Coupling Between Two LC Circuits
Quantum computation requires coupling between qubits, which can influence the parameters of the qubits. Let us first discuss two linear LC circuits coupled through a capacitor \( C_g \).
\[ \mathcal{L}_{2LC} = \frac{L_A \dot{Q}_A^2}{2} + \frac{L_B \dot{Q}_B^2}{2} - \frac{Q_A^2}{2C_A} - \frac{Q_B^2}{2C_B} - \frac{Q_g^2}{2C_g} \]
The charge \( Q_g \) on the coupling capacitor \( C_g \) can be expressed in terms of \( Q_A \) and \( Q_B \) using the properties of capacitors:
\[ Q_g = C_g (V_A - V_B) = C_g \left( \frac{Q_A}{C_A} - \frac{Q_B}{C_B} \right) \]
Substituting this into the Lagrangian and simplifying:
\[ \mathcal{L}_{2LC} = \frac{L_A \dot{Q}_A^2}{2} + \frac{L_B \dot{Q}_B^2}{2} - \frac{(C_A + C_g) }{2C_A^2} Q_A^2 - \frac{(C_B + C_g) }{2C_B^2}Q_B^2 + \frac{C_g}{C_A C_B} Q_A Q_B \]
Here, define \( C_{\Sigma A} = \frac{C_A^2}{C_A + C_g} \) and \( C_{\Sigma B} = \frac{C_B^2}{C_B + C_g} \). The canonical momentum is defined as:
\[ \Phi_A = -\frac{\partial \mathcal{L}_{2LC}}{\partial \dot{Q}_A} = -L_A \dot{Q}_A \quad \rightarrow \quad \dot{Q}_A = -\frac{\Phi_A}{L_A} \]
The Hamiltonian can then be expressed as:
\[ \bar{\mathcal{H}}_{2LC} = \frac{\Phi_A^2}{2L_A} + \frac{Q_A^2}{2C_{\Sigma A}} + \frac{\Phi_B^2}{2L_B} + \frac{Q_B^2}{2C_{\Sigma B}} - \frac{C_g}{C_A C_B} Q_A Q_B \]
Moving to canonical quantization:
\[ \begin{aligned} &a_{\Sigma A} = \sqrt{\frac{1}{2\hbar} \sqrt{\frac{L_A}{C_{\Sigma A}}}} \left(\sqrt{\frac{C_{\Sigma A}}{L_A}} \Phi_A + iQ_A \right) \\ &a_{\Sigma A}^\dagger = \sqrt{\frac{1}{2\hbar} \sqrt{\frac{L_A}{C_{\Sigma A}}}} \left(\sqrt{\frac{C_{\Sigma A}}{L_A}} \Phi_A - iQ_A \right) \end{aligned} \]
Defining the frequencies \( \omega_{\Sigma A} = \frac{1}{\sqrt{L_A C_{\Sigma A}}} \) and \( \omega_{\Sigma B} = \frac{1}{\sqrt{L_B C_{\Sigma B}}} \):
\[ \bar{\mathcal{H}}_{2LC} = \hbar \omega_{\Sigma A} \left(a_{\Sigma A}^\dagger a_{\Sigma A} + \frac{1}{2}\right) + \hbar \omega_{\Sigma B} \left(a_{\Sigma B}^\dagger a_{\Sigma B} + \frac{1}{2}\right) - \hbar \Omega_g \, i(a_{\Sigma A}^\dagger - a_{\Sigma A}) \, i(a_{\Sigma B}^\dagger - a_{\Sigma B}) \]
The coupling strength \( \Omega_g \) is defined as:
\[ \Omega_g = C_g \sqrt{\frac{\omega_{\Sigma A}}{2(C_A + C_g)}} \sqrt{\frac{\omega_{\Sigma B}}{2(C_B + C_g)}} \]
Originally written in Chinese by the author, these articles are translated into English to invite cross-language resonance.
Peir-Ru Wang