One-dimensional ultracold atomic gases: Impact of the effective range on integrability

Three identical bosons or fermions are considered in the limit of zero-range interactions and finite effective range. By using a two-channel model, we show that these systems are not integrable and that the wave function verifies specific continuity conditions at the contact of three particles. This last feature permits us to solve a contradiction brought by the contact model which can lead to an opposite result concerning the integrability issue. For fermions, the vicinity of integrability is characterized by large deviations with respect to the predictions of the Bethe ansatz.

Figure 1

Spectrum of the trimers as a function of the wave number of the dimer. Continuous line: bosons $\left(\eta =0\right)$; dashed line: fermions $\left(\eta =1\right)$. Inset: plot of $\delta q_{\eta }^{\mathrm{t}}=2\sqrt{(q_{\eta }^{\mathrm{t}}{}^2-{\kappa }_{\eta }^2)/3}$ in the region where $q_{\eta }^{\mathrm{t}}\sim {\kappa }_{\eta }$.

Figure 2

Mixed-channel wave function for the fermionic trimer $\left(\eta =1\right)$. Solution of Eq. (23) plotted in the configuration space for $(a_1={10}^3{b}_1)$ (dashed line) and $(a_1={10}^{12}{b}_1)$ (continuous line). Dotted line: BA mixed-channel state in Eq. (30) with a normalization chosen for convenience. Inset: same plot where $Z/a_1$ is in logarithmic scale. The cross indicates that the solutions of the HTCM do not vanish at the three-body contact.

Figure 3

Dimer-particle scattering lengths defined in Eq. (35) plotted as a function of the dimer binding wave number. Continuous line: bosonic system $\left(\eta =0\right)$; dashed line: fermionic system $\left(\eta =1\right)$. (a) Odd sector $({\eta }^{\prime }=1)$; (b) even sector for bosons; (c) even sector for fermions.