The Effective Single-mode model of a Binary Boson Mixture in the Quantum Droplet Region

TL;DR

This paper develops an effective single-mode model for binary quantum droplets, accurately capturing phonon energies and stability by integrating out higher excitations within a path integral framework, improving upon previous Bogoliubov theory.

Contribution

It introduces a new effective single-mode model derived from the path integral formalism that correctly describes phonon energies and stability in binary quantum droplets.

Findings

The model accurately predicts phonon energies without imaginary parts.

It extends the applicability of the LHY gas beyond the droplet region.

The extended GP equation derived is purely positive in LHY energy.

Abstract

In a binary quantum droplet, the interspecies attraction dominates over the intraspecies repulsions and the mean-field energy is unstable. The mechanical stability is restored by the repulsive Lee-Huang-Yang (LHY) energy [1]. In the Bogoliubov theory of the binary quantum droplet, there are two branches of gapless excitations. The lower branch describes the phonon excitation and its energy is imaginary in the long-wavelength limit, implying dynamical instability. Recently it is found that the phonon energy is renormalized by higher-order quantum fluctuations and the dynamical instability is removed [2]. In this work, we study a binary quantum droplet in the path integral formalism to construct an effective model with the correct phonon energy. By integrating out the upper excitation branch, we obtain an effective single-mode model describing density fluctuations, and derive the extended Gross-Pitaevskii equation. In this approach, the LHY energy in the extended-GP equation is purely positive without any assumption of neglecting the imaginary part. This effective single-mode model can be also used outside and close to the quantum droplet region such as in the LHY gas.