Can soliton compute? In this talk we explore how possible it is to use dark soliton crystals, produced by a system of weakly interacting identical cold atoms arranged in a ring geometry, to store soliton-like ”quantum bits” in the spectral memory of a free boson gas. The underlying mechanism involves a subtle combination of self-defocusing nonlinearity due to the repulsive interatomic interactions, a dispersion and the finite size of the ring structure. This combination favors the emergence of dark-type matter-wave soliton crystals that imprint quantum localized states in the spectrum of the free boson gas.

To formulate the problem we consider the Gross-Pitaevskii equation with a positive scattering length, coupled to a linear Schrödinger equation. Owing to the coupling the bound-state spectrum of the free boson gas consists in |ℓ, m⟩ quantum states whose wave functions and energy eigenvalues are unambiguously identified. Some of these eigenstates have their wave functions that are replicas of the generating dark soliton crystal.

Key references:
1. Alain M. Dikandé: Using dark solitons from a Bose-Einstein condensate necklace to imprint soliton states in the spectral memory of a free boson gas. New Journal of Physics vol. 25, p. 103017 (2023).
2. E. Aban Chenui and Alain M. Dikandé: Soliton-mode proliferation induced by cross-phase modulation of harmonic waves by a dark soliton crystal in optical media. Microwave and Optical Technology Letters vol. 63, p. 2681 (2021).
3. D. D. M. Welakuh and Alain M. Dikandé: Storage and retrieval of time-entangled soliton trains in a three-level atom system coupled to an optical cavity. Optics Communications vol. 403, p. 27 (2017).
4. Alain M. Dikandé: Induced soliton ejection from a continuous-wave source waveguided by an optical pulse soliton train. Journal of Optics vol. 13, p. 035203 (2011).
5. Alain M. Dikandé: Fundamental modes of a trapped probe photon in optical fibers conveying periodic pulse trains. Physical Reviews A vol. 81, 013821 (2010).
6. K. Steiglitz: Soliton-guided phase shifter and beam splitter. Physical Reviews A 81, 033835 (2010).
7. K. Steiglitz: Making beam splitters with dark soliton collisions. Physical Reviews A 82, 043831 (2010).
8. Ch. Anastassiou, J. W. Fleischer, T. Carmon, M. Segev and K. Steiglit: Information transfer via cascaded collisions of vector solitons. Optics Letters vol.26, p. 1498 (2001).
9. Berol Takogaing Ngassa: Matter-wave Soliton-Quantum bits (African Institute for Mathematical sciences-AIMS), Structured Masters essay, June 2026.