Metastability, collective excitations, and traveling states of one-dimensional spin-orbit-coupled quantum droplets

We study the static and traveling droplet phases of spin-orbit coupled, one-dimensional Bose-Bose mixtures with a Lee-Huang-Yang correction. For the static droplets, employing a variational analysis for an infinitely extended mixture, we show that the system can feature two ground-state droplet phases: a stripe droplet phase up to a critical Rabi coupling strength, at which it undergoes a first-order transition to a zero-momentum droplet phase. The variational results for the ground-state properties are shown to be in good agreement with numerical solutions for droplets in the thermodynamic limit as well as finite-size droplets in the "flat-top regime". For finite-size droplets, the stripe phase can feature multiple stationary states with different numbers of peaks in their stripe density patterns. The calculation of the excitation spectrum for these stationary states confirms their stability. To study the moving droplets, we investigate the ground-state phases of the system in the droplet's comoving frame. We show that in the comoving frame, droplets acquire a finite longitudinal magnetization. As a consequence, the phase diagram features two distinct phases: a traveling stripe droplet and a plane-wave droplet. The real-time evolution of these droplets in the space-fixed frame, complemented by the Bogoliubov analysis in the comoving frame, illustrates the stability of these traveling droplet phases.