First-order Bose-Einstein condensation with three-body interacting bosons

Bose-Einstein condensation, observed in either strongly interacting liquid helium or weakly interacting atomic Bose gases, is widely known to be a second-order phase transition. Here we predict a first-order Bose-Einstein condensation in a cloud of harmonically trapped bosons interacting with both attractive two-body interaction and repulsive three-body interaction, characterized respectively by an $s$-wave scattering length $a<0$ and a three-body scattering hypervolume $D>0$. It happens when the harmonic trapping potential is weak, so with increasing temperature the system changes from a low-temperature liquidlike quantum droplet to a normal gas and therefore experiences a first-order liquid-to-gas transition. At large trapping potential, however, the quantum droplet can first turn into a superfluid gas, rendering the condensation transition occurring later from a superfluid gas to a normal gas smooth. We determine a rich phase diagram and show the existence of a tricritical point, where the three phases, i.e., quantum droplet, superfluid gas, and normal gas, meet. We argue that an ensemble of spin-polarized tritium atoms could be a promising candidate to observe the predicted first-order Bose-Einstein condensation, across which the condensate fraction or central condensate density jumps to zero and the surface-mode frequencies diverge.

Figure 1

First-order BEC transition at $N^{1/3}\omega =0.5$. (a) Temperature dependence of the free energy for the condensed-state solution (red solid line) and for the normal-state solution (blue dashed line). The inset highlights the transition at $T_c\simeq 0.661$. (b) Condensate fraction as a function of temperature. The green dot-dashed line shows $N_c/N=1-{(T/T_{c0})}^3$, where $T_{c0}\simeq 0.9405N^{1/3}\omega$ is the transition temperature of an ideal Bose gas in harmonic traps [3]. The inset shows a jump in the chemical potential at $T_c$. The free energy $F$ (or the chemical potential $\mu$) and the temperature $T$ are measured in units of ${\hslash }^2/M{\xi }^2$ and ${\hslash }^2/k_{B}M{\xi }^2$, respectively. Here $\xi \equiv \sqrt{D/6{\pi }^2{a}^2}$ is the characteristic length scale of the system.

Figure 2

Second-order BEC transition at $N^{1/3}\omega =2.0$. The temperature dependence of the chemical potential is smooth and the BEC transition occurs at $T_c\simeq 1.563$. The inset shows the condensate fraction and the green dot-dashed line is the ideal gas prediction $N_c/N=1-{(T/T_{c0})}^3$. The chemical potential $\mu$ and the temperature $T$ are measured in units of ${\hslash }^2/M{\xi }^2$ and ${\hslash }^2/k_{B}M{\xi }^2$, respectively.

Figure 3

Critical temperature $T_c$ (closed and open symbols) and crossover temperature $T_{*}$ (crosses) as a function of $N^{1/3}\omega$. The thick solid line is the critical temperature predicted at small trapping frequency. The green dot-dashed line shows $T_{c0}\simeq 0.9405N^{1/3}\omega$. The dotted line that connects $T_{*}$ is a guide for the eye. A tricritical point is highlighted by the yellow circle. Typically, we take $N=1000$. The change in the reduced number of particles (i.e., the triangles for $N=125$, diamonds for $N=3375$, and hexagons for $N=8000$) does not lead to a sizable difference in $T_c$. The trapping frequency $\omega$ and the critical temperature $T_c$ are measured in units of $\hslash /M{\xi }^2$ and ${\hslash }^2/k_{B}M{\xi }^2$, respectively.

Figure 4

Mode frequencies ${\omega }_{l,n=0}$ of the breathing mode ($l=0$, solid lines), dipole mode ($l=1$, dotted lines), and surface modes ($l=2,3,4,5$, symbols from bottom to top) as a function of temperature $T$ at (a) $N^{1/3}\omega =0.5$ and (b) $N^{1/3}\omega =2.0$. The dipole mode frequency ${\omega }_{10}$ is not precisely the trapping frequency $\omega$, due to the mean-field interaction potential in the HFB-Popov theory [35]. The mode frequency ${\omega }_{l0}$ (or the trapping frequency $\omega$) and the temperature $T$ are measured in units of $\hslash /M{\xi }^2$ and ${\hslash }^2/k_{B}M{\xi }^2$, respectively.

Figure 5

Quantum depletion $N_{\text{qd}}/N$ as a function of temperature at three effective trapping frequencies $N^{1/3}\omega =0.5$, 1.0, and 2.0. The depletion is typically about $10\%$. It vanishes when we increase the temperature towards the superfluid transition. The trapping frequency $\omega$ and temperature $T$ are measured in units of $\hslash /M{\xi }^2$ and ${\hslash }^2/k_{B}M{\xi }^2$, respectively.

Figure 6

Temperature dependence of the chemical potential at different effective trapping frequency $N^{1/3}\omega$. From bottom to top, the value of $N^{1/3}\omega$ is 0.5, 1.0, 1.2, 1.5, 1.7, 2.0, 2.2, and 2.5. The chemical potential $\mu$ and temperature $T$ are measured in units of ${\hslash }^2/M{\xi }^2$ and ${\hslash }^2/k_{B}M{\xi }^2$, respectively.

Figure 7

Temperature dependence of the central condensate density at (a) $N^{1/3}\omega =0.5$ and (b) $N^{1/3}\omega =2.0$. The insets show the condensate and thermal density distributions at $T=0.95T_c$ (black thick curves) and at $T=1.05T_c$ (red thin curves), respectively. The condensate distribution is plotted as a solid line and the thermal distribution as a dashed line. The trapping frequency $\omega$ and temperature $T$ are measured in units of $\hslash /M{\xi }^2$ and ${\hslash }^2/k_{B}M{\xi }^2$, respectively. Moreover, the radius $r$ is measured in units of $\xi$ and the density is measured in units of the equilibrium density $n_0$.