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Physical Review A

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

Karen V. Kheruntsyan, P. D. Drummond

Published May 17, 2000• 38 citations
Publisher

Abstract

We consider the quantum theory of three fields interacting via parametric and repulsive quartic couplings. This can be applied to treat photonic ${\ensuremath{\chi}}^{(2)}$ and ${\ensuremath{\chi}}^{(3)}$ 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 ${\ensuremath{\chi}}^{(3)}$ 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.

DOI
10.1103/physreva.61.063816

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We consider the quantum theory of three fields interacting via parametric and repulsive quartic couplings. This can be applied to treat photonic \ensuremathχ(2){\ensuremath{\chi}}^{(2)}\ensuremathχ(2) and \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3) 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 \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3) 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.
We review recent developments in quantum and classical soliton theory, leading to the possibility of observing both classical and quantum parametric solitons in higher-dimensional environments. In particular, we consider the theory of three bosonic fields interacting via both parametric (cubic) and quartic couplings. In the case of photonic fields in a nonlinear optical medium this corresponds to the process of sum frequency generation (via (2) nonlinearity) modified by the (3) nonlinearity. Potential applications include an ultrafast photonic AND-gate. The simplest quantum solitons or energy eigenstates (bound-state solutions) of the interacting field Hamiltonian are obtained exactly in three space dimensions. They have a point-like structure - even though the corresponding classical theory is nonsingular. We show that the solutions can be regularized with the imposition of a momentum cut-off on the nonlinear couplings. The case of three-dimensional matter-wave solitons in coupled atomic/molecular Bose-Einstein condensates is discussed.
We consider the quantum field theory of two bosonic fields interacting via both parametric (cubic) and quartic couplings. In the case of photonic fields in a nonlinear optical medium, this corresponds to the process of second-harmonic generation (via ${\ensuremath{\chi}}^{(2)}$ nonlinearity) modified by the ${\ensuremath{\chi}}^{(3)}$ nonlinearity. The quantum solitons or energy eigenstates (bound-state solutions) are obtained exactly in the simplest case of two-particle binding, in one, two, and three space dimensions. We also investigate three-particle binding in one space dimension. The results indicate that the exact quantum solitons of this field theory have a singular, pointlike structure in two and three dimensions---even though the corresponding classical theory is nonsingular. To estimate the physically accessible radii and binding energies of the bound states, we impose a momentum cutoff on the nonlinear couplings. In the case of nonlinear optical interactions, the resulting radii and binding energies of these photonic particlelike excitations in highly nonlinear parametric media appear to be close to physically observable values.
We present a theoretical analysis of three-dimensional (3D) matter-wave solitons and their stability properties in coupled atomic and molecular Bose-Einstein condensates (BECs). The soliton solutions to the mean-field equations are obtained in an approximate analytical form by means of a variational approach. We investigate soliton stability within the parameter space described by the atom-molecule conversion coupling, the atom-atom sss-wave scattering, and the bare formation energy of the molecular species. In terms of ordinary optics, this is analogous to the process of sub- or second-harmonic generation in a quadratic nonlinear medium modified by a cubic nonlinearity, together with a phase mismatch term between the fields. While the possibility of formation of multidimensional spatiotemporal solitons in pure quadratic media has been theoretically demonstrated previously, here we extend this prediction to matter-wave interactions in BEC systems where higher-order nonlinear processes due to interparticle collisions are unavoidable and may not be neglected. The stability of the solitons predicted for repulsive atom-atom interactions is investigated by direct numerical simulations of the equations of motion in a full 3D lattice. Our analysis also leads to a possible technique for demonstrating the ground state of the Schr\"odinger-Newton and related equations that describe Bose-Einstein condensates with nonlocal interparticle forces.
Summary form only given. Parametric solitons or simultaneous solitary waves (simultons), involving the optical /spl chi//sup (2)/ nonlinearity, have been the topic of much recent theoretical and experimental interest in nonlinear optics. We propose a novel mechanism by which a similar phenomenon may occur in nonlinear atomic optics, in which coherent molecule formation in a Bose-Einstein condensate takes the place of second harmonic generation. This requires a coupling that converts two atoms into one molecule, thus generating coupled atomic and molecular Bose-Einstein condensates. Our model includes a coherent molecular formation process (i.e., without dissipation) in an atomic BEC vapor, either through a Feshbach resonance or Raman photo-association. The coherently coupled atom-molecular condensate could provide a route to the observation of a localized three-dimensional BEC soliton, even in the absence of a trap potential. A possible application is in the free propagation of a non-diverging atom laser pulse, thus greatly increasing the intensity in an atom laser beam.