Cascade of Solitonic Excitations in a Superfluid Fermi gas: From Planar Solitons to Vortex Rings and Lines

We follow the time evolution of a superfluid Fermi gas of resonantly interacting ${}^6\mathrm{Li}$ atoms after a phase imprint. Via tomographic imaging, we observe the formation of a planar dark soliton, its subsequent snaking, and its decay into a vortex ring, which, in turn, breaks to finally leave behind a single solitonic vortex. In intermediate stages, we find evidence for an exotic structure resembling the $\mathrm{\Phi }$ soliton, a combination of a vortex ring and a vortex line. Direct imaging of the nodal surface reveals its undulation dynamics and its decay via the puncture of the initial soliton plane. The observed evolution of the nodal surface represents dynamics beyond superfluid hydrodynamics, calling for a microscopic description of unitary fermionic superfluids out of equilibrium.

Figure 1

(a) Cascade of solitonic excitations in a unitary Fermi superfluid following the phase imprint. A planar soliton snakes and decays into a vortex ring. Shown are images of the density distribution in the central slice of the superfluid, after rapid ramp and time of flight, for the first 20 ms after the imprint. The imprint also generates two sound waves propagating towards the edges. (b) Time series of the central slice up to $t=100\mathrm{ms}$, cropped to the region around $z=0$.

Figure 2

Overview of the dynamics following the phase imprinting. (a) Representative images of the central slice at $t=4$, 50, and 95 ms showing a planar soliton ($S$), the nodal points of a vortex ring ($R$), and a vortex line ($V$). The vertical lines indicate the regions of interest for the generation of residuals in (b) and (c). (b,c) Residuals on the central slice along the central axial cut $x=0$ (b) and an outer axial cut (c). Dark (bright) color indicates density depletion (excess). Two sound waves rapidly propagate to the edges, while a sharp depletion remains at the center. Around $t=5\mathrm{ms}$, a second set of shallower sound waves is emitted. The residuals show the puncture of the soliton plane in the central slice ($t\sim 15\mathrm{ms}$) and the return of a vortex line at $t\sim 80\mathrm{ms}$.

Figure 3

Tomography of the cascade. Main panel (left): representative tomographic images at different stages of the cascade. The field of view is centered on the defect’s $z$ position. Right column: structure of the depletion due to the defect, as would be seen along the $z$ axis, reconstructed from the tomographic images. $t=3\mathrm{ms}$: sharp density depletion across the whole cloud signaling a planar dark soliton. $t=7\mathrm{ms}$: snaking of the soliton plane and first signature of a puncture. $t=9$, 16, and 20 ms: the puncture in the upper half of the soliton plane grows and yields an asymmetric vortex ring, with a nodal area left at the bottom. $t=85\mathrm{ms}$: the lower nodal area is punctured and a vortex line forms across a vortex ring. $t=95$ and 400 ms: the ring part of the defect progressively disintegrates. $t=1000\mathrm{ms}$: a single solitonic vortex remains.

Figure 4

Spectral analysis of the snaking dynamics. (a) Snapshots of the soliton’s undulation (central slice at $y=-13\mu \mathrm{m}$). (b) Extracted undulation shapes $z_s(x)$ (black dots) along with the corresponding Fourier expansions to the fifth order (solid gold line). (c) Fourier spectra at $t=3$ and 11 ms. (d) Fourier amplitudes $A_n$ as a function of time, for $n=1$ (circle), 2 (square), 3 (triangle), and 4 (diamond) and their fits to a line (solid lines). The error bars indicate the standard deviation of the mean obtained from a three-point binning of the data. (e) Growth velocities ${\dot{A}}_n$ for $n=1$ to 5, obtained from the linear fits in (d), with the error bars being the fit error.