Microscopic derivation of the extended Gross-Pitaevskii equation for quantum droplets in binary Bose mixtures

Abstract

An ultradilute quantum droplet is a self-bound liquid-like state recently observed in ultracold Bose-Einstein condensates. In all previous theoretical studies, it is described by a phenomenological low-energy effective theory, termed as the extended Gross\textendash Pitaevskii equation. Here, we microscopically derive the Gross\textendash Pitaevskii equation for the condensate and also for a pairing field in an inhomogeneous quantum droplet realized by Bose-Bose mixtures with attractive inter-species interaction. We show that the inclusion of the pairing field is essential, in order to have a consistent description of the droplet state. We clarify that, the extended Gross\textendash Pitaevskii equation used earlier should be understood as the equation of motion for the pairing field, rather than the condensate. The fluctuations of the pairing field give rise to low-energy collective excitations of the droplet. We also present the Bogoliubov equations for gapless phonon modes and gapped modes due to pairing in real space, which characterizes single-particle-like excitations of the droplet. The equations of motion derived in this work for the condensate and the pairing field serve an ideal starting point to understand the structure and collective excitations of non-uniform ultradilute quantum droplets in on-going cold-atom experiments.

6 pages, no figures; title changed; to appear in Physical Review A as a Regular Article

References in corpus (12)

• Quantum liquid droplets in a mixture of Bose-Einstein condensates
• Self-bound droplets of a dilute magnetic quantum liquid
• Quantum-fluctuation-driven crossover from a dilute Bose-Einstein condensate to a macro-droplet in a dipolar quantum fluid
• Collisions of self-bound quantum droplets
• Quantum Fluctuations in the Superfluid State of the BCS-BEC Crossover
• Equation of state of a superfluid Fermi gas in the BCS-BEC crossover
• Two-dimensional solitons and quantum droplets supported by competing self- and cross-interactions in spin-orbit-coupled condensates
• Beyond Lee-Huang-Yang description of self-bound Bose mixtures
• Comparative study of strong coupling theories of a trapped Fermi gas at unitarity
• Mean field thermodynamics of a spin-polarized spherically trapped Fermi gas at unitarity
• Microscopic pairing theory of a binary Bose mixture with interspecies attractions: bosonic BEC-BCS crossover and ultradilute low-dimensional quantum droplets
• Collective oscillations of a confined Bose gas at finite temperature in the random-phase approximation

Cited by in corpus (9)

• Few-body Bose gases in low dimensions -- a laboratory for quantum dynamics
• Quadratic fractional solitons
• Vortex-ring quantum droplets in a radially-periodic potential
• Ultradilute self-bound quantum droplets in Bose-Bose mixtures at finite temperature
• The Effective Single-mode model of a Binary Boson Mixture in the Quantum Droplet Region
• Two-dimensional quantum droplets in binary quadrupolar condensates
• Functional renormalization for repulsive Bose-Bose mixtures at zero temperature
• Breakdown of the single-mode description of ultradilute quantum droplets in binary Bose mixtures: A perspective from a microscopic bosonic pairing theory
• Rotational dynamics induced by low energy binary collisions of quantum droplets