## 1 Introduction  
As demonstrated in a seminal paper by [PERSON] [1], ultracold quantum gases can exist in the form of self-bound droplets that do not expand even in the absence of any confinement. This liquid-like behavior, which originates from the interplay of attractive mean-field interactions and the repulsive effect of quantum fluctuations [2; 3; 4], was successfully observed in a number of experiments with dipolar condensates [5; 6; 7; 8; 9], homonuclear mixtures of \({}^{39}\)K [10; 11; 12; 13], and recently in a heteronuclear mixture of \({}^{41}\)K and \({}^{87}\)Rb [14; 15]. These findings have triggered an intense research activity on droplets properties (see, e.g., the recent reviews in Refs. [16; 17]) and on the so-called Lee-Huang-Yang fluid [18; 19; 20], and have motivated further studies in low-dimensional systems [21; 22] and also beyond [PERSON]'s theory [23; 24; 25; 26; 27]. Besides being self bound, one of the most peculiar characteristics of these quantum objects is the fact that they can be self-evaporating. Namely, in certain regimes the droplet cannot sustain any collective mode, such that any initial excitation is completely dissipated (that is, evaporated) until the system relaxes into a droplet with a lower number of atoms. This remarkable feature, originally predicted in Ref. [1] for bosonic mixtures, has been so far elusive to the experimental detection. Indeed, the first pioneering experiments performed with bosonic mixtures [10; 11] have shown that the homonuclear mixtures of \({}^{39}\)K suffer from strong three-body losses that continuously drive the system out of equilibrium, and eventually lead to the depletion of the droplet. This behavior was then confirmed by the theoretical analysis reported in Ref. [28], where the complex dynamics taking place during the droplet formation is thoroughly described. There, it was shown that the evolution of the system is indeed dominated by the presence of three-body losses and by a continuous release of atoms to restore the proper population ratio, whereas the self-evaporation mechanism plays only a negligible role, if any. In this respect, the recent experiment in Ref. [14] represents a promising setup in which the self-evaporation mechanism could be properly investigated. As a matter of fact, for a \({}^{41}\)K-\({}^{87}\)Rb mixture the regime of parameters for which self-bound droplets form is such that three-body losses are expected to be significantly suppressed. Indeed, in such a mixture, droplets form at lower densities \(n\sim(\delta g/g)^{2}a^{-3}\), and this allows for a larger ratio \(\tau_{\rm life}/\tau\sim n^{-1}\), with \(\tau_{\rm life}\) and \(\tau\) being respectively the lifetime (limited by three-body losses) and the characteristic time scale of the droplet dynamics [14]. Thus, the fact that the \({}^{41}\)K-\({}^{87}\)Rb mixture is characterized by larger scattering lengths with respect to the \({}^{39}\)K mixture employed in Refs. [10; 11] allows for lower densities and, therefore, longer lifetimes. Actually, in that experiment no appreciable effect of three-body losses was observed on the timescale of several tens of milliseconds. Motivated by the previous discussion, in this paper we analyze the self-evaporation mechanism and the role of three-body losses on the collective excitations of a \({}^{41}\)K-\({}^{87}\)Rb quantum droplet. For the sake of conceptual clarity, and in view of the fact that quantum droplets at equilibrium are spherically symmetric objects, we shall restrict the analysis to the case of a spherically symmetric system. Although this assumption limits the study to the monopole collective mode, it is however sufficient to draw the general behavior of the system and to discuss the role of three-body losses. The paper is organized as follows. In Section 2 we review the general formalism for describing quantum droplets in heteronuclear bosonic mixtures. Then, in Section 3 we discuss the preparation of the initial state, namely a droplet compressed by a harmonic trapping.