviour it is reasonable to look at the behaviour ofthe superconducting phases in Figure 28.

For the chaotic behaviour two phases oscillate uniformly. Here this happens for a1

and a2.

Only the third phase oscillates di!erent to the others. Moreover, the di!erences among the "rst

NETWORK THEORY 411


Figure 26. (a) First three Ljapunov exponents and (b) bifurcation diagram for a series array ofJosephson junctions (N"3, b

C"1, p

R"2).

Figure 27. Projection of the PoincareH map of the (a) chaotic and (b) hyperchaotic attractors for the seriesarray of Josephson junctions (N"3, b

C"1, p

R"2, i"1.7).

two phases and the third phase grow with increasing time. On the other hand, there isa permutation in this behaviour, e.g. the second and third phase oscillate uniformly and the "rst isdi!erent to the others, etc. For the hyperchaotic attractor all three phases oscillate di!erent as canbe seen in Figure 28. Furthermore, no divergence among the phases can be observed. They movemore irregular which leads to the more structurated hyperchaotic attractor in comparison to thechaotic attractor. At least a system with three junctions is necessary for hyperchaotic behaviour inthe Josephson junction circuit family assumed that the dimension of the circuit is only determinedby the Josephson junctions (type (II)).

With noise we observe again jump phenomena. Under in#uence of noise only the hyperchaoticattractor is present which is demonstrated in Figure 29. Here the PoincareH map is calculated withinitial conditions in the basin of attractions of the chaotic attractor. Although the spectrum ofLjapunov exponents cannot be calculated generally for noisy systems at the moment, it can beseen by means of the PoincareH map that only the hyperchaotic attractor appears in the noisy casealready for very small noise. Here noise causes a destruction of the phase-locking mechanismbetween two of the three phases. Moreover, under in#uence of thermal noise all three supercon-ducting phases move now di!erent to each other. Thus the hyperchaotic attractor is more stableagainst thermal noise than the chaotic attractor and the chaotic attractor is no longer observablein the noisy case.

412 L. FINGER


Figure 28. Phases for (a) chaotic and (b) hyperchaotic behaviour (N"3, bC"1, p

R"2, i"1.7).

Figure 29. In#uence of noise on the chaotic attractor for the series array of Josephson junctions (N"3,bC"1, p

R"2, i"1.7D"10~8) with initial conditions of the chaotic attractor.

4.3.6. Semirotor solutions. This section deals with the exceptional case of the 2n translationproperty Equation (38) where M(N.

De,nition 4A semirotor solution is a solution where there exists a ¹'0 such that a

i(q#¹ )"a

i(q)#2n

with i"1(1)M and ai(q#¹ )"a

i, i"M#1(1)N, M(N. Hence the "rst M phases

ai, i"1(1)M wind around the cylinder and the last N!M phases perform a contractible

oscillation [53].

The considerations about the cylindrical phase space in the Josephson junction circuit familycan be applied to semirotor solutions as well. Using transformations (39) and (40) for M(N weobtain for equation (121) the new transformed system

bCy(j#yR

j#sin(y

j#y

M)"f (y, i)#i

nj, M'0, j"1(1)M!1

bCy(M#yR

M"!f (y, i)#i!

M

pR

yRM!

1

pR

N+

i/M`1

aRi, M'0 (126)

NETWORK THEORY 413


sssThere are other other models known, e.g. with an additional capacitor in series to RS

[57] which deliversa dynamical system of seventh order with very complex behaviour.

f (y, i)"1

M AM~1+j/1

sin(yj#y

M)#sinAyM!

M~1+j/1

yjB!

M+j/1

injB , M'0

bCa(j#aR

j#sin a

j"i!

1

pR

N+i/1

aRi#i

nj, j"M#1(1)N

which merges Equations (121) and (125) together and delivers a more general equation. ForM"0 we arrive at Equation (121) whereas for M"N we obtain Equation (125). Therefore, thenew transformation delivers both Equation (121) and the transformed Equation (125) as specialcases of the general system Equation (126) by changing just one parameter (M) to M"0 and N,respectively and the semirotor solutions with an arbitrary M as 0(M(N.

For such semirotor solutions very complicated semirotors can occur, even semirotor chaos.For 5 junctions (N"5) and M"3 (3 junctions wind around the cylinder and 2 junctions performa contractible motion) and the parameters p

R"5, i"1.2 with a variable b

Cwe observe

semirotor chaos by a period doubling route which can be seen in Figure 30 by the calculation ofthe Ljapunov exponent and the one-parameter bifurcation diagram. In Figure 31 the chaoticmotion is shown by the plot of the phase space for the a

1!aR

1plane (motion around the cylinder)

and for the a4!aR

4plane (contractible motion). As can be seen a

1PR for tPR whereas

0.6(a4(1.5.

4.4. A complete SQUID circuit model

Figure 32 shows the circuit model of a complete superconducting quantum interference device(SQUID). It consists of two parts, the SQUID with a double-loop structure [56] and the inputcircuitry which are coupled by two inductances described by a mutual inductance M.sss

In order to derive the di!erential equations we use Equation (58) for the Josephson junctions

'0

2n(a

1!a

2)#¸AIJ1#

'0C

P2n

(aK1!aK

2)!

I

2B!'a#MI

L"0 (127)

for the SQUID loop and the capacitor CP, and

¸SQ$

L!

M

2(IQ

2!IQ

1)"

QC

CS

"RSQQ

R(128)

for the input circuitry, we obtain after normalization the following system of stochastic di!eren-tial equations:

bCc1x(#xR #sin x cos y"

2

b(n/!x)!c

2qRS#

in1!i

n22

bCy(#yR #sin y cosx"

i

2#

in1#i

n22

(129)

c3q(S#c

4qRS#q

S"2c

5xR #2c

6(x!n/)

414 L. FINGER


Figure 30. (a) First Ljapunov exponent and (b) bifurcation diagram show period doublingroute to semirotor chaos (N"5, M"3, p

R"5, i"1.2).

Figure 31. Semirotor chaos shown by the (a) a1}aR

1plane and (b) a

4}aR

4plane (N"5, M"3,

pR"5, i"1.2, b

C"1.9).

with the parameters of the single junction and b"2n¸I0/'

0, /"'

a/'

0, k"M/J¸

S¸,

pCP

"CP/C, p

CS"C

S/C, p

LS"¸

S/¸, p

RS"R

S/R and the parameters

c1"1#2p

CP

c2"b

Ckp

CSJp

LSpRS

c3"b

Cbp

CSpLS

(1!k2)

c4"

bpLS

pRS

(1!k2) (130)

c5"

kJpLS

pRS

c6"

c5

bCpCS

pRS

NETWORK THEORY 415


Figure 32. The SQUID circuit model.

with the correlation properties SinjT"0, j"1, 2 and Si

ni(q)i

nj(q@)T"2Dd

ijd (q!q@), i, j"1, 2.

The state-space variable qS"Q

S/R

SC

SI0

is the normalized charge.System (129) without noise is of type (IIa). Using the Langevin equations (102) and relations

(73) and (74) the Fokker}Planck equation (70) for the SQUID circuit model yields

LP

Lq"A!

LLx

xR !LLy

yR !L

LqS

qRS!

1

bCA

LLxR A!sin x cos y!xR #

2

b(n/!x)!c

2qRSB#

LLyR

A!sin y cos x!yR #i

2B#L

LqRS

bC

c3

(!c4qRS#2c

5xR #2c

6(x!n/))#

D

bCA

L2

LxR 2#

L2LyR 2BBBP

(131)

For some parameters new phenomena occur for this system. For the experimentally realizedparameters [27] I

0"30 lA, R"2 ), C"1 pF, ¸"25 pH, ¸

S"2.5 nH, C

P"1 pF,

CS"20 fF, R

S"100k), k2"0.3, ¹"4.2 K(D"0.006) simulation results are shown for

I"69 lA(i"2.3), 72 lA(i"2.4). The case i"2.4 has already been discussed in Section 3.3.2 inconnection with the consideration about type II hysteresis. Figure 33 shows the /}vN character-istics of the SQUID model for i"2.3.

For i"2.3 there is a jump in the /}vN characteristic caused by noise even for the temperature4.2 K for /"0.265. A local analysis without noise shows that there exists another stable 1¹periodic solution for these parameters with almost the same stability properties but higher meanvalue of the voltage (vN+1 compared to vN+0.6). This solution is never reached in the noise-freecase. Thermal noise causes a switching to this second periodic solution, a stochastic bifurcationand, therefore, a jump in the #ux}voltage characteristic.

This is another example of erratic behaviour which can be caused by thermal #uctuations in theJosephson junction circuit family.

416 L. FINGER


Figure 33. /}vN characteristics of the SQUID model (a) without noise and (b) with noise (D"0.006, i"2.3).

5. CONCLUSION

In this paper a circuit theory for Josephson junction networks is developed. Unlike to previouspapers concerning the single junction or coupled junctions, by means of dissipative Hamiltoniansystems and ordinary di!erential equations several di!erent Josephson junction circuits areuni"ed with a single description. From the circuit theory point of view network models and thephysical mechanism for oscillations are explained. Moreover, two concepts for the description ofnoisy non-linear networks, the Langevin approach and the Fokker}Planck equation are reviewedand applied to Josephson junction networks. The circuit theory is applied to several circuitsbased on Josephson junctions and numerical simulations.

APPENDIX A: ANALYSIS OF NON-LINEAR AND STOCHASTIC SYSTEMS

All simulations in this paper were carried out by using the author's software package dynamics ofnon-linear and noisy systems (DONANS) which was developed speci"cally for the analysis notonly of non-linear but as well of noisy systems. The software package was developed using theobject oriented software technology.

For the non-linear part the program uses as integration routine for general non-linear systemsof the form

x5 "f (x, t), x3Rn (A.1)

(1) the standard Runge}Kutta-Fehlberg RKF4(5) method with step size control,(2) a Runge}Kutta triple DP5(4)T5 [58,59] with automatic step size control and an interpola-

tion algorithm for dense output,(3) the Rosenbrock method for sti! systems,(4) the Gear method for sti! systems and(5) an adaptive algorithm with switching between the Dormand}Prince solver and the Gear

method by a sti! detection algorithm.

NETWORK THEORY 417


The software package includes the following analysis methods:

(1) Time simulation with the above described integration routines.(2) Determination of stable and unstable equilibrium points by a Newton method and their

stability with continuation.(3) Searching for stable and unstable periodic solutions of non-autonomous and autonomous

circuits and systems using a quasi-Newton method and numerical calculation of theFloquet multipliers for the determination of the stability with continuation.

(4) Calculation of the PoincareH map for non-autonomous and autonomous systems usinga fast algorithm [23].

(5) Calculation of the spectrum of Ljapunov exponents with continuation.(6) Simulation of the one-parameter brute-force bifurcation diagram.(7) Calculation of the autocorrelation function.(8) Determination of the power spectra.(9) Wavelet analysis.

(10) Harmonic balance.(11) Cell mapping method. Here single cell mapping [60], interpolated cell mapping [61], and

multiple mapping [62] can be used for arbitrary dimensions of dynamical systems.(12) Calculation of static characteristics (in particular for Josephson junction circuits). Here an

algorithm based on the PoincareH map is used.

The stochastic part uses stochastic integration routines for the simulation of circuits andsystems with additive noise of the form

x5 "f (x, t)#g (t), x, g3Rn, Sgi(t)T"0, Sg

i(t)g

j(t@)T"2Dd

ijd(t!t@), i, j"1(1)n (A.2)

(1) a stochastic Runge}Kutta method third order [38] for arbitrary number of noise sources,(2) a stochastic Runge}Kutta algorithm second order [42] for arbitrary number of noise

sources and(3) a one-step collocation method [63] for one noise source.

Moreover, the following analysis methods are included in the program DONANS for noisysystems:

(1) Time simulation using the three stochastic integration routines.(2) Calculation of the PoincareH map and the probability density function of the PoincareH map

(described in Refs. [23,37]) for non-autonomous and autonomous systems.(3) Calculation of the largest Ljapunov exponent with continuation.(4) Simulation of the one-parameter brute-force bifurcation diagram.(5) Calculation of the autocorrelation function.(6) Determination of the power spectra.(7) Wavelet analysis.(8) Single cell mapping.(9) Calculation of static characteristics (in particular for Josephson junction circuits).

A model library of currently 25 systems is included in DONANS which can be easily extendedat every time. The software package DONANS is capable in simulating very large systems (e.g.cellular neural networks with a large number of cells and partial di!erential equations by the"nite di!erence method). An on-line help and graphical output are included with the possibility of

418 L. FINGER


printing, producing "gures in the Postscript (EPS) and HPGL format to include them in textprocessing systems.

ACKNOWLEDGEMENTS

The work of L. Finger was partially supported by the Alexander von Humboldt Foundation (AvH).

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420 L. FINGER


the josephson junction circuit family: network theory

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