Collective excitations of a spherical ultradilute quantum droplet

In three dimensions, exotic new state of matter of self-bound ultradilute quantum droplets can be realized in free space, when the mean-field attraction (i.e., with mean-field energy $E_{\textrm{MF}}\propto-n^{2}$ at the density $n$) is balanced by the repulsive beyond-mean-field quantum fluctuations (i.e., $E_{\textrm{BMF}}\propto n^{2+\gamma}$). The parameter $\gamma>0$ typically takes the value $1/2$ if we consider the Lee-Huang-Yang (LHY) energy functional, but it can vary when the beyond-LHY-effect becomes important or the three-body interaction becomes dominant. Here, we theoretically investigate how collective excitations of a three-dimensional quantum droplet are affected by the parameter $\gamma$ and a weak harmonic trapping potential, both of which could be tuned in experiments. We use both the approximate approach based on a Gaussian variational ansatz and the exact numerical solution of the Bogoliubov equations resulting from the linearized time-dependent extended Gross-Pitaevskii equation. We show that one of the key features of quantum droplets, i.e., the existence of the surface modes with dispersion relation $\omega_{s}\propto k^{3/2}$ is very robust with respect to the changes either in the parameter $\gamma$ or in the harmonic trapping potential. We predict the excitation spectrum of the droplet realized by binary $^{39}$K mixtures under the typical experimental conditions, which might be readily measured in current cold-atom laboratories.