Quantum Optics VII Jaynes-Cummings Model
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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=HA+HF+HAF=ω0σσ+ωaa+g(σa+σa)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 g,n+1e,n\ket{g,n+1} \leftrightarrow \ket{e,n} are coupled, and thus the Hamiltonian is block diagonal, in 2×22 \times 2 blocks, making it simple to diagonalize analytically

H=[(n+1)ωgn+1gn+1nω+ω0]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

E1=(n+12)ω+12ω0+12ΩrabiE2=(n+12)ω+12ω012Ωrabi\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

Ωrabi=42g2(n+1)+2(ωω0)2\Omega_\mathrm{rabi} = \sqrt{4 \hbar^2 g^2 (n+1) + \hbar^2(\omega - \omega_0)^2}

When resonant ω=ω0\omega = \omega_0 , the eigenstates are

+=12(e,n+g,n+1)=12(e,ng,n+1)\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}

Adiabatic population transfer

Reset zero-point energy to the midpoint of E1E_1 and E2E_2 , one could obtain

Eg=12(ωω0)Ee=12(ω0ω)E+=12ΩrabiE=12Ωrabi\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 ωω0gaa|\omega - \omega_0| \gg g \sqrt{\langle a^\dagger a\rangle} , the generalized Rabi frequency is approximately

Ωrabiωω0\Omega_\mathrm{rabi} \approx \hbar |\omega - \omega_0|

Therefore, when ωω0\omega \ll \omega_0 (blue detune), we have EgE<0E_g \approx E_- < 0 and EeE+>0E_e \approx E_+ >0. While when ωω0\omega \gg \omega_0 (red detune), we have EeE<0E_e \approx E_- < 0 and EgE+>0E_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 g\ket{g}, and the laser is blued detuned, thus g\ket{g} \sim \ket{-}. In the process of adjusting ω0\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 e\ket{e} now! The population transfers from g\ket{g} to e\ket{e}.

Collapse and Revival

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

Large Detuning

In large detuning limit, the effective Hamiltonian is[1]

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

where χ=g2/Δ\chi = g^2 / \Delta and Δ=ω0ω\Delta = \omega_0 - \omega.

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