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   "cell_type": "markdown",
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   "source": [
    "# External Flux Distribution"
   ]
  },
  {
   "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 distribution of external fluxes for each inductive element is specified by $n_L$ (number of loops) dimensional vector $\\mathbf{b}_k$. We also showed in the paper that the correct description for external fluxes depends on the capacitors associated to each inductive element (see the paper for the details)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We use the following circuit as an example to demonstrate the SQcircuit functionalities."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"pics/loopDist.png\" width=240 align=\"left\" />"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Firstly, we build our circuit by assigning three equal capacitors to each inductive elements( Josephson junctions and inductors)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import SQcircuit as sq\n",
    "\n",
    "C = sq.Capacitor(1)\n",
    "\n",
    "loop1 = sq.Loop(id_str=\"loop1\")\n",
    "\n",
    "JJ1 = sq.Junction(1, loops=[loop1], cap=C, id_str=\"JJ1\")\n",
    "JJ2 = sq.Junction(1, loops=[loop1], cap=C, id_str=\"JJ2\")\n",
    "L = sq.Inductor(1, loops=[loop1], cap=C, id_str=\"ind\")\n",
    "\n",
    "elements = {(0, 1): [JJ1],\n",
    "            (0, 2): [JJ2],\n",
    "            (1, 2): [L]}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that in the construction of circuit elements, we set ``id_str`` for the loop and inductive elements, which makes it easier later to identify the assignment of loop distributions. If we do not pick the ``id_str`` for the interested elements, SQcircuit assigns auto-generated ``id_str`` for elements."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In SQcircuit, we provided ``flux_dist`` option in circuit construction that specifies the method for assigning external fluxes to inductive elements. If ``flux_dist`` is set to ``\"all\"``, SQcircuit automatically assigns the external fluxes to junctions and inductors based on their capacitors. However, if it is set to ``\"junction\"`` or ``\"inductor\"``, SQcircuit discards the capacitor assignment of inductive elements and assumes that junction capacitors are much smaller than inductor capacitors for ``\"junction\"`` option and otherwise for ``\"inductor\"`` option."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By using ``loop_description()`` method, we can find the loop description for our circuit. Here, we expect the distribution of the flux to be equal among the elements since we set equal capacitors for each element."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "             loop1          \n",
      "----------------------------\n",
      "JJ1, b1:     0.33333        \n",
      "JJ2, b2:     0.33333        \n",
      "ind, b3:     0.33333        \n"
     ]
    }
   ],
   "source": [
    "cr = sq.Circuit(elements, flux_dist='all')\n",
    "cr.loop_description()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For junction assignment (the default option in SQcircuit):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "             loop1          \n",
      "----------------------------\n",
      "JJ1, b1:     0.5            \n",
      "JJ2, b2:     0.5            \n",
      "ind, b3:     0.0            \n"
     ]
    }
   ],
   "source": [
    "cr = sq.Circuit(elements, flux_dist=\"junctions\")\n",
    "cr.loop_description()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and for inductor assignment:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "             loop1          \n",
      "----------------------------\n",
      "JJ1, b1:     0.0            \n",
      "JJ2, b2:     0.0            \n",
      "ind, b3:     1.0            \n"
     ]
    }
   ],
   "source": [
    "cr = sq.Circuit(elements, flux_dist=\"inductors\")\n",
    "cr.loop_description()"
   ]
  }
 ],
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