Collective excitations of a spherical ultradilute quantum droplet

Ultradilute self-bound quantum droplets in Bose-Bose mixtures at finite temperature

“…In this work, we solve the static GPE numerically via a gradient method, which improves the accuracy and efficiency from our previous work [14,39,40]. The details of our numerical method are described in the Appendix.…”

“…To study the collective excitations of the Bose droplet, we consider small fluctuation modes around the condensate wave-function φ 0 (r) [14,31,40],…”

“…By substituting the above expression into the dimensionless extended GPE Eq. ( 12) and expanding it to the linear order in u j (x) and v j (x), we obtain the celebrated Bogoliubov equations [14,31,40],…”

“…2(a), there is no collective excitations in the stable droplet state below a threshold number of particles, N < N th ≃ 94.2. All collective excitations are accumulated right above the particle-emission threshold |µ|, forming an unbounded collective excitation continuum [40]. The bounded collective excitations are only possible at N > N th , where the quadruple mode frequency ω l=2,n=0 first becomes smaller than |µ| [16], as shown by the red empty squares.…”

“…6, we report the corresponding collective excitation spectrum as a function of the temperature. At zero temperature, there are a number of discrete modes, which can be well categorized as the surface modes (ω l≥2,n=0 , red open squares and red dashed curves) and bulk modes (ω 00 , the breathing mode in blue circles; and ω l,n =0 , blue dash-dotted curves) [40]. The surface modes only propagate near the edge of the Bose droplet and have an exotic dispersion relation, ω (surface) l0 ∝ σ s l(l − 1)(l + 2)/R 3/2 , with σ s being the surface tension [12,16].…”