Reproduce-PRA-30-1386-1984-Collett.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Reproduce: Squeezing of intracavity and traveling-wave light fields produced in parametric amplification"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "M. J. Collett and C. W. Gardiner, <i>Squeezing of intracavity and traveling-wave light fields produced in parametric amplification</i>, <a href=\"http://journals.aps.org/pra/abstract/10.1103/PhysRevA.30.1386\">Phys. Rev. A <b>30</b>, 1386 (1984)</a>."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Author : <b>Eunjong Kim</b> (<a href=\"mailto:ekim7206@gmail.com\">ekim7206@gmail.com</a>)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Status : Work in progress"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook, I will derive the equations governing the input-output formulation of optical cavity, which is a grueling task if done with bruteforce hand calculation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup Modules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympsi import *\n",
    "from sympsi.boson import BosonOp, MultiBosonOp, BosonVacuumKet\n",
    "from sympsi.operator import OperatorFunction\n",
    "from sympsi.expectation import Expectation, Covariance\n",
    "from sympsi.operatorordering import normal_ordered_form"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Quantum Stochastic Differential Equation (QSDE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = symbols('x', real=True)\n",
    "t, t0, t1, t_ = symbols(\"t, t_0, t_1, {t'}\", real=True)\n",
    "w, w0, w1, w_ = symbols(\"\\omega, \\omega_0, \\omega_1, {\\omega'}\", real=True)\n",
    "k_w = Function('k')(w)\n",
    "g = Function('g')\n",
    "hbar = Symbol(\"hbar\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, define the time-dependent cavity field operators $a(t)$, $a^\\dagger(t)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACAAAAAUBAMAAAAevyJ8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA50lEQVQYGUWQLW/DMBCGnySr86VKph2oImVg\naIqGCgrCRocq9R9MGshIUUHLSgamoY1lbLDT+NR/0PJOqioVDQUVFO1sJc0rnf36sX2+M9TqmTnU\nzRKntHZ/BoHsPWoezuBV3A68vCFfYq5B3dfAWYr5lliAk16NuEjw/04HuIWJDiqiF+hWcuINNcYt\niTfgzgVscY/EOZ0niDMBazolxcqCQqCAIjPvmyszFEwpNnx2tUk64NckDfLwx8NL4EblkKIOw/4d\n/hIu3yXHs4SVKV1kK7bONCdqm4u0BW379Qd9WGyH5gv/AUBSKfyShqiiAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$${{{a}}(t)}$$"
      ],
      "text/plain": [
       "OperatorFunction(a,t)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = BosonOp(\"a\")\n",
    "a_t = OperatorFunction(a, t)\n",
    "a_t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the time-dependent field operators of the external multimode field, $b_{\\omega}(t)$, $b_{\\omega}^\\dagger(t)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACoAAAAUBAMAAAAJnbK1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAELvv3c2rVESJdpki\nZjI6QXVuAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABHElEQVQYGU2RP0vEQBDFfxsvl+S8Mym0PuEK\nGwu/gJBOBAurw0qCYCMWV1hrCsEyV2hlk9oqICoq4mIvpLESwW/gH7B3ZnPKPZg3bx7LDDML03jW\nomfBDFKVDt1Fl/aF/cxJJd/CvGVBZDxSw+FV+AiiEoqksYQ3JK6k6Qqc/pvdVOS6xBYML24mfqci\n3Pneg3PMT9K3je3VEHyKfiH4Ih43bmsEs5uiDzT1J+5MDi3VH/gVhbxRqNuXEDeueUxOZM8M7VBg\n4Jg4N6tkMJei08441Gnt2qu6tampiCq4NiXcYu4eCGyvlCpM4elN+m5LQOe9nUe22VhKtyOES7tr\n91LqdQR6HcVgfJlJ8qwQ7pJONDS5+vKUpfLvh34BI6U82yvpGeoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$${{{b_{\\omega}}}(t)}$$"
      ],
      "text/plain": [
       "OperatorFunction(b,t)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b_w = MultiBosonOp(\"b\", w, 'continuous')\n",
    "b_w_t = OperatorFunction(b_w, t)\n",
    "b_w_t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The interaction Hamiltonian between the cavity field, represented by the harmonic oscillator annihilation and creation operators $a, a^\\dagger$, and the external field in the rotating wave approximation is given by"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$${H_{{int}} (t)} = i \\hbar \\int_{-\\infty}^{\\infty} g{\\left ({\\omega'} \\right )} \\left({{{a}}(t)} {{{{b_{{\\omega'}}}^\\dagger}}(t)} - {{{b_{{\\omega'}}}}(t)} {{{{a}^\\dagger}}(t)}\\right)\\, d{\\omega'}$$"
      ],
      "text/plain": [
       "                     ∞                                                        \n",
       "                     ⌠                                                        \n",
       "                     ⎮               ⎛                                      ⎛ \n",
       "{H_{int}} (t) = ⅈ⋅h̅⋅⎮  g({\\omega'})⋅⎝OperatorFunction(a,t)⋅OperatorFunction⎝b\n",
       "                     ⌡                                                        \n",
       "                     -∞                                                       \n",
       "\n",
       "                                                                 \n",
       "                                                                 \n",
       "†  ⎞                                         ⎛ †  ⎞⎞             \n",
       " ,t⎠ - OperatorFunction(b,t)⋅OperatorFunction⎝a ,t⎠⎠ d({\\omega'})\n",
       "                                                                 \n",
       "                                                                 "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hint = I*hbar * Integral((g(w) * (a_t * Dagger(b_w_t) - b_w_t * Dagger(a_t))).subs(w, w_), (w_,-oo,oo))\n",
    "Hintsym = Symbol(\"{H_{int}} (t)\")\n",
    "Eq(Hintsym, Hint)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Heisenberg Equation of Motion for the field operator $b_t(\\omega)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Heisenberg equation of motion of the operator $b_\\omega(t) = e^{-i\\omega t} b_\\omega$ can be expressed as\n",
    "\n",
    "\n",
    "$$\n",
    "\\frac{d}{dt}b_\\omega(t) = \\frac{\\partial}{\\partial t} b_\\omega(t) - \\frac{i}{\\hbar}[b_\\omega(t), H_{int}(t)]\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$g{\\left (\\omega \\right )} {{{a}}(t)} + \\frac{\\partial}{\\partial t} {{{b_{\\omega}}}(t)} + {{{{a}^\\dagger}}(t)} {{{b_{\\omega}}}(t)} \\int_{-\\infty}^{\\infty} g{\\left ({\\omega'} \\right )} {{{b_{{\\omega'}}}}(t)}\\, d{\\omega'} - {{{a}}(t)} {{{b_{\\omega}}}(t)} \\int_{-\\infty}^{\\infty} g{\\left ({\\omega'} \\right )} {{{{b_{{\\omega'}}}^\\dagger}}(t)}\\, d{\\omega'} - {{{b_{\\omega}}}(t)} {{{{a}^\\dagger}}(t)} \\int_{-\\infty}^{\\infty} g{\\left ({\\omega'} \\right )} {{{b_{{\\omega'}}}}(t)}\\, d{\\omega'} + {{{b_{\\omega}}}(t)} {{{a}}(t)} \\int_{-\\infty}^{\\infty} g{\\left ({\\omega'} \\right )} {{{{b_{{\\omega'}}}^\\dagger}}(t)}\\, d{\\omega'}$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                                                                              \n",
       "                                  ∂                                           \n",
       "g(\\omega)⋅OperatorFunction(a,t) + ──(OperatorFunction(b,t)) + OperatorFunction\n",
       "                                  ∂t                                          \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                             ∞                                                \n",
       "⎛ †  ⎞                       ⌠                                                \n",
       "⎝a ,t⎠⋅OperatorFunction(b,t)⋅⎮  g({\\omega'})⋅OperatorFunction(b,t) d({\\omega'}\n",
       "                             ⌡                                                \n",
       "                             -∞                                               \n",
       "\n",
       "                                                ∞                             \n",
       "                                                ⌠                             \n",
       "                                                ⎮                             \n",
       ") - OperatorFunction(a,t)⋅OperatorFunction(b,t)⋅⎮  g({\\omega'})⋅OperatorFuncti\n",
       "                                                ⌡                             \n",
       "                                                -∞                            \n",
       "\n",
       "                                                                              \n",
       "                                                                     ∞        \n",
       "  ⎛ †  ⎞                                                      ⎛ †  ⎞ ⌠        \n",
       "on⎝b ,t⎠ d({\\omega'}) - OperatorFunction(b,t)⋅OperatorFunction⎝a ,t⎠⋅⎮  g({\\om\n",
       "                                                                     ⌡        \n",
       "                                                                     -∞       \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "ega'})⋅OperatorFunction(b,t) d({\\omega'}) + OperatorFunction(b,t)⋅OperatorFunc\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "          ∞                                                  \n",
       "          ⌠                                                  \n",
       "          ⎮                               ⎛ †  ⎞             \n",
       "tion(a,t)⋅⎮  g({\\omega'})⋅OperatorFunction⎝b ,t⎠ d({\\omega'})\n",
       "          ⌡                                                  \n",
       "          -∞                                                 "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b_w_t_diff = b_w_t.diff(t) - I/hbar * Commutator(b_w_t, Hint.replace(w, w_))\n",
    "b_w_t_diff1 = b_w_t_diff.expand(commutator=True).expand(commutator=True).doit()\n",
    "b_w_t_diff2 = pull_outwards(b_w_t_diff1).doit()\n",
    "b_w_t_diff2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$${\\dot{b}_{\\omega}(t)} = - i \\omega {{{b_{\\omega}}}(t)} - \\int_{-\\infty}^{\\infty} g{\\left ({\\omega'} \\right )} \\left(- \\delta\\left(- \\omega + {\\omega'}\\right) {{{a}}(t)} + \\left({{{a}}(t)} {{{b_{\\omega}}}(t)} - {{{b_{\\omega}}}(t)} {{{a}}(t)}\\right) {{{{b_{{\\omega'}}}^\\dagger}}(t)} - {{{b_{{\\omega'}}}}(t)} \\left({{{{a}^\\dagger}}(t)} {{{b_{\\omega}}}(t)} - {{{b_{\\omega}}}(t)} {{{{a}^\\dagger}}(t)}\\right)\\right)\\, d{\\omega'}$$"
      ],
      "text/plain": [
       "                                                        ∞                     \n",
       "                                                        ⌠                     \n",
       "                                                        ⎮               ⎛     \n",
       "{\\dot{b}_\\omega}(t) = -ⅈ⋅\\omega⋅OperatorFunction(b,t) - ⎮  g({\\omega'})⋅⎝-Dira\n",
       "                                                        ⌡                     \n",
       "                                                        -∞                    \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "cDelta(-\\omega + {\\omega'})⋅OperatorFunction(a,t) + (OperatorFunction(a,t)⋅Ope\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "ratorFunction(b,t) - OperatorFunction(b,t)⋅OperatorFunction(a,t))⋅OperatorFunc\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "    ⎛ †  ⎞                         ⎛                ⎛ †  ⎞                    \n",
       "tion⎝b ,t⎠ - OperatorFunction(b,t)⋅⎝OperatorFunction⎝a ,t⎠⋅OperatorFunction(b,\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                \n",
       "                                                                \n",
       "                                           ⎛ †  ⎞⎞⎞             \n",
       "t) - OperatorFunction(b,t)⋅OperatorFunction⎝a ,t⎠⎠⎠ d({\\omega'})\n",
       "                                                                \n",
       "                                                                "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b_w_t_diff3 = b_w_t_diff1.subs(b_w_t, exp(-I*w*t)*b_w).doit().subs(exp(-I*w*t)*b_w, b_w_t)\n",
    "Eq(Symbol(\"{\\dot{b}_\\omega}(t)\"), b_w_t_diff3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is equivalent to\n",
    "$$\n",
    "\\frac{d}{dt}\\left( e^{i\\omega t} b_\\omega (t)\\right) = e^{i\\omega t} \\left[ \\dot{b}_\\omega(t) + i\\omega b_\\omega (t)\\right] = g(\\omega) a(t) e^{i\\omega t}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the solution to this differential equation can be written in two ways:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### a. in terms of the initial conditions at time $t_0 <t$ (the <i>input</i>)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "e^{i\\omega t}b_\\omega(t) - e^{i\\omega t_0} b_\\omega(t_0) = \\int_{t_0}^{t\\ \\ \\ } g(\\omega) a(t') e^{i\\omega t'} dt' \\\\\n",
    "\\therefore b_\\omega(t) = e^{-i\\omega(t-t_0)} b_\\omega(t_0) + g(\\omega) \\int_{t_0}^{t} e^{-i\\omega (t-t')} a(t') dt'\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYQAAABLBAMAAACCWu01AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVJl2u4kiEO8yZt2r\nRM0tcn99AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAHX0lEQVRoBe1ZX4hUVRj/zc6f68zs7AwiQil5\ny5S0sMlNUJGcLMoicFhLJSEvoiT24E0trKTRUh+kP/NQvvQwgz4EFjUSGfWQA5rUQzlSQaDhWhCI\nta2luyVqfd8595w7d/7uprvuDQ/MPef8vu98f879znfOnAu0KmdaEX1BMwZ9YWYrIwOlVlQ/0CJ3\n/pT3g52tbOxKtaL6gpa0fGFmKyN/aEX0B22BYfnD0OZWTg03p/mEcnKeTwy9YeaNGbgxA21noOMJ\nuy3PGGdYsKU8xi1sZ55xOZ5uxzPG6cH+MW5ge/PimfY8Y5xjg9/DCFhYaT7H62tJMasWGQv9JXZT\nKyIZL+l74KgXGRu9v+rNmO1AXZaXtgh42Yv8516kbmS0DhkqEL1Uz7nHgfZSra9ouvJ81xG+upUT\nNB3RWad2q0CLgHa5GrWCfzZCJXYQcK9ocnaM9BrbmrMPgRK1JFMn1VWTI8DpkjT8Z+hy0zGREuBe\n0UxCyCbW35qyD4fwtndyxNCtwxFQzZusvwn7tozjQKxMb929oln9+YwAj+upHkztxN81QAOklgP4\nyjs5gqF25dWPIiRSdOCY5TSAAk11TVk+iDIQLYWKgHtFc1pyfaCZpbSOQVRJYyIhzUpiARIp4H2U\nvJMj+INF7zA3o1cpWKN5jurWh2XddBqRZ0uRolGEGSdt7hXNgCRP0uxS2jjTzbRvvcJERpqUcM7u\nJNI7HKLVk2M8uZxeSxnGYndCqzO6a+4vWrKbG4+lNYjAg1QeRigdsGJpIz2uAtAVzQRG7cQ2yVfQ\n7CTNOIiCVZVpzylEM3kbT0/CeryIjkTGOzkhezOFJIFdZT2A4+ol+nHR5sZM0eeHmxsPkZ01JfdC\nMN9ZCVvsgr6iCZafF2zaBZYW6MfjhCppiUGFMGuganKcecADmIh92JQoeycniXE2or3kV4pHirKX\nnlOctlKAuOUgVbnR+MfWoGrcEk7c/82B28CBpK9oOtJZQT+huFhaRy8+pb7KtOGSQpjJ4wIDxDiY\nn4evsSya8U7OOsTzYL9OudZQRscs+nFRCvCM7Iunyo2BBpvz1mVYnL2jDF7O+oomcTgvBurlzNLi\n2Ugvo460LlMjBDZwIdCb/W7y2fGmUfJOjnCBt52VLE0UXi742OkIBeO7J4DPDRn6rau4uTF83uEy\nls62nKaqwqZqVdXTqL26Z67N0k4ueSxtMU1k2tVLCymNEKhdeK57OjNRMb6wQjs7XgM+8U6OCKQO\nMuvX7lslJ4ImEkfOH3Z6pCCcifyBRynZExtyaUBNZtcFh+mEFck6TVUlSqpVVZ+hdbcLv4Ol6Vhl\naYTSiVFFLwHKhcRBrK0SIJr3eAGxnLvIxSt2zpIkDgFaapgzh/ukYIqNczhAwcv9cApQuTF5jhHa\nMrZjY1423SeHY03h97uliNvB0nSssjRCZ7kIAcqFdSljRo0UhLyAMXMZcDMNGUQyK0m8EGmpxbKU\na4W5OxAdwOvAjwDBFu1okhEbBmSjc3f3uw7kVpwUagrnhrttfASWpmOVpRFKoauiN9zXd/aNvr4y\nEb7cN9+mylMCRU+XO4YpbM45LnA6jGcRr4B3ygKiFxEssdI0oibCAqMHlSn9sk6WZe15hixPlzuc\noXfAoFMFSTPU+YpdIPSyixCg3kKDozAot9aWIPlJ+eBUShLYhVwFuTy2E1BAbAAhk1+9iWBKRKbK\njat65Qj1/mTPeUYyni53HiLDL4ggJWmJfpBeKiRNohohULmgVpvgbP1IFullHufzmcjop2DQzvmZ\nUEDrgjYNWoCPoGCCtlW9nJeYUui4LL+i2rK+FuAYNM4h1DuRpYV7A3nBQatNohohWLmwEwjWymnS\nT1aM3Sjz+Uxk9KdwU0EELZlrDOCYxalvRc/2SyuKJGCaI+TNlGwEe7HRcrB21c94r5xlaR3lTZKX\npQlUIwQoF+6DMb2dSIfeWQyZ8nyGsAncZaTpLbxKRFJwcs5K8bo3X8nPZUj933A359nzi4wPpWyd\ns+dwhYMn8mhR8p+hSqAaIUC5EJy6X3K1fxpTZ0Kez8AZfc1+Xgu84FgBFtECt7ghitj7qJVosDk7\nLO2qBtK8Q5QLXrRtT57PAJnRZUYic6MiVcRMPV6dm4IN04Vma9VoIM3Lblje/hB78nwGyIweNHlf\nIHNX2SFqcDpxijq9dl5UyPDremnDl9FohDifAU5G7+mxRSqfvE8cPdboEUp9vO5vo2Zp26iX1nbI\ncBgiGZdbmdvoj2ey3+UbbitSdEZwph2B4mb0lgoKmRHQPboiF2ZHV98IaDtWGQGhoyvy0OiqGwlt\nV0ZC6KjKjA7jFDmqhg1dWewqtoWhaxlRzs5tIyp+NIQ3/KMzGoqvnY5T+Wsn67pIWokj10XvNVS6\nOLrrGkq7LqLW3uv3OLou03ZD6f9wBrr8vxRytu/fyyTfe0Bfkn3vw2mIj6V+9oOuu/ljqY9LYhvE\nNy4fu0BfksWXRh+7QF+S/e4CfUn2eyBRAPl9OZML4mOpL9bCv83ypT62CYvcAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$g{\\left (\\omega \\right )} \\int_{t_{0}}^{t} e^{- i \\omega \\left(t - {t'}\\right)} {{{a}}({t'})}\\, d{t'} + e^{- i \\omega \\left(t - t_{0}\\right)} {{{b_{\\omega}}}(t_{0})}$$"
      ],
      "text/plain": [
       "          t                                                                   \n",
       "          ⌠                                                                   \n",
       "          ⎮   -ⅈ⋅\\omega⋅(t - {t'})                                     -ⅈ⋅\\ome\n",
       "g(\\omega)⋅⎮  ℯ                    ⋅OperatorFunction(a,{t'}) d({t'}) + ℯ       \n",
       "          ⌡                                                                   \n",
       "          t₀                                                                  \n",
       "\n",
       "                                  \n",
       "                                  \n",
       "ga⋅(t - t₀)                       \n",
       "           ⋅OperatorFunction(b,t₀)\n",
       "                                  \n",
       "                                  "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b_w_t_input = exp(-I*w*(t-t0)) * b_w_t(t0) + g(w) * Integral(exp(-I*w*(t-t_)) * a_t(t_), (t_, t0, t))\n",
    "b_w_t_input"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### b. in terms of the final conditions at time $t_1>t$ (the <i>output</i>)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "$$\n",
    "\\displaystyle e^{i\\omega t_1}b_\\omega(t_1) - e^{i\\omega t} b_\\omega(t) = \\int_{t}^{t_1\\ \\ } g(\\omega) a(t') e^{i\\omega t'} dt'\\\\\n",
    "\\therefore b_\\omega(t) = e^{-i\\omega(t-t_1)} b_\\omega(t_1) - g(\\omega) \\int_{t}^{t_1} e^{-i\\omega (t-t')} a(t') dt'\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZQAAABJBAMAAADoPM3WAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nk0Dw+F78F18HAcn/4usgFPlfvAdC+Q8fgaHU+KLEqAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- g{\\left (\\omega \\right )} \\int_{t}^{t_{1}} e^{- i \\omega \\left(t - {t'}\\right)} {{{a}}({t'})}\\, d{t'} + e^{- i \\omega \\left(t - t_{1}\\right)} {{{b_{\\omega}}}(t_{1})}$$"
      ],
      "text/plain": [
       "            t₁                                                                \n",
       "            ⌠                                                                 \n",
       "            ⎮   -ⅈ⋅\\omega⋅(t - {t'})                                     -ⅈ⋅\\o\n",
       "- g(\\omega)⋅⎮  ℯ                    ⋅OperatorFunction(a,{t'}) d({t'}) + ℯ     \n",
       "            ⌡                                                                 \n",
       "            t                                                                 \n",
       "\n",
       "                                    \n",
       "                                    \n",
       "mega⋅(t - t₁)                       \n",
       "             ⋅OperatorFunction(b,t₁)\n",
       "                                    \n",
       "                                    "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b_w_t_output = exp(-I*w*(t-t1)) * b_w_t(t1) - g(w) * Integral(exp(-I*w*(t-t_))*a_t(t_), (t_, t, t1))\n",
    "b_w_t_output"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Heisenberg equation of motion of the cavity field operator $a(t)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Heisenberg equation of motion of the cavity field operator can be expressed as\n",
    "\n",
    "\n",
    "$$\n",
    "\\dot{a} =  - \\frac{i}{\\hbar}[a, H_{sys}] - \\frac{i}{\\hbar}[a, H_{int}] \n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWMAAABGBAMAAADr+ujjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$i \\hbar \\int_{-\\infty}^{\\infty} g{\\left (\\omega \\right )} \\left({{{a}}(t)} {{{{b_{\\omega}}^\\dagger}}(t)} - {{{b_{\\omega}}}(t)} {{{{a}^\\dagger}}(t)}\\right)\\, d\\omega$$"
      ],
      "text/plain": [
       "     ∞                                                                        \n",
       "     ⌠                                                                        \n",
       "     ⎮            ⎛                                      ⎛ †  ⎞               \n",
       "ⅈ⋅h̅⋅⎮  g(\\omega)⋅⎝OperatorFunction(a,t)⋅OperatorFunction⎝b ,t⎠ - OperatorFunc\n",
       "     ⌡                                                                        \n",
       "     -∞                                                                       \n",
       "\n",
       "                                           \n",
       "                                           \n",
       "                          ⎛ †  ⎞⎞          \n",
       "tion(b,t)⋅OperatorFunction⎝a ,t⎠⎠ d(\\omega)\n",
       "                                           \n",
       "                                           "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hint = Hint.replace(w_, w)\n",
    "Hint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAG0AAAAtBAMAAABWjvgJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nff217p1/BCyc6z29x8zfvfHWb3+5VwkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{i \\left[H_{sys},{{{a}}(t)}\\right]}{\\hbar}$$"
      ],
      "text/plain": [
       "ⅈ⋅[H_sys,OperatorFunction(a,t)]\n",
       "───────────────────────────────\n",
       "               h̅              "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hsyssym = Operator('H_sys') # Hamiltonian of the system\n",
    "term1 = -I/hbar * Commutator(a_t, Hsyssym)\n",
    "term1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$- \\int_{-\\infty}^{\\infty} g{\\left (\\omega \\right )} \\left(\\left({{{a}}(t)} {{{b_{\\omega}}}(t)} - {{{b_{\\omega}}}(t)} {{{a}}(t)}\\right) {{{{a}^\\dagger}}(t)} - {{{a}}(t)} \\left({{{a}}(t)} {{{{b_{\\omega}}^\\dagger}}(t)} - {{{{b_{\\omega}}^\\dagger}}(t)} {{{a}}(t)}\\right) + {{{b_{\\omega}}}(t)}\\right)\\, d\\omega$$"
      ],
      "text/plain": [
       " ∞                                                                            \n",
       " ⌠                                                                            \n",
       " ⎮            ⎛                                                               \n",
       "-⎮  g(\\omega)⋅⎝(OperatorFunction(a,t)⋅OperatorFunction(b,t) - OperatorFunction\n",
       " ⌡                                                                            \n",
       " -∞                                                                           \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                             ⎛ †  ⎞                         ⎛ \n",
       "(b,t)⋅OperatorFunction(a,t))⋅OperatorFunction⎝a ,t⎠ - OperatorFunction(a,t)⋅⎝O\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                     ⎛ †  ⎞                   ⎛ †  ⎞          \n",
       "peratorFunction(a,t)⋅OperatorFunction⎝b ,t⎠ - OperatorFunction⎝b ,t⎠⋅OperatorF\n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                \n",
       "                                                \n",
       "            ⎞                        ⎞          \n",
       "unction(a,t)⎠ + OperatorFunction(b,t)⎠ d(\\omega)\n",
       "                                                \n",
       "                                                "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "term2_0 = -I/hbar * Commutator(a_t, Hint)\n",
    "term2_1 = term2_0.expand(commutator=True).expand(commutator=True).doit(independent=True)\n",
    "term2_1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "substituting the solution in terms of the input field,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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2GQyssKstyo43knXPTZnBl7qxbDAcM6eBOKHevbvOAFcZphbh7VUbL8tAkrvhvz5nHFuE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\n",
      "text/latex": [
       "$$- \\int_{-\\infty}^{\\infty} g^{2}{\\left (\\omega \\right )} \\int_{t_{0}}^{t} e^{- i \\omega t} e^{i \\omega {t'}} {{{a}}({t'})}\\, d{t'} - g^{2}{\\left (\\omega \\right )} \\left(\\int_{t_{0}}^{t} e^{- i \\omega t} e^{i \\omega {t'}} {{{a}}({t'})}\\, d{t'}\\right) {{{a}}(t)} {{{{a}^\\dagger}}(t)} + g^{2}{\\left (\\omega \\right )} {{{a}}(t)} \\left(\\int_{t_{0}}^{t} e^{- i \\omega t} e^{i \\omega {t'}} {{{a}}({t'})}\\, d{t'}\\right) {{{{a}^\\dagger}}(t)} + g{\\left (\\omega \\right )} {{{a}}(t)} {{{{b_{\\omega}}^\\dagger}}(t)} {{{a}}(t)} - g{\\left (\\omega \\right )} \\left({{{a}}(t)}\\right)^{2} {{{{b_{\\omega}}^\\dagger}}(t)} + g{\\left (\\omega \\right )} e^{- i \\omega t} e^{i \\omega t_{0}} {{{a}}(t)} {{{b_{\\omega}}}(t_{0})} {{{{a}^\\dagger}}(t)} + g{\\left (\\omega \\right )} e^{- i \\omega t} e^{i \\omega t_{0}} {{{b_{\\omega}}}(t_{0})} - g{\\left (\\omega \\right )} e^{- i \\omega t} e^{i \\omega t_{0}} {{{b_{\\omega}}}(t_{0})} {{{a}}(t)} {{{{a}^\\dagger}}(t)}\\, d\\omega$$"
      ],
      "text/plain": [
       " ∞                                                                            \n",
       " ⌠                                                                            \n",
       " ⎮  ⎛           t                                                             \n",
       " ⎮  ⎜           ⌠                                                             \n",
       " ⎮  ⎜ 2         ⎮   -ⅈ⋅\\omega⋅t  ⅈ⋅\\omega⋅{t'}                                \n",
       "-⎮  ⎜g (\\omega)⋅⎮  ℯ           ⋅ℯ             ⋅OperatorFunction(a,{t'}) d({t'}\n",
       " ⎮  ⎜           ⌡                                                             \n",
       " ⎮  ⎝           t₀                                                            \n",
       " ⌡                                                                            \n",
       " -∞                                                                           \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "               ⎛t                                                             \n",
       "               ⎜⌠                                                             \n",
       "     2         ⎜⎮   -ⅈ⋅\\omega⋅t  ⅈ⋅\\omega⋅{t'}                                \n",
       ") - g (\\omega)⋅⎜⎮  ℯ           ⋅ℯ             ⋅OperatorFunction(a,{t'}) d({t'}\n",
       "               ⎜⌡                                                             \n",
       "               ⎝t₀                                                            \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       " ⎞                                                                            \n",
       " ⎟                                                                            \n",
       " ⎟                                       ⎛ †  ⎞    2                          \n",
       ")⎟⋅OperatorFunction(a,t)⋅OperatorFunction⎝a ,t⎠ + g (\\omega)⋅OperatorFunction(\n",
       " ⎟                                                                            \n",
       " ⎠                                                                            \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "     ⎛t                                                              ⎞        \n",
       "     ⎜⌠                                                              ⎟        \n",
       "     ⎜⎮   -ⅈ⋅\\omega⋅t  ⅈ⋅\\omega⋅{t'}                                 ⎟        \n",
       "a,t)⋅⎜⎮  ℯ           ⋅ℯ             ⋅OperatorFunction(a,{t'}) d({t'})⎟⋅Operato\n",
       "     ⎜⌡                                                              ⎟        \n",
       "     ⎝t₀                                                             ⎠        \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "         ⎛ †  ⎞                                                   ⎛ †  ⎞      \n",
       "rFunction⎝a ,t⎠ + g(\\omega)⋅OperatorFunction(a,t)⋅OperatorFunction⎝b ,t⎠⋅Opera\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                  2                 ⎛ †  ⎞    \n",
       "torFunction(a,t) - g(\\omega)⋅OperatorFunction(a,t) ⋅OperatorFunction⎝b ,t⎠ + g\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "          -ⅈ⋅\\omega⋅t  ⅈ⋅\\omega⋅t₀                                            \n",
       "(\\omega)⋅ℯ           ⋅ℯ           ⋅OperatorFunction(a,t)⋅OperatorFunction(b,t₀\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                  ⎛ †  ⎞              -ⅈ⋅\\omega⋅t  ⅈ⋅\\omega⋅t₀                \n",
       ")⋅OperatorFunction⎝a ,t⎠ + g(\\omega)⋅ℯ           ⋅ℯ           ⋅OperatorFunctio\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                     -ⅈ⋅\\omega⋅t  ⅈ⋅\\omega⋅t₀                                 \n",
       "n(b,t₀) - g(\\omega)⋅ℯ           ⋅ℯ           ⋅OperatorFunction(b,t₀)⋅OperatorF\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                              \n",
       "                                              \n",
       "                                   ⎞          \n",
       "                                   ⎟          \n",
       "                             ⎛ †  ⎞⎟          \n",
       "unction(a,t)⋅OperatorFunction⎝a ,t⎠⎟ d(\\omega)\n",
       "                                   ⎟          \n",
       "                                   ⎠          \n",
       "                                              \n",
       "                                              "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "term2_2 = term2_1.replace(b_w_t, b_w_t_input).expand()\n",
    "term2_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAkQAAABKBAMAAACoW1YwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$- \\int_{-\\infty}^{\\infty} g^{2}{\\left (\\omega \\right )} e^{- i \\omega t} \\int_{t_{0}}^{t} e^{i \\omega {t'}} {{{a}}({t'})}\\, d{t'}\\, d\\omega - \\int_{-\\infty}^{\\infty} g{\\left (\\omega \\right )} e^{- i \\omega t} e^{i \\omega t_{0}} {{{b_{\\omega}}}(t_{0})}\\, d\\omega$$"
      ],
      "text/plain": [
       "  ∞                                                                           \n",
       "  ⌠                                                                           \n",
       "  ⎮                          t                                                \n",
       "  ⎮                          ⌠                                                \n",
       "  ⎮   2          -ⅈ⋅\\omega⋅t ⎮   ⅈ⋅\\omega⋅{t'}                                \n",
       "- ⎮  g (\\omega)⋅ℯ           ⋅⎮  ℯ             ⋅OperatorFunction(a,{t'}) d({t'}\n",
       "  ⎮                          ⌡                                                \n",
       "  ⎮                          t₀                                               \n",
       "  ⌡                                                                           \n",
       "  -∞                                                                          \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "              ∞                                                               \n",
       "              ⌠                                                               \n",
       "              ⎮             -ⅈ⋅\\omega⋅t  ⅈ⋅\\omega⋅t₀                          \n",
       ") d(\\omega) - ⎮  g(\\omega)⋅ℯ           ⋅ℯ           ⋅OperatorFunction(b,t₀) d(\n",
       "              ⌡                                                               \n",
       "              -∞                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "       \n",
       "       \n",
       "       \n",
       "       \n",
       "       \n",
       "\\omega)\n",
       "       \n",
       "       \n",
       "       \n",
       "       "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "term2_3 = pull_outwards(term2_2)\n",
    "term2_3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Markov quantum stochastic process"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose that $g(\\omega)$ is independent of the frequency over a wide range of frequencies around $\\omega=0$ and has the value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGcAAAAlBAMAAACt/+qtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnszv6+AQym6GZJkyRjvUoSNXpaL/uDl85tFvIbwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$g^{2}{\\left (\\omega \\right )} = \\frac{\\gamma}{2 \\pi}$$"
      ],
      "text/plain": [
       " 2           \\gamma\n",
       "g (\\omega) = ──────\n",
       "              2⋅π  "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gamma = Symbol(\"\\gamma\", real=True); Eq(g(w)**2, gamma/(2*pi))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$- \\frac{\\sqrt{2} \\sqrt{\\gamma}}{2 \\sqrt{\\pi}} \\int_{-\\infty}^{\\infty} e^{- i \\omega t} e^{i \\omega t_{0}} {{{b_{\\omega}}}(t_{0})}\\, d\\omega - \\frac{\\gamma}{2 \\pi} \\int_{-\\infty}^{\\infty} e^{- i \\omega t} \\int_{t_{0}}^{t} e^{i \\omega {t'}} {{{a}}({t'})}\\, d{t'}\\, d\\omega$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                   ∞                                                          \n",
       "                   ⌠                                                          \n",
       "    ___   ________ ⎮   -ⅈ⋅\\omega⋅t  ⅈ⋅\\omega⋅t₀                               \n",
       "  ╲╱ 2 ⋅╲╱ \\gamma ⋅⎮  ℯ           ⋅ℯ           ⋅OperatorFunction(b,t₀) d(\\omeg\n",
       "                   ⌡                                                          \n",
       "                   -∞                                                         \n",
       "- ────────────────────────────────────────────────────────────────────────────\n",
       "                                         ___                                  \n",
       "                                     2⋅╲╱ π                                   \n",
       "\n",
       "            ∞                                                                 \n",
       "            ⌠                                                                 \n",
       "            ⎮               t                                                 \n",
       "            ⎮               ⌠                                                 \n",
       "            ⎮   -ⅈ⋅\\omega⋅t ⎮   ⅈ⋅\\omega⋅{t'}                                 \n",
       "     \\gamma⋅⎮  ℯ           ⋅⎮  ℯ             ⋅OperatorFunction(a,{t'}) d({t'})\n",
       "            ⎮               ⌡                                                 \n",
       "a)          ⎮               t₀                                                \n",
       "            ⌡                                                                 \n",
       "            -∞                                                                \n",
       "── - ─────────────────────────────────────────────────────────────────────────\n",
       "                                             2⋅π                              \n",
       "                                                                              \n",
       "\n",
       "          \n",
       "          \n",
       "          \n",
       "          \n",
       "          \n",
       " d(\\omega)\n",
       "          \n",
       "          \n",
       "          \n",
       "          \n",
       "──────────\n",
       "          \n",
       "          "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "term2_4 = pull_outwards(term2_3.replace(g(w), sqrt(gamma/(2*pi))))\n",
    "term2_4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Defining an <i>input field</i> as "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "a_{in}(t) = -\\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} d\\omega\\ e^{-i\\omega(t-t_0)} b_\\omega(t_0)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQoAAABGBAMAAADGCMdZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nj+My//EhwU8Uv0d94o8P8pH8P/748C9CqG4dYhZiJwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\frac{\\sqrt{2}}{2 \\sqrt{\\pi}} \\int_{-\\infty}^{\\infty} e^{- i \\omega \\left(t - t_{0}\\right)} {{{b_{\\omega}}}(t_{0})}\\, d\\omega$$"
      ],
      "text/plain": [
       "       ∞                                                       \n",
       "       ⌠                                                       \n",
       "   ___ ⎮   -ⅈ⋅\\omega⋅(t - t₀)                                  \n",
       "-╲╱ 2 ⋅⎮  ℯ                  ⋅OperatorFunction(b,t₀) d(\\omega) \n",
       "       ⌡                                                       \n",
       "       -∞                                                      \n",
       "───────────────────────────────────────────────────────────────\n",
       "                                ___                            \n",
       "                            2⋅╲╱ π                             "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_in_t = -1/sqrt(2*pi) * Integral(exp(-I*w*(t-t0)) * b_w_t(t0), (w, -oo, oo))\n",
    "a_in_t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This operator satisfies the commutation relation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$[a_{in}(t), a_{in}^\\dagger(t')] = $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgQAAABLCAMAAAA4TW80AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\noYOYGReH/GTSwTp+xf68B+DDcngU/wd5+17u+j6taQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\frac{1}{2 \\pi} \\left[\\int_{-\\infty}^{\\infty} e^{i {\\omega'} \\left(- t_{0} + {t'}\\right)} {{{{b_{{\\omega'}}}^\\dagger}}(t_{0})}\\, d{\\omega'},\\int_{-\\infty}^{\\infty} e^{- i \\omega \\left(t - t_{0}\\right)} {{{b_{\\omega}}}(t_{0})}\\, d\\omega\\right]$$"
      ],
      "text/plain": [
       " ⎡∞                                                                 ∞         \n",
       " ⎢⌠                                                                 ⌠         \n",
       " ⎢⎮   ⅈ⋅{\\omega'}⋅(-t₀ + {t'})                 ⎛ †   ⎞              ⎮   -ⅈ⋅\\om\n",
       "-⎢⎮  ℯ                        ⋅OperatorFunction⎝b ,t₀⎠ d({\\omega'}),⎮  ℯ      \n",
       " ⎢⌡                                                                 ⌡         \n",
       " ⎣-∞                                                                -∞        \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                             2⋅π              \n",
       "\n",
       "                                             ⎤ \n",
       "                                             ⎥ \n",
       "ega⋅(t - t₀)                                 ⎥ \n",
       "            ⋅OperatorFunction(b,t₀) d(\\omega)⎥ \n",
       "                                             ⎥ \n",
       "                                             ⎦ \n",
       "───────────────────────────────────────────────\n",
       "                                               "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "comm = Commutator(a_in_t, Dagger(a_in_t.replace(t, t_)).replace(w, w_))\n",
    "comm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFoAAAAXBAMAAAB5QEWDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARHZmMlS7qxCJIs3d\n75nrAFFbAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABjklEQVQ4EbVTMUjDQBR9UVqStI1OuggtuktB\nnCqa3SEOjoKBDoJLi4ODU3BVqCi4CBIQnCN0S4cOLhYHBzeXbiIuFUR08/+7f5BAFRx8w7v33r1c\n75oL8B+w3v+yqvcpbWvBZ3UldvxQGWIy5qnlGf4VN2T9I4IYOODZNiLiUkwkOI6N0qM9wDmpI3Zn\nKlpVrGlbdKGmRWGEW1IO22pMhAGT4FVG0/YSN6HIfiAq9YlcJoM7EaZdPkHM0Q315r9ITAzZKlj7\nH3taSXu2tVNTwSWwHrVIFuu6wEzb1DBrm3M0YD9izQfKG9KgwUtEm7Y5xxa8EAG1K9P0D90zuqBt\nEpw07T2laZukOUeASh1BpNvcYXToSQVZ234TH6DjY4VMdidV2Lm2NUKkgjlM+WiSzJ7yApu5tpMU\nfBU06AVZIUlnqLyiZ7umjezEax9q/wKcLrK0+jpgvl4SLW23WddBV/L8mzehtI3NvO/srTLTdmyU\nGtWt0kkxP5OriVE3Vms3HFfIZbsZ9/uXRkX60r4BcA9KzEMfbEkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\delta\\left(- t + {t'}\\right)$$"
      ],
      "text/plain": [
       "DiracDelta(-t + {t'})"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "replace_dirac_delta(pull_outwards(push_inwards(comm.expand(commutator=True).expand(commutator=True)).doit().args[1][0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The operator $a_{in}(t)$ satisfies the commutation relations:\n",
    "$$\n",
    "[a_{in}(t),a_{in}^\\dagger (t')] = \\delta (t-t'), \\quad [a_{in}(t),a_{in}(t')] =[a_{in}^\\dagger(t),a_{in}^\\dagger (t')] =0\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So it is possible to treat them as multimode Bosonic Operators with continuous mode $t$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADEAAAAUBAMAAADW08iqAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABLUlEQVQoFW2RMUvEQBCFvyReLrmYI7WCBrSw\njCKCYBFEsNVGsbM4OzUWWolH/AenjY3F2XmNRqyV+A8s7bzCH5AqYudsEk64+GBmd99L3u7MwBi0\nniKuxlh1nCq4tleXLsF8woxriu2Dm8FyTXHERpcfDmrKtDBOANYJaHPzO38frEP38SHE8OHMs8RX\nmAIDyZ8Sdoy5h96Ht4KHF1kXJNwEPccJK1qWXYlnicmMRp8olW0FUcxv2YsSBcr3dMj17YZfuDUz\n9mkmRO/cu95Kzs0P2yAvMGI3VS+wQvvV0I4SrZOQw5IUOuxICjC/1mY2aYW0w+YWnEuBg55UmlYX\nR4c0Ut1PUd1RUL0o8GEwi3Mh/n5JjDp6vEiXiVVh7wrlnylUk2tVpqVBmUfT/gWx6DrOYhIEZAAA\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$${{a_{in}}(t)}$$"
      ],
      "text/plain": [
       "OperatorFunction(a_in,t)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_in = Operator('a_in')\n",
    "a_in_t = OperatorFunction(a_in, t)\n",
    "a_in_t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXwAAABLBAMAAACBwSFYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/8GT7tCVbdzird0tc2MgWfNfYv/oYLERahoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sqrt{\\gamma} {{a_{in}}(t)} - \\frac{\\gamma}{2 \\pi} \\int_{-\\infty}^{\\infty} e^{- i \\omega t} \\int_{t_{0}}^{t} e^{i \\omega {t'}} {{{a}}({t'})}\\, d{t'}\\, d\\omega$$"
      ],
      "text/plain": [
       "                                             ∞                                \n",
       "                                             ⌠                                \n",
       "                                             ⎮               t                \n",
       "                                             ⎮               ⌠                \n",
       "                                             ⎮   -ⅈ⋅\\omega⋅t ⎮   ⅈ⋅\\omega⋅{t'}\n",
       "                                      \\gamma⋅⎮  ℯ           ⋅⎮  ℯ             \n",
       "                                             ⎮               ⌡                \n",
       "                                             ⎮               t₀               \n",
       "                                             ⌡                                \n",
       "  ________                                   -∞                               \n",
       "╲╱ \\gamma ⋅OperatorFunction(a_in,t) - ────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "                                           \n",
       "                                           \n",
       "                                           \n",
       "                                           \n",
       "                                           \n",
       "⋅OperatorFunction(a,{t'}) d({t'}) d(\\omega)\n",
       "                                           \n",
       "                                           \n",
       "                                           \n",
       "                                           \n",
       "───────────────────────────────────────────\n",
       "2⋅π                                        "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "term2_5 = term2_4.subs(Integral(exp(-I*w*(t-t0)) * b_w_t(t0), (w, -oo, oo)).expand(), -sqrt(2*pi)* a_in_t)\n",
    "term2_5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Applying the <b>Leibniz integral rule</b>(http://en.wikipedia.org/wiki/Leibniz_integral_rule),"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZ8AAABLCAMAAACY5HorAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\n70yyKaT1NgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\sqrt{\\gamma} {{a_{in}}(t)} - \\frac{\\gamma}{2 \\pi} \\int_{t_{0}}^{t} \\left(\\int_{-\\infty}^{\\infty} e^{- i \\omega t} e^{i \\omega {t'}}\\, d\\omega\\right) {{{a}}({t'})}\\, d{t'}$$"
      ],
      "text/plain": [
       "                                             t                                \n",
       "                                             ⌠                                \n",
       "                                             ⎮  ⎛∞                            \n",
       "                                             ⎮  ⎜⌠                            \n",
       "                                             ⎮  ⎜⎮   -ⅈ⋅\\omega⋅t  ⅈ⋅\\omega⋅{t'\n",
       "                                      \\gamma⋅⎮  ⎜⎮  ℯ           ⋅ℯ            \n",
       "                                             ⎮  ⎜⌡                            \n",
       "                                             ⎮  ⎝-∞                           \n",
       "                                             ⌡                                \n",
       "  ________                                   t₀                               \n",
       "╲╱ \\gamma ⋅OperatorFunction(a_in,t) - ────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "                                             \n",
       "                                             \n",
       "           ⎞                                 \n",
       "           ⎟                                 \n",
       "}          ⎟                                 \n",
       "  d(\\omega)⎟⋅OperatorFunction(a,{t'}) d({t'})\n",
       "           ⎟                                 \n",
       "           ⎠                                 \n",
       "                                             \n",
       "                                             \n",
       "─────────────────────────────────────────────\n",
       " 2⋅π                                         "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "term2_6 = pull_outwards(exchange_integral_order(push_inwards(term2_5)))\n",
    "term2_6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and replacing the integrand with Dirac Delta function,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUYAAAAsBAMAAADr4QxLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nHllZ+Svk9g+iVFy8lIPQYwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\sqrt{\\gamma} {{a_{in}}(t)} + \\gamma \\theta\\left(- t + t_{0}\\right) {{{a}}(t)} - \\frac{\\gamma {{{a}}(t)}}{2}$$"
      ],
      "text/plain": [
       "  ________                                                                    \n",
       "╲╱ \\gamma ⋅OperatorFunction(a_in,t) + \\gamma⋅Heaviside(-t + t₀)⋅OperatorFuncti\n",
       "                                                                              \n",
       "\n",
       "          \\gamma⋅OperatorFunction(a,t)\n",
       "on(a,t) - ────────────────────────────\n",
       "                       2              "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "term2_7 = replace_dirac_delta(term2_6).doit().expand()\n",
    "term2_7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To use our assumption $t>t_0$, we introduce the function <tt>assume_positive</tt>,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def assume_positive(e, p_number):\n",
    "    \"\"\"\n",
    "    Assume p_number positive in the expression e\n",
    "    \"\"\"\n",
    "    positive = Symbol(\"epsilon\",positive=True)\n",
    "    return e.subs(p_number, positive).subs(positive, p_number)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJsAAAAsBAMAAACAgNdyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nG3vUJ9RSCVMngY/h/gG/ycR6MTltkQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\sqrt{\\gamma} {{a_{in}}(t)} - \\frac{\\gamma {{{a}}(t)}}{2}$$"
      ],
      "text/plain": [
       "  ________                            \\gamma⋅OperatorFunction(a,t)\n",
       "╲╱ \\gamma ⋅OperatorFunction(a_in,t) - ────────────────────────────\n",
       "                                                   2              "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "term2 = assume_positive(term2_7, t-t0)\n",
    "term2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, the Heisenberg equation of motion for cavity field $a(t)$ reduces to"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWUAAAAtBAMAAACE12c9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nond6+mz9Ce4H+rfXH/Y/jvgPndfScOfsen0AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\dot{a} (t) = \\sqrt{\\gamma} {{a_{in}}(t)} - \\frac{\\gamma {{{a}}(t)}}{2} + \\frac{i \\left[H_{sys},{{{a}}(t)}\\right]}{\\hbar}$$"
      ],
      "text/plain": [
       "                ________                            \\gamma⋅OperatorFunction(a,\n",
       "\\dot{a} (t) = ╲╱ \\gamma ⋅OperatorFunction(a_in,t) - ──────────────────────────\n",
       "                                                                 2            \n",
       "\n",
       "t)   ⅈ⋅[H_sys,OperatorFunction(a,t)]\n",
       "── + ───────────────────────────────\n",
       "                    h̅              "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_t_diff1 = term1 + term2\n",
    "a_t_diffsym = Symbol(\"\\dot{a} (t)\")\n",
    "Eq(a_t_diffsym, a_t_diff1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a <i>quantum stochastic differential equation</i> (qsde) for the intracavity field, $a(t)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Alternatively, we can define the output field operator as\n",
    "$$\n",
    "a_{out}(t) = \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} d\\omega \\ e^{-i\\omega(t-t_1)} b_\\omega(t_1)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By the same token, $a_{out}(t)$ also satisfied the commutation relations\n",
    "\n",
    "\n",
    "$$\n",
    "[a_{out}(t),a_{out}^\\dagger (t')] = \\delta (t-t'), \\quad [a_{out}(t),a_{out}(t')] =[a_{out}^\\dagger(t),a_{out}^\\dagger (t')] =0\n",
    "$$\n",
    "and hence can also be considered a multimode Bosonic Operators with continuous mode $t$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAUBAMAAADFBIheAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABSElEQVQoFXWSMUvDUBDHf23sa9o0MbNCKbSD\nIEiQgotD0MFFxElxEBwqji1IJ6FU5w71G9RNF404K/Eb9BPYii5udamr99LQVogH98jd772743+B\nBEt1dfI6gejUUpR33GR8CeoR1U6kuRLYI6gmUktKpuXhaSJdlqzlgdmAVLly8PfSFrQe7n2MEjRd\nU3rM250EA/FcG3VEujfHOvAs4Yq4HZAeY/kzqsZwKOGTeGFEpkc9lM/Y5L6m6kdioXUv6lEtcz4k\nSK1ehbpydsQJ2YB6n1vbtRpO1xsX9jH7IFMZbTvUU5l+7sVgDeP9LFgYsujCuogxrMnhoT43izsc\nk/ccz/FpSr8LEeGuK2qE8TDfMlqmkXllT7loJbVpzSLr8IEVDmpUDFGhNElOt1Dc7mN/vW3Q2hVy\nE9F/NhhvPx83mBSandM/5xfMmkNPjqbKxQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$${{a_{out}}(t)}$$"
      ],
      "text/plain": [
       "OperatorFunction(a_out,t)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_out = Operator('a_out')\n",
    "a_out_t = OperatorFunction(a_out, t)\n",
    "a_out_t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Repeating the same procedure we did just above,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAV4AAAAsBAMAAADfmM1XAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiJ2q0SZVIlm\n77uIHctDAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFYklEQVRYCdVXXWgcVRT+Zjf7O5tkSbEVpWTZ\ntv48JSoqKppF6INPWe1DKGjZB/GHIt0ijQ9Vu6BQQaurQkoN6kqt4oN0qBoS48NW8aFooY2IIiqr\nIPpQNEGr/ZHGc869d2dmf4MNZPaDnXvuOd/c+ebunbnfAL2ASJVVft4LUkXjDjnG8r0ieAtg3Qwr\n0yN6U2UguQjM9IjeMC2EBE3uxh7Ru410hh0gVAqu4OTYT0imsX3usTQeBfb/8WsBNq2LoOLk3rsQ\nRfxrjABnSORR+qUyQVWLSBEDpSPYWMUbwO0k8xX6JUcDqzeVhuU4eDONW4BvSeYd9OtbDKxeFjab\nw4+w/hK91rng652HdVHeu7Qe4otIIx7c9cDzOwPrAgYyR/h5szPJXJCfN9a7G7gGnxUdzNF2UVxP\nB4fTPkR2LTNaLuz7mZnK+/hux/pSYiG52VaRz2d1sF5lYMPOrZMlPANEzlRpv8ihEbFSY6bej9Qk\n3FNP+IMXr6LxoUn+UkPP57M6WC8eT8D7MYP3uQbc19D3dEN01ro8hjwpT9h3GkvUZZIBkX0giyWw\nMqqVY3vrFa8aGvsdRgu/85qqtDqeoCTtMnZBF9/xkcJVXKAEkwx4S2IkC9LIoy6R57odrJft3u4v\ncpqVkcZ7sEa9PX/Ms0O7jHVKp/16TwL/UkFPoVCILDB62WIJPD6rg/WyNZsatWgG3BswpUTFRE1t\nhG+Fd5nrdMmv9zb00fwKyZzKZIbRyxZL4PFZK7NebR/KdTS6GKPI9NRpPbpuomXEn740CX67CIze\nw5fSeCt+DlHagohkoMjc03rFYklV+azVsF6fAMoYHciHFs2VVTtQpStz7lOd13pT33xQQyH2zxNP\nUZFJBkLmjplfs57Ve381rBctX2WMrO+RqJkLqzZcUSYfh3Ve650AfkvmwxkMZshVE8mgvlyNXl7P\nd3NVfNb/tF5DNzE2q4tEq1DGKLGEcAEuqN5fUiYf45w+ls3emc1eS9Fx4Gobgw7GK4pkRlTL1c5m\nNy1ks0Vi0nq+53Fqlc9qYb2sTazlxhxzuuI5YoQVazbXX8Nwzj3Folcr6x2mn9JLrZ5fiuxnMVzB\n6xQxyUDI3NHzKxZL/hzxhZ2t16Bss80HMziiCxQ+pLrzGHbE0Jsq/4H8V78EC7w7Cjx6Hb6TvynL\nJAMhc0frFYsletlnXa71undXCaBxGDM8XT8n85iZxvNFjEZevT4nj9IxvN30vBHfLtDE2nyy93kT\nMg+n9YrFEr38XXPZ1mu4hpTDw7MxChVSt9oIl2JVZ6nvFEIVElUGDloFYJopBHd+Y2kkCg/kKafe\nVFJWZA61XrFYold8VlfrpUZpe0ycRTitqmVYkx9OPIKDsPdcORotYpC08J+4gz2YfjY9ehM0XVMv\n87lMMhAyd7ResViiN5SjbFfrRZxOiJzHF7pe1u13GHBiTqyAA5zQW219C3Pnl/QaePdjk9N6pSt6\nPT6rg/Uyp7dr96VvUKW6MTpLL4n+Uv9WPGzRBJ9QVVqsClfoFnihHhmSm6CITzYQvQ1+h2uelKHq\nNnLo91xDSnXHP9bzapvxN2Mvwrmj6zHF3kPbjaHmkz9yUy08iVuk6H36WRk6GLS1XoawATZ/azYj\ntJBWSRYnmNhZQfLJd2ex/0HqR2qSVBeQ0BwcE9RJbsIfbR8jA+K7p7bWy5y4G/jBxL42ed7Xbe50\n/h7S/FX8HlIjfgXsyzWLocyWltm1To6k2+lda2Vtrz+mF2pbQrAK8YvB0tNNTbjYjRGs+qFgyemm\nxvON1Y0aiPp7iOQDIWRlIvrKiPaS3m3zc/IRtbK7W3vWyPLyn66K/wBqlFpfQ7pMbQAAAABJRU5E\nrkJggg==\n",
      "text/latex": [
       "$$- \\sqrt{\\gamma} {{a_{out}}(t)} + \\gamma \\theta\\left(- t + t_{1}\\right) {{{a}}(t)} - \\frac{\\gamma {{{a}}(t)}}{2}$$"
      ],
      "text/plain": [
       "    ________                                                                  \n",
       "- ╲╱ \\gamma ⋅OperatorFunction(a_out,t) + \\gamma⋅Heaviside(-t + t₁)⋅OperatorFun\n",
       "                                                                              \n",
       "\n",
       "             \\gamma⋅OperatorFunction(a,t)\n",
       "ction(a,t) - ────────────────────────────\n",
       "                          2              "
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "term2_2 = term2_1.replace(b_w_t, b_w_t_output).expand()\n",
    "term2_3 = pull_outwards(term2_2)\n",
    "term2_4 = pull_outwards(term2_3.replace(g(w), sqrt(gamma/(2*pi))))\n",
    "term2_5 = term2_4.subs(Integral(exp(-I*w*(t-t1)) * b_w_t(t1), (w, -oo, oo)).expand(), sqrt(2*pi)* a_out_t)\n",
    "term2_6 = pull_outwards(exchange_integral_order(push_inwards(term2_5)))\n",
    "term2_7 = replace_dirac_delta(term2_6).doit().expand()\n",
    "term2_7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To use our assumption $t_1>t$, we employ <tt>assume_positive</tt> function, as we did before:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALMAAAAsBAMAAADcCpRWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\sqrt{\\gamma} {{a_{out}}(t)} + \\frac{\\gamma {{{a}}(t)}}{2}$$"
      ],
      "text/plain": [
       "    ________                             \\gamma⋅OperatorFunction(a,t)\n",
       "- ╲╱ \\gamma ⋅OperatorFunction(a_out,t) + ────────────────────────────\n",
       "                                                      2              "
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "term2_8 = assume_positive(term2_7, t1-t)\n",
    "term2_8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX0AAAAtBAMAAACwrqYhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\dot{a} (t) = - \\sqrt{\\gamma} {{a_{out}}(t)} + \\frac{\\gamma {{{a}}(t)}}{2} + \\frac{i \\left[H_{sys},{{{a}}(t)}\\right]}{\\hbar}$$"
      ],
      "text/plain": [
       "                  ________                             \\gamma⋅OperatorFunction\n",
       "\\dot{a} (t) = - ╲╱ \\gamma ⋅OperatorFunction(a_out,t) + ───────────────────────\n",
       "                                                                    2         \n",
       "\n",
       "(a,t)   ⅈ⋅[H_sys,OperatorFunction(a,t)]\n",
       "───── + ───────────────────────────────\n",
       "                       h̅              "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_t_diff2 = term1 + term2_8\n",
    "Eq(a_t_diffsym, a_t_diff2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To summarize, the Heisenberg equation of motion for the intracavity operator $a(t)$ are given by"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWUAAAAtBAMAAACE12c9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nond6+mz9Ce4H+rfXH/Y/jvgPndfScOfsen0AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\dot{a} (t) = \\sqrt{\\gamma} {{a_{in}}(t)} - \\frac{\\gamma {{{a}}(t)}}{2} + \\frac{i \\left[H_{sys},{{{a}}(t)}\\right]}{\\hbar}$$"
      ],
      "text/plain": [
       "                ________                            \\gamma⋅OperatorFunction(a,\n",
       "\\dot{a} (t) = ╲╱ \\gamma ⋅OperatorFunction(a_in,t) - ──────────────────────────\n",
       "                                                                 2            \n",
       "\n",
       "t)   ⅈ⋅[H_sys,OperatorFunction(a,t)]\n",
       "── + ───────────────────────────────\n",
       "                    h̅              "
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(a_t_diffsym, a_t_diff1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX0AAAAtBAMAAACwrqYhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\dot{a} (t) = - \\sqrt{\\gamma} {{a_{out}}(t)} + \\frac{\\gamma {{{a}}(t)}}{2} + \\frac{i \\left[H_{sys},{{{a}}(t)}\\right]}{\\hbar}$$"
      ],
      "text/plain": [
       "                  ________                             \\gamma⋅OperatorFunction\n",
       "\\dot{a} (t) = - ╲╱ \\gamma ⋅OperatorFunction(a_out,t) + ───────────────────────\n",
       "                                                                    2         \n",
       "\n",
       "(a,t)   ⅈ⋅[H_sys,OperatorFunction(a,t)]\n",
       "───── + ───────────────────────────────\n",
       "                       h̅              "
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(a_t_diffsym, a_t_diff2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and hence the input and output fields are then seen to be related by"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPsAAAAyBAMAAABou7DpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nUX570EdNTSo+SM2qr9HRvup/IT4GB/0MJfenVDoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$${{{a}}(t)} = \\frac{1}{\\sqrt{\\gamma}} \\left({{a_{in}}(t)} + {{a_{out}}(t)}\\right)$$"
      ],
      "text/plain": [
       "                        OperatorFunction(a_in,t) + OperatorFunction(a_out,t)\n",
       "OperatorFunction(a,t) = ────────────────────────────────────────────────────\n",
       "                                               ________                     \n",
       "                                             ╲╱ \\gamma                      "
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(a_t, solve(a_t_diff2 - a_t_diff1, a_t)[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear Systems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a linear system, the Heisenberg equation of motion for $a(t)$ can be simply expressed as\n",
    "\n",
    "\n",
    "$$\n",
    "\\frac{d\\mathbf{a}(t)}{dt} = \\left( \\mathbf{A} - \\frac{\\gamma}{2}\\mathbb{I}\\right) \\mathbf{a}(t) + \\sqrt{\\gamma} \\mathbf{a}_{in}(t) \\\\\n",
    "= \\left( \\mathbf{A} + \\frac{\\gamma}{2}\\mathbb{I}\\right) \\mathbf{a}(t) - \\sqrt{\\gamma} \\mathbf{a}_{out}(t)\n",
    "$$\n",
    "\n",
    "where\n",
    "$$\\mathbf{a}(t) = \\begin{bmatrix} a(t) \\\\ a^\\dagger (t) \\end{bmatrix}\n",
    "$$\n",
    "and $\\mathbf{A}$ is a matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In terms of the frequency components, defined by $\\tilde{a}(\\omega) \\equiv \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} e^{i\\omega t} a(t) dt$,"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\mathbf{a}(t) = \\begin{bmatrix}\n",
    "\\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} e^{-i\\omega t} \\tilde{a}(\\omega) d\\omega \\\\\n",
    "\\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} e^{i\\omega t} \\tilde{a}^\\dagger(\\omega) d\\omega \n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "\\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} e^{-i\\omega t} \\tilde{a}(\\omega) d\\omega \\\\\n",
    "\\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} e^{-i\\omega t} \\tilde{a}^\\dagger(-\\omega) d\\omega \n",
    "\\end{bmatrix}\n",
    "= \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} e^{-i\\omega t}  \\tilde{\\mathbf{a}}(\\omega) d\\omega\n",
    "$$\n",
    "where\n",
    "$$\\tilde{\\mathbf{a}}(\\omega) = \\begin{bmatrix} a(\\omega) \\\\ a^\\dagger (-\\omega) \\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The equation becomes\n",
    "$$\n",
    "-i\\omega \\tilde{\\mathbf{a}}(\\omega) = \\left(\\mathbf{A} - \\frac{\\gamma}{2}\\mathbb{I}\\right) \\tilde{\\mathbf{a}}(\\omega)+ \\sqrt{\\gamma} \\tilde{\\mathbf{a}}_{in}(\\omega) \\\\\n",
    "= \\left( \\mathbf{A} + \\frac{\\gamma}{2}\\mathbb{I}\\right) \\tilde{\\mathbf{a}}(\\omega) - \\sqrt{\\gamma} \\tilde{\\mathbf{a}}_{out}(\\omega)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rearranging to eliminate the intermal mode,$$\n",
    "\\tilde{\\mathbf{a}}_{out}(\\omega) = \\left[ \\mathbf{A} + \\left(\\frac{\\gamma}{2} + i\\omega\\right) \\mathbb{1}\\right]\n",
    "\\left[-\\mathbf{A} + \\left(\\frac{\\gamma}{2} - i\\omega\\right) \\mathbb{1}\\right]^{-1} \\tilde{\\mathbf{a}}_{in}(\\omega)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A One-Sided Cavity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the system Hamiltonian is assumed to be a free Harmonic oscillator, we get a model of a single mode in a one-sided cavity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALMAAAAZCAMAAABNTe9JAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADxgEwMAAAAEHRSTlMAZqvNmRDdRHYy\niVS7Iu8g4cVo4gAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAyJJREFUWAnVV9sSoyAMRQTBiC7//7Wb\nG4gttN22M53lpRgPSUhOEmvMT5YNPzH7kdH/0WfwH135B4fTnNyx/cDwiybXcO+cN+BLoPepKpr2\nur1supDJ5jw7Y+S3aLuce/fBw7F0ztpVhXtq3tqu0yNInuVo7hlotL6xXaBzKJXgz+KmX+jXqxfX\nAwNIyJFxa8Zof3VtLlsAOAkg2ovPk9rbMstB3Lh4MIJAllzFXO5/OfbJw5Z7XLNqh8OL6p1E2B/3\npkaQWa5pUtfAvZ5/kLgu27Q/7+VlIdB8N2uGkKyV8AGdSy3tx9Xs4iaIiVm7O3Dx4LA7zStTI9iU\nk2VagHLlhMYBJOTZ0kqv0BnmZqkJbDnFU0+lESFq5flMJWQtZiYcuNOMgviclOdKZ1QjHGmgI8gX\n6FydN7CZCR3eyE10ItNlEmbBH+Sgk3xEIfmitFY64ykmSwsdQMrlP6Bz4DBzkHAyLPQkpWHZiQNj\n71iwSGfQ9lx4VOhsVi6sFjqAmErnbn9E+08Xe7JysMF47kIynA/iyE6CgyIsr6o6dchzMkgqPrfQ\nAeTszkpCiBGTFoDSu1AWRUBKcb80SwiAQaQ3dMjsgIap5Baiws7eAAbbZ1JeiYt7XJr4QNFltgRK\nywXah5iTzlJILpgVgwKU3pU6pgpwN1oJAykDCUtuPVkwca4x2NMfbv1pNhv3EFGkBQaY343pwjXo\nW2gfYpab7pzShp8wu6dLR8qoCEb+khwOwC81HHlUaBJn4rCxzDYMNpgZn6fF4u5c2tqcNV547vht\nC+1BvJ1zpuboUs6JT674yUS3tqiA22YRnMZudz7leTXxYEeFtMyKmZsZDm9vvI2w7jZKLkUBUwEZ\nlCbGYSvkaLfQAeTGAUoeu0rlf6CSKrgBDh+bvjHEyIsymBVWh15z7AUIFga6yXMKN1wdVdBoeril\n2aX9+SEOyXL2dUI++kZSTT0I6pkmSWCwEWumEeix5z+AU1qmxjOsTHVF9b9FX4BUKx45UiZnFX57\nc/mgv3hXLb0AqVgswfD2kKlanm26f5yuh16AlANhgrYvFfHvf/8CS3UZOAarjOsAAAAASUVORK5C\nYII=\n",
      "text/latex": [
       "$$H_{sys} = \\omega_{0} \\hbar {{{{a}^\\dagger}}(t)} {{{a}}(t)}$$"
      ],
      "text/plain": [
       "                                   ⎛ †  ⎞                      \n",
       "H_sys = \\omega₀⋅h̅⋅OperatorFunction⎝a ,t⎠⋅OperatorFunction(a,t)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hsys = hbar * w0 * Dagger(a_t) * a_t\n",
    "Eq(Hsyssym, Hsys)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAUBAMAAADVZMaCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nR4GE63n/DVIwlpvD337bh0P+DXA5WlcjlQOOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- i \\omega_{0} {{{a}}(t)}$$"
      ],
      "text/plain": [
       "-ⅈ⋅\\omega₀⋅OperatorFunction(a,t)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "term1_ = term1.subs(Hsyssym, Hsys).expand(commutator=True).doit()\n",
    "term1_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The equation for the intermal-mode operator is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATMAAAAsBAMAAAD7gUKXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$\\dot{a} (t) = \\sqrt{\\gamma} {{a_{in}}(t)} - \\frac{\\gamma {{{a}}(t)}}{2} - i \\omega_{0} {{{a}}(t)}$$"
      ],
      "text/plain": [
       "                ________                            \\gamma⋅OperatorFunction(a,\n",
       "\\dot{a} (t) = ╲╱ \\gamma ⋅OperatorFunction(a_in,t) - ──────────────────────────\n",
       "                                                                 2            \n",
       "\n",
       "t)                                  \n",
       "── - ⅈ⋅\\omega₀⋅OperatorFunction(a,t)\n",
       "                                    "
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(a_t_diffsym, a_t_diff1.subs(Hsyssym, Hsys).expand(commutator=True).doit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, the matrix $\\mathbf{A}$ becomes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGwAAAAyBAMAAABLzCN5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- i \\omega_{0} & 0\\\\0 & i \\omega_{0}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-ⅈ⋅\\omega₀      0    ⎤\n",
       "⎢                     ⎥\n",
       "⎣    0       ⅈ⋅\\omega₀⎦"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[-I*w0, 0], [0, I*w0]])\n",
    "A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the coefficient matrix connecting $\\tilde{\\mathbf{a}}_{out}(\\omega)$ and $\\tilde{\\mathbf{a}}_{out}(\\omega)$ becomes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left(\\left[\\begin{matrix}- i \\omega_{0} & 0\\\\0 & i \\omega_{0}\\end{matrix}\\right] + \\left(\\frac{\\gamma}{2} + i \\omega\\right) \\mathbb{I}\\right) \\left(\\left[\\begin{matrix}i \\omega_{0} & 0\\\\0 & - i \\omega_{0}\\end{matrix}\\right] + \\left(\\frac{\\gamma}{2} - i \\omega\\right) \\mathbb{I}\\right)^{-1}$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "⎛                          ⎛\\gamma           ⎞  ⎞ ⎛                          ⎛\n",
       "⎜⎡-ⅈ⋅\\omega₀      0    ⎤ + ⎜────── + ⅈ⋅\\omega⎟⋅I⎟⋅⎜⎡ⅈ⋅\\omega₀      0     ⎤ + ⎜\n",
       "⎜⎢                     ⎥   ⎝  2              ⎠  ⎟ ⎜⎢                     ⎥   ⎝\n",
       "⎝⎣    0       ⅈ⋅\\omega₀⎦                        ⎠ ⎝⎣    0      -ⅈ⋅\\omega₀⎦    \n",
       "\n",
       "                     -1\n",
       "\\gamma           ⎞  ⎞  \n",
       "────── - ⅈ⋅\\omega⎟⋅I⎟  \n",
       "  2              ⎠  ⎟  \n",
       "                    ⎠  "
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Id = Identity(2)\n",
    "T = (A + (gamma/2 + I*w) *  Id) * (-A + (gamma/2 - I*w)*Id).inverse()\n",
    "T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANAAAABLCAMAAAA22/nGAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{\\frac{\\gamma}{2} + i \\omega - i \\omega_{0}}{\\frac{\\gamma}{2} - i \\omega + i \\omega_{0}} & 0\\\\0 & \\frac{\\frac{\\gamma}{2} + i \\omega + i \\omega_{0}}{\\frac{\\gamma}{2} - i \\omega - i \\omega_{0}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡\\gamma                                                      ⎤\n",
       "⎢────── + ⅈ⋅\\omega - ⅈ⋅\\omega₀                               ⎥\n",
       "⎢  2                                                         ⎥\n",
       "⎢─────────────────────────────                0              ⎥\n",
       "⎢\\gamma                                                      ⎥\n",
       "⎢────── - ⅈ⋅\\omega + ⅈ⋅\\omega₀                               ⎥\n",
       "⎢  2                                                         ⎥\n",
       "⎢                                                            ⎥\n",
       "⎢                               \\gamma                       ⎥\n",
       "⎢                               ────── + ⅈ⋅\\omega + ⅈ⋅\\omega₀⎥\n",
       "⎢                                 2                          ⎥\n",
       "⎢              0                ─────────────────────────────⎥\n",
       "⎢                               \\gamma                       ⎥\n",
       "⎢                               ────── - ⅈ⋅\\omega - ⅈ⋅\\omega₀⎥\n",
       "⎣                                 2                          ⎦"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T.subs(Id, eye(2)).doit(deep=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We obtain the following relations:\n",
    "$$\\tilde{a}(\\omega) = \\frac{\\sqrt{\\gamma}}{\\frac{\\gamma}{2} - i(\\omega-\\omega_0)} \\tilde{a}_{in} (\\omega),\\quad\n",
    "\\tilde{a}_{out} (\\omega) = \\frac{\\frac{\\gamma}{2}+i(\\omega-\\omega_0)}{\\frac{\\gamma}{2}-i(\\omega-\\omega_0)} \\tilde{a}_{in} (\\omega) .$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}