Vacuum Rabi oscillation

A vacuum Rabi oscillation is a damped oscillation of an initially excited atom coupled to an electromagnetic resonator or cavity in which the atom alternately emits photon(s) into a single-mode electromagnetic cavity and reabsorbs them. The atom interacts with a single-mode field confined to a limited volume V in an optical cavity. Spontaneous emission is a consequence of coupling between the atom and the vacuum fluctuations of the cavity field.

Mathematical treatment
A mathematical description of vacuum Rabi oscillation begins with the Jaynes–Cummings model, which describes the interaction between a single mode of a quantized field and a two level system inside an optical cavity. The Hamiltonian for this model in the rotating wave approximation is


 * $$\hat{H}_{\text{JC}} = \hbar \omega \hat{a}^{\dagger}\hat{a}

+\hbar \omega_0 \frac{\hat{\sigma}_z}{2} +\hbar g \left(\hat{a}\hat{\sigma}_+ +\hat{a}^{\dagger}\hat{\sigma}_-\right)$$

where $$\hat{\sigma_z}$$ is the Pauli z spin operator for the two eigenstates $$|e \rangle$$ and $$|g\rangle$$ of the isolated two level system separated in energy by $$\hbar \omega_0$$; $$\hat{\sigma}_+ = |e \rangle \langle g |$$ and $$\hat{\sigma}_- = |g \rangle \langle e |$$ are the raising and lowering operators of the two level system; $$\hat{a}^{\dagger}$$ and $$\hat{a}$$ are the creation and annihilation operators for photons of energy $$\hbar \omega$$ in the cavity mode; and


 * $$g=\frac{\mathbf{d}\cdot\hat{\mathcal{E}}}{\hbar}\sqrt{\frac{\hbar \omega}{2 \epsilon_0 V}}$$

is the strength of the coupling between the dipole moment $$\mathbf{d}$$ of the two level system and the cavity mode with volume $$V$$ and electric field polarized along $$\hat{\mathcal{E}}$$. The energy eigenvalues and eigenstates for this model are
 * $$E_{\pm}(n) = \hbar\omega \left(n+\frac{1}{2}\right) \pm \frac{\hbar}{2} \sqrt{4g^2 (n+1) + \delta^2}=\hbar \omega_n^\pm $$


 * $$|n,+\rangle= \cos \left(\theta_n\right)|g,n+1\rangle+\sin \left(\theta_n\right)|e,n\rangle$$


 * $$|n,-\rangle= \sin \left(\theta_n\right)|g,n+1\rangle-\cos \left(\theta_n\right)|e,n\rangle$$

where $$\delta = \omega_0 - \omega$$ is the detuning, and the angle $$\theta_n$$ is defined as


 * $$\theta_n = \tan^{-1}\left(\frac{g \sqrt{n+1}}{\delta}\right).$$

Given the eigenstates of the system, the time evolution operator can be written down in the form


 * $$\begin{align}

e^{-i\hat{H}_{\text{JC}}t/\hbar} & = \sum_{|n,\pm \rangle} \sum_{|n',\pm \rangle} |n,\pm \rangle \langle n,\pm| e^{-i\hat{H}_{\text{JC}}t/\hbar} |n',\pm \rangle \langle n',\pm|\\ &= ~e^{i(\omega-\frac{\omega_0}{2})t} |g,0\rangle \langle g,0| \\ & + \sum_{n=0}^\infty{e^{-i\omega_n^+ t} ( \cos{\theta_n}|g,n+1\rangle+\sin{\theta_n}|e,n\rangle) ( \cos{\theta_n}\langle g,n+1|+\sin{\theta_n}\langle e,n|)} \\ & + \sum_{n=0}^\infty{e^{-i\omega_n^- t} (-\sin{\theta_n}|g,n+1\rangle+\cos{\theta_n}|e,n\rangle) (-\sin{\theta_n}\langle g,n+1|+\cos{\theta_n}\langle e,n|)} \\ \end{align}.$$

If the system starts in the state $$|g,n+1\rangle$$, where the atom is in the ground state of the two level system and there are $$n+1$$ photons in the cavity mode, the application of the time evolution operator yields


 * $$\begin{align}

e^{-i\hat{H}_{\text{JC}}t/\hbar} |g,n+1\rangle &= (e^{-i\omega_n^+ t}(\cos^2{(\theta_n)}|g,n+1\rangle+\sin{\theta_n}\cos{\theta_n}|e,n\rangle) + e^{-i\omega_n^- t} (-\sin^2{(\theta_n)}|g,n+1\rangle-\sin{\theta_n}\cos{\theta_n}|e,n\rangle)\\ &= (e^{-i\omega_n^+ t}+e^{-i\omega_n^- t}) \cos{(2 \theta_n)}|g,n+1\rangle + (e^{-i\omega_n^+ t}-e^{-i\omega_n^- t}) \sin{(2 \theta_n)}|e,n\rangle\\ &= e^{-i \omega_c(n+\frac{1}{2})}\Biggr[\cos{\biggr(\frac{t}{2}\sqrt{4g^2(n+1)+\delta^2} \biggr)} \biggr[\frac{\delta^2-4g^2(n+1)}{\delta^2+4g^2(n+1)}\biggr]|g,n+1\rangle + \sin{\biggr(\frac{t}{2}\sqrt{4g^2(n+1)+\delta^2}\biggr)}\biggr[\frac{8 \delta^2 g^2(n+1)}{\delta^2+4g^2(n+1)}\biggr]|e,n\rangle\Biggr] \end{align}.$$

The probability that the two level system is in the excited state $$|e,n\rangle$$ as a function of time $$t$$ is then
 * $$ \begin{align}

P_e(t) & =|\langle e,n|e^{-i\hat{H}_{\text{JC}}t/\hbar} |g,n+1\rangle |^2\\ &= \sin^2{\biggr(\frac{t}{2}\sqrt{4g^2(n+1)+\delta^2}\biggr)}\biggr[\frac{8 \delta^2 g^2(n+1)}{\delta^2+4g^2(n+1)}\biggr]\\ &= \frac{4g^2(n+1)}{\Omega_n^2} \sin^2{\bigr(\frac{\Omega_n t}{2}\bigr)} \end{align}$$

where $$\Omega_n=\sqrt{4g^2(n+1)+\delta^2}$$ is identified as the Rabi frequency. For the case that there is no electric field in the cavity, that is, the photon number $$n$$ is zero, the Rabi frequency becomes $$\Omega_0=\sqrt{4g^2+\delta^2}$$. Then, the probability that the two level system goes from its ground state to its excited state as a function of time $$t$$ is
 * $$ P_e(t) =\frac{4g^2}{\Omega_0^2} \sin^2{\bigr(\frac{\Omega_0 t}{2}\bigr).}$$

For a cavity that admits a single mode perfectly resonant with the energy difference between the two energy levels, the detuning $$\delta$$ vanishes, and $$P_e(t)$$ becomes a squared sinusoid with unit amplitude and period $$\frac{2 \pi}{g}.$$

Generalization to N atoms
The situation in which $$N$$ two level systems are present in a single-mode cavity is described by the Tavis–Cummings model , which has Hamiltonian


 * $$\hat{H}_{\text{JC}} = \hbar \omega \hat{a}^{\dagger}\hat{a}

+\sum_{j=1}^N{\hbar \omega_0 \frac{\hat{\sigma}_j^z}{2} +\hbar g_j \left(\hat{a}\hat{\sigma}_j^+ +\hat{a}^{\dagger}\hat{\sigma}_j^-\right)}. $$

Under the assumption that all two level systems have equal individual coupling strength $$g$$ to the field, the ensemble as a whole will have enhanced coupling strength $$g_N=g\sqrt{N}$$. As a result, the vacuum Rabi splitting is correspondingly enhanced by a factor of $$\sqrt{N}$$.