Jaynes-Cummings model dynamics

Express excited and ground state in DressdState§

From

$$\begin{cases} \ket{n, +}=\sin\theta\ket{0}+e^{i\phi}\cos\theta\ket{1} \\ \ket{n, -}=\cos\theta\ket{0}-e^{i\phi}\sin\theta\ket{1} \\ \end{cases}$$

We get

$$\begin{cases} \ket{0}=\sin\theta\ket{n,+}+\cos\theta\ket{n,-}\\ \ket{1}=e^{-i\phi}\left(\cos\theta\ket{n,+}-\sin\theta\ket{n,-}\right) \end{cases}$$

Time evolution of DressedState§

$$E_{n,\pm}=\hbar\left(\omega n + \frac{\Delta}{2} \pm \frac{1}{2}\Omega\right)$$

Thus

$$\ket{\psi(t)}=e^{-i\omega nt}e^{-i\frac{\Delta}{2}t}\left(c_- e^{\frac{i}{2}\Omega t}\ket{n,-} + c_+ e^{-\frac{i}{2}\Omega t}\ket{n,+}\right)$$

But for what we are trying to do next, let's use the $\ket{0},\ket{1}$ basis.

$$\begin{align*} \ket{\psi(t)}&=e^{-i\omega nt}e^{-i\frac{\Delta}{2}t}\left(c_- e^{\frac{i}{2}\Omega t}(\cos\theta\ket{0}-e^{i\phi}\sin\theta\ket{1}) + c_+ e^{-\frac{i}{2}\Omega t}(\sin\theta\ket{0}+e^{i\phi}\cos\theta\ket{1})\right) \\ &=e^{-i\omega nt}e^{-i\frac{\Delta}{2}t}\left((c_+ e^{-\frac{i}{2}\Omega t}\sin\theta + c_- e^{\frac{i}{2}\Omega t}\cos\theta)\ket{0} + (c_+ e^{-\frac{i}{2}\Omega t}e^{i\phi}\cos\theta - c_- e^{\frac{i}{2}\Omega t}e^{i\phi}\sin\theta)\ket{1}\right) \\ \end{align*}$$

Probabilities of§

Being on excited state§

With $\delta = \arg(c_+c_-^*)$

$$\braket{1|\phi}^2=c_+^2\cos^2\theta+c_-^2\sin^2\theta - 2|c_+||c_-|\cos\theta\sin\theta \cos\left(\Omega t - \delta\right)$$

Which we want to link back to Rabi frequency and Detuning, thus we need $2\theta$ everywhere, and note $c_+^2+c_-^2=1$ as a normalization condition

$$\begin{align*} \braket{1|\phi}^2&=\frac{c_+^2}{2}(1+\cos2\theta)+\frac{c_-^2}{2}(1-\cos2\theta)-|c_+||c_-|\sin2\theta\cos\left(\Omega t - \delta \right) \\ &=\frac{c_+^2 + c_-^2}{2}+\cos2\theta\left(\frac{c_+^2-c_-^2}{2}\right) - |c_+||c_-|\sin2\theta\cos\left(\Omega t - \delta \right)\\ &=\frac{1}{2}+\cos2\theta\left(\frac{c_+^2-c_-^2}{2}\right) - |c_+||c_-|\sin2\theta\cos\left(\Omega t - \delta\right) \end{align*}$$

Thus

$$P_e(t)=\frac{1}{2}-\frac{\Delta}{\Omega}\left(\frac{c_+^2-c_-^2}{2}\right)-|c_+||c_-|\frac{\Omega_R}{\Omega}\cos\left(\Omega t - \delta\right)$$

Being on Ground state§

Easily, as the algebra is extremely similar,

$$P_g(t)=\frac{1}{2} - \frac{\Delta}{\Omega}\left(\frac{c_+^2-c_-^2}{2}\right) + |c_+||c_-|\frac{\Omega_R}{\Omega}\cos\left(\Omega t - \delta\right)$$

Which also can be seen as

$$P_g(t) = 1 - P_e(t)$$

Rabi Frequency§

Notational Note§

When $\Delta=0$ we obtain $\Omega_R=\Omega$, thus one is called Rabi frequency one is the Rabi frequency on resonant since $\Delta=0$ means on resonant.

Physical significant§

As seen in the formula, it is the frequency that drive the state between excited and ground state, which we can carry out the calculation also for ground state and other states as well and the Rabi frequency will give showing up.

Rabi Splitting§

The fact that purely due to an existing field that didn't need to be controled, as it can also be from vacuum fluctuation, will cause the coupled system having splitted eigenstate which it is not the ground or excited state, but rather the dressedstate, is tied to the concept of rabi splitting, which is literraly the energy gap of this two eigenstate.

$$E_n = E_{n,+} - E_{n,-} = \hbar\Omega$$