Critical velocity of superfluid flow through single-barrier and periodic potentials

We investigate the problem of an ultracold atomic gas in the superfluid phase flowing in the presence of a potential barrier or a periodic potential. We use a hydrodynamic scheme in the local density approximation (LDA) to obtain an analytic expression for the critical current as a function of the barrier height or the lattice intensity, which applies to both Bose and Fermi superfluids. In this scheme, the stationary flow becomes energetically unstable when the local superfluid velocity is equal to the local sound velocity at the point where the external potential is maximum. We compare this prediction with the results of the numerical solutions of the Gross-Pitaevskii and Bogoliubov-de Gennes equations. We discuss the role of long-wavelength excitations in determining the critical velocity. Our results allow one to identify the different regimes of superfluid flow, namely, the LDA hydrodynamic regime, the regime of quantum effects beyond LDA for weak barriers and the regime of tunneling between weakly coupled superfluids for strong barriers. We finally discuss the relevance of these results in the context of current experiments with ultracold gases.

Figure 1

(Color online) Critical velocity for energetic instability of a dilute bosonic superfluid subject to a 1D external potential. The critical velocity is given in units of the sound velocity of a uniform gas, $c_s^{\left(0\right)}$, and is plotted as a function of the maximum of the external potential in units of the chemical potential ${\mu }_{j=0}$ of the superfluid at rest. Top panel: the case of a single potential barrier. Bottom panel: the case of a periodic potential. Thick solid lines: prediction of the hydrodynamic theory within the LDA, calculated from Eq. (3). Symbols: results obtained from the numerical solution of the GP equation (4) for various values of $L/\xi$. Filled circles: $L/\xi =0.5$ $\left(g{n}_0/E_{\text{R}}=0.1\right)$; open squares: $L/\xi =1$ $\left(g{n}_0/E_{\text{R}}=0.4\right)$; filled triangles: $L/\xi =1.57$ $\left(g{n}_0/E_{\text{R}}=1\right)$; open triangles: $L/\xi =3$ $\left(g{n}_0/E_{\text{R}}=3.65\right)$; filled squares: $L/\xi =5$ $\left(g{n}_0/E_{\text{R}}=10\right)$; open circles: $L/\xi =10$ $\left(g{n}_0/E_{\text{R}}=40\right)$. The thinner black solid lines are the tight-binding prediction (9) for $L/\xi =1$ and 1.57. Dashed lines are guides to the eye.

Figure 2

(Color online) Same as in Fig. 1 but for a superfluid of dilute fermions at unitarity. Thick solid lines: prediction of the LDA hydrodynamic theory, from Eq. (3). Symbols in the top panel: BdG results of Ref. [11]. with $L{k}_{\text{F}}=4$. Symbols in the bottom panel represent our BdG results for a periodic lattice. The $s\ne 0$ values are shown as follows: filled circles for $L{k}_{\text{F}}=0.497$ $\left(E_{\text{F}}/E_{\text{R}}=0.1\right)$ and $s=0.1$, 0.5, 1, 2.5, 3, and 3.5; open squares for $L{k}_{\text{F}}=0.886$ $\left(E_{\text{F}}/E_{\text{R}}=0.3178\right)$ and $s=0.1$, 0.5, 1, 1.5, 2.5, 3, and 4; filled triangles for $L{k}_{\text{F}}=1.11$ $\left(E_{\text{F}}/E_{\text{R}}=0.5\right)$ and $s=0.1$, 0.5, 1, 1.5, and 2.5; open triangles for $L{k}_{\text{F}}=1.57$ $\left(E_{\text{F}}/E_{\text{R}}=1\right)$ and $s=0.1$, 0.5, 1, 2.5, and 4; filled squares for $L{k}_{\text{F}}=1.92$ ($E_{\text{F}}/E_{\text{R}}=1.5$) and $s=0.5$ and 1; open circle for $L{k}_{\text{F}}=2.48$ $\left(E_{\text{F}}/E_{\text{R}}=2.5\right)$ and $s=2.5$. The result for $L{k}_{\text{F}}=0.497$ and $s=0.1$ being larger than unity is due to the numerical uncertainty, which is estimated to be on the order of 3% or less for all BdG points in the bottom panel of this figure. The thinner black solid lines are the tight-binding prediction (9) for $L{k}_{\text{F}}=0.497$ and 0.886. Dashed lines are guides to the eye.

Figure 3

(Color online) Velocity-phase diagrams in the LDA hydrodynamic regime for a bosonic superfluid (top panel) and for a fermionic superfluid at unitarity (bottom panel) through a single barrier. The thick solid lines show the prediction of the LDA hydrodynamic theory. The symbols in the top panel are the results obtained from the GP equation; those in the bottom panel are taken from the BdG results of Ref. [11]. The barrier heights for bosonic case are $V_{\text{max}}/{\mu }_{j=0}=0.05$ (open circles), 0.2 (filled squares), and 0.4 (filled circles). Those for fermionic case are $V_{\text{max}}/{\mu }_{j=0}=0.042$ (open circles), 0.17 (filled squares), and 0.68 (filled circles). The barrier width is $L/\xi =10$ for the bosonic case, and $L{k}_{\text{F}}=4$ for the fermionic case; these parameters are such that the corresponding curves for $v_c/c_s^{\left(0\right)}$ in the top panels of Figs. 1, 2 are qualitatively similar. The dashed lines are guides to the eyes.

Figure 4

(Color online) Comparison between theoretical results and the experimental data for the critical velocity of unitary fermions in an optical lattice. The filled triangles are our BdG results calculated for $E_{\text{F}}/E_{\text{R}}=0.5$. The solid line is the prediction of hydrodynamics in LDA. The open squares are the experimental result of Ref. [7], normalized by using the density at the $e^{-2}$ waist of the lattice, which corresponds to $E_{\text{F}}/E_{\text{R}}=0.56$.

Figure 5

(Color online) Schematic picture for the determination of the critical quasimomentum $P_c$ for the energetic instability using the hydrodynamic relation (6). Here, we take the unitary Fermi gas at $E_{\text{F}}/E_{\text{R}}=1$ as an example. Dashed lines correspond to the uniform case $\left(s=0\right)$ while solid lines to $s=1$. The value of $P_c$ is given by the abscissa of the crossing point between the curve representing ${\partial }_n{\partial }_{P}e$ (red line) and ${\left({\partial }_n^{2}e{\partial }_P^{2}e\right)}^{1/2}$ (blue line). The critical quasimomentum $P_{c,\text{dyn}}$ for the dynamical instability of long-wavelength excitations is given by a zero of ${\left({\partial }_n^{2}e{\partial }_P^{2}e\right)}^{1/2}$, where ${\partial }_P^{2}e=0$.

Figure 6

(Color online) Energy density for fermions at unitarity as a function of the quasimomentum $P$ for various values of the lattice strength $s$. Here, we take $E_{\text{F}}/E_{\text{R}}=1$ $\left(k_{\text{F}}L=1.57\right)$ as an example. The inset shows the magnification of the result for $s=2.5$; the dotted line is a sinusoidal curve with the same amplitude.