Synchronization in disordered Josephson junction arrays: Small-world connections and the Kuramoto model

B. R. Trees, V. Saranathan, and D. Stroud

Phys. Rev. E 71, 016215 – Published 14 January 2005

Abstract

We study synchronization in disordered arrays of Josephson junctions. In the first half of the paper, we consider the relation between the coupled resistively and capacitively shunted junction (RCSJ) equations for such arrays and effective phase models of the Winfree type. We describe a multiple-time-scale analysis of the RCSJ equations for a ladder array of junctions with non-negligible capacitance in which we arrive at a second order phase model that captures well the synchronization physics of the RCSJ equations for that geometry. In the second half of the paper, motivated by recent work on small-world networks, we study the effect on synchronization of random, long-range connections between pairs of junctions. We consider the effects of such shortcuts on ladder arrays, finding that the shortcuts make it easier for the array of junctions in the nonzero voltage state to synchronize. In two-dimensional (2D) arrays we find that the additional shortcut junctions are only marginally effective at inducing synchronization of the active junctions. The differences in the effects of shortcut junctions in 1D and 2D can be partly understood in terms of an effective phase model.

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Received 12 May 2004

DOI:https://doi.org/10.1103/PhysRevE.71.016215

Authors & Affiliations

B. R. Trees^* and V. Saranathan

Department of Physics and Astronomy, Ohio Wesleyan University, Delaware, Ohio 43015, USA

D. Stroud

Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA

^*Electronic address: brtrees@owu.edu

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Vol. 71, Iss. 1 — January 2005

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Figure 1
Ladder array with periodic boundary conditions and $N = 8$ plaquettes. A uniform, dc bias current $I_{B}$ is inserted into and extracted from each rung as shown. The gauge-invariant phase difference across the rung junctions is denoted by $γ_{j}$ where $1 ⩽ j ⩽ N$ , while the corresponding quantities for the off-rung junctions along the outer (inner) edge are $ψ_{1, j}$ $(ψ_{2, j})$ . The rung junctions are assumed to be disordered while the off-rung junctions are uniform.Reuse & Permissions
Figure 2
(Color online) Frequency synchronization order parameter $f$ , plotted versus nearest-neighbor coupling strength $α$ for a ladder with $N = 10$ plaquettes and bias current $i_{B} = 5$ . Rung junction critical currents are assigned nonrandomly with $Δ = 0.05$ . $β_{c} = (a)$ 1, (b) 5, (c) 10, and (d) 20. For each plot the phase differences and voltages are reset to zero with each new value of $α$ .Reuse & Permissions
Figure 3
(Color online) Frequency synchronization order parameter $f$ , plotted versus nearest-neighbor coupling strength $α$ for a ladder with $N = 15$ plaquettes, bias current $i_{B} = 5$ , and $β_{c} = 20$ . Rung junction critical currents are assigned randomly with $Δ = 0.025$ , and results are averaged over ten realizations of critical currents. Error bars represent the standard deviation of the mean value of $f$ and for clarity only a few, representative error bars are shown. Numerical solution of the RCSJ model are denoted by filled circles, and the results from the LKM2 are denoted by crosses.Reuse & Permissions
Figure 4
(Color online) Phase synchronization order parameter ${⟨ ∣ r ∣ ⟩}_{τ}$ , plotted versus nearest-neighbor coupling strength $α$ for a ladder with $N = 10$ plaquettes, $i_{B} = 5$ , and nonrandom critical currents with $Δ = 0.05$ . $β_{c} = (a)$ 1, (b) 5, (c) 10, and (d) 20. For each plot phase differences and voltages across rung junctions are reset to zero with each new value of $α$ .Reuse & Permissions
Figure 5
(Color online) (a) Frequency synchronization order parameter $f$ , plotted versus dc bias current $i_{B}$ for fixed coupling strength $α = 0.25$ for both the RCSJ model and the LKM2. $N = 10$ , $β_{c} = 5$ , and nonrandom critical currents with $Δ = 0.05$ . (b) Standard deviation of time-averaged rung junction voltages versus bias current for the same model parameters as in (a). The difference between the two models evident at large bias currents is probably due to phase slips in the off-rung junctions that violate Eq. (13).Reuse & Permissions
Figure 6
Time-averaged voltage across $j = 1$ rung junction, plotted versus bias current $i_{B}$ for $N = 10$ , $α = 0.01 ⪡ α_{c}$ , and $β_{c} = 1$ . The solid circles are the numerical result from the RCSJ model. They agree well with the analytical result for the weak-coupling limit, ${⟨ v_{1} ⟩}_{τ} = Ω_{1}$ , where $Ω_{1}$ is given by Eq. (26), which is represented by a dashed line in the graph. Deviations from the large bias current result, ${⟨ v_{1} ⟩}_{τ} \approx i_{B} ∕ i_{c 1}$ (solid line), do not appear until $i_{B} ≲ 2.5$ .Reuse & Permissions
Figure 7
(a) (Color online) Frequency synchronization order parameter $f$ , plotted versus coupling strength $α$ for a ladder with $N = 10$ , $i_{B} = 5$ , and nonrandom critical currents with $Δ = 0.05$ . Results based on the RCSJ model for ten different $β_{c}$ values are shown. In these simulations, $α$ is initialized to a value greater than $α_{c}$ and then gradually decreased. With each new value of $α$ the phase differences and voltages across the rung junctions are reset to zero. Note the appearance of a discontinuity in $f$ as $β_{c}$ is increased. There is also some evidence of additional discontinuities in $f$ for $α < α_{c}$ (see, for example, the data for $β_{c} = 8, 12$ ). (b) Discontinuity in the frequency synchronization order parameter $Δ f$ at $α = α_{c}$ for different values of $β_{c}$ . This graph is produced from the data for $f$ versus $α$ seen in (a). The discontinuity is well fitted by an exponential rise to a maximum, $Δ f = Δ f_{\max} [1 - \exp - b (β_{c} - β_{c}^{*})]$ , where we find parameter values of $Δ f_{\max} = 0.679 \pm 0.020$ , $b = 0.140 \pm 0.020$ , and $β_{c}^{*} = 7.750 \pm 0.294$ for this array.Reuse & Permissions
Figure 8
Hysteresis in the frequency synchronization order parameter $f$ , plotted as a function of the coupling strength $α$ for a ladder with $N = 10$ , $i_{B} = 5$ , $β_{c} = 30$ , and nonrandom critical currents with $Δ = 0.05$ . To produce these plots, $α$ is initialized to a value greater than $α_{c}$ and then gradually decreased until the system jumps discontinuously to the unsynchronized state; the coupling $α$ is then increased until the system jumps back to the synchronized state. The final values of the phase differences and voltages across the junctions, as obtained from the previous value of $α$ , are used as the initial values for each new value of $α$ . (a) Results based on the RCSJ model. (b) Results based on the LKM2. Note that the models agree on the value of $α_{c}$ at which the jump occurs from the synchronized to the unsynchronized state for decreasing $α$ . The solid lines are a guide to the eye.Reuse & Permissions
Figure 9
(Color online) Real part of the minimum-magnitude Floquet exponent for an array with $N = 10$ and $i_{B} = 5$ . (a) Dependence of exponents on $β_{c}$ for fixed coupling strength, $α = 1$ . Symbols are results of a numerical calculation based on the RCSJ model with either no disorder (open squares) or nonrandom critical currents and $Δ = 0.05$ (solid circles). The solid line is an analytic result [Eq. (39)] from the LKM2. (b) Dependence of exponents on coupling strength $α$ for fixed $β_{c} = 1$ .Reuse & Permissions
Figure 10
A ladder array with periodic boundary conditions, $N = 8$ plaquettes, and one long-range connection. Rung junctions $j = 1$ and $j = 4$ are shown as connected by a pair of off-rung junctions, which have phase differences of $ψ_{1 : 1, 4}$ and $ψ_{2; 1, 4}$ . The additional off-rung junctions are uniform and identical to the off-rung junctions in the pristine ladder. Uniform dc bias currents are applied along the rungs as in Fig. 1.Reuse & Permissions
Figure 11
Scaled average path length $l ∕ N$ (solid circles and left vertical axis) and cluster coefficient $C$ (hollow circles and right vertical axis), plotted versus the product $p N$ , where $p$ is the probability of making a shortcut connection between pairs of nodes and $N$ is the total number of nodes in the network. This graph corresponds to $N = 50$ . The solid line is $C = 2 ∕ N$ , which is the expected cluster coefficient for a random network of size $N = 50$ in which each node has two neighbors in the pristine limit. Note that $C \to 0$ as $p N \to 0$ , since each node’s neighbors are not connected to each other in that limit. Also note that $C \to 1$ as $p \to 1$ , which is the limit of a fully connected, regular network in which each node’s neighbors are all connected to one another.Reuse & Permissions
Figure 12
Time dependence of several combinations of voltages or voltage derivatives for a ladder with $N = 10$ , $i_{B} = 5$ , and nonrandom critical currents with $Δ = 0.05$ . The array was chosen to have three shortcuts between the following pairs of rung junctions: (1,3), (2,7), (4,6). In both plots the following quantities are compared (note that $v_{1} = d γ_{1} ∕ d τ$ ): $i_{c 1} v_{1} + i_{c 1} β_{c} d v_{1} ∕ d τ$ (dot-dashed line), $(α ∕ 2) (v_{1} - v_{3})$ (solid line), and $(α ∕ 2) β_{c} (d v_{1} ∕ d τ - d v_{3} ∕ d τ)$ (dashed line). The time-average value of the latter two quantities is negligible. (a) $α = 1$ , $β_{c} = 10$ . (b) $α = 1$ , $β_{c} = 1$ .Reuse & Permissions
Figure 13
(Color online) Frequency synchronization order parameter $f$ , plotted versus coupling strength $α$ for overdamped arrays with bias current $i_{B} = 5$ and nonrandom critical currents with $Δ = 0.05$ . Solid symbols represent numerical results based on the LKM model, Eq. (49), while hollow symbols are from the RSJ model. The data for nonzero $p$ represent an average over ten realizations of randomly assigned shortcuts. (a) $N = 30$ , (b) $N = 50$ . Note that shortcuts improve the frequency synchronization behavior in that the critical coupling needed for $f = 1$ is clearly reduced as $p$ is increased from zero. In fact $α_{c}$ is clearly growing with increasing $N$ in the case of $p = 0$ , while the same is not true of the array with shortcuts.Reuse & Permissions
Figure 14
(Color online) Frequency synchronization order parameter $f$ , plotted versus coupling strength $α$ for an underdamped ladder array with $N = 30$ , bias current $i_{B} = 5$ , and nonrandom critical currents with $Δ = 0.05$ . Solid symbols represent numerical results based on the LKM2 model, Eq. (48), while hollow symbols are from the RCSJ model. The data for nonzero $p$ represent an average over ten realizations of randomly assigned shortcuts. (a) $β_{c} = 1$ , (b) $β_{c} = 30$ . Agreement between the two models is excellent, except for large $β_{c}$ and $p = 0$ near $α_{c}$ , which is not unexpected based on Fig. 2.Reuse & Permissions
Figure 15
Floquet exponent of minimum magnitude $λ_{\min}$ , plotted versus coupling strength $α$ for the LKM with $N = 100$ . Exponents are calculated based on a technique described in Ref. 36. For $p = 0$ , the exponents can be calculated analytically, with the result $Re (λ_{\min} t_{c}) = - 4 α \sin^{2} (π ∕ N)$ . For each nonzero $p$ , the results are averaged over 100 realizations of shortcuts. Note the logarithmic scale on both axes.Reuse & Permissions
Figure 16
A two-dimensional array of junctions with $M = 2$ columns, or “ladders,” each with $N = 5$ plaquettes. A dc bias current $I_{B}$ is injected along the left side and extracted along the ride side of the array. We assume periodic boundary conditions in the $y$ direction. A junction with a label such as 1, 2 means the junction is in row 1 and column 2.Reuse & Permissions
Figure 17
A cluster diagram of the time-averaged voltages ${⟨ v_{i j} ⟩}_{τ}$ across the horizontal junctions, plotted versus coupling strength $α$ for an array with $M = 2$ columns and $N = 5$ plaquettes per column. A bias current $i_{B} = 5$ is applied and extracted along each row, and the horizontal junctions are assigned critical currents randomly according to Eq. (11) with $Δ = 0.05$ . We assume periodic boundary conditions in the vertical direction. The array is pristine $(p = 0)$ . The symbols with dots correspond to voltages across the junctions in column 2. The diagram demonstrates that the two columns are each frequency synchronized but that the synchronized voltages for each column are different. The numbers in the legend denote the coordinates of each horizontal junction. For example, the coordinates 1, 2 denote the horizontal junction in the first row (starting from the bottom row) and second column (starting from the left column).Reuse & Permissions
Figure 18
(Color online) Scaled standard deviation $s_{v} (α) ∕ s_{v} (0)$ of time-averaged voltages across the horizontal junctions, plotted versus coupling strength $α$ for an array with $M = 2$ columns and $N = 5$ plaquettes per column. The bias current is $i_{B} = 5$ , and the horizontal junctions are assigned critical currents randomly according to Eq. (11) with $Δ = 0.05$ . We assume periodic boundary conditions in the vertical direction. The solid circles are for the pristine array, $p = 0$ . The hollow circles and squares represent two different realizations of shortcuts at $p = 0.25$ in which the added shortcut junctions are critical current disordered. The hollow up (down) triangles denote the value of $s_{v} (α) ∕ s_{v} (0)$ for only the junctions in column one (two). The arrows pointing to the right axis denote the average values of $s_{v} (α) ∕ s_{v} (0)$ over the interval $4 < α < 10$ for the hollow circles and both the up and down triangles.Reuse & Permissions
Figure 19
Cluster diagrams for the time-averaged voltages ${⟨ v_{i j} ⟩}_{τ}$ across the horizontal junctions, plotted versus coupling strength $α$ for an array with $M = 2$ columns and $N = 5$ plaquettes per column. The bias current is $i_{B} = 5$ , applied and extracted along each row, and the horizontal junctions are assigned critical currents randomly according to Eq. (11) with $Δ = 0.05$ . We assume periodic boundary conditions in the vertical direction. The legend provides the coordinates for each horizontal junction as in Fig. 17. (a) $p = 0.25$ , first shortcut realization. (b) $p = 0.25$ , second shortcut realization.Reuse & Permissions
Figure 20
Cluster diagrams for the time-averaged voltages ${⟨ v_{i j} ⟩}_{τ}$ across the horizontal junctions, plotted versus coupling strength $α$ for an array with $M = 2$ and $N = 5$ . The bias current is $i_{B} = 5$ , and the horizontal junctions are assigned critical currents randomly according to Eq. (11) with $Δ = 0.05$ . We assume periodic boundary conditions in the vertical direction. All shortcut junctions are identical to the uniform vertical junctions in the pristine array, i.e., the shortcut junctions are not disordered. The legend provides the coordinates for each horizontal junction as in Fig. 17. (a) $p = 0.25$ , first shortcut realization. For $2.2 ≲ α ≲ 3.6$ , the horizontal junctions are weakly entrained, and for $α ≳ 3.6$ the array is in the zero-voltage state. The inset shows $s_{v} (α) ∕ s_{v} (0)$ versus $α$ . The range of $α$ values between the two vertical lines denotes the region of weak entrainment. (b) $p = 0.25$ , second shortcut realization. There is no evidence of entrainment up to $α \approx 4.4$ , beyond which the array is in the zero-voltage state.Reuse & Permissions
Figure 21
Cluster diagram for the time-averaged voltages ${⟨ v_{i j} ⟩}_{τ}$ across the horizontal junctions, plotted versus coupling strength $α$ for an array with $M = 2$ and $N = 5$ . The bias current is $i_{B} = 5$ , and the horizontal junctions are assigned critical currents randomly according to Eq. (11) with $Δ = 0.05$ . We assume periodic boundary conditions in the vertical direction. All shortcut junctions are identical to the uniform vertical junctions in the pristine array, i.e., the shortcut junctions are not disordered. The legend provides the coordinates for each horizontal junction as in Fig. 17. Shortcut junctions are added according to $p = 1$ .Reuse & Permissions

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