**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 *L* of a single branch of the device to be negligible, so that β = *LIJ*/Φ<sup>0</sup> ≈0, where Φ0 is the elementary flux quantum and *IJ* 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 differential equation (ODE) written in terms of the phase variable *φ*, which represents the average of the two gauge-invariant superconducting phase differences, *φ*1 and *φ*2, 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* can be written as follows (Barone & Paternò, 1982):

© 2012 De Luca, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 De Luca, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

$$\frac{\text{d}\,\phi}{\text{d}\,\pi} + \left(-1\right)^{n} \cos\pi\nu\prime\_{ex} \sin\phi = \frac{i\_{\text{B}}}{2} \tag{1}$$

Effective Models of Superconducting Quantum Interference Devices 223

values in the *N* current loops of a one-dimensional array containing *N+1* 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 currentphase relation. As specified before, the critical current, the *I*-*V* characteristics and the fluxvoltage curves of the array can be determined analytically by means of the effective model*.* Furthermore, a one dimensional array of *N* 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

Finally, a single-junction model for a d.c. SQUID is derived when we consider the effect of

applied magnetic flux have, in addition to a constant term *A*, an a. c. component, we can find, by similar reasoning as in the case of a constant applied magnetic flux, the effective

component of the applied magnetic flux in a closed analytic form. From the analysis of the

The work will thus be organized as follows. In Section 2 the derivation of an effective singlejunction 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

Let us consider a symmetric two junction interferometer with equal junctions of negligible

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

*n B*

πψ

+ − + =

<sup>d</sup> 1 cos sin ; (a) d 2

 φ

<sup>d</sup> 1 sin cos . (b) <sup>d</sup> 2 2

*n ex*

 φ

 ψ

β

ψ

> β

capacitance, as shown in fig. 1. The dynamical equations for the variables

() ( )

() ( )

 *i* πψ

+ − =

ω

is comparable with <sup>1</sup>

ω

β =

ω

ψτ

and the amplitude *B* of the a. c.

<sup>−</sup> . By letting the

0, the critical current of

and *B* can play the role

φ and ψ

(4)

existing experiments (Scharinger et al., 2010).

rapidly varying applied fields whose frequency

of additional control parameters in the device.

further investigations are suggested.

reduced single-junction model for the system. In particular, for

voltage vs. applied flux curves it can be argued that the quantities

the device is seen to depend on *A*, and on the frequency

**2. Two-junction quantum interference devices** 

φ

τ

π ψ

τ

where *n* is an integer denoting the number of fluxons initially trapped in the superconducting interference loop, τ=2π*RIJt*/Φ0=*t*/τφ , *R* being the intrinsic resistive junction parameter, ψ*ex*=Φ*ex*/Φ0 is the externally applied flux normalized to Φ0 and *iB= IB*/*IJ*, is the bias current normalized to *IJ*. In what follows we shall consider zero-field cooling conditions, thus taking *n*=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 *IJF* (*IJF*=|cosπψ*ex*|) in which a normalized bias current *iB*/2 flows. This strict equivalence comes from the hypothesis that the total normalized flux ψ=Φ/Φ0 linked to the interferometer loop can be taken to be equal to ψ*ex*. However, being

$$
\Psi = \Psi\_{ex} + \mathcal{B}(i\_1 - i\_2) \,\prime \tag{2}
$$

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 β, 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):

$$\frac{\mathbf{d}\,\phi}{\mathbf{d}\,\pi} + X\_{ex}\sin\phi + \pi\beta Y\_{ex}\,^2\sin 2\phi = \frac{\dot{\mathbf{l}}\_{\text{B}}}{2} \tag{3}$$

where *Xex*= cosπψ*ex* and *Yex*= sinπψ*ex*. A second-order harmonic in φ thus appears in addition to the usual sinφ term. The sin2φ addendum, however, arises solely from electromagnetic coupling between the externally applied flux and the system, as described by Eq. (2), when β≠0. Therefore, the non-conventional CPR of the equivalent JJ in the SQUID model cannot be considered as a strict consequence of 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 sin2φ term for finite values of β. In order to understand how the reduction in the dimensional order of the dynamical equations is possible, it is noted that the quantities τφ *=* Φ0/2π*RIJ* and τψ *= L*/*R*, denoting the characteristic time scales of the variables φ and of the number of fluxons ψ in the superconducting SQUID loop, respectively, are intimately linked to the parameter β, since τψ/τφ *=* 2πβ. In this way, for constant applied magnetic fields, the flux dynamics for small values of β can be considered very fast with respect to the equivalent junction dynamics given in Eq. (3). As a consequence, the superconducting phase φ can be assumed to be adiabatic and the equation of motion for ψ in terms of the quasi-static variable φ can be solved by perturbation analysis. When the information for ψ is substituted back into the effective dynamical equation for φ, Eq. (3) is finally obtained.

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 *N+1* 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 currentphase relation. As specified before, the critical current, the *I*-*V* characteristics and the fluxvoltage curves of the array can be determined analytically by means of the effective model*.* Furthermore, a one dimensional array of *N* 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).

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 <sup>1</sup> ψτ <sup>−</sup> . By letting the applied magnetic flux have, in addition to a constant term *A*, 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 β = 0, the critical current of the device is seen to depend on *A*, and on the frequency ω and the amplitude *B* 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 ω and *B* can play the role of additional control parameters in the device.

The work will thus be organized as follows. In Section 2 the derivation of an effective singlejunction 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.
