Ground state of a resonantly interacting Bose gas

We show that a two-channel mean-field theory for a Bose gas near a Feshbach resonance allows for an analytic computation of the chemical potential, and therefore the universal constant $\beta$, at unitarity. To improve on this mean-field theory, which physically neglects condensate depletion, we study a variational Jastrow ansatz for the ground-state wave function and use the hypernetted-chain approximation to minimize the energy for all positive values of the scattering length. We also show that other important physical quantities such as Tan’s contact and the condensate fraction can be directly obtained from this approach.

Figure 1

The chemical potential $\mu$ in units of the Fermi energy ${\epsilon }_{\mathrm{F}}$ as a function of the inverse scattering length $1/k_{\mathrm{F}}a$. The solid line shows the mean-field result from Eq. (5). The dashed line shows the Bogoliubov result from Eq. (8) with LHY correction, while the dotted line is without this correction. Our two-channel mean-field approach stays finite in the unitarity limit, while the Bogoliubov theory diverges in that limit.

Figure 2

The scattering-length $a$ in units of $R_{\mathrm{c}}$ as a function of the dimensionless interaction strength $C_6/({\hslash }^2{R}_{\mathrm{c}}^4/m)$. At certain values of $C_6$ the scattering length diverges and the system is at a resonance.

Figure 3

(Top) The minimized radial distribution function $g$ (solid line) and $f{\left(r\right)}^2$ (dashed line) as a function of the radius in units of the interparticle distance $R_{\mathrm{i}}$. (Bottom) The structure factor $S\left(k\right)$ (solid line) and the dispersion relation in units of ${\epsilon }_{\mathrm{F}}$ (dashed line) as a function of the momentum $k/k_{\mathrm{F}}$.

Figure 4

The energy as a function of the inverse scattering length $1/k_{\mathrm{F}}a$ calculated with the variational approach of HNC/0 (solid line). The (red-shaded) area depicts the estimated accuracy of the result, which shows the large error for large $k_{\mathrm{F}}a$. The dash-dotted line shows the Gross-Pitaevskii energy while the dashed line also includes the LHY correction in Eq. (8). This shows that the HNC calculation correctly includes this term.

Figure 5

The condensate fraction as a function of the inverse scattering length $1/k_{\mathrm{F}}a$ calculated with the variational approach of HNC/0 (solid line). The (red) dashed line shows the condensate fraction for Bogoliubov theory. The condensate fraction for HNC/0 is comparable to the Bogoliubov result for small $k_{\mathrm{F}}a$, but for larger scattering lengths it is significantly smaller.

Figure 6

The contact as a function of the inverse scattering length $1/k_{\mathrm{F}}a$ calculated with the variational approach of HNC/0 (solid line). The (red) dashed line shows the contact for the Bogoliubov theory in Eq. (32).