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RWA Hamiltonian

$$ \hat{H}_{\rm RWA}=\frac{\hbar}{2} \begin{bmatrix} 0 && \Omega \\ \Omega^* && -2\Delta \end{bmatrix} $$

Optical Bloch equation

$$ \partial_t \hat{\rho} = \frac{1}{i\hbar}[\hat{H},\hat{\rho}] \\

• \frac{1}{T_1}\left(\rho_{ee}\left|g\right>\left<g\right|- \rho_{ee}\left|e\right>\left<e\right|\right) - \frac{1}{T_2} \left(\rho_{ge}\left|g\right>\left<e\right| + \rho_{eg}\left|e\right>\left<g\right| \right) $$

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Quantum dynamics of a two-level system

RWA Hamiltonian

$$ \hat{H}_{\rm RWA}=\frac{\hbar}{2} \begin{bmatrix} 0 && \Omega \\ \Omega^* && -2\Delta \end{bmatrix} = \frac{\hbar}{2}(|\Omega|e^{i\phi}\left|g\right>\left<e\right|+({\rm h.c})) -\hbar\Delta \hat{n} $$

Dressed state

$$ \hat{H}{\lambda} = \begin{bmatrix} \lambda+ && i\hbar\frac{\dot{\theta}}{2}\\ -i\hbar\frac{\dot{\theta}}{2} && \lambda_-\end{bmatrix} $$

$$ \left|\lambda_-\right> = -\sin\frac{\theta}{2}\left|g\right>+\cos\frac{\theta}{2}\left|e\right> $$

where the mixing angle is $\theta = \tan^{-1} (\Omega/\Delta)$.

Quenching

Rabi oscillation

$$ P_e (t) = \frac{|\Omega|^2}{\Delta^2+|\Omega|^2} \sin^2\left(\frac{\sqrt{\Delta^2+|\Omega|^2}}{2} t\right) $$

Annealing

Rapid adiabatic passage (RAP)

$$ \left|\lambda_-\right>: \left|g\right> \rightarrow \left|e\right> \\ \Delta: -\infty \rightarrow +\infty $$

LZ-path: $\Delta(t)=\dot{\Delta} t$

$$ P_{\rm LZ}= \exp\left(-2\pi \frac{|\Omega|^2}{ 4\dot{\Delta}}\right) $$

Spin-1/2 system

$$ \hat{R}(\hat{a},\theta) =\exp\left(-\frac{i}{2} \vec{a}\cdot\vec{\sigma}\theta\right) = \begin{bmatrix} \cos\frac{\theta}{2} +ia_z\sin\frac{\theta}{2} && (-ia_x+a_y)\sin\frac{\theta}{2}\\ (-ia_x-a_y) \sin\frac{\theta}{2}&& \cos\frac{\theta}{2} - i a_z \sin\frac{\theta}{2}\end{bmatrix} $$

Single qubit gate

Open quantum system