Rapid ramps across the BEC-BCS crossover: A route to measuring the superfluid gap

We investigate the response of superfluid Fermi gases to rapid changes of the three-dimensional $s$-wave scattering length $a$ by solving the time-dependent Bogoliubov–de Gennes equations. In general the magnitude of the order parameter $\left|\Delta \right|$ performs oscillations, which are sometimes called the “Higgs” mode, with the angular frequency $2{\Delta }_{\text{gap}}/\hslash$, where ${\Delta }_{\text{gap}}$ is the gap in the spectrum of fermionic excitations. First, we excite the oscillations with a linear ramp of $1/a$ and study the evolution of $\left|\Delta \right|$. Second, we continously drive the system with a sinusoidal modulation of $1/a$. In the first case, the oscillations in $\left|\Delta \right|$ damp according to a power law. In the second case, the continued driving causes revivals in the oscillations. In both cases, the excitation of the oscillations causes a reduction in the time-averaged value of $\left|\Delta \right|$. We propose two experimental protocols, based around the two approaches, to measure the frequency and damping of the oscillations, and hence ${\Delta }_{\text{gap}}$.

Figure 1

The equilibrium magnitude of the order parameter ${\Delta }_{\text{eq}}\left(1/k_{f}a\right)$ obtained by solving the stationary Bogoliubov–de Gennes equations for a uniform Fermi gas.

Figure 2

(a) The response of $\left|\Delta \left(t\right)\right|$ following a ramp from $1/k_f{a}_0=-0.2$ to $1/k_f{a}_1=0$ over the time $t_1=2.2\hslash /E_f$. Inset: $\ln \left(A\right)$ (see text), evaluated at the positions of the crosses in the main figure, against $\ln (t{E}_f/\hslash )$. The dotted line is a fit with a gradient of $-0.5$. (b) As (a), but $1/k_f{a}_0=0.8$, $1/k_f{a}_1=1$, and $t_1=1.2\hslash /E_f$. The gradient of the dotted line in the inset is $-1.5$. The two insets are plotted with the same scale to aid comparison.

Figure 3

(a)–(f) The filled circles indicate the occupation of the energy levels $O_{\eta }=\int |v_{\eta }\text{(}\mathbf{r},t{\left)\right|}^{2}d\mathbf{r}$ at the points indicated by the crosses in Fig. 2a, in chronological order from the first maximum. Hence the left-hand (right-hand) column corresponds to maximums (minimums) in $\left|\Delta \right|$. The dashed curve shows the equilibrium distribution function for comparison.

Figure 4

(a) Solid (dashed) curve: the response of $\left|\Delta \left(t\right)\right|$ to an abrupt ramp at $t=0$ from $1/k_f{a}_0=0.2$ ($1.0$) to $1/k_f{a}_1=0$. (b) Solid (dashed/dashed-dotted) curve: as (a), but $1/k_f{a}_0=-0.5$ (0/$0.5$) and $1/k_f{a}_1=1.0$. Dotted curve: response of $\left|\Delta \left(t\right)\right|$ to an abrupt ramp at $t=0$ from $1/k_f{a}_0=0$ to $1/k_f{a}_1=-0.5$ and a second at $t=0.90\hslash /E_f$ to $1/k_f{a}_2=1.0$.

Figure 5

The response of $\left|\Delta \left(t\right)\right|$ to a sinusoidal modulation of $1/k_{f}a$ at the resonant angular frequency ${\omega }_m=2{\Delta }_{\mathrm{gap}}(t=0)/\hslash$ [see Eq. (2)] with $1/k_f{a}_0=1.0$ (top curve), $1/k_f{a}_0=0.5$ (middle curve), and $1/k_f{a}_0=0.0$ (bottom curve).

Figure 6

Experimental protocol A to detect the oscillations in $\left|\Delta \right|$. (a) Dotted curve: response of $\left|\Delta \left(t\right)\right|$ in a uniform superfluid to an abrupt ramp at $t=0$ from $1/k_f{a}_0=0.5$ to $1/k_f{a}_1=0$ and a second at $t=t_2=7.1$$\hslash /E_f$ to $1/k_f{a}_2=1.0$. Solid curve: as dotted curve, but for a ${}^{40}K$ superfluid in a trap with ${\omega }_x=2\pi 50$ rad s${}^{-1}$ and peak density $1.8\times {10}^{18}$ m${}^{-3}$. Inset: profile of $\left|\Delta \right(x,t=16\hslash /E_f\left)\right|$ in the trapped superfluid. The field of view is $120k_f^{-1}$. (b) As (a), but with $t_2=4.9$$\hslash /E_f$. (c) ${\Delta }_{\infty }/{\Delta }_{\text{eq}}$ in a uniform superfluid as a function of $t_2$.

Figure 7

Experimental protocol B to detect the oscillations in $\left|\Delta \right|$. (a) Dotted curve: response of $\left|\Delta \left(t\right)\right|$ in a uniform superfluid to a resonant periodic modulation of $1/a$ [${\omega }_m=2{\Delta }_{\mathrm{gap}}\left(0\right)/\hslash$, see Eq. (2)] followed by a ramp to $1/k_{f}a=1$ from $t=t_m=47.1$$\hslash /E_f$ to $t=t_{m2}=49.1$$\hslash /E_f$. Solid curve: as dotted curve, but for a ${}^{40}\text{K}$ superfluid in a trap with ${\omega }_x=2\pi 50$ rad s${}^{-1}$ and peak density $1.8\times {10}^{18}$ m${}^{-3}$. Inset: profile of $\left|\Delta \right(x,t=60\hslash /E_f\left)\right|$ in the trapped superfluid. The field of view is $120k_f^{-1}$. (b) As (a), but for an off-resonant modulation of $1/a$ [${\omega }_m=1.25{\Delta }_{\mathrm{gap}}\left(0\right)/\hslash$]. (c) ${\Delta }_{\infty }/{\Delta }_{\mathrm{eq}}$ in a uniform superfluid as a function of ${\omega }_m$.