Dynamics of a strongly interacting Fermi gas: The radial quadrupole mode

We report on measurements of an elementary surface mode in an ultracold, strongly interacting Fermi gas of ${}^6\mathrm{Li}$ atoms. The radial quadrupole mode allows us to probe hydrodynamic behavior in the crossover from Bose-Einstein condensation (BEC) to the Bardeen-Cooper-Schrieffer (BCS) regime without being influenced by changes in the equation of state. We examine the frequency and damping of this mode, along with its expansion dynamics. In the unitarity limit and on the BEC side of the resonance, the observed frequencies agree with standard hydrodynamic theory. However, on the BCS side of the crossover, a striking downshift of the oscillation frequency is observed in the hydrodynamic regime as a precursor to an abrupt transition to collisionless behavior; this indicates coupling of the oscillation to fermionic pairs.

Figure 1

Illustration of the radial quadrupole mode as an elementary collective excitation of an elongated, trapped atom cloud.

Figure 2

Schematic illustration of the scanning system. A wide collimated beam passes through an AOM. The resulting deflection angle depends on the driving frequency of the AOM. The beam passes through a lens at the distance of one focal length behind the AOM. The lens focuses the beam for atom trapping. A change in deflection angle results in a parallel shift of the beam position in the focal plane. The solid and dashed lines show the beam path for different deflection angles. The zeroth-order beam is not shown.

Figure 3

Timing scheme for the excitation of the radial quadrupole mode. The ellipticity of the trap is slowly ramped up within $100\mathrm{ms}$. This results in a change of $\alpha$ in the trap frequencies, where $\alpha$ characterizes the ellipticity and sets the initial, normalized deformation $\mathrm{\Delta }W∕W_0=-2\alpha$. $W_0$ is defined as the width of the cloud in the trap without excitation. At $t=0$, the elliptic deformation is switched off and the oscillation in the trap begins. (Shown here is an oscillation in the hydrodynamic regime.) The oscillation continues until the trap is turned off at $t=t_{\mathrm{trap}}$, which is usually between 0 and $10\mathrm{ms}$. At $t=t_{\mathrm{trap}}$, the cloud is released from the trap and expands for the time $t_{\mathrm{TOF}}$, which is typically $2\mathrm{ms}$.

Figure 4

Typical radial quadrupole oscillations in the hydrodynamic (a) and collisionless (b) regimes. The solid lines show fits to our data according to Eq. (3). The dashed lines indicate $\mathrm{\Delta }W=0$. The expansion time $t_{\mathrm{TOF}}$ is $2\mathrm{ms}$. In (a), the oscillation in the unitarity limit $(B=834\mathrm{G})$ is shown, whereas (b) shows the oscillations for $B=1132\mathrm{G}$ $(1∕k_{\mathrm{F}}a\approx -1.34)$.

Figure 5

Frequency ${\omega }_{\mathrm{q}}$ (upper plot) and damping rate $\kappa$ (lower plot) of the radial quadrupole mode. Both quantities are normalized to the radial trap frequency ${\omega }_{\mathrm{r}}$ and plotted versus the interaction parameter $1∕k_{\mathrm{F}}a$. The dashed lines indicate the theoretical predictions in the hydrodynamic $({\omega }_{\mathrm{q}}∕{\omega }_{\mathrm{r}}=\sqrt{2})$ and in the collisionless limit $({\omega }_{\mathrm{q}}∕{\omega }_{\mathrm{r}}=2)$. The shaded area marks the transition from hydrodynamic to collisionless behavior between $1∕k_{\mathrm{F}}a\approx -0.72$ $(B\approx 930\mathrm{G})$ and $1∕k_{\mathrm{F}}a\approx -0.85$ $(B\approx 960\mathrm{G})$.

Figure 6

(a) Phase shift $\varphi$ and (b) relative amplitude of the quadrupole mode versus interaction parameter $1∕k_{\mathrm{F}}a$ after $t_{\mathrm{TOF}}=2\mathrm{ms}$ expansion. The horizontal lines show calculations from our theoretical model: the solid lines in the collisionless limit, the dotted lines in the hydrodynamic regime at unitarity $(\gamma =2∕3)$, and the dashed lines in the hydrodynamic regime in the BEC limit $(\gamma =1)$. These calculated values can be read off from Fig. 10 for the phase and Fig. 9 for the amplitude. The shaded area marks the transition between hydrodynamic and collisionless behavior between $1∕k_{\mathrm{F}}a\approx -0.72$ and $1∕k_{\mathrm{F}}a\approx -0.85$ (see also Fig. 5).

Figure 7

(Color online) Oscillations of the quadrupole surface mode at a magnetic field of $920\mathrm{G}$ and $1∕k_{\mathrm{F}}a=-0.66$. The solid circles correspond to a cold ensemble, whereas the open triangles correspond to a heated ensemble. The solid lines are fits to the data according to Eq. (3).

Figure 8

Phase-space dynamics for the quadrupole mode in the collisionless regime. Shown are phase-space contours of an ensemble of particles which is held in a round trap (i.e., ${\omega }_x={\omega }_y={\omega }_{\mathrm{r}}$). In (a) and (b) the situation during the oscillation in the trap is shown for two different times $t$. The solid line indicates the equilibrium phase-space contour (without excitation), whereas the dotted (dashed) line shows the contour in the $x$ $\left(y\right)$ direction after excitation of the oscillation mode. (c) After long times, residual thermalization finally damps out the oscillations and leads to a circular phase space contour.

Figure 9

(Color online) Calculated relative amplitude of a surface mode oscillation versus reduced time of flight ${\omega }_{\mathrm{r}}{t}_{\mathrm{TOF}}$ after release from the trapping potential. The values are calculated for the hydrodynamic (dashed curve, $\gamma =1$; dotted curve, $\gamma =2∕3$) and collisionless regime (solid curve). The vertical dotted line marks the typical expansion time in our experiments.

Figure 10

(Color online) Phase $\varphi$ of the collective surface mode as detected by fits according to Eq. (3) versus reduced expansion time ${\omega }_{\mathrm{r}}{t}_{\mathrm{TOF}}$ at unitarity (open circles) and at $1∕k_{\mathrm{F}}a=-1.34$ (solid triangles). The lines are numerical simulations for the hydrodynamic (dashed line) and collisionless regime (solid line). The vertical dotted line marks the typical expansion time in our experiments.

Figure 11

Calculated quadrupole oscillations in the near-collisionless regime. The lines show the relative difference in width $\mathrm{\Delta }W$ as a function of the reduced time ${\omega }_{\mathrm{r}}{t}_{\mathrm{trap}}$. The oscillation is modeled according to Eqs. (B1, B2). The dark line shows the result of the calculation when ${\omega }_{\mathrm{r}}{\tau }_R=2.3$, and the gray line shows the oscillation in the collisionless limit at ${\omega }_{\mathrm{r}}{\tau }_R=1000$.