Anderson localization of quantum droplets in disordered potentials

TL;DR

This work investigates Anderson localization of a 1D quantum droplet in a disordered potential using the time-dependent generalized Gross-Pitaevskii equation (TDGGPE). By modeling the disorder as a speckle-like pattern built from Gaussian spikes, the authors compute the droplet width, density profiles, diffusion exponent $\alpha$, generalized diffusion coefficient $D_{\alpha}$, and localization length $L_{\text{loc}}$ for both small and large droplets. They observe anomalous diffusion ranging from superdiffusion to subdiffusion as disorder strength increases and identify a disorder-induced transition to Anderson localization above a critical threshold, with $L_{\text{loc}}$ and the localization time $t_{\text{loc}}$ depending on the droplet size. The results are directly relevant to current cold-atom experiments and illuminate the interplay between disorder, interactions, and beyond-mean-field effects in quantum liquids.

Abstract

We study Anderson localization of a one-dimensional quantum droplet in a speckle-like potential employing the generalized Gross-Pitaevskii equation. We compute the droplet width, density profiles, diffusion exponent and coefficient, and the localization length for both small and large droplets. Interesting classes of anomalous diffusions are obtained in transport dynamics ranging from superdiffusion to subdiffusion for a strong disorder strength. We find that above a certain critical disorder strength the droplet exhibits a transition to Anderson localization. Our results can be redibly probed with recent experiments.

Figures (6)

• Figure 1: Stationary density profiles of the droplet after $t=200$ subjected to a speckle-like potential for different values of $N$. Parameters are: $U_0 = 3$ and $\sigma =0.5$. Here we set $\mu=-0.1493$, $\mu=-0.1954$, $\mu =-0.222222$, and $\mu = -0.2222222222222$ to obtain $N=0.1$, $N=1$, $N=10$, and $N = 20$, respectively.
• Figure 2: Time evolution of quantum droplets (on a log scale) in the speckle-like potential for $U_0=2$ and $\sigma=0.5$. (a) Large droplet with $N=20$. (b) Small droplet with $N=1$.
• Figure 3: (a) Time evolution of the width, $q^2(t)-q^2(0)$, of a large droplet, $N=20$, for several values of $U_0$ with $\sigma =0.5$. (b) The same but for a small droplet, $N=1$.
• Figure 4: Critical time as a function of the strength of disorder for small (triangles) and large (circles) droplets.
• Figure 5: (a) The exponent $\alpha$ as a function of time taken from Fig. \ref{['size']} for several values of $U_0$ with $\sigma =0.5$. (b) Time dependence of the diffusive cofficient, $D_{\alpha}$, for several values of $U_0$ with $\sigma =0.5$. (c) and (d) are the same as (a) and (b) but for a small droplet.