AtomicFountainTheoryBeamsplitter.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Theory for Atomic Fountain Interferometer - Beamsplitter"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:53.528044Z",
     "start_time": "2019-06-02T16:24:53.523753Z"
    }
   },
   "outputs": [],
   "source": [
    "from functools import partial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:56.785143Z",
     "start_time": "2019-06-02T16:24:53.531275Z"
    }
   },
   "outputs": [],
   "source": [
    "import sympy\n",
    "import numpy as np\n",
    "import matplotlib\n",
    "import matplotlib.pylab as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:56.795450Z",
     "start_time": "2019-06-02T16:24:56.788509Z"
    }
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.147710Z",
     "start_time": "2019-06-02T16:24:56.801443Z"
    }
   },
   "outputs": [],
   "source": [
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.162516Z",
     "start_time": "2019-06-02T16:24:57.152840Z"
    }
   },
   "outputs": [],
   "source": [
    "from sympy import Wild, symbols, Function, exp, cos, I, DiracDelta, Abs, oo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.171923Z",
     "start_time": "2019-06-02T16:24:57.166731Z"
    }
   },
   "outputs": [],
   "source": [
    "from sympy import UnevaluatedExpr as protect"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.192610Z",
     "start_time": "2019-06-02T16:24:57.176889Z"
    }
   },
   "outputs": [],
   "source": [
    "from symbolic_equation import Eq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.200983Z",
     "start_time": "2019-06-02T16:24:57.197236Z"
    }
   },
   "outputs": [],
   "source": [
    "# local imports\n",
    "from dirac_tools import normalize_dirac_delta_terms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Symbols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.210817Z",
     "start_time": "2019-06-02T16:24:57.204714Z"
    }
   },
   "outputs": [],
   "source": [
    "z, t, tc, k, Δ, μ12, p, p0, α, β = symbols('z, t, t_c, k, Delta, mu_12, p, p0, alpha, beta', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.218976Z",
     "start_time": "2019-06-02T16:24:57.213610Z"
    }
   },
   "outputs": [],
   "source": [
    "ω0, m, ħ, ωk, ωL = symbols('omega_0, m, hbar, omega_k, omega_L', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.240955Z",
     "start_time": "2019-06-02T16:24:57.230052Z"
    }
   },
   "outputs": [],
   "source": [
    "a, b, b̃ = symbols('a, b, b̃', cls=Function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.252828Z",
     "start_time": "2019-06-02T16:24:57.245795Z"
    }
   },
   "outputs": [],
   "source": [
    "δ0, δ0p = symbols(\"delta_0, delta_0^'\", real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.264380Z",
     "start_time": "2019-06-02T16:24:57.255558Z"
    }
   },
   "outputs": [],
   "source": [
    "ϕ1, ϕ2, ϕ3, ϕp, ϕm, ϕd, η = symbols(\"phi_1, phi_2, phi_3, phi_+, phi_-, phi_d, eta\", cls=Function, real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.275325Z",
     "start_time": "2019-06-02T16:24:57.266863Z"
    }
   },
   "outputs": [],
   "source": [
    "δ12, δ13, δ23 = symbols(\"delta12, delta_13, delta_23\", cls=Function, real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.285904Z",
     "start_time": "2019-06-02T16:24:57.278197Z"
    }
   },
   "outputs": [],
   "source": [
    "ω1, ω2, ω3, Enp, Enm = symbols(\"omega_1, omega_2, omega_3, E_n>0, E_n<0\", cls=Function, real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.295972Z",
     "start_time": "2019-06-02T16:24:57.291346Z"
    }
   },
   "outputs": [],
   "source": [
    "class _OmegaSym:\n",
    "    def __getitem__(self, key):\n",
    "        repl = {\n",
    "            '-k^(+)': '+',\n",
    "            '-k^(-)': '-',\n",
    "        }\n",
    "        key = repl.get(key, key)\n",
    "        return symbols('Omega_%s' % key, cls=Function, positive=True)\n",
    "Ω = _OmegaSym()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.302121Z",
     "start_time": "2019-06-02T16:24:57.298740Z"
    }
   },
   "outputs": [],
   "source": [
    "n, n0, nprime = symbols(\"n, n_0, n'\", integer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.310182Z",
     "start_time": "2019-06-02T16:24:57.304758Z"
    }
   },
   "outputs": [],
   "source": [
    "_w = Wild('w')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.316262Z",
     "start_time": "2019-06-02T16:24:57.312587Z"
    }
   },
   "outputs": [],
   "source": [
    "defs = {}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.323220Z",
     "start_time": "2019-06-02T16:24:57.318771Z"
    }
   },
   "outputs": [],
   "source": [
    "def definition(sym):\n",
    "    return Eq(sym, defs[sym])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.331320Z",
     "start_time": "2019-06-02T16:24:57.325698Z"
    }
   },
   "outputs": [],
   "source": [
    "def standard_phase_form(expr):\n",
    "    \"\"\"Combine exponentials and separate out $i$ and $t$.\n",
    "    This ensures the standard form $e^{i ω t)$\n",
    "    \"\"\"\n",
    "    return (\n",
    "        expr\n",
    "        .powsimp()\n",
    "        .replace(exp, lambda arg: exp(arg.collect(I)))\n",
    "        .replace(exp, lambda arg: exp(arg.collect(t)))\n",
    "    )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Schrödinger equation in position space"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**NOTE**: The original notes on the derivation of the Hamiltonian used a convention where the signs for the phases in $\\Omega_{-}$ and $\\Omega_{+}$ used opposite signs. Here, we have used a sign convention that leads to fully symmetric equations. The only things that changes is that any $\\phi_3(t)$ and any $\\delta_0^\\prime$ change signs. Any lines that are affected by this are marked with a comment `# SIGN`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.419247Z",
     "start_time": "2019-06-02T16:24:57.333867Z"
    }
   },
   "outputs": [],
   "source": [
    "defs[b̃(z, t)] = (\n",
    "    1 / (2 * Δ) * (\n",
    "        Ω['k'](t) * exp(-I * (ϕ1(t) + k*z))\n",
    "        + Ω['-k^(+)'](t) * exp(-I * (ϕ2(t) + δ0 * t - k * z))\n",
    "        + Ω['-k^(-)'](t) * exp(-I * (ϕ3(t) + δ0p * t - k * z))  # SIGN\n",
    "    ) * a(z, t)\n",
    ").expand().collect(a(z, t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.458671Z",
     "start_time": "2019-06-02T16:24:57.427015Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  \\operatorname{b̃}{\\left(z,t \\right)} = \\left(\\frac{\\Omega_{+}{\\left(t \\right)} e^{- i \\phi_{2}{\\left(t \\right)}} e^{- i \\delta_{0} t} e^{i k z}}{2 \\Delta} + \\frac{\\Omega_{-}{\\left(t \\right)} e^{- i \\phi_{3}{\\left(t \\right)}} e^{- i \\delta^{'}_{0} t} e^{i k z}}{2 \\Delta} + \\frac{\\Omega_{k}{\\left(t \\right)} e^{- i \\phi_{1}{\\left(t \\right)}} e^{- i k z}}{2 \\Delta}\\right) a{\\left(z,t \\right)}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "b̃(z, t) = (Omega_+(t)*exp(-I*phi_2(t))*exp(-I*delta_0*t)*exp(I*k*z)/(2*Delta) + Omega_-(t)*exp(-I*phi_3(t))*exp(-I*delta_0^'*t)*exp(I*k*z)/(2*Delta) + Omega_k(t)*exp(-I*phi_1(t))*exp(-I*k*z)/(2*Delta))*a(z, t)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "definition(b̃(z, t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.513202Z",
     "start_time": "2019-06-02T16:24:57.464162Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{\\partial}{\\partial t} a{\\left(z,t \\right)} = - \\left(\\Omega_{+}{\\left(t \\right)} \\cos{\\left(k z - \\omega_{2}{\\left(t \\right)} \\right)} + \\Omega_{-}{\\left(t \\right)} \\cos{\\left(k z - \\omega_{3}{\\left(t \\right)} \\right)} + \\Omega_{k}{\\left(t \\right)} \\cos{\\left(k z + \\omega_{1}{\\left(t \\right)} \\right)}\\right) b{\\left(t \\right)} + \\frac{p^{2} a{\\left(z,t \\right)}}{2 m}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(z, t), t) = -(Omega_+(t)*cos(k*z - omega_2(t)) + Omega_-(t)*cos(k*z - omega_3(t)) + Omega_k(t)*cos(k*z + omega_1(t)))*b(t) + p**2*a(z, t)/(2*m)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_a = Eq(\n",
    "    I * ħ * a(z, t).diff(t),\n",
    "    p**2/(2*m) * a(z, t) - (\n",
    "        Ω['k'](t) * cos(ω1(t) + k * z)\n",
    "        + Ω['-k^(+)'](t) * cos(ω2(t) - k * z)\n",
    "        + Ω['-k^(-)'](t) * cos(ω3(t) - k * z)\n",
    "    ) * b(t)\n",
    ")\n",
    "TDSE_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.963861Z",
     "start_time": "2019-06-02T16:24:57.516293Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{\\partial}{\\partial t} a{\\left(z,t \\right)} = \\left(\\frac{p^{2}}{2 m} - \\frac{\\Omega_{+}^{2}{\\left(t \\right)}}{4 \\Delta} - \\frac{\\Omega_{+}{\\left(t \\right)} \\Omega_{-}{\\left(t \\right)} e^{i \\left(t \\left(- \\delta_{0} + \\delta^{'}_{0}\\right) - \\phi_{2}{\\left(t \\right)} + \\phi_{3}{\\left(t \\right)}\\right)}}{4 \\Delta} - \\frac{\\Omega_{+}{\\left(t \\right)} \\Omega_{-}{\\left(t \\right)} e^{i \\left(t \\left(\\delta_{0} - \\delta^{'}_{0}\\right) + \\phi_{2}{\\left(t \\right)} - \\phi_{3}{\\left(t \\right)}\\right)}}{4 \\Delta} - \\frac{\\Omega_{+}{\\left(t \\right)} \\Omega_{k}{\\left(t \\right)} e^{i \\left(- \\delta_{0} t + 2 k z + \\phi_{1}{\\left(t \\right)} - \\phi_{2}{\\left(t \\right)}\\right)}}{4 \\Delta} - \\frac{\\Omega_{+}{\\left(t \\right)} \\Omega_{k}{\\left(t \\right)} e^{i \\left(\\delta_{0} t - 2 k z - \\phi_{1}{\\left(t \\right)} + \\phi_{2}{\\left(t \\right)}\\right)}}{4 \\Delta} - \\frac{\\Omega_{-}^{2}{\\left(t \\right)}}{4 \\Delta} - \\frac{\\Omega_{-}{\\left(t \\right)} \\Omega_{k}{\\left(t \\right)} e^{i \\left(- \\delta^{'}_{0} t + 2 k z + \\phi_{1}{\\left(t \\right)} - \\phi_{3}{\\left(t \\right)}\\right)}}{4 \\Delta} - \\frac{\\Omega_{-}{\\left(t \\right)} \\Omega_{k}{\\left(t \\right)} e^{i \\left(\\delta^{'}_{0} t - 2 k z - \\phi_{1}{\\left(t \\right)} + \\phi_{3}{\\left(t \\right)}\\right)}}{4 \\Delta} - \\frac{\\Omega_{k}^{2}{\\left(t \\right)}}{4 \\Delta}\\right) a{\\left(z,t \\right)}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(z, t), t) = (p**2/(2*m) - Omega_+(t)**2/(4*Delta) - Omega_+(t)*Omega_-(t)*exp(I*(t*(-delta_0 + delta_0^') - phi_2(t) + phi_3(t)))/(4*Delta) - Omega_+(t)*Omega_-(t)*exp(I*(t*(delta_0 - delta_0^') + phi_2(t) - phi_3(t)))/(4*Delta) - Omega_+(t)*Omega_k(t)*exp(I*(-delta_0*t + 2*k*z + phi_1(t) - phi_2(t)))/(4*Delta) - Omega_+(t)*Omega_k(t)*exp(I*(delta_0*t - 2*k*z - phi_1(t) + phi_2(t)))/(4*Delta) - Omega_-(t)**2/(4*Delta) - Omega_-(t)*Omega_k(t)*exp(I*(-delta_0^'*t + 2*k*z + phi_1(t) - phi_3(t)))/(4*Delta) - Omega_-(t)*Omega_k(t)*exp(I*(delta_0^'*t - 2*k*z - phi_1(t) + phi_3(t)))/(4*Delta) - Omega_k(t)**2/(4*Delta))*a(z, t)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_a_2 = (\n",
    "    TDSE_a\n",
    "    .apply_mtd(\n",
    "        'subs',\n",
    "        {\n",
    "            b(t): defs[b̃(z, t)] * exp(-I * ω0 * t),\n",
    "            ω1(t): ω0 * t + ϕ1(t),\n",
    "            ω2(t): ω0 * t + ϕ2(t) + δ0 * t,\n",
    "            ω3(t): ω0 * t + ϕ3(t) + δ0p * t,  # SIGN\n",
    "        })\n",
    "     .apply_mtd('rewrite', exp, cont=False)\n",
    "     .apply_mtd('expand')\n",
    "     .apply_mtd('subs', {exp(-2 * I * ω0 * t): 0}) # neglect counter-rotating term\n",
    "     .apply(standard_phase_form)\n",
    "     .apply_mtd('collect', a(z, t))\n",
    ")\n",
    "TDSE_a_2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduce relative phases"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:57.973677Z",
     "start_time": "2019-06-02T16:24:57.967754Z"
    }
   },
   "outputs": [],
   "source": [
    "defs.update({\n",
    "    δ12(t): ϕ1(t) - ϕ2(t),\n",
    "    δ13(t): ϕ1(t) - ϕ3(t),  # SIGN\n",
    "    δ23(t): ϕ2(t) - ϕ3(t),  # SIGN\n",
    "})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:58.145828Z",
     "start_time": "2019-06-02T16:24:57.975861Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{\\partial}{\\partial t} a{\\left(z,t \\right)} = \\left(\\frac{p^{2}}{2 m} - \\frac{\\Omega_{+}^{2}{\\left(t \\right)}}{4 \\Delta} - \\frac{\\Omega_{+}{\\left(t \\right)} \\Omega_{-}{\\left(t \\right)} e^{i \\left(t \\left(- \\delta_{0} + \\delta^{'}_{0}\\right) - \\delta_{23}{\\left(t \\right)}\\right)}}{4 \\Delta} - \\frac{\\Omega_{+}{\\left(t \\right)} \\Omega_{-}{\\left(t \\right)} e^{i \\left(t \\left(\\delta_{0} - \\delta^{'}_{0}\\right) + \\delta_{23}{\\left(t \\right)}\\right)}}{4 \\Delta} - \\frac{\\Omega_{+}{\\left(t \\right)} \\Omega_{k}{\\left(t \\right)} e^{i \\left(- \\delta_{0} t + 2 k z + \\delta_{12}{\\left(t \\right)}\\right)}}{4 \\Delta} - \\frac{\\Omega_{+}{\\left(t \\right)} \\Omega_{k}{\\left(t \\right)} e^{i \\left(\\delta_{0} t - 2 k z - \\delta_{12}{\\left(t \\right)}\\right)}}{4 \\Delta} - \\frac{\\Omega_{-}^{2}{\\left(t \\right)}}{4 \\Delta} - \\frac{\\Omega_{-}{\\left(t \\right)} \\Omega_{k}{\\left(t \\right)} e^{i \\left(- \\delta^{'}_{0} t + 2 k z + \\delta_{13}{\\left(t \\right)}\\right)}}{4 \\Delta} - \\frac{\\Omega_{-}{\\left(t \\right)} \\Omega_{k}{\\left(t \\right)} e^{i \\left(\\delta^{'}_{0} t - 2 k z - \\delta_{13}{\\left(t \\right)}\\right)}}{4 \\Delta} - \\frac{\\Omega_{k}^{2}{\\left(t \\right)}}{4 \\Delta}\\right) a{\\left(z,t \\right)}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(z, t), t) = (p**2/(2*m) - Omega_+(t)**2/(4*Delta) - Omega_+(t)*Omega_-(t)*exp(I*(t*(-delta_0 + delta_0^') - delta_23(t)))/(4*Delta) - Omega_+(t)*Omega_-(t)*exp(I*(t*(delta_0 - delta_0^') + delta_23(t)))/(4*Delta) - Omega_+(t)*Omega_k(t)*exp(I*(-delta_0*t + 2*k*z + delta12(t)))/(4*Delta) - Omega_+(t)*Omega_k(t)*exp(I*(delta_0*t - 2*k*z - delta12(t)))/(4*Delta) - Omega_-(t)**2/(4*Delta) - Omega_-(t)*Omega_k(t)*exp(I*(-delta_0^'*t + 2*k*z + delta_13(t)))/(4*Delta) - Omega_-(t)*Omega_k(t)*exp(I*(delta_0^'*t - 2*k*z - delta_13(t)))/(4*Delta) - Omega_k(t)**2/(4*Delta))*a(z, t)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_a_3 = TDSE_a_2.apply_mtd('subs', {defs[sym]: sym for sym in [δ12(t), δ13(t), δ23(t)]})\n",
    "TDSE_a_3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define common light shift and effective envelopes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:58.176585Z",
     "start_time": "2019-06-02T16:24:58.149692Z"
    }
   },
   "outputs": [],
   "source": [
    "defs.update(\n",
    "    {\n",
    "        Ω['Δ'](t): (\n",
    "            (\n",
    "                Ω['k'](t) ** 2\n",
    "                + Ω['-k^(+)'](t) ** 2\n",
    "                + Ω['-k^(-)'](t) ** 2\n",
    "                + (\n",
    "                    2\n",
    "                    * Ω['-k^(+)'](t)\n",
    "                    * Ω['-k^(-)'](t)\n",
    "                    * cos(δ23(t) + (δ0 - δ0p) * t)  # SIGN\n",
    "                )\n",
    "            )\n",
    "            / (4 * Δ)\n",
    "        ).expand(),\n",
    "        Ω['-eff'](t): ((Ω['k'](t) * Ω['-k^(-)'](t)) / (4 * Δ)),\n",
    "        Ω['+eff'](t): ((Ω['k'](t) * Ω['-k^(+)'](t)) / (4 * Δ)),\n",
    "        ϕp(t): δ13(t) - δ0p * t, # SIGN\n",
    "        ϕm(t): δ12(t) - δ0 * t,\n",
    "    }\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:58.207500Z",
     "start_time": "2019-06-02T16:24:58.179760Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  \\Omega_{Δ}{\\left(t \\right)} = \\frac{\\Omega_{+}^{2}{\\left(t \\right)}}{4 \\Delta} + \\frac{\\Omega_{+}{\\left(t \\right)} \\Omega_{-}{\\left(t \\right)} \\cos{\\left(\\delta_{0} t - \\delta^{'}_{0} t + \\delta_{23}{\\left(t \\right)} \\right)}}{2 \\Delta} + \\frac{\\Omega_{-}^{2}{\\left(t \\right)}}{4 \\Delta} + \\frac{\\Omega_{k}^{2}{\\left(t \\right)}}{4 \\Delta}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "Omega_Δ(t) = Omega_+(t)**2/(4*Delta) + Omega_+(t)*Omega_-(t)*cos(delta_0*t - delta_0^'*t + delta_23(t))/(2*Delta) + Omega_-(t)**2/(4*Delta) + Omega_k(t)**2/(4*Delta)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "definition(Ω['Δ'](t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:58.222614Z",
     "start_time": "2019-06-02T16:24:58.209903Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  \\Omega_{-eff}{\\left(t \\right)} = \\frac{\\Omega_{-}{\\left(t \\right)} \\Omega_{k}{\\left(t \\right)}}{4 \\Delta}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "Omega_-eff(t) = Omega_-(t)*Omega_k(t)/(4*Delta)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "definition(Ω['-eff'](t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:58.243361Z",
     "start_time": "2019-06-02T16:24:58.226769Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  \\Omega_{+eff}{\\left(t \\right)} = \\frac{\\Omega_{+}{\\left(t \\right)} \\Omega_{k}{\\left(t \\right)}}{4 \\Delta}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "Omega_+eff(t) = Omega_+(t)*Omega_k(t)/(4*Delta)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "definition(Ω['+eff'](t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:58.266560Z",
     "start_time": "2019-06-02T16:24:58.247210Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{align}\n",
       "  \\phi_{+}{\\left(t \\right)} &= - \\delta^{'}_{0} t + \\delta_{13}{\\left(t \\right)}\\\\\n",
       "   &= - \\delta^{'}_{0} t + \\phi_{1}{\\left(t \\right)} - \\phi_{3}{\\left(t \\right)}\n",
       "\\end{align}\n"
      ],
      "text/plain": [
       "phi_+(t) = -delta_0^'*t + delta_13(t)\n",
       "         = -delta_0^'*t + phi_1(t) - phi_3(t)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "definition(ϕp(t)).apply_mtd('subs', definition(δ13(t)).as_dict, cont=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:58.282790Z",
     "start_time": "2019-06-02T16:24:58.268832Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{align}\n",
       "  \\phi_{-}{\\left(t \\right)} &= - \\delta_{0} t + \\delta_{12}{\\left(t \\right)}\\\\\n",
       "   &= - \\delta_{0} t + \\phi_{1}{\\left(t \\right)} - \\phi_{2}{\\left(t \\right)}\n",
       "\\end{align}\n"
      ],
      "text/plain": [
       "phi_-(t) = -delta_0*t + delta12(t)\n",
       "         = -delta_0*t + phi_1(t) - phi_2(t)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "definition(ϕm(t)).apply_mtd('subs', definition(δ12(t)).as_dict, cont=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**These definitions are counter-intuitive, ase $\\phi_3(t)$ is associated with $\\Omega_{-}(t)$, driving the negative branch. Yet, it determines $\\phi_{+}(t)$**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:58.718777Z",
     "start_time": "2019-06-02T16:24:58.285996Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{\\partial}{\\partial t} a{\\left(z,t \\right)} = \\left(\\left(- \\Omega_{+eff}{\\left(t \\right)} e^{- i \\phi_{-}{\\left(t \\right)}} - \\Omega_{-eff}{\\left(t \\right)} e^{- i \\phi_{+}{\\left(t \\right)}}\\right) e^{- 2 i k z} + \\left(- \\Omega_{+eff}{\\left(t \\right)} e^{i \\phi_{-}{\\left(t \\right)}} - \\Omega_{-eff}{\\left(t \\right)} e^{i \\phi_{+}{\\left(t \\right)}}\\right) e^{2 i k z} - \\Omega_{Δ}{\\left(t \\right)} + \\frac{p^{2}}{2 m}\\right) a{\\left(z,t \\right)}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(z, t), t) = ((-Omega_+eff(t)*exp(-I*phi_-(t)) - Omega_-eff(t)*exp(-I*phi_+(t)))*exp(-2*I*k*z) + (-Omega_+eff(t)*exp(I*phi_-(t)) - Omega_-eff(t)*exp(I*phi_+(t)))*exp(2*I*k*z) - Omega_Δ(t) + p**2/(2*m))*a(z, t)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_a_4 = (\n",
    "    TDSE_a_3.apply_mtd(\n",
    "        'subs',\n",
    "        {standard_phase_form(defs[Ω['Δ'](t)].rewrite('exp').expand()): Ω['Δ'](t)},\n",
    "    )\n",
    "    .apply_mtd(\n",
    "        'subs',\n",
    "        {defs[sym]: sym for sym in [Ω['-eff'](t), Ω['+eff'](t), ϕp(t), ϕm(t)]},\n",
    "    )\n",
    "    .apply_mtd('expand')\n",
    "    .apply_mtd('collect', a(z, t))\n",
    "    .apply_mtd('collect', exp(2 * I * k * z))\n",
    "    .apply(standard_phase_form)\n",
    ")\n",
    "TDSE_a_4\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Transformation to momentum space"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mathematically, we take a Fourier transform:\n",
    "\n",
    "\\begin{align}\n",
    "\\int \\textrm{d}z\\, e^{-i \\frac{p}{\\hbar} z} a(z, t) &= a(p, t) \\\\\n",
    "\\int \\textrm{d}z\\, e^{-i \\frac{p}{\\hbar} z \\pm i 2 k z} a(z, t) &= a(p\\mp 2 \\hbar k, t)\n",
    "\\end{align}\n",
    "\n",
    "Thus, we can get the result of the Fourier transform by a simple substitution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:58.944512Z",
     "start_time": "2019-06-02T16:24:58.724659Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{\\partial}{\\partial t} a{\\left(p,t \\right)} = \\left(- \\Omega_{+eff}{\\left(t \\right)} e^{- i \\phi_{-}{\\left(t \\right)}} - \\Omega_{-eff}{\\left(t \\right)} e^{- i \\phi_{+}{\\left(t \\right)}}\\right) a{\\left(2 \\hbar k + p,t \\right)} + \\left(- \\Omega_{+eff}{\\left(t \\right)} e^{i \\phi_{-}{\\left(t \\right)}} - \\Omega_{-eff}{\\left(t \\right)} e^{i \\phi_{+}{\\left(t \\right)}}\\right) a{\\left(- 2 \\hbar k + p,t \\right)} + \\left(- \\Omega_{Δ}{\\left(t \\right)} + \\frac{p^{2}}{2 m}\\right) a{\\left(p,t \\right)}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(p, t), t) = (-Omega_+eff(t)*exp(-I*phi_-(t)) - Omega_-eff(t)*exp(-I*phi_+(t)))*a(2*hbar*k + p, t) + (-Omega_+eff(t)*exp(I*phi_-(t)) - Omega_-eff(t)*exp(I*phi_+(t)))*a(-2*hbar*k + p, t) + (-Omega_Δ(t) + p**2/(2*m))*a(p, t)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_mom = (TDSE_a_4\n",
    " .apply_mtd('expand')\n",
    " .apply_mtd('subs', {\n",
    "     a(z,t) * exp(2*I*k*z): a(p - 2*ħ*k, t),  # note the sign!\n",
    "     a(z,t) * exp(-2*I*k*z): a(p + 2*ħ*k, t)\n",
    " })\n",
    "  .apply_mtd('subs', {\n",
    "     a(z,t): a(p, t),\n",
    " })\n",
    " .apply(standard_phase_form)\n",
    " .apply_mtd('collect',  a(p + 2*ħ*k, t))\n",
    " .apply_mtd('collect',  a(p - 2*ħ*k, t))\n",
    " .apply_mtd('collect',  a(p, t))\n",
    ")\n",
    "TDSE_mom"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:59.094800Z",
     "start_time": "2019-06-02T16:24:58.947830Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{\\partial}{\\partial t} a{\\left(p,t \\right)} = \\left(- \\Omega_{+eff}{\\left(t \\right)} e^{- i \\phi_{-}{\\left(t \\right)}} - \\Omega_{-eff}{\\left(t \\right)} e^{- i \\phi_{+}{\\left(t \\right)}}\\right) a{\\left(- 2 \\hbar k + p,t \\right)} + \\left(- \\Omega_{+eff}{\\left(t \\right)} e^{i \\phi_{-}{\\left(t \\right)}} - \\Omega_{-eff}{\\left(t \\right)} e^{i \\phi_{+}{\\left(t \\right)}}\\right) a{\\left(2 \\hbar k + p,t \\right)} + \\left(- \\Omega_{Δ}{\\left(t \\right)} + \\frac{p^{2}}{2 m}\\right) a{\\left(p,t \\right)}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(p, t), t) = (-Omega_+eff(t)*exp(-I*phi_-(t)) - Omega_-eff(t)*exp(-I*phi_+(t)))*a(-2*hbar*k + p, t) + (-Omega_+eff(t)*exp(I*phi_-(t)) - Omega_-eff(t)*exp(I*phi_+(t)))*a(2*hbar*k + p, t) + (-Omega_Δ(t) + p**2/(2*m))*a(p, t)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_mom = (TDSE_a_4\n",
    " .apply_mtd('expand')\n",
    " .apply_mtd('subs', {\n",
    "     a(z,t) * exp(2*I*k*z): a(p + 2*ħ*k, t),  # note the sign!\n",
    "     a(z,t) * exp(-2*I*k*z): a(p - 2*ħ*k, t)\n",
    " })\n",
    "  .apply_mtd('subs', {\n",
    "     a(z,t): a(p, t),\n",
    " })\n",
    " .apply(standard_phase_form)\n",
    " .apply_mtd('collect',  a(p + 2*ħ*k, t))\n",
    " .apply_mtd('collect',  a(p - 2*ħ*k, t))\n",
    " .apply_mtd('collect',  a(p, t))\n",
    ")\n",
    "TDSE_mom"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ansatz"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The sum is explicit in the following"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:59.114125Z",
     "start_time": "2019-06-02T16:24:59.097798Z"
    }
   },
   "outputs": [],
   "source": [
    "ansatz_positive = a(n, t) * DiracDelta(p-p0-2*n*ħ*k) * exp(I * n * ϕp(t))\n",
    "ansatz_negative = a(n, t) * DiracDelta(p-p0-2*n*ħ*k) * exp(I * n * ϕm(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:24:59.919326Z",
     "start_time": "2019-06-02T16:24:59.118227Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a{\\left(n,t \\right)} e^{i n \\phi_{+}{\\left(t \\right)}} \\delta\\left(- 2 \\hbar k n + p - p_{0}\\right)$"
      ],
      "text/plain": [
       "         ⅈ⋅n⋅φ₊(t)                      \n",
       "a(n, t)⋅ℯ         ⋅δ(-2⋅h̅⋅k⋅n + p - p₀)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ansatz_positive"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:00.472859Z",
     "start_time": "2019-06-02T16:24:59.923321Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle a{\\left(n,t \\right)} e^{i n \\phi_{-}{\\left(t \\right)}} \\delta\\left(- 2 \\hbar k n + p - p_{0}\\right)$"
      ],
      "text/plain": [
       "         ⅈ⋅n⋅φ₋(t)                      \n",
       "a(n, t)⋅ℯ         ⋅δ(-2⋅h̅⋅k⋅n + p - p₀)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ansatz_negative"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plug in the positive ansatz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:01.375566Z",
     "start_time": "2019-06-02T16:25:00.476271Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  \\left(- \\hbar n a{\\left(n,t \\right)} \\frac{d}{d t} \\phi_{+}{\\left(t \\right)} + i \\hbar \\frac{d}{d t} a{\\left(n,t \\right)}\\right) e^{i n \\phi_{+}{\\left(t \\right)}} \\delta\\left(- 2 \\hbar k n + p - p_{0}\\right) = \\left(- \\Omega_{Δ}{\\left(t \\right)} a{\\left(n,t \\right)} + \\frac{p^{2} a{\\left(n,t \\right)}}{2 m}\\right) e^{i n \\phi_{+}{\\left(t \\right)}} \\delta\\left(- 2 \\hbar k n + p - p_{0}\\right) + \\left(- \\Omega_{+eff}{\\left(t \\right)} a{\\left(n - 1,t \\right)} e^{i \\left(\\left(n - 1\\right) \\phi_{+}{\\left(t \\right)} - \\phi_{-}{\\left(t \\right)}\\right)} - \\Omega_{-eff}{\\left(t \\right)} a{\\left(n - 1,t \\right)} e^{i \\left(\\left(n - 1\\right) \\phi_{+}{\\left(t \\right)} - \\phi_{+}{\\left(t \\right)}\\right)}\\right) \\delta\\left(- 2 \\hbar k n + p - p_{0}\\right) + \\left(- \\Omega_{+eff}{\\left(t \\right)} a{\\left(n + 1,t \\right)} e^{i \\left(\\left(n + 1\\right) \\phi_{+}{\\left(t \\right)} + \\phi_{-}{\\left(t \\right)}\\right)} - \\Omega_{-eff}{\\left(t \\right)} a{\\left(n + 1,t \\right)} e^{i \\left(\\left(n + 1\\right) \\phi_{+}{\\left(t \\right)} + \\phi_{+}{\\left(t \\right)}\\right)}\\right) \\delta\\left(- 2 \\hbar k n + p - p_{0}\\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "(-hbar*n*a(n, t)*Derivative(phi_+(t), t) + I*hbar*Derivative(a(n, t), t))*exp(I*n*phi_+(t))*DiracDelta(-2*hbar*k*n + p - p0) = (-Omega_Δ(t)*a(n, t) + p**2*a(n, t)/(2*m))*exp(I*n*phi_+(t))*DiracDelta(-2*hbar*k*n + p - p0) + (-Omega_+eff(t)*a(n - 1, t)*exp(I*((n - 1)*phi_+(t) - phi_-(t))) - Omega_-eff(t)*a(n - 1, t)*exp(I*((n - 1)*phi_+(t) - phi_+(t))))*DiracDelta(-2*hbar*k*n + p - p0) + (-Omega_+eff(t)*a(n + 1, t)*exp(I*((n + 1)*phi_+(t) + phi_-(t))) - Omega_-eff(t)*a(n + 1, t)*exp(I*((n + 1)*phi_+(t) + phi_+(t))))*DiracDelta(-2*hbar*k*n + p - p0)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_mom_pos = (\n",
    "    TDSE_mom\n",
    "    .apply_mtd('subs', {a(p, t): ansatz_positive})\n",
    "    .apply_mtd('subs', {a(p - 2*ħ*k, t): ansatz_positive.subs({p:p - 2*ħ*k})})\n",
    "    .apply_mtd('subs', {a(p + 2*ħ*k, t): ansatz_positive.subs({p:p + 2*ħ*k})})\n",
    "    .apply_mtd_to_lhs('doit')  # apply the derivative (chain rule)\n",
    "    .apply_mtd('expand')\n",
    "    .apply_mtd('collect', DiracDelta(_w))\n",
    "    .apply_mtd('collect', exp(_w))\n",
    "    .apply(standard_phase_form)\n",
    "    .apply_mtd('replace', DiracDelta, lambda arg: DiracDelta(arg.collect(2*ħ*k)))\n",
    "    .apply(lambda term: normalize_dirac_delta_terms(term, target=(-2 * ħ * k * n + p - p0)))\n",
    ")\n",
    "TDSE_mom_pos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integrate over momentum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:08.393336Z",
     "start_time": "2019-06-02T16:25:01.378834Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{d}{d t} a{\\left(n,t \\right)} = \\left(- \\Omega_{+eff}{\\left(t \\right)} e^{i \\left(- \\phi_{+}{\\left(t \\right)} - \\phi_{-}{\\left(t \\right)}\\right)} - \\Omega_{-eff}{\\left(t \\right)} e^{- 2 i \\phi_{+}{\\left(t \\right)}}\\right) a{\\left(n - 1,t \\right)} + \\left(- \\Omega_{+eff}{\\left(t \\right)} e^{i \\left(\\phi_{+}{\\left(t \\right)} + \\phi_{-}{\\left(t \\right)}\\right)} - \\Omega_{-eff}{\\left(t \\right)} e^{2 i \\phi_{+}{\\left(t \\right)}}\\right) a{\\left(n + 1,t \\right)} + \\left(\\frac{2 \\hbar^{2} k^{2} n^{2}}{m} + \\frac{2 \\hbar k n p_{0}}{m} + \\hbar n \\frac{d}{d t} \\phi_{+}{\\left(t \\right)} - \\Omega_{Δ}{\\left(t \\right)} + \\frac{p_{0}^{2}}{2 m}\\right) a{\\left(n,t \\right)}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(n, t), t) = (-Omega_+eff(t)*exp(I*(-phi_+(t) - phi_-(t))) - Omega_-eff(t)*exp(-2*I*phi_+(t)))*a(n - 1, t) + (-Omega_+eff(t)*exp(I*(phi_+(t) + phi_-(t))) - Omega_-eff(t)*exp(2*I*phi_+(t)))*a(n + 1, t) + (2*hbar**2*k**2*n**2/m + 2*hbar*k*n*p0/m + hbar*n*Derivative(phi_+(t), t) - Omega_Δ(t) + p0**2/(2*m))*a(n, t)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_discrete_pos = (\n",
    "    TDSE_mom_pos\n",
    "    .apply(lambda side: sympy.Integral(side, (p, -oo, oo)).doit())\n",
    "    .apply(lambda side: side * exp(-I * n * ϕp(t)))\n",
    "    .apply_mtd('expand')\n",
    "    .apply(lambda side: side + ħ * n * a(n, t) * ϕp(t).diff(t))\n",
    "    .apply(standard_phase_form)\n",
    "    .apply_mtd('collect', a(n-1, t))\n",
    "    .apply_mtd('collect', a(n+1, t))\n",
    "    .apply_mtd('collect', a(n, t))\n",
    ")\n",
    "TDSE_discrete_pos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Define p₀ and ωₖ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:08.416873Z",
     "start_time": "2019-06-02T16:25:08.400206Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  \\omega_{k} = \\frac{2 \\hbar k^{2}}{m}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "omega_k = 2*hbar*k**2/m"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "defs[ωk] = (2 * ħ * k**2) / m\n",
    "definition(ωk)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**NOTE: this is the two-photon recoil frequency, different from $\\omega_r = \\omega_k/4$ as defined by Kovachy et al.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:08.709585Z",
     "start_time": "2019-06-02T16:25:08.420112Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{d}{d t} a{\\left(n,t \\right)} = \\left(- \\Omega_{+eff}{\\left(t \\right)} e^{i \\left(- \\phi_{+}{\\left(t \\right)} - \\phi_{-}{\\left(t \\right)}\\right)} - \\Omega_{-eff}{\\left(t \\right)} e^{- 2 i \\phi_{+}{\\left(t \\right)}}\\right) a{\\left(n - 1,t \\right)} + \\left(- \\Omega_{+eff}{\\left(t \\right)} e^{i \\left(\\phi_{+}{\\left(t \\right)} + \\phi_{-}{\\left(t \\right)}\\right)} - \\Omega_{-eff}{\\left(t \\right)} e^{2 i \\phi_{+}{\\left(t \\right)}}\\right) a{\\left(n + 1,t \\right)} + a{\\left(n,t \\right)} \\left(\\beta^{2} \\hbar \\omega_{k} + n^{2} \\hbar \\omega_{k} + n \\left(2 \\beta \\hbar \\omega_{k} + \\hbar \\frac{d}{d t} \\phi_{+}{\\left(t \\right)}\\right) - \\Omega_{Δ}{\\left(t \\right)}\\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(n, t), t) = (-Omega_+eff(t)*exp(I*(-phi_+(t) - phi_-(t))) - Omega_-eff(t)*exp(-2*I*phi_+(t)))*a(n - 1, t) + (-Omega_+eff(t)*exp(I*(phi_+(t) + phi_-(t))) - Omega_-eff(t)*exp(2*I*phi_+(t)))*a(n + 1, t) + a(n, t)*(beta**2*(hbar*omega_k) + n**2*(hbar*omega_k) + n*(2*beta*(hbar*omega_k) + hbar*Derivative(phi_+(t), t)) - Omega_Δ(t))"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_discrete_p0_pos = (\n",
    "    TDSE_discrete_pos\n",
    "    .apply_mtd('subs', {p0: β*2*ħ*k})\n",
    "    .apply_mtd('subs', {defs[ωk]: ωk})\n",
    "    .apply_mtd('subs', {ħ * ωk: protect(ħ * ωk)}) # keep ħωₖ together\n",
    "    .apply_mtd('collect', n)\n",
    "    .apply_mtd('collect', a(n-1, t))\n",
    "    .apply_mtd('collect', a(n+1, t))\n",
    "    .apply_mtd('collect', a(n, t))\n",
    ")\n",
    "TDSE_discrete_p0_pos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:08.757513Z",
     "start_time": "2019-06-02T16:25:08.714437Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  \\phi_{+}{\\left(t \\right)} - \\phi_{-}{\\left(t \\right)} = \\delta_{0} t - \\delta^{'}_{0} t + \\phi_{2}{\\left(t \\right)} - \\phi_{3}{\\left(t \\right)}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "phi_+(t) - phi_-(t) = delta_0*t - delta_0^'*t + phi_2(t) - phi_3(t)"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pos_branch_phase = (defs[ϕp(t)] - defs[ϕm(t)]).subs({δ12(t): defs[δ12(t)], δ13(t): defs[δ13(t)]})\n",
    "Eq(ϕp(t) - ϕm(t), pos_branch_phase)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Use identical pulse envelopes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:08.943343Z",
     "start_time": "2019-06-02T16:25:08.763794Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{d}{d t} a{\\left(n,t \\right)} = \\left(\\left(- e^{i \\left(- \\phi_{+}{\\left(t \\right)} - \\phi_{-}{\\left(t \\right)}\\right)} - e^{- 2 i \\phi_{+}{\\left(t \\right)}}\\right) a{\\left(n - 1,t \\right)} + \\left(- e^{i \\left(\\phi_{+}{\\left(t \\right)} + \\phi_{-}{\\left(t \\right)}\\right)} - e^{2 i \\phi_{+}{\\left(t \\right)}}\\right) a{\\left(n + 1,t \\right)}\\right) \\Omega_{}{\\left(t \\right)} + a{\\left(n,t \\right)} \\left(\\beta^{2} \\hbar \\omega_{k} + n^{2} \\hbar \\omega_{k} + n \\left(2 \\beta \\hbar \\omega_{k} + \\hbar \\frac{d}{d t} \\phi_{+}{\\left(t \\right)}\\right) - \\Omega_{Δ}{\\left(t \\right)}\\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(n, t), t) = ((-exp(I*(-phi_+(t) - phi_-(t))) - exp(-2*I*phi_+(t)))*a(n - 1, t) + (-exp(I*(phi_+(t) + phi_-(t))) - exp(2*I*phi_+(t)))*a(n + 1, t))*Omega_(t) + a(n, t)*(beta**2*(hbar*omega_k) + n**2*(hbar*omega_k) + n*(2*beta*(hbar*omega_k) + hbar*Derivative(phi_+(t), t)) - Omega_Δ(t))"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_discrete_p0_pos = (\n",
    "    TDSE_discrete_p0_pos\n",
    "    .apply_mtd('subs', {Ω['-eff'](t): Ω[''](t), Ω['+eff'](t): Ω[''](t)})\n",
    "    .apply_mtd('collect', Ω[''](t))\n",
    ")\n",
    "TDSE_discrete_p0_pos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Energy levels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:08.985126Z",
     "start_time": "2019-06-02T16:25:08.950286Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  \\operatorname{E_{n>0}}{\\left(t \\right)} = \\beta^{2} \\hbar \\omega_{k} + n^{2} \\hbar \\omega_{k} + n \\left(2 \\beta \\hbar \\omega_{k} + \\hbar \\frac{d}{d t} \\phi_{+}{\\left(t \\right)}\\right) - \\Omega_{Δ}{\\left(t \\right)}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "E_n>0(t) = beta**2*(hbar*omega_k) + n**2*(hbar*omega_k) + n*(2*beta*(hbar*omega_k) + hbar*Derivative(phi_+(t), t)) - Omega_Δ(t)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq_Enp = Eq(Enp(t), TDSE_discrete_p0_pos.rhs.coeff(a(n,t)))\n",
    "Eq_Enp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plug in the negative ansatz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:09.722144Z",
     "start_time": "2019-06-02T16:25:08.988207Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  \\left(- \\hbar n a{\\left(n,t \\right)} \\frac{d}{d t} \\phi_{-}{\\left(t \\right)} + i \\hbar \\frac{d}{d t} a{\\left(n,t \\right)}\\right) e^{i n \\phi_{-}{\\left(t \\right)}} \\delta\\left(- 2 \\hbar k n + p - p_{0}\\right) = \\left(- \\Omega_{Δ}{\\left(t \\right)} a{\\left(n,t \\right)} + \\frac{p^{2} a{\\left(n,t \\right)}}{2 m}\\right) e^{i n \\phi_{-}{\\left(t \\right)}} \\delta\\left(- 2 \\hbar k n + p - p_{0}\\right) + \\left(- \\Omega_{+eff}{\\left(t \\right)} a{\\left(n - 1,t \\right)} e^{i \\left(\\left(n - 1\\right) \\phi_{-}{\\left(t \\right)} - \\phi_{-}{\\left(t \\right)}\\right)} - \\Omega_{-eff}{\\left(t \\right)} a{\\left(n - 1,t \\right)} e^{i \\left(\\left(n - 1\\right) \\phi_{-}{\\left(t \\right)} - \\phi_{+}{\\left(t \\right)}\\right)}\\right) \\delta\\left(- 2 \\hbar k n + p - p_{0}\\right) + \\left(- \\Omega_{+eff}{\\left(t \\right)} a{\\left(n + 1,t \\right)} e^{i \\left(\\left(n + 1\\right) \\phi_{-}{\\left(t \\right)} + \\phi_{-}{\\left(t \\right)}\\right)} - \\Omega_{-eff}{\\left(t \\right)} a{\\left(n + 1,t \\right)} e^{i \\left(\\left(n + 1\\right) \\phi_{-}{\\left(t \\right)} + \\phi_{+}{\\left(t \\right)}\\right)}\\right) \\delta\\left(- 2 \\hbar k n + p - p_{0}\\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "(-hbar*n*a(n, t)*Derivative(phi_-(t), t) + I*hbar*Derivative(a(n, t), t))*exp(I*n*phi_-(t))*DiracDelta(-2*hbar*k*n + p - p0) = (-Omega_Δ(t)*a(n, t) + p**2*a(n, t)/(2*m))*exp(I*n*phi_-(t))*DiracDelta(-2*hbar*k*n + p - p0) + (-Omega_+eff(t)*a(n - 1, t)*exp(I*((n - 1)*phi_-(t) - phi_-(t))) - Omega_-eff(t)*a(n - 1, t)*exp(I*((n - 1)*phi_-(t) - phi_+(t))))*DiracDelta(-2*hbar*k*n + p - p0) + (-Omega_+eff(t)*a(n + 1, t)*exp(I*((n + 1)*phi_-(t) + phi_-(t))) - Omega_-eff(t)*a(n + 1, t)*exp(I*((n + 1)*phi_-(t) + phi_+(t))))*DiracDelta(-2*hbar*k*n + p - p0)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_mom_neg = (\n",
    "    TDSE_mom\n",
    "    .apply_mtd('subs', {a(p, t): ansatz_negative})\n",
    "    .apply_mtd('subs', {a(p - 2*ħ*k, t): ansatz_negative.subs({p:p - 2*ħ*k})})\n",
    "    .apply_mtd('subs', {a(p + 2*ħ*k, t): ansatz_negative.subs({p:p + 2*ħ*k})})\n",
    "    .apply_mtd_to_lhs('doit')    # apply the derivative (chain rule)\n",
    "    .apply_mtd('expand')\n",
    "    .apply_mtd('collect', DiracDelta(_w))\n",
    "    .apply_mtd('collect', exp(_w))\n",
    "    .apply(standard_phase_form)\n",
    "    .apply_mtd('replace', DiracDelta, lambda arg: DiracDelta(arg.collect(2*ħ*k)))\n",
    "    .apply(lambda term: normalize_dirac_delta_terms(term, target=(-2 * ħ * k * n + p - p0)))\n",
    ")\n",
    "TDSE_mom_neg"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integrate over momentum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:15.976028Z",
     "start_time": "2019-06-02T16:25:09.725752Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{d}{d t} a{\\left(n,t \\right)} = \\left(- \\Omega_{+eff}{\\left(t \\right)} e^{- 2 i \\phi_{-}{\\left(t \\right)}} - \\Omega_{-eff}{\\left(t \\right)} e^{i \\left(- \\phi_{+}{\\left(t \\right)} - \\phi_{-}{\\left(t \\right)}\\right)}\\right) a{\\left(n - 1,t \\right)} + \\left(- \\Omega_{+eff}{\\left(t \\right)} e^{2 i \\phi_{-}{\\left(t \\right)}} - \\Omega_{-eff}{\\left(t \\right)} e^{i \\left(\\phi_{+}{\\left(t \\right)} + \\phi_{-}{\\left(t \\right)}\\right)}\\right) a{\\left(n + 1,t \\right)} + \\left(\\frac{2 \\hbar^{2} k^{2} n^{2}}{m} + \\frac{2 \\hbar k n p_{0}}{m} + \\hbar n \\frac{d}{d t} \\phi_{-}{\\left(t \\right)} - \\Omega_{Δ}{\\left(t \\right)} + \\frac{p_{0}^{2}}{2 m}\\right) a{\\left(n,t \\right)}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(n, t), t) = (-Omega_+eff(t)*exp(-2*I*phi_-(t)) - Omega_-eff(t)*exp(I*(-phi_+(t) - phi_-(t))))*a(n - 1, t) + (-Omega_+eff(t)*exp(2*I*phi_-(t)) - Omega_-eff(t)*exp(I*(phi_+(t) + phi_-(t))))*a(n + 1, t) + (2*hbar**2*k**2*n**2/m + 2*hbar*k*n*p0/m + hbar*n*Derivative(phi_-(t), t) - Omega_Δ(t) + p0**2/(2*m))*a(n, t)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_discrete_neg = (\n",
    "    TDSE_mom_neg\n",
    "    .apply(lambda side: sympy.Integral(side, (p, -oo, oo)).doit())\n",
    "    .apply(lambda side: side * exp(-I * n * ϕm(t)))  # using ϕ₋(t) instead of ϕ₊(t)\n",
    "    .apply_mtd('expand')\n",
    "    .apply(lambda side: side + ħ * n * a(n, t) * ϕm(t).diff(t))  # using ϕ₋(t) instead of ϕ₊(t)\n",
    "    .apply(standard_phase_form)\n",
    "    .apply_mtd('collect', a(n-1, t))\n",
    "    .apply_mtd('collect', a(n+1, t))\n",
    "    .apply_mtd('collect', a(n, t))\n",
    ")\n",
    "TDSE_discrete_neg"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Define p₀ and ωₖ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:16.297439Z",
     "start_time": "2019-06-02T16:25:15.978930Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{d}{d t} a{\\left(n,t \\right)} = \\left(- \\Omega_{+eff}{\\left(t \\right)} e^{- 2 i \\phi_{-}{\\left(t \\right)}} - \\Omega_{-eff}{\\left(t \\right)} e^{i \\left(- \\phi_{+}{\\left(t \\right)} - \\phi_{-}{\\left(t \\right)}\\right)}\\right) a{\\left(n - 1,t \\right)} + \\left(- \\Omega_{+eff}{\\left(t \\right)} e^{2 i \\phi_{-}{\\left(t \\right)}} - \\Omega_{-eff}{\\left(t \\right)} e^{i \\left(\\phi_{+}{\\left(t \\right)} + \\phi_{-}{\\left(t \\right)}\\right)}\\right) a{\\left(n + 1,t \\right)} + a{\\left(n,t \\right)} \\left(\\beta^{2} \\hbar \\omega_{k} + n^{2} \\hbar \\omega_{k} + n \\left(2 \\beta \\hbar \\omega_{k} + \\hbar \\frac{d}{d t} \\phi_{-}{\\left(t \\right)}\\right) - \\Omega_{Δ}{\\left(t \\right)}\\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(n, t), t) = (-Omega_+eff(t)*exp(-2*I*phi_-(t)) - Omega_-eff(t)*exp(I*(-phi_+(t) - phi_-(t))))*a(n - 1, t) + (-Omega_+eff(t)*exp(2*I*phi_-(t)) - Omega_-eff(t)*exp(I*(phi_+(t) + phi_-(t))))*a(n + 1, t) + a(n, t)*(beta**2*(hbar*omega_k) + n**2*(hbar*omega_k) + n*(2*beta*(hbar*omega_k) + hbar*Derivative(phi_-(t), t)) - Omega_Δ(t))"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_discrete_p0_neg = (\n",
    "    TDSE_discrete_neg\n",
    "    .apply_mtd('subs', {p0: β*2*ħ*k})\n",
    "    .apply_mtd('subs', {defs[ωk]: ωk})\n",
    "    .apply_mtd('subs', {ħ * ωk: protect(ħ * ωk)})\n",
    "    .apply_mtd('collect', n)\n",
    "    .apply_mtd('collect', a(n-1, t))\n",
    "    .apply_mtd('collect', a(n+1, t))\n",
    "    .apply_mtd('collect', a(n, t))\n",
    ")\n",
    "TDSE_discrete_p0_neg"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Use identical pulse envelopes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:16.442029Z",
     "start_time": "2019-06-02T16:25:16.302845Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{d}{d t} a{\\left(n,t \\right)} = \\left(\\left(- e^{i \\left(- \\phi_{+}{\\left(t \\right)} - \\phi_{-}{\\left(t \\right)}\\right)} - e^{- 2 i \\phi_{-}{\\left(t \\right)}}\\right) a{\\left(n - 1,t \\right)} + \\left(- e^{i \\left(\\phi_{+}{\\left(t \\right)} + \\phi_{-}{\\left(t \\right)}\\right)} - e^{2 i \\phi_{-}{\\left(t \\right)}}\\right) a{\\left(n + 1,t \\right)}\\right) \\Omega_{}{\\left(t \\right)} + a{\\left(n,t \\right)} \\left(\\beta^{2} \\hbar \\omega_{k} + n^{2} \\hbar \\omega_{k} + n \\left(2 \\beta \\hbar \\omega_{k} + \\hbar \\frac{d}{d t} \\phi_{-}{\\left(t \\right)}\\right) - \\Omega_{Δ}{\\left(t \\right)}\\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(n, t), t) = ((-exp(I*(-phi_+(t) - phi_-(t))) - exp(-2*I*phi_-(t)))*a(n - 1, t) + (-exp(I*(phi_+(t) + phi_-(t))) - exp(2*I*phi_-(t)))*a(n + 1, t))*Omega_(t) + a(n, t)*(beta**2*(hbar*omega_k) + n**2*(hbar*omega_k) + n*(2*beta*(hbar*omega_k) + hbar*Derivative(phi_-(t), t)) - Omega_Δ(t))"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_discrete_p0_neg = (\n",
    "    TDSE_discrete_p0_neg\n",
    "    .apply_mtd('subs', {Ω['-eff'](t): Ω[''](t), Ω['+eff'](t): Ω[''](t)})\n",
    "    .apply_mtd('collect', Ω[''](t))\n",
    ")\n",
    "TDSE_discrete_p0_neg"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Energy levels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:16.516682Z",
     "start_time": "2019-06-02T16:25:16.450215Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  \\operatorname{E_{n<0}}{\\left(t \\right)} = \\beta^{2} \\hbar \\omega_{k} + n^{2} \\hbar \\omega_{k} + n \\left(2 \\beta \\hbar \\omega_{k} + \\hbar \\frac{d}{d t} \\phi_{-}{\\left(t \\right)}\\right) - \\Omega_{Δ}{\\left(t \\right)}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "E_n<0(t) = beta**2*(hbar*omega_k) + n**2*(hbar*omega_k) + n*(2*beta*(hbar*omega_k) + hbar*Derivative(phi_-(t), t)) - Omega_Δ(t)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq_Enm = Eq(Enm(t), TDSE_discrete_p0_neg.rhs.coeff(a(n,t)))\n",
    "Eq_Enm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As expected, we get the same equation, just with $\\phi_{-}(t)$ instead of $\\phi_{+}(t)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rapid adiabatic passage"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:16.568740Z",
     "start_time": "2019-06-02T16:25:16.523656Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{align}\n",
       "  \\phi_{-}{\\left(t \\right)} &= - \\delta_{0} t + \\delta_{12}{\\left(t \\right)}\\\\\n",
       "   &= - \\delta_{0} t + \\phi_{1}{\\left(t \\right)} - \\phi_{2}{\\left(t \\right)}\n",
       "\\end{align}\n"
      ],
      "text/plain": [
       "phi_-(t) = -delta_0*t + delta12(t)\n",
       "         = -delta_0*t + phi_1(t) - phi_2(t)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "definition(ϕm(t)).apply_mtd('subs', {δ12(t): defs[δ12(t)]}, cont=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:16.615075Z",
     "start_time": "2019-06-02T16:25:16.581496Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{align}\n",
       "  \\phi_{+}{\\left(t \\right)} &= - \\delta^{'}_{0} t + \\delta_{13}{\\left(t \\right)}\\\\\n",
       "   &= - \\delta^{'}_{0} t + \\phi_{1}{\\left(t \\right)} - \\phi_{3}{\\left(t \\right)}\n",
       "\\end{align}\n"
      ],
      "text/plain": [
       "phi_+(t) = -delta_0^'*t + delta_13(t)\n",
       "         = -delta_0^'*t + phi_1(t) - phi_3(t)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "definition(ϕp(t)).apply_mtd('subs', {δ13(t): defs[δ13(t)]}, cont=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:16.626577Z",
     "start_time": "2019-06-02T16:25:16.618489Z"
    }
   },
   "outputs": [],
   "source": [
    "rap_phases = {\n",
    "    ϕp(t): α * protect(t - tc)**2 / 2,\n",
    "    ϕm(t): -α * protect(t - tc)**2 / 2,\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:17.391715Z",
     "start_time": "2019-06-02T16:25:16.630664Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ \\phi_{+}{\\left(t \\right)} : \\frac{\\alpha \\left(t - t_{c}\\right)^{2}}{2}, \\  \\phi_{-}{\\left(t \\right)} : - \\frac{\\alpha \\left(t - t_{c}\\right)^{2}}{2}\\right\\}$"
      ],
      "text/plain": [
       "⎧                  2                     2 ⎫\n",
       "⎪       α⋅(t - t_c)          -α⋅(t - t_c)  ⎪\n",
       "⎨φ₊(t): ────────────, φ₋(t): ──────────────⎬\n",
       "⎪            2                     2       ⎪\n",
       "⎩                                          ⎭"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rap_phases"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:17.408388Z",
     "start_time": "2019-06-02T16:25:17.397440Z"
    }
   },
   "outputs": [],
   "source": [
    "def plug_in_phases(phases_dict):\n",
    "    def _doit(expr):\n",
    "        expr_expanded = standard_phase_form(\n",
    "            expr\n",
    "            .subs(phases_dict)\n",
    "            .doit()  # do the derivatives\n",
    "            .expand()\n",
    "        )\n",
    "        return (\n",
    "            expr_expanded\n",
    "            .collect(a(n-1, t))\n",
    "            .collect(a(n+1, t))\n",
    "            .collect(a(n, t))\n",
    "        )\n",
    "    return _doit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:17.424239Z",
     "start_time": "2019-06-02T16:25:17.412142Z"
    }
   },
   "outputs": [],
   "source": [
    "def rap_rewriting(expr):\n",
    "    \"\"\"Write things a bit more intuitively, specifically for the RAP case\"\"\"\n",
    "    return (\n",
    "        expr\n",
    "        .subs({(α * (t - tc)**2).expand(): (α * (t - tc)**2)})\n",
    "        .subs({(α * ħ * n * (t - tc)).expand(): n * protect(ħ*α * (t - tc))})\n",
    "        .subs({tc - t: sympy.Mul(-1, protect(t - tc), evaluate=False)})\n",
    "        .subs({ħ * ωk: protect(ħ * ωk)})\n",
    "        .collect((t-tc))\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:17.794465Z",
     "start_time": "2019-06-02T16:25:17.428241Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{d}{d t} a{\\left(n,t \\right)} = a{\\left(n,t \\right)} \\left(\\beta^{2} \\hbar \\omega_{k} + 2 \\beta n \\hbar \\omega_{k} + n^{2} \\hbar \\omega_{k} + n \\alpha \\hbar \\left(t - t_{c}\\right) - \\Omega_{Δ}{\\left(t \\right)}\\right) + a{\\left(n - 1,t \\right)} \\left(- \\Omega_{}{\\left(t \\right)} - \\Omega_{}{\\left(t \\right)} e^{- i \\alpha \\left(t - t_{c}\\right)^{2}}\\right) + a{\\left(n + 1,t \\right)} \\left(- \\Omega_{}{\\left(t \\right)} - \\Omega_{}{\\left(t \\right)} e^{i \\alpha \\left(t - t_{c}\\right)^{2}}\\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(n, t), t) = a(n, t)*(beta**2*(hbar*omega_k) + 2*beta*n*(hbar*omega_k) + n**2*(hbar*omega_k) + n*(alpha*hbar*(t - t_c)) - Omega_Δ(t)) + a(n - 1, t)*(-Omega_(t) - Omega_(t)*exp(-I*alpha*(t - t_c)**2)) + a(n + 1, t)*(-Omega_(t) - Omega_(t)*exp(I*alpha*(t - t_c)**2))"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_rap_pos = (\n",
    "    TDSE_discrete_p0_pos\n",
    "    .apply(plug_in_phases(rap_phases))\n",
    "    .apply(rap_rewriting)\n",
    ")\n",
    "TDSE_rap_pos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:18.047414Z",
     "start_time": "2019-06-02T16:25:17.797255Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{d}{d t} a{\\left(n,t \\right)} = a{\\left(n,t \\right)} \\left(\\beta^{2} \\hbar \\omega_{k} + 2 \\beta n \\hbar \\omega_{k} + n^{2} \\hbar \\omega_{k} - \\Omega_{Δ}{\\left(t \\right)} + \\left|{n}\\right| \\alpha \\hbar \\left(t - t_{c}\\right)\\right) + a{\\left(n - 1,t \\right)} \\left(- \\Omega_{}{\\left(t \\right)} - \\Omega_{}{\\left(t \\right)} e^{i \\alpha \\left(t - t_{c}\\right)^{2}}\\right) + a{\\left(n + 1,t \\right)} \\left(- \\Omega_{}{\\left(t \\right)} - \\Omega_{}{\\left(t \\right)} e^{- i \\alpha \\left(t - t_{c}\\right)^{2}}\\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(n, t), t) = a(n, t)*(beta**2*(hbar*omega_k) + 2*beta*n*(hbar*omega_k) + n**2*(hbar*omega_k) - Omega_Δ(t) + Abs(n)*(alpha*hbar*(t - t_c))) + a(n - 1, t)*(-Omega_(t) - Omega_(t)*exp(I*alpha*(t - t_c)**2)) + a(n + 1, t)*(-Omega_(t) - Omega_(t)*exp(-I*alpha*(t - t_c)**2))"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_rap_neg = (\n",
    "    TDSE_discrete_p0_neg\n",
    "    .apply(plug_in_phases(rap_phases))\n",
    "    .apply(rap_rewriting)\n",
    "    .apply_mtd('subs', {-n: Abs(n)})\n",
    ")\n",
    "TDSE_rap_neg"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dynamic Frame"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:18.077109Z",
     "start_time": "2019-06-02T16:25:18.051998Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  \\eta{\\left(t \\right)} = - \\frac{\\alpha^{2} \\left(t - t_{c}\\right)^{2} + 1}{4}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "eta(t) = -(alpha**2*(t - t_c)**2 + 1)/4"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "defs[η(t)] = -protect((α * (t - tc))**2 + 1) / 4\n",
    "definition(η(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:18.139553Z",
     "start_time": "2019-06-02T16:25:18.080464Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{align}\n",
       "  \\phi_{d}{\\left(t \\right)} &= \\int \\left(- \\frac{\\alpha^{2} \\left(t - t_{c}\\right)^{2} + 1}{4}\\right)\\, dt\\\\\n",
       "   &= - \\frac{\\alpha^{2} t^{3}}{12} + \\frac{\\alpha^{2} t^{2} t_{c}}{4} + t \\left(- \\frac{\\alpha^{2} t_{c}^{2}}{4} - \\frac{1}{4}\\right)\n",
       "\\end{align}\n"
      ],
      "text/plain": [
       "phi_d(t) = Integral(-(alpha**2*(t - t_c)**2 + 1)/4, t)\n",
       "         = -alpha**2*t**3/12 + alpha**2*t**2*t_c/4 + t*(-alpha**2*t_c**2/4 - 1/4)"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "defs[ϕd(t)] = sympy.Integral(defs[η(t)], t)\n",
    "definition(ϕd(t)).apply_mtd('doit', cont=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:18.150196Z",
     "start_time": "2019-06-02T16:25:18.143113Z"
    }
   },
   "outputs": [],
   "source": [
    "def to_dynamic_frame(H, U):\n",
    "    # this is valid only because U is a constant\n",
    "    return (\n",
    "        I*ħ*U.diff(t).doit()*U.conjugate().transpose().doit()\n",
    "        + (U * H * U.conjugate().transpose()).doit()\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:18.227614Z",
     "start_time": "2019-06-02T16:25:18.154470Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  \\operatorname{E_{n>0}}{\\left(t \\right)} = \\alpha \\hbar n \\left(t - t_{c}\\right) + \\hbar n^{2} \\omega_{k} + \\frac{\\hbar \\left(\\alpha^{2} \\left(t - t_{c}\\right)^{2} + 1\\right)}{4}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "E_n>0(t) = alpha*hbar*n*(t - t_c) + hbar*n**2*omega_k + hbar*(alpha**2*(t - t_c)**2 + 1)/4"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(Enp(t), to_dynamic_frame(\n",
    "    H=TDSE_rap_pos.rhs.coeff(a(n, t)).subs({β: 0, Ω['Δ'](t): 0}),\n",
    "    U=exp(I * sympy.Integral(defs[η(t)], t))\n",
    "))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Scheme with π pulses"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We choose a laser frequency that targets the transition $n_0 \\rightarrow n_0+1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:18.258978Z",
     "start_time": "2019-06-02T16:25:18.231163Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  \\omega_{L} = \\omega_{k} \\left(2 n_{0} + 1\\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "omega_L = omega_k*(2*n_0 + 1)"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "defs[ωL] = (((n0+1)**2 - n0**2) * ωk).expand().collect(ωk)\n",
    "definition(ωL)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:18.272929Z",
     "start_time": "2019-06-02T16:25:18.262367Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  n' = - n_{0} + \\left|{n}\\right|\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "n' = -n_0 + Abs(n)"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "defs[nprime] = Abs(n) - n0\n",
    "definition(nprime)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:18.289839Z",
     "start_time": "2019-06-02T16:25:18.277393Z"
    }
   },
   "outputs": [],
   "source": [
    "pipulse_phases = {\n",
    "    ϕp(t): (-ωL * t).subs(definition(ωL).as_dict),\n",
    "    ϕm(t): (ωL * t).subs(definition(ωL).as_dict),\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:18.991022Z",
     "start_time": "2019-06-02T16:25:18.293294Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ \\phi_{+}{\\left(t \\right)} : - \\omega_{k} t \\left(2 n_{0} + 1\\right), \\  \\phi_{-}{\\left(t \\right)} : \\omega_{k} t \\left(2 n_{0} + 1\\right)\\right\\}$"
      ],
      "text/plain": [
       "{φ₊(t): -ωₖ⋅t⋅(2⋅n₀ + 1), φ₋(t): ωₖ⋅t⋅(2⋅n₀ + 1)}"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pipulse_phases"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:19.024023Z",
     "start_time": "2019-06-02T16:25:19.016469Z"
    }
   },
   "outputs": [],
   "source": [
    "def rewrite_with_nprime(abs_n=None):\n",
    "    E_n = ((nprime**2 - nprime) - (n0 * (n0+1))) * protect(ħ*ωk)\n",
    "    E_n_expanded = (\n",
    "        E_n\n",
    "        .subs(definition(nprime).as_dict)\n",
    "        .expand()\n",
    "        .doit() # un-protect ħωₖ\n",
    "    )\n",
    "    if abs_n is not None:\n",
    "        E_n_expanded = E_n_expanded.subs({Abs(n): abs_n})\n",
    "        \n",
    "    def _doit(expr):\n",
    "        return expr.subs({E_n_expanded: E_n})\n",
    "    \n",
    "    return _doit        "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:19.334310Z",
     "start_time": "2019-06-02T16:25:19.034680Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{d}{d t} a{\\left(n,t \\right)} = \\left(- \\Omega_{}{\\left(t \\right)} e^{i t \\left(- 4 n_{0} \\omega_{k} - 2 \\omega_{k}\\right)} - \\Omega_{}{\\left(t \\right)}\\right) a{\\left(n + 1,t \\right)} + \\left(- \\Omega_{}{\\left(t \\right)} e^{i t \\left(4 n_{0} \\omega_{k} + 2 \\omega_{k}\\right)} - \\Omega_{}{\\left(t \\right)}\\right) a{\\left(n - 1,t \\right)} + a{\\left(n,t \\right)} \\left(\\beta^{2} \\hbar \\omega_{k} + 2 \\beta \\hbar n \\omega_{k} + \\left(n'^{2} - n' - n_{0} \\left(n_{0} + 1\\right)\\right) \\hbar \\omega_{k} - \\Omega_{Δ}{\\left(t \\right)}\\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(n, t), t) = (-Omega_(t)*exp(I*t*(-4*n_0*omega_k - 2*omega_k)) - Omega_(t))*a(n + 1, t) + (-Omega_(t)*exp(I*t*(4*n_0*omega_k + 2*omega_k)) - Omega_(t))*a(n - 1, t) + a(n, t)*(beta**2*hbar*omega_k + 2*beta*hbar*n*omega_k + (n'**2 - n' - n_0*(n_0 + 1))*(hbar*omega_k) - Omega_Δ(t))"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_pipulse_pos = (\n",
    "    TDSE_discrete_p0_pos\n",
    "    .apply(plug_in_phases(pipulse_phases))\n",
    "    .apply(rewrite_with_nprime(abs_n=n))\n",
    ")\n",
    "TDSE_pipulse_pos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:19.499940Z",
     "start_time": "2019-06-02T16:25:19.337917Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "  i \\hbar \\frac{d}{d t} a{\\left(n,t \\right)} = \\left(- \\Omega_{}{\\left(t \\right)} e^{i t \\left(- 4 n_{0} \\omega_{k} - 2 \\omega_{k}\\right)} - \\Omega_{}{\\left(t \\right)}\\right) a{\\left(n - 1,t \\right)} + \\left(- \\Omega_{}{\\left(t \\right)} e^{i t \\left(4 n_{0} \\omega_{k} + 2 \\omega_{k}\\right)} - \\Omega_{}{\\left(t \\right)}\\right) a{\\left(n + 1,t \\right)} + a{\\left(n,t \\right)} \\left(\\beta^{2} \\hbar \\omega_{k} + 2 \\beta \\hbar n \\omega_{k} + \\left(n'^{2} - n' - n_{0} \\left(n_{0} + 1\\right)\\right) \\hbar \\omega_{k} - \\Omega_{Δ}{\\left(t \\right)}\\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "I*hbar*Derivative(a(n, t), t) = (-Omega_(t)*exp(I*t*(-4*n_0*omega_k - 2*omega_k)) - Omega_(t))*a(n - 1, t) + (-Omega_(t)*exp(I*t*(4*n_0*omega_k + 2*omega_k)) - Omega_(t))*a(n + 1, t) + a(n, t)*(beta**2*hbar*omega_k + 2*beta*hbar*n*omega_k + (n'**2 - n' - n_0*(n_0 + 1))*(hbar*omega_k) - Omega_Δ(t))"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TDSE_pipulse_neg = (\n",
    "    TDSE_discrete_p0_neg\n",
    "    .apply(plug_in_phases(pipulse_phases))\n",
    "    .apply(rewrite_with_nprime(abs_n=-n))\n",
    ")\n",
    "TDSE_pipulse_neg"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Effective pulse amplitude"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:20.180968Z",
     "start_time": "2019-06-02T16:25:19.505900Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - e^{i t \\left(4 n_{0} + 2\\right)} - 1$"
      ],
      "text/plain": [
       "   ⅈ⋅t⋅(4⋅n₀ + 2)    \n",
       "- ℯ               - 1"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "effective_pi_amplitude = TDSE_pipulse_neg.rhs.coeff(a(n+1, t)).coeff(Ω[''](t)).subs({ωk: 1})\n",
    "effective_pi_amplitude"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:20.192990Z",
     "start_time": "2019-06-02T16:25:20.184959Z"
    }
   },
   "outputs": [],
   "source": [
    "def plot_effective_amplitude(n0_val, tlist):\n",
    "    y = np.abs([complex(effective_pi_amplitude.subs({t: tval, n0: n0_val})) for tval in tlist])\n",
    "    fig, ax = plt.subplots()\n",
    "    ax.plot(tlist, y)\n",
    "    plt.show(fig)\n",
    "    print(\"average: %.2f\" % np.average(y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:22.254194Z",
     "start_time": "2019-06-02T16:25:20.196277Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "average: 1.27\n"
     ]
    }
   ],
   "source": [
    "plot_effective_amplitude(0, np.linspace(0, 50, 500))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:24.272574Z",
     "start_time": "2019-06-02T16:25:22.269857Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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LUcOgbfFif0qhc0xRRIiOSI9TGW1LUcwGbRhzyD+7ktT5Ox+1aC66PItErsG6cGNU60I7c35OD2ZmdN6gbiIAZB2wNNZR7sfO/2YI1r6wF7KPvK6tILu02a0vdpxMcVlndlbtXBl65k4lLy+NmJ0Inmy3CAKDptEOXrg5a+Tfmuwnj7hcrhUFYOrHGfqAupHXjjhLZwTnPmPWhuPSccOgLUGuAXVjERob+hYNFs2Rlctlgeil05LXjhy9N9r174hiiBE9dvL2WaK5t8+5WTCv0++y+/30O3zPo5ZBC6JWRVGcVC6bwVCieIzTs2HqRo5luVxmYGIBW8AAtBF9ReeRjllKs4KZ0zk2Ylc7X4Z+NbGbcrJtFl2244IQsHEgiifbUjeRUmuervbZDmNR0AjtEGkFlePamFrqK0EyNjKCrcoijV4w/+kWLi8c6wCPsUFrOcDKX7YoLTyuyKDlrO8v5Rc5BwcMttggKVkGTvtqHKB1Dkp+gXOQ8y+RvzUoEZe9aRi0qBoponsUMAjkknPsXKL5t/qyUXqFo2kbtfK1fdS4O5qPyiuJYh23a79EJ3tEf/K2Log+CN8CRG/pFuUcosl2hh4n1xRFEfGiARc5PQv+XYR2rTOTstgO0RixXIhig2wTxmw4PUd/cupGzotsEaW1CYye7RNV/cjFPow5BAAa1WH+NaKIokS8fRZbQiqbcmgBRREDg6uQi90wFPDKCgAdY0cVJWNbv5P3z7nOy+WAhrH/bpadzr/bjBYw4UhXrQMzL0tAntKXBVV9p93OpaFXdEWAloi0kCXnGSH66N9EyyZbGafGgorC901wXaI4GbsJUWAc1urFhjc2hWGt5SIXGGS7cCrdEJeraQcco/MrgUFcYhxa+iKPIbgcIDr5zJKvtvdcSt1sVRQi6UmMXFvAIDL0ETAgkuWVsVzkfEdRC5FGshIMRDqiK64aDlD2bzjNguibIA+vAyuXqM9xMMYjFRnr5zqrdi4N/SZAro7WCEJ5Tfdg42rPG4lojSgU34z4uraPMvQNuoEVeTNkVQoW9VFJJ4fQpFJjwxMptB1LtHBb1A1fzyYddaSzjNI6ChCaNrS7nbz9e6MWcQO5BmWkkkpZAgym58S/k2O8shBMRInt42Be5e+OtnHBgVwX0XzbZ5bj3IZjjB2gnL8wGev0cuqTcwvkBZGGWfsnjc5kFLFPxl5FK8lYYGj7LjkkshlGWvFOTWPQDvm8FYAWDvrOcZ7bIdM1a44ovIJcu+4NoszlWoiLXPd+0wzfv++SjwI2YzlozdJCPEYpF1boVZcA5zpguRwP1CX/zPJwOHs+yGbIZTcucoCHq87RPduhjl+h7e1QDqFC6HwF5HJlMxS5HIHE/LpP2pnN8j7oO0hpoYQ/3/Oate9zvB2LXqg+Y70WirQO+g5SFCwXFAWkZHMXWcjFo1CWJXLa6z6RRcqbIdPhqnMFB0fbEc4xI9/DVQerkfiIDQWS1DN7uay65CPALV7HHN10CRcplHVhHP0KyqXOC4rAy9oHsjxcdXruhe7tqZuraGuQjOUJvuHaNUxU8nkzFrny6XIqzJp/c+Gwd159M4x03cFU3omQzHWHvQr5tuJaiL98zrUH87W8QbvuoFd0xTiXSt5w4UCNmWhSKh6jVWiiae+BD4Uz7MPb2+3CLWM86J1BmeTSq9/xGImm3cxycQ5jpjETXQCynOSynp/fL6gLB16WVzZSLt5pX7vuYZL62dd6uRxvR7rusFf95b+vO1ipUDznaV5YL6SObYdMFw6BXOZ/P+fC2p0btNnmIheL6FMievY1a+gALxz0Pum4HYosjwCiv3CwUvrKnz/rGqzjfFSAdAC8Rq47XME8FJdDb8y6WPeJDlcdlMsNF9Ywd4H0lef7hgsHMEnNfWxlGu/HOQIRwQWzjvmZefyIbrpw0KtIZyvlsjf0J2+Io2eE9pwLa2Wc2KAgZdts8cRVg7ZyXn071oVrJ7tL00FZ0jko4wASq6xsGtFPz2gXTlkE88K9YpxTMQ7AAT772jVEoWXhGlpk3Xd0sMKI/rrDlXOA2yEXGSPO2DoH6czk3yyX54Axcv/rD/28XAn6HM8o8GDVw9D/2deu4MagC8Voe77cLtztmClnKnqxMRz9dchos0E9XDmnuR1HKJcr25EOV52LGouMgaG9fDxUfQFRwHUHvXbAM1VTHLDTMQ+YNsLQoQQuO82tcZqrrqP1qnNzTzQdXHhlM6od45tBOg2v48+5dg0B0/Wwz1hBHpSLl+V2wA74eDtFbQcrzQCoa+0N/ckbrqOfEZJB9MdCCac+eoHy50vQOdGk1GjhHg9TWDvRGn6yL8wLynKeRXFBAs8itKLQF9ig6YWDxlgM2jUAuQ4jXQDGeTuOtJrRFgr3rztYOc4zNGjzPa87XLmFTkT1/lu9CK9Z93Sw6lTSsRgUYBwvHwvkajjrg1VHB4CiIJqMgJ0XieitXHgsCNEhRL8Zcr1WIBfrtI63dV6sQbtm3dM16w5GLRcOengtGLVu5bzUsQ9CX+34j7eD+Fyg87E6YMTRs1w2Zl2s+zTRg8YAExFdf830zBo0CTCx9X2mtQ8iHSDLTYDOpb2wstyMdb3aSPdg1dGq6yDIu2Cc6Vm2c2XoUXklG4FnXaO9ukQ7rs9MUVjkWkMxb1A2Y1UQu3APVz2tuqTCN6lsiPO8PkA7/MyWuyaiALlmOlz3jtfnZ7n+0CPXyWnx/fUiXHUxor8Q0BotiuK6g5UO9wWitNc62k65k3WXzHPJ6KD+PudMR8Kg2Shg3U/IUVEqY0VbOWsjNFF9DURvkKuMdOTv+JmvXfeUEubbrzvslVy4f0SDHa46Olz1Grlua6QxGgd8PIyQhikAxOhYRfR9uaccZ9Fx5LQBoj+SNNiI1x6qsuJ14cEEpmGIfNQqqUYiMpTqlIdYmZyenEsH8oJ5OR4GOlh1cx4IR4Cb4fTetdtq58zQe45+M9ZFiNAOQhXbGVU4Lnqoi9BRN0OGKJQR/ao3CZlCN2CDhgyKRM6ojAzypPP9p+QaUrbeV5cIikKX2zF/qg2KRa42OroeRAcyOoLUDXKac9Jr1WvjXDl6LRcWw7Pma9ndoJNc7Lzo+9uQH3H0W2Ec8RiRXEZa9Z1DrixLRENtIhpuO4MJY1COnY4Lwy2iVkj3GKfJhhXlTqqOJ6V7W2HQ7Hk+x4OkB+3a6+akp6+4wlEA1lcutrh23ZmKKz0WF7V2cx4K2YtDDyaGMQt90evyoJ/1VXL04lr2/mfVzpehh5n/abLDBCJauHKyYfim6QY+iAqhiqOZP7XVHRthnOwzH21wcqsYlANtnPk3XMVg73PQp9mgeeM00Q06gagNmg7FV32iw7WNdDjcxwjpAkKhc+7imnXflouR5UGJjrDTQhtu+JWINqI66Cfkdgzkcn2Qb8HObKSUyBkUi0IVqtsKigLI0iZDi46B6GgzCDABjt9ABmUzZLr+0FNaMoEo554N/bUIuc5c9NqMRSJ6GzVkkR9zHH2f5qjRg4nrr2nTYBuzLlZ9muWCjbb8m6816Xiv9yQE0bzMD8nnJJrkfbhmuWCO3sryrNqSVwm+LqV0f0rpD4Pvv0O8eeoPU0pDSul583efTin9wfzdxdN+eNsQRy8newDIDRmUMtmrHtbRX3+IJxsa7cLTBWgnMIJRooqNI6Jh+EUhqFLGloUVXnlWXF7InEAMKa2uc6WHMklNpENxWXViQ/EJhXah0bZyYYNikaucS43OtAP0cpkMCpLLhWDhXgcc8HGhG3qI6C8AuWzGgB5k5Hq4Mo6pOmY7lu2QadUlWnWJBsgFA+Q8jnTtAaA6h2q0MaJnB+i56LVBriWaNhy9izSMLOv9JZ05qD5W/heCyHzVdbNcfKRT+my1XEplmXTaQaRT9AXmLngshrYVOR3+3Vm3JYj+9UT08ujLnPMP55xfnHN+MRF9FxH9jnld4DfO39/09B51d0McPU/2uk9NtGMXASojVMktUC5VUaBZBHP4BtEOoDU2w0jXrHylRjUonVNoIiq1vIMZ55oXztb3sWjXGieLghGitxSFLSNFHP3xdkY7HU6GIoM2jONs0GKEZkv1tFzMImTj1EJoplKnJLa3WsfWHDWCkjyc78CG3lYjMX8rdU/+TSTATGfpuUbUOmQ66HvqEtIXHwF6Qx+AGSB/rlThsciEs/wdP1dF9MBpg/HLCBQVD6w6PcetSIedph1/xNFbB2z1BclFViPZ+59V22noc863EtHS97y+moje8LSe6Gm0vkvUJcSFJup3efUTTrasLrGTbdH54bqfvbpfhCiBOyk7oHu2uSAEu9CJBHJViJqTjjgZe81KOwdX3micwwohemecjRGcn9nRMA10jiIdNgI2uSWdk9xNy3JpRTr2ubivlQv3KYZ21AZtxUYb5Q4ig9Yl16fsrzjQSccW1VdKEk3uotVHGkErF5S7qDoGqktEH0SDXTjsVcGBzHXJ3/H9OQJFm694Xqzh5OfSchEgryEXVIjh6cGq49Jptfh2nZ/z6xWt/bNqp8bRp5Qu0IT8f018nInoHSml21NKN+/of3NK6WJK6eIDDzxw1c9hlY0ne1JoRDcg/nKe7B47B8vTboUSENmyrIEO+25GFT4UR159K5CrVbb1ynOhfN3DNV44h2AR8v2vMRtNagUNiDTGTOtVo44ebFrhUNwuXK56sYkqS4M55NqlsI91NDXS8TTYkYx0gBGwcuFx1sS2kcvMt9tkHJGo1wZz6Sm16TdlLiMwYZzTqkNgRtODCMxMOq7ncg3AxFCeyxva4+1Ih72vLrHrheef/1/Q+WhlOVVDIRrOyoVlg+ZlGKfoxMrFR436Pn2RCwIT2gHzc9T9BXr+K3Xj74/yHWfVTjMZ+z8S0e8Z2ubrc85fQ0SvIKJvTyn9t1HnnPMtOeebcs433XjjjVf9EPZ4Aq4UWfd2shltIP4Sh8JHZrJ5wppGe8i0XiWAkIzTmJ95nDdyTXkFj0Q4rEaLgA2a5WnZCKFkLFNEg0VbYAPMdhhpXRyQN/RFlqN+NhSKD+NIfZdm6iamG46MQ1v1IBS2UUiZl1HJBVcjLZPLOE7J0IO+d057O9TkPUwGc0RhHd0cNVrkysdMoLGERnvWcbTV/oKhwTixu+q7uezWOObCt0vaDMuFqDoHX11iaI35eXzxgH5mlKTeGlnyfXLO83ESuOCgh5H5DgfYJ+pNpHMsigekLO0cq7OZAh2zyVhb9XYW7TQN/avI0DY553vn/99PRL9ORC85xfvBZnfUDbOH7jvLxe5YOF3nEZJIrvDvZN8LgIudEIJPyNjacz5LgxcBU0eWD+TcATJOMOm4rdGJdTQpUTmjhBdSq/a7LoKIBtN0zzjTKKj2XibJW3QDRvQeIaFFWIwTo1CLAgEKdYltzl3Mz4FKVSO58LVQ1Y+ke2wFFRtgvraSC6L6RqYbbKSDdZzlwI4G6fiqT+74Zh7/JBfsHI63XsfqWT840rKUItRxA2a2Yo5zphKdKUc7RwdxMhZVlk0FB+veJ7aZb5+uHYzFzPEhb5hCuQsQUZxVOxVDn1J6DhF9AxH9hvjsupTSs/jfRPQyIoKVO6fZcFjdMLQhT4coghlRr4zicli58shxGAXdgJI4xquzQvJi9/cHtd/FOLFBE+MUdI9FFazQ8plt1QuqFInqpS+YUJz/HyFnNtqbEcgFJHBrpAXGMsuF5STlEKNQHx3x/bkixUY6KFHJ9BRXEFX+NnbAnDtxDnAYZyevDUoTmAi6wVajEPnKMn6uSMcOVr6+nzdM8WF/g0HuaI2Veem0oy0OEERaxyI/hXIXtlKIZVooRZC8762ONda+dNp2EyPPl7zGxjzXMOi133ecU9I2iQjr+Fm11a4fpJTeQEQvJaLnp5TuIaLvIaI1EVHO+bXzz/4nInpHzvkp0fWLiOjXU0p8n1/OOf/m6T06bj4hNBb+cswTyuxE7TREroK6sYtwvaqKyxPGk8h9LEItFAWo1LE15vybVUBrsKHhpGPfpaJcFu3wdVGNM1Mnfac5V+cAXTI2OYQ0WEc3aEPHfZRz4EgH0CDq/oAi8TsNK6+sZNlAjjF1o51DMSjbHU5rnnuiaaNWn1BEYZHrRFGotwXDuqQAACAASURBVA8Vo22T5NppuCR15yktt1ekUCdax2wfdpoasEx9EEfPY/ER6ET1cXTCeurkYuR/0Ki6ucaAqeqAfb5jw9SNy4/FiLrokq3UEehc3r/oGAB52zma7Wxid6ujRrmWzqrtNPQ551cv+M3raSrDlJ/dRURfdbUPdrXN7qjbFITAxnmkw64nm1yxk83UjZ+4CTkR1QljZVt1PoE7zIm6KRT2i5ANGvcZpHEEtAZX0PDffdc3EZI8O8SebS+NUymvnO93uOqoS+TQ3oQCcUWGXbgSOa4MRcCJslWvHXBFrmjXIjsaHGkd2EhrqHRDl8gt3JJ03IKxmKRbibQ6XynEG+zYoNV5mR3gDkrN60ulCCx1U9G2vT/SF0aOGszUeZnQLkps8nMVMDFfNopa+9mZO5A1j5FI0B0tuQy1gqhF3djcxUHJURhgNt8fgzxQWSYS25ZqQ2CCf7PuPdW7nUHeyhxHbOlBe9TFWbRztTOWaEb06pTKiuinv/XCiWqMJXKu16rJHaI6Ybxw1r2vlBlm5OoSVZyQOjAIYf5NzzypCWsZuUx/a+R6CBBSRa6+vHItFuHWGDROrllDWxPbOjogEmiT+X6xCO1ZP1sjy8qFzzRQQN30PeboJXVTKAqBXO3GLBnpoAoei9A2xdDPiW3gTFdGxyx1pNCuiBo1mBi1XIxBY14/TMYauRD56KhEjYxcDd2zAvfnsTB1w32mndS1HNlSFFz1ouTC66VD1FEuVCNKBh+ubaRTDa1LbI8WzOCo1dXRI3rS6qtZe0z3bIyOrziiADpmE8tn2c6foV/ZpKOe7IJq2NCuEXXDyNXXq68Er12vFXv1zexoHA2z5YWDkeMahJzboR57Kq9RQ2FtnLhSBC4CQcPw33IsPRs0U8EURTrq/uZaKyCX8P6GOhmzvv+61ItLo204+q2e44hSw3JhtKUpCo60GKFtDELjqEXe125Ks2jzIJDLWkSN1ThVMGFfWCFRqC0j7EQy1HL0TKtIuXBEcWDGwraocPRzH75dLTgAUYuhB6te4E1e/FxoJ7vVMZ5rjtpxHkgDM5eHUs5pLE5b6hjn2tZujuvaX5lnHtj2BHtFrI6dZTt/ht7xdJNxsGjLnt3BfRihrDtfkmkRPfLqFrkOQ+VcUULGVoRsjUGxC3e9SmXhRCVeZREIB2SrkUode4ejk1rdovuswCLgksAo0kEcPVfjlEVox3LgndaYac4r+LJTjlqIZKQjKDV0f1SSaJwWRyvSOKEoQDotp2OGbuB3IfD+DocceznH+v4HM3U2GAe46sFekWFURttSWuycLDDqhaOxvL6tY1dGewei9jSYT2yzA0bzxeBDjkUlyVeI0hI0mHlmOy/jOFXwlCo9gOhZLpW6iXW8Vtzhss+DlY5OzrKdS0N/pJKOtYyPSPDHxtBadMxctDW0EiFYQ8uI3pZXlkoNRzfUBV0NinAaZhEeD5q/5cVuEb0t/ap19DZqmThaeV++Fqo/5tIzy7dzSeDBig3tjEIFrwwjnd5Xl3DfcmyBicAY0XrqxhuBwcjS1fejMQ6m6mWwxokpLX1/HiNfQz6zdeZFX1aoJFNHjdYBIuQoKTV/cFpcjYTOx2Gq01JqNeGux6KNNpiXrhM6pvWyVqSYiAbIhROrNtKRDnDd2ZJIQ92YHMW1prxSASNLTw51V7i8v6TUfB4qh3JR0eze0J+8oY1RvVi4RUFL0lFvabfh/mbUTqNXRsAjV8QtohIrTu7YeumhLML53HWpuFud9BuKgraNY5dAomqsFQn8nGr8nT8qVlaXbIFxqCG6lYtHmzWBZ8J6Q904gzIjV++Aq1z47UMuopBh9VDRliqJHOu+CyQXrqBCxxk45OgiLR1prEukAWRpoqPjbdVLRJ1x1GgT3kwdTNdAYMKXymod1/I/XDXk0luqEXPkUsesAz4eKgpv6VithmIA4EGWLG2WsnTACKxjBNhUpGNpsN4ncLnizh6BwNG01bGzbOfO0FsUzklHW0YolV2WZdlFaJGTrOBBaAvxpGzoxqzL5WQFjy2jY2Xz55BIVMPKpo3jxhpHoGzMd8tKkalP5ejxztxOlaqyXBTaMnKBm4kGpm70/WV1iewjy079c9XKJn4eOy/WAU/PLDYACbQnoyYXHQR0g+TorRG01SWq6sVx0bk4E9mnVpdwRYy5/6yvMBlYnovUtVBJZqW0cKRrK5tUlRg4N6cJZnqfu9rOVT/9vF5Gs15spHMsdAxRnSsxl3btl2Oqt8bQA31hpxlHR/6FOAXRz2OsYKLmAeTznGU7d4be1nhvrYKMfuHK5JpchLbqpSCETtMNlVaIuUVk0GS474wzK5tDwQLRuz466VhoGA73rVwEdeIQPXPR1tEB5GoTVXaTTzm8zDhgPS/G0RnkKg2KrXGWu58juaCkY995g741KNghV9YLQJ0guRBR+c6BCSCXUl7ZW2BSKQKL6KteduqwvWHM1CVZ9eJRqC1VZbnYfAvThAd9p0683AodQ5sVJTDyUSMuIVYASNCTcsewK16YwZx15rzxUY5bghkEJtZAX2rVjXXAAtH3gIYDAKCWKet5Oct27gw9CmsVqghCTlR6hvhTlYw1u+OQQeGElDdoFekTVSqpoi0QVo9zRUYXGEdzbIN0ZlYuG7Nw5Jbycv/Ov5QDK642DhtEESBZihxFRGvwIrBlp9Y4oaSbLIm0TlvWnltZShqoGu06L/7cIE0dWY7e7lrl54gO3FLAwFSKoA1rsiRQyrBU46SAOoEOWAMju2O763QJsdp8ZaOmQW+YQhy9nBc+g2cNDHoFBhpM8DW7lNQmNe6r8h0GmPVJJ1Al1Wh5dVd1g5x212FgCMAMWsdn2c6dobfJpbJpobNevU7qqksFOcjSM3t4lU1UWUNb64Ix2uLn4ftw0kleS6NzHQpHyFXuzJXHI8hSyT75zRxrMBap7LJem1//ppFrpXuwQtew1p41NIy1sknKkCt4OuOcZNmpLcmrcrHIVYbV9cUzOYsKnkTq+uUQvCDfUIy2oVsgFy0Nh8hRbK1xMJGGvv90jVHohUeuel54LOPsTDkZ6vSlw2WEcr0g0LAS+SZntO05/bJ4wDiaugNVOzMUHUXluLzUkVw4OkBrLAkdG6y+tvIgthrK0nBjXS+2UkhWg61XHmSdZTt3ht4hV7cItVdnozLYxQloGJ+oMgal15usGKH0na+9Px7q4UlEnqOHCdyRnVaU3NIlkQrRg6QbK6ccQ6V79CLQIbLnPFfCONmw9mCFUWgvIho5L4yyV2BeUNJRvnNAycXRQEgunlZA1UgygThVUGm64QBEOsM4HeplnYOs1IiixprvANGB6CPLTj3doyMdRJ0gp8H6gvowFYToKSuX40HPS03s6jyQA1+9LziwNIjlyFkuI9BxX6qai0ykLm3FWDzI48P5dGWZrNSRTlOtSbteBgP+9oj+5M1TBFwS6JVt1SVKSS8cydPZ3ZQuUWWqXkrmH6FQw3mOWaNQv3A6eH+IXGcHkGwoaozD6Ba0P9TMKiji7i3dIcvIiCQKFoje0C1bEwXIeWGqQRohlQcxXLCXCzLOSUQ61ZlZI8glgf4gLumA/fb8qLpEGhS7YYc3Rlm6Qxpnm7vpjaFV6NjQPVYuLtLq/cttKqL30cHUB+tF2TA1aB2T5Z2o4EC+57YY7YSBGcrpeDBj58WDqXEe4/Tccr2y0wA5rblPlxqVXb0HRr3KEQhEL6i2PUd/Fa0Doaj06lJB0WTrkkA02X6COFxdl0VgUFgvub16HxQibwRCsa/Zs8lgjdxSeW6Lzgt1MuoyQg53p78rDcP371JyxgFtPmNZ2goev5Esdpr1mUmMpVak8PdcEurC6uSNk9zM08pdyN/aSGcwRmA1y0WuTVveKeefP5PVJcUIGGcq5VKjA5PvMLkLjWitjuuSRBsdMZhxCcQu5vX7pPd36JyOP70TyVgVHHSgeKDD9KRcx7siHTl+G1FwpMPjGYADtide2vJOT/fMZ0BZByyjeQkAQAR6lu3cGfpVl9S2eVvGWJRqyNqguImrxkGWReFFoENxtAjsrk1GTvMjVFRhEIKv5QWJKjkW0UfRHfP3rFOcdEROQ97fU1reoPNmksT1+s6g+hpnu5tUoiqmDTByrc+VRUUGRq51scnyUolCeS61cQTVOEaWloaSvLLMN7BhkDuTZR7GOmArF74WI8i+TwrM2CStmpdAx6xxVHtFBq1jo5nLOn5Mncjfyl2uWsbiWpC6kZVtEkz4hLt2DoCelBVERi7cj7GUKq+0ZdozMIsoLQvy5JEZqFRVjkWW155VO3eG3nGOtpYWIHp5jKicOEdrDNqgjKZPZBwVF81KnbOhjgByC6gjz0WPiteGYa1DrvX0SHkti6pspIFCYaaOuF8xDlkoe4cOiKuc67Ewwmx8Zb24lqWle3DUpuq1QVgtAYCkFdazMe0S+T49yHeMuI6ea6/LvBhn2iXvgGulSLDvwzgaNBYb6aWkE4XybCYZNZU+O/UC61gHjDBywDIZLHl9ew8iUhGdnJeoSkzqGO+Yjip4eD5tZRcGefPasyAv2JNgo0knlzl3Iz8/y3buDL2bbBMKy/BRGscRTJzdZDWM+jQ+i8I5fLZIXxpUi7a4n48OmO7QnKOui65KyMZXcsEFbUFln5NbARdtqStZjdSDPjy+PnnOEyaDx1E7U0aueYFcnEHVKNQienZajgvuAiNQ9MLnCDiiG8y8oOoOS6kVjjjrSEc+03aOaOymOL6fTYZuBXWAdLwTsnTItcN7RSQX7TjnTu+v2CjdZ6elwYw/1Ez3cZ8nP34JzFbQOfg6eh6LpU7CtSciQJQLUNF01uOXkaYfY6fkwlFjdcB7Q3/iNlEEplwtoCjqZHfOOKBKHTfZFrkadG7DXSIx2YNQXIGqHOdp8w2dR672Wpg/RUlHyff7Z+6SR25yLMppKoNiIxotF/nMqOwUUWqoikFuWFNjNHTLOuCCEXW0GSwNhvId9blyng7C6lSOoN4fRo3AOCoH3PkKJjv+MevPow1bWpY1GchyscdJWF5/a3Q83qsB9h4Mk6OxOa1hrPMiKTWI6IXMWCZyB2oU6cj34rrzpCIdU7St37CGdGwYp3LgyWh7jn4CAKSuNY4EHc1Ztp2GPqX0upTS/Skl+BrAlNJLU0qPpZQ+NP/33eK7l6eUPpFSujOl9J2n+eBRs5PN3tuHoiOcbKm4aHckMrTDqOtyC9pphbUBcpWbn+ROQzkWh1xnhEBEIarwyUUtF8fFs0GxiqtKMmufktwSshyzvv/WLsIOV1FUFNo544hojSIXMMdc3hg5IItc+QwiOy9R0g8ZJ5T0iwCApzX00RBsfEalY52SPcuqDWb8WHgu5emNY471hYio65ge1FGjSmAbehLRpkRTabN0WoijtzTQ9L0sONDRmU3SrsG8DGMuuqIKDoTTdDvpZ+fQWsfyULVB6GsPEL0Cec+QqpvXE9HLd/zmPTnnF8//fR8RUUqpJ6LXENEriOgriejVKaWvfDoPu6RJikBvwPB0A+bofXJpYwxKxx7aOAAiUuVyA1iEMrlVjXMnFpSkKOq12OkjXpk56ul7gJBSgFwF2pHljcWgiHK1stCTPzZBh8L1CN3IOJax9F1ZOBKhYkSv5UIkooC5JNMmVjfiWmuAHNFRB3IskqIYjEGzSFcm3RBH33eJODiTfWxCzpadYhoIlwPbc1iUjoHoiGkNG5lGOjZ9p9+wBaPGIEkunSnnDlB+rEu+6mYjZCkT/hbMOAeMKt7U2otoW5Dv6H0ENo65gAUJ8rQDJPWsQ9aIXm7+O6u209DnnG8looev4tovIaI7c8535ZyPiehXiOiVV3GdEzUd1gr+stPGwYa1PNkShfpDqnCJla7L7ZxxWqMkzqj5U4/oUwmrJwQ8c7QKBfpQVNYyS+TW2cU26mSsTAYXGiZF6NxuctJJRzZa46j72A1DigYSKLCJQjtBa5RFbfIQuS7cdqQj8jDAoKJ5YYPC8mAwJnMnCEzIZKhCoWhPAkwsC4MCHDCMzkZMT5aduZ3esSz57i4ZHRMoHO1v6DukY4aeg8DErz10nIVer52KdMr41XzVMfp8RyAXcy17f+W0jDMjMiDP6L78LGITzrKdFkf/X6eUPpxSeltK6b+cP/sSIvqs+M0982ewpZRuTildTCldfOCBB676QWQyUPF3ACFpg8JKQKWPNag1UUXl73KtVMM3tAj5WkWpHUJr85Qtvp03pnA/6YCmz8DCKRw5jlrKc5mwWkUHSymCsgg9R+4WzoDpjq0Kha3TtvypjFo8cmrJUlJHU6IQc/QuSayu5R0woo40raEdLevYTkQPUKjdFDjdX0QNIg+17hNZYy6fqzq0aY5TSjNFUeeL749kqegptTO11rGj3EVvZMkRKH8Pk+QS5KFcm4marCyV00A7pju/Y3fIWW3wq/PSWK+D0DFTWXdW7TQM/QeJ6Mtyzl9FRP+CiP7t1Vwk53xLzvmmnPNNN95441U/TN9NJ/iNYzYLyvC6zqDUxUGkk2uywoBDThsmKuQGapx3oT3Pn+oMP+Lb5QKRdIM9VXMlUIUsfeyTH4tahGIsKBkrFVf2sZwrJ+RGnhc0xijSAdRJhFytXCQKlO/tlPScRa6jWLitfAfrWEH0neeCrUGLchcswzqXtSJjDHTMIkdJ3chIrzptT6mx4eJ5GYChVc5cUBR+TwJCrhnShnz+FPeLavXl/bXTivugdYSqkcr9E3Z0dS51HxuZS74fgzwfHY0Zg7yzbE/b0OecH885Pzn/+61EtE4pPZ+I7iWiF4iffun82Zk2idxreaEP3yJDy4heenUdvgkjuDB8U0ggV8UtBkWhPVGSKIzgKJTQ76bVpaLWOLZCUf7ebpv3cpFoJzaoqI/lb6MxEplF0EvkCqIAc3/vtESp5I46el5rOqLpBD0EkHvOouKKmhx95DTQngT+TJeq4jOAUHmjTVLz99bRdJ18/WNdL9HO4B7MS5OLno2g3ZOwHWs5sNKXHM/xdjDjz/penPQsIA/MV0ypWbngeeFkbJdIgRnpAGWuheVidSyibc+yPW1Dn1L64pSmkaaUXjJf8yEiuo2IviKl9OUppQMiehURvfnp3m9X08Zx/gygc5coM8rWz9UFfC0ivwhlH+kAXGIXVSQIJIARvUbOxTj0oIxtHMuLKhDnqBeON+gro6AKOWUtF676kOPj2u/Sx9zfJtcUPeV2pmL+tBjnXuxmFQgVLRy5CFcdOOtGRQGMtkhRN+44i05z0Yhqgwn/5B1Nr2Tpo7Ou00nqLsVyQUf72jl2Tjvp8UNELQCQdIDLuGgpf92nRzomgUkjGd2JealRowF5Cx0gPJ8meR2PANCYJQ2DdoX7qFHnG/7jcPSrXT9IKb2BiF5KRM9PKd1DRN9DRGsiopzza4norxHR/5ZS2hLRZSJ6VZ62k21TSn+XiN5ORD0RvS7nfMeZjEI0yXkqo23DWmvoeOKUccLhG/eRNBCjGXzWDE7ioOSS5KJl6R/iyGW9eDV08bZ1Hjc/R2QE0OcjuL9cOCvVp12vztfU78wFyLFDXHSM6Gufmm+pCxqfAYR4ZRWdiDlGY5EoEM2Lei5bRthpZy5rv/n5EHWBIkBJQw3COescAUjggnmRMpZGuAdzLDcZIaeNchR8/EaRsXBy/Bk66wbpBYyaR5vTAslg8FxSLutOr5doLNaZoajN5jsip3GWbaehzzm/esf3P0lEPxl891YieuvVPdrVNUR3SL59A4xTONlBHT3RhNJ1SWA7rEZctAr5hKGTtd/cRy1Cg3YmukPu5gROQxhnucmHiBNC3jh1Aukj6gah8Bi51ohG1n5zqaoM0VvUkaUV6v4CbzgmtEV1XkyirutIoK16nyoXXEcPo5OE6vjrvMjzaXSkU4EJr/cif0URkJaL4cg1DSVlKeQyVLmkROX4Df6t5JWjIzvstUqf3oxFXIufT76uElEnqOxUyp/Xo9SxKssI5NVyaPvqS34uqS88V72QyzhO66UXa6zqeDAvsHggcA7PhPLKP21NIiRkaNBkI56wM+GbnWyZxFHb9gFHLw92kggZ8fqy9lshFBCKohC965I6N4WfSaIKKReWmbpW7xcBWoSllnnAvPpkNGeDUrahj4Y/7dT1o0QV36tPmjqxxjFCW12HUaA1jpMsqXxvI500v8mozAsbGmicdNmprGBhWcm5lGWn0/d6f4UqYbX62uFE4S659CLSlU7DOuBhrFGjfOcCrNQZpWOkMlYFmBAFCpwWOgnUAiOZn+HraJCndUw5YHV/Ks+kcxd1vqY+9VkdmBBzz8/qoyNdDSaPOTmrdu4MvUJbsrrBhrWjDB/FOSRgEW4HP9l9pxUXGU1U3TECZbeGVoa1/MwyuYNQKL6/iAKEEZJy4e9hpYio1NDhvtlSPuLStyGDsQjjONX3k7p+5IBtvTT3kTJ2shT8aZ/wvOyktMDLKuTLSkoFTUI0VLuyyvL6FRhU5AoTiD2m1KxB2RqnLSkdKZfpOoYjh8YRRVreoE3RiUb0Nq8gI2OHzpHTFHsi+k4nY+V8FVkCRA/XK9gw1nVVn7aDzl1MfWoErNYr4OijoykQyDvLdg4N/fT/rfHqXZcopTh8c9yaOd+8hYLthqntOPOtqLxTod0avtlz6uW9pEHpEkD0OaJOwMIdRUmgQLuojA9xoYpuQKVvSZdX8iKTL/Kwm8LUWHbcXzntUXOx/H3pY2ioMfNrBKUD9JvfFHVT5kVz5Pb+ymmYclz3XErH6v0lpcX3UfMCqrRQ1BqVJMox1s+n+ZElvCgPIpPBiqMHfUZVDkz1mYVcZGQs5yucl0wiGa2PGpDRrJ2XDuyY1nLxIE8nqXUFT70/lWvOH6sTWqPkPX+H5vIs2zk09BjREzHdsmMRyORaoRv8ZNtjE+SGKb6/Qm6WInCJMhEKzoojFbS1CIdRLgJyyqYQ/SDlwjIT/LWQC9Md0jhKjh6izU6fh+8QveGCeayYbugUr8zPhHIX1TkI5Ji1cSp9ZO7ERkdmZ66s7JF6xLIazIJOyVZX+I1skjqS84IMCq5UqWBCAgCbjNVgplM5gkJpiUgTbViqB3GJCLRPbvOV7SPnmL+vJby1es1GGvxb6zSn82HIyzJjRF8q7pDTGkzU5taLPo/fgjx5eqmN2ni9KHrQ3H8Chj7h/k/e9jF6xY+/h86inTtDr8I34dWJZsOV/WRLvl2irWgR8H2koZVhtb3/WpQEloUrUM3KbGmXYR1/Ju9v68UnxaH5e0y3IOqkOAfDk9qFM2bNX0q0w32kcSpcdPZj2RqOvrUxCCJXycXmrCgd/j+OtGofSR31hguWkZZNLEt6aro+QRpMAYAGFy2PI5bApAOylLwunJfOJ2NlFGDl0hm5SH2d6EFSctHFA+iUxrjqha+JqJOuq5vPoAPkPNhIkIazwIDHJ8FMSv7dAqvez3FhADqtL1szFktPWjAxRa1oU54cf503vv9jlzb00JNHdBbt3Bn6qmyj8upEPhQufDuopY2Qo0W7fC9rHC1/LBWHKJ5sS8PwteQi5D4YhXpEb+9vjWPX2QoiawRGIReqdIOoMIj2ESAaCiGkXVy0yrc05sUmkK0sZaJQJhDZYFrnMCJ03AO5gBI7aYTghilbdSOcqZellnG5vzBo6DwlvVdjdHLB0VFXX4gDSwKlMxEbg5RxnHeYo3kRTkP3qWOBuZOEZSmpLv7MgRk3FrHBUMiFfwtLmwUwk0eZWJBnHR1O+LdB3mm3c2fo5cJ1FEWyobAviRzRZCsU6Pto6qJy0RCFDnKxSWUTaKu5COT9R9cHIkdl0KpxkLw6QvQYockNSzSPNTryWTtZJ5cAuUZjJNI5ArQIFX8LZKnpFv+eV0WpmXxHcfLCCCAwsVVcNECOAUWBKDWNzkmNZbq/l4us7sE5Am0AuY9F4SF1lDSlxZ/JI3xZLlUvOrEpD/PqI5wXf38VaQ0emAwqGSx0TKxXnQwmdS8rS7mHg8caJe/5+rJ4A9GDcF5EdHDa7dwZeo1CGVHXWmY42Uqh5+vIyR59dYesIrAlgURT8g4japrv0yh9M4tgO+jqDvfMWTsau2HKcsEyauH/y7cPrQxCkrRC35GqS+YzzOEuY5G7kC9ykCWsZSzB/gY3LwZtwbBa8PpWlhbRo5JELReA6IUDtsbZ7tpUDtDU8Uu9sMUDRJ5StPkO6zSlceTNV8ppCgcgwU+VpdVxvY+iB3KZnKmulFHrBejrCGS5NXMp6/ghPQfzEEAuSpZ+LAhM9J1M4GrAVO6vxu8BAFp7Y678PYp0pFxOu507Q684TxAKw/NpAkQPed0y2XpjiuU8NULRRxD4ycYJxFL6lnflCCR/qkN3onkzCQgrZZ9o154di0LU0tABLjiiOywK9ogejDEHYxnai7AudBJzKSg1s/lL9zEcuZGLcppIx4yjqUZzLGOR5+Ngg0LxvBinuWt/gToKWuSt7P0VABFjUZVVCjB5B+QinR5TR9o4VrqnB7kLrWPT9e2rJ3leKt+OnPaIwYS4jyzJlOi83j/WC3lwn6Q6J8Dm56XqeL3GabdzZ+gRddKJUBhNdvSaN4QcdyF6ZAR1SSQ1JxtxfjZR5u5vUWiuBqUkPJO/VlRjLhXa9gn5S7AIZKWGuj+gKNS5OSkYYyI1FskFK+dQkKuY+2BjkIxapioWnQzF9Jw0qFovJlqBxDP752KjKTf52N2c9f5VLjYZK6OAvtuxv6DX8+Lq6DOqYDLIFTmAMav1Fd3fJj1RNZQ8Ihzy7Upf/f4GlNjeORaYjNXUjUTndV6wXIhI58Gc07D6Wl+Iwzp+Fu38GfoEFuGC6hJZ9khE8wsW/GQjxVGc4zxRY64GXW61H0TVieJ1C9oRC11W/SCDBhCaHAsyTmP2cpHJWIQcZRWD4i9hdKBriSUK4z6OolDIlfQmHzEv75zETgAAIABJREFUSwyt5Ty9LHUorlDoPF9aLvNzoVLRHERH4P72TUoydC9jMXKZnDaV7+WhZvwZSiAqYND5eYlkKc+W5+9QZdVEgU7PFVJ9IDqRB+TB+6PiBcS3q+IJmfAHIA9EWp6jx1ELPxeKpiXIQ6Wqcvxy89kuGmiP6Bc2FFajUFhz9KIuGSHHsSJ9lKiTKFxONjoILOT8RL0yzzVcOIijN6VnsoRTJsNCuYixqJdjA+Qok0uaow4WgZELVHaB0ND+hjov3jj4qhv9xiS5v4Dvr84hgdUVND9XnavtOCqqjftYikAmKuX9ldOSG7lgpCP7+CSxyhEIg6Y5citjnYew1A1C9L0BDSrhn+scs76qSK+Z8CfotFQ1EnBaCLDZmvgyLyDS2lXCKx2wlAuitJpUpwBTXacR/S56bJ+MXdgW1yUPHlWN2aKw6ZqjUByNQmPkalFN5UJHN9n2eNPeKBtCSFMoLs/aIdWHF7ulgeQiQPyluj+irgQXjNB5FyzCiuhJRQdVllRkicP64LkaHL3mb7kkVNNNGtG35bIyeiETyxrtegDAiDrPzhFGTRaF7pClHMuqM7kTJ5cOzjFKOsr3D0vqrNSeB/qCdvl2QmajoLTkjtnyzGJe5Bn2ZSc3Mo5AL7bj6ICZLe+MQJ50JmUsgG6p+Q7t5LmPdE5SLnaOJwck5bI39ItaxJPxdzJ8dsm1ccTIMQi5orOqy/2N4WCe1l4rpUQZeHW0CGAdvVBQy23KctD6OWm5JFkvrl8wwX3sWPpgLC5JXH5Pbl7QlnZVEihzFJKeQvXKYtemmheE6IVxlBu2EHKsdAM5RC9RMDRCiDrKDX1BSWqRjJVj5LmS/HVK085cHOlElV31+rDqJkD0PFdqjABMRMjVgZkBo10V6aQqH0lbIo7eAjNbcSePYKjzQn5eIACoTnFyDqS+txvp2HbjXFu9lpTLabdzZ+hVvbgxQl3CmxYU2oRoJ0D000dqEUKeTiibLCND1ImigYRxbi1cm4wlqkbIo2ApF6pjKSiYhFxARYhCm+Sips4saEtR6F2L0gFSGWvfA/mPPoEojTaKtAYpF1DdoZ02pi7QgkbnKSFaYRQUhXVObbnUeZFOC1FXlqJgvXCHivWaUkOVXW16EJ+PY/dwsFysA9yZjAVzyXQLpPpa1IkEMwZk2bGoajBJQ3VSLqTGZ1/LWNeLlguPAYKJIDr5glE3KaXXpZTuTyn9YfD9/5xS+khK6Q9SSu9NKX2V+O7T8+cfSildPM0Hj5rk6K1xVrsWsw8fufLBGlqZwERvH4KLMOCvFXcPDAragKEVpCtjgrsGlbLjg7gQ2pFOC23MGezCnUsy7W7OVqWGvdau2nMrS/e5REho85mSi0fh+plHNy9xkpodIAATnU06kpK1paHQWCCYABz9ZvBGqJuNIAYGjQg0MEKI17aUIo5OPDpHyVhJaaI+CFHLTU5L9n1wX6Zo0DPHII+E7lPpo2vfSc0bcjQOTBhnRqSpxtNuSxD964no5Y3vP0VE35Bz/nNE9P1EdIv5/htzzi/OOd90dY94sqb5sOkzqVSI7rDKjhC9W1CdqBQByJVRqPyME6V2EXZJnyFvN2AgBbHnvnueEnP0dtfedE1SyubvP/pQPJlFWNBuPdoXIVdFaQHnIPMdOjqiHU7DI0cVHUhefZxevGET2K68MWkH7HMH5CO9ZBxdgFwdF5zJoXNJacljJopcFAChKssB7y+owIBcBBhRjdELtfn+at8HX8tsFkRy8cCEcB8UaaS204o2nyFEHYM8cnLRdfzxcRIsl+n+pJ55F6Lne5x222noc863EtHDje/fm3N+ZP7z/TS9BPwL1uRk+1CYUahBCLKPNDQQ0U7XWplFaK81IQT9TFzK5ZGb4PyCRF1Bm+CQLPneStlHLgKIzgHdISmSlRiLXYSckLNIf0JuVOSJxuIcMDsNE7UoimLHvKCXcKs+xjjKBdX3OtJCcsnAAU4L14xlBgDZOADJRY/CAcmEP6JO5IapVkSjAEjW7y/g7yENlcC8JD0vRAxmgnlxiJ7wcwEdk2BicoxTzor7jIGOVWcuz5lqVJbNfRA9J8cC8xDwWnXebJ96rTqW1dzH6qumtKo+nHY77cv+LSJ6m/g7E9E7Ukq3p5RuPuV7waYXgT4CgSsP0E47olpd44xm4NXxhqnpOQbg1VdG2VQVA/DqMoHqooDO0AoAOaKoRSZWpaFVYTX3UbtZvRFEcglRoESO5lpsnJFcuI/aZSvmuHU0haLBzI5pyYUy3WMRvY20/LUoBBMRcuSxoJp8WKopNgYhFB4ZNOfMgwgQ8f0FzJg+1jhPSUcwRllZJhwN0jELJlbC0jHaxYi6Rid1LFTlgnJqQPctyIPOtEXbQhpKJ6n5u0Fs1lPrCOjrabed74xd2lJK30iTof968fHX55zvTSn9J0T0zpTSx+cIAfW/mYhuJiJ64QtfeNXPIY0jtxryYb7ZVqr4EiuCCwptTLLUjUIoSS9CmfmvJXl1oemEkDFoSdSYC2XvzMLdVfrF16wLmoI+BO+P5UKln6Ub0GLjfIMzTsKgyFe2oZr8SimJ6EgYdElpyWvJ8Xu51GupMrqiLyPNQ3HUjXMaog8jVysXRx3J6CT7zTwMTJJDjlFNvqehIA0EwAw0grneX8ulHTUiGo7pQWUcU61SUnJRwKjmoVgGXCqJ+rBZQCBPOTMJ8kB0AingpPtYMCGdVn2VISkw8Yyuo08p/Xki+lkiemXO+SH+POd87/z/+4no14noJdE1cs635JxvyjnfdOONN171s8Aab1Nd4r265vUtT6c4t84vAhS+ch+puGURIkMLElUaueJQ2C0CY9BRHgKeA5IldVSft1zLRCccUaCyS13eWGVvZWmNkNtkIxEailoGUKlhIx2zPX8012KZTbxuHZuTy+jpDiT/Ihe+FkCuA0D0u+voK0VRjbPndTk6Q9dq6asCM8Zp8FgdDTWa2nMwFh5nSlScotQxOS/y3cNlLCPWsSqXIKcG8x31uVhkCuRlfC02xHIuZZLaMwA6auJnH0a570N8rtbeM9TQp5ReSERvIqJvzjn/kfj8upTSs/jfRPQyIoKVO6fZZOkdnGyU3JHlcgLtWc5N9ZGLUC5ch+jBZINzQHKmetgZQKHlmU0ojMbI95fIiV++AM9USaa6InAOMjqpiF7Lijn6Wt1g5SLOFLFOyyQQbelh80wR0QeFwpbX7sSCYo4e7eZEVScwrA8RPak+fNYOMtqwVFI4LbthazBy4T7ToV76vuFGOuNM5W9VZdcIIooyx/rzsFRVjMUmcCdefdTz0gU6lrQDivh2NC8e6esSYqf7Si71uaKjIcr9s6cHNcj00bx12qfZdlI3KaU3ENFLiej5KaV7iOh7iGhNRJRzfi0RfTcR/Rki+pezEdjOFTZfRES/Pn+2IqJfzjn/5hmMQTWJwhlByMnbbEaH3Oxkt15oLXnaFtqxyVAiEb6Jw5vsfaRXly8sQFHA8UAOOfprVdmwEYZhtUCuKEkN+VNFaWlZV1lq4xRVPhxvxZun7J6ErJGj3BgEK2XkWBwNlx1y4h2g8vREJxdBd0i+nwlCu2s0oqGG+T7V0E/9J4OiZdh1CZ7hbikChegZzFhHIwAQOppCImepf0cbnyPohaNRVVqdmBdrUIVxVnX80qBmj+h1HoJKnwImMqZbUKSFkT6V8UmqsyWXqBCCH50jXQnO2TnYXJtzgGeE6Hca+pzzq3d8/21E9G3g87uI6Kt8j7NtcoJ4EdpNG3anna1IYVmzQYn4S2no7YmPrLh2EW4BonchnzEOKFHoDR25+9vqEhsKaxRMpR9CNZbXLnX0Vi6qjNO/YWqKQkiNz0ZayGlBpwkMqi17jRC9lovJQ4iISkZaCLlyk3Mmk9fWaU9zJudxirR0dUm9pnI0QdQm5TLlSLBc0PjttfS8dLQdh/l7/OKVcF5AjgIfDRFfi8eC1h73wccWkNOxVZdoM/iiAjcWdw9AadnENnTmfizs5OV8SB2z6/U027nbGbsr6Yh4ZVvipSbIUhRiscvwDSmORMdElduzaIPnlpEIrPoBRhBysWbhpISUTctKJR3F+OsrAzGvDSMd+czCoKk6emvQjHGwoXBFSAbtjVkkQ/lalQJBFEF1WkgugdMcMXUEdambHTZwpnVeRm8EMkCO5u1Hrry0gRyj6hKrY3B/AzjUDVEk7LSlLGvBASge4HkJHLDk26tccDKWrxPWvgMdg3sl5v/nrKlOGWm1cgRSLiqaz5Ghn/6OSmVln9Nsp1Z180xpdRGIWmZhhNRLCYwR4DMqHK+OQmH5JqfsKxJQMtZWl1jOs9AKkL8k9VnIOTYoCl5s6AgAhBw7EdbasNLKxTtNjegt3y7vX8pOzUFYtuqphehhWI2Q48xf20hHLWjkHIShlfXafBlbe+5KSMW1rEHjSMsBkzSV5Llz8qUsATBBlSK7dCyu49/Nhff19mJe6t9EOtKS1VA2garlovUFRTToWkjHIhlbR8frG0VaOmrj9RLri7QjrJdRND+YOT7tdu4MvVRcJm/UZGfk1ae+ZUHbhYMWbq+3VGPjZBIyvQmrEUJBCEEa1PlyMedoEIJxWtGGpVYytmwyasjFVsSM8yKMnkvKshjHbA0N1T4WOVpHI9AuSsaqkkyDgqfwGfC6xXCYa4nnSmI+uC/SsVJeOR/Nse5tviNGjiOguqpcPHKUp2pyRFejxsBoB+NXte9dUte0yVh+dpzvIHMtUvdq0R27nBZ3sVELj4HI58dcRDPqVy+W+wO56HcujGUsLUrLR41Ax4zTPM12fqmbjCtCEKWjFBeGXD58rMZp+h2qiJDnYPD9oo05RJz40RGIvFaXdNVLROnUPvV5+LtR0QpijAXRCxrEKO5qgeKWKGRGLy5qybXqpiD6fsciyB45OiPQablM8tSOuch4DKgbZ2hpfmaNHAuil87JRBSO1xVGSCZWS5/s+0SUkpWLQo6dLu90dIvRMZXTyromP0T0EsyYCNhtCgSRFubCUe4kMrTYoCajr/K33ewAa9Sm5TOO5HQsLu+s9mACM6ZUtoCJJetl+p7HeVaI/twZemVoLecYeHUbCmsuGCcdeQcoKpUkqhUBPViEu3j1cuypoEEcog4Qvdo1CULhATgHRzegiMLJRSPXOhZdwWRpIBVWO0Rv5qWB9lpJP80Fk7qXrZeufcghR5VvEddS+yvgGUi4Jp/vj8puo2okxBHX+3tKy+sFqWuWqNEYupKHALQlf+/2NxQwI+/fKR2T8zICHZOUFsqdKOoGgZmMcyeIbtkVzcconNS11F4NoGNcQuz0FTyX0zEh/9Ns5466kYaWDYdDSNEimIWNkKtfbLpUEXJuADlKVAGNs+ijTq/MOrHq7g+oE2icAxSODPCSKgKLHG0FE6zgAYtdc+Skvq90i0F7I3Za4+jr+K3T0ollu8nHo2ApSwkMyBh/b1DIzwty2tCgB5SW4bUtckSHeqkcyQioxjwZKJtwZ6Q/ZnAtGAGT2sim8g3Z65jefKXvz2Aiik4GM5fK0GZzcF0f60soS/PMcme22pRXxkLiWhQ4DWwvUL7lNNu5Q/RysqMT9BBHTFQTZS6sRzTQ/P+NeZONVRy0oKNr8VZ/l4xl5GSNUyM6WRIF2Dr6IheEqCHa8zSQXLjR0RAIUauDqIxcytt/AKIvzlyG1cDJ22OaW1U3cGNWxpUqu+gW6xy2M0L0ToucXrRKSImq01r1Ui901AgBSMb7G1wJbdJHEMB9H2NgHIEskYzt7msJaCMdiyI9HWkFVWIO0cc6xlGIPaDOFkJEVXLSZluQt6Qa7DTbuTP0qvRsNLx2pzk371W1oSUSKNh59en//AZ3V8GDlM1FB/w5lWe2nGOXKkWgaRgcncgSL1iSiMLqGQVHkUbOPhRlFOxzBPNYxqn0ESMXjdxWnaG0nHMg6By0EeTnavPa9jyjMn4Q1lsUbFEoio440op0rNId5fbELxTHyJG80bSRlnXAyJmLPuhUUYuO5bXcS0RMpIWMo3UOTHegPRw8L8t1rI4lZzzGATnzDJzmImDmo5BROqAFEXAtO63Xlve3Onba7dwZeiJNRdiJU8cJGOOEqggqRYFDTjb0KFHlkWPXDB8rqvLPbOul7YJCZXwWOTqKxqBgzx9SvVYmRWl1zqBitCsNMH9ukaMco5bxfH+AHGP+ui42+bd1NLb2u5k7ifRCjLEmyU0yEBgnWMGUvY6xvI9Zx1iWjeiEo4DomGR22tDQjUb3jFzgoWpAx1REJfoguajd31EEaqMTR7cAuQCQNSVc9W9VeeWI6MHY0RaQZ3QvSsYiSq+lY6fZzqehF8jVJb2AcUjSOGVvBFqZ/802SJQBD12MQAPtQbQJEEIZY4DCK0IScul00tGHoqSuoRGKPockKm9UBlXSHQYF+0WIjQNRsCfBGgGDqqpx9A4wZyAXFOmJ8Y/ZGDQ2ArsijQAFy3xLCnSsggn8eStq9MiRyljkM1tKq0X1wejEApCkIxp+tBLpNKhGmQfha7V0zCJn3snOm5/setHnPGn52BJa/g1mAKbvee3D/TABMEA0EBHNMtM6dprtXBr6riNoHAuva8+HUQjFl/HBEqvAoGgPrZ1GxN+q5BZ4ZsS38waUyNAhhNSlWt7H15ZjKhVEYnGq5zKGLuIvpz4EURUqIytlp6M2DnpBI0ML6JYg0pJIP8w3gOqKSZZc3UKiT1DemAxFEeoYcnQ4Cjnempp0SWlFOrYDOcYVXx4YRYfNIRTapXiTERHRZoyjE0n1KbkEIIvljIAZioDhG8EEbep25hYHXP8mqsBwYyrueqNjlrpBETiLu9orOpN2Lg29NAIWHeM3T2GEwN9Fb8whEsYRnXgJ0U6MwsfskaOsvNDHGZiNIQhtAYQEN2yVULQRIgODosoLbXSEQvFUF5uWcQcRtc23QBQeOM2yyzahsVjjTEYuVcZ8f8v5hpGWjTQWGIFIx+q7YTWYsK+5s3s1VGUTogcBCkZGs09234UxtFxHv2j8pMeC5DL6BKaq1DF9UL6lRFoAMKF9J5Y2dY5OOeD6XHIsUZIcUVq79OKskrHnrrySSISPFjl2wYFPAlE75JwqOuZr8OdEgroJJhvz/fPfzgh55CgTlaiKwCHHXQalYVAjzrGG9ULGbJyiUDhPCE2XhEZ0Q8BrN6KTOhZ939Cg9HYsC+RiQ3ErfyN7lguiB5WOuaqbeNu+HIt1WhnI0kV61qAAGkxSd715rpbRLg4Y0C1c3lh35s5jcVSnWC820gpQcDH0wwyM5Hrt6rXsHG8H/M7Wcv+Abgmjk20EjDyYYb4/itqQLE+znUtEX0LO7L064rWrcSS/azFAjsWrj3gRVkRvFqFAKJWioNIHISQuybPUzTiKemV0eiZYhGiTT6EILHKcu+ZMwNG0q24sCuV/o4QzyyUqV0PRid2Awpfj3xwPEd0RywUdIUsU5wh4/BYFK47cOWByTjPSMU5UWuqmszpmdVxw0Sjp6hyN0EsJKMMKmh0OGDkgu15QpUrEa0e5E1urL+UPo1kBTKBcjNPwFVzaOVkdk5EWAjMj0IuWLE+znUtEX8NHTIM4Gmb+CeJcazK2Xps/JyLabGMuGFZXiAVtr8X3d7z2ACpVTIiOEH22imuQq33BiqM7rNMA13KLwNFA9ZllYhs5gLC8MTC0ldLSJbTT/ePciT0aolBq1mkJuWSbjO2Se14oY+vMQVi/U8fAW4mI2uWVIa9dko51/FIvnaFTzhQ5YBud1EjL5i6msdgzcISOBSjYOc3eXMs4J0TBrsqGKSzjGmnqa3F0pnRs/o3TMUkDweNHRGLZRMDb0evYabbzieiFEXT85eiRow0fUUJocAtXL0K0aQKhmhbnGCZxghAdbowxyNUtgka+wVE3FgWj58rYCNhrlWcG0UlFbnxtLeNCaQHkFiLHrTaO1dDFNFCE3OwxFzxedHCdpA2RLKMS2raOtfdqWEQ/zhUcsg//BKNNXEFk5WJfCFMBQOlSdMxtvrLzAhCtdQ5d0u9/tcDERjpFlgENNIxy7WkZs45FiN6uPSJJD9pr+WieK6tc2asBWWeF6BcZ+pTS61JK96eU4KsA09R+IqV0Z0rpIymlrxHffUtK6ZPzf99yWg/eanIDijeOILnUCOuLERp1vTTPRxyKEkaOoy9vlCWR1qurKgJraEcfijY5+kTwlEZGKMeB4pZSSWcc4ugEUjedHIuWy0QPmeTW3LVEWmZBc2LdykWOxaHz4gBjvbB9IHIURkgfmXHyEtqdOrZjr4Z1gLKM0L5ovh5RXSegSxiYdN00L/a9vE2nmeLSYiKpY1YuYFNeZ40zBhM43+BLJVGSWuUbgL0oB7cBHav3B8dsBNG84/s7ra9fUENPRK8nopc3vn8FEX3F/N/NRPRTREQppefR9OrBr6XpxeDfk1J67tU+7NKmPLGIxWwVAw5FQblWgLSJJHUzfy43XzmEVLlAed9KEQDF7QSvnZDimmvZscCwOqA7ttg41ujILAIQnURVL/X+AJ2b6Mgi6iJL06dsW4fXwpQal4Ra49SiO2zUxt8Noz+JssjFOnOHgk+gY4MFBlTlAu4vwYQFIMcm0pFj8WCC1FjcKzYB1SmjACuvaSw2aiQlF+VMjSxtFFD0xSDnyWnbddQ1qUY2wuo4ia4WFcjnKsbZRo0CMEK5SL0oOkZaLl9I6ibnfCsRPdz4ySuJ6Ofz1N5PRDeklP4sEX0TEb0z5/xwzvkRInontR3GqTSpIMgIHG9PZlBK6VnDOC0q43NGAF/L0xoN6iIKRRu8dhyKRnIBYX3S+YYl1I2MtJpGAMqSsAO2Ser5nwjtlQSuDesD5OhC9DDSKh+rhLO8lkTBiO+HNfkc6bXkYvIwfKpqVF2C5yXOnWjjWMde748pPbTvA8lSvuDDrTEjS+sctg0d4+hIXouo0nAhDbaEHnSRngFZQb4BHjPiogM6k3Zal/0SIvqs+Pue+bPoc9dSSjenlC6mlC4+8MADT+thChWwI+Qqhpa9ep5CyL73kx0twlZdMCqxwoqLFYefEW7bL/exdEfD0SRcjVTQ3jCov+X9o4oIb1BovpbeGFNkCVC4V3Ysy5UZP3PBEgRF1+I+0aFiigsOqBsUaYV5mKjqpjgaIxe+FnCAMd3hN/i5jUGRQTHjb+nY1tA9LeMY5ZS8LMX4paNp6VhEdwAdm96F0Ilr0dwnrrqBlVUgD+I5en2tEehYLSHW8ohAzmm3Z0wyNud8S875ppzzTTfeeOPTupbkj6WyRdRNCR/HTMM4GrTLOyMtcsTXkhtQYiOwi6cDY8k+sUzkoxOJHEPO077KzkQ6CO1tgXGecgT6eXyliFk4wKCVeWlFWnYsAgVC42iSfvzvKhctS01R8P1Jy8XNJQYTKt9gUDAa/6Qv5HNKTpb8e7nV3+/7yA0di5xWW8dM7qYYx5hXH7PdQ2F0DDmaKA9l6I7igAClJhP+Wo87MxasYwjRo701SpbWmRe5CFl2ulSUbxOt49Nup2Xo7yWiF4i/v3T+LPr8TFuXAi408sQtgyImSM5B4aJHYBwFQov4fnV/ozi49K6d+fe8+mQ8PH/sDYpDWxDtWbpF89rWoOGwWsoSjKVEJ/q5eOFEXHCLo48SuJ7S8nPZQvTszOMcAZ5jxIWXPMwuXjuijqxxArmTCNHaayEdqwCEk47T9ygZWyIt4EwjWfJc5mwcc1flpcbSiICljtkTQolkHkw/F3JaEdVZgRmmlGLqBhQv/ClD9G8mor85V9/8BSJ6LOf8OSJ6OxG9LKX03DkJ+7L5szNtZeG4RTD933p1lURBVS8wrJ7+HxrHDCoiOj4fpv6OKHZA/JucsXFAfewilHojNzktuRb/u6AaFJ0EXPRmBCg4kqWZF5t0rAunjkVywc2KCEND5YycRt0VDceCnEbC0clOtMcOOJBlK2q0dBva6s/GiXWMb8PoupXwH0Z7zIZ2Wu48+ozq5eO3q7VkiZKxu2iwVtWNA0YsS6OXXsfMHLd0bNZXlovKN0SUFutYRN2cEaJftGEqpfQGInopET0/pXQPTZU0ayKinPNrieitRPSXiOhOIrpERN86f/dwSun7iei2+VLfl3NuJXVPpcXGWSuuNbS1D6k+cMOOCWt1zfCyLdUWuVrqovQBTssZFLQIARKJNh/Ja0WLECGUqLqkVCMZhDjd3yNtOBZDd6Cql12IHlJXYEEj/rSVjA3npWODiukGNBbFkTcMrb6PODYgcEDymR0KdnqJtu1P/w+rsYpBI9FH5LQaOub0oiEXu8GvrD1QdSM3ObWMM9IxdOTy0fbkUSN0Gl2Vi+zTAian2RYZ+pzzq3d8n4no24PvXkdErzv5o119qwbN83dEsXGMFk47GYu54BIdOMWNzyGxtd/8zKjqJiy9E4vQKlsSBk3eo557rh3Q1GdGwa5czezMXYiCowQiGkvlPAGtwAgpyJ3YsFo+M6LB9Pk09XnltVBE4aMTPZfW0Naa9HotKUt9j+n/tupGynIYwasvYZJcAxOUcI90jMdSNkyZ9QLpjjDhH1WWNaLGHfkGWxJaThttGWdzrToWqtcqaw/r2C6qE0X5HjCSkQudSXvGJGNPs3WBQYm4aBb2yNyiNbQoRC/IDVd3VLQp7s+171nXsbdD9HhLNepTFHcEZXwcijqDYsYCkOt2HJ2yq3xDWThknkvcp8N0g0/6TX+zfMJ68TzRBPb1d9FYeBFm6IAbBmULDG0xaAujE6svycoSJGMb0UmrVJbL+JCOHRsaRo5ll45ZRA+PhpAOUMmLQlmW8e+IGm2+w/Lt5VoZRAeBLNVpq2MmuZFMJvyRjkXl0HiDn6SGSf1ejuWZztE/oxobgcg4V7Q3f25QsFoEgluzykkUh2+oLCusy11AEaBQEPWxXGRIXZwwFEVhraqjNwsHJf16STc0DBqoa5vjAAAgAElEQVRCW4gL5rHgSMtTapLuQYjaG5QdhnYGBvbNW0iWbX0JcieNiCIyKF2KK76i+xfjGOjYseGifdml17GcaXHZa33BB3DAGTitIAIscgERYAV5OGqEB6QFzrQVgZfcCXDaaINZSy6n2c6loecXZ0fG2aJwSRFYtCe5tVYoGpZlAYOGKliI8A5MhVBa1E1xWnwtbJw4amndP0JVsPY5CNEjuqFGFKDPqMdiF+dSRKv6LAqfMQqOEO307yma2Jpy3IjSq8Yhpi52OXNI3QCDVmr10RxvPQpm4xjpWFTHH81LBiDLRzri/glXtsnjq1tVYqjqZlfxhNMxIBeV8Ec6to3WGCjvVOuooS97Q7+8hcY52G7cLC/k6MCVKk7/h6F41+ZvNwMu1bT1yjwW3raPkCuiO7oU0A0JRy0usbwgoqj5Bo+CiARCEn1Sakcnlj/uzHxZWaKkW0QD8fXwDlQqewXsuTXT/eMSWneekRmLdcAtSinMw4QGhZxBKYZ20Neqb0WK9gTgDX7o/lbG1gHC3ccNYNRy2thpxSi4UopaLqXqyCB3q2MQ5AU6Fm7+YpAXAaNWpLWnbpY3nmx3DkkQvtl6YT2puLogQqH8HVIQSVEsnWxZloWdBqZ7UNKPHdDWIfr2wilRADCOOZudqdbQ2bAaha/B/XcitwU0GKTOgqqbnWE1Mk4L56U44PlzKTNpaNG82EPFpn/LDWPiWkIvkb42jWPAq1tE7x2wvg8CWTanheYSUVdlg1sjaot0DOXnotLqqIKndZRKpUflAXG4T0qplvZCm7Snbk7cymTv4rXNd6i8MUrGWp7O85TtzP/SyZaZf2Qc0VGtXUrwubpg4bQjmkqD2dwFlEtjLJGhdTuWQ4qgdIkRrV3QwcKFJZEnpBvgbspWFNIlTN0sNLSOugkqiPj+0JkDudT1gmUZvWISApOu7nI9kV4EaJfr6DFt2tCxHXPpdSyIGkdPw/AjYlm28y0bkB+a7u+d1mm2c2vol3B7tsSsvhzbTDYIReur0RrGMeT1R2fMp+dCpZryOAM5RlJjsVQMpm6CmnirbAChDQOOTvzRCNP/6+mVJPrU4ySa+QaBglOKKYIx80mUUi67F65Dm4yCTaQVPRdft4bojT52XgJnnhtOK6RukEGRUWMQTdrn0hENuWvZZGyLV+5TLYSwr5FUcgHOAclykstoqM7p/7DqpqOSwG1GjULH5LViSkvIxTo6tF6c06rPDG0CmOPTbOfS0E+H/PukZ51Ub9DkIlyUjF2ycLJX6KnPjuQSuH+4a3L0BkVSNz468YugFZ1InjR65lYZG67g2WGczX12RQfWAUi5eBqqkSMYsNFEckmJikFrVWTY76LELuJ1faSnZYbmpYw/0DFMUcg3TzV03DjgOALl4wzE8y5A4aNNYPL9Qxool+eRMoNJzyA64X/HEfCCOUbrpQGmOiP76XM/x6fZzqWhVwdeLZygvkt0ZJALUeW10ZkiREF1h5ps/VxEk1fHaCdI1LXKsoINMDDfEEQarWtF1QIq3xCEqGgsOBk7/R/vMk5BYjWoFOnMWKCjiyKKERu61puMIkQN7t+nBN8k1HeYhrFcMKy6Ceby2DlgKmOM7m91rJTK8rowp7pGTqtZXRIVHJSomWCfJnWD6J6o6iigwdChYn03RflRrg0enNfVvRoJ2AsbzUfO9LTb+TT0YhEigxJNdqu6wu+y1deC3F6jWgCHgnHte4TC43wDNtpLNvksqjwIEUrD0HU7nFZAqYXlhSgPY6MjEz7X+5P6nPssfS6ZcMd0D+avK99Oqs8IZGwdsKM7WgnckIZqVaosT/h3XYr1FVXd8HMhp9UFVTcyamzRMA6F+x3DLpqX8u8aDhAher7WCNYLswkRpbgdnR5PY9mXV564dUlQJ2jhBPW/UaKqUDcIbURVFI1qgWMbVu8IBVtn7UT8cbQzdURjaVyrvLFn9MlYfuYWDRPWfgdOw46/T6m9+StMeDeSnoEsj7fRWHYk6fv4/tqgNyqIGoge0XBxArfev2m0F8ly+n9EtyBEXdZeIz9lr9V1CZ4Cy/+01I2X8QJZtsYvohMUTduozTpN7VDax0lMVCfVz+3a3yP65U0h+oZxjvhTv2kCb8+ernUCLlggNMhrI7ohcXmjV2iiYOGkunAcfwn5Q2rIZUqGhfmGrUWO87VGhJzaSfLjYVQblrhPdEBaZrkAXhsaoa7u2oyiI/WyCnMt67TQDtQWctY65o1jvJEOVFZ1goYKjFAysifC58GvBDBB1zoKojOeFwtyMtiU5mgwJct4TXIfjIIjWaJNgTt0bEc0a3MaRPF7ChDS53/a4gVpE2yf02zn0tBLg9LyxHYi8GYSvNW+GdbPCMUalJ28NjQo0amaZPqI8QcIJQqrPXUj7pNwYpcdaFgqGnDUcNu+uL/lKPtgEXYpOFp2x1gi4zD1MWeaNOg5ru5o3d8ZlIRLeJnXjWig6CCyAVWJRTpmQU4ETBZGR13CkUaX6r4TO/bo/tMc4yolNJZmHX8nzsmP1p7RsWm9gBLexPPinzcai7pWMC+t4oU9dXOCVlBwsHC2Zmcqfxcm8HJcdQNDzi4Rejm2TFRiQ9ugCELqZio9s+eIx8cZzDRMD54ruD9KhoYUgTO0HiFFBmVrKC1+NrQIJUduHbO8fxTp2DFyn2hx2vsXI+AiLb4/GEuXystV7HzhbfsNp9VFxgnf36Ngfa0JTHg9JorzLa2qm9FW3TSdRgrLG8tYgO6jqpsqS8+d87UsalZgwkY687yoqGVh1IYNetuZ7pOxJ2gaBe9GO/zdrsRqk/M0KDx6LSARh6L1c0c3gI054THJ2+zGIpNLlgpobRiCVQwpqK4QfaBBQVU3Hd7kpNGOGopJYPr727GkpBeODbmj3EG9P3KmLYOCSyJt1Qt/F3LBI/mEf2ssKcHKqi4aS6MQoZZE7irv1OOHu58TrtJqbv7q5EYycX+hY1KUrbWXUk1S2/p6Ip+k5mdp5S5cfb91wHYsQTTLfWwEpMayR/TLm+KCobJl5znlwnGTDUvP5msFO1PbXn15iZVUNkgdjYDuSAmG+6pSpOG07FjKeeRBnxYKRDRUtAHFOg3+LnLARJxAXTYWvQjFPYReNBE9kGWLuoGIfguik06+KEb/Xo3FRW0t6mZHHX0AZpADjJKuzVLNXVRnhIIDB4wMbXMsQe7kePDAqO/indStPAy6f+jM+Zm3Y5vS+kIi+pTSy1NKn0gp3ZlS+k7w/T9PKX1o/u+PUkqPiu8G8d2bT/Pho9anVF9nFqAdhOjRZPMZFbu3x+tJ3SKeTtEty7y6UraFY5HJJYSQLEXSMmhdR8U4hU4LGLSakNL333U+TSusjjjXVr4hMk7QCO1wGm5eRhBp7dCx+l5c4DSCOY7eS9ym1HB00toB6iO96f/RfeAu17Tw9MpFToPqWJDTbIyltWPY2tKpgqhd2QRpoF1yAX22LtdWHfPUh86k7XzDVEqpJ6LXENFfJKJ7iOi2lNKbc84f5d/knP9P8fu/R0RfLS5xOef84tN75N0tVFyh7D58ixfU1AdTJ/CNQTsVN9N1h9qZpCQMqg2Fs9/ko6gbFJ0EYfV0/wiF4122m5L002PkPp6iiLnYjJKx8lp2LIGj0fyt/H17LPiAsCqXtbiYu5ZxzigZLPvYKDyUy/zvyNBGyeCm0xpGumYNxhIlY0f/cmzrtFxiuZHYtgfnyTmWY+P+UcUZ95E5JX621lj8jmGx9pvAyIIskOvbsYemOjM9Ru5z3UGvfh9d6zTbEv/xEiK6M+d8V875mIh+hYhe2fj9q4noDafxcFfbSulZo/7WGacwFKa5j66xttUlUVgdLUKH9pIwqEpxd1fdOBSckjtzn8dI1CjxCqsIsKFBcuE+IXXUqO6wb7EqcoEocPp/eHZIlIwNNvnwWJxBCYxA39Wy28VzfEK9cMjR6dg0FvnMuuDA3x+jc/zqyYgGKtcKql6IpiigGWlZHYMVb7EsJwCA79/Oz2FgVCKdAOQtjk66+MgMIgpLq58JdfRfQkSfFX/fM3/mWkrpy4joy4not8XH16SULqaU3p9S+ivRTVJKN8+/u/jAAw8seKy4cQIz3BgTGdqg6gb16cvnwearHQYlWWVroQrmTxeOpevwwpHleshoh+ezIOpG3r9J3eixDCMK66f/H219iN6FDrDKv0U32MPeYAJTAgA7loQNWq26wWfYw3mR1BHSixGfOGnfiFbGAkFGvT8aS2s3q60gSoG+cJ9Wkn6i9PTz8nOhZ45omGgs0jmgJHnrDHlMD6KSSKr3h9cKopPmHAfVWKDPabbTZoReRUS/mnMexGdflnO+iYj+BhH9WErpP0Udc8635JxvyjnfdOONNz6th2AUHG0yijg/+ELphPsURAlPvAwSm4xcB5104/4RdVMqIhaPJa66KX2CMU59zHM1k35RRIH3JKBt+xrRarl0gaORUUBET6H7R4lVoshpNpLkOX7nwU5EjRD91lIEDeSYqjNdnruJE6u1skw/byiXRHhPQETdlKoX7GgQOk+NsaioFY3FJlBlBOqcRrxjt/bxzxUmqdEZOIEd4e9Qn9NsSwz9vUT0AvH3l86fofYqMrRNzvne+f93EdG7SfP3Z9IkCoZ0xxbTDTurGJASBAdendSgtCiCkhCC4X52Y1HRCUL0RnFdQsigumgLfLk/iE6iUs2d2/YhCm3Mi41OhKFxzxwYlL5hBCIU3KVdZwBFc4xQIME+lTqI6MHGxpxtNJa4UibcsbwD0UcAoEkPWuPY2ncS6Fg4L7l9/MnS6ETeP5pje/8InUuQ50uInxkc/W1E9BUppS9PKR3QZMxd9UxK6T8noucS0fvEZ89NKR3O/34+EX0dEX3U9j3tVrhgo7jydWqtLD5OCAXIcfSKozjq4BwYH4rG2/bRSyFaC7oLDYo0jo1EXYSCA7SDxoKig1KpEixCe4Y59zlG15KLEKHgQmvoPlGlCPdBdfzwULEIOQpniugGFAFGfZwDNjJDIKOMfwQGRUZa1jiizwMHxH0QDRahcBkd2D6a79e6R4Tnpe/w/fuujgU5oImG0deSsoRU6zZwWlu/+1lRfUtBXiDL02w7q25yztuU0t8lorcTUU9Er8s535FS+j4iuphzZqP/KiL6lZxzFt3/CyL66ZTSSJNT+aeyWues2sQRtypo2sYJTralCIShWxvNUcgNLGi7pZr7XN4M8zOLzxNGbso4gEW4aW6Px3XsEbcYHRBGhBFKqw96rhU4FKz+XZ9LnUOjaAWCn3cJHEHQojuA0+67RJePBz8Wrn2PHPBORB/1qb9v5pRU0k983qRuqo5ZvYjyM3ytVY90HEV6NPexLx5pyAXcc9f4owhcOiAky3jDFD4Cge+Pq6F0fobH2bQjyPYE0dFptp2Gnogo5/xWInqr+ey7zd/fC/q9l4j+3NN4vqtqUx17rLjI0OpQUH9OBMr45uvmTBgFNiZb9pfftUrv7HPJcN8qW5+mWmZ7rRQsnIie4uccTloRIfpElSLIONnn5e9aW/1tn5ah6zssl2a+I6A7eE9A9PxYlvLfu/uoMTYiUAQm7DlLrs+CBKJM3h+uaklg7RPrhX8XgxgjMHTcbGJ16oPzUDyXlgaLkD6RL1Pm76pz8GAiesfy8TDS4co4wISdOf87ByDv0vHZ1tGfy52xU9XN9G8kbCJsUMrvQMjlN7MQ/D33wRyx6LPQOKr7gEVoEeX0O9wnGqOkTpBBQf+WpZpo4XCzqG4AkVYXyNX12fH80++n/6MFjSpK5P3RIuRa6qmPGOMOFDjlVMxYdiBXm4fR8vbXGoHTQvSW/Ls4OktDZd+nxWsnBQDEPQJ9kw7ARYBgvclrjZkgmEHj7IQzRzu57TO2nlnub0BzLMcl/965joG9QH1Os51LQx8Z9EjwzT4LJrhlHCOD5KpuIocULIKruT+iOPy99XMtGTNC9Dv7LBhX67twjM1FGFxX0kit+4dGePczuu8WjIs30hFhFHqSZ5z+DvqDSFH+5ngA+wuS/519TgRM7DPa3y2N9NDbm+y/EQBA91dloAt0ZOlYoogMUTfld9YwnVI7n4ZeyGoVCX4pQgiUTS3CpYauMaGR8kSLpYUqwmuF/yb4+/ZzNfoE30UKHY2x9d0SubajE/y8TYMaUQwL5G37hzICaA+OZYlcl/ZZYMQWg4kTOsbpu/rvXVHPrustAQOojh7/WzzXAt0lsk4jekbVpRltnFY7l4a+Fdahz6fvRP8TLpwWclpCN9j7L1Gq5YsAf26NFv/ZWoRhdNRymgsMWmsskXOOZIkoGfjdQhS8hAZbIu/WM7cQHX/XGkts9FUX1ce+F/npjGWZjsbX2pWMhX2WRHcLDLjrs0BHWmNZoiOt3M2eujlBW0LdtBR3FSpIsAgXosCWoV9Cy0Th8mLqZAGt0FpQS+Rqf7eIOrqKBd1auOFYFhinltOyZacnecalzx8982IaZqEslzi6pzsW6VgkmFiuL/G8RMZxCbBqRa1LogM1lqtwWnvq5pTakkW81KC1KRL+nMznJzMI7tlOGJY3qaOlEUVAESyhlJ6uQY1Qc+uZm6F42u2Ar44/PpnTayWDl0Z6fO2rGctiALKg/1KnEem+GktDX5Zw3P755ef4N23qaLfTi/q0n2v32t313Wm182noF6AVuzFnEbcYUTdL+cuGQYvpDtxHLwJ1qfA+UdQin3kxf7mQ81yCqqKopXX/lhFg+S0N0ZvIdVEov9tQ2edZoiPyGovphibdIr5bMP7FeaClTnPBelkKJhbRYAvmq3Wf1ljYfiymblpOs/HdabVzaegjnrJlHBdNNqiXJ/JOY9ki0NdapOxLjVPYxz+7/Z3Vs0U0kOmDaqFbz7LU0D59Bxzc82pQ8KIxLp2XhnFchOiDz5eizatxgE8HBTeM49Ik9RLQsOS609/+GXeOhddLM7FaP5e3bBcc0Jm082noA6E2Q9ElPF2wcJpoZ+EiWOIcFieWF9ACkUFpUjdRktktaPk7fw/bX3Ke6AiE+u/o82AsfXwttfCC0N8+ZzSWJVVS9u9o4S+loRYZughRNwyt7NKqutHPvPu51FgW1tEvycPY54/6RM/r+yxbL0uom8VjESDL6v9ptXNp6JcklJpVNwtRcLRwllAnEdpKKUbES9C9u8+C2mn53dWUvkUL2o0lQE7y71YZYbiIFi7CRdUV4DiL6TfxWK6qjn4hmChOa+G1mmV8oYzxdVs00NVw4RGYuCrqaAHIWhKNLu3z/7d3rcGWFdX5W+c+Z+YO8x4e82BmYHgjA1x5qLxfo1BAFBUNBhNx4oOggZQBMZpgTKGpMpqKSUkhiZVKRIJRJykiopDSxKgzBBJEtATEADGCggQCDHNnOj/27nP27t1r7dV99pkzd09/VbfOPX12v7tXf73W6t7VnW65HCF5auvSJFop6GO21TEMxf5WGQT9TGhR0BXCIwSKyIIVbI9bKNiBq5zQUv4xDKnbL8KujZ+ETloqQVufh7b8XJml/EMJQAyj9b2QxZ9nNb/e92p53bRYksMsTjGLhrg70vYLO8b9cdwdrC/OoAyxQEsFPdvBSuHouzzLjZ/9xqSlYQiVOP5wzY4gZlvPDVBpoQn2elHWJctfEUfRr8XfRDc+7Radqs/IZamWozaOtDgUdhRFFJN2X65Sl3+Ue6OouuEEdbnMKnWHkkz0xks5j+Jj/dobNMZokRgF7oATow+EzGrsJy/QtDpPjRsfvxV1yqxgCNqtoIZVcIuTtBXlbplUe1co8q8YyZmJy4Vnact5SGXhmJs4XpSMWrWjqAjh+rQ0qrosjn1GVy6Nl1YlvoKAaImJhsyIi2mpLoWyi3EKz2l24No+1qihEqMPg4a5Sysxf0rVn4+kBtKu6qy7FjsJ/M9I+WjYpjQJ1GyHEU6ioVIhUNW6cHYBLv5fL6iKaUj9wh2wk+xA7AKmrkt9W3BxfFfr+uJo2tiNr2HOoh0oUD0pkglHXdIlecwCWMlT0S+iSk059ziS1SRaKeg5nSkgsYpifOXEVXR2MYo78EpxuuUq14XT7ZVP5/FxtPpbdhIqJrRkWObS0uavObDEeeq4eXBXIMhGR6ZfmPoX70DS9ku5j/z5qPXKChbs1pFfqOANr5S/8JNO3QFveDX/+rS046UUR+hL3vZRjsPt9HR18eefVDeB0Og8RXctJXPtCTRe0JcHjv+Zcrl0aRXLJrFNzQIGFBkKyuEKQyO3OMl1YeIoGVr5f7fMzKKlWKi0vt+iMVpT/0DmrGeOfBuzaigmTnnREvqFGQvcvJB09Oxi2KfqRB1H2S+sSo1pS5EYMeSzSagEPRFtJKIfEtGDRHS15/e3EtGTRHRv/ndZ4bdLiehH+d+lTRaeg8pQqGVIoqC1afL5q9NiBqGGCahtBJr8JdWNkm1xE1o2xsrlyv6v5uHNv7towBue5e9Pq6LW0CymfY6xEiNWMteYRUtHDLiFpjz4VOoeZlxKdeHVnqUorJE8RqWnYeFag7+GMPJkAgND7RumiGgEwKcAnAXgMQBbiGiz55WAnzfGXO7EXQzgQwCmARgAd+dxn26k9AxkXXj2KU4cZrBxg0pie1pPGc3A5SYhJ5zc/0Wvm/y79pQvt4AU09Ia/bI6dPxxFCcl3TgcC1V58HDCSRIolQUNwI4AFi70S28BRjlctdPxpxUyxjodAnYaz66JiS/u2phycYJWOs3KCEeNZxc3Loh0p7qL32VipktrdzHGHgfgQWPMw8aYlwDcDOACZfrnALjDGPNULtzvALAxrqh6cPpXQGCb1OtsDYssxonbHTDlClAR8AzJH0cSzn15EVQWU9supWBWaGf523AnTh9CgFtMpXRj1GP8jkISqEx4ZQHkhHOhLApjZlYef7iqLu5BMsXiwi6aMSRLyai5di3+xnppMcLcm38gyZPr4g9vEhpBvwLAo4Xvj+VhLl5HRP9JRLcS0arAuCCiTUS0lYi2Pvnkk4pi8dBsHzmhqXUjA3ghELM7CHU9y35jwpn8i49xunitukW6cZJNq1QW/6TidLEVthWjolAwLG7iSv3CehApBarUx5xwVO20Ase+FEf01GHJjE44c55hmnksXk6oPf3N7EBjPGV4g7u/jMXvbns1iaa0Qv8AYI0x5mXIWPtnQxMwxtxgjJk2xkwvW7asr8JoDGWc1w3XoYCkWxMmjsKnXUqreHiLNW4pB5vKIMToQt34MbpY0YuiZhKG6fvD2qVYZ26hl41uTByGOYoqAmVaLFuU2liz01KyTU44lcmEsl8Uu04toxYNuDVjaUzs49JPrEpNQyDc9t9dVDePA1hV+L4yD+vCGPMLY8y2/OuNAI7Vxh0EYgwf9jmJuWiNoTqvH93A1QgBtY9voVxjTmVU+bMCpRQlyuumTqDL+n4nrb7qwggHhulLddEuWqK9Q0UmdAJFswBq5wW/mPW+c8RE9KPnVEIVAlQ/xoPJhJAWRwxlMsGF+/MZtupmC4D1RLSWiMYBXAxgc/EBItq38PV8AA/k/98O4GwiWkREiwCcnYcNFLK6Q+7UkM7uGVDL+ccYcTg/+vJp1PJvdQyFYw5y/gJDUhjgir/FsHCtLrbYLlrmplIRcIup25YlgcItmvCGizsdt55cHMWixcUZdfXtUr/UqMG0N1EW05YMy8EePMLc490YnfxtuzDkx023nH94v2h3bU2i1uvGGDNDRJcjE9AjAG4yxtxPRNcB2GqM2QzgCiI6H8AMgKcAvDWP+xQRfRjZYgEA1xljnhpAPUooNqR7GZMdmDH6Sy07l404AHbE5c8KNCZ/Al8ubuciMZTSHUCKCe2OW5FtdevixGHcLrn7iIrfY4Rjhbmyqj7FYq4UNOJOpysEnHBODUX+Z4plrlxQpmDBLDGSBF3MTosZ725fjuX9r1UDSfnXzSP3/2IcbVtKixZHAJpEraAHAGPMbQBuc8I+WPj/GgDXMHFvAnBTH2UMRnGAuCqKOrcoceA4HUScEFAwerdP7WNuWtypxeJ3ziPDwHif96YVyIJFn3y2LavP1NZFs9NhttVVt1N//uV2CVdpcZ5drF5ZamNtuzA7Ek73n+WTfWqvQCiWLVQNIqUl2cHK8atl734fUaTFjD+uLSVHDG7XKntD+etSbbNq3KbRypOxkkCr06tLVny1bk/wSKllSCGMvst2S8EVYWVRjF5ZAANZcOkZZrCHTUJ50eTcDr1l7rItQUXBTMIoRs8IFLWRWjCGsqobgW36nimmIaluQnetbs6qdpEWGqVKjXM7ld1Lw+aehrCJ9o7C/6W7dpxG212MsbMOxfk95g5qjlUwLaFioVJnq5mAf+AUJyXPtsqFH8spwo6dekbPsk1h4nDhUWow4sL9eRaFONeWY67ajukXiQWHGqmL36MWc64tmTpmceAFl39FnSmwcPtVGheltCSSY+eechdQuptIOcaK8ceZhV5LMiRiwrdL/KI5bGPsrENJ58oMavb9r05akncHJxxV+lvtYFMI5/FRV9Bn3x05L+ro6w7TEPGvOYvxY6+qqOp1ruXn+X7pCnpHApbiMHXhdmeipwhTZre9WOEgjBeuXUp2KEbSc8JR2ulo9eo2bWeIiTtQzaLFkgknjh3jXB2Lz1TiKNWD8lkRZtGSduAsyfS3S5PY4wQ920GKic953YTob0NdvCSBZvJZNj7inwQuNIYqtxm6ahCBbfDGWD1DqjOSu7lzL9soldlpl+KCGMqCxRegs4u5P+3qAlC/mLvjRYrTjVsRXP6+1OjVtTstlddNAAHwPVNMQ/Lg4fqfJXkBCzAnnGNcdXeFMbaVgr60qjszutvZnEBhJoebLlBmu75wbz4Me1Bt0Z3frGqmylzqJ2HFj75GrcBNwOwZ97t/Eorqjg7TLvlzEnPkyswxOl/+FlpvJGnRqutL7c2pxfJIXjcuuJ2DzUY8/a2tP6B9htsAABj0SURBVCOUyuoWp8wRKhLfM4DgRy8w+nGG0XfnsZunRNg4NVSpL/3pab20mkQrBb20qk+M+rfCvc7Wr+q8XjtcCPQGTrku3FudAEnQhzN6m4+WhfmeqYvDHSaR4tgobp0kRt3pxvEzOl8+XDhv0ynWxS2zXwho/Pv1Nh1v8ctlZupSmRNj9XYg6eI8X3mLcd04VUZdfMabLL8DFearq6Mf5Tx17GNO3ppzH1K7VC8ItOHlfDj1WJNopaCX9Jd2sleFU/bJdQKgV7e4d3qX8qnR+YpqII7RMzp6F+V2cRfAEaZc9WyD1WtHLIBcHHfSFqH1uikKes7ewKn6pEWLU2vFed3405AM/i64BYXzupkQFsDu7kDJ6OVFiwkv7HK5fplg7VDlvV6xau5CX9eXbs6Sa3MnsF00+SfVTSCKk8WduHVbPrepVR40AZOw7qCJFbi98nYqz1jM5IJ+Qqm6KUZ36zI5NuItF2eMLD/jr4vo+63VBVvhxNRJiuPeXSItFhacS2LIAuzG7capqaMvrRFmpyUJFFY4M+y8OObYHSjTxy46AsmpW8yleTM5Xp4XNk6ISq9OPemWt9guesMqWwVB0Oe/D07Ot1PQS2zLsjztwJUmIacLFdUcnODI05jDDGhfPpbNjI2Ww1nVTaldys9Mjvm9GDTG2DjVjU6g2jhcnbJnyt8tyZMYPQeJ7XJ5atVAdWc4fGXUXIFQLZs/DrdoTkqqm5rx6iJG0GqMkZMVApQLepfRC6qT+oUG3nBfmTl7i2anpW3jJtFKQV9sMLezLdutqGjs9o1R6bjpFr9rJ0E5n3K47fx54/4BDVT12qE6+pLqxpnscyyjVxqdfM9081GoG3gjNSOcAvLvLoCMMU6C25dTE9nh8Rde2lHOU1iAufCeSovPc/7EmPc3TqXjg03Ozae3mLqLGc/oeRbM5S30MSdomV1uERwBchm9ajftlN3OeVff74vbzYdZnKJUN4xKrUm0UtBLnd31S2Y6zoXGjY7b7vtQ19lzx0e9z/vy38kIek7NIfnRW9WNcX3vO/WTMOYKV9euwLKqPF+Jjbttac8PSMZYDm7bWUH/vCPoyyem/WlpzxcU05qaHPX+Jtk7uHy1Rurid24RctOyi8NLMzudPPh2qVNdSXWyRMTCjt/KWRFFu3ALjTDE+bowaXnTKNgifHEWzh1zozSGdgp6obM5HT3nKllO15+P5C5XSaNmgBQ9IIrl9aW7I5fKLlON8bqxE2n7Dv/EDfG6qTsUBlR3LvVsJ5zRu3Ek9Y9bjm45c0H/4vYd7HNuHKtOYF9aLhCDeRNOuzCeItI45WxHXL+IJ1CZOEumxgEA2xxBL7WLzYc92yLUqSLouxc6Oae/RZLHLTR8vm4ZLexX6d4gF9xZERtnwZwk6IMgdnadi5WULjMJJJ0rl4Y7IKxwco2xkhrKqm5cpqpRUVR09ON+htYrbzUNOzC5iSMdQNHrT/PyCnWqLIB5u8To6N2xYxn2C9t51Y3G1bCYtjQ+qsb4+l2AizqvG41q0YIz7C6ZN8Hkzaf19PMvAQD2WzCnnKdGRz/m3wG6jF7SfoTec1XKT6m6mXQWJF8abltacjV/UnXHZBRaKeilzrb6yRhGX+1s++kXWv6y+SebZUbugJZOQHI6eslDhXtmMheCL+1wBT285QWAFQuzCcv5kXPt5UOdAVdavKqqG/9OhzOsSmlxqhvJdtP1YFLehCmXxz9eJRZsw9iFRqFadL+74UtzRu+ieHmXKzgfe/p5AMD+S+Y65a1fgFyvm64xVrih1QV7viD/LvUKu9Nxynzw3vNr03DTeu7FGQDA/MnE6IMgDRirj3St9b3OrhfS3e9MZ1vD3SkHVV+J2H1ZidPy22ayOC6jK8ct5zPTFfTl8BgVxRyG0RMjgAFgv1zQv+jqryNYKOfG2fO6kRhs+XvP6ya8Xdw488b9jF46SPeHFx6Bd516AM48dLk3Toi/dFd/7IRLiwZ7YEsj0FhjeLntFs3zC/piGm7ZHn3qBQDA/kvmefOQ2sVV3XSNscJ9Ti44e4O0a+XS5crsGo19abhpPbctE/SWVAwCg0t5iNBYvmfcmx2FATI1MYrnts3wbllOfsfuvwibTl6Hd5xyAJu/m9aL2zMBK7FObiDGeJdUjLH5AlPRuQqCfsXCSQDAfz/zojdOjEdCxVDVsYJecq/0M3rp8i4ObrvY7XTF6FhU3ThxjlixAEesWFAtZ3d3VFuMXnksMXHCpX7pMnpmTIsqCo7MOM0v9keHgJ2mSoDyxdLuBLt5Khg951nmqm40dYvxutGerwCAj7/hKDz05HNs/m5az764HcBgVTeqlIloI4BPAhgBcKMx5nrn9ysBXIbsDVNPAvgNY8xP8t92ALgvf/S/jDHnN1R2FhIzsAPEFfS9i42qceZPZoKePdLthI+PdvD+1xzqLxszQS2j1+j4XFR8rxWShGP0rjGWG5wAcNlJ6/CvD/0C5x65bymc87qxE8llZ8X03Tpy6impLt04Acy5l1Y5n3kMy+IOH0nQGLar5cme3clcOe1zySueNC3C7mKl7LmFlnvHgbfM3YW+HH7Lb56ILY88VRmv3D31Yh5dRh+uuuF2AUGOGMJi+tpjVvrTYNrl2Vx143pcNYnalIloBMCnAJwF4DEAW4hoszHm+4XH7gEwbYx5nojeCeBjAN6Y//aCMWZDw+UWoTGGune1c8eggd6W6sUZTkWhLxu3fbRMWmL03CDWet0UUT0FmOvoGXc5X96rFs/F1648pVpOKse1mDsxgqVTE/i986qLILc7mtmZlUdW3ZR/M133ynDNpNsvrhdMN0+FuqGSdpdMBMSxwsnNX2L0rMG/HNcH7hyJrymXTk1g8byqXplT3R23djGOW7uYfT6gWdTXThRh22NHZXHIPn0xx0c7eGlmZ5S9o5q/v4xW0O81ZB39cQAeNMY8bIx5CcDNAC4oPmCMucsY83z+9dsA/EvaLoK4Bcw7pspcs0/fJLzo2Kw6rp+r7eOYye7G2bbd6uj5LuEERPWum3AmO84aY+NZqFvHsZEOtn7gTFywYUUlDsfot8/41TC+/Cx2dFU3EYy+cgkeI+hr1CPetBUqChfcoz1G71Pd+PvMdAW9OvtCWtX233LtGfjqb1cXegpsG0kNxcaxOx3hrptqnOyTI3m+xp5gbrvVuP1W8mfquenkdQCANUvnVeI0Bc1eYQWARwvfHwNwvPD82wD8U+H7JBFtRabWud4Y8yVfJCLaBGATAKxevVpRLB5SZ1uBMbODPzrtYtPJ6/Dm41dXrOKcu5ZYNsbFzuroJdUNhxhjbDWNLI4VrBbcRVQSOK8bCdyCYheeENUNdzJWA27iLp9fdieMWQDrvG4O328vNq6rbugxbZ+gt5/+dgmhztKBOY54hO5cNG6fLqzKKuTAlI3DvXnNF3NidATPYqYSHsfo/XEuPHoFLjy6Sn6aRKNKISK6BMA0gOIyv78x5nEiWgfgTiK6zxjzkBvXGHMDgBsAYHp62t2pBkHjR799p5+5+mISkdf1KYahceoea6BawritSYhR3VTSYBh9P0I7ZBKMdoVDOXxmR73qxo1j9dkhbMvCJyi+8t6TsHSqLOi5U46atH3tcs/vnSV6bLhuhBKjZBl993d1keNcQgMZeozQHGW85zR3zVQYPeMIAAD7LJjAz5/bVnXeiFicNOcoBgWNoH8cwKrC95V5WAlEdCaAawGcYozZZsONMY/nnw8T0T8DOBpARdA3CXlVzzvbYfTdDmto4nLgGP2Hzj8MZx++Nw7fr+qtUQfti0ck2C0q+3LsiInehEDZvkNhjK0w1+zTVWlp4BOch+xTZdpRjL67aFZ/49wVbRzOgOiqmgBedWKFouRd4iJGrRLK0OMYfd4uTrjs2ZV9svY5T/43/trL8ZXv/bTrSmzRcztWFzmqnk1BI+i3AFhPRGuRCfiLAby5+AARHQ3g0wA2GmOeKIQvAvC8MWYbES0F8EpkhtqBQtoy2u2bu0L3XKz0kCYuB04VMnd8FGccundA7j1IL+XQYsXCObjqrINw/ob9SuExzKUvFuh63SjUMKzqJuKSKO2C1o9eOYTRcY/2rnmo1pHz7tip8LqpppV9xvSltvklwvTFd73Cm3c3LMCPnjXGCnN/nwWTeOsr13ryz9OMYfRDOL1UK+iNMTNEdDmA2wGMALjJGHM/EV0HYKsxZjOAPwYwBeDvciFj3SgPBfBpItqJzPB7veOts8tht3wznOomQt8+6MlehxgG74KI8FtnrK+Ex53mzONGLA7uZL/khP3x8JPP4Z2nVs8kuHEtut4lERNKq+6Jsc/E7I4sXBWFZaWiEHTQa5fq7x++4HDcfv/PKuExRsfQxUHaAR69epE3Tk9H7xK2+nxcV1U7TkKmZD8LfUicpqDS0RtjbgNwmxP2wcL/ZzLxvgXgyH4K2DROP2Q5Pvjl+/GG6VWl8Ai1NnuaUwKnuukHMQxei6ite8wkYBj91MQoPnbRUaq4FnYix7RxWD1DVRp5HiGMPv90VRR2R+pLi1P3SLbYt5y4Bm85cU0lvB/Vjbb9Y/zoLWGLOTDlmKEKrtWD3bWGeiM1iVaejJWwctFcPHL9uZVwyY+eQ4zObZiregxitu5xO508btSC61f3+ATN/MnRrt+yDyF3go90aOCMjhihbXXGb89d84qwVagK+ggdfdSOLkzQxxgpubtuJHSNsU7DdL/uIkY/SGLGYY8T9BxiBG8/hsqYHYSLDasW4t5Hf9l/QgKiVBSMB03T+XTjMm6Evjbecu2ZFQFYTiss3zjddTgLdLFgzpiXsNhyAVW1xm7rddPHYiL1pYtVi7PL1PZdMFkKt4tFVLtEqO52W9XNngDN7ZVcnChddAOr+ufefkL3noxBIWZh0txGyOcTI+jL33sHg6pp1Z1TCDq12qEoI3VIFXuqG71EI4a5WpXWoHXRofUkIhCFet1kA5IT9IfuW/WUet0xK7BkahynOpcNdlVaIaqbPvzok+pmiOhPT6fPp8lVfc74COt7fes7Tqz4fscgzhjbR5wG9OrLpibw45//n+r++X4wQhS3mwupY5ed66NYV1lXCGquQHDRVd0FxulQ+FUPYTvA7NM1UgPAnVedgmXzq2OfiHDawcsr4RHnyHrec23xutnTEMZ2ss9hMXoJ02uqd4rEIMZLqBPTLn14pLgC5c8vOQZ3/eAJrFw0l4nRDEJZaEwdu4w+QNB/4o0bcNO//BhHr1pYCreqnCDPssjzAqFEphNo7+ganD2/rVs2FZS33S3F7OZb5XUzG3HF6Qdiw+qF9Q/2gbhDM/nnLDHGRt3SGLE4WC+KJibB0qkJvN7xqhoERjphjD7K6yaiOfZbOAcfOO+wSrjm9koXsUbHUIPjSODiYJN37RAxiFHdRB2W3EUkz5v3Ls9xF+HKsw/G6YfoDyCF6EAt+vK6GUJnx4AiFqZ+9PohQujKsw6KEoRNIfO6CXseiF3k+xdol520DoftuxfOP2q/+odzSJenSXFCx/dIh8IOkuVCuQE5323ZGEYfc75gd70CYY9AT083YIPMEPV0ALB68VzVK/Usou70iBBoMSqiK85Yjys8h7x2FToUJtCiroZg3CtjsGrxXNz2npOC4sT5i0eobigsD3t3+1mHxZ0mL8L0odKKOTCXVDe7AUK6oK9LzQLjuPdzxOIb7zst6Pm+DkzNkkkQi3DVTYTQzD+bUFHEIOYcxQiFLWY2/RChOTUxiu++/wwsFl5pqMW6pZlOf+Ph+6jj7Io7gJpEEvQOony/Y7bvARltvfbMyiv+dhViXrARdchqiPpLIPNUCW3jDlHUfeQhdbS3pkrvaB0kYnX0oQt2JuiDomD5XpP1DymweslcPHDdRkyOhex0y5+qOMnrZviwxsAFc/RveYm60yVC3TOsSW4x0gkUaBEnAGPuvW8S3/zd0/DM82FnEjqdOI+rkDq++oh98Ee/ciRee8xg7yvnsMu8biLiNAnpimgfhn2GJhRJ0Oc4eO/5+MC5h3rffsShn6P+s0lFkflFx2xRQ/IYLqNfPn8Sy+eHMcRQT5EoL60O4c3H9/cinn4QswuJ8roJVIMNG7Nt7idBn4OIcNlJ1XtDJPSli54lXjdA/FH/kDjHrF6EMw9djtWLB+v73iRCfb81L6He3fDqI/dFp0NBbz4LNVIDcZ46w0TUocAhkpkk6PtAX760s8ix9ZzD98HL1/ivi/Uhpl1WLZ6LGy99eXDZhokRCjMgxvjRDxsHLp/CgcsPDIoTugACwPrlU1i3bHDvTG0asd5IoXGaQhL0fSBm+7Zi4RwsnZoIuiVx2PjTNx0d9LyVY7NJoMVgV+x0ZiNGKJzI/OWvHzeYwgwIB+09hQOWzcO+e82pfzhHzxFjUKXioeoOItpIRD8kogeJ6GrP7xNE9Pn89+8Q0ZrCb9fk4T8konOaK/rwEeNB87pjVuJbV5/e6sk+iJer7I5YNn8CywLuExrpECbHOpg30W5+NdvUMDE4ZJ+98PWrTsWCuSHOG9l9PrvlNcVENALgUwDOAvAYgC1EtNl5U9TbADxtjDmQiC4G8FEAbySiw5C9evBwAPsB+BoRHWSM2dF0RYaBObneMsQtq9MhjLdYyANFN7J21/MvLjkmaGdGRPjCO18xq+wQMegEXva2p2A00Hut0bwVzxwH4EFjzMMAQEQ3A7gAQFHQXwDg9/P/bwXwZ5QtWxcAuDl/WfiPiejBPL1/a6b4w8WJByzBp99yLA7zXIm6J6NnqBpyQQYM6+MegpiXv882jI12MN72zo/A66dX4YDAC9eagkbQrwDwaOH7YwCO557J3zH7DIAlefi3nbhe/0Ui2gRgEwCsXj08d7IQjHQI5wScpttTsP+SeXjXqQfg5PXL6h9OaB2uOH09nnlhsO9JmI1Yu3Qe1i4djsF5t1EWGmNuAHADAExPTw/nvHdCIxjpEN638ZBhFyNhSDh4n/nDLkKCA83+6nEAxTtfV+Zh3meIaBTAAgC/UMZNSEhISBggNIJ+C4D1RLSWiMaRGVc3O89sBnBp/v9FAO402ZVwmwFcnHvlrAWwHsB3myl6QkJCQoIGtaqbXOd+OYDbAYwAuMkYcz8RXQdgqzFmM4DPAPjr3Nj6FLLFAPlztyAz3M4AeHdbPG4SEhISZgvI987FYWN6etps3bp12MVISEhImDUgoruNMdO+35IPVEJCQkLLkQR9QkJCQsuRBH1CQkJCy5EEfUJCQkLLsVsaY4noSQA/iYy+FMDPGyzObECqc/uxp9UXSHUOxf7GGO9x9N1S0PcDItrKWZ7bilTn9mNPqy+Q6twkkuomISEhoeVIgj4hISGh5WijoL9h2AUYAlKd2489rb5AqnNjaJ2OPiEhISGhjDYy+oSEhISEApKgT0hISGg5WiPo615g3gYQ0U1E9AQRfa8QtpiI7iCiH+Wfi4ZZxqZBRKuI6C4i+j4R3U9E78nDW1tvIpokou8S0X/kdf6DPHwtEX0nH+Ofz68Nbw2IaISI7iGif8y/t7q+AEBEjxDRfUR0LxFtzcMaH9utEPSFF5i/GsBhAN6Uv5i8bfgrABudsKsBfN0Ysx7A1/PvbcIMgKuMMYcBOAHAu/O+bXO9twE43RhzFIANADYS0QkAPgrgT4wxBwJ4GsDbhljGQeA9AB4ofG97fS1OM8ZsKPjPNz62WyHoUXiBuTHmJQD2BeatgjHmG8ju+y/iAgCfzf//LIALd2mhBgxjzE+NMf+e//8sMkGwAi2ut8nwXP51LP8zAE4HcGse3qo6E9FKAOcCuDH/TmhxfWvQ+Nhui6D3vcDc+xLyFmJvY8xP8///B8DewyzMIEFEawAcDeA7aHm9czXGvQCeAHAHgIcA/NIYM5M/0rYx/gkA7wOwM/++BO2ur4UB8FUiupuINuVhjY/t3ebl4An9wxhjiKiV/rJENAXgCwDea4z534zwZWhjvfM3sW0gooUAvgigtW9bJ6LzADxhjLmbiE4ddnl2MV5ljHmciJYDuIOIflD8samx3RZGvye/hPxnRLQvAOSfTwy5PI2DiMaQCfm/Mcb8fR7c+noDgDHmlwDuAnAigIVEZMlZm8b4KwGcT0SPIFO7ng7gk2hvfbswxjyefz6BbEE/DgMY220R9JoXmLcVxRezXwrgy0MsS+PIdbWfAfCAMebjhZ9aW28iWpYzeRDRHABnIbNN3AXgovyx1tTZGHONMWalMWYNsrl7pzHmV9HS+loQ0Twimm//B3A2gO9hAGO7NSdjieg1yPR89gXmHxlykRoHEX0OwKnIrjL9GYAPAfgSgFsArEZ2tfMbjDGuwXbWgoheBeCbAO5DT3/7fmR6+lbWm4hehswIN4KMjN1ijLmOiNYhY7yLAdwD4BJjzLbhlbR55Kqb3zHGnNf2+ub1+2L+dRTA3xpjPkJES9Dw2G6NoE9ISEhI8KMtqpuEhISEBAZJ0CckJCS0HEnQJyQkJLQcSdAnJCQktBxJ0CckJCS0HEnQJyQkJLQcSdAnJCQktBz/D7qopzMzSn8IAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "average: 1.27\n"
     ]
    }
   ],
   "source": [
    "plot_effective_amplitude(1, np.linspace(0, 50, 500))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-06-02T16:25:26.197209Z",
     "start_time": "2019-06-02T16:25:24.277605Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "average: 1.27\n"
     ]
    }
   ],
   "source": [
    "plot_effective_amplitude(2, np.linspace(0, 50, 500))"
   ]
  }
 ],
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