Optical Bloch Equations

JC to the semi classical limit§

From the JC model, and then by taking classical limit for light ( laser ) with $\hat{b}=\alpha e^{-i\omega_k t}$ where we recognise the Rabi frequency at resonant, so $\Omega_R=2|g|\sqrt{n}=|2g\alpha|$. Where from now onward, since the phase of $\alpha g$ is constant and can be absorbed into the universal phase of the state of the system which have no physical significant in the current model. We assume $g$ and $\alpha$ to be real.

$$\begin{align*} \hat{H}_\text{JC} &= \hbar\omega_{eg}\hat{\sigma}_e+\hbar\omega\hat{n} + \hbar g\hat{A}+\hbar g^*\hat{A}^\dagger \\ &= \hbar\omega_{eg}\hat{\sigma}_e+\hbar\omega|\alpha|^2 + \frac{\hbar\Omega_R}{2} \left(\hat{\sigma}^+ e^{-i\omega_kt} +\hat{\sigma}^- e^{i\omega_kt} \right) \\ \end{align*}$$

Where since $\alpha$ is now a constant, drop it and let's work in the interaction frame ( explicit laser energy )

$$\hat{H}=\hbar\Delta\hat{\sigma}_e + \frac{\hbar\Omega_R}{2}\left(\hat{\sigma}^+ + \hat{\sigma}^-\right)$$

Where $\Delta = \omega_{eg} - \omega_L$

The Hamiltonian Matrix§

Since the photon number is fixed, $\ket{0}=\ket{e}$, $\ket{1}=\ket{g}$, this trivially gives

$$H=\hbar\begin{bmatrix} \Delta & \Omega_R/2\\ \Omega_R/2 & 0 \end{bmatrix}$$

Master equation§

Notice that the JC model is a very simplified model, it doesn't even contain a dephasing term which we can now add it back to the master equation as a dissipation. The justification is simply because in order to gain a dissipation, all it needs is for the dephasing operator coupled to the infinite mode of field which is very reasonable. Thus, the master equation reads

$$\dot{\rho} = -\frac{i}{\hbar}\left[H,\rho\right]+\gamma\mathcal{D}_{\sigma^-}(\rho)+\gamma_\phi\mathcal{D}_{\sigma_z}(\rho)$$

Commutator term§

$$\begin{align*} \text{first term}&=-\frac{i}{\hbar}\left[H,\rho\right] \\ &=-i\left[ \begin{bmatrix} \Delta & \Omega/2\\ \Omega/2 & 0 \end{bmatrix}, \begin{bmatrix} \rho_{ee} & \rho_{eg}\\ \rho_{ge} & \rho_{gg} \end{bmatrix}\right]\\ &=-i\left\{ \begin{bmatrix} \Delta & \Omega/2\\ \Omega/2 & 0 \end{bmatrix} \begin{bmatrix} \rho_{ee} & \rho_{eg}\\ \rho_{ge} & \rho_{gg} \end{bmatrix} - \begin{bmatrix} \rho_{ee} & \rho_{eg}\\ \rho_{ge} & \rho_{gg} \end{bmatrix} \begin{bmatrix} \Delta & \Omega/2\\ \Omega/2 & 0 \end{bmatrix}\right\} \\ &=-i \begin{bmatrix} \frac{\Omega}{2}\left(\rho_{ge}-\rho_{eg}\right) & \Delta\rho_{eg}+\frac{\Omega}{2}\left(\rho_{gg}-\rho_{ee}\right)\\ -\Delta \rho_{ge} + \frac{\Omega}{2}\left(\rho_{ee} - \rho_{gg}\right) & \frac{\Omega}{2}\left(\rho_{eg} - \rho_{ge}\right) \end{bmatrix} \end{align*}$$

Spontaneous emission term§

$$\begin{align*} \text{second term} &= \gamma\mathcal{D}_{\sigma^-}(\rho)\\ &=\gamma\left(\sigma^-\rho\sigma^+ - \frac{1}{2}\left\{\sigma^+\sigma^-,\rho\right\}\right)\\ &=\gamma\left( \begin{bmatrix} 0 & 0 \\ 0 & \rho_{ee} \end{bmatrix} - \begin{bmatrix} \rho_{ee} & 1/2\rho_{eg} \\ 1/2\rho_{ge} & 0 \end{bmatrix} \right) \\ &= -\gamma\begin{bmatrix} \rho_{ee} & 1/2\rho_{eg} \\ 1/2\rho_{ge} & -\rho_{ee} \end{bmatrix} \end{align*}$$

Dephasing term§

$$\begin{align*} \text{third term} &= \gamma_\phi\mathcal{D}_{\sigma_z}(\rho)\\ &=\gamma_\phi\left(\sigma^z\rho\sigma^z - \frac{1}{2}\left\{\sigma_z^2,\rho\right\}\right)\\ &=\gamma_\phi\left( \begin{bmatrix} \rho_{ee} & -\rho_{eg} \\ -\rho_{ge} & \rho_{gg} \end{bmatrix} - \begin{bmatrix} \rho_{ee} & \rho_{eg} \\ \rho_{ge} & \rho_{gg} \end{bmatrix} \right) \\ &= -2\gamma_\phi\begin{bmatrix} 0 & \rho_{eg} \\ \rho_{ge} & 0 \end{bmatrix} \end{align*}$$

All together§

$$\begin{cases} \dot{\rho}_{ee} &= -\cfrac{i\Omega}{2}\left(\rho_{ge}-\rho_{eg}\right) - \gamma \rho_{ee} \\ \dot{\rho}_{eg} &= -\left(\cfrac{\gamma}{2} + 2\gamma_\phi + i\Delta \right)\rho_{eg} + \cfrac{i\Omega}{2}\left(\rho_{ee}-\rho_{gg}\right) \\ \dot{\rho}_{ge} &= -\left(\cfrac{\gamma}{2} + 2\gamma_\phi - i\Delta \right)\rho_{ge} - \cfrac{i\Omega}{2}\left(\rho_{ee}-\rho_{gg}\right) \\ \dot{\rho}_{gg} &= -\cfrac{i\Omega}{2}\left(\rho_{eg}-\rho_{ge}\right) + \gamma\rho_{ee} \end{cases}$$