Analytical limits for cold-atom Bose gases with tunable interactions

We discuss the equilibrium properties of dilute Bose gases using a nonperturbative formalism based on auxiliary fields related to the normal and anomalous densities. We show analytically that for a dilute Bose gas of weakly interacting particles at zero temperature, the leading-order auxiliary field (LOAF) approximation leads to well-known analytical results. Close to the critical point the LOAF predictions are the same as those obtained using an effective field theory in the large-$N$ approximation. We also report analytical approximations for the LOAF results in the unitarity limit, which compare favorably with our numerical results. LOAF predicts that the equation of state for the Bose gas in the unitarity limit is $E/\left(pV\right)=1$, unlike the case of the Fermi gas when $E/\left(pV\right)=3/2$.

Figure 1

Comparisons of the exact zero-temperature values of the auxiliary field ${\Delta }_0$, condensate fraction ${\rho }_0/\rho$, pressure $p_0$, and their respective first- and second-order approximations as a function of $\xi ={\rho }^{1/3}{a}_0$ in the weakly interacting regime. The LOAF approximations for the auxiliary field ${\Delta }_0$, condensate fraction ${\rho }_0/\rho$, and pressure $p_0$ in the weakly interacting regime are given in Eqs. (58), (61), and (63), respectively.

Figure 2

Comparisons of the exact critical values of the auxiliary field ${\Delta }_c$, ratio $(T_c-T_0)/T_0$, and pressure $p_c$, and their respective first- and second-order approximations, as a function of $\xi ={\rho }^{1/3}{a}_0$ in the weakly interacting regime. The LOAF approximations for the critical values of the auxiliary field ${\Delta }_c$, ratio $(T_c-T_0)/T_0$, and pressure $p_c$ in the weakly interacting regime are given in Eqs. (87), (85), and (92), respectively.

Figure 3

Zero-temperature values of the auxiliary field ${\Delta }_0$, condensate fraction ${\rho }_0/\rho$, and chemical potential ${\mu }_0$ as a function of $\xi ={\rho }^{1/3}{a}_0$. The condensate fraction ${\rho }_0/\rho$ and chemical potential ${\mu }_0/\left(\lambda \rho \right)$ plots depicted in the bottom two panels were first shown in Fig. 2 of Ref. [22]. The reader is directed to Ref. [22] for further studies and derivations. Here, we emphasize that the numerical values of the auxiliary field ${\Delta }_0$, condensate fraction ${\rho }_0/\rho$, and chemical potential ${\mu }_0$ in the unitarity limit compare well with the exact solutions given in Eqs. (96), (97), and (98), respectively.

Figure 4

Critical values of the auxiliary field ${\Delta }_c$, ratio $(T_c-T_0)/T_0$, and chemical potential ${\mu }_c$ as a function of $\xi ={\rho }^{1/3}{a}_0$. The critical auxiliary field ${\Delta }_c$ and the ratio $(T_c-T_0)/T_0$ depicted in the top two panels were reported previously in Fig. 6 of Ref. [22], and further discussion regarding these plots can be found in Ref. [22]. Here, we emphasize that the numerical values of the critical auxiliary field ${\Delta }_c$ and the ratio $(T_c-T_0)/T_0$ in the unitarity limit compare well with the semianalytical approximations given in Table . The numerical value of the chemical potential ${\mu }_c$ agrees with the exact result obtained in Eq. (98).

Figure 5

Comparison of the exact $z$ dependence of $I_1\left(z\right)$ and the first- and second-order approximations in powers of $z$.

Figure 6

Comparison of the exact $z$ dependence of $I_2\left(z\right)$ and the first- and second-order approximations in powers of $z$.

Figure 7

Comparison of the exact $z$ dependence of $I_3\left(z\right)$ and the first- and second-order approximations in powers of $z$.