Jaynes-Cummings Model

A fully quantized model of a two-level atom interacting with light.

Notes

1. Hamiltonian

$$\begin{array}{rl}\hat{H}& ={\hat{H}}_{atom}+{\hat{H}}_{field}+{\hat{H}}_{int}\\ & =\frac{1}{2}\hslash {\omega }_a\widehat{{\sigma }_z}+\hslash {\omega }_f{\hat{a}}^{†}\hat{a}+\hslash \lambda (\hat{a}+{\hat{a}}^{†})({\hat{\sigma }}^{+}+{\hat{\sigma }}^{-})\\ & =\frac{1}{2}\hslash {\omega }_a\widehat{{\sigma }_z}+\hslash {\omega }_f{\hat{a}}^{†}\hat{a}+\hslash \lambda (\hat{a}{\hat{\sigma }}^{+}+{\hat{a}}^{†}{\hat{\sigma }}^{-}+\hat{a}{\hat{\sigma }}^{-}+{\hat{a}}^{†}{\hat{\sigma }}^{+})\end{array}\text{\hspace{1em}(1)}$$

For how to write this Hamiltonian, see Quantization of electromagnetic field and Hamiltonian Function and n-level quantum systems. With the RWA, we could neglect $\hat{a}{\hat{\sigma }}^{-}$ & ${\hat{a}}^{†}{\hat{\sigma }}^{+}$. 2. Dynamics Rewrite the Hamiltonian into $H_1+H_2$, where $[H_1,H_2]=0$. A method here is using conserved quantity. The particle number is conserved, and the excitation number $\widehat{P_e}=|e⟩⟨e|+|g⟩⟨g|$ and $\widehat{N_e}=|e⟩⟨e|+{\hat{a}}^{†}\hat{a}$ We could represent H by this two conserved operator.

$$\begin{array}{rl}\hat{H}& =\widehat{H_1}+\widehat{H_2}\\ & =\frac{1}{2}\hslash {\omega }_a(|e⟩⟨e|-|g⟩⟨g|)+\hslash {\omega }_f{\hat{a}}^{†}\hat{a}+\hslash \lambda ({\hat{a}}^{†}{\hat{\sigma }}^{-}+\hat{a}{\hat{\sigma }}^{+})\\ & =\hslash {\omega }_f(\widehat{N_e}-\widehat{P_e})+\frac{1}{2}\hslash ({\omega }_a-{\omega }_f){\sigma }_z+\hslash \lambda ({\hat{a}}^{†}{\hat{\sigma }}^{-}+\hat{a}{\hat{\sigma }}^{+})\end{array}\text{\hspace{1em}(2)}$$

When ${\omega }_a={\omega }_f$ , the only thing important is $\hslash \lambda ({\hat{a}}^{†}{\hat{\sigma }}^{-}+\hat{a}{\hat{\sigma }}^{+})$ We set the origin state is $|e⟩|n⟩$ denote as $|i⟩$, it could generate to $|g⟩|n+1⟩$ denote as $|f⟩$. Due to Rabi oscillation, as $V=\hslash \lambda \sqrt{n+1}$ the result is:

$$C(t)=\left(\begin{array}{c}cos(\lambda \sqrt{n+1})\\ -isin(\lambda \sqrt{n+1})\end{array}\right)$$

For a generalized case, where the field is a coherent state:

Assumptions

Conclusions

1. Hamiltonian

$$\begin{array}{rl}\hat{H}& ={\hat{H}}_{atom}+{\hat{H}}_{field}+{\hat{H}}_{int}\\ & =\hslash {\omega }_a(\widehat{{\sigma }_z}+\frac{1}{2})+\hslash {\omega }_f{\hat{a}}^{†}\hat{a}+\hslash \lambda (\hat{a}+{\hat{a}}^{†})({\hat{\sigma }}^{+}+{\hat{\sigma }}^{-})\\ & =\hslash {\omega }_a(\widehat{{\sigma }_z}+\frac{1}{2})+\hslash {\omega }_f{\hat{a}}^{†}\hat{a}+\hslash \lambda (\hat{a}{\hat{\sigma }}^{+}+{\hat{a}}^{†}{\hat{\sigma }}^{-}+\hat{a}{\hat{\sigma }}^{-}+{\hat{a}}^{†}{\hat{\sigma }}^{+})\end{array}$$

For how to write this Hamiltonian, see Quantization of electromagnetic field and Hamiltonian Function 2. Dynamics

Techniques

Phenomenons

Related Problems