Two-mode approximation for a Bose gas ($\hat{\psi }(\mathbf{r})\simeq {\phi }_L(\mathbf{r})a_L+{\phi }_R(\mathbf{r})a_R$)

Project a quantized field onto two low-energy orbitals. A sufficient regime has temperature, hopping and interaction shifts small compared with the energy gap to discarded orbitals. Small interwell contact interaction integrals can then be neglected while retaining a small quantum tunnelling matrix element. The projected field's operator commutator is the two-mode projection operator kernel, rather than a complete spatial Dirac delta function.