cr门的一些简易推导

实验坐标系下哈密顿量：

\[\mathcal{H}= \frac{1}{2} \omega_1 \sigma_1^z +\Omega_1 \cos \left(w_2 t\right) \sigma_1^x +\frac{1}{2} \omega_2 \sigma_2^z +\frac{1}{2} \omega_{x x} \sigma_1^x \sigma_2^x\]

两个比特都变换到以\(w_2\)转动的参考系下： \[2 \mathcal{H}_{\mathrm{DF}}= \delta_1 \sigma_1^z +\Omega_1 \sigma_1^x +\omega_{x x} [\sigma_1^x \cos \left(\omega_2 t\right) -\sigma_1^y \sin \left( \omega_2 t\right)]\left[\sigma_2^x \cos \left(\omega_2 t\right) -\sigma_2^y \sin \left( \omega_2 t\right)\right],\]

其中\(\delta_1=w1-w2\) 为了方便处理想把1的静态项去掉，先做旋转变换：

\[\mathcal{U}_1 =\exp \left[-i\left(\xi_1 \sigma_1^y\right) / 2\right]\]

其中\(\xi =\arctan \delta_1/\Omega_1\)，\(\eta_1 =\sqrt{\delta_1^2+\Omega_1^2}\)

\[2 \mathcal{H}^{\prime}= \eta_1 \sigma_1^{x\prime} +\omega_{x x} [(\cos\xi \sigma_1^{x\prime} -sin\xi \sigma_1^{z\prime})\cos \left(\omega_2 t\right) -\sigma_1^y \sin \left( \omega_2 t\right)]\left[\sigma_2^x \cos \left(\omega_2 t\right) -\sigma_2^y \sin \left( \omega_2 t\right)\right],\]

再做含时变换：

\[\mathcal{U}_d =\exp \left[-i t\left(\eta_1 \sigma_1^{x\prime}\right) / 2\right],\]

最终得到:

\[\begin{aligned} 2 \mathcal{H}^{\prime}= \omega_{x x} [(\cos\xi \sigma_1^{x\prime} -sin\xi (\sigma_1^{z\prime}\cos\eta_1 t+ \sigma_1^{y\prime}\sin\eta_1 t)) \cos \left(\omega_2 t\right)\\ -(\sigma_1^{y\prime}\cos\eta_1 t - \sigma_1^{z\prime}\sin\eta_1 t) \sin \left( \omega_2 t\right)]\left[\sigma_2^x \cos \left(\omega_2 t\right) -\sigma_2^y \sin \left( \omega_2 t\right)\right], \end{aligned}\]

省去高阶振荡项：

\[2 \mathcal{H}^{\prime}\approx w_{xx} \cos \xi \sigma_x^{1\prime} \sigma_x^{2}\]

转回旋转参考系：

\[2 \mathcal{H}^{\prime}\approx w_{xx} \cos \xi (\sigma_x^{1}\cos\xi +\sigma_z^{1}\sin\xi ) \sigma_x^{2}\]

as \(\xi \rightarrow \pi/2\)，进一步的：

\[2 \mathcal{H}^{\prime}\approx w_{xx}\sin\xi \cos \xi \sigma_z^{1} \sigma_x^{2}\]