## 3 Fluorophore Coupled to a Resonant Cavity in the Strong-Coupling Regime  
### Strong-Coupling with a Perfect Lossless Cavity  
Strong-coupling naturally appears if we consider a two-level system (atom, fluorophore) resonantly coupled to an ideal, lossless and isolated, cavity mode. The Hamiltonian describing such a system is well-known. It is the so-called Jaynes-Cumming (J-C) Hamiltonian. It is obtained after performing various approximations (such as dipolar coupling and rotating wave approximation). In an empty cavity, for an atom dipole moment \(\mathbf{d}=d\vec{u}_{d}\) and a perfect polarization and position matching with the mode, the dipolar interaction is equal to the product \(-dE_{0}\) between the dipole moment and the effective cavity electric field (in the absence of photon). The electric field of the empty cavity is \(E_{0}=\sqrt{\hbar\omega_{c}/(2e_{0}V_{c})}\), where \(V_{c}\) is the cavity volume and \(\omega_{c}\) is the cavity mode frequency. The coupling strength is characterized by \(g=dE_{0}/\hbar=\Omega_{0}/2\). \[\hbar g=d\sqrt{\frac{\hbar\omega_{c}}{2e_{0}V_{c}}} \tag{1}\]  
The J-C Hamiltonian [12] then reads  
\[\hat{H}_{\text{JC}}=\hbar\omega_{0}\hat{\sigma}^{\dagger}\phi+\hbar\omega_{c} \hat{a}^{\dagger}\hat{a}+\hbar g\Big{(}\hat{\sigma}\hat{a}^{\dagger}+\hat{ \sigma}^{\dagger}\hat{a}\Big{)} \tag{2}\]  
Figure 1: Scheme of _Quantum Filigran_ experiment. The blue arrow represents the light emitted by the laser, resonant with the peak of absorption of the fluorophore (440 nm). The green arrow represents the emitted photon. Note that in the strong coupling regime, photons are emitted at two possible wavelengths (530 \(\pm\) 25 nm), a manifestation of frequency splitting. The parameters of this Hamiltonian are the atomic transition frequency \(\omega_{0}\), the cavity mode frequency \(\omega_{c}\), and the coupling constant \(g\). When a single dipole (atom/fluorophore) is considered, the state of the system is  
\[|\psi(t)\rangle=\alpha(t)|e,0_{c}\rangle+\beta(t)|g,1_{c}\rangle \tag{3}\]  
where the state \(|e,0_{c}\rangle\) stands for dipole in the excited state and no photon in the cavity and \(|g,1_{c}\rangle\) stands for dipole in the ground state and one photon in the cavity. The J-C Hamiltonian then reduces to a \(2\times 2\) matrix  
\[\hat{H}_{\mathrm{JC}}=\hbar \begin{bmatrix}\omega_{0}&g\\ g&\omega_{c}\end{bmatrix}  \tag{4}\]  
It is worth noting that \(\hat{H}_{\mathrm{JC}}\) is Hermitian. In what follows, we shall consider a nearly resonant cavity (\(\omega_{c}\approx\omega_{0}\) so that the detuning \(\delta=\omega_{c}-\omega_{0}\) is small). The new eigenenergies of the coupled system are [13]  
\[E_{\pm}=\hbar\omega_{c}\pm\frac{1}{2}\hbar\sqrt{\delta^{2}+\Omega_{0}^{2}} \tag{5}\]  
where we defined the so-called vacuum Rabi frequency:  
\[\Omega_{0}\equiv\sqrt{4g^{2}} \tag{6}\]  
At resonance (\(\delta=0\)), the energy difference between the two modes is minimal and equal to \(\hbar\Omega_{0}\). The energy eigenstates are given by  
\[ \begin{split}|+\rangle&=\sin\theta\;|e,0_{c}\rangle+ \cos\theta\;|g,1_{c}\rangle\\ |-\rangle&=\cos\theta\;|e,0_{c}\rangle-\sin\theta\;|g,1_{c} \rangle\end{split}  \tag{7}\]  
The parameter \(\theta\) defines the entanglement strength [3] between the dipole and the cavity mode (\(\tan 2\theta=\Omega_{0}/\delta\)). At resonance, entanglement is maximal (\(\theta=\pi/4\)) and the dressed states have matter and field components of equal weight. The separation between two distinct matter-radiation subsystems is then impossible. Far from resonance (\(\theta\approx 0\) and \(\theta\approx\pi/2\)), these states exhibit dominant matter or electromagnetic components. These polaritonic states inherit dispersion relations from their electromagnetic component. Let us note that the pair of eigenenergies \(E_{\pm}\) and eigenstates \(|\pm\rangle\) are only the lowest of an infinite ladder of pairs.