We study the decay rate $\mathrm{\Gamma }\left(k\right)$ of density excitations of two-component Bose-Einstein condensates at zero temperature. Those excitations, where the two components oscillate in phase, include the Goldstone mode resulting from condensation. While within Bogoliubov approximation the density sector and the spin (out-of-phase) sector are independent, they couple at the three-phonon level. For a Bose-Bose mixture we find that the Belyaev decay is slightly modified due to the coupling with the gapless spin mode. At the phase separation point the decay rate changes instead from the standard $k^5$ to a $k^{5/2}$ behavior due to the parabolic nature of the spin mode. If instead a coherent coupling between the two components is present, the spin sector is gapped and, away from the ferromagnetic-like phase transition point, the decay of the density mode is not affected. On the other hand, at the transition point, when the spin fluctuations become critical, the Goldstone mode is not well defined anymore since $\mathrm{\Gamma }\left(k\right)\propto k$. As a consequence, we show that the friction induced by a moving impurity is enhanced—a feature which could be experimentally tested. Our results apply to every nonlinear 2-component quantum hydrodynamic Hamiltonian which is time-reversal invariant and possesses an $U\left(1\right)\times {\mathbb{Z}}_2$ symmetry.

• Received 7 September 2016

DOI:https://doi.org/10.1103/PhysRevB.99.064505