The condensate fraction x(T) and the transition temperature Tc of a charged Bose gas at high density are obtained in the scheme of the density-phase operator approach by introducing a trial Hamiltonian based on the thermal-Hartree approximation.
A phase transition is conventionally described by a single critical temperature T_c. We analyze Bose–Einstein condensation of a quasi-one-dimensional interacting gas and find the transition presents itself as a cluster of separately locatable sub-events, quantified by a five-temperature constellation: condensate half-height T_R, peak-interaction handover T_I, cloud-size minimum T_B, momentum-width minimum T_D, and overlap peak T_S. Across 17 interaction strengths and three thermal-cloud models (Boltzmann, Bose, Hartree–Fock), we find: (i) the handover offset ΔT_I(g) is a monotonic crossover—hump at weak coupling (+0.15 T_c at g = 30), zero at g* ≈ 39 (Bose) or 44 (Boltzmann), deepening to −0.10 T_c—with a statistics double valley at the crossover; (ii) the constellation expands (separation 0.21 → 0.53 T_c) and its internal order reverses (T_I crosses T_R near g ≈ 32), these two markers bracketing the crossover region; (iii) the two cloud-sensitive dimensions carry different signatures (T_I: zero-crossing; T_S: deepening −0.13 → −0.18 T_c). The density profile remains center-peaked at all temperatures—an observed central dip would falsify the model family. Ground states use multi-start Krotov minimization (μ(g) strictly monotone, dominant-peak fractions > 0.994). All features map onto laboratory parameters (N ≈ 200–1600 for ⁸⁷Rb) and are falsifiable predictions for Feshbach experiments with simultaneous condensate-fraction and in-situ density measurements.
We investigate condensation and finite-temperature magnetic phase transitions in a coherently (Rabi) coupled Bose gas confined in a three-dimensional harmonic trap. For the noninteracting system, we derive analytical expressions for the critical temperature and condensate fraction using a semiclassical description of the single-particle spectrum and density of states. At fixed particle number, coherent coupling enhances the critical temperature relative to the uncoupled system, and this enhancement decreases with increasing particle number N. The finite-size correction, in contrast, lowers the transition temperature, with its effect diminishing for larger N. We then incorporate repulsive interactions within the Hartree--Fock--Bogoliubov--Popov framework to investigate the finite-temperature phase diagram in the temperature--Rabi coupling plane. The system undergoes successive transitions from a ferromagnetic to a paramagnetic condensate and, at higher temperature, to a thermal gas. The transition boundaries obtained from the vanishing of the condensate fraction show excellent agreement with the analytical Hartree-Fock predictions. We find that the interactions substantially suppress the variation in the condensation temperature with an increase in the coherent coupling.