Unbound-to-bound transition of two-atom polaritons in an optical cavity

We consider two spin-1/2 fermions inside an optical cavity which supports a single-mode quantized light field. We demonstrate that the atom-light coupling (ALC) gives rise to the two-atom polariton states, where the two atoms are highly entangled with cavity photons. We focus on the case where the cavity light is on resonant with the bare atomic transition. We show that in the absence of interatomic interaction, the polariton is unbound, has finite center-of-mass momentum, and contains no atomic spin-singlet fraction in its ground state. When strong attractive interatomic contact interaction is present, a stable bound polariton state exists when the ALC strength is below a critical value. When the ALC strength exceeds the critical value, a first-order transition is observed and the bound polariton becomes unbound. The first-order transition is characterized by abrupt changes of various quantities associated with the polariton and should be readily detectable.

Figure 1

(a) Energy $E_{\mathrm{UP}}$, (b) COM momentum $q_{\mathrm{UP}}$, and (c) atom-photon entanglement entropy of the ground state of an unbound two-atom polariton (UP) as functions of the normalized ALC strength $\sqrt{N}{g}_0/E_c$ ($E_c\equiv k_c^2/2$) for $\delta =0$, with excitation number $N=1$ (blue solid line), $N=2$ (red dashed line), and $N=3$ (greed dash-dotted line), respectively. For comparison, the momentum of the ground state of the single-atom polariton is also plotted in (b) as a black dotted line.

Figure 2

(a) Energy $E_{\mathrm{BP}}-E_0$, where $E_0$ is the energy of the system in the limit $g_0=0$, (b) COM momentum $q_{\mathrm{BP}}$, and (c) atom-photon entanglement entropy of the bound two-atom polariton (BP) as functions of the normalized ALC strength $\sqrt{N}{g}_0/E_c$ for $\delta =0$ and ${\left(k_c{a}_s\right)}^{-1}=1$, with excitation number $N=1$ (blue solid line), $N=2$ (red dashed line), and $N=3$ (green dotted line).

Figure 3

(a) ${\epsilon }_b$ (the energy of the BP with respect to the continuum threshold) as a function of the normalized ALC strength $\sqrt{N}{g}_0/E_c$ for $\delta =0$ and ${\left(k_c{a}_s\right)}^{-1}=1$. (b) Ground-state phase diagram for the bound (BP) to unbound (UP) transition of the two-atom polaritons in the $g_0\text{−}{\left(a_s\right)}^{-1}$ plane. The photon number $n_{\mathrm{ph}}$ and the singlet population in the ground state for $\delta =0$ and ${\left(k_c{a}_s\right)}^{-1}=1$ as functions of $\sqrt{N}{g}_0/E_c$ are plotted in (c) and (d), respectively. Both of these quantities exhibit a discontinuous jump across the transition point. Different lines correspond to different excitation numbers $N$ as indicated in the figure. The photon number $n_{\mathrm{ph}}$ plotted in (c) is downshifted by $N-1$.