Threshold for creating excitations in a stirred superfluid ring

We have measured the threshold for creating long-lived excitations when a toroidal Bose-Einstein condensate is stirred by a rotating (optical) barrier of variable height. When the barrier height is on the order of or greater than half of the chemical potential, the critical barrier velocity at which we observe a change in the circulation state is much less than the speed for sound to propagate around the ring. In this regime we primarily observe discrete jumps (phase slips) from the noncirculating initial state to a simple, well-defined, persistent current state. For lower barrier heights, the critical barrier velocity at which we observe a change in the circulation state is higher, and approaches the effective sound speed for vanishing barrier height. The response of the condensate in this small-barrier regime is more complex, with vortex cores appearing in the bulk of the condensate. We find that the variation of the excitation threshold with barrier height is in qualitative agreement with the predictions of an effective one-dimensional hydrodynamic model.

Figure 1

(a) in situ absorption image of the ring condensate without a barrier, viewed from above. (b) Schematic showing the orientation of the blue-detuned ($\lambda =532$ nm) “barrier” beam used to create a barrier in the ring condensate. The arrow indicates the azimuthal movement of the barrier around the ring axis. (c) in situ absorption image showing the effect of the barrier beam on the ring condensate. The peak barrier height in (c) is $\approx$40% of the chemical potential. (a) and (c) are the average of five partial-transfer absorption images [38] of different condensates, with a 96$\times$96 $\mu$m field of view.

Figure 2

Time-of-flight (TOF) absorption images showing vertical column density profiles observed after 1 s of stirring, adiabatic relaxation of the radial trap confinement (see text), and 10 ms of expansion. The upper row (a)–(c) shows representative results for stirring at low speeds $\Omega /2\pi <5$ Hz, while the lower row shows results for high speeds $\Omega /2\pi >5$ Hz. For sufficiently small barrier height $U_b$ (a) and (d), the condensate remains in the noncirculating state, and the density profile is peaked in the center after TOF expansion, with no evidence of vortices. For higher $U_b$ the ring can be excited to a persistent current state [(b), (c), (e), and (f)], causing a central hole to appear in the TOF density profile. The size of the central hole increases with the phase winding number $l$: in (b) $l=1$, in (c) $l=2$, and in (e) $l=3$. At high $\Omega$, the stirring may produce off-axis vortices, as seen in (e) and (f). At high $\Omega$ and sufficiently high $U_b$ (f), many vortices appear and the central hole associated with the persistent current may be distorted. The stirring conditions ($\Omega /2\pi$, $U_b/h$) for each image are as follows: (a) 1 Hz, 1080 Hz; (b) 2 Hz, 930 Hz; (c) 2 Hz, 1010 Hz; (d) 30 Hz, 40 Hz; (e) 8 Hz, 370 Hz; (f) 25 Hz, 60 Hz. The field of view for each TOF image is 220 by 220 $\mu$m.

Figure 3

Fraction of experimental runs where an excitation was observed, as a function of the peak height of the potential barrier $U_b$, with the barrier moving at an angular velocity $\Omega /2\pi$ = 1 Hz. The experiment was repeated six to eight times for each value of $U_b$ (black dots). The vertical error bars are the statistical uncertainty in the measured excitation fraction for each barrier height $U_b$. The black curve is a least-squares fit of a sigmoidal function [Eq. (2)] to the data points. The gray region is a 68% confidence band for the sigmoidal fit (see text).

Figure 4

Comparison of data for each of the selected values of $\Omega$ used in the experiment, as plotted in Fig. 3, with barrier height $U_b$ on the horizontal axis. The value of $\Omega /2\pi$ in Hertz for each plot is shown on the left. The black dots are the observed excitation fraction (ranging from zero to one for each subplot) for a given value of $U_b$ and $\Omega$, each dot representing six to eight repetitions. The vertical error bars are the statistical uncertainty in the measured excitation fraction for each barrier height $U_b$. The black curves are least-squares fits of a sigmoidal function [Eq. (2)] to the black dots. The gray regions are 68% confidence bands for the sigmoidal fits (see text).

Figure 5

Critical barrier height $U_c$ as a function of angular velocity of the rotating potential barrier. The black circles are the values of $U_c$ extracted from fits to the data sets of Fig. 4 at each angular speed $\Omega$. The error bars are the combined statistical uncertainty of the fits and the calibration of the barrier height. The dashed (red) line is the prediction for harmonic confinement in the transverse direction, using Eq. (16). The normalized theory curve was calculated using ${\mu }_{\infty }$ = $g{n}_{\infty }$ = 1.98(25) kHz, which assumes excitations were created at the least dense azimuth around the ring. This value of ${\mu }_{\infty }$ gives $c_{\infty }^{*}=\sqrt{g{n}_{\infty }/2m}$ = 4.24(24) m/s, which corresponds to an angular velocity $c_{\infty }^{*}/2\pi R$ = 30.0(1.7) Hz.