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

Abstract

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.

Figures (8)

• Figure 1: Finite-temperature phase diagram of a homogeneous coherently coupled BEC with $g n=1.0$ and $g_{\uparrow \downarrow}/g=1.1$. The phase boundary $\Omega_{\rm cr}(T)$ separates ferromagnetic and paramagnetic regions. HFB-Popov results (black solid line) are obtained from the ferromagnetic spin gap. Dotted lines are the "mean-field estimates" of the phase boundary, $\Omega_{\rm cr}(T) \approx \Omega_{\rm cr}(0) n_c(T)/n$, using either the condensate fraction $n_c(T)/n$ obtained from the ideal Bose gas approximation (blue dots) or that calculated from the HFB-Popov formalism (magenta dashed line). Figures \ref{['sg']}(a) and \ref{['sg']}(b) illustrate the typical temperature dependence of the spin gap and the magnetization $s_z/n$, respectively.
• Figure 2: Dispersion relations across the ferro–paramagnetic transition. (a)–(c): Excitation spectra at $T = 0$, where solid circles correspond to the numerical solutions of Eq. \ref{['matrix1']} with color denoting mode character $Q$, which has values $-1$ and $+1$ for pure spin and density mode, respectively. The dot-dashed and continuous gray lines are analytic results in Eq. (\ref{['gendspsn']}). As the parameter $\Omega$ decreases, the dispersion changes from (a) featuring a gapped spin branch in the paramagnetic phase, to (b) experiencing gap closure at the critical point, and then (c) reopening in the ferromagnetic phase. An avoided crossing at small $k$ indicates the presence of spin-density hybridization. (d)-(f): Excitation spectra at $T = 0.2T_c$ with similar features to those at $T = 0$ in the paramagnetic and ferromagnetic phase, but with the critical $\Omega = 0.96\Omega_{\rm cr}(0)$ accompanied by spin gap softening rather than closing in (e).
• Figure 3: Temperature dependence of the ferromagnetic spin gap for different $\Omega$. The gap decreases with temperature, signaling the ferro–paramagnetic transition. Points are HFB-Popov results; lines are guides to the eye, with values extracted from Fig. \ref{['findisp']}. (b) Magnetization $s_z/n$ versus temperature for the same parameters. Its rapid drop marks the transition, consistent with the spin-gap data. (Inset): $s_z/n$ fitted with $s_z \propto (1-T/T_{\rm ferro})^{\beta}$ for $T\leqslant T_{\rm ferro}$ (dashed lines), yielding $T_{\rm ferro}$ in agreement with (a) and $\beta \approx 0.5$, as expected from mean-field theory. Here $T_{\rm ferro}$ and $\beta$ are fitting parameters
• Figure 4: Total density profiles of a trapped coherently coupled condensate across the ferro-paramagnetic transition. (a)–(c) Ground-state density profiles at $T=0$, showing the transition from paramagnetic to ferromagnetic phase; GP results $(n_\uparrow,n_\downarrow)$ agree well with TF predictions $(n^{\rm TF}_\uparrow,n^{\rm TF}_\downarrow)$. (d)–(f) Finite-temperature HFB-Popov profiles, where thermal effects shift the critical point and produce elongated density tails. Dimensionless interaction parameters are $g_{\uparrow \downarrow}=1.1g$ and $g=0.06$. Here, length is expressed in units of the oscillator length $a_{x} = (\hbar/m\omega_{x})^{1/2}$.
• Figure 5: Excitation spectrum of a trapped coherently coupled BEC at $T=0$ from BdG calculations, compared with dynamical results (colored markers). The density and spin character of the modes is identified via the mode parameter $Q$. The softening of the spin-breathing mode marks the ferro–paramagnetic transition.