Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Three-dimensional quantum solitons with parametric coupling

Karen V. Kheruntsyan, P. D. Drummond

Published Sep 1, 1998• 28 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.58.2488

Related papers

Novel solitons in parametric amplifiers and atom lasers

P. D. Drummond, Karen V. Kheruntsyan, Hao He

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.

Journal of Optics B Quantum and Semiclassical Optics•Aug 1, 1999•21 citations

Multidimensional parametric quantum solitons

Karen V. Kheruntsyan, P. D. Drummond

We consider the parametric quantum field theory involving cubic and quartic couplings of two bosonic fields. This is exactly soluble for the two-particle energy eigenstates (or quantum solitons) in one, two, and three space dimensions. We estimate the binding energies and corresponding radii in the case of photonic fields in nonlinear optical materials, and Bose-Einstein condensates.

Physical Review A•Oct 1, 1998•25 citations

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

Karen V. Kheruntsyan, P. D. Drummond

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.

Physical Review A•May 17, 2000•38 citations

Quantum field dynamics

In a realistic treatment of a three-dimensional nonlinear optical experiment, the complete Maxwell equations in (3 + 1) space-time dimensions should be employed. It is then necessary to utilize a multi-mode Hamiltonian that correctly describes the propagating modes. There is an important difference between these experiments and traditional particle scattering. Quantum field dynamics in nonlinear media is dominated by multiple scattering, which is the reason why perturbation theory is less useful. Nevertheless, it is interesting to make a link to conventional perturbation theory. Accordingly, we start by considering a perturbative theory of propagation in a one-dimensional nonlinear optical system in a χ (3) medium. While this calculation cannot treat long interaction times, it does give a qualitative understanding of the important features. This problem is the ‘hydrogen atom’ of quantum field theory: it has fully interacting fields with exact solutions for their energy levels. This is because a photon in a waveguide is an elementary boson in one dimension. The interactions between these bosons are mediated by the Kerr effect, which in quantum field theory is a quartic potential, equivalent to a delta-function interaction. The quantum field theory involved is the simplest model of a quantum field that has an exact solution. This elementary model is still nontrivial in terms of its dynamics, as the calculation of quantum dynamics using standard eigenfunction techniques would require exponentially complex sums over multi-dimensional overlap integrals. This is not practicable, and accordingly we use other methods including quantum phase-space representations to solve this problem.

Cambridge University Press eBooks•Mar 27, 2014•0 citations

Three-dimensional solitons in coupled atomic-molecular Bose-Einstein condensates

Timothy G. Vaughan, Karen V. Kheruntsyan, P. D. Drummond

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 $s$-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.

arXiv:cond-mat/0410234•Physical Review A•Dec 14, 2004•13 citations

Solitons in quantum nonlinear optics

R. Y. Chiao, Ivan Deutsch, J. C. Garrison +1 more

Alkali vapors excited by light near resonance are known to possess very large nonlinear optical coefficients n 2 , which can have either sign depending on the sign of the detuning. Spatial solitons in a Fabry-Perot resonator filled with such vapors can result from these nonlinearities. The nonlinear Schrodinger equation (NLS) obeyed by these solitons is formally equivalent to a nonrelativistic many-body field theory, in which photons behave like nonrelativistic bosons with pairwise interactions. When n 2 is positive, these interactions are attractive; when is negative, they are repulsive. With n 2 positive, it is predicted that a self-trapped, 1D spatial soliton (the hyperbolic secant solution to the ID NLS) should be observable, and we plan to perform soliton-soliton collision experiments with such solitons. At the quantum level, the light inside such a soliton should be self-squeezed, and we plan to observe this using the Shirasaki-Haus method. The possibility of observing the two-photon bound state, the diphoton, will be explored. With n 2 negative, a ID spatial dark soliton with antibunching of photons should result. Moreover, with n 2 negative, a 2D vortex soliton should form, and we plan to examine the dynamics of these photonic vortices at both classical and quantum levels. The possibility that these vortices obey anyonic statistics will be explored. These optical systems thus constitute an experimentally accessible arena in which many-body phenomena analogous to those in condensed matter physics, especially topological ones, such as the Kosterlitz-Thouless phase transition, superfluidity, and anyonic statistics, may be observed.

CRC Press eBooks•Jul 16, 2021•1 citation
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 \ensuremathχ(2){\ensuremath{\chi}}^{(2)}\ensuremathχ(2) nonlinearity) modified by the \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(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 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 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.
In a realistic treatment of a three-dimensional nonlinear optical experiment, the complete Maxwell equations in (3 + 1) space-time dimensions should be employed. It is then necessary to utilize a multi-mode Hamiltonian that correctly describes the propagating modes. There is an important difference between these experiments and traditional particle scattering. Quantum field dynamics in nonlinear media is dominated by multiple scattering, which is the reason why perturbation theory is less useful. Nevertheless, it is interesting to make a link to conventional perturbation theory. Accordingly, we start by considering a perturbative theory of propagation in a one-dimensional nonlinear optical system in a χ (3) medium. While this calculation cannot treat long interaction times, it does give a qualitative understanding of the important features. This problem is the ‘hydrogen atom’ of quantum field theory: it has fully interacting fields with exact solutions for their energy levels. This is because a photon in a waveguide is an elementary boson in one dimension. The interactions between these bosons are mediated by the Kerr effect, which in quantum field theory is a quartic potential, equivalent to a delta-function interaction. The quantum field theory involved is the simplest model of a quantum field that has an exact solution. This elementary model is still nontrivial in terms of its dynamics, as the calculation of quantum dynamics using standard eigenfunction techniques would require exponentially complex sums over multi-dimensional overlap integrals. This is not practicable, and accordingly we use other methods including quantum phase-space representations to solve this problem.
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.
Alkali vapors excited by light near resonance are known to possess very large nonlinear optical coefficients n 2 , which can have either sign depending on the sign of the detuning. Spatial solitons in a Fabry-Perot resonator filled with such vapors can result from these nonlinearities. The nonlinear Schrodinger equation (NLS) obeyed by these solitons is formally equivalent to a nonrelativistic many-body field theory, in which photons behave like nonrelativistic bosons with pairwise interactions. When n 2 is positive, these interactions are attractive; when is negative, they are repulsive. With n 2 positive, it is predicted that a self-trapped, 1D spatial soliton (the hyperbolic secant solution to the ID NLS) should be observable, and we plan to perform soliton-soliton collision experiments with such solitons. At the quantum level, the light inside such a soliton should be self-squeezed, and we plan to observe this using the Shirasaki-Haus method. The possibility of observing the two-photon bound state, the diphoton, will be explored. With n 2 negative, a ID spatial dark soliton with antibunching of photons should result. Moreover, with n 2 negative, a 2D vortex soliton should form, and we plan to examine the dynamics of these photonic vortices at both classical and quantum levels. The possibility that these vortices obey anyonic statistics will be explored. These optical systems thus constitute an experimentally accessible arena in which many-body phenomena analogous to those in condensed matter physics, especially topological ones, such as the Kosterlitz-Thouless phase transition, superfluidity, and anyonic statistics, may be observed.