Multidimensional quantum solitons with nondegenerate parametric interactions: Photonic and Bose-Einstein condensate environments

We consider the quantum theory of three fields interacting via parametric and repulsive quartic couplings. This can be applied to treat photonic ${\chi }^{\mathrm{}\left(2\right)\mathrm{}}$ and ${\chi }^{\mathrm{}\left(3\right)\mathrm{}}$ interactions, and interactions in atomic Bose-Einstein condensates or quantum Fermi gases, describing coherent molecule formation together with s-wave scattering. The simplest two-particle quantum solitons or bound-state solutions of the idealized Hamiltonian, without a momentum cutoff, are obtained exactly. They have a pointlike structure in two and three dimensions—even though the corresponding classical theory is nonsingular. We show that the solutions can be regularized with a momentum cutoff. The parametric quantum solitons have much more realistic length scales and binding energies than ${\chi }^{\mathrm{}\left(3\right)\mathrm{}}$ quantum solitons, and the resulting effects could potentially be experimentally tested in highly nonlinear optical parametric media or interacting matter-wave systems. N-particle quantum solitons and the ground state energy are analyzed using a variational approach. Applications to atomic/molecular Bose-Einstein condensates (BEC’s) are given, where we predict the possibility of forming coupled BEC solitons in three space dimensions, and analyze “superchemistry” dynamics.