Cooperative stabilization of persistent currents in superfluid ring networks

TL;DR

The paper addresses the stabilization of persistent currents in annular superfluids with periodic barriers by modeling the ring as a network of locally coupled Kuramoto-like oscillators. It derives an analytic stability diagram for phase-locked states with circulation $w$, showing that stability requires $0 < \frac{4 w}{N} < 1$ and that increasing the number of barriers $N$ enhances robustness, effectively stabilizing higher-circulation states. Unstable configurations relax into cluster states that conserve topological charge, and noise robustness is quantified via a survival probability $P_{\rm surv}(T)$ with a rapid, $N$-dependent decay of instability. The authors validate the framework with cold-atom experiments and find good agreement, including a $\sqrt{N}$ scaling of winding-number dispersion and enhanced robustness for larger $N$, demonstrating a universal cooperative stabilization mechanism in ring networks. The results have broad implications for stable persistent-current states in various platforms, including superconducting circuits, due to the intrinsic cooperative topology of rings.

Abstract

Cooperative effects in oscillator networks are often associated with enhanced stability of phase-locked solutions, which increases with system size. We show that the stabilization of persistent currents in annular atomic superfluids with periodic barriers is a concrete manifestation of this phenomenon. Under the simplifying assumption of continuity of atomic flow across identical barriers, the system reduces to a ring of locally coupled Kuramoto-like oscillators. We analytically derive the stability diagram of phase-locked configurations and quantify their robustness to noise and small random initial imperfections, finding excellent agreement with experimental observations. These results are inherent to the ring topology and independent of the specific physical platform.

Figures (6)

• Figure 1: (a) A directed graph representation of a bidirectional ring network of $8$ identical nodes. (b) In-situ density distribution of our annular atomic superfluids with $8$ periodic barriers. c) Equivalent RSJ circuit representation, in which each node is modeled as an ideal Josephson junction shunted by a resistor (see SM).
• Figure 2: (a) Eigenvalues $\lambda$ of persistent current states $C_{w}^0$ ($0 < w < N$), as a function of $N$. Blue (red) regions where $\lambda$ is negative (positive) indicate stable (unstable) states. Solid lines represent the stability boundaries ($\lambda=0$) given by Eq. (\ref{['eq7']}): $w=N/4$ (green) and $w=3 N/4$ (magenta). (b-c) Time traces of $\tilde{\theta}_n=\theta_n/2 \pi$ for $N=10$ (upper panel) and $\tilde{\phi}_n=\phi_n/2 \pi$ (lower panel), (b) near the stable state $C_{2}^0$ and (c) near the unstable state $C_{4}^0$. Inset: evolution of $\phi_n$ on the unit circle. The system is initialized with random initial conditions around $C_{w}^0$.
• Figure 3: Final-state probabilities $P(w_f \vert w)$ obtained by integrating Eq. (\ref{['eq1']}) for $20$ random initial conditions in the vicinity of $C_{w}^0$, for $w=1,2,3,4$, shown as a function of $N$: (a) deterministic evolution; (b) evolution in the presence of zero-mean spatiotemporal additive Gaussian noise $\zeta(n,t)$ with correlation $\langle \zeta(n',t') \zeta(n,t)\rangle = 2 D \,\delta(n'-n)\delta(t'-t)$ and $D=8.45 \times 10^{-3}$. The winding number is calculated via Eq. \ref{['wind']}. (c) Experimental final-state probabilities from $15$-$20$ realizations for each $N$ and $w$.
• Figure 4: Robustness of persistent current states $C_{w}^0$ ($0 < w < N$), as a function of $N$ quantified by the indicator $\mathcal{S}$ (colormap). Blue (red) regions indicate more (less) robust states. Dashed lines marks the stability boundary $w=N/4$. (a) $\mathcal{S}$ calculated from Eq. \ref{['toy']}, using $\langle w_f \rangle_{esc} = \frac{N}{2\pi} \arccos(-\Lambda)$, $\sigma=6\times10^{-2}$, $r=\sigma$ and $T=3$; (b) $\mathcal{S}$ computed from numerical solutions of Eq. \ref{['eq1']}, with parameters as in Fig. \ref{['figu2']}(b), averaged over $20$ realizations; (c) $\mathcal{S}$ extracted averaging over $15$-$20$ experimental realizations for each $N$ and $w$.
• Figure 5: Final-state probabilities $P(w)$ for different $N$ calculated by integrating Eq. (4) of the main text over a numerical time $t_{f}=80$, starting from $1000$ different initial conditions with $\theta_n(0)$ distributed uniformly in the interval $[0,2\pi)$. Lower panel: measured standard deviation of the probability distributions as a function of $N$. The winding number is calculated via Eq. (7).