Dynamics of the BCS-BEC Crossover in a Degenerate Fermi Gas

We study the short-time dynamics of a degenerate Fermi gas positioned near a Feshbach resonance following an abrupt jump in the atomic interaction resulting from a change of magnetic field. We investigate the dynamics of the condensate order parameter and pair wave function for a range of field strengths. When the jump is sufficient to span the BCS to Bose-Einstein condensation crossover, we show that the rigidity of the momentum distribution precludes any atom-molecule oscillations in the entrance channel dominated resonances observed in ${}^{40}\mathrm{K}$ and ${}^6\mathrm{Li}$. Focusing on material parameters tailored to the ${}^{40}\mathrm{K}$ Feshbach resonance at 202.1 G, we comment on the integrity of the fast sweep projection technique as a vehicle to explore the condensed phase in the crossover region.

Figure 1

Variation of the chemical potential $\mu$ with field $B$ at $T=0$ for a mixture of fermionic ${}^{40}\mathrm{K}$ atoms prepared in the $(f=9/2,m_f=-9/2)$ and $(f=9/2,m_f=-7/2)$ Zeeman states at a density of $n=1.5\times {10}^{13}{\mathrm{cm}}^{-3}$ (i.e., $T_{\mathrm{F}}=0.35\mu \mathrm{K}$). The FR takes place at $B_0=202.1\mathrm{G}$ while the BEC-BCS crossover ($\mu =0$) occurs when $B-B_0\simeq -0.3\mathrm{G}$. Note that, since the density is nonzero, $\mu$ reaches zero below the two-body FR. The inset indicates the scattering length $a(B)=a_{\mathrm{bg}}(1-\frac{\Delta B}{B-B_0})$ in the universal regime, where $a_{\mathrm{bg}}$ is the background potential scattering length and $\Delta B$ the width of the resonance. Note that, in the present theory, we use finite-range potentials with parameters which also describe the FR far beyond the universal region [10].

Figure 2

Time dependence of the order parameter $|{\Delta }_{\mathbf{k}=0}|/{\Delta }_{\mathrm{eq}}$ following an abrupt switch from a field of $(B_{\mathrm{I}}-B_0)/\mathrm{G}=-0.5$ (top), $0.0$, $0.5$, $1.0$ (bottom) to $(B_{\mathrm{F}}-B_0)/\mathrm{G}=-10$, and from $(B_{\mathrm{I}}-B_0)/\mathrm{G}=-0.5$ (top inset) and $(B_{\mathrm{I}}-B_0)/\mathrm{G}=0.5$ (bottom inset) to $(B_{\mathrm{F}}-B_0)/\mathrm{G}=-1.0$. In addition to the constant phase velocity of ${\Delta }_{\mathbf{k}=0}$ (found numerically to be set by the final state equilibrium $\mu$), there is an additional time-dependent phase modulation whose characteristics mirror closely that of the amplitude oscillations.

Figure 3

The pair wave function $|{\kappa }_{\mathbf{k}}|$ (upper panel) and the density distribution ${\Phi }_{\mathbf{k}}$ (lower panel) shown as a function of $k=|\mathbf{k}|$ with $B_{\mathrm{I}}-B_0=0.5\mathrm{G}$ (dashed lines) and $B_{\mathrm{F}}-B_0=-10\mathrm{G}$ after $50\mu \mathrm{s}$ (dash-dotted line) and $B_{\mathrm{F}}-B_0=-1.0\mathrm{G}$ after $800\mu \mathrm{s}$ (solid line) following the abrupt switch as in Fig. 2. The dotted lines signify the ground state equilibrium distributions at $B_{\mathrm{F}}-B_0=-1.0\mathrm{G}$ included for comparison. The upper inset shows oscillations of $|{\kappa }_{\mathbf{k}=0}|$ for $B_{\mathrm{F}}-B_0=-10\mathrm{G}$. The lower two insets refer to a weak perturbation on the BCS side from $B_{\mathrm{I}}-B_0=0.3\mathrm{G}$ (dashed line) to $B_{\mathrm{F}}-B_0=0.2\mathrm{G}$. The dotted line shows the equilibrium distribution while the solid line provides a snapshot of the distribution at 3 ms after the switch. Note that the harmonic modulations visible in the distribution functions translate to a single energy scale of the same order of magnitude as the period of oscillations seen in $|{\Delta }_{\mathbf{k}=0}|$.

Figure 4

Time dependence of the condensed molecule density $[n_{\mathrm{mc}}(t)-n_{\mathrm{mc}}(0)]/n_{\mathrm{mc}}(0)$. Here we have used the same field values as that used in Fig. 2 (main) with $(B_{\mathrm{I}}-B_0)/\mathrm{G}=-0.5$ (top), $0.0$, $0.5$, and $1.0$ (bottom) and $(B_{\mathrm{F}}-B_0)/\mathrm{G}=-10$. When normalized to one-half of the total atomic density, $n_{\mathrm{mc}}(0)=0.3$, $0.1$, $0.012$, and $0.0005$, respectively.