驅動 LC 電路
現在我們來討論如何驅動 LC 電路與 Transmon。透過耦合電容 \( C_d \) 與外部電壓源 \( V_d \) 耦合，系統的 Lagrangian 為：
\[ \mathcal{L}_d(\dot{Q}, Q) = T - U = \frac{L\dot{Q}^2}{2} - \frac{Q^2}{2C} - \frac{Q_d^2}{2C_d} \]
其中耦合電容 \( C_d \) 所攜帶的能量為：
\[ \frac{Q_d^2}{2C_d} = \frac{C_d (V - V_d)^2}{2} = \frac{C_d \left(\frac{Q}{C} - V_d\right)^2}{2} \] \[ = \frac{C_d Q^2}{2C^2} - \frac{C_d Q}{C}V_d + \frac{C_d V_d^2}{2} \]
將上述式子代入並整理得：
\[ \mathcal{L}_d(\dot{Q}, Q) = \frac{L\dot{Q}^2}{2} - \frac{Q^2}{2C} - \frac{C_d}{C} \frac{Q^2}{2C} + \frac{C_d}{C} QV_d - \frac{C_d V_d^2}{2} \] \[ = \frac{L\dot{Q}^2}{2} - \frac{Q^2}{2C_\Sigma} + \frac{C_d}{C} QV_d - \frac{C_d V_d^2}{2} \]
其中 \( C_\Sigma = \frac{C^2}{C + C_d} \)。定義正則動量：
\[ \Phi = -\frac{\partial \mathcal{L}_d}{\partial \dot{Q}} = -L\dot{Q} \quad \rightarrow \quad \dot{Q} = -\frac{\Phi}{L} \]
接著計算 Hamiltonian：
\[ \bar{\mathcal{H}}_d = -\mathcal{L}_d - \Phi \dot{Q}\] \[ = -\frac{L\dot{Q}^2}{2} + \frac{Q^2}{2C_\Sigma} - \frac{C_d}{C} QV_d + \frac{\Phi^2}{L}\] \[ = \frac{\Phi^2}{2L} + \frac{Q^2}{2C_\Sigma} - \frac{C_d}{C} QV_d \]
完整推導過程
接著進行正則量子化：
\[ \begin{aligned} &a_d = \sqrt{\frac{1}{2\hbar} \sqrt{\frac{L}{C_\Sigma}}} \left(\sqrt{\frac{C_\Sigma}{L}} \Phi + iQ\right) \\ &a_d^\dagger = \sqrt{\frac{1}{2\hbar} \sqrt{\frac{L}{C_\Sigma}}} \left(\sqrt{\frac{C_\Sigma}{L}} \Phi - iQ\right) \end{aligned} \] \[\quad \leftrightarrow \quad \begin{aligned} &\Phi_d = \sqrt{\frac{\hbar}{2} \sqrt{\frac{L}{C_\Sigma}}} (a_d^\dagger + a_d) \\ &Q_d = i\sqrt{\frac{\hbar}{2} \sqrt{\frac{C_\Sigma}{L}}} (a_d^\dagger - a_d) \end{aligned} \]
將上述代換代入
\[ =\hbar \sqrt{\frac{1}{LC_\Sigma}} \left( a_d^\dagger a_d + \frac{1}{2} \right) - \frac{C_d}{C} \sqrt{\frac{\hbar}{2 C_\Sigma \sqrt{LC_\Sigma}}} \cdot i (a_d^\dagger - a_d) V_d \]
定義頻率\(\omega_\Sigma = \frac{1}{\sqrt{L C_\Sigma}}\)
\[ = \hbar \omega_\Sigma \left( a_d^\dagger a_d + \frac{1}{2} \right) - C_d V_d \sqrt{\frac{\omega_\Sigma}{2\hbar (C + C_d)}} \cdot i (a_d^\dagger - a_d) \]
最後結果為：
\[ \bar{\mathcal{H}} _d = \hbar \omega_\Sigma \left(a_d^\dagger a_d + \frac{1}{2}\right) - i \Omega (a_d^\dagger - a_d) \]
其中 \( \Omega = C_d V_d \sqrt{\frac{\omega_\Sigma}{2\hbar (C + C_d)}} \) 稱為拉比頻率（Rabi Frequency）。
王培儒