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\begin{document} 

{\small
{\bf Phase Locking Dynamics in 2D Josephson Junction Networks with Small 
Loop Inductances}

Wolfram Krech

\tiny
{\it Friedrich Schiller University, Institute of Solid State Physics,\\
Helmholtzweg 5, D-07743 Jena, Germany}

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For about a decade there is continuous interest in 2D Josephson junction 
networks because according to basic estimates the radiation output 
generated by these arrays is expected to be much larger than from linear 
chains. However, experimental results show that one of the serious 
problems in 2D networks could be the presence of flux in the loops. 
Magnetic fields can influence the phase locking of the voltage 
oscillations and, consequently, reduce the output power immensely.

Here we will study problems connected with external flux in hybrid 
networks with junctions attached to the lines in bias direction only. 
Furthermore, assuming very small loop inductances, the junctions within the 
interferometers perpendicular to the bias current are strongly coupled. 
This way, the synchronization of the Josephson oscillations in 2D hybrid 
arrays reveals a certain similarity to the well-known locking dynamics in 
linear chains.

We start with an analysis of phase locking in elementary SQUID cells 
within the RSJ model, taking into account the flux quantization. We 
develop a systematic perturbation method allowing  the investigation of 
locking in cells with small but nonvanishing loop inductance. The junction 
voltages are found to be locked with very small phase differences for 
almost all values of external flux. Strongly coupled cells show 
quasi-uniform synchronization even for large parameter splitting. Next, we 
consider phase locking in simple 2D arrays consisting of two 
small-inductance loops coupled via a joint line transverse to bias 
current. It is shown that in the stable dynamic regime both cells 
oscillate antiphase. This result may already hint at the low radiation 
output obtained so far in 2D networks experimentally. 

Based on the analytical procedure combining ideas from small-induc\-tance 
approximation with those from slowly varying phase we deduce rigorous 
results on ladder arrays and inductively coupled multi-junction 
interferometers. These are excellently confirmed by numerical simulations. 
Finally, considering hybrid Josephson junction networks with external load 
impedance and solving the problem of the interplay of long-range and 
next-neighbouring interferometer interactions, we are able to look for 
effective multi-junction microwave sources.

}

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