Synchronization in a one-dimensional array of point Josephson junctions coupled to a common load

We study the synchronization in a one-dimensional array of point Josephson junctions coupled to a common capacitor, which establishes a long-range interaction between junctions and synchronizes them. The stability diagram of synchronization in a noise-free system is obtained. The current when junctions transform from resistive state into zero-voltage state is then calculated and its dependence on the shunt parameters and the dissipation of junctions is revealed. In the presence of thermal noise, the synchronized oscillations are destroyed at a critical temperature and the system undergoes a continuous phase transition of desynchronization. A possible stability diagram of the synchronized oscillations with respect to thermal noise, current, dissipations, and shunt capacitance is then constructed. Finally we investigate the dynamic relaxation from random oscillations into a synchronized state. The relaxation time increases with the system size and temperature, but is reduced by the shunt capacitor.

Figure 1

(a) Schematic view of an array of Josephson junctions shunted with a capacitor. The junctions are biased by a dc current $I_B$. (b) The Josephson junction is modeled as a shunt circuit of a capacitor, a resistor, and a nonlinear Josephson current.

Figure 2

The largest Floquet exponent calculated by $\det \mathbf{D}=0$ with $\mathbf{D}$ given by Eq. (18) (red line), and by numerical calculation using the Floquet theory in Eq. (21) (symbols). For the exponent smaller than 0, the uniform oscillations are stable.

Figure 3

Stability diagram of the uniform solution in the absence thermal fluctuations. Light blue/pink/orange region denotes complete synchronization/zero-voltage state/partial synchronization. The blue line is the stability boundary of the uniform solution calculated by the Floquet theory $I_s$, and the open red circle is the retrapping current $I_r$ determined by direct calculations of Eqs. (5, 6, 7). The dashed line shows the retrapping current determined by Eq. (8) while the dotted line shows the retrapping current for a single junction. Here $C_s=3/N$.

Figure 4

I-V curve (black) and dependence of the order parameter on the voltage (red) for $\beta =2.0$ and $C_s=3/N$.

Figure 5

Dependence of the boundary retrapping current $I_r$ on the shunt capacitance for several $\beta$'s. Lines are obtained with the Floquet theory and symbols are direction simulations of Eqs. (5, 6, 7). The region above the line corresponds to the uniform oscillations while zero-voltage state below the line. Here $N=200$.

Figure 6

I-V curves for (a) $\beta =0.02$ and (b) $\beta =1.0$ for several typical values of $C_s$. Here $N=200$.

Figure 7

Same as Fig. 5 but with a shunt RC circuit. Parameters are indicated in the figures.

Figure 8

Stability of the uniform oscillations when an array of Josephson junctions are shunted by an LRC circuit. The regions below lines are stable.

Figure 9

Dependence of the order parameter on temperature with different system sizes. Inset is a double-logarithm plot of the reduced temperature $T_m-T$ and order parameter. Here $I_B=1.5$, $\beta =1.0$, and $C_s=3.0/N$.

Figure 10

Dependence of the fluctuations ${\sigma }_r$ on the temperature. Here $I_B=1.5$, $\beta =1.0$, and $C_s=3.0/N$.

Figure 11

(a1), (b1), and (c1): Dependence of $T_m$ on the bias current, shunt capacitance, and $\beta$, respectively. (a2), (b2), and (c2) are the corresponding largest Floquet exponent. Other parameters used are shown in the figure.

Figure 12

Possible stability diagram of the uniform solution (a) at a given $C_s$, and (b) at a given $\beta$. The region inside the green surface corresponds to stable uniform oscillations.

Figure 13

Time evolution of the distribution of phase difference Eq. (29), starting from completely random state at several temperatures. Here $\beta =0.02$, $I_B=1.5$, and $C_s=3/N$.

Figure 14

(a) Dependence of the relaxation time on the system size with different temperatures. Symbols are numerics and lines are the best fitting. (b) Dependence of the relaxation time on the temperature with the system size of $N=200$ and with different shunt capacitance. (c) Speedup of the relaxation by increasing the shunt capacitance. Here the system size is $N=200$ and temperature $T=0.002$. All these results are obtained with $\beta =0.02$ and $I_B=0.15$.