Momentum Distribution and Condensate Fraction of a Fermion Gas in the BCS-BEC Crossover

By using the diffusion Monte Carlo method we calculate the one- and two-body density matrix of an interacting Fermi gas at $T=0$ in the BCS to Bose-Einstein condensate (BEC) crossover. Results for the momentum distribution of the atoms, as obtained from the Fourier transform of the one-body density matrix, are reported as a function of the interaction strength. Off-diagonal long-range order in the system is investigated through the asymptotic behavior of the two-body density matrix. The condensate fraction of pairs is calculated in the unitary limit and on both sides of the BCS-BEC crossover.

Figure 1

OBDM for different values of the interaction strentgth $1/k_{F}a$ (solid lines). The dotted line (blue online) corresponds to $e^{-r/a}$ for $1/k_{F}a=4$ and the dashed line (red online) is the OBDM of a non interacting gas.

Figure 2

Momentum distribution $n_{\mathbf{k}}$ for different values of $1/k_{F}a$ (solid lines). The dashed lines (red online) correspond to $n_{\mathbf{k}}$ calculated using the BCS mean-field approach [15]. Inset: $n_{\mathbf{k}}$ for $1/k_{F}a=4$. The dotted line (blue online) corresponds to the momentum distribution of the molecular state $n_{\mathbf{k}}=4(k_{F}a{)}^3/[3\pi (1+k^2{a}^2{)}^2]$.

Figure 3

Projected TBDM, ${\rho }_2^P(r)$, (solid symbols), $(N/2)[2{\rho }_1(r)/n){]}^2$ (lines) and $h(r)={\rho }_2^P(r)-(N/2)[2{\rho }_1(r)/n{]}^2$ (open symbols) for different values of $1/k_{F}a$: $1/k_{F}a=1$ [solid line and triangles (red online)], $1/k_{F}a=0$ [dashed line and circles (blue online)] and $1/k_{F}a=-1$ [dotted line and diamonds (green online)]. The condensate fraction of pairs $\alpha$ corresponds to the asymptotic behavior of $h(r)$.

Figure 4

Condensate fraction of pairs $\alpha$ as a function of the interaction strength: FN DMC results (symbols), Bogoliubov quantum depletion of a Bose gas with $a_m=0.6a$ [dashed line (red online)], BCS theory using Eq. (10) [dot-dashed line (blue online)] and self-consistent mean-field theory [solid line (green online)].