[Paper Review] Propagation of matter wave solitons in periodic and random nonlinear potentials

F. Kh. Abdullaev, Josselin Garnier|arXiv (Cornell University)|Nov 10, 2005

Cold Atom Physics and Bose-Einstein Condensates19 references100 citations

TL;DR

This paper investigates bright matter wave solitons in periodic and random nonlinear potentials generated by spatially varying atomic scattering lengths via Feshbach resonance. Using perturbation theory and collective coordinate methods, it derives analytical conditions for soliton radiation, decay, and stable propagation, showing that solitons can be stabilized via radiative emission in critical parameter regimes, with results validated by Gross-Pitaevskii equation simulations.

ABSTRACT

We study the motion of bright matter wave solitons in nonlinear potentials, produced by periodic or random spatial variations of the atomic scattering length. We obtain analytical results for the soliton motion, the radiation of matter wave, and the radiative soliton decay in such configurations of the Bose-Einstein condensate. The stable regimes of propagation are analyzed. The results are in remarkable agreement with the numerical simulations of the Gross-Pitaevskii equation with periodic or random spatial variations of the mean field interactions.

Motivation & Objective

• To understand the dynamics of bright matter wave solitons in nonlinear potentials with spatially varying atomic scattering lengths.
• To identify conditions under which solitons emit radiation or decay radiatively in such potentials.
• To determine stable propagation regimes for solitons in periodic and random nonlinear potentials.
• To validate analytical predictions against numerical simulations of the Gross-Pitaevskii equation with spatially modulated nonlinearity.

Proposed method

• Formulation of the Gross-Pitaevskii equation with a spatially varying nonlinear coefficient, g1D(x) = 2ℏω⊥(as0 + as1f(x))/m.
• Use of dimensionless variables based on healing length ξ and sound speed c to derive Eq. (2): iut + uxx + 2|u|2u = −V(x)|u|2u.
• Application of the collective coordinate ansatz to model the soliton as a classical particle with effective potential V(ζ) = −Anl cos(Kζ).
• Perturbation theory based on the Inverse Scattering Transform to calculate radiation emission rates and soliton parameter evolution.
• Derivation of evolution equations (4) for soliton amplitude ν and velocity parameter µ, with radiation-induced decay terms F(ν,µ) and G(ν,µ).
• Numerical simulations of the perturbed GP equation with random stepwise potentials to validate analytical predictions.

Experimental results

Research questions

• RQ1 Under what conditions does a bright matter wave soliton emit radiation when propagating through a nonlinear periodic potential?
• RQ2 How does the radiative decay rate of a soliton depend on the modulation wavenumber K and soliton parameters ν0, µ0?
• RQ3 Can solitons be stabilized in nonlinear periodic or random potentials despite radiative decay?
• RQ4 What is the role of the nonlinear potential strength V0 and correlation length lc in determining soliton transmission or trapping?
• RQ5 How does the soliton's mass and velocity evolve over time in the presence of random nonlinear disorder?

Key findings

• Solitons in nonlinear periodic potentials experience an enhanced effective potential barrier Anl, with enhancement factors up to K²/12 for broad solitons (K/ν ≫1) and 4ν²/3 for narrow solitons (K/ν ≪1).
• Radiation emission occurs when the modulation wavenumber K > ν₀²/µ₀, leading to significant soliton mass and velocity decay governed by the system of ODEs (4).
• For K ≈ ν₀²/µ₀, the soliton reaches a stable equilibrium state after emitting a small amount of radiation, enabling long-term propagation.
• In the critical regime where µ₀K = ν₀², the soliton mass decays as ν(t) ≈ ν₀(1 + t/Tc)⁻¹/⁴ with decay time Tc = 3µ₀/(32d(4µ₀)ν₀⁴), inversely proportional to the fourth power of soliton amplitude.
• In the regime µ₀ ≪ ν₀, radiation is weak and broadband, with soliton velocity decaying logarithmically; the soliton remains transmittable over long times.
• Numerical simulations of the randomly perturbed GP equation show excellent agreement with analytical predictions, particularly in power-law decay regimes and transmission behavior.