{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Hamiltonian and Transformation of Coordinates"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{equation}\n",
    "{{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}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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`. \n",
    "\n",
    "The $\\mathbf{C}$ in farad unit, $\\mathbf{L}^*$ in henry unit, and $\\mathbf{W}$ are accessible via"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# capacitance matrix \n",
    "cr.C\n",
    "# susceptance matrix\n",
    "cr.L\n",
    "# vector assigning fluxes to junctions\n",
    "cr.W"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SQcircuit transforms the charge and flux operators via following transformation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align*}\n",
    "\\hat{\\tilde{\\mathbf{Q}}} = \\mathbf{R}^{-1}\\hat{\\mathbf{Q}},\\\\\n",
    "\\hat{\\tilde{\\mathbf{\\Phi}}} = \\mathbf{S}^{-1}\\hat{\\mathbf{\\Phi}},\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $\\mathbf{R}^{-1}$ and $\\mathbf{S}^{-1}$ can be obtained via `coord_transform()` method such as "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# transformation of cooridante for charge operators\n",
    "R_inv = cr.coord_transform(var_type=\"charge\")\n",
    "# transformation of cooridante for flux operators\n",
    "S_inv = cr.coord_transform(var_type=\"flux\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The transformed Hamiltonian is "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\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),\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where the transformed inverse capacitance and susceptance matrices are:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align*}\n",
    "\\tilde{\\mathbf{C}}^{-1} = \\mathbf{R}^T\\mathbf{C}^{-1}\\mathbf{R},\\\\\n",
    "\\tilde{\\mathbf{L}}^{*} = \\mathbf{S}^T\\mathbf{L}^{*}\\mathbf{S},\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the vector assigning fluxes to junctions is given by"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{equation*}\n",
    "\\tilde{\\mathbf{W}} = \\mathbf{W}\\mathbf{S}.\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The $\\tilde{\\mathbf{C}}^{-1}$, $\\tilde{\\mathbf{L}}^{*}$, and $\\tilde{\\mathbf{W}}$ are accessible via"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# transformed inverted capacitance matrix \n",
    "cr.cInvTrans\n",
    "# transformed susceptance matrix\n",
    "cr.lTrans\n",
    "# transformed vector assigning fluxes to junctions\n",
    "cr.wTrans"
   ]
  }
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