Composite-boson approach to molecular Bose-Einstein condensates in mixtures of ultracold Fermi gases

We show that an ansatz based on independent composite bosons [Phys. Rep. 463, 215 (2008)] accurately describes the condensate fraction of molecular Bose-Einstein condensates in ultracold Fermi gases. The entanglement between the fermionic constituents of a single Feshbach molecule then governs the many-particle statistics of the condensate, from the limit of strong interaction to close to unitarity. This result strengthens the role of entanglement as the indispensable driver of composite-boson behavior. The condensate fraction of fermion pairs at zero temperature that we compute matches excellently previous results obtained by means of fixed-node diffusion Monte Carlo methods and the Bogoliubov depletion approximation. This paves the way towards the exploration of the BEC-BCS crossover physics in mixtures of cold Fermi gases with an arbitrary number of fermion pairs as well as the implementation of Hong-Ou-Mandel–like interference experiments proposed within coboson theory.

Figure 1

Binding energy as a function of the magnetic field $B$ (solid line). Horizontal lines are the confining energies $E_{\text{h.o.}}/2$ for the frequencies $\omega =2\pi \times {10}^2$ Hz (dashed), $3\times 2\pi \times {10}^3$ Hz (dotted dashed), and $2\pi \times {10}^5$ Hz (dotted). For these trapping frequencies, the BEC-BCS crossover extends to the strong-binding regime $(E_m>E_{\text{h.o.}}/2)$, i.e., from the vertical lines given by $k_{F}a=1$ with $N=1$ to the right, as the arrows indicate. The unitary limit is found in the resonant position $B_{\text{res}}$ (vertical solid line).

Figure 2

Entanglement between the atoms comprising a Feshbach molecule (red dots, solid line is a guide for the eye) as a function of the dimensionless parameter ${(16/9\pi )}^{1/3}\sigma /a={\left(k_{F}a\right)}^{-1}$ with $N=1$. The shaded area depicts the strong-binding region $2E_m>E_{\text{h.o.}}$, which is bounded by the vertical dotted line $\left(2E_m=E_{\text{h.o.}}\right)$. The insets show the Schmidt coefficient distributions ${\lambda }_j={\lambda }_{nl}$ of the state $|\mathrm{\Psi }\rangle$ for ${\left(k_{F}a\right)}^{-1}=3$ and 0.5 (dots indicated by the arrows). The upper panel shows the finite entanglement in the limit of weak binding $k_{F}a\gg 1$.

Figure 3

Condensate fraction for $N=33$ predicted by coboson theory $\langle N|{\widehat{c}}^{†}\widehat{c}|N\rangle /N$ as a function of the dimensionless parameter ${\left(k_{F}a\right)}^{-1}$ (red dots, solid line is a guide for the the eye). Black dots correspond to the FN DMC results [16], and the black dashed line to the Bogoliubov depletion of a Bose gas with $a_{dd}=0.6a$. The shaded area depicts the strong-binding region.

Figure 4

Condensate fraction, $\langle N|{\widehat{c}}^{†}\widehat{c}|N\rangle /N$, as a function of ${\left(k_{F}a\right)}^{-1}$ in the dilute regime with few fermion pairs, $N=2,3,5$ (yellow triangles, blue shaded squares and red squares, respectively), and with many fermion pairs $N=33$ (black dots). Different $N$ yield different curves, which reflect that the condensate fraction loses its universal character for few pairs. The inset shows the universality of the normalization ratio ${\chi }_{N+1}/{\chi }_N$ as a function of ${\left(k_{F}a\right)}^{-1}$ for $N=2,3,5$, and 10.

Figure 5

Normalization of the wave function $|\mathrm{\Psi }\rangle$ using the discretization method of Ref. [26] with $n_{\text{max}}=350,l_{\text{max}}=79,r_{\text{max}}=83338a_0$, and $\omega =2\pi \times {10}^4$ Hz, as a function of $1/k_{F}a={(16/9\pi )}^{1/3}\sigma /a$ for $N=1$.