Controlled generation of 3D vortices in driven atomic Josephson junctions

Abstract

We propose an ac-driven atomic Josephson junction as a clean and tunable source of three dimensional (3D) solitary waves in quantum fluids. Depending on the height of the junction barrier, the emitted excitations appear as vortex rings at low velocity or vorticity-free rarefaction pulses near the sound velocity, thus spanning the complete Jones-Roberts family of solitons. The Shapiro-step phenomenon renders the emission deterministic: on the first, second, third Shapiro steps, the junction ejects one, two, and three solitary excitations per drive cycle. This enables controlled generation of single- and multi-excitation configurations, allowing detailed studies of the full crossover between vortex rings and rarefaction pulses and their interaction dynamics. In particular, deterministic multi-ring emission provides insights into leapfrogging dynamics of two and three coaxial rings and their decay via boundary-assisted, sound-mediated processes. This ac-driven protocol establishes a compact and reproducible platform for generating, classifying, and controlling 3D solitonic excitations, paving the way for precision studies of nonlinear vortex dynamics, dissipation, and quantum turbulence in trapped superfluids.

Figures (5)

• Figure 1: On-demand generation of Jones-Roberts excitations in trapped 3D BEC. (a) Simulation of the atomic Josephson junction (AJJ), which is created by separating two 3D clouds with a Gaussian barrier of height $V_0$ and width $w$ (indicated by two vertical dotted lines). $N_L$ ($N_R$) represents the atom number of the left (right) reservoir. We use $w=1.1\um$, and $\tilde{V}_0 \equiv V_0/\mu$ in the range $0.4-1.65$, where $\mu$ is the transverse trap-averaged chemical potential. dc and ac drives are implemented via the barrier position $x(t) = v t + x_1 \sin(2\pi f t)$, where $v$ is the dc velocity, $f$ is the ac frequency, and $x_1$ is the ac amplitude related with the ac velocity $v_{ac}= 2\pi f x_1$. (b) Velocity-imbalance $(v-\Delta z)$ characteristics for $\tilde{V}_0 =0.5$ and $1.1$. (c) Semi-transparent isosurface $n({\bf r}, t)/n_0=0.2$ shows the emission of one, two, three, and four vortex rings (VRs) at first, second, third, and fourth Shapiro steps, respectively, for $\tilde{V}_0=0.5$. The oscillating barrier (blue disc, labeled B) emits both VRs and phonons per drive cycle. Extracted vortex polylines overlaid as black tubes; closed loops correspond to VRs and open curves to vortex lines (VLs). (d) At $\tilde{V}_0=1.1$, the barrier nucleates rarefaction pulses (RPs), visible as hallow reduced density shells in the background subtracted isosurfaces. One and two RPs per drive cycle occur at first and second Shapiro steps, respectively. The density ripples (reduction as blue and increase as red) depict phonons.
• Figure 2: Dynamics of JR excitations at the first Shapiro step. (a1-d1) Time evolution of the column density $\Delta n (x, t)= n (x, t) - n_0(x)$ at different barrier heights $\tilde{V}_0$, where $n_0(x)$ is the density without the barrier. Time is given in units of the drive period $T_f$. The oscillating barrier (thick depletion) generates both phononic (linear) and nonlinear excitations, with the latter propagating below the sound velocity. Density pulses include vortex ring (VR), vortex line (VL), rarefaction pulse (RP), and phononic pulse (PP). (a2-d2) Corresponding compressible $E_k^c$ and incompressible $E_k^i$ kinetic energy components. (a3) For $\tilde{V}_0=0.67$, the isosurface $\Delta n({\bf r}, t)/n_0=0.2$ shows nucleation of a VR (closed polyline) and phonons (density excess in red) at $t/T_f=1.3$. During evolution, the VR bends toward the transverse boundary and generally decays into two VLs (open loops) propagating in opposite directions. B marks the density reduction at the barrier. (b3) For $\tilde{V}_0=0.82$, a small-radius VR is nucleated, which dissipates its energy and momentum into an RP (hollow ring) and a VL (open polyline). Their different characteristic velocities allow them to separate over time. (c3, d3) At $\tilde{V}_0=1$ and $1.3$, the barrier mainly emits an RP (hollow ring) and phonons.
• Figure 3: Characterization of single vortex rings. (a, b) Time evolution of the VR radius $R_\mathrm{VR}(t)$ and velocity $v_\mathrm{VR}(t)$ for a single sample in the ensemble at different $\tilde{V}_0$. The healing length $\xi \simeq 0.3\um$ (horizontal shaded region) defines the lower bound of $R_\mathrm{VR}$, close to which the ring becomes unstable as $R_\mathrm{VR}$ approaches the core radius ($\sim \xi$). The vertical shaded region indicates the initial build-up time near the barrier. (c, d) Averaged $R_\mathrm{VR}$ and $v_\mathrm{VR}$ as a function of $\tilde{V}_0$. The inset shows the corresponding ring lifetime $\tau$, averaged over four samples; error bars represent the standard deviation. In (d), circles represent predictions of the ring velocity $v_\mathrm{ring}$ from Eq. \ref{['eq:vring']}, using $R_\mathrm{VR}$ values from panel (c); see text.
• Figure 4: Vortex-ring (VR) to rarefaction-pulse (RP) crossover. (a) Solitary-wave velocity $v_e$ (squares), normalized by the sound velocity $c_s$, as a function of $\tilde{V}_0$. The crossover between VR and RP excitations occurs near $\tilde{V}_0 \sim 1$ (vertical red shaded region). Triangles show the VR velocity $v_\mathrm{VR}$ from Fig. \ref{['Fig:vel']}(d). The Bogoliubov estimate of the sound velocity, $v_\mathrm{B}/c_s = \sqrt{n_e/n_\mathrm{col}}$ (circles), is based on the reduced density $n_e$ at the dip location, where $n_\mathrm{col}$ is the column density without the excitation. The horizontal dashed line at $v_\mathrm{cp}/c_s \approx 0.66$ marks the minimum of the energy-momentum dispersion (the cusp) of GPE solutions, where the low-velocity (VR) and high-velocity (RP) branches merge Jones1986. (b) $v_e$ and $v_\mathrm{VR}$ as a function of momentum $p$. The inset shows the VR (triangles) and RP (squares) momenta at varying $\tilde{V}_0$; see text. Error bars in momentum indicate the uncertainty arising from atom-number estimation within the pulse.
• Figure 5: Leapfrogging dynamics at higher Shapiro steps for $\tilde{V}_0=0.5$. (a1, b1) Time evolution of the column density $\Delta n (x, t)$ between cycles $1$ and $2$, showing the nucleation of two and three VRs at the second and third Shapiro steps, respectively. The barrier location is indicated as the dashed line. When the velocity fields of neighboring VRs overlap, they exhibit leapfrogging dynamics, leading to intricate dynamics for systems with more than two rings. (a2, b2) Corresponding compressible $E_k^c$ and incompressible $E_k^i$ components of the kinetic energy per particle. (a3) Isosurface snapshots of the density difference showing leapfrogging between two sequentially created VRs, which subsequently decay into vortex lines (open loops). (b3) Sequential creation and interaction of three VRs at the third Shapiro step. (c-f) Time evolution of the positions and velocities of the dominant nonlinear density dips at second and third steps, determined after averaging $\Delta n (x, t)$ over $32$ samples. The shaded region and dotted lines display the scattering time window, where two (c, d) and three (e, f) VRs interact and their subsequent decay dynamics follows. Small amplitude oscillations of solid lines and large amplitude oscillations of dashed lines are artifacts of numerical noise and derivatives.