Strongly interacting Bose gas: Nozières and Schmitt-Rink theory and beyond

We calculate the critical temperature for Bose-Einstein condensation in a gas of bosonic atoms across a Feshbach resonance and show how medium effects at negative scattering lengths give rise to pairs reminiscent of the ones responsible for fermionic superfluidity. We find that the formation of pairs leads to a large suppression of the critical temperature. Within the formalism developed by Nozières and Schmitt-Rink, the gas appears mechanically stable throughout the entire crossover region, but when interactions between pairs are taken into account we show that the gas becomes unstable close to the critical temperature. We discuss prospects of observing these effects in a gas of ultracold ${}^{133}\text{C}\text{s}$ atoms where recent measurements indicate that the gas may be sufficiently long lived to explore the many-body physics around a Feshbach resonance.

Figure 1

Universal phase diagram for bosons as a function of the interaction parameter $1/n^{1/3}a$, where $n$ is the total density of atoms. The solid line shows $T_c$ calculated within the NSR formalism, while the dashed line is the two-body result. The dashed-dotted line is the critical temperature $T_c^a$ for the condensation of unpaired bosons in units of the critical temperature $T_a$ of an ideal atomic gas. The phase with a pair condensate (PC), which lies between $T_c$ and $T_c^a$, extends all the way to $-1/n^{1/3}a=+\infty$. Both an atomic condensate and a pair condensate $\left(\text{AC}+\text{PC}\right)$ exist below $T_c^a$.

Figure 2

Spectral function of a pair evaluated at $1/n^{1/3}a=2.18$, $T=T_c$, and $\left|\mathbf{k}\right|\simeq 0$. The two-body (dashed line) and many-body (solid line) calculations agree at high energy, but differ substantially at low energy. Also, a delta function is present at positive frequency just below the continuum in the many-body case (not shown). At $\left|\mathbf{k}\right|=0$ this delta function corresponds to the Thouless pole at $\omega =0$. Inset: binding energy of a pair at $T=T_c$. The dashed line is the ideal gas limit ${\epsilon }_m=-{\hslash }^2/m{a}^2$, while the solid line is the NSR result given by ${\epsilon }_m=2\mu$.

Figure 3

The pressure of the gas in its normal phase as a function of density in the NSR formalism for $-1/n^{1/3}a\simeq -{10}^3$ (dashed line), 0 (solid line), and ${10}^3$ (dashed-dotted line). Here, $\Lambda \left(T\right)=\sqrt{2\pi {\hslash }^2/m{k}_{\text{B}}T}$ is the thermal de Broglie wavelength. The dotted lines are the ideal gas law for atoms and molecules.

Figure 4

The renormalization factor $Z$ (solid line). Inset: the dimer-dimer scattering length $a_{\text{mm}}$ (solid line). The respective BEC limits which are recovered as $a\to 0^{+}$ are shown as dashed lines.

Figure 5

Universal phase diagram beyond NSR for the same parameters as in Fig. 1. The solid line is $T_c$ computed including molecular interactions. The dashed-dotted and dashed lines are the same many-body and two-body $T_c$ results, respectively, shown in Fig. 1. The shading denotes the mechanical instability region.

Figure 6

The pressure of the gas in the NSR formalism (dashed lines) and with molecular interactions beyond NSR included (solid lines). The curves beginning at $P\Lambda {\left(T\right)}^3/k_{\text{B}}T=0.2$ and 0.4 correspond to $-1/n^{1/3}a\simeq {10}^{-3}$ and ${10}^3$, respectively.

Figure 7

Binding energy of the $6s$ and $4d$ bare states (dashed lines) giving rise to the avoided crossing between the two dressed states (solid lines). In the close-up we show the Wigner threshold behavior of the binding energy near the continuum.

Figure 8

Critical temperature for pair condensation $T_c$ in a gas of ${}^{133}\text{C}\text{s}$ for particle densities $n={10}^{11}{\text{cm}}^{-3}$ (dash-dotted line) and ${10}^{14}{\text{cm}}^{-3}$ (solid line) and the corresponding instability regions (dark and light gray, respectively). For $n={10}^{14}{\text{cm}}^{-3}$ we have $T_a=79\text{nK}$ and $T_m=25\text{nK}$. Note that the curve has no kink at $B-B_0=\Delta B=0.16\text{G}$ although this is not visible on this scale. The dashed line is the two-body result for $n={10}^{11}{\text{cm}}^{-3}$.