Finite-temperature phase diagram and collective modes of coherently coupled Bose mixtures

We investigate the ferromagnetic-paramagnetic phase transition in coherently (Rabi) coupled Bose-Einstein condensates at zero and finite temperatures, exploring different routes to the transition by tuning the Rabi coupling or increasing the temperature at a fixed coupling. Using the Hartree-Fock-Bogoliubov theory within the Popov approximation, we map out the finite-temperature phase diagram of a three-dimensional homogeneous condensate and identify the critical line through the softening of the spin gap. Magnetization and the spin dispersion branch reveal the progressive suppression of the ferromagnetic order with increasing temperature. In quasi-one-dimensional harmonic traps, the transition, driven by Rabi coupling, is inferred through the softening of the spin breathing mode with its minimum shifting to lower coupling values with increasing temperature. Notably, the thermally driven transition causes monotonic hardening of all the spin modes. For both coupling and temperature-driven transition, the hybridized density modes in the ferromagnetic phase acquire more density character while approaching the critical point.