Universal and nonuniversal effective $N$-body interactions for ultracold harmonically trapped few-atom systems

We derive the ground-state energy for a small number of ultracold atoms in an isotropic harmonic trap using effective quantum field theory (EFT). Atoms are assumed to interact through pairwise energy-independent and energy-dependent delta-function potentials with strengths proportional to the scattering length $a$ and effective range volume $V$, respectively. The calculations are performed systematically up to the order of $l^{-4}$, where $l$ denotes the harmonic-oscillator length. The effective three-body interaction contains a logarithmic divergence in the cutoff energy, giving rise to a nonuniversal three-body interaction in the EFT. Our EFT results are confirmed by nonperturbative numerical calculations for a Hamiltonian with finite-range two-body Gaussian interactions. For this model Hamiltonian, we explicitly calculate the nonuniversal effective three-body contribution to the energy.

Figure 1

(a), (b) The effective four- and five-body contributions $U_4^{(4,0,0)}$ and $U_5^{(4,0,0)}$, respectively, as a function of $a/l$. The interaction energies are scaled by ${(a/l)}^{-4},$ such that the EFT predictions at order $K=4$ are given by the solid horizontal lines. Circles show numerical values for a model Hamiltonian with a pairwise Gaussian interaction with $r_0=0.01l$. The numerical data is unreliable in the regime (a) $|a/l|\lesssim 0.007$ and (b) $|a/l|\lesssim 0.11$, as the numerical uncertainty becomes comparable to or larger than $0.3$ times the quantity of interest. The dashed line in panel (a) includes the scaled $K=5$ effective-range-volume-dependent contribution, which is proportional to $c_4^{(2,1,0)}V/a^2$. We use the numerically obtained effective range volume, as a function of $a,$ for the Gaussian potential with $r_0=0.01l$, and $c_4^{(2,1,0)}=25.422472$ as determined within the EFT. The dotted line in panel (b) shows a linear fit of the form $c_5^{(4,0,0)}+c_5^{(5,0,0)}a/l$ to the numerically determined energies in the regime $|a/l|>0.11$, with $c_5^{(4,0,0)}$ fixed at our EFT value of $-11.12$. We find $c_5^{(5,0,0)}\approx 210$. The error bars, which are one standard deviation, are estimated from the basis set extrapolation errors of the numerically determined $N=3,4,$ and 5 energies $E_N$.

Figure 2

Diagrammatic representation of the divergent sums that contribute to the effective three-body interaction $U_3^{(4,0,0)}$. Diagrams (a)–(d) represent the quantities $D_{\text{a}},D_{\text{b}},D_{\text{c}},$ and $D_{\text{d}}$, respectively (see Table 1 and text). The dot represents the two-body interaction with coupling constant $g_2^{\left(0\right)}$. The circled dot represents the two-body counterterm with coupling constant $g_{2,\text{ct}}^{\left(0\right)}$.

Figure 3

Diagrammatic representation of the nonuniversal three-body interaction. The square represents the three-body interaction with coupling constant $g_3^{\left(0\right)}$.

Figure 4

Scaled three-body interaction ${\overline{U}}_3^{K=4}/{(a/l)}^4$ as a function of $a/l$, extracted from numerical $N$-body ground-state energies for the Gaussian two-body potential with width $r_0=0.01l$ and using Eq. (16). Circles, squares, and diamonds are determined from Eq. (16) for $N=3,4,$ and 5, respectively. The numerical data is unreliable for $|a/l|\lesssim 0.005,$ as the numerical uncertainty becomes comparable to or larger than $0.3$ times the quantity of interest. The error bars, which are one standard deviation, are estimated from the basis set extrapolation errors of the numerically determined $N=3,4,$ and 5 energies $E_N$.