Formation of Atomic Tritium Clusters and Bose-Einstein Condensates

We present an extensive study of the static and dynamic properties of systems of spin-polarized tritium atoms. In particular, we calculate the two-body $|F,m_F⟩=|0,0⟩$ $s$-wave scattering length and show that it can be manipulated via a Feshbach resonance at a field strength of about 870 G. Such a resonance might be exploited to make and control a Bose-Einstein condensate of tritium in the $|0,0⟩$ state. It is further shown that the quartet tritium trimer is the only bound hydrogen isotope and that its single vibrational bound state is a Borromean state. The ground state properties of larger spin-polarized tritium clusters are also presented and compared with those of helium clusters.

Figure 1

Tritium dimer triplet $b^3{\Sigma }_u^{+}$ potential (solid line) together with the He dimer potential (dotted line) as a function of the interparticle distance $r$. The inset compares the tritium dimer triplet potential ($S=1$, solid line) with the tritium dimer singlet ground state potential ($S=0$, dashed line). Note the different vertical scales of the main figure and the inset. See text for a detailed discussion.

Figure 2

$a(00+00)$ scattering length (diamonds; using the reduced atomic mass in the coupled-channel calculation, see text) as a function of the magnetic field strength $B$ (dotted lines are shown to guide the eye). The solid line describes the behavior for the range $B\in [400,1300\mathrm{G}]$ well using the following parametrization, $a(00,00)=a_{\mathrm{B}\mathrm{G}}[1-\Delta /(B-B_R)]$ with $a_{\mathrm{B}\mathrm{G}}=-37.9\mathrm{a}\mathrm{.}\mathrm{u}\mathrm{.}$, $\Delta =-1238\mathrm{G}$, and $B_R=870.8\mathrm{G}$; however, the fit is inaccurate at low fields. Inset: Threshold energies in GHz as a function of magnetic field $B$ in Gauss. The assignment of quantum numbers is approximate, except for $|1,1⟩+|1,-1⟩$, which is an exact eigenstate without admixtures.

Figure 3

DMC ground state energy per particle $E_0/N$ for $\mathrm{(}\mathrm{T}\uparrow {)}_N$ clusters (plusses) and ${{}^{\mathrm{4}}\mathrm{H}\mathrm{e}}_N$ clusters (diamonds) as a function of $N$. Dotted lines are shown to guide the eye. Inset: Pair distribution $P(r)$ for $(\mathrm{T}\uparrow {)}_5$ cluster (solid line) and ${{}^{\mathrm{4}}\mathrm{H}\mathrm{e}}_5$ cluster (dotted line), calculated by the DMC technique with importance sampling. $P(r)$ is normalized such that ${\int }_0^{\infty }P(r)r^{2}dr=1$. This figure illustrates that $\mathrm{(}\mathrm{T}\uparrow {)}_N$ clusters are even more weakly bound than ${\mathrm{H}\mathrm{e}}_N$ clusters.