Modulational instability and discrete quantum droplets in a deep quasi-one-dimensional optical lattice

TL;DR

This work analyzes modulational instability and discrete quantum droplets in a deep quasi-1D optical lattice described by a discrete GPE with cubic-quintic nonlinearity arising from Lee-Huang-Yang (LHY) corrections. It derives the discrete model $i \psi_{n,t}+ \kappa (\psi_{n+1}+\psi_{n-1}-2 \psi_n) + \gamma |\psi_n|^2 \psi_n-\delta |\psi_n|^3\psi_n=0$, characterizes MI via the plane-wave ansatz and its gain spectrum, and maps instability regions in parameter spaces. The study then constructs a variety of strongly localized discrete breathers (even/odd, topological/flat-top and LHY-only modes) using the Page method, analyzes their linear stability, and validates results with numerics; it further develops a quasi-continuous variational approach to describe soliton- and flat-top-like droplets and their interactions. The findings show that quantum fluctuations can both modify MI landscapes and sustain novel localized states, with stability strongly dependent on parity, coupling, and the presence of LHY terms, offering insights for experiments on BECs in deep optical lattices. The combination of linear stability analysis, Page method, and variational treatment provides a comprehensive framework for understanding quasi-1D discrete quantum droplets in lattice-confinement settings.

Abstract

We study the properties of modulational instability and discrete breathers arising in a quasi-one-dimensional discrete Gross-Pitaevskii equation with Lee-Huang-Yang corrections. Conditions for modulation instability and instability regions of nonlinear plane waves are determined in parameter space. We analytically investigate the existence of different quantum droplet solutions, including intersite, onsite, front-like, flat-top and dark localized modes, using the Page method and variational approach. Their stability is checked using linear stability analyses and numerical simulations. The analytical predictions corroborated with the numerical simulations.

Figures (21)

• Figure 1: Evaluation of slightly perturbed plane wave solutions. (a) For $Q=\pi>Q_{\mathrm{cr}}$. (b) For $Q=\pi/2<Q_{\mathrm{cr}}$. Other parameters are $A=0.5$, $\kappa=0.1$, $q=\pi$, $\gamma=0$ and $\delta=1$.
• Figure 2: Typical gain spectrum of MI. Lines are found from Eq. (\ref{['gainQ=pi']}) and points are found from direct numerical simulations of Eq. (\ref{['moddnlseQ1D']}). (a) For $A=0.3$ and the bottom to top curves is for the different values of $(\gamma, \delta)$ that correspond to $(0,1)$, $(-1,0)$ and $(-1,1)$, respectively. (b) The same plot as in the (a) panel, but for $A=0.5$. Other parameters are $\kappa=0.1$, $q=\pi$.
• Figure 3: Modulational instability regions in the $(\gamma,\delta)$ plane for different signs of hopping rate. (a) $\kappa=0.1$. (b) $\kappa=-0.1$. In both figures color bar represents maximum values of gain. Other parameters are $A=0.3$ and $q=\pi$.
• Figure 4: Modulational instability regions in the $(Q,q)$ plane. (a) $\kappa=0.1$. (b) $\kappa=-0.1$. In both figures color bar represents maximum values of gain. Other parameters are $A=0.3$, $\gamma=0$ and $\delta=1$.
• Figure 5: Even strongly localized modes (a) s=-1. (b) s=1. The solid line represents the solution of Eq. (\ref{['discStatEQ']}) by the Newton method, and the points represents the approximate solution by the Page method given by Eq. (\ref{['discMU']}). Other parameters are $A=1$, $\kappa=0.1$, $\gamma=1$ and $\delta=0.3$.