Harmonically trapped four-boson system

Four identical spinless bosons with purely attractive two-body short-range interactions and repulsive three-body interactions under external spherically symmetric harmonic confinement are considered. The repulsive three-body potential prevents the formation of deeply bound states with molecular character. The low-energy spectrum with vanishing orbital angular momentum and positive parity for infinitely large two-body $s$-wave scattering length is analyzed in detail. Using the three-body contact, states are classified as universal, quasiuniversal, or strongly nonuniversal. Connections with the zero-range interaction model are discussed. The energy spectrum is mapped out as a function of the two-body $s$-wave scattering length $a_s, a_s>0$. In the weakly to medium-strongly interacting regime, one of the states approaches the energy obtained for a hard-core interaction model. This state is identified as the energetically lowest-lying “BEC state.” Structural properties are also presented.

Figure 1

Energy spectrum for three harmonically trapped identical bosons as a function of $a_{\text{ho}}/a_s$. Circles show the relative energy of the seven energetically lowest-lying states with $L^{\mathrm{\Pi }}=0^{+}$ symmetry for model II with $V_0=97700E_{\text{ho}}$. The solid line shows the relative energy of the lowest state for two-body hard-core interactions (model I).

Figure 2

Dependence of the energy spectrum for three identical harmonically trapped bosons on the height $V_0$ of the three-body Gaussian potential. The circles, diamonds, and squares show the relative energy as a function of $a_{\text{ho}}/a_s$ for $V_0=97700E_{\text{ho}}$ (same as in Fig. 1), $V_0=40000E_{\text{ho}}$, and $V_0=10000E_{\text{ho}}$, respectively. Only states with $L^{\mathrm{\Pi }}=0^{+}$ symmetry are considered.

Figure 3

Energy spectrum for four harmonically trapped identical bosons as a function of $a_{\text{ho}}/a_s$. Only states with $L^{\mathrm{\Pi }}=0^{+}$ symmetry are considered. The filled circles (neighboring points are connected by solid lines) show the relative energies for model II with $V_0=97700E_{\text{ho}}$. The number of states considered is larger for $a_{\text{ho}}/a_s\gtrsim 1.4$ than for $a_{\text{ho}}/a_s\lesssim 1.4$. The squares show the relative energies for model II with $V_0=97700E_{\text{ho}}$ obtained using the “target state approach” [these energies are not necessarily upper bounds; moreover, the error bar (not shown) for $a_{\text{ho}}/a_s=2.5$ is comparatively large]. For comparison, the solid line shows the relative ground-state energy for two-body hard-core interactions (model I).

Figure 4

Blowup of a portion of the energy spectrum shown in Fig. 3. It can be seen that the avoided crossings are quite narrow.

Figure 5

Structural properties of the harmonically trapped four-boson system at unitarity for interaction model II with $V_0=97700E_{\text{ho}}$. The hyperradial density $P\left(R\right)$ (left column) and subhyperradial density $P_{\text{sub}}\left(r_{jkl}\right)$ (right column) are shown for different $\nu$ families (the first, second, third, and fourth row are for $\nu =0, \nu =1, \nu =4$, and $\nu =5$, respectively). The density of the lowest state ($q=0$) in a given family is shown by a dotted line, that of the second-lowest state ($q=1$) in a given family by a solid line, and that of the third-lowest state ($q=2$) in a given family by a dashed line.

Figure 6

Scaled pair distribution function $r_{jk}^2{P}_{\text{pair}}\left(r_{jk}\right)$ for the lowest BEC state for various $a_s$ for (a) $N=3$ and (b) $N=4$. The calculations are for the interaction model II with $V_0=97700E_{\text{ho}}$. The curves from bottom to top at $r_{jk}/a_{\text{ho}}=1.5$ correspond to $a_{\text{ho}}/a_s=0, 1/2, \approx 1.3500$, and $\approx 1.7012$.