Increasing the stability of a superfluid in a rotating necklace potential

TL;DR

The paper investigates stability of a 1D ring Bose-Einstein condensate with rotating barriers, addressing both hydrodynamic and Josephson regimes. By combining analytical GPE solutions for square barriers with self-consistent, numerically solved co-rotating dynamics, it shows the critical rotation $\omega_c$ increases nearly linearly with the number of barriers $n$, with a slope set by barrier height $V_0$, width $\sigma$, and interactions $g$. Upon crossing $\omega_c$, the system emits $n$ solitons simultaneously, enabling controlled changes and even reversal of the circulation, showcasing a topological stabilization mechanism. The stabilization persists under barrier imperfections and even strengthens with moderate speckle disorder, highlighting potential for robust ring-superfluid devices and motivating further extensions to fermionic superfluids and higher-dimensional toroidal geometries.

Abstract

Recent experiments have probed the stability of ring superfluids in the presence of Josephson barriers or Gaussian impurities. Here we present a theoretical analysis that extends beyond the regimes explored so far. We study the onset of dynamical instabilities induced by a one-dimensional potential rotating at an effective angular velocity $ω$, addressing both the tunneling and the hydrodynamic regimes. We show that the critical angular velocity $ω_c$ increases almost linearly with the number of barriers, with a slope set by their height and width. When the system is quenched into the dynamically unstable regime, it emits multiple solitons, which can switch or even reverse the direction of circulation. The stabilization mechanism is robust against imperfections of the potential and does not require a perfectly periodic array of barriers. In particular, we find that adding a disordered speckle potential to an ordered array of barriers can further increase $ω_c$: disorder can therefore make a ring superfluid more resilient to dynamical instabilities.

Figures (8)

• Figure 1: Schematic summary of the main results of this work. Our system is a 1D ring BEC including potential barriers rotating at an effective frequency $\omega$. We distinguish a regime where the BEC supports stationary (stable) solutions (green region) and a regime where the system becomes dynamically unstable due to soliton emission (yellow region). These are separated by a critical frequency $\omega_c$ (black line). (a) Clean necklace. The potential consists of $n$ identical barriers of height $V_0$ and half-width $\sigma$ (inset). We find that $\omega_c(n)$ increases monotonically with $n$, with a slope sets by the interaction strength and the the barrier height and width. (b) Dirty necklace. Here a disordered background potential of maximum amplitude $V_{\rm dis}$ is superposed to the clean necklace (inset). We find that $\omega_c$ increases further with $V_{\rm dis}$, reaching a maximum for $V_{\rm dis} \approx V_0$.
• Figure 2: Top panels show the analytical density profiles of the ground (solid black) and excited (dotted pink) states, Eqs. (\ref{['solutionsins']}) and (\ref{['solutionsout']}), for $n = 1$ (a) and $n = 4$ (b), respectively. The density is plotted as a function of both the azimuthal angle $\theta$ and the effective rotation frequency $\omega$. Blue dots indicate the numerical ground-state density obtained by solving Eq. (\ref{['1DGPERes']}). The gray shaded areas represent the external potential of Eq. (\ref{['Barriers']}). Panel (c) displays the mean-field energy, Eq. (\ref{['GPEEN']}), of the ground (solid black) and excited (dotted pink) states as a function of $\omega$. Different curves correspond to different values of $n$ and are vertically offset for clarity. Vertical lines and filled dots highlight the critical frequency $\omega_c$. Panel (d) shows $\omega_c(n)$ as a function of the number of barriers $n$ (blue dots); the solid line is a guide to the eye. Green (yellow) shading indicates the stable (unstable) region, as in Fig. \ref{['Fig1']}(a). In all panels, the parameters are $g = 300$, $V_{0}/\mu \approx 0.4$, and $\sigma/\xi \approx 0.5$.
• Figure 3: Critical effective rotation frequency $\omega_{c}(n)$ as a function of the number of barriers $n$ for different system's parameters. In panels \ref{['Fig3']}(a)–(c), the critical frequency is normalized to its value at $n=1$, i.e., $\omega_c(n)/\omega_c(1)$. We vary the barrier height $V_{0}/\mu$ in (a), the barrier width $\sigma/\xi_b$ in (b), and the coupling constant $g$ (reporting the ratio $g/V_0$) in (c). Dots show the numerical solutions of Eq. (\ref{['1DGPERes']}), while solid lines correspond to the prediction of Eq. (\ref{['critfreq2']}). Panels \ref{['Fig3']}(d)–(f) display the critical frequency at a single barrier, $\omega_c(1)$, as a function of $V_{0}/\mu$ (d), $\sigma/\xi_b$ (e), and $g/V_0$ (f) (orange triangles). The black solid line is Eq. (\ref{['instcrit']}). Parameters are: (a) $g=5000$ and $\sigma/\xi_b\simeq 1.2$; (b) $g=10^4$ and $V_0/\mu\simeq 0.7$; (c) $\sigma/\xi_b\simeq 0.6$ and $V_0/\mu\simeq 2.2$; (d) $g=10^4$ and $\sigma/\xi_b\simeq 0.9$; (e) $g=10^4$ and $V_0/\mu\simeq 0.12$; (f) $V_0/\mu\simeq 0.12$ and $\sigma/\xi_b\simeq 0.9$.
• Figure 4: Joint emission of solitons in the clean necklace. Each panel shows the time evolution of the density profile (top) and the winding number $\nu(t)$ (bottom). Different panels correspond to different numbers of barriers: $n=1$ (a), $n=2$ (b), and $n=4$ (c). In each case, the rotation frequency is quenched to a value $\omega = \omega_c(n) + \delta\omega$, with $\delta\omega \simeq 0.05$. In the lower panels, the red dashed line indicates the initial winding number $\nu(0) = 1$, while the blue dashed line marks the value $\nu = \nu(0) - n$ observed immediately after soliton emission. The inset in panel (c) shows the superfluid phase as a function of $\theta$ at different times: $t=0.5$ (solid line), corresponding to a circulation $\nu=1$, and $t=1$ (dotted line), corresponding to a circulation $\nu=-3$. Parameters are the same as in Fig. \ref{['Fig2']}.
• Figure 5: Panels (a)–(c) show the critical frequency in the dirty necklace with non-uniform square barriers; examples of the corresponding potentials are displayed in the insets. Panels (d)–(f) present the associated density dynamics (top) and winding number (bottom) as a function of time. In all cases, the dynamics is triggered by quenching the rotation frequency to $\omega = \omega_c(n) + \delta\omega$, with $\delta\omega \simeq 0.05$. (a), (d): randomly distributed barriers (see inset). Panel (a) shows $\omega_c(n)$ (dots) as a function of $n$, obtained by averaging over 30 random realizations of the potential (the rms fluctuations are smaller than the symbol size). The dashed line corresponds to a uniform necklace with the same $n$, $V_0$, and $\sigma$. Here, $g = 10^4$, $V_0/\mu \sim 1$, $\sigma/\xi \sim 1$, and $\nu(0) = 0$. (b), (e): non-uniform barrier height. One barrier has height $V_0$, while the remaining $n-1$ barriers have height $V_0 - \Delta V$. Panel (b) shows the critical frequency as a function of $\Delta V/V_0$. (c), (f): non-uniform barrier width. One barrier has width $\sigma$, while the remaining $n-1$ barriers have width $\sigma - \Delta\sigma$. Panel (c) reports the critical frequency as a function of $\Delta\sigma/\sigma$. In panels (b) and (c), the upper dashed line indicates $\omega_c$ for $n$ identical barriers, while the lower dashed line gives $\omega_c$ for a single barrier ($n=1$). In panels (b),(c),(e),(f), the parameters are $n=9$, $g = 5000$, $V_0/\mu \sim 0.6$, $\sigma/\xi_b \sim 6$, and $\nu(0) = 1$.