## 1. Introduction

It is well known that the electrodynamic properties of SQUIDs (Superconducting Quantum Interference Devices) are obtained by means of the dynamics of the Josephson junctions in these superconducting system (Barone & Paternò, 1982; Likharev, 1986; Clarke & Braginsky, 2004). Due to the intrinsic macroscopic coherence of superconductors, r. f. SQUIDs have been proposed as basic units (qubits) in quantum computing (Bocko et al., 1997). In the realm of quantum computing non-dissipative quantum systems with small (or null) inductance parameter and finite capacitance of the Josephson junctions (JJs) are usually considered (Crankshaw & Orlando, 2001). The mesoscopic non-simply connected classical devices, on the other hand, are generally operated and studied in the overdamped limit with negligible capacitance of the JJs and small (or null) values of the inductance parameter. Nonetheless, r. f. SQUIDs find application in a large variety of fields, from biomedicine to aircraft maintenance (Clarke & Braginsky, 2004), justifying actual scientific interest in them.

As for d. c. SQUIDs, these systems can be analytically described by means of a single junction model (Romeo & De Luca, 2004). The elementary version of the single-junction model for a d. c. SQUID takes the inductance

*of a single branch of the device to be negligible, so that*L

β

*=*

*/Φ*LI

_{J}

_{0}≈0, where Φ

_{0}is the elementary flux quantum and

*is the average value of the maximum Josephson currents of the junctions. In this way, the Josephson junction dynamics is described by means of a nonlinear first-order ordinary*I

_{J}

*differential equation (ODE) written in terms of the phase variable*

*, which represents the average of the two gauge-invariant superconducting phase differences,*φ

*and*φ

_{1}

*, across the junctions in the d. c. SQUID. By considering a device with equal Josephsons junction in each of the two symmetric branches, the dynamical equation of the variable*φ

_{2}

*can be written as follows (Barone & Paternò, 1982):*φ

where * n* is an integer denoting the number of fluxons initially trapped in the superconducting interference loop, τ=2π

*/Φ*RI

_{J}t

_{0}=

*/*t

*,*τ

_{ϕ}

*being the intrinsic resistive junction parameter,*R

*=Φ*ψ

_{ex}

*/Φ*

_{ex}

_{0}is the externally applied flux normalized to Φ

_{0}and

*/*i

_{B}= I

_{B}

*, is the bias current normalized to*I

_{J}

*. In what follows we shall consider zero-field cooling conditions, thus taking*I

_{J}

*=0. Eq. (1) is similar to the non-linear first-order ODE describing the dynamics of the gauge-invariant superconducting phase difference across a single overdamped JJ with field-modulated maximum current*n

*(*I

_{JF}

*=|cosπ*I

_{JF}

*|) in which a normalized bias current*ψ

_{ex}

*/2 flows. This strict equivalence comes from the hypothesis that the total normalized flux*i

_{B}

*=Φ/Φ*ψ

_{0}linked to the interferometer loop can be taken to be equal to

*. However, being*ψ

_{ex}

we may say that the above hypothesis may be stated merely by means of the following identity: * β=0*.

*Therefore, for finite values of the parameter*

*, Eq. (1) is not anymore valid and the device behaves as if the equivalent Josephson junction possessed a non-conventional current-phase relation (CPR). In fact, for small finite values of*β

*, one can see that the following model may be adopted (Romeo & De Luca, 2004):*β

where * X*= cosπ

_{ex}

*and*ψ

_{ex}

*= sinπ*Y

_{ex}

*.*ψ

_{ex}

*A second-order harmonic in*

*thus appears in addition to the usual sin*ϕ

*term. The sin2*ϕ

*addendum, however, arises solely from electromagnetic coupling between the externally applied flux and the system, as described by Eq. (2), when*ϕ

*0. Therefore, the non-conventional CPR of the equivalent JJ in the SQUID model cannot be considered as a strict consequence of*β

*an intrinsic non-conventional CPR of the single JJs. The Josephson junctions in the device, in fact,*

*could behave in the most classical way, obeying strictly to the Josephson current-phase relation; the*

*interferometer, however, would still show the additional sin2*

*term for finite values of*ϕ

*. In order to understand how the reduction in the dimensional order of the dynamical equations is possible, it is noted that the quantities*β

*Φ*τ

_{ϕ}

_{}=

_{0}/2π

*and*RI

_{J}

τ

_{ψ}

*/*= L

*, denoting the characteristic time scales of the variables*R

*and of the number of fluxons*ϕ

*in the superconducting SQUID loop, respectively, are intimately linked to the parameter*ψ

*, since*β

τ

_{ψ}/τ

_{ϕ}

=

*2π*

*. In this way, for constant applied magnetic fields, the flux dynamics for small values of*β

*can be*β

*considered very fast with respect to the equivalent junction dynamics given in Eq. (3). As a consequence,*

*the superconducting phase*

*can be assumed to be adiabatic and the equation*ϕ

*of motion for*

*in terms of the quasi-static variable*ψ

*can be solved by perturbation analysis. When the information for*ϕ

*is substituted back into the effective dynamical equation for*ψ

*, Eq. (3) is finally obtained.*ϕ

The single-junction model can be adopted also when dealing with more complex systems, as one-dimensional Josephson junction arrays. In fact, by assuming small inductance values in the * N* current loops of a one-dimensional array containing

*identical overdamped Josephson junctions, the dynamical equations for the gauge-invariant superconducting phase differences can be reduced to a single non linear differential equation (Romeo & De Luca, 2005). The resulting time-evolution equation is seen to be similar to the single-junction dynamical equation with an appropriately defined current-phase relation. As specified before, the critical current, the*N+1

*-*I

*characteristics and the flux-voltage curves of the array can be determined analytically by means of the effective model*V

.

*Furthermore, a one dimensional array of*

*cells of 0- and π-junctions in parallel can be considered (De Luca, 2011). In this case, by assuming that junctions parameters and effective loop areas alternate as one moves along the longitudinal direction of the array, going from 0- to π-junctions, an effective single junction model for the system can be derived. It can be shown that, by this model, interference patterns of the critical current as a function of the applied magnetic flux can be analytically found and compared with existing experiments (Scharinger et al., 2010).*N

Finally, a single-junction model for a d.c. SQUID is derived when we consider the effect of rapidly varying applied fields whose frequency * ω* is comparable with

*, an a. c. component, we can find, by similar reasoning as in the case of a constant applied magnetic flux, the effective reduced single-junction model for the system. In particular, for*A

β =

*0, the critical current of the device is seen to depend on*

*, and on the frequency*A

*and the amplitude*ω

*of the a. c. component of the applied magnetic flux in a closed analytic form. From the analysis of the voltage vs. applied flux curves it can be argued that the quantities*B

*and*ω

*can play the role of additional control parameters in the device.*B

The work will thus be organized as follows. In Section 2 the derivation of an effective single-junction model for a symmetric d. c. SQUID containing two equal junctions will be briefly reviewed. In Section 3 the extension to this model to Josephson junction arrays with equal junctions in all branches will be considered. In Section 4 the case of the alternate presence of 0- and π-junctions in the array is considered, the system being similar to multifacets Josepshon junctions. In Section 5 the effective single-junction model for a d. c. SQUID in the presence of rapidly varying field is derived. Finally, in Section 6 conclusions are drawn and further investigations are suggested.

## 2. Two-junction quantum interference devices

Let us consider a symmetric two junction interferometer with equal junctions of negligible capacitance, as shown in fig. 1. The dynamical equations for the variables * ϕ* and

*characterizing this system, can be written in the following form (Romeo & De Luca, 2004):*ψ

where * n* is the number of initially trapped fluxons in the superconducting loop. Let us consider a new time variable

*a and*ϕ

ψ

*in the following form:*

*=0.*n

This approach allows us not only to account for the regular part of the solution, as seen in (Grønbech-Jensen et al., 2003) and in (Romeo & De Luca, 2004), but also to consider its singular part. Moreover, as we shall see, the role of the two time variables will become evident in what follows, since one time scale is defined for Eq. (4a) and one for Eq. (4b). Consider then * β*, we can obtain a system of equations for the functions

k

*= 0, 1, describing the k-th order solutions for*

*and*ϕ

*, respectively. These approximate solutions are determined according to the following sequential scheme. As a first step, we use Eq. (4b) to determine*ψ

As for initial conditions, from Eq. (4b) we may notice that

Furthermore, we may also notice that = 0, 1.

By the general procedure described above we get the following differential equations for the superconducting phase variables:

and the following for the flux number variables:

In Eqs. (7a-b) and (8a-b) we may notice the appearance of two different time scales the first, * ϕ*. We have already noticed that

*could be studied by taking asymptotic solutions of*ϕ

*. In this case, therefore, we may first let the flux variable evolve, so that a stationary magnetic state is reached, and then solve for the superconducting phase difference time evolution of the system. This is exactly what is done, under the assumptions of negligible value of the ratio*ψ

*and*ϕ

*, one would follow the more general perturbation analysis described above, where the ratio*ψ

*might not a priori be considered as negligible. Furthermore, considering that this ratio is proportional to the perturbation parameter*r

*, one might wish to generalize the procedure described above to higher order in*β

*to acquire a wider range of validity of the analysis.*β

Despite the fact that the more general approach allows extension to higher order approximations of the perturbation solutions, we wish to limit our analysis to the study of the electrodynamic properties of a two-junction or a multi-junction quantum interferometer with very small parameter * β*. Therefore, while in the present section we shall only be concerned with a single time scale, namely

By considering, for the time being, only the time scale

where the term * β*, and the subscript

*in*β

*in this limit.*β

In order to obtain some preliminary results, we start by considering

and the following for the flux number variables:

According to the scheme described above, by having already set

where k is an integer,

On the other hand, in the case

where

Finally, in the case

where

Solutions for

As a simple application, let us calculate, to first order in the parameter β, the circulating current iS in the circuit, normalized to

For an arbitrary value n, which represents the number of fluxons initially trapped in the superconducting ring, we have

Graphs of circulating currents are shown in figs. 4a, 4b, 4c for n even,

Notice that the lowest value of the period is obtained for

The above results have been obtained for the magnetic response of the system in the presence of a constant applied flux. In the following we shall analyze the electrodynamic response of the two junction quantum interferometer in the presence of a time-dependent external flux. For this purpose, we shall take a sinusoidal forcing term, in such a way that

Now, since

where* B* of the oscillating signal is much less than one (

*<<*B

*). The perturbation analysis is then carried out in a way at all similar as done above.*1

We start by setting, by Eq. (9a) and (9b),

By noticing, however, that

where

Notice then that the solutions to the above equations can be found by exactly the same procedure described for the case of constant applied fields. Once the solution for

As before, the above expression is equal to * i* in the circuit. In figs. 5a, 5b, and 5c the time dependence of the current

_{S}

*for normalized frequency values*i

_{S}

*,*A=0

*0.01 and*β=

Another important quantity to be measured in these systems is the critical current

Therefore, we have

Noticing that the time-averaged value <* i*> of the critical current does not depend on the normalized frequency, it can be calculated in terms of solely

_{c}

*and*A

*, the results being shown in fig. 6a and fig. 6b for null values of the initially trapped flux. In particular, in fig. 6a <*B

*> is shown as a function of the applied magnetic field amplitude*i

_{c}

*, for*B

*and*A=0.1

*, while in fig. 6b, <*A=0.4

*> vs.*i

_{c}

*curves are shown for*A

*and*B=0.1

*. In the curves in fig. 6a we notice Fraunhofer-like oscillations, while ordinary cosinusoidal oscillations are present in fig. 6b.*B=0.2

For what seen above, the electrodynamic properties of a symmetric quantum interferometer containing two identical junctions with negligible capacitance can be studied by means of a perturbation approach in the parameter * β*, whose value gives the strength of the electromagnetic coupling between the two junction in the system. The analysis is rather similar to what done in other works in the literature (Grønbech-Jensen et al., 2003; Romeo & De Luca, 2004). However, in the present section we have presented a rather general procedure to obtain the solution to the problem to first order in the parameter

*. Considering at first transient solutions, we have noticed that the function*β

*is the ordinary time*θ

*, normalized to the characteristic circuital time constant*t

*is different from the fluxon dynamics characteristic time*ϕ

*is sufficiently small to allow, for finite values of*β

*, an asymptotic evaluation of*τ

The perturbation analysis has been first carried out for a constant applied magnetic flux. Successively, since it could be experimentally possible to force the system with a time-dependent magnetic field, it is noted that the perturbed solution for the flux number * ψ*, obtained for a sinusoidal magnetic flux, needs careful evaluation. In order to exhibit experimentally detectable quantities, the circulating current

*is evaluated as a function of time, for different values of the frequency of the forcing field, whose a. c. component is assumed to be small. Finally, the time average <*i

_{S}

*> of the critical current of the device has been studied both as a function of the d. c. component*i

_{c}

*and of the small amplitude*A

*of the oscillating part of the applied flux. In these curves two characteristic behaviors have been detected: A Fraunhofer-like pattern in <*B

*> vs.*i

_{c}

*curves; independence of <*B

*> from*i

_{c}

*is not assumed to be small.*B

## 3. Multi-junction quantum interference devices

In this section we shall consider the one-dimensional Josephson junction array (1D-JJA) represented in fig. 7, consisting of * N*+1 identical overdamped junctions connected in parallel. In this figure we notice that the bias current

*and maximum Josephson current*R

*of the horizontal upper branches to be such that*L

where

When an external magnetic field

for

where

Define now the partial sum

By fluxoid quantization (Eq. (25)), setting all

for

where

where

The above analysis has been carried out essentially to write the dynamical equations in terms of the effective superconducting phase difference

We shall now develop a reduction of these variables to one, by assuming small values of the parameter * β*. Therefore, start by considering the dynamical equations of the system as written in Eqs. (32a-b). For small values of the parameter

*we can set:*β

By substituting the above expression in Eq. (32b) and, after having multiplied both members by * β*, by setting the coefficients of

For the first order corrections, on the other hand, we need to solve the following set of equations:

where

Substitution of the above results into Eq. (32a) gives:

The above differential equation represents the effective model for the 1D-JJA represented in Fig. 7, containing * β*, does not explicitly contain a

We can now explicitly perform the sum in Eq. (37), so that we write:

where the absolute value of

In this way, we can find the I-V characteristics by simply integrating Eq. (39), recalling the well-known procedure for a single overdamped junction. Indeed, noticing that

where T is the period for the instantaneous voltage curve and a pseudo-period for the superconducting phase difference

The pseudo-period T of the above solution can be found by inspection, so that:

Therefore, by Eqs. (40) and (42), the * I-V* characteristics are given by the following expression:

where only the positive branch has been chosen.

In Figs. 8a-b

In Figs. 9a-b * I-V* characteristics for different externally applied flux values are shown. In Fig. 9a the number of JJ’s is taken to be equal to 10, while in Fig. 9b it is set equal to 15. Starting from Fig. 9a, we notice that, as the normalized flux approaches the first zero in the

For fixed bias current values, the voltage versus flux curves can be obtained by Eq. (43) and is given by the following:

The above expression is similar to the homologous d. c. SQUID

In Figs. 10a and 10b we report the

In conclusion, by considering the dynamical equations of one-dimensional arrays containing * N*+1 identical overdamped Josephson junctions, the system of

*+1 nonlinear first-order ordinary differential equation equations can be broken into two coupled subsystems, one consisting of only one equation for the superconducting phase of one junction in the array (arbitrarily chosen to be the first), the second describing the time evolution of*N

*opportunely defined normalized flux variables.*N

When a solution of the latter * N* equations is found, by means of a perturbative approach to first order in the parameter

*, the dynamical properties of the system are described by a single time-evolution equation. In this way, we may affirm that, for small values of*β

*, the system may be described by an equivalent single-junction model, where the maximum Josephson current is appropriately defined in terms of the normalized applied flux*β

*is one order less than the first-order perturbation analysis carried out for the simplest two-junction system. This is a consequence of the approach followed in the present work, where we had to appropriately define partial sums of flux variables in order to separate the dynamical equations into two subsystems. When we refer to the SQUID case, then, we might state that the present analysis corresponds exactly to the*β

*.*β

Even though part of the present analysis reproduces known results, as, for example, the expression for the maximum Josephson current, it still represents a simple way of approaching the problem of the electrodynamic response of one-dimensional arrays of overdamped Josephson junctions by an equivalent single-junction model. In fact, by starting with the simple representation of the dynamics of the system given in Eq. (38), all the results obtained for a single Josephson junction can be reproduced for an array of * N*+1 equal overdamped Josephson junctions. In addition, in case the solutions of the normalized flux variable would be extended to second order in the parameter

*, following the same analysis as in the present section, effects due to finite*β

*values in the electrodynamic properties of the system would be detected. Finally, considering that the present analysis has been carried out in the absence of flux fluctuations, its extension to noise effects can be obtained by means of already known results obtained for a single overdamped Josephson junction (Ambegaokar & Halperin, 1969; Bishop & Trullinger, 1978). In this case, however, care must be taken in considering the stochastic terms on all branches of the array.*β

## 4. Parallel connections of N × (0-π) overdamped Josephson junctions

In the previous section we have considered an array of * N*+1 overdamped 0-junctions, without considering the possibility of inserting π-junctions in the system. We briefly recall that π-junctions (Bulaevskii et al., 1977; Geshkenbein et al., 1987; Baselman et al., 1999; Ryazanov et al., 2001 M. Weides et al., 2006), when compared to 0-junctions, possess an intrinsic phase difference exactly equal to π. By inserting a 0-junction and a π-junction in the same superconducting loop, π-SQUIDs may be realized. These non-conventional SQUIDs can be fabricated either by exploiting the symmetry properties of d-wave superconductors (Chesca, 1999; Schultz et al., 2000) or by utilizing both s-wave and d-wave superconductors (Wollman et al., 1993; Smilde et al., 2004). A π-SQUID can thus be viewed as an elementary cell of a

*×(0-π) one-dimensional array of overdamped Josephson junctions shown in fig. 11.*N

Therefore, π-SQUIDs can be viewed as the building block of discretized models of multifacets Josephson junctions (MJJs) (Scharinger et al., 2010), in which the critical current density alternates between two opposite values along the junction length. However, even though some characteristic features of MJJs can be qualitatively reproduced by * N*×(0-π) one-dimensional arrays, one should bear in mind that the latter are, in general, less complex systems than MJJs.

For conventional arrays of overdamped Josephson junctions we have already shown that, for small enough values of the characteristic parameter * β* a series solution for the magnetic flux variable can be found by perturbation analysis. In this way, the multi-junction interferometer model reduces to a single non-linear ordinary differential equation. The same perturbation approach will be proposed again in the present section to derive the equivalent single-junction model of

*×(0-π) one-dimensional arrays of overdamped Josephson junctions. Therefore, we start by considering the model system represented in fig. 11,*N

*consisting of identical overdamped junctions connected in parallel. In this system one half of the bias current*

*equal to the length of the array. In order to have well focused bias, the penetration length of the superconducting bar would be much smaller than its width*w

*.*w

### 4.1. The homogeneous case

Consider, as a first approach to the problem, the loop areas

By fluxoid quantization, the normalized magnetic flux

where

for* k*-th branch and

where

By fluxoid quantization, setting all

for

where

where now

Considering the dynamical equations of the system as written in Eqs. (51a-d), for small values of the parameter * β* we may assume that the solution, to first order in this perturbation parameter, can be written as follows:

By substituting the above expression in Eqs. (51b-d) and, after having multiplied both members by * β*, by setting to zero the coefficients of

For the first order corrections, on the other hand, we need to solve the following set of equations:

where

Substitution of the above result into Eq. (51a) gives:

It is now possible to explicitly calculate the finite sum in Eq. (56) to get, in terms of the number

where* N* cells, each one containing one 0-junction and one π-junction.

## 4.2 The non-homogeneous case

Consider, next, a non-homogeneous array with alternating 0-π Josephson junctions. We shall take the parameters of all 0-junctions equal. The parameters of π-junctions, even being equal among them, are assumed to be different from those of the 0-junctions. In this case we can omit some of the calculations, having already treated the problem in detail in the previous subsection.

Considering again the 1D-JJA represented in fig. 11, we now assume that the loop areas

for all allowed values of * k*. In this way, the additional parameters

*,*α

*, and*ε

*are implicitly defined. As before, we set*σ

Fluxoid quantization give the same relation as in Eq. (45) between the superconducting phases and the normalized magnetic flux

for

where * n* runs over all allowed

*-values also in all following equations.*k

By adopting a first-order perturbation analysis in the parameter * β*, we set:

By substituting the above expression in Eqs. (49) to obtain the superconducting phases

Where here * α*,

*, and*ε

*all equal to one, the ordinary differential equation (62) reduces to Eq. (57).*σ

### 4.3. Critical current

In order to find the critical current of arrays with alternating 0-π Josephson junctions, we proceed as follows. First, consider the homogeneous array described in Section 4.1. We look for the maximum value of the bias current

we can express the critical current of the homogeneous device as follows:

In order to understand the origin of the patterns we are going to show for the non homogeneous case, let us consider the result in Eq. (64) as the product of the envelop function

(

with

One notices that regular primary peaks of

By proceeding in the same way for the non-homogeneous case, starting from Eq. (62) we find that the critical current of the non homogeneous array described in Section 3 can be written as follows:

Notice that the resulting pattern does not depend on * α*, which can be absorbed, in the effective dynamical equation, by a rescaling of the normalized time

*. Notice also that the periodicity of the envelop function*τ

*=12,*N

In fig. 13a the envelop curve * N*=12,

*is a rational number, one has periodicity in the*σ

We have seen that a reduced single-junction model can be adopted to describe the overall dynamics of a one-dimensional arrays with alternating parameters of *N*×(0-π) Josephson junctions. This effective model is very useful, since it allows to obtain the critical current vs. normalized magnetic flux curves in closed analytic form. The interference patterns are seen to be qualitatively similar to recently obtained experimental results on multifacets Josephson junctions (Scharinger et al., 2010) in which the critical current density alternates many times between two opposite values along the junction length. Discrete Josephson junction arrays, even presenting some analogies with the latter devices, are much too simple systems to describe the complete behaviour of MJJs. As a matter of fact, shielding current effects is not taken into account by analysis carried out in this section in the lowest order approximation in

*. Nevertheless, the analytic results obtained for the interference patterns shed some light on the causes of the presence or absence of periodicity and on the nature of primary and secondary peaks in the*β

## 5. Quantum interferometers in the presence of rapidly varying fields

In Section 2 we have analyzed the effective model for a two-junction quantum interferometer in the presence of an oscillating magnetic flux under the hypothesis that the frequency of oscillation is comparable with the inverse of the characteristic time evolution * τ* of the superconducting time variable

_{ϕ}

*. In this way, the quasi-static approach described in Section 2 has been proven to be applicable. In the present section, on the other hand, we shall consider rapidly varying externally applied fluxes, whose frequency*ϕ

*is comparable with*ω

Let us therefore consider an externally applied flux having d. c. component * A* and a. c. amplitude

*, so that*B

where * n*=0, as follows:

where the normalization * τ*-dependence of the externally applied flux as written in Eq. (18) with a normalized frequency

*in terms of the superconducting phase by perturbation analysis on*Ψ

*for arbitrary values of*β

*and*A

*. In this way, the effective dynamics for*B

*can be found when the solution for*φ

*is substituted in the cosine term of (69a).*Ψ

Start by considering the cosine term in (69b) as a quasi-static quantity (i. e., it does not vary appreciably over an interval of time of the order of* φ* varies on a characteristic time interval

*, the variable*Δτ

_{ϕ}

*varies within a time interval*Ψ

*=*Δτ

_{Ψ}

*2π*

*«*βΔτ

_{φ}

*. Within the former time interval*Δτ

_{φ}

*it is then possible to choose a subinterval, of the order of*Δτ

_{φ}

*, in which the variable*Δτ

_{ψ}

*does not vary appreciably. We can thus solve (69b), by perturbation analysis, by first setting*φ

*2π*τ=

βθ

*and by rewriting it as follows:*

By now setting

we again find the ODEs for _{0}_{} and * ψ* in (8a) and (8b). Recall that an ODE of the type

_{1}

has solution * f(θ)*=

e

^{−θ}

^{}

*d*

^{θ}g(x)e

^{x}

*. By now considering (68), by taking the non-decaying solutions of the system of ordinary differential equations (8a-b) and by considering the non-vanishing solution of (8a) for*x

*at large values of*ψ

_{0}

*, we have*θ

Having found the solution to (8a), we can find the solution to (8b) by the same type of reasoning. After some rather long calculations one finds

where

with* n*, and

*, by the following ODE:*β

where

Equation (76) thus represents the differential equation describing the dynamics of the superconducting phase difference * ϕ* in a d. c. SQUID in the presence of a time-varying externally applied flux, whose frequency

*is considered to be comparable with*ω

τ

_{ψ}

^{-1}, in such a way that

For* ϕ* and

*, still occurs with two completely different time scales. In fact, as already stated, one has*ψ

*=*τ

_{ψ}

*2π*

*«*βτ

_{ϕ}

*. Therefore, flux motion is very fast with respect to the dynamics of the phase variable*τ

_{ϕ}

*. The only difference, here, is that the externally applied flux*ϕ

*. Therefore, we may write*ψ

where the symbol <* x*> stands for the time average of the variable

*. Equation (77) can thus be considered an effective single-junction model for a d. c. SQUID in the presence of a rapidly varying magnetic field (*x

Let us next express the exponential of a cosine and a sine terms in

In this way, (78) becomes

It is now easy to show that

where the symbol

with

Proceeding in a similar way in finding the effective coefficient of the

where

Having expressed the effective time-averaged terms in (77) in a closed analytic form, we can understand the effect of a high-frequency field on the electrodynamic behaviour a d. c. SQUID with extremely small value of the parameter * β*, for instance. The critical current

β =

*0) can be expressed as follows:*

where the quantity * A*, as

β =

*0 and getting the maximum stationary value for*

*.*ϕ

In fig. 15a-b we thus show the critical current * B* of the applied flux. In particular, in fig. 15a, we report the

*curves for various values of the d. c. component*B

*of the applied flux. normalized frequency*A

*curves are shown for various values of the normalized frequency*B

*.*B

The particular shape of these patterns in figs. 15a-b depends on the value of the normalized frequency* A* and the time-varying portion of the magnetic flux attain fixed values, and we let the normalized frequency vary with continuity, the

*=0.1 and for various values of*A

*. In this respect, we notice that, while for small fixed*B

*values the quantity*B

*approaching 1.0, this character is lost (see brown and cyan lines in fig. 16).*B

We can now calculate, for * β=0*, the flux-voltage curves (v vs.

*) by the following well known expression (Barone & Paternò, 1982):*A

for* B* and

*curves can be varied and*A

*and*B

*curves for*A

*. In particular, in fig. 17a we notice that the amplitude of the v vs.*B

*curves obtained at*A

*=0 and*B

*increase to 0.15 first (brown line) and to 0.30 next (cyan line). The same decreasing behavior is detected in fig. 17b for*B

*increases from 0.0 (orange line) to 0.15 (brown line) and to 0.30 (cyan line).*B

In fig. 18a-b, finally, by fixing the value of * B* first to 0.5 (a) and then to 1.0 (b), we notice that the amplitude of the v vs.

*curves increases for increasing values of*A

## 6. Conclusion

We have studied the dynamical properties of quantum interferometers consisting of single or multiple superconducting loops, each containing two Josephson junctions.

A symmetric quantum interferometer containing two identical junctions with negligible capacitance has been considered first. The analysis of the system has been carried out by means of a perturbation approach in the parameter * β*, whose value gives the strength of the electromagnetic coupling between the two junction in the system. We have noticed that the flux-number function

*normalized to the characteristic circuital time constant*t

*is different from the characteristic time*φ

*and the time average of the critical current <*i

_{S}

*> as a function of the d. c. and a. c. components of the applied flux are evaluated in the adiabatic limit, assuming that the oscillation frequency*i

_{c}

*> vs.*i

_{c}

*curves are shown to be independent from the normalized frequancy*B

Next, the dynamical equations of one-dimensional arrays containing * N*+1 identical overdamped Josephson junctions are considered. It has been noticed that the system of

*+1 nonlinear first-order ordinary differential equations can be broken into two coupled subsystems, one consisting of only one equation for the superconducting phase of one junction in the array (arbitrarily chosen to be*N

*), the second describing the time evolution of*φ

_{0}

*opportunely defined normalized flux variables. When a solution of the latter*N

*equations is found, by means of a perturbative approach to first order in the parameter*N

*, the dynamical properties of the system are described by a single time-evolution equation for*β

*. In this way, we may affirm that, for small values of*φ

_{0}

*, the system may be described by an equivalent single-junction model, where the maximum Josephson current is appropriately defined. The analysis represents a simple way of approaching the problem of the electrodynamic response of one-dimensional arrays of overdamped Josephson junctions by an equivalent single-junction model.*β

The same approach is followed for one-dimensional arrays with alternating parameters of * N*×(0-π) Josephson junction. Even in this case an effective single-junction model can be adopted to describe the overall dynamics of the system. This model is very useful, since it allows us to obtain the critical current vs. normalized magnetic flux curves in closed analytic form. The interference patterns are seen to be qualitatively similar to recently obtained experimental results on multifacets Josephson junctions (Scharinger et al., 2010) in which the critical current density alternates many times between two opposite values along the junction length. The analytic results for the interference patterns clarify the presence or absence of periodicity and the nature of primary and secondary peaks in these curves. Further investigation on the dependence of

*on different distributions of the JJs in the array can be of interest.*i

_{c}

Finally, by allowing the magnetic flux, applied to a two-junction superconducting quantum interference device, to have an a. c. component in addition to a constant term * A*, we derive the effective reduced single-junction model describing the dynamics of the average superconducting phase difference

*of the two junctions in the device. The difference between this case and the one previously treated is that the alternating flux now varies with a frequency*ϕ

*of the same order of magnitude of*ω

*. Averaging of the rapidly varying quantities in the differential equation for*β

*gives the effective dynamics of the two junctions in the system. In particular, for*ϕ

*, the critical current of the device is seen to depend on*β = 0

*, on the frequency*A

*of the a. c. component of the applied magnetic flux in a closed analytic form. From the analysis of the voltage vs. applied flux curves it can be argued that the quantities*B

*is necessary. Experimental work confirming the predictions of the present analysis needs to be performed. As far as non-normalized quantities are concerned, for direct experimental confirmation of the present results, we finally notice that the junction dynamics evolves with characteristic frequencies the order of 1 THz. Therefore one needs to run the experiment with very rapidly oscillating signals (10 THz or more) in such a way that normalized frequencies of*β

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