From Kardar-Parisi-Zhang scaling to soliton proliferation in Josephson junction arrays

Abstract

We propose Josephson junction arrays as realistic platforms for observing nonequilibrium scaling laws characterizing the Kardar-Parisi-Zhang (KPZ) universality class, and space-time soliton proliferation. Focusing on a two-chain ladder geometry, we perform numerical simulations for the roughness function. Together with analytical arguments, our results predict KPZ scaling at intermediate time scales, extending over sufficiently long time scales to be observable, followed by a crossover to the asymptotic long-time regime governed by soliton proliferation.

Figures (4)

• Figure 1: (a) Sketch of a JJA ladder with two 1D chains ($A$ and $B$) of $N$ superconducting grains (blue squares), $n=1,\ldots,N$. Neighboring grains are connected via Josephson junctions (crosses), with Josephson coupling $E_J$ ($E_X$) along (across) the chains. By attaching leads to each superconducting grain, the current $I$ is driven through every rung across the $E_{X}$ junction. (b) Schematic scaling regimes in the $\tilde{g}_2$--$\tilde{g}_3$ plane for the roughness function ${\cal D}_l(t)$ in Eq. \ref{['Ddef']} at intermediate times. The dimensionless couplings $\tilde{g}_{1,2,3}$ entering our continuum model are defined below Eq. (\ref{['eq:coupling_constants']}), where $\tilde{g}_1\propto E_X$ encodes the nonlinearity strength, $\tilde{g}_{2}\propto I$ the drive current, and $\tilde{g}_3\propto \sqrt{TR}$ the noise level. In the gray dashed regions, our theory does not apply, either because superconductivity is lost (top region), the Markovian noise assumption breaks down (bottom region), or the critical current is exceeded, $|I|>I_c=2eE_{J}/\hbar$ (right region, $\tilde{g}_2>2$). Blue regions correspond to KPZ scaling regimes spanning at least one decade in time, where more intense color implies a longer KPZ time window. Similarly, red and orange regions indicate soliton proliferation (SP) and diffusive (EW) regimes, respectively. In the narrow white region, $k_{B}T \ll 2E_{J}$ and ${\cal D}_l(t)$ is constant. The dashed vertical line separates regions with local minima ($\tilde{g}_2<\tilde{g}_1$) of the tilted washboard potential associated to Eq. \ref{['langevin']} from those without minima.
• Figure 2: Numerical results for the roughness function ${\cal D}_{l}$ in Eq. \ref{['Ddef']} vs dimensionless time $s$ for a JJA ladder with $N=256$ grains per chain, $l= L/2$, $\tilde{g}_1=0.2$, and several $\tilde{g}_{2,3}$, see Eq. \ref{['eq:coupling_constants']}. Note the double-logarithmic scales. Red lines show the expected scaling ${\cal D}_l\sim s^{2\beta}$ for the diffusive ($\beta^{(i)}=1/4$, dotted), KPZ ($\beta^{(ii)}=1/3$, solid), and soliton proliferation regimes ($\beta^{(iii)}=1/2$, dashed), respectively. Main panel: Results for $\tilde{g}_3\approx 0.0076$ and three values of $\tilde{g}_2$, with time-averaging window $\Delta s=0.01$. Triangles and stars mark the approximate crossover times $s_{1}$ and $s_{2}$, respectively. Inset: Results for $\tilde{g}_{2}\approx 0.2054$ and $\tilde{g}_{3} \approx 0.0002$, with $\Delta s=0.1$, where the KPZ regime persists for more than two decades. Circles mark the time window used for extracting $\beta$.
• Figure 3: Numerical results for the soliton density $\rho$ (in units of $1/a$) vs inverse temperature ($T$ in Kelvin) in the overdriven regime $\tilde{g}_2>\tilde{g}_1=0.2$ for several $\tilde{g}_2$ on a log-linear scale. For small noise levels $\tilde{g}_3$ corresponding to low $T$, see Eq. \ref{['eq:coupling_constants']}, we find activated behavior, $\rho \sim \exp(-E_A/k_B T)$. Numerical fits (dashed lines) yield $E_{A}\approx 109, 124$ and $133~\mu$eV with increasing $\tilde{g}_{2}$, respectively.
• Figure A1: Color-scale plot of $\Theta(s,x) = \varphi(s,x)-\varphi_{0}(s,x)$ in the space-time plane restricted to a central segment $l=L/2$ of the ladder. We consider the soliton proliferation regime with $\tilde{g}_1=0.2$, $\tilde{g}_2\approx 1.027$ and $\tilde{g}_3\approx 0.0132$, where $\varphi(s,x)$ and $\varphi_{0}(s,x)$ are numerical solutions of Eq. (\ref{['langevin-dimensionless']}), with $\varphi$ obtained for a single noise trajectory and $\varphi_0$ without noise ($\tilde{g}_3=0$). Space-time solitons are identified as explained in the main text.