Phase Imprinting in Equilibrating Fermi Gases: The Transience of Vortex Rings and Other Defects

We present numerical simulations of phase imprinting experiments in ultracold trapped Fermi gases, which were obtained independently and are in good agreement with recent experimental results. Our focus is on the sequence and evolution of defects using the fermionic time-dependent Ginzburg-Landau equation, which contains dissipation necessary for equilibration. In contrast to other simulations, we introduce small, experimentally unavoidable symmetry breaking, particularly that associated with thermal fluctuations and with the phase-imprinting tilt angle, and we illustrate their dramatic effects. As appears consistent with experiment, the former causes vortex rings in confined geometries to move to the trap surface and rapidly decay into more stable vortex lines. The latter aligns the precessing and relatively long-lived vortex filaments, rendering them difficult to distinguish from solitons.

Figure 1

(a) Depictions of axes used. (i) 3D rendering of initial depletion plane (red) with axes used in the paper: $y$ is vertical, and $z$ is the long axis of the trap. (ii) Side view of axes at $t=20$ after phase imprinting for the run in (b), along with the tilt of the imaging beam, $\alpha$, considered later in this work. (b) TDGL simulation results for one time sequence, focusing on the vortex ring decay, of a phase imprinting applied at $t=0$ with $\lambda =3.3$ and parameters as in the text. (i) Cloud density integrated along $y$ and (ii) the phase in the $y=0$ plane [from 0 (black) to $2\pi$ (white)]. Here, phase imprinting has formed a ring ($t=50$), which is just slightly off center. The ring moves toward the cloud edge ($t=100$) and then impacts the edge and decays to a vortex line ($t=175$).

Figure 2

(a) Lifetimes of defects as a function of the imprinting phase, using the same parameters as in Fig. 1. Red squares indicate lifetimes of vortex lines, which are substantially longer than those of vortex rings (black circles). Error bars reflect run-to-run variations and limitations on time estimates. Lines are cubic polynomial fits provided as a guide to the eye. (b) Plots of the trajectory of the precessing single vortex near the end stage of equilibration for the same run. Here, frames (cuts of ${|\psi |}^2$ along the $y=0$ plane) with $\mathrm{\Delta }t=5$ were combined and the minimum density among all frames selected, displaying the vortex trajectory during $t=[180,1160]$. (c) A series of density plots for the $\lambda =3.3$ run in Fig. 1, integrated along the $x$ axis, during the interval in which only one vortex is present. The density effectively shows a line depletion due to the orientation of the vortex, which oscillates along the cloud. (d) Comparison of a (i) perfectly orthogonal phase imprinting and (ii) phase imprinting with a beam tilted by 1° relative to the vertical axis, leaving it orthogonal to the $x$ (short horizontal) axis but not to the $z$ (loosely confined horizontal) axis. Each cell is integrated along $x$ at $t=400$ for a run with different random noise but conditions otherwise unchanged. This shows that without a tilt, the vortex can be oriented in many different directions (i), but a small tilt stabilizes the orientation in different runs (ii).