Persistent currents supported by solitary waves in toroidal Bose-Einstein condensates

We analyze the nucleation of persistent currents in Bose-Einstein condensates of ultracold gases confined in a ring. This phenomenon has been recently investigated in an experiment [S. Eckel, J. G. Lee, F. Jendrzejewski, N. Murray, C. W. Clark, C. J. Lobb, W. D. Phillips, M. Edwards, and G. K. Campbell, Nature (London) 506, 200 (2014)], where hysteresis loops have been observed in the activation of quantized persistent currents by rotating weak links. In this work, we demonstrate the existence of three-dimensional stationary currents with nonquantized angular momentum. They are generated by families of solitary waves that exhibit a continuous variation in the angular momentum and provide a bridge between different winding numbers. We show that the size of hysteresis loops is determined by the range of existence within the weak link region of solitary waves which configure the energy barrier preventing phase slips. The barrier vanishes when the critical rotation leads winding number and solitonic states to a matching configuration. At this point, Landau and Feynman criteria for phase slips meet: the fluid flow reaches the local speed of sound, and stationary vortex lines (which are the building blocks of multidimensional solitons) can be excited inside the system.

Figure 1

Soliton connection between winding number states in a 1D ring for condensates with $\widehat{g}=100$. Only winding numbers 0 and 1 are considered, without (a) and with (b) an external weak link of depth $U_{\mathrm{wl}}=0.56\mu$ and width $w_{\theta }=R$ along the angular coordinate. For each case, angular momentum per particle is shown above and energy per particle is shown below. The thick dotted lines of the top graph (a) indicate the values of the speed of sound, $c_0$ and $c_1$, relative to the winding number states, 0 (right) and 1 (left). The hysteresis loop is displayed in panel (b) by thick gray lines and arrows.

Figure 2

Phases (left column) and dimensionless azimuthal densities $a{n}_1\left(\theta \right)$ (right column), obtained from numerical solutions of the time-independent 1D GPE (4), corresponding to winding number $q=0$ (above) and solitonic states (below) for the same parameters as in Fig. 1, rotating at angular frequencies $\widehat{\mathrm{\Omega }}=$ 0.5, 1.0, and 1.4. Thick arrows (in the left column) indicate the sequence of both types of states for increasing $\mathrm{\Omega }$, approaching convergence near $\widehat{\mathrm{\Omega }}=1.4$.

Figure 3

Same as Fig. 1 for a 3D system, showing numerical results of time-independent 3D GPE (1) for condensates with $5.3\times {10}^4$ atoms of ${}^{23}\mathrm{Na}$ in an isotropic harmonic trap with $\omega =2\pi \times 100$ Hz centered at $R=6a_{\perp }$. For comparison with Fig. 1, energies have been shifted by $m{\mathrm{\Omega }}^2\langle r_{\perp }^2\rangle /2$. Two solitary waves can be excited in the absence of a weak link (a), a solitonic vortex (SV) and a dark soliton (DS), whereas only a dark soliton is possible at the center of a weak link (b) with depth $U_{\mathrm{wl}}=0.5\mu$ and width $w_{\theta }=0.6R$ [30]. The insets represent density isocontours (at $5\%$ of maximum density), colored by phase, of solitonic vortices belonging to two separated branches containing either vortices or antivortices. The external diameter of the tori is $17a_{\perp }$.

Figure 4

Evolution in real time, obtained by solving the 3D time-dependent GPE with $\widehat{\mathrm{\Omega }}=0$, of a dark soliton state, with the same parameters as in Fig. 3, after an initial Gaussian-noise perturbation. The dimensionless azimuthal density $a{n}_1\left(\theta \right)$ is plotted as a function of time. Density isocontours (at $5\%$ of maximum density) are shown for the following times (from left to right): 2, 32, and 160 ms. Shallow density depletions (light gray lines on the background) after soliton decay correspond to the emission of phonons traveling at the speed of sound.

Figure 5

Density isocontours at $5\%$ of maximum density, colored by phase, of winding number $q=0$ (above) and solitonic states (below), obtained by solving the time-independent GPE (1), in 3D condensates with the same trap and rotating weak link as those of Fig. 3. Two rotation rates are considered: $\widehat{\mathrm{\Omega }}=0.5$ (left) and $\widehat{\mathrm{\Omega }}=1.87$ (right), the latter near the critical frequency. The angular momentum per particle $\langle L_z\rangle /N\hslash$ is indicated in the inner part of the tori. Background graphs show the dimensionless density $a{n}_1\left(\theta \right)$, calculated after integration over $(z,r_{\perp })$, with $n_1$ normalized along the radius $R$.

Figure 6

Real time evolution, after an initial Gaussian-noise perturbation, of solitonic vortex states containing $1.325\times {10}^5{}^{23}\mathrm{Na}$ atoms trapped by an isotropic harmonic potential, with $\omega =2\pi \times 100$ Hz centered at $R=6a_{\perp }$, in the presence of a weak link with a depth of $U_{\mathrm{wl}}=0.45\mu$. The dimensionless azimuthal density $a{n}_1\left(\theta \right)$ is plotted as a function of time. (a) Weak link waist $w_{\theta }=1.9R$ at $\widehat{\mathrm{\Omega }}=0$. (b) Weak link with $w_{\theta }=0.6R$ [30] rotating at $\widehat{\mathrm{\Omega }}=0.36$. Density isocontours (at $5\%$ of maximum density) are shown for the following times (from left to right): 0, 24, and 160 ms. Images correspond to the reference frame comoving with the weak link.