Quantum Lattice Solitons

TL;DR

This paper uses the number state method to analyze quantum lattice solitons in three anharmonic systems, deriving eigenfunctions, energies, and properties like binding energy and effective mass, linking quantum states to classical solitons.

Contribution

It provides exact solutions at the second quantum level and asymptotic expressions for key soliton properties across different anharmonicities in quantum lattices.

Findings

Exact eigenfunctions and energies for n=2 quantum states.

Expressions for binding energy, effective mass, and maximum group velocity.

Connection between quantum eigenstates and classical soliton wave packets.

Abstract

The number state method is used to study soliton bands for three anharmonic quantum lattices: i) The discrete nonlinear Schr\"{o}dinger equation, ii) The Ablowitz-Ladik system, and iii) A fermionic polaron model. Each of these systems is assumed to have $f$-fold translational symmetry in one spatial dimension, where $f$ is the number of freedoms (lattice points). At the second quantum level $(n=2)$ we calculate exact eigenfunctions and energies of pure quantum states, from which we determine binding energy $(E_{\rm b})$, effective mass $(m^{*})$ and maximum group velocity $(V_{\rm m})$ of the soliton bands as functions of the anharmonicity in the limit $f \to \infty$. For arbitrary values of $n$ we have asymptotic expressions for $E_{\rm b}$, $m^{*}$, and $V_{\rm m}$ as functions of the anharmonicity in the limits of large and small anharmonicity. Using these expressions we discuss and describe wave packets of pure eigenstates that correspond to classical solitons.