Quantum Optics VII Jaynes-Cummings Model The Jaynes-Cummings model (sometimes abbreviated JCM) is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity (or a bosonic field).

## Two-Level Atom Interacting with a Quantum Field

### Static aspect

The Hamiltonian of Jaynes–Cummings model is

$H=H_\mathrm{A}+H_\mathrm{F}+H_\mathrm{AF}=\hbar \omega_0 \sigma^{\dagger} \sigma+\hbar \omega a^{\dagger} a+\hbar g\left(\sigma a^{\dagger}+\sigma^{\dagger} a\right)$

Only pairs of eigenstates $\ket{g,n+1} \leftrightarrow \ket{e,n}$ are coupled, and thus the Hamiltonian is block diagonal, in $2 \times 2$ blocks, making it simple to diagonalize analytically

$H = \begin{bmatrix} (n+1) \hbar \omega & \hbar g\sqrt{n+1} \\ \hbar g\sqrt{n+1} & n \hbar \omega +\hbar \omega_0 \end{bmatrix}$

The eigenvalues are

$\begin{gathered} E_1 = \left( n+\frac{1}{2} \right) \hbar \omega + \frac{1}{2} \hbar \omega_0 + \frac{1}{2} \Omega_\text{rabi}\\ E_2 = \left( n+\frac{1}{2} \right) \hbar \omega + \frac{1}{2} \hbar \omega_0 - \frac{1}{2}\Omega_\text{rabi} \end{gathered}$

in which the generalized Rabi frequency is

$\Omega_\mathrm{rabi} = \sqrt{4 \hbar^2 g^2 (n+1) + \hbar^2(\omega - \omega_0)^2}$

When resonant $\omega = \omega_0$ , the eigenstates are

$\begin{gathered} \ket{+} = \frac{1}{\sqrt{2}} \left(\ket{e,n}+\ket{g,n+1} \right) \\ \ket{-} = \frac{1}{\sqrt{2}} \left(\ket{e,n}-\ket{g,n+1} \right) \end{gathered}$

Reset zero-point energy to the midpoint of $E_1$ and $E_2$ , one could obtain

$\begin{gathered} E_g = \frac{1}{2} \hbar(\omega - \omega_0) \qquad E_e = \frac{1}{2} \hbar(\omega_0 - \omega) \\ E_+ = \frac{1}{2} \Omega_\text{rabi}\qquad E_- = -\frac{1}{2} \Omega_\text{rabi} \end{gathered}$

For large detuning $|\omega - \omega_0| \gg g \sqrt{\langle a^\dagger a\rangle}$ , the generalized Rabi frequency is approximately

$\Omega_\mathrm{rabi} \approx \hbar |\omega - \omega_0|$

Therefore, when $\omega \ll \omega_0$ (blue detune), we have $E_g \approx E_- < 0$ and $E_e \approx E_+ >0$. While when $\omega \gg \omega_0$ (red detune), we have $E_e \approx E_- < 0$ and $E_g \approx E_+ >0$. Now, to perform adiabatic population transfer, we adjust the laser gradually and slowly from blue detune to red detune. Assume the atom was initially at $\ket{g}$, and the laser is blued detuned, thus $\ket{g} \sim \ket{-}$. In the process of adjusting $\omega_0$, due to the adiabatic theorem, the atom will remain in the state $\ket{-}$. However, when the adjustment ended, we find that $\ket{-}$ sims $\ket{e}$ now! The population transfers from $\ket{g}$ to $\ket{e}$. ### Collapse and Revival

When an atom was initially at $\ket{g,\alpha}$, where $\ket{\alpha}$ is the coherent state, the phenomenon of collapse and revival occurs. ### Large Detuning

In large detuning limit, the effective Hamiltonian is

$H_\text{eff} = \hbar \chi \left[(a^\dagger a +1)\ket{e}\bra{e}-a^\dagger a\ket{g}\bra{g}\right]$

where $\chi = g^2 / \Delta$ and $\Delta = \omega_0 - \omega$.

1. Gerry, C., & Knight, P. (2004). Introductory Quantum Optics. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511791239 ↩︎