Conventional character of the BCS-BEC crossover in ultracold gases of ${}^{40}\mathrm{K}$

We use the standard fermionic and boson-fermion Hamiltonians to study the BCS–BEC crossover near the 202 G resonance in a two-component mixture of fermionic ${}^{40}\mathrm{K}$ atoms employed in the experiment of Regal et al. [Phys. Rev. Lett. 92, 040403 (2004)]. Our mean-field analysis of many-body equilibrium quantities shows virtually no differences between the predictions of the two approaches, provided they are both implemented in a manner that properly includes the effect of the highest excited bound state of the background scattering potential, rather than just the magnetic-field dependence of the scattering length. Consequently, we rule out the macroscopic occupation of the molecular field as a mechanism behind the fermionic pair condensation and show that the BCS-BEC crossover in ultracold ${}^{40}\mathrm{K}$ gases can be analyzed and understood on the same basis as in the conventional systems of solid state physics.

Figure 1

(Color online) Binding energies of the highest excited multichannel vibrational molecular bound states (solid and dashed curves), associated with a pair of unlike ${}^{40}\mathrm{K}$ atoms in the asymptotic $(f=9∕2,m_f=-9∕2)$ and $(f=9∕2,m_f=-7∕2)$ Zeeman states, versus the magnetic-field strength $B$. The Feshbach molecular state ${\varphi }_{\mathrm{b}}\left(B\right)$ and its energy $E_{\mathrm{b}}\left(B\right)$ emerge at the position of the zero-energy resonance $B_0$. The linearly varying energy $E_{\mathrm{res}}\left(B\right)$ of the Feshbach resonance level ${\varphi }_{\mathrm{res}}$ and the energy $E_{-1}$ of the highest excited vibrational state of the background scattering potential are indicated by the dot-dashed and dotted lines, respectively. We note that the measurable resonance position $B_0$ is significantly shifted with respect to the magnetic-field strength $B_{\mathrm{res}}$ at which the closed-channel resonance-state energy crosses the dissociation threshold of the entrance channel. Here and throughout this paper, we have chosen the zero of the energy, at each magnetic-field strength $B$, as the sum of the Zeeman energies of a pair of unlike fermions.

Figure 2

(Color online) Binding energy $E_{\mathrm{b}}\left(B\right)$ of the Feshbach molecular bound state ${\varphi }_{\mathrm{b}}\left(B\right)$ associated with a pair of unlike ${}^{40}\mathrm{K}$ atoms in the asymptotic $(f=9∕2,m_f=-9∕2)$ and $(f=9∕2,m_f=-7∕2)$ Zeeman states versus the magnetic-field strength $B$. The figure shows a comparison between the results of the two-channel approach (solid curve), the effective single-channel approach (dashed curve), and analytic estimates of the binding energy (dotted and dot-dashed curves). While the dotted curve refers to the universal estimate of Eq. (20), just depending on the scattering length $a\left(B\right)$, the dot-dashed curve also accounts for the nonretarded van der Waals interaction $-C_6∕r^6$ of the background scattering potential, at asymptotically large interatomic distances $r$, in terms of the mean scattering length $\overline{a}$ [21, 37] via Eq. (24).

Figure 3

(Color online) The fraction of the population associated with the Feshbach resonance level of the closed channel as a function of the magnetic-field strength $B$ and the density $n$. The solid curve corresponds to the density characteristic for the experiment of Ref. 9 while the result for the lowest density studied (the dot-dashed curve) is practically indistinguishable from the closed-channel admixture to the coupled-channel bound-state wave function of the two-body problem. The inset enlarges the region in the vicinity of the zero-energy resonance. We note that, regardless of the density, the maximum population associated with the bosonic field operator $b_0$ is only 8% for $B-B_0\approx -7.2\mathrm{G}$. Given the virtual density independence of this maximum population, this result also coincides with the maximum closed-channel admixture to the highest excited vibrational diatomic bound state of the two-channel two-body Hamiltonian (2).

Figure 4

(Color online) The single-particle excitation spectrum (the upper panel), given by Eq. (54), and the fermionic momentum distribution function (the lower panel), determined by Eq. (53), for the density $n=1.5\times {10}^{13}{\mathrm{cm}}^{-3}$ and a range of magnetic field strengths $B$, relevant to the experiment of Ref. 9. We note that the minimum of the excitation spectrum evolves from its position at $k=k_{\mathrm{F}}$ on the BCS side to $k=0$ at $B-B_0=-0.32\mathrm{G}$.

Figure 5

(Color online) The spatial form of the pair wave function for the density $n=1.5\times {10}^{13}{\mathrm{cm}}^{-3}$ and a range of magnetic field strengths $B$. We note that for $B-B_0$ smaller than about $-0.3\mathrm{G}$ the wave function loses its oscillatory form characteristic for the Cooper pairs and decays as a function of the interatomic distance. Such a virtually exponential decay is characteristic for diatomic bound states.

Figure 6

(Color online) The chemical potential $\mu$ as a function of the magnetic-field strength $B$ for different densities $n$. Away from $B_0$, in the BEC limit, $2\mu$ approaches the energy of the highest excited vibrational molecular bound state $E_{\mathrm{b}}$, as depicted in Fig. 2. We note that for the density $n=1.5\times {10}^{13}{\mathrm{cm}}^{-3}$ of the experiment [9] the chemical potential crosses zero at $B-B_0=-0.32\mathrm{G}$. The inset enlarges the region in the vicinity of the zero-energy resonance.

Figure 7

(Color online) The coherence length $\xi$, characterizing the size of the fermion pairs, as a function of the magnetic-field strength $B$ for different densities $n$. The circles indicate the values of the magnetic-field strength at which the size of the pairs equals the mean interatomic spacing. We note that for the density $n=1.5\times {10}^{13}{\mathrm{cm}}^{-3}$ of the experiment of Ref. 9 (solid curve) the size of the pairs equals the mean interatomic spacing at $B\approx B_0+0.5\mathrm{G}$.

Figure 8

(Color online) The critical temperature $T_{\mathrm{c}}$ for the BCS transition as a function of the magnetic field strength $B$ for a range of densities $n$. The density of $n=1.5\times {10}^{13}{\mathrm{cm}}^{-3}$ corresponds to the Fermi temperature of $T_{\mathrm{F}}=0.35\mu \mathrm{K}$ quoted in Ref. 9. We note the slight differences between the predictions of the boson-fermion (thin solid curve) and standard fermionic (dotted curve) models for the highest density of $n=5\times {10}^{14}{\mathrm{cm}}^{-3}$. This high-density critical temperature curve reflects the only calculation for which we were able to identify any visible differences between the two approaches.

Figure 9

(Color online) The fermionic condensate fraction $f_{\mathrm{c}}$ and its overlap $f_{\mathrm{mol}}$ with the target molecular bound state at 10 G below the zero-energy resonance for $T=0$ (dashed and solid curves, respectively) and $0.08T_{\mathrm{F}}$ (dotted and dot-dashed curves, respectively). The density is $n=1.5\times {10}^{13}{\mathrm{cm}}^{-3}$ corresponding to the Fermi temperature of $T_{\mathrm{F}}=0.35\mu \mathrm{K}$ quoted in Ref. 9.