Hamiltonian and Transformation of Coordinates

In the SQcircuit original paper, we show that the Hamiltonian of any circuit (without any transformation of coordinates) can be written as the following form

\begin{equation} {{H}} = \frac{1}{2} \hat{{\mathbf{Q}}}^T {\mathbf{C}}^{-1}\hat{{\mathbf{Q}}} + \frac{1}{2} \hat{{\mathbf\Phi}}^T {\mathbf{L}}^{*} \hat{{\mathbf\Phi}} +\sum_{k\in \mathcal{S}_L}\left(\frac{\Phi_0}{2\pi}\frac{\mathbf{b}_k^T\mathbf{\varphi}_{\text{ext}}}{l_k}\right){\mathbf{w}}^T_k{\hat{\mathbf{\Phi}}}-\sum_{k\in \mathcal{S}_J} E_{J_k} \cos \left(\frac{2\pi}{\Phi_0}{\mathbf{w}}^T_k{\hat{\mathbf{\Phi}}}+\mathbf{b}_k^T\mathbf{\varphi}_{\text{ext}}\right)\end{equation}

where \(\mathbf{C}\) and \(\mathbf{L}^*\) are respectively capacitance and susceptance matrices, and \(k\)th row of \(\mathbf{W}\) matrix is \({\mathbf{w}}^T_k\). Consider cr as an object of Circuit class, all three matrices of \(\mathbf{C}\), \(\mathbf{L}^*\), and \(\mathbf{W}\) are the attributes of cr.

The \(\mathbf{C}\) in farad unit, \(\mathbf{L}^*\) in henry unit, and \(\mathbf{W}\) are accessible via

# capacitance matrix
cr.C
# susceptance matrix
cr.L
# vector assigning fluxes to junctions
cr.W

SQcircuit transforms the charge and flux operators via following transformation

\begin{align*} \hat{\tilde{\mathbf{Q}}} = \mathbf{R}^{-1}\hat{\mathbf{Q}},\\ \hat{\tilde{\mathbf{\Phi}}} = \mathbf{S}^{-1}\hat{\mathbf{\Phi}}, \end{align*}

where \(\mathbf{R}^{-1}\) and \(\mathbf{S}^{-1}\) can be obtained via coord_transform() method such as

# transformation of cooridante for charge operators
R_inv = cr.coord_transform(var_type="charge")
# transformation of cooridante for flux operators
S_inv = cr.coord_transform(var_type="flux")

The transformed Hamiltonian is

\begin{equation}\hat{\tilde{H}} = \frac{1}{2} \hat{\tilde{\mathbf{Q}}}^T \tilde{\mathbf{C}}^{-1}\hat{\tilde{\mathbf{Q}}} + \frac{1}{2} \hat{\tilde{\mathbf\Phi}}^T \tilde{\mathbf{L}}^{*} \hat{\tilde{\mathbf\Phi}}+\sum_{k\in \mathcal{S}_L}\left(\frac{\Phi_0}{2\pi}\frac{\mathbf{b}_k^T\mathbf{\varphi}_{\text{ext}}}{l_k}\right)\tilde{\mathbf{w}}^T_k\hat{\tilde{\mathbf{\Phi}}}-\sum_{k\in \mathcal{S}_J} E_{J_k} \cos \left(\frac{2\pi}{\Phi_0}\tilde{\mathbf{w}}^T_k\hat{\tilde{\mathbf{\Phi}}}+\mathbf{b}_k^T\mathbf{\varphi}_{\text{ext}}\right), \end{equation}

where the transformed inverse capacitance and susceptance matrices are:

\begin{align*} \tilde{\mathbf{C}}^{-1} = \mathbf{R}^T\mathbf{C}^{-1}\mathbf{R},\\ \tilde{\mathbf{L}}^{*} = \mathbf{S}^T\mathbf{L}^{*}\mathbf{S}, \end{align*}

and the vector assigning fluxes to junctions is given by

\begin{equation*} \tilde{\mathbf{W}} = \mathbf{W}\mathbf{S}. \end{equation*}

The \(\tilde{\mathbf{C}}^{-1}\), \(\tilde{\mathbf{L}}^{*}\), and \(\tilde{\mathbf{W}}\) are accessible via

# transformed inverted capacitance matrix
cr.cInvTrans
# transformed susceptance matrix
cr.lTrans
# transformed vector assigning fluxes to junctions
cr.wTrans