Collective excitations of a one-dimensional quantum droplet

We calculate the excitation spectrum of a one-dimensional self-bound quantum droplet in a two-component bosonic mixture described by the Gross-Pitaevskii equation (GPE) with cubic and quadratic nonlinearities. The cubic term originates from the mean-field energy of the mixture proportional to the effective coupling constant $\delta g$, whereas the quadratic nonlinearity corresponds to the attractive beyond-mean-field contribution. The droplet properties are governed by a control parameter $\gamma \propto \delta g{N}^{2/3}$, where $N$ is the particle number. For large $\gamma >0$, the droplet features the flat-top shape with the discrete part of its spectrum consisting of plane-wave Bogoliubov phonons propagating through the flat-density bulk and reflected by edges of the droplet. With decreasing $\gamma$, these modes cross into the continuum, sequentially crossing the particle-emission threshold at specific critical values. A notable exception is the breathing mode, which we find to be always bound. The balance point $\gamma =0$ provides implementation of a system governed by the GPE with an unusual quadratic nonlinearity. This case is characterized by the ratio of the breathing-mode frequency to the particle-emission threshold equal to 0.8904. As $\gamma$ tends to $-\infty$, this ratio tends to 1 and the droplet transforms into the soliton solution of the integrable cubic GPE.

Figure 1

(a) The droplet wave functions for different values of the control parameter $\gamma =-\infty$, 0, 2, and 6, ranging from the soliton to flat-top shapes. For better comparison, we normalize the corresponding density profiles to 1 and rescale $x$ so that they have the same rms widths. (b) The ratio of discrete Bogoliubov frequencies ${\omega }_{\eta }$ to the particle-emission threshold $-\mu$ as a function of $\gamma =\mathrm{sgn}\left(\delta g\right){\tilde{N}}^{2/3}$. Red crosses indicate branching points where the discrete modes cross the particle-emission threshold and enter the continuum of excitations. The dashed and dotted curves represent asymptotic approximations (18) and (19), correspondingly.