Fermion-mediated BCS-BEC crossover in ultracold ${}^{40}\mathrm{K}$ gases: Mean-field theory and collective modes

Microscopic theories of Feshbach resonance phenomena in fermionic alkali-metal gases have, in the main, relied upon the intuition afforded by a “Fermi-Bose” theory which presents the Feshbach molecule as a featureless pointlike Bose particle. While this model may provide a suitable platform to explore the condensation phenomena in the ${}^6\mathrm{Li}$ system, it has been argued recently that its application to the ${}^{40}\mathrm{K}$ system, where the hyperfine structure is inverted, is inappropriate. Here, the formation of a Feshbach resonance involves the formation of a Feshbach molecule which shares a spin state with the open channel. In such an arrangement, the mechanism of condensate formation cannot escape the effects of Pauli exclusion. Building on a model three-state fermion Hamiltonian recently introduced by Parish et al. to describe the ${}^{40}\mathrm{K}$ system, we elucidate the mean-field structure of the model, investigate the internal structure of the condensate wave function in the crossover region, and develop a field theory of the transition.

Figure 1

Atomic states $\mid F,m_F⟩$ involved in the Feshbach resonance for the ${}^{40}\mathrm{K}$ system. The Fermi gas is initially prepared in the two lowest spin states. In contrast to ${}^6\mathrm{Li}$, the selection rules implied by the interatomic interaction between states allows the hyperfine exchange coupling of only a single spin state (denoted by the dotted line). This leads to the formation of a Feshbach molecule between the spin state $\mid 7∕2,-7∕2⟩$ and the entrance channel state $\mid 9∕2,-9∕2⟩$ (denoted by the solid line).

Figure 2

Zero-temperature phase diagram showing the critical value of the detuning ${\nu }_c$ between the BCS-like phase (white) and BEC-like phase (gray) in the low density system. The points marked by $\times$ on the upper panel show the results of the numerical analysis discussed in the text. The upper panel shows ${\nu }_c\left(u_0\right)$ at ${\gamma }_0=0.2$ and $u_a=0$ while the lower panel shows ${\nu }_c\left(u_a\right)$ at ${\gamma }_a=1$ and $u_0=0$. Note that, for $u_a>1$, the scattering states can always accommodate a bound state. Here the system remains in the BEC phase regardless as to the strength of ${\nu }_c>0$. The behavior of the system along the dotted line in the upper panel is shown in more depth in Figs. 3, 4.

Figure 3

Variation of $\alpha$ (solid line) and $-\mu ∕E_0=E_B∕2E_0$ ($\times 5$, dashed line) as a function of $\nu ∕E_0$ as inferred from Eq. (12), for $u_a=0$, $u_0=4.1$, and ${\gamma }_0=0.2$ in the low-density limit $({ϵ}_F\to 0)$, i.e., along the dotted line in the upper panel of Fig. 2. The points ($\times$ and $+$) are numerical results obtained using the algorithm discussed in the text. The numerical simulations are performed at a small but nonvanishing density $({ϵ}_F∕E_0=0.0536)$ accounting for the small systematic deviations between simulation and theory. The vertical dashed line marks the onset of the BCS phase at $\nu ={\nu }_c$. Inset: The dependence of the inverse scattering length ${\left(k_{F}a\right)}^{-1}$ on $\Delta \nu ∕E_0\equiv (\nu -{\nu }_c)∕E_0$ for the same parameters employed in the main figure.

Figure 4

(Color online) Distribution of the condensate wave function ${\kappa }_{\mathbf{k},\uparrow \downarrow }{E}_0∕{\Delta }_{\uparrow \downarrow }$ (black: solid line and $\times$) and ${\kappa }_{\mathbf{k},c\downarrow }{E}_0∕{\Delta }_{\uparrow \downarrow }$ (blue: dashed line and $+$), in the BEC phase (upper panel, $\nu ∕E_0=1.701$) and the BCS phase (lower panel, $\nu ∕E_0=2.382$) for $u_a=0$, $u_0=4.1$, ${\gamma }_0=0.2$, and ${ϵ}_F∕E_0=0.0536$ (i.e., at the points marked by black circles in upper panel of Fig. 2). The lines are found from Eqs. (19, 22). The points are from the numerical simulations discussed in the text. The dotted regions of the ${\kappa }_{\mathbf{k},c\downarrow }{E}_0∕{\Delta }_{\uparrow \downarrow }$ curve is where the high- and low-energy expansions have been extrapolated into the intermediate regime. Note that, in the low-density phase, the distribution function is characterized by two characteristic length scales. The deviations between theory and simulation at large $k$ reflect the use of a contact interaction in the analytics whereas the simulations use a Gaussian potential.

Figure 5

Variation of $T_c∕{ϵ}_F$, the critical temperature (solid line), and $T_{\mathrm{diss}}∕{ϵ}_F$, the dissociation temperature (dashed line), with $\nu ∕E_0$ for $E_0∕{ϵ}_F=10$, $U_a=0$, $u_0=4.1$, and ${\gamma }_0=0.2$ in the limit ${ϵ}_F\to 0$. In this limit, $T_c$ and $T_{\mathrm{diss}}$ coincide below ${\nu }_c$ (marked by the dotted vertical line). $T_c$ is not shown in the region immediately below ${\nu }_c$ because the approximations used to obtain Eq. (25) become unreliable.