Faraday waves in elongated superfluid fermionic clouds

We use hydrodynamic equations to study the formation of Faraday waves in a superfluid Fermi gas at zero temperature confined in a strongly elongated cigar-shaped trap. First, we treat the role of the radial density profile in the limit of an infinite cylindrical geometry and analytically evaluate the wavelength of the Faraday pattern. The effect of the axial confinement is fully taken into account in the numerical solution of hydrodynamic equations, and shows that the infinite cylinder geometry provides a very good description of the phenomena.

Figure 1

Time evolution of the transverse width $\Delta r$ for a Fermi superfluid in the BCS limit (see text). The top panel depicts $\Delta r$ [in units of $a_{\mathrm{ho}}=\sqrt{\hslash ∕\left(m{\omega }_{\perp }\right)}$] as a function of $t$ (in units of ${\omega }_{\perp }^{-1}$) and the bottom panel depicts its Fourier transform (in arbitrary units and logscale) as a function of $\omega ∕{\omega }_{\perp }$.

Figure 2

Idem Fig. 1 for the axial width $\Delta z$. The Fourier transform in the bottom panel is displayed in arbitrary units and logscale, and the frequency $\omega$ in units of ${\omega }_z$.

Figure 3

Time evolution of the column density at the trap center $n_{1\mathrm{D}}^0=n_{1\mathrm{D}}(z=0)$. The top panel shows $n_{1\mathrm{D}}^0$ (in units of $a_{\mathrm{ho}}^{-1}$) as a function of time $t$ (in units of ${\omega }_{\perp }$). The bottom panel displays its temporal Fourier transform (in arbitrary units and logscale) as a function of $\omega ∕{\omega }_{\perp }$.

Figure 4

(Color online) Density plots $n(x,y=0,z,t)$ of a Fermi gas in the BCS regime at several times $t$ during the transverse frequency modulation.

Figure 5

Fraction ${\delta }_N$ (in logscale) as a function of the time $t$ (in units of ${\omega }_{\perp }^{-1}$) for $a∕a_0=-{10}^{-1}$ (circles), $-{10}^5$ (triangles), and ${10}^3$ (squares). The empty symbols corresponds to the continuous driving while the full ones to the three-cycle driving. The lines are guides to the eyes for better visualizing the growth rates.

Figure 6

(Color online) Density plots and integrated density $n_{1\mathrm{D}}$ as a function of position (in units of $a_{\mathrm{ho}}$) after the Faraday instability has been set in. Top, middle, and bottom panels correspond to $a∕a_0=120$, $-{10}^5$, and $-{10}^{-1}$, respectively.

Figure 7

Comparison of the ratio $q∕\Omega$ between the observed Faraday wave vector $q$ (in units of $a_{\mathrm{ho}}$) and the driving frequency $\Omega$ (in units of ${\omega }_{\perp }$) with that predicted for a soundlike excitation [Eq. (11)] as functions of $1∕\left(k_F^{0}a\right)$, where $k_F^0$ is the Fermi momentum of the noninteracting gas at the trap center. The error bars have been estimated from the variance of $d$ during several time steps. The cross corresponds to a result obtained by modulating at $\Omega ∕{\omega }_{\perp }=1.5$ corrected by taking into account the induced excitation at $\Omega =2{\omega }_{\perp }$ (see text). The dashed line corresponds to the prediction of Eq. (11) with the BCS mean-field EOS.