Reproduce-Nat-Phys-advance-online-publication-2016-Liu.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Reproduce: Quantum electrodynamics near a photonic bandgap (Preliminary)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Yanbing Liu and Andrew A Houck, Quantum electrodynamics near a photonic bandgap, Nat. Phys., (2016)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Reproduced by <strong>Eunjong Kim</strong> (eunjongkim@caltech.edu)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Import Modules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Setup for symbolic calculation\n",
    "import sympy as sp\n",
    "sp.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Setup for numerical calculation\n",
    "import numpy as np\n",
    "import numpy.linalg as la\n",
    "from scipy.optimize import fsolve, brentq\n",
    "from scipy.integrate import quad"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Setup for plots\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import rcParams, cm\n",
    "%matplotlib inline\n",
    "rcParams.update({\"text.usetex\": True, \"font.size\": 16})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  Dispersion relation and the band structure"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Symbolic Calculation using the transfer matrix method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this section, we derive the equation (1) of this paper using the SymPy module (symbolic calulation)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define symbolic variables corresponding to frequency,\n",
    "# wavevector, and size of the unit cell\n",
    "omega_sym = sp.Symbol(r'\\omega')\n",
    "k_sym = sp.Symbol(r'k')\n",
    "d_sym = sp.Symbol(r'd')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Define a list of phase velocity, characteristic impedance,\n",
    "# and length of each section of the unit cell\n",
    "v_sym = [sp.Symbol(r'v_{p1}'), sp.Symbol(r'v_{p2}')]\n",
    "Z_sym = [sp.Symbol(r'Z_{1}'), sp.Symbol(r'Z_{2}')]\n",
    "l_sym = [sp.Symbol(r'l_{1}'), sp.Symbol(r'l_{2}')]\n",
    "Z0_sym = sp.Symbol(r'Z_{0}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# qubit parameters\n",
    "gamma_sym = sp.Symbol(r'\\gamma')\n",
    "omega_a_sym = sp.Symbol(r'\\omega_a')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define the transfer matrix (ABCD) of each coplanar waveguide (CPW) section,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def CPW_ABCD_sym(omega, idx):\n",
    "    \"\"\"\n",
    "    Construct an ABCD matrix (symbolic) of the coplanar waveguide of specified index\n",
    "    \"\"\"\n",
    "    A = sp.cos(omega * l_sym[idx] / v_sym[idx])\n",
    "    B = sp.I * sp.sin(omega * l_sym[idx] / v_sym[idx]) * Z_sym[idx]\n",
    "    C = sp.I * sp.sin(omega * l_sym[idx] / v_sym[idx]) / Z_sym[idx]\n",
    "    D = sp.cos(omega * l_sym[idx] / v_sym[idx])\n",
    "    return sp.Matrix([[A, B], [C, D]])\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The transfer matrix $M_\\textrm{unit}$ is the product of that of each section:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{Z_{1}}{Z_{2}} \\sin{\\left (\\frac{\\omega l_{1}}{v_{p1}} \\right )} \\sin{\\left (\\frac{\\omega l_{2}}{v_{p2}} \\right )} + \\cos{\\left (\\frac{\\omega l_{1}}{v_{p1}} \\right )} \\cos{\\left (\\frac{\\omega l_{2}}{v_{p2}} \\right )} & i Z_{1} \\sin{\\left (\\frac{\\omega l_{1}}{v_{p1}} \\right )} \\cos{\\left (\\frac{\\omega l_{2}}{v_{p2}} \\right )} + i Z_{2} \\sin{\\left (\\frac{\\omega l_{2}}{v_{p2}} \\right )} \\cos{\\left (\\frac{\\omega l_{1}}{v_{p1}} \\right )}\\\\\\frac{i}{Z_{2}} \\sin{\\left (\\frac{\\omega l_{2}}{v_{p2}} \\right )} \\cos{\\left (\\frac{\\omega l_{1}}{v_{p1}} \\right )} + \\frac{i}{Z_{1}} \\sin{\\left (\\frac{\\omega l_{1}}{v_{p1}} \\right )} \\cos{\\left (\\frac{\\omega l_{2}}{v_{p2}} \\right )} & \\cos{\\left (\\frac{\\omega l_{1}}{v_{p1}} \\right )} \\cos{\\left (\\frac{\\omega l_{2}}{v_{p2}} \\right )} - \\frac{Z_{2}}{Z_{1}} \\sin{\\left (\\frac{\\omega l_{1}}{v_{p1}} \\right )} \\sin{\\left (\\frac{\\omega l_{2}}{v_{p2}} \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡           ⎛\\omega⋅l_{1}⎞    ⎛\\omega⋅l_{2}⎞                                  \n",
       "⎢  Z_{1}⋅sin⎜────────────⎟⋅sin⎜────────────⎟                                  \n",
       "⎢           ⎝   v_{p1}   ⎠    ⎝   v_{p2}   ⎠      ⎛\\omega⋅l_{1}⎞    ⎛\\omega⋅l_\n",
       "⎢- ───────────────────────────────────────── + cos⎜────────────⎟⋅cos⎜─────────\n",
       "⎢                    Z_{2}                        ⎝   v_{p1}   ⎠    ⎝   v_{p2}\n",
       "⎢                                                                             \n",
       "⎢       ⎛\\omega⋅l_{2}⎞    ⎛\\omega⋅l_{1}⎞        ⎛\\omega⋅l_{1}⎞    ⎛\\omega⋅l_{2\n",
       "⎢  ⅈ⋅sin⎜────────────⎟⋅cos⎜────────────⎟   ⅈ⋅sin⎜────────────⎟⋅cos⎜───────────\n",
       "⎢       ⎝   v_{p2}   ⎠    ⎝   v_{p1}   ⎠        ⎝   v_{p1}   ⎠    ⎝   v_{p2}  \n",
       "⎢  ───────────────────────────────────── + ───────────────────────────────────\n",
       "⎣                  Z_{2}                                   Z_{1}              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "{2}⎞             ⎛\\omega⋅l_{1}⎞    ⎛\\omega⋅l_{2}⎞              ⎛\\omega⋅l_{2}⎞ \n",
       "───⎟  ⅈ⋅Z_{1}⋅sin⎜────────────⎟⋅cos⎜────────────⎟ + ⅈ⋅Z_{2}⋅sin⎜────────────⎟⋅\n",
       "   ⎠             ⎝   v_{p1}   ⎠    ⎝   v_{p2}   ⎠              ⎝   v_{p2}   ⎠ \n",
       "                                                                              \n",
       "}⎞                                                        ⎛\\omega⋅l_{1}⎞    ⎛\\\n",
       "─⎟                                               Z_{2}⋅sin⎜────────────⎟⋅sin⎜─\n",
       " ⎠            ⎛\\omega⋅l_{1}⎞    ⎛\\omega⋅l_{2}⎞            ⎝   v_{p1}   ⎠    ⎝ \n",
       "──         cos⎜────────────⎟⋅cos⎜────────────⎟ - ─────────────────────────────\n",
       "              ⎝   v_{p1}   ⎠    ⎝   v_{p2}   ⎠                     Z_{1}      \n",
       "\n",
       "                 ⎤\n",
       "                 ⎥\n",
       "   ⎛\\omega⋅l_{1}⎞⎥\n",
       "cos⎜────────────⎟⎥\n",
       "   ⎝   v_{p1}   ⎠⎥\n",
       "                 ⎥\n",
       "omega⋅l_{2}⎞     ⎥\n",
       "───────────⎟     ⎥\n",
       "  v_{p2}   ⎠     ⎥\n",
       "────────────     ⎥\n",
       "                 ⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M_unit_sym = CPW_ABCD_sym(omega_sym, 0) * CPW_ABCD_sym(omega_sym, 1)\n",
    "M_unit_sym"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Also, the ABCD matrix $M_a$ for qubit can be described as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def qubit_ABCD_sym(omega):\n",
    "    \"\"\"\n",
    "    Construct an ABCD matrix (symbolic) of an atom in the single photon regime\n",
    "    \"\"\"\n",
    "    A = 1\n",
    "    B = 0\n",
    "    C = - sp.I * gamma_sym / (omega - omega_a_sym) / Z0_sym\n",
    "    D = 1\n",
    "    return sp.Matrix([[A, B], [C, D]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIwAAAA1BAMAAABhKx6wAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nccmY2MGYGIUKTmz8HxNWwVn1pRaUAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0\\\\- \\frac{i \\gamma}{Z_{0} \\left(\\omega - \\omega_a\\right)} & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡           1              0⎤\n",
       "⎢                           ⎥\n",
       "⎢       -ⅈ⋅\\gamma           ⎥\n",
       "⎢────────────────────────  1⎥\n",
       "⎣Z_{0}⋅(\\omega - \\omegaₐ)   ⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qubit_ABCD_sym(omega_sym)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we calculate the band structure for the bare photonic crystal (in the absence of qubit). For a periodic system, the voltage $V(x, t)$ and current $I(x, t)$ should be of the Bloch form,\n",
    "$$\n",
    "\\begin{bmatrix} V(0, t) \\\\ I(0, t) \\end{bmatrix}\n",
    "= M_\\textrm{unit}\n",
    "\\begin{bmatrix} V(d, t) \\\\ I(d, t) \\end{bmatrix} = M_\\textrm{unit}e^{ikd}\\begin{bmatrix} V(0, t) \\\\ I(0, t) \\end{bmatrix},\n",
    "$$\n",
    "so that\n",
    "$$\n",
    "M_\\textrm{unit}\\begin{bmatrix} V(0, t) \\\\ I(0, t) \\end{bmatrix}\n",
    "= e^{-ikd}\\begin{bmatrix} V(0, t) \\\\ I(0, t) \\end{bmatrix}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We require that the determinant of $ (e^{ikd} M_\\textrm{unit} - I)$ be zero for non-trivial solutions, and we get the following dispersion relation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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alzHGnIHT+xINtb4iri+VsTF8vd5K/vesM1bg5yXRzcJInyhyxk97Hvc+gsykDBy+zXAk\nDjD5zBzA6YAbjtupXfQfwDgM30e1HbMPg2Yh6NC8jDHmDNxoGVMfyxv0zXalKtFNtkXgYv+FRtbz\nPPJN5FFwCoiv7KP/ACbefhglK2bwYdAsOB0eL3FR0HIh1/riDT19vRfGwo/lDTQhwNnOw03Aiw5T\nqm20U4o1Uiq8XbwBxFf250jlSHIrZvBhiFsIOnxeCiJs7sT7Gi5j4cfyJpLbpeo7u+6SfzvwV6mz\n5PmqUGY+vbMJK+hbqHLN0nCIKhtmP4aYhaDD52X4JwGmG7G+RENfX0NiLBkIaJsUxHRFiRbhdKHL\n91Xl+UoajWDBvJ9/e2ix8HvpcWTJfQ2JsZ8KjNcrrJc3lWgR8nWuzDVkZfZkKY0qNzHvD4Np4ffS\n58lS+xouY/kFb7hMRzaPPFwGhiVjhqVbGIMqxZL7Goix/wNY3TZ3zBqmfgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{Z_{1}}{2 Z_{2}} \\sin{\\left (\\frac{\\omega l_{1}}{v_{p1}} \\right )} \\sin{\\left (\\frac{\\omega l_{2}}{v_{p2}} \\right )} + \\cos{\\left (k \\left(l_{1} + l_{2}\\right) \\right )} - \\cos{\\left (\\frac{\\omega l_{1}}{v_{p1}} \\right )} \\cos{\\left (\\frac{\\omega l_{2}}{v_{p2}} \\right )} + \\frac{Z_{2}}{2 Z_{1}} \\sin{\\left (\\frac{\\omega l_{1}}{v_{p1}} \\right )} \\sin{\\left (\\frac{\\omega l_{2}}{v_{p2}} \\right )} = 0$$"
      ],
      "text/plain": [
       "         ⎛\\omega⋅l_{1}⎞    ⎛\\omega⋅l_{2}⎞                                     \n",
       "Z_{1}⋅sin⎜────────────⎟⋅sin⎜────────────⎟                                     \n",
       "         ⎝   v_{p1}   ⎠    ⎝   v_{p2}   ⎠                               ⎛\\omeg\n",
       "───────────────────────────────────────── + cos(k⋅(l_{1} + l_{2})) - cos⎜─────\n",
       "                 2⋅Z_{2}                                                ⎝   v_\n",
       "\n",
       "                                      ⎛\\omega⋅l_{1}⎞    ⎛\\omega⋅l_{2}⎞    \n",
       "                             Z_{2}⋅sin⎜────────────⎟⋅sin⎜────────────⎟    \n",
       "a⋅l_{1}⎞    ⎛\\omega⋅l_{2}⎞            ⎝   v_{p1}   ⎠    ⎝   v_{p2}   ⎠    \n",
       "───────⎟⋅cos⎜────────────⎟ + ───────────────────────────────────────── = 0\n",
       "{p1}   ⎠    ⎝   v_{p2}   ⎠                    2⋅Z_{1}                     "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Expr = sp.simplify(sp.det(M_unit_sym * sp.exp(sp.I * k_sym * d_sym) - sp.eye(2))\n",
    "                   / (2 * sp.exp(sp.I * k_sym * d_sym))).subs(d_sym, sum(l_sym))\n",
    "sp.Eq(Expr, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assuming the same phase velocity in all sections, $v_{p} \\equiv v_{p1} = v_{p2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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WlRGPeQOXTJfhCPJYUrQj+Hq9lf7vmacfhZ+XmJuFhk4kydpTn8fdjz4usHnCtxmepIyN\nCJh/AOMR3I9swpx+eh7gNJDbniEMJBEBDZaVEY95A/eePZYcreEq9bG8wRPkbFezUt10m9ci/0Kj\n2xkV+SZSCZh/ABPtP4iaiRmOgJPFGCEMpoQCHbASFwUtH/Kl6op4LBFaPbL+WF7zJGUCnO08JK60\njIif05kxVhtcJXC2wRw8aWIuvl26XowYwmBKKNAhK4HmHteaJesyPJYMrR5TfyyveUTdSgTAd3bc\nKSp/1TyHGq1pFglU6po3DMrAnPv0dnGyhTEYEgQaQlbqnwS49ixVl+GxhGjdMWM46b5ibpRUtqpY\nMcRpmkcC52nWUCgD82hPjhjGYEgQ6LCVATuWrEt7LCHawOBG0+2Kvk5Sl4lzWPGjxGhD1aVA/iTF\nGRKhMReaYsh9YdASyawMGbJUXdpjSdGGRnfbRncFA7YfBFwV75uzaaknzZKtDEBcsq4EaP8PHuX2\nwLsCEPUAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{Z_{1}}{2 Z_{2}} \\sin{\\left (\\frac{\\omega l_{1}}{v_{p}} \\right )} \\sin{\\left (\\frac{\\omega l_{2}}{v_{p}} \\right )} + \\cos{\\left (k \\left(l_{1} + l_{2}\\right) \\right )} - \\cos{\\left (\\frac{\\omega l_{1}}{v_{p}} \\right )} \\cos{\\left (\\frac{\\omega l_{2}}{v_{p}} \\right )} + \\frac{Z_{2}}{2 Z_{1}} \\sin{\\left (\\frac{\\omega l_{1}}{v_{p}} \\right )} \\sin{\\left (\\frac{\\omega l_{2}}{v_{p}} \\right )} = 0$$"
      ],
      "text/plain": [
       "         ⎛\\omega⋅l_{1}⎞    ⎛\\omega⋅l_{2}⎞                                     \n",
       "Z_{1}⋅sin⎜────────────⎟⋅sin⎜────────────⎟                                     \n",
       "         ⎝     vₚ     ⎠    ⎝     vₚ     ⎠                               ⎛\\omeg\n",
       "───────────────────────────────────────── + cos(k⋅(l_{1} + l_{2})) - cos⎜─────\n",
       "                 2⋅Z_{2}                                                ⎝     \n",
       "\n",
       "                                      ⎛\\omega⋅l_{1}⎞    ⎛\\omega⋅l_{2}⎞    \n",
       "                             Z_{2}⋅sin⎜────────────⎟⋅sin⎜────────────⎟    \n",
       "a⋅l_{1}⎞    ⎛\\omega⋅l_{2}⎞            ⎝     vₚ     ⎠    ⎝     vₚ     ⎠    \n",
       "───────⎟⋅cos⎜────────────⎟ + ───────────────────────────────────────── = 0\n",
       "vₚ     ⎠    ⎝     vₚ     ⎠                    2⋅Z_{1}                     "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v_p_sym = sp.Symbol('v_p')\n",
    "Expr = Expr.subs({v_sym[0]: v_p_sym, v_sym[1]: v_p_sym})\n",
    "sp.Eq(Expr, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, the Equation (1) is obtained."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Evaluation of the band structure"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From this dispersion relation, it is possible to numerically evaluate the band structure. First, we put the values of parameters. \n",
    "The eigenfrequency of the second band at $k=\\pi/d$ is given by $\\omega_0 /2\\pi= 7.7\\textrm{ GHz}$. The parameters corresponding to lengths $l$ and the impedances $Z$ are: $l_1 = 0.45\\mathrm{\\ mm}$, $l_2 = 8\\mathrm{\\ mm}$, $Z_1 = 28\\mathrm{\\ \\Omega}$, $Z_2 = 125\\mathrm{\\ \\Omega}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# values of parameters in SI units\n",
    "omega0 = 2 * np.pi * 7.7e9\n",
    "\n",
    "l = [0.45e-3, 8e-3]\n",
    "Z = [28, 125]\n",
    "d = sum(l)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "dict_items = []\n",
    "for idx in range(2):\n",
    "    dict_items.append((l_sym[idx], l[idx]))\n",
    "    dict_items.append((Z_sym[idx], Z[idx]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can compute the value of the phase velocity $v_p$ (assuming $v_{p1} = v_{p2}$) using the dispersion relation and $\\omega = \\omega_0$ at $k = \\pi/d$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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LqTONAI3JRINzIVsIaMxmIxVP68CwEUbuHi29XcfLSZZQ9c6bqgqHfoZ+ixCi6L93crRQ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\ngg==\n",
      "text/latex": [
       "$$\\frac{16409}{7000} \\sin{\\left (\\frac{21771237.0893773}{v_{p}} \\right )} \\sin{\\left (\\frac{387044214.922263}{v_{p}} \\right )} - \\cos{\\left (\\frac{21771237.0893773}{v_{p}} \\right )} \\cos{\\left (\\frac{387044214.922263}{v_{p}} \\right )} - 1.0$$"
      ],
      "text/plain": [
       "         ⎛21771237.0893773⎞    ⎛387044214.922263⎞                             \n",
       "16409⋅sin⎜────────────────⎟⋅sin⎜────────────────⎟                             \n",
       "         ⎝       vₚ       ⎠    ⎝       vₚ       ⎠      ⎛21771237.0893773⎞    ⎛\n",
       "───────────────────────────────────────────────── - cos⎜────────────────⎟⋅cos⎜\n",
       "                       7000                            ⎝       vₚ       ⎠    ⎝\n",
       "\n",
       "                       \n",
       "                       \n",
       "387044214.922263⎞      \n",
       "────────────────⎟ - 1.0\n",
       "       vₚ       ⎠      "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Expr_ = Expr.subs(dict_items)\n",
    "expr_ = Expr_.subs({omega_sym: omega0, k_sym: np.pi/d}); expr_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAPBAMAAAA/sQ3hAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$124756072.413$$"
      ],
      "text/plain": [
       "124756072.413"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "func = sp.lambdify(v_p_sym, expr_, \"numpy\")\n",
    "v_p = fsolve(func, omega0/(np.pi/d))[0]; v_p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus the value of the phase velocity is estimated as $v_p = 1.248 \\times 10^{8} \\mathrm{\\ m/s}$. Using this, we calculate the band structure from the dispersion relation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = Expr_.subs(v_p_sym, v_p)\n",
    "func = sp.lambdify((omega_sym, k_sym), expr, \"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Number of k-points to evaluate\n",
    "N_kPts = 100\n",
    "# Number of bands to evaluate\n",
    "N_band = 4\n",
    "\n",
    "kPts = np.linspace(- np.pi / d, np.pi / d, N_kPts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "freq = np.zeros((N_band, N_kPts))\n",
    "\n",
    "for n in range(N_band):\n",
    "    for idx, k_ in enumerate(kPts):\n",
    "        # search for eigenfrequency between\n",
    "        # n v_p k_0 and (n+1) v_0 k_0 (k_0 = \\pi / d)\n",
    "        freq[n, idx] = brentq(func, n * (v_p * np.pi / d),\n",
    "                              (n + 1) * (v_p * np.pi / d),\n",
    "                              args=(k_)) / (2 * np.pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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H2LEDfv1VPNLRoxtELSMDHnkEYgKUCWgtXL4sdu3fL0K/d694y6NGwaOPwhNP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\n/DB+rOMPAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c73f438>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1, 1, figsize=(5, 4))\n",
    "fig_colors = ['green','red','blue', 'purple']\n",
    "for n in range(N_band):\n",
    "    ax.plot(kPts / (np.pi / d), freq[n, :] / 1e9, color=fig_colors[n])\n",
    "\n",
    "ax.set_xlabel(r'Wavevector $k/(\\pi/d)$');\n",
    "ax.set_ylabel('Frequency [GHz]');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# characteristic impedance of the measurement chain\n",
    "Z0 = 50"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def ABCDtoS(ABCD):\n",
    "    A, B = ABCD[0]\n",
    "    C, D = ABCD[1]\n",
    "    denom = (A + B / Z0 + C * Z0 + D)\n",
    "    S11 = (A + B / Z0 - C * Z0 - D) / denom\n",
    "    S12 = 2 * (A * D - B * C) / denom\n",
    "    S21 = 2 / denom\n",
    "    S22 = (- A + B / Z0 - C * Z0 + D) / denom\n",
    "    return np.array([[S11, S12], [S21, S22]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The transmission amplitude of the bare photonic crystal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, I calculate the transmission amplitude of the bare photonic crystal (14 cells) using the transfer matrix method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "M_unit_sym_ = M_unit_sym.subs(dict_items).subs({v_sym[0]: v_p, v_sym[1]: v_p})\n",
    "M_unit = sp.lambdify(omega_sym, M_unit_sym_, \"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/local/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/site-packages/ipykernel/__main__.py:8: ComplexWarning: Casting complex values to real discards the imaginary part\n"
     ]
    }
   ],
   "source": [
    "N_cell = 14    # Number of unit cells to calculate\n",
    "N_freqPts = 100\n",
    "freqPts = np.linspace(5.5, 8.5, N_freqPts)   # frequency points in units of GHz\n",
    "\n",
    "S12dB = np.zeros(N_freqPts)\n",
    "for idx, freq_ in enumerate(freqPts):\n",
    "    S12dB[idx] = 20 * np.log10(ABCDtoS(la.matrix_power(\n",
    "                    M_unit(2e9*np.pi*freq_), N_cell))[0, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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K46ab4Cc/sWVst94KffrEbZETA4mfbQmTeKOx9JhRJ1ojJshRWgEJ59eRg8mT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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c9555c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1, 1, figsize=(5, 3))\n",
    "ax.plot(freqPts, S12dB, color='red', lw=2)\n",
    "ax.set_xlabel('Frequency [GHz]')\n",
    "ax.set_ylabel('$S_{12}$ [dB]')\n",
    "ax.set_ylim([-40, 5]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numerical Calculation of the band and mode structures from the differential equation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### PDE Solving using the discretization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To analyze the mode structure (Bloch wavefunction) of an infinite microwave photonic crystal, we solve the following series of differential equations\n",
    "\n",
    "$$\n",
    "\\frac{\\partial}{\\partial x} V(x, t) = -l(x) \\frac{\\partial}{\\partial t} I(x,t), \\quad \\frac{\\partial}{\\partial x} I(x, t) = -c(x) \\frac{\\partial}{\\partial t} V(x,t)\n",
    "$$\n",
    "\n",
    "which reduces to a single partial differential equation of $V(x, t)$\n",
    "\n",
    "$$\n",
    "\\frac{\\partial}{\\partial x} \\left[ \\frac{v_p}{Z_c(x)} \\frac{\\partial V(x,t)}{\\partial x}\\right]\n",
    "= \\frac{1}{v_p Z_c(x)} \\frac{\\partial^2}{\\partial t^2} V(x,t)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The characteristic impedance $Z_c(x) = \\sqrt{l(x)/c(x)}$ inside the unit cell ($-d/2< x < d/2$) is let to be\n",
    "$$\n",
    "Z_c (x) = \n",
    "$$\n",
    "while the phase velocity $v_p = 1/\\sqrt{l(x)c(x)}$ is a constant."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The time-independent ODE is obtained by separation of variables $V(x, t) = V(x) e^{i\\omega t}$\n",
    "$$\n",
    "\\frac{d}{dx} \\left[\\frac{1}{Z_c(x)} \\frac{dV(x)}{dx}\\right] = - \\left(\\frac{\\omega}{v_p}\\right)^2 \\frac{V(x)}{Z_c(x)}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We solve the partial differential equation\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For numerical calculation of the mode structure, we impose the periodic boundary condition $Z_c(x) = Z_c(x+d)$ Fourier series expand $1/Z_c(x)$ as\n",
    "$$\n",
    "\\frac{1}{Z_c (x)} = \\sum_{m} \\eta_{m} e^{i2\\pi m x/d}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here the Fourier coefficients $\\eta_m$ are written as\n",
    "$$\\eta_m = \\frac{1}{d} \\int_{-d/2}^{d/2} dx \\frac{e^{-i 2\\pi m x/d}}{Z_c(x)}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Characteristic impedance inside the unit cell\n",
    "def Zc(x):\n",
    "    return Z[1] * ((x > - l[1] / 2) * (x < l[1] / 2)) \\\n",
    "           + Z[0] * ((x < - l[1] / 2) + (x > l[1] / 2))\n",
    "\n",
    "def eta(n_):\n",
    "    \"\"\"\n",
    "    The n-th discrete Fourier coefficient of 1/Zc(x).\n",
    "    Imaginary part vanishes due to the even parity of Zc(x)\n",
    "    with respect to the x = 0 plane.\n",
    "    \"\"\"\n",
    "    integrand = lambda x: (1 / Zc(x) *\n",
    "                           np.exp(- 2j * np.pi * n_ * x / d))\n",
    "    return quad(integrand, - d / 2, d / 2)[0] / d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "N_xPts = 100\n",
    "xPts = np.linspace(- d / 2, d / 2, N_xPts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/local/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/site-packages/scipy/integrate/quadpack.py:380: ComplexWarning: Casting complex values to real discards the imaginary part\n",
      "  return _quadpack._qagse(func,a,b,args,full_output,epsabs,epsrel,limit)\n"
     ]
    }
   ],
   "source": [
    "# discrete Fourier space dimension\n",
    "L = 10\n",
    "n = np.arange(- 2 * L, 2 * L + 1)\n",
    "\n",
    "# \n",
    "eta_array = np.zeros(4 * L + 1)\n",
    "for n_ in n:\n",
    "    eta_array[n_ + 2 * L] = eta(n_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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QnjezXanHnZ1xT5rGINJeVU5jiPm+TwAfAjvS29z9Q0LA+wPghJn9aekNyODu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\niIwtBbsiIiIiMrb+fy/zqvE/YUePAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c9c8ef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(1, 2, figsize=(10, 4))\n",
    "\n",
    "axes[0].plot(xPts / d, Zc(xPts), color=fig_colors[0], lw=2)\n",
    "axes[0].set_xlabel('Position in unit cell $x/d$')\n",
    "axes[0].set_ylabel(r'Characteristic Impedance $Z_c$   [$\\Omega$]')\n",
    "axes[0].set_xlim([- 1/2, 1/2]);\n",
    "\n",
    "axes[1].bar(n, eta_array, color=fig_colors[2]);\n",
    "axes[1].set_xlabel(r'$m$')\n",
    "axes[1].set_xlim([- 2 * L, 2 * L]);\n",
    "axes[1].set_ylabel(r'Discrete Fourier Coefficient $\\eta_m$  [$\\Omega^{-1}$]')\n",
    "fig.tight_layout();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To solve this differential equation, we introduce eigenfunctions of the Bloch form $V_k(x) = u_k(x) e^{ikx}$ which satisfies $u_k(x+d) = u_k(x)$ and thus can be Fourier series expanded as\n",
    "$$ u_k(x) = \\sum_{m} c_{k,m} e^{i2\\pi m x/d}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{d}{dx} \\left[\\sum_{m} \\eta_{m} e^{i2\\pi m x/d} \\frac{d}{dx}\\sum_{m'} c_{k,m'} e^{i(k+2\\pi m' /d)x}\\right] = - \\left(\\frac{\\omega}{v_p}\\right)^2 \\sum_{m} \\eta_{m} e^{i2\\pi m x/d}\\sum_{m'} c_{k,m'} e^{i(k+2\\pi m' /d)x}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{d}{dx} \\left[\\sum_{m,m'} \\eta_{m} i\\left(k+\\frac{2\\pi m'}{d}\\right)c_{k,m'} e^{i[k+2\\pi (m+m') /d]x}\\right] = - \\left(\\frac{\\omega}{v_p}\\right)^2 \\sum_{m,m'} \\eta_{m} c_{k,m'} e^{i[k+2\\pi (m+m') /d)]x}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\sum_{m,m'} \\eta_{m} i\\left(k+\\frac{2\\pi m'}{d}\\right)i\\left(k+\\frac{2\\pi (m+m')}{d}\\right) c_{k,m'} e^{i[k+2\\pi (m+m') /d]x} = - \\left(\\frac{\\omega}{v_p}\\right)^2 \\sum_{m,m'} \\eta_{m} c_{k,m'} e^{i[k+2\\pi (m+m') /d)]x}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We introduce a new index $n$ to replace $m = n-m'$\n",
    "$$\n",
    "-\\sum_{m', n} \\eta_{n-m'} \\left(k+\\frac{2\\pi m'}{d}\\right)\\left(k+\\frac{2\\pi n}{d}\\right) c_{k,m'} e^{i(k+2\\pi n /d)x} = - \\left(\\frac{\\omega}{v_p}\\right)^2 \\sum_{m',n} \\eta_{n-m'} c_{k,m'} e^{i(k+ 2\\pi n /d)x}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Applying the inverse Fourier transform $\\frac{1}{d}\\int_{-d/2}^{d/2} dx \\ e^{-i(k+2\\pi m/d)x}$,\n",
    "$$\n",
    "-\\sum_{m', n} \\eta_{n-m'} \\left(k+\\frac{2\\pi m'}{d}\\right)\\left(k+\\frac{2\\pi n}{d}\\right) c_{k,m'}\\left[ \\frac{1}{d}\\int_{-d/2}^{d/2}dx\\ e^{i2\\pi (n-m)x /d }\\right] = - \\left(\\frac{\\omega}{v_p}\\right)^2 \\sum_{m',n} \\eta_{n-m'} c_{k,m'}\\left[ \\frac{1}{d}\\int_{-d/2}^{d/2}dx\\ e^{i2\\pi (n-m)x /d }\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The delta function identity\n",
    "$$\n",
    "\\frac{1}{d}\\int_{-d/2}^{d/2} dx\\ e^{i2\\pi (n-m)x /d} = \\delta_{n,m}\n",
    "$$\n",
    "reduces the equation into a summation only in one index $m'$\n",
    "$$\n",
    "\\sum_{m'} \\eta_{m-m'} \\left(k + \\frac{2\\pi m}{d}\\right)\\left(k + \\frac{2\\pi m'}{d}\\right) c_{k, m'}\n",
    "=  \\left(\\frac{\\omega}{v_p}\\right)^2 \\sum_{m'} \\eta_{m-m'} c_{k,m'}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By identifying the matrices $\\mathbf{A}$, $\\mathbf{B}$, and $\\mathbf{c}$ as\n",
    "\n",
    "$$[\\mathbf{A}]_{m, m'} = \\eta_{m-m'},\\quad\n",
    "[\\mathbf{B}]_{m, m'} = \\eta_{m-m'} \\left(k + \\frac{2\\pi m}{d}\\right) \\left(k + \\frac{2\\pi m'}{d}\\right),\\quad\n",
    "[\\mathbf{c}]_m' = c_{k, m'},$$\n",
    "\n",
    "This is a matrix equation $\\mathbf{B}\\mathbf{c} = (\\omega/v_p)^2 \\mathbf{A}\\mathbf{c}$ which can be viewed as an eigenequation\n",
    "$$(\\mathbf{A}^{-1}\\mathbf{B})\\mathbf{c} = \\left(\\frac{\\omega}{v_p}\\right)^2 \\mathbf{c}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "A = np.zeros((2 * L + 1, 2 * L + 1))\n",
    "for i in range(2 * L + 1):\n",
    "    for j in range(2 * L + 1):\n",
    "        m, m_ = i - L, j - L   # index offset\n",
    "        A[i, j] = eta_array[m - m_ + 2 * L]\n",
    "\n",
    "def B(k):\n",
    "    temp = np.zeros((2 * L + 1, 2 * L + 1))\n",
    "    for i in range(2 * L + 1):\n",
    "        for j in range(2 * L + 1):\n",
    "            m, m_ = i - L, j - L   # index offset\n",
    "            temp[i, j] = (eta_array[m - m_ + 2 * L]\n",
    "                          * (k + 2 * np.pi * m / d)\n",
    "                          * (k + 2 * np.pi * m_ / d))\n",
    "    return temp\n",
    "\n",
    "def Op(k):\n",
    "    return np.dot(la.inv(A), B(k))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def solve_eigenequation(k):\n",
    "    eig_vals, eig_vecs = la.eig(Op(k))\n",
    "    # sorting eigenvalues and eigenvectors\n",
    "    idx = eig_vals.argsort()\n",
    "    eig_vals = eig_vals[idx]\n",
    "    eig_vecs = eig_vecs[:,idx]\n",
    "    \n",
    "    eig_freq = np.sqrt(eig_vals) * v_p / (2 * np.pi)\n",
    "    return eig_freq, eig_vecs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def Bloch_wavefunc(x, c):\n",
    "    \"\"\"`1\n",
    "    Given the amplitudes [c],\n",
    "    Calculate the Bloch wavefunction\n",
    "\n",
    "        u_k (x) = sum_{n} c_{k, n} exp(i2\\pi n x/d)\n",
    "\n",
    "    and return its absolute value.\n",
    "    \"\"\"\n",
    "    V = np.zeros(len(x))\n",
    "    for n in range(2 * L + 1):\n",
    "        V = V + c[n] * np.exp(2j * np.pi * (n - L) * x / d)\n",
    "    return abs(V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X, Y = np.meshgrid(kPts, xPts)\n",
    "\n",
    "N_modes = 2  # Number of modes to consider\n",
    "eig_funcs = []\n",
    "eig_freqs = []\n",
    "for n in range(N_modes):\n",
    "    eig_funcs.append(np.zeros((len(kPts), len(xPts))))\n",
    "    eig_freqs.append(np.zeros(len(kPts)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "for idx, k_ in enumerate(kPts):\n",
    "    eig_vals, eig_vecs = solve_eigenequation(k_)\n",
    "    for n in range(N_modes):\n",
    "        eig_funcs[n][:, idx] = Bloch_wavefunc(xPts, eig_vecs[:, n])\n",
    "        eig_freqs[n][idx] = eig_vals[n]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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hBmWMKRSRZK/3qcB4Eenjp66Om4qiBKQ6jJnRiCr1enOKK0SEd756h5c/fZklXyxh9/7d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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10cb89b38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(2, 2, figsize=(11, 8))\n",
    "\n",
    "# Band structure\n",
    "for n in range(N_modes):\n",
    "    axes[0, 0].plot(kPts/(np.pi/d), eig_freqs[n]/1e9, label=r'$n=%d$' % n, lw=2)\n",
    "axes[0, 0].set_xlabel(r'Wavevector  $k/(\\pi/d)$')\n",
    "axes[0, 0].set_ylabel(r'Frequency $\\omega/2\\pi$   [GHz]')\n",
    "axes[0, 0].set_title('Band Structure')\n",
    "axes[0, 0].legend(loc=0)\n",
    "\n",
    "# Transmission Amplitude\n",
    "axes[0, 1].plot(freqPts, S12dB, color='red', lw=2)\n",
    "axes[0, 1].set_xlabel('Frequency [GHz]')\n",
    "axes[0, 1].set_ylabel('$S_{12}$ [dB]')\n",
    "axes[0, 1].set_ylim([-40, 5])\n",
    "axes[0, 1].set_title('Transmission Amplitude')\n",
    "\n",
    "# Mode structure\n",
    "p0 = axes[1, 0].pcolor(X/(np.pi/d), Y/d, eig_funcs[0], cmap=cm.jet, vmin=0, vmax=1.5)\n",
    "p1 = axes[1, 1].pcolor(X/(np.pi/d), Y/d, eig_funcs[1], cmap=cm.jet, vmin=0, vmax=1.5)\n",
    "axes[1, 0].set_xlabel(r'Wavevector  $k/(\\pi/d)$'); axes[1, 1].set_xlabel(r'Wavevector  $k/(\\pi/d)$')\n",
    "axes[1, 0].set_ylabel(r'Position in unit cell  $x/d$')\n",
    "axes[1, 0].set_xlim([-1, 1]); axes[1, 1].set_xlim([-1, 1])\n",
    "axes[1, 0].set_ylim([-0.5, 0.5]); axes[1, 1].set_ylim([-0.5, 0.5])\n",
    "axes[1, 0].set_title(r'Mode structure ($n=0$ band)')\n",
    "axes[1, 1].set_title(r'Mode structure ($n=1$ band)');\n",
    "fig.tight_layout()\n",
    "\n",
    "fig.subplots_adjust(right=0.85)\n",
    "cbar_ax = fig.add_axes([0.9, 0.15, 0.02, 0.3])\n",
    "fig.colorbar(p1, cax=cbar_ax);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All panels of Figure S1 has been reproduced."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Transmission amplitude in the presence of the qubit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "M_a_sym = qubit_ABCD_sym(omega_sym).subs({gamma_sym: 2e9 * np.pi * 0.5, Z0_sym: 28})\n",
    "M_a = sp.lambdify((omega_a_sym, omega_sym), M_a_sym, \"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/local/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/site-packages/ipykernel/__main__.py:9: ComplexWarning: Casting complex values to real discards the imaginary part\n"
     ]
    }
   ],
   "source": [
    "omega_a = 2e9 * np.pi * 8.5\n",
    "freqPts = np.linspace(7.2, 8.9, N_freqPts)   # Frequency Points in units of GHz\n",
    "S12dB = np.zeros(N_freqPts)\n",
    "\n",
    "for idx, freq in enumerate(freqPts):\n",
    "    omega_ = 2e9 * np.pi * freq\n",
    "    M_PhC = la.matrix_power(M_unit(omega_), 7)\n",
    "    M = np.dot(M_PhC, np.dot(M_a(omega_a, omega_), M_PhC))\n",
    "    S12dB[idx] = 20 * np.log10(ABCDtoS(M)[0][1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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DAVT1isQxi7BECV3h86rVLPKAizl9Is/cbmF2rsUMXBkXr4isjWcFDYkSHtBQ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\ncDE7Tkn4/8YyftRFPbvRAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c9bd9e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1, 1, figsize=(3, 5))\n",
    "ax.plot(freqPts, S12dB, lw=2, color='blue')\n",
    "ax.set_xlabel(r'Frequency [GHz]')\n",
    "ax.set_ylabel(r'$S_{12}$ [dB]');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Version Information"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "application/json": {
       "Software versions": [
        {
         "module": "Python",
         "version": "3.4.5 64bit [GCC 4.2.1 Compatible Apple LLVM 7.0.2 (clang-700.1.81)]"
        },
        {
         "module": "IPython",
         "version": "5.0.0"
        },
        {
         "module": "OS",
         "version": "Darwin 15.6.0 x86_64 i386 64bit"
        },
        {
         "module": "sympy",
         "version": "1.0"
        },
        {
         "module": "numpy",
         "version": "1.11.1"
        },
        {
         "module": "scipy",
         "version": "0.18.0"
        },
        {
         "module": "matplotlib",
         "version": "1.5.2"
        }
       ]
      },
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>3.4.5 64bit [GCC 4.2.1 Compatible Apple LLVM 7.0.2 (clang-700.1.81)]</td></tr><tr><td>IPython</td><td>5.0.0</td></tr><tr><td>OS</td><td>Darwin 15.6.0 x86_64 i386 64bit</td></tr><tr><td>sympy</td><td>1.0</td></tr><tr><td>numpy</td><td>1.11.1</td></tr><tr><td>scipy</td><td>0.18.0</td></tr><tr><td>matplotlib</td><td>1.5.2</td></tr><tr><td colspan='2'>Tue Aug 23 00:33:34 2016 PDT</td></tr></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{|l|l|}\\hline\n",
       "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
       "Python & 3.4.5 64bit [GCC 4.2.1 Compatible Apple LLVM 7.0.2 (clang-700.1.81)] \\\\ \\hline\n",
       "IPython & 5.0.0 \\\\ \\hline\n",
       "OS & Darwin 15.6.0 x86\\_64 i386 64bit \\\\ \\hline\n",
       "sympy & 1.0 \\\\ \\hline\n",
       "numpy & 1.11.1 \\\\ \\hline\n",
       "scipy & 0.18.0 \\\\ \\hline\n",
       "matplotlib & 1.5.2 \\\\ \\hline\n",
       "\\hline \\multicolumn{2}{|l|}{Tue Aug 23 00:33:34 2016 PDT} \\\\ \\hline\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "Software versions\n",
       "Python 3.4.5 64bit [GCC 4.2.1 Compatible Apple LLVM 7.0.2 (clang-700.1.81)]\n",
       "IPython 5.0.0\n",
       "OS Darwin 15.6.0 x86_64 i386 64bit\n",
       "sympy 1.0\n",
       "numpy 1.11.1\n",
       "scipy 0.18.0\n",
       "matplotlib 1.5.2\n",
       "Tue Aug 23 00:33:34 2016 PDT"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%reload_ext version_information\n",
    "\n",
    "%version_information sympy, numpy, scipy, matplotlib"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}