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Using dark solitons from a Bose-Einstein condensate necklace to imprint soliton states in the spectral memory of a free boson gas

Alain M. Dikande

TL;DR

The paper addresses imprinting spectral memory in a free boson gas using a matter-wave dark soliton crystal generated in a finite ring-shaped BEC. By coupling a repulsive BEC to a non-interacting 1D boson gas and solving the resulting equations, the free-gas spectrum is mapped to a Lamé equation of order $\nu$, yielding a discrete set of bound states with $2\nu+1$ modes per order; explicit results are presented for $\nu=1$ and $\nu=2$, including replicas of the mother soliton crystal and degeneracy patterns in the $\kappa \to 1$ limit. The framework provides a tunable, qudit-like memory resource in cold-atom mixtures with potential applications to quantum information processing. The authors discuss possible extensions to other particle statistics and coupling schemes, illustrating the broad relevance of soliton-induced spectral memory in quantum media.

Abstract

A possible use of matter-wave dark-soliton crystal produced by a Bose-Einstein condensate with ring geometry, to store soliton states in the quantum memory of a free boson gas, is explored. A self-defocusing nonlinearity combined with dispersion and the finite size of the Bose-Einstein condensate, favor the creation of dark-soliton crystals that imprint quantum states with Jacobi elliptic-type soliton wavefunctions in the spectrum of the free boson gas. The problem is formulated by considering the Gross-Pitaevskii equation with a positive scattering length, coupled to a linear Schrödinger equation for the free boson gas. With the help of the matter-wave dark soliton-crystal solution, the spectrum of bound states created in the free boson gas is shown to be determined by the Lamé eigenvalue problem. This spectrum consists of $\vert ν, \mathcal{L} \rangle$ quantum states whose wave functions and energy eigenvalues can be unambiguously identified. Among these eigenstates some have their wave functions that are replicas of the generating dark soliton crystal.

Using dark solitons from a Bose-Einstein condensate necklace to imprint soliton states in the spectral memory of a free boson gas

TL;DR

The paper addresses imprinting spectral memory in a free boson gas using a matter-wave dark soliton crystal generated in a finite ring-shaped BEC. By coupling a repulsive BEC to a non-interacting 1D boson gas and solving the resulting equations, the free-gas spectrum is mapped to a Lamé equation of order , yielding a discrete set of bound states with modes per order; explicit results are presented for and , including replicas of the mother soliton crystal and degeneracy patterns in the limit. The framework provides a tunable, qudit-like memory resource in cold-atom mixtures with potential applications to quantum information processing. The authors discuss possible extensions to other particle statistics and coupling schemes, illustrating the broad relevance of soliton-induced spectral memory in quantum media.

Abstract

A possible use of matter-wave dark-soliton crystal produced by a Bose-Einstein condensate with ring geometry, to store soliton states in the quantum memory of a free boson gas, is explored. A self-defocusing nonlinearity combined with dispersion and the finite size of the Bose-Einstein condensate, favor the creation of dark-soliton crystals that imprint quantum states with Jacobi elliptic-type soliton wavefunctions in the spectrum of the free boson gas. The problem is formulated by considering the Gross-Pitaevskii equation with a positive scattering length, coupled to a linear Schrödinger equation for the free boson gas. With the help of the matter-wave dark soliton-crystal solution, the spectrum of bound states created in the free boson gas is shown to be determined by the Lamé eigenvalue problem. This spectrum consists of quantum states whose wave functions and energy eigenvalues can be unambiguously identified. Among these eigenstates some have their wave functions that are replicas of the generating dark soliton crystal.
Paper Structure (6 sections, 9 equations, 4 figures, 2 tables)

This paper contains 6 sections, 9 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The model and matter-wave periodic dark soliton solution for the repulsive BEC necklace
  3. From linear Schrödinger to Lamé eigenvalue equation for the gas of free bosons
  4. Three-level system with basis states $\vert \nu=1, \mathcal{L}(=1, 2, 3)\rangle$:
  5. Five-level system with basis states $\vert \nu=2, \mathcal{L}(=1, 2, 3, 4, 5)\rangle$:
  6. Summary and concluding remarks

Figures (4)

  • Figure 1: (Colour online) Distribution of the three energy levels $E_1\equiv \vert 1, 1\rangle$, $E_2\equiv \vert 1, 2\rangle$ and $E_3\equiv \vert 1, 3\rangle$, in the potential induced by the matter-wave soliton crystal from the repulsive BEC. Left graph: $\kappa=0.99$ (Periodic lattice of "$sech^2$" potential wells). Right graph: $\kappa=1$ (single "$sech^2$" potential well).
  • Figure 2: (Colour online) Wave functions $\phi_{\beta}$ of the three bound states of the first-order Lamé equation, plotted versus $x$ for $\kappa=0.99$ (left column) and $\kappa=1$ (right column).
  • Figure 3: (Colour online) Five-level spectrum for the trapped boson gas when $\nu=2$: $E_1\equiv \vert 2, 1\rangle$, $E_2\equiv \vert 2, 2\rangle$, $E_3\equiv \vert 2, 3\rangle$, $E_4\equiv \vert 2, 4\rangle$ and $E_5\equiv \vert 2, 5\rangle$, in the potential induced by the matter-wave soliton crystal from the repulsive BEC. Left graph: $\kappa=0.99$ (Periodic lattice of "$sech^2$" potential wells). Right graph: $\kappa=1$ (single "$sech^2$" potential well).
  • Figure 4: (Colour online) Wave functions $\phi_{\beta}$ of the five bound states of the second-order Lamé equation, plotted versus $z$ for $\kappa=0.99$ (left column) and $\kappa=1$ (right column). The notations $M_1$, $M_2$, $M_3$, $M_4$ and $M_5$ in the graphs refer to the wave functions given respectively in (\ref{['m21']}), (\ref{['m22']}), (\ref{['m23']}), (\ref{['s24']}) and (\ref{['s25']}).