Phase transition, phase separation and mode softening of a two-component Bose-Einstein condensate in an optical cavity

TL;DR

This work analyzes a two-component BEC with distinct atomic detunings inside a single-mode optical cavity driven by a transverse pump, combining perturbation theory and self-consistent GP/numerical methods. The phase transition to the superradiant state is dominated by the red-detuned component, yielding a phase diagram similar to the red-detuned single-component case, with a minimum cavity detuning threshold and a driven-dissipative instability boundary. A roton-type mode softening in the Bogoliubov spectrum signals a superfluid-to-lattice supersolid transition, accompanied by spontaneous phase separation where the two components form alternating stripes in the normal phase and distinct Bragg gratings in the superradiant phase. These results demonstrate detuning engineering as a control knob for collective quantum phenomena in cavity QED and suggest avenues for quantum simulation and optical switching applications, with concrete proposals for experimental realization using mixed species BECs such as $^{87}$Rb and $^{88}$Sr.

Abstract

We investigate the superradiant phase transition in a two-component Bose-Einstein condensate with distinct atomic detunings, confined in an optical cavity and driven by a transverse pump laser. By combining perturbation theory and numerical simulations, we demonstrate that the phase transition is dominated by the red-detuned component, resulting in a phase diagram completely different from that of a single-component case under blue-detuned condition. The system exhibits spontaneous phase separation between the two components, manifested as alternating stripe patterns in the normal phase and distinct Bragg gratings in the superradiant phase. Furthermore, the Bogoliubov excitation spectrum reveals roton-type mode softening, indicating that the phase transition also corresponds to the superfluid-to-lattice supersolid transition. Our findings provide insights into the interplay between atomic detunings and collective quantum many-body phenomena, offering potential applications in quantum simulation and optical switching technologies.

Figures (6)

• Figure 1: (a) Schematic illustration of the two-component BEC trapped inside a high-finesse optical cavity (the cavity field is drew by orange stripes) and driven by a transverse pump (the pump field is drew by green long-line), where red and blue balls represent single-component BECs with red ($\Delta_{a,1}<0$) and blue detunings ($\Delta_{a,2}>0$), respectively. The cavity field decays at a rate of $\kappa$. Here the angle between the cavity beam and pump beam is $60^\circ$, and both the pump and cavity fields are polarized orthogonal to the $x$-$y$ plane. (b) The scattering paths of momentum state can be visualized for two components. Light scattering between the pump field and the cavity mode induces Raman couplings between the zero momentum state $|\mathbf{p}\rangle=|0\rangle$ and the excited state $|\pm\hbar(\mathbf{k}_p-\mathbf{k}_c)\rangle$ at energy $E_R=\hbar\omega_R$. $\Omega_j$ and $g_j$ represent respectively the Rabi frequencies of pump laser and cavity mode, for components $j=1,2$. $\omega_{a,j}$ is the transition frequency of the two-level atoms, $\Delta_{a,j}$ is the atom-pump detuning, and $\Delta_c$ is the cavity-pump detuning. The colors in the energy level correspond to the colors in the schematic diagram. It is not assumed here that the ground states corresponding to $|0\rangle$ of components 1 and 2 are at the same level.
• Figure 2: (a) Numerical results for individual components $f_j$ and total susceptibility $\mathcal{F}$ as a function of $V_0/E_R$, where $\mathcal{F}=\sum_jf_j$. The inset shows the details for all curves. The curves of $\mathcal{F}$ and $f_1$ are very similar, both increase monotonically with $V_0/E_R$. In contrast, $f_2$ reaches its maximum at $V_0\approx 4E_R$ and subsequently decreases slowly for $V_0>4E_R$. (b) The evolution of the ratios $f_1/\mathcal{F}$, $f_2/\mathcal{F}$ and $(f_1-f_2)/\mathcal{F}$ with respect to $V_0/E_R$. The rhombus indicates a relationship $f_2=f_1-f_2$ at $V_0\approx1.4E_R$. The red and blue lines correspond to the components of $\Delta_{a,1}<0$ and $\Delta_{a,2}>0$. $N$ represents the atomic number in the corresponding system.
• Figure 3: (a) Phase diagram as a function of the tuning parameter $V_0/E_R$ and the effective cavity detuning $\tilde{\Delta}_c$, calculated from Eq (\ref{['eq:critical_effective_cavity_detuning']}). Phase boundaries are shown for the binary mixture (black), a red-detuned single-component BEC (red), and a blue-detuned single-component BEC (blue). The inset shows a magnified view near $V_0\sim0.03E_R$. (b) Phase diagrams from the mean-field approach (green line) and the self-consistent numerical approach. The unstable region arises because the ground-state energy diverges with increasing order parameter $\langle \hat{a} \rangle$. The superradiant phase is defined by the condition $\langle \hat{a} \rangle \neq 0$.
• Figure 4: The evolution of the order parameters as a function of the tuning parameter $V_0$. we decompose the order parameter $\Theta$ into a set of components $\theta_j$, where $\Theta=\sum_j\theta_j$. (a) Here, we set $\Delta_c=2\pi\times-15 \mathrm{MHz}$ and (b) we set $\Delta_c=2\pi\times-24 \mathrm{MHz}$.
• Figure 5: The density modulations of the two components are presented as follows: (a) the left column is in the normal phase, $V_0=1E_R$ and (b) the right column is in the supperradiant phase, $V_0=5E_R$. The cavity detuning is fixed at $\Delta_c=2\pi \times-17 \mathrm{MHz}$. Red and blue colors represent density modulations under red-detuned and blue-detuned component conditions, respectively.