06 - Three level atom, ladder (symbolic).ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from fast import *\n",
    "from matplotlib import pyplot\n",
    "from sympy import sin,cos,exp,sqrt,pi,zeros,I\n",
    "from numpy import array\n",
    "%matplotlib inline\n",
    "init_printing()\n",
    "print_ascii=True\n",
    "print_ascii=False"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "path=\"folder_06___Three_level_atom_ladder_symbolic/\" \n",
    "name='suite'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will be deriving the optical Bloch equations for a three level system in a ladder configuration as that in the figure."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f1693039350>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig=pyplot.figure(); ax=fig.add_subplot(111,aspect=\"equal\")\n",
    "\n",
    "p1=[0.5,1]; p2=[1.5,3]; p3=[2.5,5]\n",
    "draw_state(ax,p1,text=r\"$|1\\rangle$\",l=1.0,alignment='right',label_displacement=0.05,fontsize=25,linewidth=4.0)\n",
    "draw_state(ax,p2,text=r\"$|2\\rangle$\",l=1.0,alignment='right',label_displacement=0.05,fontsize=25,linewidth=4.0)\n",
    "draw_state(ax,p3,text=r\"$|3\\rangle$\",l=1.0,alignment='right',label_displacement=0.05,fontsize=25,linewidth=4.0)\n",
    "\n",
    "excitation(ax,[p1[0]+0.25,p1[1]],[p2[0]+0.25,p2[1]], fc=\"r\", ec=\"r\",width=0.01, head_width=0.2, head_length=0.2)\n",
    "excitation(ax,[p2[0]+0.25,p2[1]],[p3[0]+0.25,p3[1]], fc=\"b\", ec=\"b\",width=0.01, head_width=0.2, head_length=0.2)\n",
    "\n",
    "decay(     ax,[p1[0]-0.25,p1[1]],[p2[0]-0.25,p2[1]], 0.05,10.0,color=\"r\",linewidth=1.0)\n",
    "decay(     ax,[p2[0]-0.25,p2[1]],[p3[0]-0.25,p3[1]], 0.05,10.0,color=\"b\",linewidth=1.0)\n",
    "\n",
    "pyplot.axis('off')\n",
    "pyplot.savefig(path+name+'_diagram.png',bbox_inches=\"tight\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define the number of states and of radiation fields."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Ne=3\n",
    "Nl=2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define the variables related to the laser field."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAAUBAMAAACT0mUoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n7RuIB8MsIrOD3l9xJPYN2j1vpCyz+gEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ E_{01}, \\quad E_{02}\\right ]$$"
      ],
      "text/plain": [
       "[E_{01}, E_{02}]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E0,omega_laser=define_laser_variables(Nl)\n",
    "fprint(E0,print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFsAAAAUBAMAAAAQFlwTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdt3NMolEVO9mqyIQ\nmbvrVuQsAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABQElEQVQoFWMQMmEgFnAaJTCEEasYqI51yCsv\n+g8GDdg8za4PlvsBlgN7lXPZsjQgyNrA4IypYVYWSC5tNUPs2wOQkDnKPQGkjJXBdxGGcp6AnWCx\nANYNXAbQgGRlYOAFKWcowlDOwBDAwHABSDJtYPkNVQ40oJSB4SBW5UCb2QMYeASYEri/QZXPZmAw\nZGCIwaqccwMDswAD6wEGBvY/UOUiDFyxDGxAK7E4htGBYekFhlygK5mAusCJYCrL0xN7JgNFQMrP\nGwAJBOAM8IvpPgkSi4WlmfP/BXj/2wBFQMqZdBBqgSye/x8Z7n8+wMDACbQdmsQ2MDDsBikCO6Ya\nxEIAbgYGbqA8QypDCkw5TLICxAiA8VBoFvPyN2jK9+ovBNougKIMxuH4//8DmnKwFCdMARYa6HYh\nFSzi2IUYlRIAQNJXLbta5Z8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\varpi_{1}, \\quad \\varpi_{2}\\right ]$$"
      ],
      "text/plain": [
       "[varpi₁, varpi₂]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint(omega_laser,print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define a few important symbols."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAAUBAMAAACABSXcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdt3NMolEECJUmau7\nZu/DRrifAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABMklEQVQoFX2SMUvDQBTH/xbiJY1pY8aiUESr\nOGVwFTI5R6FxE3F1adcu3jeoo1tBP4AFcXcXMSDi6NShg1AoZDZ3h8i9d/q2+/3un/fuLkgO4Kij\nawes0VsPhcuIs64L16xQAbHHbCNnyAAdCBbUBveP/ZJCvdaBZsrcZMCQATrg6H/7x359hsvX55j6\n/fcNSZk47khz6AlVEEsZTim9kH5uAndUIaqw9kSoP8coM4EdooCwixYdM/xa/zDvICoWaOUYZ4S2\n9YzqllYXGBA5PscJQWjniqiAlwYZsZ/ALCJfqbsi1oHmdIir1Io8AJUnlhZbSTEy1+q/lGjsWvIQ\n6J9iy+7R2SzNSHrrjRUwi0ByqH8NhesBWXmM/HYQsUMOHeynQ+RwqEdmVSDZZvAfkPS+AcYOOnmA\nUCmHAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ t, \\quad \\hbar, \\quad e\\right ]$$"
      ],
      "text/plain": [
       "[t, h̅, e]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t,hbar,e=symbols(\"t hbar e\",positive=True)\n",
    "fprint([t,hbar,e],print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We write two electric fields propagating trough the $\\hat{x}$ direction polarized in the $\\hat{z}$ direction. First the wave vectors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI4AAABLCAMAAACRBk89AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nPg+Gsc8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\left[\\begin{matrix}1\\\\0\\\\0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}-1\\\\0\\\\0\\end{matrix}\\right]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡1⎤, ⎡-1⎤⎤\n",
       "⎢⎢ ⎥  ⎢  ⎥⎥\n",
       "⎢⎢0⎥  ⎢0 ⎥⎥\n",
       "⎢⎢ ⎥  ⎢  ⎥⎥\n",
       "⎣⎣0⎦  ⎣0 ⎦⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi1=0 ; theta1=pi/2; alpha1=pi/2; beta1=0\n",
    "phi2=pi; theta2=pi/2; alpha2=pi/2; beta2=0\n",
    "\n",
    "k1=Matrix([cos(phi1)*sin(theta1),sin(phi1)*sin(theta1),cos(theta1)])\n",
    "k2=Matrix([cos(phi2)*sin(theta2),sin(phi2)*sin(theta2),cos(theta2)])\n",
    "\n",
    "fprint([k1,k2],print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The polarization vectors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUQAAABLCAMAAAA/FQ6VAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n9bjFYatZ/QvTeZaLg2DwMwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\left [ \\left[\\begin{matrix}0\\\\0\\\\1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\0\\\\1\\end{matrix}\\right]\\right ], \\quad \\left [ \\left[\\begin{matrix}0\\\\0\\\\1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\0\\\\1\\end{matrix}\\right]\\right ]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡⎡0⎤, ⎡0⎤⎤, ⎡⎡0⎤, ⎡0⎤⎤⎤\n",
       "⎢⎢⎢ ⎥  ⎢ ⎥⎥  ⎢⎢ ⎥  ⎢ ⎥⎥⎥\n",
       "⎢⎢⎢0⎥  ⎢0⎥⎥  ⎢⎢0⎥  ⎢0⎥⎥⎥\n",
       "⎢⎢⎢ ⎥  ⎢ ⎥⎥  ⎢⎢ ⎥  ⎢ ⎥⎥⎥\n",
       "⎣⎣⎣1⎦  ⎣1⎦⎦  ⎣⎣1⎦  ⎣1⎦⎦⎦"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ep1=polarization_vector(phi1,theta1,alpha1,beta1, 1)\n",
    "ep2=polarization_vector(phi2,theta2,alpha2,beta2, 1)\n",
    "\n",
    "em1=ep1.conjugate()\n",
    "em2=ep2.conjugate()\n",
    "\n",
    "ep=[ep1,ep2]\n",
    "em=[em1,em2]\n",
    "\n",
    "fprint([ep,em],print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The electric field (evaluated in $\\vec{R}=0$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAeoAAABNCAMAAABe3dfqAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\left[\\begin{matrix}0\\\\0\\\\\\frac{E_{01}}{2} e^{- i t \\varpi_{1}} + \\frac{\\overline{E_{01}}}{2} e^{i t \\varpi_{1}}\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\0\\\\\\frac{E_{02}}{2} e^{- i t \\varpi_{2}} + \\frac{\\overline{E_{02}}}{2} e^{i t \\varpi_{2}}\\end{matrix}\\right]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡                   0                    ⎤, ⎡                   0            \n",
       "⎢⎢                                        ⎥  ⎢                                \n",
       "⎢⎢                   0                    ⎥  ⎢                   0            \n",
       "⎢⎢                                        ⎥  ⎢                                \n",
       "⎢⎢        -ⅈ⋅t⋅varpi₁    ⅈ⋅t⋅varpi₁ ______⎥  ⎢        -ⅈ⋅t⋅varpi₂    ⅈ⋅t⋅varpi\n",
       "⎢⎢E_{01}⋅ℯ              ℯ          ⋅E_{01}⎥  ⎢E_{02}⋅ℯ              ℯ         \n",
       "⎢⎢─────────────────── + ──────────────────⎥  ⎢─────────────────── + ──────────\n",
       "⎣⎣         2                    2         ⎦  ⎣         2                    2 \n",
       "\n",
       "        ⎤⎤\n",
       "        ⎥⎥\n",
       "        ⎥⎥\n",
       "        ⎥⎥\n",
       "₂ ______⎥⎥\n",
       " ⋅E_{02}⎥⎥\n",
       "────────⎥⎥\n",
       "        ⎦⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zero_vect=Matrix([0,0,0])\n",
    "E_cartesian = [(E0[l]*ep[l]*exp(-I*omega_laser[l]*t) + E0[l].conjugate()*em[l]*exp( I*omega_laser[l]*t))/2 \n",
    "                    for l in range(Nl)]\n",
    "\n",
    "fprint(E_cartesian,print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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VYXh4mP+jYgCjkX4vbogLiDvfEXJF2kGJayyXy1FTUyN5Mz0ADAwMgGGYc+dlksgPGPIl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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f168fad5f50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "l1=PlaneWave(phi1,theta1,alpha1,beta1,color=\"red\")\n",
    "l2=PlaneWave(phi2,theta2,alpha2,beta2,color=\"blue\")\n",
    "\n",
    "laseres=[l1,l2]\n",
    "Nl=len(laseres)\n",
    "\n",
    "fig = pyplot.figure(); ax = fig.gca(projection='3d')\n",
    "draw_lasers_3d(ax,laseres,path+'lasers.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We write the electric fields in the helicity basis (see notebook \"Vectors in the helicity basis and the electric field\")."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAeoAAABLCAMAAACIhDT3AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$\\left [ \\left[\\begin{matrix}0\\\\\\frac{E_{01}}{2} e^{- i t \\varpi_{1}} + \\frac{\\overline{E_{01}}}{2} e^{i t \\varpi_{1}}\\\\0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\\\frac{E_{02}}{2} e^{- i t \\varpi_{2}} + \\frac{\\overline{E_{02}}}{2} e^{i t \\varpi_{2}}\\\\0\\end{matrix}\\right]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡                   0                    ⎤, ⎡                   0            \n",
       "⎢⎢                                        ⎥  ⎢                                \n",
       "⎢⎢        -ⅈ⋅t⋅varpi₁    ⅈ⋅t⋅varpi₁ ______⎥  ⎢        -ⅈ⋅t⋅varpi₂    ⅈ⋅t⋅varpi\n",
       "⎢⎢E_{01}⋅ℯ              ℯ          ⋅E_{01}⎥  ⎢E_{02}⋅ℯ              ℯ         \n",
       "⎢⎢─────────────────── + ──────────────────⎥  ⎢─────────────────── + ──────────\n",
       "⎢⎢         2                    2         ⎥  ⎢         2                    2 \n",
       "⎢⎢                                        ⎥  ⎢                                \n",
       "⎣⎣                   0                    ⎦  ⎣                   0            \n",
       "\n",
       "        ⎤⎤\n",
       "        ⎥⎥\n",
       "₂ ______⎥⎥\n",
       " ⋅E_{02}⎥⎥\n",
       "────────⎥⎥\n",
       "        ⎥⎥\n",
       "        ⎥⎥\n",
       "        ⎦⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E=[cartesian_to_helicity(E_cartesian[l]) for l in range(Nl)]\n",
    "fprint(E,print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define the position operator."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAxcAAABMBAMAAADtr1emAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$\\left [ \\left[\\begin{matrix}0 & - r_{+1;21} & 0\\\\r_{-1;21} & 0 & - r_{+1;32}\\\\0 & r_{-1;32} & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & r_{0;21} & 0\\\\r_{0;21} & 0 & r_{0;32}\\\\0 & r_{0;32} & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & - r_{-1;21} & 0\\\\r_{+1;21} & 0 & - r_{-1;32}\\\\0 & r_{+1;32} & 0\\end{matrix}\\right]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡    0      -r_{+1;21}      0     ⎤, ⎡   0      r_{0;21}     0    ⎤, ⎡    0  \n",
       "⎢⎢                                 ⎥  ⎢                            ⎥  ⎢       \n",
       "⎢⎢r_{-1;21}      0       -r_{+1;32}⎥  ⎢r_{0;21}     0      r_{0;32}⎥  ⎢r_{+1;2\n",
       "⎢⎢                                 ⎥  ⎢                            ⎥  ⎢       \n",
       "⎣⎣    0      r_{-1;32}       0     ⎦  ⎣   0      r_{0;32}     0    ⎦  ⎣    0  \n",
       "\n",
       "    -r_{-1;21}      0     ⎤⎤\n",
       "                          ⎥⎥\n",
       "1}      0       -r_{-1;32}⎥⎥\n",
       "                          ⎥⎥\n",
       "    r_{+1;32}       0     ⎦⎦"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r=define_r_components(Ne,helicity=True,explicitly_hermitian=True)\n",
    "#Ladder means that r_{p;31}=0\n",
    "r=[ri.subs({r[0][2,0]:0,r[1][2,0]:0,r[2][2,0]:0}) for ri in r]\n",
    "fprint(r,print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The frequencies of the energy levels, the resonant frequencies, and the decay frequencies."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIYAAAAUBAMAAAC6+VjqAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\npqxeLHkXLOgAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\omega_{1}, \\quad \\omega_{2}, \\quad \\omega_{3}\\right ]$$"
      ],
      "text/plain": [
       "[ω₁, ω₂, ω₃]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "omega_level,omega,gamma=define_frequencies(Ne,explicitly_antisymmetric=True)\n",
    "#Ladder means gamma31=0\n",
    "gamma=gamma.subs({gamma[2,0]:0})\n",
    "\n",
    "fprint(omega_level,print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALUAAABLCAMAAAABFTbyAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & - \\omega_{21} & - \\omega_{31}\\\\\\omega_{21} & 0 & - \\omega_{32}\\\\\\omega_{31} & \\omega_{32} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 0   -ω₂₁  -ω₃₁⎤\n",
       "⎢               ⎥\n",
       "⎢ω₂₁   0    -ω₃₂⎥\n",
       "⎢               ⎥\n",
       "⎣ω₃₁  ω₃₂    0  ⎦"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint(omega, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAK8AAABLCAMAAAAxmSfTAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & - \\gamma_{21} & 0\\\\\\gamma_{21} & 0 & - \\gamma_{32}\\\\0 & \\gamma_{32} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 0   -γ₂₁   0  ⎤\n",
       "⎢               ⎥\n",
       "⎢γ₂₁   0    -γ₃₂⎥\n",
       "⎢               ⎥\n",
       "⎣ 0   γ₃₂    0  ⎦"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint(gamma, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The atomic hamiltonian is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ4AAABLCAMAAAC2qM7VAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nhQPMTbMfKNT9vtNy87HqpjF/AbriOA6j9IahAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\hbar \\omega_{1} & 0 & 0\\\\0 & \\hbar \\omega_{2} & 0\\\\0 & 0 & \\hbar \\omega_{3}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡h̅⋅ω₁    0      0  ⎤\n",
       "⎢                   ⎥\n",
       "⎢  0    h̅⋅ω₂    0  ⎥\n",
       "⎢                   ⎥\n",
       "⎣  0      0    h̅⋅ω₃⎦"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H0=Matrix([[hbar*omega_level[i]*KroneckerDelta(i,j) for j in range(Ne)] for i in range(Ne)])\n",
    "fprint(H0, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "The interaction hamiltonian is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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4kMqiEGsT8ovSp95guz/Uz+xwpXmeTTrhOVEeSUNFRriOH5KSS5Sfwwk5qS1O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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & e r_{0;21} \\left(\\frac{E_{01}}{2} e^{- i t \\varpi_{1}} + \\frac{\\overline{E_{01}}}{2} e^{i t \\varpi_{1}}\\right) + e r_{0;21} \\left(\\frac{E_{02}}{2} e^{- i t \\varpi_{2}} + \\frac{\\overline{E_{02}}}{2} e^{i t \\varpi_{2}}\\right) & 0\\\\e r_{0;21} \\left(\\frac{E_{01}}{2} e^{- i t \\varpi_{1}} + \\frac{\\overline{E_{01}}}{2} e^{i t \\varpi_{1}}\\right) + e r_{0;21} \\left(\\frac{E_{02}}{2} e^{- i t \\varpi_{2}} + \\frac{\\overline{E_{02}}}{2} e^{i t \\varpi_{2}}\\right) & 0 & e r_{0;32} \\left(\\frac{E_{01}}{2} e^{- i t \\varpi_{1}} + \\frac{\\overline{E_{01}}}{2} e^{i t \\varpi_{1}}\\right) + e r_{0;32} \\left(\\frac{E_{02}}{2} e^{- i t \\varpi_{2}} + \\frac{\\overline{E_{02}}}{2} e^{i t \\varpi_{2}}\\right)\\\\0 & e r_{0;32} \\left(\\frac{E_{01}}{2} e^{- i t \\varpi_{1}} + \\frac{\\overline{E_{01}}}{2} e^{i t \\varpi_{1}}\\right) + e r_{0;32} \\left(\\frac{E_{02}}{2} e^{- i t \\varpi_{2}} + \\frac{\\overline{E_{02}}}{2} e^{i t \\varpi_{2}}\\right) & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                      0                      \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢           ⎛        -ⅈ⋅t⋅varpi₁    ⅈ⋅t⋅varpi₁ ______⎞              ⎛        -\n",
       "⎢           ⎜E_{01}⋅ℯ              ℯ          ⋅E_{01}⎟              ⎜E_{02}⋅ℯ \n",
       "⎢e⋅r_{0;21}⋅⎜─────────────────── + ──────────────────⎟ + e⋅r_{0;21}⋅⎜─────────\n",
       "⎢           ⎝         2                    2         ⎠              ⎝         \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                      0                      \n",
       "⎣                                                                             \n",
       "\n",
       "                                             ⎛        -ⅈ⋅t⋅varpi₁    ⅈ⋅t⋅varpi\n",
       "                                             ⎜E_{01}⋅ℯ              ℯ         \n",
       "                                  e⋅r_{0;21}⋅⎜─────────────────── + ──────────\n",
       "                                             ⎝         2                    2 \n",
       "                                                                              \n",
       "ⅈ⋅t⋅varpi₂    ⅈ⋅t⋅varpi₂ ______⎞                                              \n",
       "             ℯ          ⋅E_{02}⎟                                              \n",
       "────────── + ──────────────────⎟                                              \n",
       "2                    2         ⎠                                              \n",
       "                                                                              \n",
       "                                             ⎛        -ⅈ⋅t⋅varpi₁    ⅈ⋅t⋅varpi\n",
       "                                             ⎜E_{01}⋅ℯ              ℯ         \n",
       "                                  e⋅r_{0;32}⋅⎜─────────────────── + ──────────\n",
       "                                             ⎝         2                    2 \n",
       "\n",
       "₁ ______⎞              ⎛        -ⅈ⋅t⋅varpi₂    ⅈ⋅t⋅varpi₂ ______⎞             \n",
       " ⋅E_{01}⎟              ⎜E_{02}⋅ℯ              ℯ          ⋅E_{02}⎟             \n",
       "────────⎟ + e⋅r_{0;21}⋅⎜─────────────────── + ──────────────────⎟             \n",
       "        ⎠              ⎝         2                    2         ⎠             \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "          0                                                        e⋅r_{0;32}⋅\n",
       "                                                                              \n",
       "                                                                              \n",
       "₁ ______⎞              ⎛        -ⅈ⋅t⋅varpi₂    ⅈ⋅t⋅varpi₂ ______⎞             \n",
       " ⋅E_{01}⎟              ⎜E_{02}⋅ℯ              ℯ          ⋅E_{02}⎟             \n",
       "────────⎟ + e⋅r_{0;32}⋅⎜─────────────────── + ──────────────────⎟             \n",
       "        ⎠              ⎝         2                    2         ⎠             \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                           0                                  \n",
       "                                                                              \n",
       "                                                                              \n",
       "⎛        -ⅈ⋅t⋅varpi₁    ⅈ⋅t⋅varpi₁ ______⎞              ⎛        -ⅈ⋅t⋅varpi₂  \n",
       "⎜E_{01}⋅ℯ              ℯ          ⋅E_{01}⎟              ⎜E_{02}⋅ℯ             \n",
       "⎜─────────────────── + ──────────────────⎟ + e⋅r_{0;32}⋅⎜─────────────────── +\n",
       "⎝         2                    2         ⎠              ⎝         2           \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                           0                                  \n",
       "                                                                              \n",
       "\n",
       "                    ⎤\n",
       "                    ⎥\n",
       "                    ⎥\n",
       "                    ⎥\n",
       "                    ⎥\n",
       "  ⅈ⋅t⋅varpi₂ ______⎞⎥\n",
       " ℯ          ⋅E_{02}⎟⎥\n",
       " ──────────────────⎟⎥\n",
       "         2         ⎠⎥\n",
       "                    ⎥\n",
       "                    ⎥\n",
       "                    ⎥\n",
       "                    ⎥\n",
       "                    ⎦"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zero_matrix=zeros(Ne,Ne)\n",
    "H1=sum([ e*helicity_dot_product(E[l],r) for l in range(Nl)],zero_matrix)\n",
    "fprint(H1,print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the complete hamiltonian is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "H=H0+H1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Rotating wave approximation\n",
    "Notice that the electric field can be separated by terms with positive and negative frequency:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\left [ \\left [ \\left[\\begin{matrix}0\\\\\\frac{E_{01}}{2} e^{- i t \\varpi_{1}}\\\\0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\\\frac{E_{02}}{2} e^{- i t \\varpi_{2}}\\\\0\\end{matrix}\\right]\\right ], \\quad \\left [ \\left[\\begin{matrix}0\\\\\\frac{\\overline{E_{01}}}{2} e^{i t \\varpi_{1}}\\\\0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\\\frac{\\overline{E_{02}}}{2} e^{i t \\varpi_{2}}\\\\0\\end{matrix}\\right]\\right ]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡⎡         0         ⎤, ⎡         0         ⎤⎤, ⎡⎡        0         ⎤, ⎡     \n",
       "⎢⎢⎢                   ⎥  ⎢                   ⎥⎥  ⎢⎢                  ⎥  ⎢     \n",
       "⎢⎢⎢        -ⅈ⋅t⋅varpi₁⎥  ⎢        -ⅈ⋅t⋅varpi₂⎥⎥  ⎢⎢ ⅈ⋅t⋅varpi₁ ______⎥  ⎢ ⅈ⋅t⋅\n",
       "⎢⎢⎢E_{01}⋅ℯ           ⎥  ⎢E_{02}⋅ℯ           ⎥⎥  ⎢⎢ℯ          ⋅E_{01}⎥  ⎢ℯ    \n",
       "⎢⎢⎢───────────────────⎥  ⎢───────────────────⎥⎥  ⎢⎢──────────────────⎥  ⎢─────\n",
       "⎢⎢⎢         2         ⎥  ⎢         2         ⎥⎥  ⎢⎢        2         ⎥  ⎢     \n",
       "⎢⎢⎢                   ⎥  ⎢                   ⎥⎥  ⎢⎢                  ⎥  ⎢     \n",
       "⎣⎣⎣         0         ⎦  ⎣         0         ⎦⎦  ⎣⎣        0         ⎦  ⎣     \n",
       "\n",
       "   0         ⎤⎤⎤\n",
       "             ⎥⎥⎥\n",
       "varpi₂ ______⎥⎥⎥\n",
       "      ⋅E_{02}⎥⎥⎥\n",
       "─────────────⎥⎥⎥\n",
       "   2         ⎥⎥⎥\n",
       "             ⎥⎥⎥\n",
       "   0         ⎦⎦⎦"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E_cartesian_p=[E0[l]/2*ep[l]*exp(-I*omega_laser[l]*t) for l in range(Nl)]\n",
    "E_cartesian_m=[E0[l].conjugate()/2*em[l]*exp( I*omega_laser[l]*t) for l in range(Nl)]\n",
    "\n",
    "E_p=[cartesian_to_helicity(E_cartesian_p[l]) for l in range(Nl)]\n",
    "E_m=[cartesian_to_helicity(E_cartesian_m[l]) for l in range(Nl)]\n",
    "\n",
    "fprint([E_p,E_m], print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAABLCAMAAABZRmeuAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRIlmzd0i77ts7uXj/QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAMtJREFUOBHtVcsS\nwiAMDOVhBVqK+f9/lRClZETPztBcSndZCLAQUFhjgRY7IwAKtSlhGwOR/hckSp3o2YqCcjrE8O4n\nqa0M6vbESkHFg8CwDahck7ToKidUWCmPnGtPOdTU3aP5UCUMhCn+QK9KrBpRPwYETsMO0oCcaa44\nSp6XrEdLhp026nhtYp9hgUM5i/H20lQtpKrB1Jiemsait/XOJy8t+li7u3xZVDxff3JTprFoqynS\nouKNuixaLfq1ZDsq0Mb407u1ZBsDT+aoGdQmGjgBAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0⎤\n",
       "⎢ ⎥\n",
       "⎢0⎥\n",
       "⎢ ⎥\n",
       "⎣0⎦"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint( simplify(sum([E[l] for l in range(Nl)],zero_vect)-(sum([E_p[l]+E_m[l] for l in range(Nl)],zero_vect) )), print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The position operator can also be separated in this way. We go to the interaction picture (with $\\hat{H}_0$ as the undisturbed hamiltonian)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAaMAAABMCAMAAAD3LiglAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & - r_{+1;21} e^{- i \\omega_{21} t} & 0\\\\r_{-1;21} e^{i \\omega_{21} t} & 0 & - r_{+1;32} e^{- i \\omega_{32} t}\\\\0 & r_{-1;32} e^{i \\omega_{32} t} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                -ⅈ⋅ω₂₁⋅t                      ⎤\n",
       "⎢        0           -r_{+1;21}⋅ℯ                   0          ⎥\n",
       "⎢                                                              ⎥\n",
       "⎢           ⅈ⋅ω₂₁⋅t                                    -ⅈ⋅ω₃₂⋅t⎥\n",
       "⎢r_{-1;21}⋅ℯ                  0            -r_{+1;32}⋅ℯ        ⎥\n",
       "⎢                                                              ⎥\n",
       "⎢                                ⅈ⋅ω₃₂⋅t                       ⎥\n",
       "⎣        0            r_{-1;32}⋅ℯ                   0          ⎦"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_I=[ Matrix([[exp(I*omega[i,j]*t)*r[p][i,j] for j in range(Ne)] for i in range(Ne)]) for p in range(3)]\n",
    "fprint(r_I[0], print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAVwAAABMCAMAAAAWaf8xAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nMD/CISX7HwnQV5RavZCIAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & r_{0;21} e^{- i \\omega_{21} t} & 0\\\\r_{0;21} e^{i \\omega_{21} t} & 0 & r_{0;32} e^{- i \\omega_{32} t}\\\\0 & r_{0;32} e^{i \\omega_{32} t} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                             -ⅈ⋅ω₂₁⋅t                    ⎤\n",
       "⎢        0          r_{0;21}⋅ℯ                  0         ⎥\n",
       "⎢                                                         ⎥\n",
       "⎢          ⅈ⋅ω₂₁⋅t                                -ⅈ⋅ω₃₂⋅t⎥\n",
       "⎢r_{0;21}⋅ℯ                 0           r_{0;32}⋅ℯ        ⎥\n",
       "⎢                                                         ⎥\n",
       "⎢                             ⅈ⋅ω₃₂⋅t                     ⎥\n",
       "⎣        0          r_{0;32}⋅ℯ                  0         ⎦"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint(r_I[1], print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAaMAAABMCAMAAAD3LiglAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & - r_{-1;21} e^{- i \\omega_{21} t} & 0\\\\r_{+1;21} e^{i \\omega_{21} t} & 0 & - r_{-1;32} e^{- i \\omega_{32} t}\\\\0 & r_{+1;32} e^{i \\omega_{32} t} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                -ⅈ⋅ω₂₁⋅t                      ⎤\n",
       "⎢        0           -r_{-1;21}⋅ℯ                   0          ⎥\n",
       "⎢                                                              ⎥\n",
       "⎢           ⅈ⋅ω₂₁⋅t                                    -ⅈ⋅ω₃₂⋅t⎥\n",
       "⎢r_{+1;21}⋅ℯ                  0            -r_{-1;32}⋅ℯ        ⎥\n",
       "⎢                                                              ⎥\n",
       "⎢                                ⅈ⋅ω₃₂⋅t                       ⎥\n",
       "⎣        0            r_{+1;32}⋅ℯ                   0          ⎦"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint(r_I[2], print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which can be decomposed in positive and negative frequencies as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUgAAABMCAMAAAA4LN6jAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nS/2D5EIEP2F4b8fv5p35QXch2L8hlUQaL9EkxgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & - r_{+1;21} e^{- i \\omega_{21} t} & 0\\\\0 & 0 & - r_{+1;32} e^{- i \\omega_{32} t}\\\\0 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡               -ⅈ⋅ω₂₁⋅t                      ⎤\n",
       "⎢0  -r_{+1;21}⋅ℯ                   0          ⎥\n",
       "⎢                                             ⎥\n",
       "⎢                                     -ⅈ⋅ω₃₂⋅t⎥\n",
       "⎢0           0            -r_{+1;32}⋅ℯ        ⎥\n",
       "⎢                                             ⎥\n",
       "⎣0           0                     0          ⎦"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_I_p=[ Matrix([[ delta_greater(j,i)*exp(-I*omega[j,i]*t)*r[p][i,j] for j in range(Ne)]for i in range(Ne)]) for p in range(3)]\n",
    "fprint(r_I_p[0], print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQ4AAABMCAMAAACriKlEAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nGjWNXyCmlPsyirfr5k91yH6NmlLiPz5cQi4ncbvRAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & r_{0;21} e^{- i \\omega_{21} t} & 0\\\\0 & 0 & r_{0;32} e^{- i \\omega_{32} t}\\\\0 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡             -ⅈ⋅ω₂₁⋅t                    ⎤\n",
       "⎢0  r_{0;21}⋅ℯ                  0         ⎥\n",
       "⎢                                         ⎥\n",
       "⎢                                 -ⅈ⋅ω₃₂⋅t⎥\n",
       "⎢0          0           r_{0;32}⋅ℯ        ⎥\n",
       "⎢                                         ⎥\n",
       "⎣0          0                   0         ⎦"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint(r_I_p[1], print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUkAAABMCAMAAADX7rWdAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nZWAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & - r_{-1;21} e^{- i \\omega_{21} t} & 0\\\\0 & 0 & - r_{-1;32} e^{- i \\omega_{32} t}\\\\0 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡               -ⅈ⋅ω₂₁⋅t                      ⎤\n",
       "⎢0  -r_{-1;21}⋅ℯ                   0          ⎥\n",
       "⎢                                             ⎥\n",
       "⎢                                     -ⅈ⋅ω₃₂⋅t⎥\n",
       "⎢0           0            -r_{-1;32}⋅ℯ        ⎥\n",
       "⎢                                             ⎥\n",
       "⎣0           0                     0          ⎦"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint(r_I_p[2], print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQ4AAABLCAMAAAC2jZn8AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 0\\\\r_{-1;21} e^{i \\omega_{21} t} & 0 & 0\\\\0 & r_{-1;32} e^{i \\omega_{32} t} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡        0                   0           0⎤\n",
       "⎢                                         ⎥\n",
       "⎢           ⅈ⋅ω₂₁⋅t                       ⎥\n",
       "⎢r_{-1;21}⋅ℯ                 0           0⎥\n",
       "⎢                                         ⎥\n",
       "⎢                               ⅈ⋅ω₃₂⋅t   ⎥\n",
       "⎣        0           r_{-1;32}⋅ℯ         0⎦"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_I_m=[ Matrix([[ delta_lesser( j,i)*exp( I*omega[i,j]*t)*r[p][i,j] for j in range(Ne)]for i in range(Ne)]) for p in range(3)]\n",
    "fprint(r_I_m[0],print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPQAAABLCAMAAABwbVpsAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\ndFmfeoZh88ddpGR/APuMQOYZf5DMAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 0\\\\r_{0;21} e^{i \\omega_{21} t} & 0 & 0\\\\0 & r_{0;32} e^{i \\omega_{32} t} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡        0                  0          0⎤\n",
       "⎢                                       ⎥\n",
       "⎢          ⅈ⋅ω₂₁⋅t                      ⎥\n",
       "⎢r_{0;21}⋅ℯ                 0          0⎥\n",
       "⎢                                       ⎥\n",
       "⎢                             ⅈ⋅ω₃₂⋅t   ⎥\n",
       "⎣        0          r_{0;32}⋅ℯ         0⎦"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint(r_I_m[1], print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQ4AAABLCAMAAAC2jZn8AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n0/6iDfMzakKw/wBL+EF4zJ7GBQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 0\\\\r_{+1;21} e^{i \\omega_{21} t} & 0 & 0\\\\0 & r_{+1;32} e^{i \\omega_{32} t} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡        0                   0           0⎤\n",
       "⎢                                         ⎥\n",
       "⎢           ⅈ⋅ω₂₁⋅t                       ⎥\n",
       "⎢r_{+1;21}⋅ℯ                 0           0⎥\n",
       "⎢                                         ⎥\n",
       "⎢                               ⅈ⋅ω₃₂⋅t   ⎥\n",
       "⎣        0           r_{+1;32}⋅ℯ         0⎦"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint(r_I_m[2], print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "that summed equal $\\vec{\\hat{r}}_I$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYAAAABLCAMAAABOQaBOAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n5JZyIZn5CxenlHXZOsAdAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡0  0  0⎤, ⎡0  0  0⎤, ⎡0  0  0⎤⎤\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎢⎢0  0  0⎥  ⎢0  0  0⎥  ⎢0  0  0⎥⎥\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎣⎣0  0  0⎦  ⎣0  0  0⎦  ⎣0  0  0⎦⎦"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint( [r_I[p]-(r_I_p[p]+r_I_m[p]) for p in range(3)] , print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus the interaction hamiltonian in the interaciton picture is\n",
    "\\begin{equation}\n",
    "    \\hat{H}_{1I}=e\\vec{E}\\cdot \\vec{\\hat{r}}_I= e(\\vec{E}^{(+)}\\cdot \\vec{\\hat{r}}^{(+)}_I + \\vec{E}^{(+)}\\cdot \\vec{\\hat{r}}^{(-)}_I + \\vec{E}^{(-)}\\cdot \\vec{\\hat{r}}^{(+)}_I + \\vec{E}^{(-)}\\cdot \\vec{\\hat{r}}^{(-)}_I)\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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xs1CbbXfmKA7sZhtVaI8RB7ANoK1KsfWUmg4RgQMJUBhxIGySOieB5ontekEA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\nT+Auoofq/1j7AEXTexxvAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & e r_{0;21} \\left(\\frac{E_{01}}{2} e^{- i t \\varpi_{1}} + \\frac{\\overline{E_{01}}}{2} e^{i t \\varpi_{1}}\\right) e^{- i \\omega_{21} t} + e r_{0;21} \\left(\\frac{E_{02}}{2} e^{- i t \\varpi_{2}} + \\frac{\\overline{E_{02}}}{2} e^{i t \\varpi_{2}}\\right) e^{- i \\omega_{21} t} & 0\\\\e r_{0;21} \\left(\\frac{E_{01}}{2} e^{- i t \\varpi_{1}} + \\frac{\\overline{E_{01}}}{2} e^{i t \\varpi_{1}}\\right) e^{i \\omega_{21} t} + e r_{0;21} \\left(\\frac{E_{02}}{2} e^{- i t \\varpi_{2}} + \\frac{\\overline{E_{02}}}{2} e^{i t \\varpi_{2}}\\right) e^{i \\omega_{21} t} & 0 & e r_{0;32} \\left(\\frac{E_{01}}{2} e^{- i t \\varpi_{1}} + \\frac{\\overline{E_{01}}}{2} e^{i t \\varpi_{1}}\\right) e^{- i \\omega_{32} t} + e r_{0;32} \\left(\\frac{E_{02}}{2} e^{- i t \\varpi_{2}} + \\frac{\\overline{E_{02}}}{2} e^{i t \\varpi_{2}}\\right) e^{- i \\omega_{32} t}\\\\0 & e r_{0;32} \\left(\\frac{E_{01}}{2} e^{- i t \\varpi_{1}} + \\frac{\\overline{E_{01}}}{2} e^{i t \\varpi_{1}}\\right) e^{i \\omega_{32} t} + e r_{0;32} \\left(\\frac{E_{02}}{2} e^{- i t \\varpi_{2}} + \\frac{\\overline{E_{02}}}{2} e^{i t \\varpi_{2}}\\right) e^{i \\omega_{32} t} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                               0             \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢           ⎛        -ⅈ⋅t⋅varpi₁    ⅈ⋅t⋅varpi₁ ______⎞                       ⎛\n",
       "⎢           ⎜E_{01}⋅ℯ              ℯ          ⋅E_{01}⎟  ⅈ⋅ω₂₁⋅t              ⎜\n",
       "⎢e⋅r_{0;21}⋅⎜─────────────────── + ──────────────────⎟⋅ℯ        + e⋅r_{0;21}⋅⎜\n",
       "⎢           ⎝         2                    2         ⎠                       ⎝\n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                               0             \n",
       "⎣                                                                             \n",
       "\n",
       "                                                               ⎛        -ⅈ⋅t⋅v\n",
       "                                                               ⎜E_{01}⋅ℯ      \n",
       "                                                    e⋅r_{0;21}⋅⎜──────────────\n",
       "                                                               ⎝         2    \n",
       "                                                                              \n",
       "        -ⅈ⋅t⋅varpi₂    ⅈ⋅t⋅varpi₂ ______⎞                                     \n",
       "E_{02}⋅ℯ              ℯ          ⋅E_{02}⎟  ⅈ⋅ω₂₁⋅t                            \n",
       "─────────────────── + ──────────────────⎟⋅ℯ                                   \n",
       "         2                    2         ⎠                                     \n",
       "                                                                              \n",
       "                                                                ⎛        -ⅈ⋅t⋅\n",
       "                                                                ⎜E_{01}⋅ℯ     \n",
       "                                                     e⋅r_{0;32}⋅⎜─────────────\n",
       "                                                                ⎝         2   \n",
       "\n",
       "arpi₁    ⅈ⋅t⋅varpi₁ ______⎞                        ⎛        -ⅈ⋅t⋅varpi₂    ⅈ⋅t\n",
       "        ℯ          ⋅E_{01}⎟  -ⅈ⋅ω₂₁⋅t              ⎜E_{02}⋅ℯ              ℯ   \n",
       "───── + ──────────────────⎟⋅ℯ         + e⋅r_{0;21}⋅⎜─────────────────── + ────\n",
       "                2         ⎠                        ⎝         2                \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                      0                                       \n",
       "                                                                              \n",
       "                                                                              \n",
       "varpi₁    ⅈ⋅t⋅varpi₁ ______⎞                       ⎛        -ⅈ⋅t⋅varpi₂    ⅈ⋅t\n",
       "         ℯ          ⋅E_{01}⎟  ⅈ⋅ω₃₂⋅t              ⎜E_{02}⋅ℯ              ℯ   \n",
       "────── + ──────────────────⎟⋅ℯ        + e⋅r_{0;32}⋅⎜─────────────────── + ────\n",
       "                 2         ⎠                       ⎝         2                \n",
       "\n",
       "⋅varpi₂ ______⎞                                                               \n",
       "       ⋅E_{02}⎟  -ⅈ⋅ω₂₁⋅t                                                     \n",
       "──────────────⎟⋅ℯ                                                             \n",
       "    2         ⎠                                                               \n",
       "                                                                              \n",
       "                                      ⎛        -ⅈ⋅t⋅varpi₁    ⅈ⋅t⋅varpi₁ _____\n",
       "                                      ⎜E_{01}⋅ℯ              ℯ          ⋅E_{01\n",
       "                           e⋅r_{0;32}⋅⎜─────────────────── + ─────────────────\n",
       "                                      ⎝         2                    2        \n",
       "                                                                              \n",
       "⋅varpi₂ ______⎞                                                               \n",
       "       ⋅E_{02}⎟  ⅈ⋅ω₃₂⋅t                                                      \n",
       "──────────────⎟⋅ℯ                                                             \n",
       "    2         ⎠                                                               \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "             0                                                                \n",
       "                                                                              \n",
       "                                                                              \n",
       "_⎞                        ⎛        -ⅈ⋅t⋅varpi₂    ⅈ⋅t⋅varpi₂ ______⎞          \n",
       "}⎟  -ⅈ⋅ω₃₂⋅t              ⎜E_{02}⋅ℯ              ℯ          ⋅E_{02}⎟  -ⅈ⋅ω₃₂⋅t\n",
       "─⎟⋅ℯ         + e⋅r_{0;32}⋅⎜─────────────────── + ──────────────────⎟⋅ℯ        \n",
       " ⎠                        ⎝         2                    2         ⎠          \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "             0                                                                \n",
       "                                                                              \n",
       "\n",
       "⎤\n",
       "⎥\n",
       "⎥\n",
       "⎥\n",
       "⎥\n",
       "⎥\n",
       "⎥\n",
       "⎥\n",
       "⎥\n",
       "⎥\n",
       "⎥\n",
       "⎥\n",
       "⎥\n",
       "⎦"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H1I=sum([ e*helicity_dot_product(E[l],r_I) for l in range(Nl)],zero_matrix)\n",
    "fprint(H1I,print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since both $\\omega^l$ and $\\omega_{ij}$ are in the order of THz, the terms that have frequencies with the same sign are summed, and thus also of the order of THz. The frequencies in the terms with oposite signs however, are detunings of the order of MHz. Since we are only interested in the coarse-grained evolution of the density matrix, we may omit the fast terms and approximate\n",
    "\n",
    "\\begin{equation}\n",
    "    \\hat{H}_{1I} \\simeq \\hat{H}_{1I,RWA}= e( \\vec{E}^{(+)}\\cdot \\vec{\\hat{r}}^{(-)}_I + \\vec{E}^{(-)}\\cdot \\vec{\\hat{r}}^{(+)}_I )\n",
    "\\end{equation}\n",
    "\n",
    "That is known as the rotating wave approximation (RWA)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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c4pDkmn16llmPSyYURxazUYtZjQTKdzdINmpxZ0hRj/s458rj1g2hWGPsovFs1GJmV+QD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\nBvwfTFetpRX3VnMAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & \\frac{e r_{0;21}}{2} e^{- i \\omega_{21} t} e^{i t \\varpi_{1}} \\overline{E_{01}} + \\frac{e r_{0;21}}{2} e^{- i \\omega_{21} t} e^{i t \\varpi_{2}} \\overline{E_{02}} & 0\\\\\\frac{e r_{0;21}}{2} E_{01} e^{i \\omega_{21} t} e^{- i t \\varpi_{1}} + \\frac{e r_{0;21}}{2} E_{02} e^{i \\omega_{21} t} e^{- i t \\varpi_{2}} & 0 & \\frac{e r_{0;32}}{2} e^{- i \\omega_{32} t} e^{i t \\varpi_{1}} \\overline{E_{01}} + \\frac{e r_{0;32}}{2} e^{- i \\omega_{32} t} e^{i t \\varpi_{2}} \\overline{E_{02}}\\\\0 & \\frac{e r_{0;32}}{2} E_{01} e^{i \\omega_{32} t} e^{- i t \\varpi_{1}} + \\frac{e r_{0;32}}{2} E_{02} e^{i \\omega_{32} t} e^{- i t \\varpi_{2}} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                        0                                    \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                   ⅈ⋅ω₂₁⋅t  -ⅈ⋅t⋅varpi₁                      ⅈ⋅ω₂₁⋅t  -ⅈ⋅t⋅va\n",
       "⎢E_{01}⋅e⋅r_{0;21}⋅ℯ       ⋅ℯ              E_{02}⋅e⋅r_{0;21}⋅ℯ       ⋅ℯ       \n",
       "⎢─────────────────────────────────────── + ───────────────────────────────────\n",
       "⎢                   2                                         2               \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                        0                                    \n",
       "⎣                                                                             \n",
       "\n",
       "                  -ⅈ⋅ω₂₁⋅t  ⅈ⋅t⋅varpi₁ ______               -ⅈ⋅ω₂₁⋅t  ⅈ⋅t⋅varp\n",
       "      e⋅r_{0;21}⋅ℯ        ⋅ℯ          ⋅E_{01}   e⋅r_{0;21}⋅ℯ        ⋅ℯ        \n",
       "      ─────────────────────────────────────── + ──────────────────────────────\n",
       "                         2                                         2          \n",
       "                                                                              \n",
       "rpi₂                                                                          \n",
       "                                                                              \n",
       "────                                          0                               \n",
       "                                                                              \n",
       "                                                                              \n",
       "                         ⅈ⋅ω₃₂⋅t  -ⅈ⋅t⋅varpi₁                      ⅈ⋅ω₃₂⋅t  -ⅈ\n",
       "      E_{01}⋅e⋅r_{0;32}⋅ℯ       ⋅ℯ              E_{02}⋅e⋅r_{0;32}⋅ℯ       ⋅ℯ  \n",
       "      ─────────────────────────────────────── + ──────────────────────────────\n",
       "                         2                                         2          \n",
       "\n",
       "i₂ ______                                                                     \n",
       "  ⋅E_{02}                                                                     \n",
       "─────────                                          0                          \n",
       "                                                                              \n",
       "                                                                              \n",
       "                       -ⅈ⋅ω₃₂⋅t  ⅈ⋅t⋅varpi₁ ______               -ⅈ⋅ω₃₂⋅t  ⅈ⋅t\n",
       "           e⋅r_{0;32}⋅ℯ        ⋅ℯ          ⋅E_{01}   e⋅r_{0;32}⋅ℯ        ⋅ℯ   \n",
       "           ─────────────────────────────────────── + ─────────────────────────\n",
       "                              2                                         2     \n",
       "                                                                              \n",
       "⋅t⋅varpi₂                                                                     \n",
       "                                                                              \n",
       "─────────                                          0                          \n",
       "                                                                              \n",
       "\n",
       "              ⎤\n",
       "              ⎥\n",
       "              ⎥\n",
       "              ⎥\n",
       "              ⎥\n",
       "⋅varpi₂ ______⎥\n",
       "       ⋅E_{02}⎥\n",
       "──────────────⎥\n",
       "              ⎥\n",
       "              ⎥\n",
       "              ⎥\n",
       "              ⎥\n",
       "              ⎥\n",
       "              ⎦"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H1IRWA=sum( [ (e*(helicity_dot_product(E_p[l],r_I_m)+helicity_dot_product(E_m[l],r_I_p))) for l in range(Nl)],zero_matrix)\n",
    "fprint(H1IRWA,print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The matrix element $(\\hat{H}_{1I,RWA})_{21}$ element is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZgAAAAlBAMAAACNNuJ5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n5ZqT+R9RWs3AWP5m/wAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{e r_{0;21}}{2} E_{01} e^{i \\omega_{21} t} e^{- i t \\varpi_{1}} + \\frac{e r_{0;21}}{2} E_{02} e^{i \\omega_{21} t} e^{- i t \\varpi_{2}}$$"
      ],
      "text/plain": [
       "                   ⅈ⋅ω₂₁⋅t  -ⅈ⋅t⋅varpi₁                      ⅈ⋅ω₂₁⋅t  -ⅈ⋅t⋅var\n",
       "E_{01}⋅e⋅r_{0;21}⋅ℯ       ⋅ℯ              E_{02}⋅e⋅r_{0;21}⋅ℯ       ⋅ℯ        \n",
       "─────────────────────────────────────── + ────────────────────────────────────\n",
       "                   2                                         2                \n",
       "\n",
       "pi₂\n",
       "   \n",
       "───\n",
       "   "
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint(H1IRWA[1,0].expand(),print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But if the detuning $\\omega_{21}-\\omega^1 \\ll \\omega_{21}-\\omega^2$ (the second field is far detuned from the $1 \\rightarrow 2$ transition), then $\\omega_{21}-\\omega^2$ may be also considered too high a frequency to be relevant to coarse-grained evolution. So we might neclect that term in $(\\hat{H}_{1I,RWA})_{21}$ and similarly neglect the $\\omega_{32}-\\omega^1$ for term in $(\\hat{H}_{1I,RWA})_{32}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZgAAAAlBAMAAACNNuJ5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nUM1rHf1Wrngx/wFd89gLLewSzgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{e r_{0;32}}{2} E_{01} e^{i \\omega_{32} t} e^{- i t \\varpi_{1}} + \\frac{e r_{0;32}}{2} E_{02} e^{i \\omega_{32} t} e^{- i t \\varpi_{2}}$$"
      ],
      "text/plain": [
       "                   ⅈ⋅ω₃₂⋅t  -ⅈ⋅t⋅varpi₁                      ⅈ⋅ω₃₂⋅t  -ⅈ⋅t⋅var\n",
       "E_{01}⋅e⋅r_{0;32}⋅ℯ       ⋅ℯ              E_{02}⋅e⋅r_{0;32}⋅ℯ       ⋅ℯ        \n",
       "─────────────────────────────────────── + ────────────────────────────────────\n",
       "                   2                                         2                \n",
       "\n",
       "pi₂\n",
       "   \n",
       "───\n",
       "   "
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint(H1IRWA[2,1].expand(),print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In other words, if the detunings in our experiments allow the approximmation, we might choose which frequency components $\\omega^l$ excite which transitions. Let us say that $L_{ij}$ is the set of $l$ such that $\\omega^l$ excites the transition $i\\rightarrow j$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[[] [1] []]\n",
      " [[1] [] [2]]\n",
      " [[] [2] []]]\n"
     ]
    }
   ],
   "source": [
    "Lij=[[1,2,[1]],[2,3,[2]]]\n",
    "Lij=formatLij(Lij,Ne)\n",
    "print array(Lij)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus the interacion hamiltonian in the interaction picture can be approximated as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAmgAAABNBAMAAADgP4XWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDVyOvVQul/4P+PFISFBrO3UAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & \\frac{e r_{0;21}}{2} e^{- i \\omega_{21} t} e^{i t \\varpi_{1}} \\overline{E_{01}} & 0\\\\\\frac{e r_{0;21}}{2} E_{01} e^{i \\omega_{21} t} e^{- i t \\varpi_{1}} & 0 & \\frac{e r_{0;32}}{2} e^{- i \\omega_{32} t} e^{i t \\varpi_{2}} \\overline{E_{02}}\\\\0 & \\frac{e r_{0;32}}{2} E_{02} e^{i \\omega_{32} t} e^{- i t \\varpi_{2}} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                                     -ⅈ⋅ω₂₁⋅t  ⅈ⋅t⋅varpi₁ ___\n",
       "⎢                                         e⋅r_{0;21}⋅ℯ        ⋅ℯ          ⋅E_{\n",
       "⎢                   0                     ────────────────────────────────────\n",
       "⎢                                                            2                \n",
       "⎢                                                                             \n",
       "⎢                   ⅈ⋅ω₂₁⋅t  -ⅈ⋅t⋅varpi₁                                      \n",
       "⎢E_{01}⋅e⋅r_{0;21}⋅ℯ       ⋅ℯ                                                 \n",
       "⎢───────────────────────────────────────                     0                \n",
       "⎢                   2                                                         \n",
       "⎢                                                                             \n",
       "⎢                                                            ⅈ⋅ω₃₂⋅t  -ⅈ⋅t⋅var\n",
       "⎢                                         E_{02}⋅e⋅r_{0;32}⋅ℯ       ⋅ℯ        \n",
       "⎢                   0                     ────────────────────────────────────\n",
       "⎣                                                            2                \n",
       "\n",
       "___                                         ⎤\n",
       "01}                                         ⎥\n",
       "───                     0                   ⎥\n",
       "                                            ⎥\n",
       "                                            ⎥\n",
       "                 -ⅈ⋅ω₃₂⋅t  ⅈ⋅t⋅varpi₂ ______⎥\n",
       "     e⋅r_{0;32}⋅ℯ        ⋅ℯ          ⋅E_{02}⎥\n",
       "     ───────────────────────────────────────⎥\n",
       "                        2                   ⎥\n",
       "                                            ⎥\n",
       "pi₂                                         ⎥\n",
       "                                            ⎥\n",
       "───                     0                   ⎥\n",
       "                                            ⎦"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H1IRWA =sum([ e*( helicity_dot_product( E_p[l],vector_element(r_I_m,i,j)) ) * ket(i+1,Ne)*bra(j+1,Ne) \n",
    "            for l in range(Nl) for j in range(Ne) for i in range(Ne) if l+1 in Lij[i][j] ],zero_matrix)\n",
    "H1IRWA+=sum([ e*( helicity_dot_product( E_m[l],vector_element(r_I_p,i,j)) ) * ket(i+1,Ne)*bra(j+1,Ne) \n",
    "            for l in range(Nl) for j in range(Ne) for i in range(Ne) if l+1 in Lij[i][j] ],zero_matrix)\n",
    "\n",
    "fprint(H1IRWA, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Returning to the Schrödinger picture we have."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAqUAAABMBAMAAACmItNiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ \\left[\\begin{matrix}0 & - r_{+1;21} & 0\\\\0 & 0 & - r_{+1;32}\\\\0 & 0 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & r_{0;21} & 0\\\\0 & 0 & r_{0;32}\\\\0 & 0 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & - r_{-1;21} & 0\\\\0 & 0 & - r_{-1;32}\\\\0 & 0 & 0\\end{matrix}\\right]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡0  -r_{+1;21}      0     ⎤, ⎡0  r_{0;21}     0    ⎤, ⎡0  -r_{-1;21}      0  \n",
       "⎢⎢                         ⎥  ⎢                     ⎥  ⎢                      \n",
       "⎢⎢0      0       -r_{+1;32}⎥  ⎢0     0      r_{0;32}⎥  ⎢0      0       -r_{-1;\n",
       "⎢⎢                         ⎥  ⎢                     ⎥  ⎢                      \n",
       "⎣⎣0      0           0     ⎦  ⎣0     0         0    ⎦  ⎣0      0           0  \n",
       "\n",
       "   ⎤⎤\n",
       "   ⎥⎥\n",
       "32}⎥⎥\n",
       "   ⎥⎥\n",
       "   ⎦⎦"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_p=[ Matrix([[ delta_greater(j,i)*r[p][i,j] for j in range(Ne)]for i in range(Ne)]) for p in range(3)]\n",
    "fprint(r_p, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAmQAAABMBAMAAAAxX7b9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left [ \\left[\\begin{matrix}0 & 0 & 0\\\\r_{-1;21} & 0 & 0\\\\0 & r_{-1;32} & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & 0\\\\r_{0;21} & 0 & 0\\\\0 & r_{0;32} & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & 0\\\\r_{+1;21} & 0 & 0\\\\0 & r_{+1;32} & 0\\end{matrix}\\right]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡    0          0      0⎤, ⎡   0         0      0⎤, ⎡    0          0      0⎤\n",
       "⎢⎢                       ⎥  ⎢                     ⎥  ⎢                       ⎥\n",
       "⎢⎢r_{-1;21}      0      0⎥  ⎢r_{0;21}     0      0⎥  ⎢r_{+1;21}      0      0⎥\n",
       "⎢⎢                       ⎥  ⎢                     ⎥  ⎢                       ⎥\n",
       "⎣⎣    0      r_{-1;32}  0⎦  ⎣   0      r_{0;32}  0⎦  ⎣    0      r_{+1;32}  0⎦\n",
       "\n",
       "⎤\n",
       "⎥\n",
       "⎥\n",
       "⎥\n",
       "⎦"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_m=[ Matrix([[ delta_lesser( j,i)*r[p][i,j] for j in range(Ne)]for i in range(Ne)]) for p in range(3)]\n",
    "fprint(r_m, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYAAAABLCAMAAABOQaBOAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n5JZyIZn5CxenlHXZOsAdAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡0  0  0⎤, ⎡0  0  0⎤, ⎡0  0  0⎤⎤\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎢⎢0  0  0⎥  ⎢0  0  0⎥  ⎢0  0  0⎥⎥\n",
       "⎢⎢       ⎥  ⎢       ⎥  ⎢       ⎥⎥\n",
       "⎣⎣0  0  0⎦  ⎣0  0  0⎦  ⎣0  0  0⎦⎦"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint( [r[p]-(r_p[p]+r_m[p]) for p in range(3)] , print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus the interaction hamiltonian in the Schrödinger picture in the rotating wave approximation is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAckAAABMCAMAAAAx67ycAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & \\frac{e r_{0;21}}{2} e^{i t \\varpi_{1}} \\overline{E_{01}} & 0\\\\\\frac{e r_{0;21}}{2} E_{01} e^{- i t \\varpi_{1}} & 0 & \\frac{e r_{0;32}}{2} e^{i t \\varpi_{2}} \\overline{E_{02}}\\\\0 & \\frac{e r_{0;32}}{2} E_{02} e^{- i t \\varpi_{2}} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                            ⅈ⋅t⋅varpi₁ ______                \n",
       "⎢                                e⋅r_{0;21}⋅ℯ          ⋅E_{01}                \n",
       "⎢              0                 ─────────────────────────────                \n",
       "⎢                                              2                              \n",
       "⎢                                                                             \n",
       "⎢                   -ⅈ⋅t⋅varpi₁                                              ⅈ\n",
       "⎢E_{01}⋅e⋅r_{0;21}⋅ℯ                                             e⋅r_{0;32}⋅ℯ \n",
       "⎢──────────────────────────────                0                 ─────────────\n",
       "⎢              2                                                              \n",
       "⎢                                                                             \n",
       "⎢                                                   -ⅈ⋅t⋅varpi₂               \n",
       "⎢                                E_{02}⋅e⋅r_{0;32}⋅ℯ                          \n",
       "⎢              0                 ──────────────────────────────               \n",
       "⎣                                              2                              \n",
       "\n",
       "                ⎤\n",
       "                ⎥\n",
       " 0              ⎥\n",
       "                ⎥\n",
       "                ⎥\n",
       "⋅t⋅varpi₂ ______⎥\n",
       "         ⋅E_{02}⎥\n",
       "────────────────⎥\n",
       " 2              ⎥\n",
       "                ⎥\n",
       "                ⎥\n",
       "                ⎥\n",
       " 0              ⎥\n",
       "                ⎦"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H1RWA =sum([ e*( helicity_dot_product( E_p[l],vector_element(r_m,i,j)) ) * ket(i+1,Ne)*bra(j+1,Ne) \n",
    "            for l in range(Nl) for j in range(Ne) for i in range(Ne) if l+1 in Lij[i][j] ],zero_matrix)\n",
    "H1RWA+=sum([ e*( helicity_dot_product( E_m[l],vector_element(r_p,i,j)) ) * ket(i+1,Ne)*bra(j+1,Ne) \n",
    "            for l in range(Nl) for j in range(Ne) for i in range(Ne) if l+1 in Lij[i][j] ],zero_matrix)\n",
    "\n",
    "fprint(H1RWA, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And the complete hamiltonian in the Schrödinger picture in the rotating wave approximation is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAckAAABMCAMAAAAx67ycAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\hbar \\omega_{1} & \\frac{e r_{0;21}}{2} e^{i t \\varpi_{1}} \\overline{E_{01}} & 0\\\\\\frac{e r_{0;21}}{2} E_{01} e^{- i t \\varpi_{1}} & \\hbar \\omega_{2} & \\frac{e r_{0;32}}{2} e^{i t \\varpi_{2}} \\overline{E_{02}}\\\\0 & \\frac{e r_{0;32}}{2} E_{02} e^{- i t \\varpi_{2}} & \\hbar \\omega_{3}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                            ⅈ⋅t⋅varpi₁ ______                \n",
       "⎢                                e⋅r_{0;21}⋅ℯ          ⋅E_{01}                \n",
       "⎢            h̅⋅ω₁               ─────────────────────────────                \n",
       "⎢                                              2                              \n",
       "⎢                                                                             \n",
       "⎢                   -ⅈ⋅t⋅varpi₁                                              ⅈ\n",
       "⎢E_{01}⋅e⋅r_{0;21}⋅ℯ                                             e⋅r_{0;32}⋅ℯ \n",
       "⎢──────────────────────────────              h̅⋅ω₂               ─────────────\n",
       "⎢              2                                                              \n",
       "⎢                                                                             \n",
       "⎢                                                   -ⅈ⋅t⋅varpi₂               \n",
       "⎢                                E_{02}⋅e⋅r_{0;32}⋅ℯ                          \n",
       "⎢              0                 ──────────────────────────────              h\n",
       "⎣                                              2                              \n",
       "\n",
       "                ⎤\n",
       "                ⎥\n",
       " 0              ⎥\n",
       "                ⎥\n",
       "                ⎥\n",
       "⋅t⋅varpi₂ ______⎥\n",
       "         ⋅E_{02}⎥\n",
       "────────────────⎥\n",
       " 2              ⎥\n",
       "                ⎥\n",
       "                ⎥\n",
       "                ⎥\n",
       "̅⋅ω₃            ⎥\n",
       "                ⎦"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "HRWA=H0+H1RWA\n",
    "fprint(HRWA, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Rotating Frame\n",
    "Next we will make a phase transformation in order to eliminate the explicit time dependance of the equations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARAAAABLCAMAAACP6iinAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nqfklLgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\left[\\begin{matrix}\\operatorname{c_{1}}{\\left (t \\right )}\\\\\\operatorname{c_{2}}{\\left (t \\right )}\\\\\\operatorname{c_{3}}{\\left (t \\right )}\\end{matrix}\\right], \\quad \\left[\\begin{matrix}\\tilde{c}_{1}{\\left (t \\right )}\\\\\\tilde{c}_{2}{\\left (t \\right )}\\\\\\tilde{c}_{3}{\\left (t \\right )}\\end{matrix}\\right], \\quad \\left[\\begin{matrix}\\theta_{1}\\\\\\theta_{2}\\\\\\theta_{3}\\end{matrix}\\right]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡c₁(t)⎤, ⎡\\tilde{c}_{1}(t)⎤, ⎡θ₁⎤⎤\n",
       "⎢⎢     ⎥  ⎢                ⎥  ⎢  ⎥⎥\n",
       "⎢⎢c₂(t)⎥  ⎢\\tilde{c}_{2}(t)⎥  ⎢θ₂⎥⎥\n",
       "⎢⎢     ⎥  ⎢                ⎥  ⎢  ⎥⎥\n",
       "⎣⎣c₃(t)⎦  ⎣\\tilde{c}_{3}(t)⎦  ⎣θ₃⎦⎦"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c,ctilde,phase=define_psi_coefficients(Ne)\n",
    "fprint([c,ctilde,phase], print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGEAAABMCAMAAABK6il5AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\tilde{c}_{1}{\\left (t \\right )} e^{i t \\theta_{1}}\\\\\\tilde{c}_{2}{\\left (t \\right )} e^{i t \\theta_{2}}\\\\\\tilde{c}_{3}{\\left (t \\right )} e^{i t \\theta_{3}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                  ⅈ⋅t⋅θ₁⎤\n",
       "⎢\\tilde{c}_{1}(t)⋅ℯ      ⎥\n",
       "⎢                        ⎥\n",
       "⎢                  ⅈ⋅t⋅θ₂⎥\n",
       "⎢\\tilde{c}_{2}(t)⋅ℯ      ⎥\n",
       "⎢                        ⎥\n",
       "⎢                  ⅈ⋅t⋅θ₃⎥\n",
       "⎣\\tilde{c}_{3}(t)⋅ℯ      ⎦"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi=Matrix([ exp(I*phase[i]*t)*ctilde[i] for i in range(Ne)])\n",
    "fprint(psi, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Schrödinger equation $i\\hbar \\partial_t |\\psi\\rangle=\\hat{H}_{RWA}$ is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI8AAABMCAMAAABjwRS7AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nVlzPRE8aAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\hbar \\theta_{1} \\tilde{c}_{1}{\\left (t \\right )} e^{i t \\theta_{1}}\\\\- \\hbar \\theta_{2} \\tilde{c}_{2}{\\left (t \\right )} e^{i t \\theta_{2}}\\\\- \\hbar \\theta_{3} \\tilde{c}_{3}{\\left (t \\right )} e^{i t \\theta_{3}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                         ⅈ⋅t⋅θ₁⎤\n",
       "⎢-h̅⋅θ₁⋅\\tilde{c}_{1}(t)⋅ℯ      ⎥\n",
       "⎢                               ⎥\n",
       "⎢                         ⅈ⋅t⋅θ₂⎥\n",
       "⎢-h̅⋅θ₂⋅\\tilde{c}_{2}(t)⋅ℯ      ⎥\n",
       "⎢                               ⎥\n",
       "⎢                         ⅈ⋅t⋅θ₃⎥\n",
       "⎣-h̅⋅θ₃⋅\\tilde{c}_{3}(t)⋅ℯ      ⎦"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lhs=Matrix([(I*hbar*Derivative(psi[i],t).doit()).expand() for i in range(Ne)])\n",
    "fprint(lhs, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "rhs=HRWA*psi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We multiply each of these equations by $e^{-i \\theta_i t}$ and substracting $i \\theta_i \\tilde{c}_i$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAABLCAMAAABZRmeuAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRIlmzd0i77ts7uXj/QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAMtJREFUOBHtVcsS\nwiAMDOVhBVqK+f9/lRClZETPztBcSndZCLAQUFhjgRY7IwAKtSlhGwOR/hckSp3o2YqCcjrE8O4n\nqa0M6vbESkHFg8CwDahck7ToKidUWCmPnGtPOdTU3aP5UCUMhCn+QK9KrBpRPwYETsMO0oCcaa44\nSp6XrEdLhp026nhtYp9hgUM5i/H20lQtpKrB1Jiemsait/XOJy8t+li7u3xZVDxff3JTprFoqynS\nouKNuixaLfq1ZDsq0Mb407u1ZBsDT+aoGdQmGjgBAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0⎤\n",
       "⎢ ⎥\n",
       "⎢0⎥\n",
       "⎢ ⎥\n",
       "⎣0⎦"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lhs_new=Matrix([simplify(  lhs[i]*exp(-I*phase[i]*t) +hbar*phase[i]*ctilde[i] ) for i in range(Ne)])\n",
    "fprint(lhs_new, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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CSdqathp1jGKhT4UTS2dP9DyBpsxJF3QlGAGFAhhFJ+xDUdJR45Z5aMZ/kWvNwFw2Zwej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\nunH59QNSf3MAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{e r_{0;21}}{2} \\tilde{c}_{2}{\\left (t \\right )} e^{- i t \\theta_{1}} e^{i t \\theta_{2}} e^{i t \\varpi_{1}} \\overline{E_{01}} + \\hbar \\omega_{1} \\tilde{c}_{1}{\\left (t \\right )} + \\hbar \\theta_{1} \\tilde{c}_{1}{\\left (t \\right )}\\\\\\frac{e r_{0;21}}{2} E_{01} \\tilde{c}_{1}{\\left (t \\right )} e^{i t \\theta_{1}} e^{- i t \\theta_{2}} e^{- i t \\varpi_{1}} + \\frac{e r_{0;32}}{2} \\tilde{c}_{3}{\\left (t \\right )} e^{- i t \\theta_{2}} e^{i t \\theta_{3}} e^{i t \\varpi_{2}} \\overline{E_{02}} + \\hbar \\omega_{2} \\tilde{c}_{2}{\\left (t \\right )} + \\hbar \\theta_{2} \\tilde{c}_{2}{\\left (t \\right )}\\\\\\frac{e r_{0;32}}{2} E_{02} \\tilde{c}_{2}{\\left (t \\right )} e^{i t \\theta_{2}} e^{- i t \\theta_{3}} e^{- i t \\varpi_{2}} + \\hbar \\omega_{3} \\tilde{c}_{3}{\\left (t \\right )} + \\hbar \\theta_{3} \\tilde{c}_{3}{\\left (t \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                                              -ⅈ⋅t⋅θ₁  ⅈ⋅t⋅θ₂\n",
       "⎢                                 e⋅r_{0;21}⋅\\tilde{c}_{2}(t)⋅ℯ       ⋅ℯ      \n",
       "⎢                                 ────────────────────────────────────────────\n",
       "⎢                                                                2            \n",
       "⎢                                                                             \n",
       "⎢                                    ⅈ⋅t⋅θ₁  -ⅈ⋅t⋅θ₂  -ⅈ⋅t⋅varpi₁             \n",
       "⎢E_{01}⋅e⋅r_{0;21}⋅\\tilde{c}_{1}(t)⋅ℯ      ⋅ℯ       ⋅ℯ              e⋅r_{0;32}\n",
       "⎢──────────────────────────────────────────────────────────────── + ──────────\n",
       "⎢                               2                                             \n",
       "⎢                                                                             \n",
       "⎢                                                                     ⅈ⋅t⋅θ₂  \n",
       "⎢                                 E_{02}⋅e⋅r_{0;32}⋅\\tilde{c}_{2}(t)⋅ℯ      ⋅ℯ\n",
       "⎢                                 ────────────────────────────────────────────\n",
       "⎣                                                                2            \n",
       "\n",
       "  ⅈ⋅t⋅varpi₁ ______                                                           \n",
       "⋅ℯ          ⋅E_{01}                                                           \n",
       "─────────────────── + h̅⋅ω₁⋅\\tilde{c}_{1}(t) + h̅⋅θ₁⋅\\tilde{c}_{1}(t)         \n",
       "                                                                              \n",
       "                                                                              \n",
       "                   -ⅈ⋅t⋅θ₂  ⅈ⋅t⋅θ₃  ⅈ⋅t⋅varpi₂ ______                         \n",
       "⋅\\tilde{c}_{3}(t)⋅ℯ       ⋅ℯ      ⋅ℯ          ⋅E_{02}                         \n",
       "───────────────────────────────────────────────────── + h̅⋅ω₂⋅\\tilde{c}_{2}(t)\n",
       "                     2                                                        \n",
       "                                                                              \n",
       "-ⅈ⋅t⋅θ₃  -ⅈ⋅t⋅varpi₂                                                          \n",
       "       ⋅ℯ                                                                     \n",
       "──────────────────── + h̅⋅ω₃⋅\\tilde{c}_{3}(t) + h̅⋅θ₃⋅\\tilde{c}_{3}(t)        \n",
       "                                                                              \n",
       "\n",
       "                         ⎤\n",
       "                         ⎥\n",
       "                         ⎥\n",
       "                         ⎥\n",
       "                         ⎥\n",
       "                         ⎥\n",
       "                         ⎥\n",
       " + h̅⋅θ₂⋅\\tilde{c}_{2}(t)⎥\n",
       "                         ⎥\n",
       "                         ⎥\n",
       "                         ⎥\n",
       "                         ⎥\n",
       "                         ⎥\n",
       "                         ⎦"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rhs_new=Matrix([simplify(  rhs[i]*exp(-I*phase[i]*t) +hbar*phase[i]*ctilde[i] ) for i in range(Ne)])\n",
    "fprint(rhs_new, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It can be seen that the equations loose their explicit time dependance only if $\\omega^{1} - \\theta_{1} + \\theta_{2}=0$ and $\\omega^{2} - \\theta_{2} + \\theta_{3}=0$. Which is satisfied if"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATsAAAAVBAMAAADLI1QGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left \\{ \\theta_{2} : \\theta_{1} - \\varpi_{1}, \\quad \\theta_{3} : \\theta_{1} - \\varpi_{1} - \\varpi_{2}\\right \\}$$"
      ],
      "text/plain": [
       "{θ₂: θ₁ - varpi₁, θ₃: θ₁ - varpi₁ - varpi₂}"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phase_transformation=solve([omega_laser[0]-phase[0]+phase[1],omega_laser[1]-phase[1]+phase[2]],[phase[1],phase[2]],\n",
    "                           dict=True)[0]\n",
    "fprint(phase_transformation, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a free parameter $\\theta_1$, which is to be expected, since state vetors $|\\psi\\rangle$ always have a global phase invariance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALMAAABNCAMAAADaUqWoAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n/O1J/qfG4T+qNI3fv/yrZdP/1MAfP/8Hp9RLN6cz7PEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\tilde{c}_{1}{\\left (t \\right )} e^{i t \\theta_{1}}\\\\\\tilde{c}_{2}{\\left (t \\right )} e^{i t \\left(\\theta_{1} - \\varpi_{1}\\right)}\\\\\\tilde{c}_{3}{\\left (t \\right )} e^{i t \\left(\\theta_{1} - \\varpi_{1} - \\varpi_{2}\\right)}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                            ⅈ⋅t⋅θ₁          ⎤\n",
       "⎢          \\tilde{c}_{1}(t)⋅ℯ                ⎥\n",
       "⎢                                            ⎥\n",
       "⎢                      ⅈ⋅t⋅(θ₁ - varpi₁)     ⎥\n",
       "⎢    \\tilde{c}_{2}(t)⋅ℯ                      ⎥\n",
       "⎢                                            ⎥\n",
       "⎢                  ⅈ⋅t⋅(θ₁ - varpi₁ - varpi₂)⎥\n",
       "⎣\\tilde{c}_{3}(t)⋅ℯ                          ⎦"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint(psi.subs(phase_transformation), print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus the equations become"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAABLCAMAAABZRmeuAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRIlmzd0i77ts7uXj/QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAMtJREFUOBHtVcsS\nwiAMDOVhBVqK+f9/lRClZETPztBcSndZCLAQUFhjgRY7IwAKtSlhGwOR/hckSp3o2YqCcjrE8O4n\nqa0M6vbESkHFg8CwDahck7ToKidUWCmPnGtPOdTU3aP5UCUMhCn+QK9KrBpRPwYETsMO0oCcaa44\nSp6XrEdLhp026nhtYp9hgUM5i/H20lQtpKrB1Jiemsait/XOJy8t+li7u3xZVDxff3JTprFoqynS\nouKNuixaLfq1ZDsq0Mb407u1ZBsDT+aoGdQmGjgBAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0⎤\n",
       "⎢ ⎥\n",
       "⎢0⎥\n",
       "⎢ ⎥\n",
       "⎣0⎦"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fprint(lhs_new, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAkwAAABMCAMAAACoJRjYAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\nmBhS/wPzrOBcrRShJgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{e r_{0;21}}{2} \\tilde{c}_{2}{\\left (t \\right )} \\overline{E_{01}} + \\hbar \\omega_{1} \\tilde{c}_{1}{\\left (t \\right )} + \\hbar \\theta_{1} \\tilde{c}_{1}{\\left (t \\right )}\\\\\\frac{e r_{0;21}}{2} E_{01} \\tilde{c}_{1}{\\left (t \\right )} + \\frac{e r_{0;32}}{2} \\tilde{c}_{3}{\\left (t \\right )} \\overline{E_{02}} + \\hbar \\omega_{2} \\tilde{c}_{2}{\\left (t \\right )} + \\hbar \\theta_{1} \\tilde{c}_{2}{\\left (t \\right )} - \\hbar \\varpi_{1} \\tilde{c}_{2}{\\left (t \\right )}\\\\\\frac{e r_{0;32}}{2} E_{02} \\tilde{c}_{2}{\\left (t \\right )} + \\hbar \\omega_{3} \\tilde{c}_{3}{\\left (t \\right )} + \\hbar \\theta_{1} \\tilde{c}_{3}{\\left (t \\right )} - \\hbar \\varpi_{1} \\tilde{c}_{3}{\\left (t \\right )} - \\hbar \\varpi_{2} \\tilde{c}_{3}{\\left (t \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                                             ______          \n",
       "⎢                                 e⋅r_{0;21}⋅\\tilde{c}_{2}(t)⋅E_{01}          \n",
       "⎢                                 ────────────────────────────────── + h̅⋅ω₁⋅\\\n",
       "⎢                                                 2                           \n",
       "⎢                                                                             \n",
       "⎢                                                                 ______      \n",
       "⎢E_{01}⋅e⋅r_{0;21}⋅\\tilde{c}_{1}(t)   e⋅r_{0;32}⋅\\tilde{c}_{3}(t)⋅E_{02}      \n",
       "⎢────────────────────────────────── + ────────────────────────────────── + h̅⋅\n",
       "⎢                2                                    2                       \n",
       "⎢                                                                             \n",
       "⎢    E_{02}⋅e⋅r_{0;32}⋅\\tilde{c}_{2}(t)                                       \n",
       "⎢    ────────────────────────────────── + h̅⋅ω₃⋅\\tilde{c}_{3}(t) + h̅⋅θ₁⋅\\tild\n",
       "⎣                    2                                                        \n",
       "\n",
       "                                                                         ⎤\n",
       "                                                                         ⎥\n",
       "tilde{c}_{1}(t) + h̅⋅θ₁⋅\\tilde{c}_{1}(t)                                 ⎥\n",
       "                                                                         ⎥\n",
       "                                                                         ⎥\n",
       "                                                                         ⎥\n",
       "                                                                         ⎥\n",
       "ω₂⋅\\tilde{c}_{2}(t) + h̅⋅θ₁⋅\\tilde{c}_{2}(t) - h̅⋅varpi₁⋅\\tilde{c}_{2}(t)⎥\n",
       "                                                                         ⎥\n",
       "                                                                         ⎥\n",
       "                                                                         ⎥\n",
       "e{c}_{3}(t) - h̅⋅varpi₁⋅\\tilde{c}_{3}(t) - h̅⋅varpi₂⋅\\tilde{c}_{3}(t)    ⎥\n",
       "                                                                         ⎦"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rhs_new=simplify(rhs_new.subs(phase_transformation)).expand()\n",
    "fprint(rhs_new, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It can be seen that this is the Schrödinger equation derived from an effective hamiltonian $\\tilde{H}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgIAAABMCAMAAAAoVi/LAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\hbar \\omega_{1} + \\hbar \\theta_{1} & \\frac{e r_{0;21}}{2} \\overline{E_{01}} & 0\\\\\\frac{e r_{0;21}}{2} E_{01} & \\hbar \\omega_{2} + \\hbar \\theta_{1} - \\hbar \\varpi_{1} & \\frac{e r_{0;32}}{2} \\overline{E_{02}}\\\\0 & \\frac{e r_{0;32}}{2} E_{02} & \\hbar \\omega_{3} + \\hbar \\theta_{1} - \\hbar \\varpi_{1} - \\hbar \\varpi_{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                  ______                                     \n",
       "⎢                       e⋅r_{0;21}⋅E_{01}                                     \n",
       "⎢  h̅⋅ω₁ + h̅⋅θ₁        ─────────────────                        0            \n",
       "⎢                               2                                             \n",
       "⎢                                                                             \n",
       "⎢                                                                   ______    \n",
       "⎢E_{01}⋅e⋅r_{0;21}                                       e⋅r_{0;32}⋅E_{02}    \n",
       "⎢─────────────────  h̅⋅ω₂ + h̅⋅θ₁ - h̅⋅varpi₁            ─────────────────    \n",
       "⎢        2                                                       2            \n",
       "⎢                                                                             \n",
       "⎢                       E_{02}⋅e⋅r_{0;32}                                     \n",
       "⎢        0              ─────────────────      h̅⋅ω₃ + h̅⋅θ₁ - h̅⋅varpi₁ - h̅⋅\n",
       "⎣                               2                                             \n",
       "\n",
       "      ⎤\n",
       "      ⎥\n",
       "      ⎥\n",
       "      ⎥\n",
       "      ⎥\n",
       "      ⎥\n",
       "      ⎥\n",
       "      ⎥\n",
       "      ⎥\n",
       "      ⎥\n",
       "      ⎥\n",
       "varpi₂⎥\n",
       "      ⎦"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Htilde=Matrix([ [Derivative(rhs_new[i],ctilde[j]).doit() for j in range(Ne)] for i in range(Ne)])\n",
    "fprint(Htilde, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that it is convenient to choose $\\theta_1=-\\omega_1$ to simplify the hamiltonian. Also, we can recognize $\\omega^1-\\omega_2+\\omega_1=\\delta^1$ as the detuning of the first field relative to the atomic transition $\\omega_{21}=\\omega_2-\\omega_1$, and the same for $\\omega^2-\\omega_3+\\omega_2=\\delta^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAT4AAABMCAMAAADDeyt8AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & \\frac{e r_{0;21}}{2} \\overline{E_{01}} & 0\\\\\\frac{e r_{0;21}}{2} E_{01} & - \\delta_{1} \\hbar & \\frac{e r_{0;32}}{2} \\overline{E_{02}}\\\\0 & \\frac{e r_{0;32}}{2} E_{02} & - \\delta_{1} \\hbar - \\delta_{2} \\hbar\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                              ______                   ⎤\n",
       "⎢                   e⋅r_{0;21}⋅E_{01}                   ⎥\n",
       "⎢        0          ─────────────────          0        ⎥\n",
       "⎢                           2                           ⎥\n",
       "⎢                                                       ⎥\n",
       "⎢                                                 ______⎥\n",
       "⎢E_{01}⋅e⋅r_{0;21}                     e⋅r_{0;32}⋅E_{02}⎥\n",
       "⎢─────────────────       -δ₁⋅h̅        ─────────────────⎥\n",
       "⎢        2                                     2        ⎥\n",
       "⎢                                                       ⎥\n",
       "⎢                   E_{02}⋅e⋅r_{0;32}                   ⎥\n",
       "⎢        0          ─────────────────   -δ₁⋅h̅ - δ₂⋅h̅  ⎥\n",
       "⎣                           2                           ⎦"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "delta1,delta2=symbols(\"delta1 delta2\",real=True)\n",
    "Htilde=Htilde.subs({phase[0]:-omega_level[0]})\n",
    "Htilde=Htilde.subs({omega_laser[0]:delta1+omega_level[1]-omega_level[0]})\n",
    "Htilde=Htilde.subs({omega_laser[1]:delta2+omega_level[2]-omega_level[1]})\n",
    "\n",
    "Htilde=Htilde.expand()\n",
    "\n",
    "fprint(Htilde, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we define the Rabi frequencies $\\Omega_1 =e E_0^1 r_{0;21}/\\hbar$ and $\\Omega_2 =e E_0^2 r_{0;32}/\\hbar$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO4AAABXCAMAAAA09YmtAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & \\frac{\\hbar \\overline{\\Omega_{1}}}{2} & 0\\\\\\frac{\\Omega_{1} \\hbar}{2} & - \\delta_{1} \\hbar & \\frac{\\hbar \\overline{\\Omega_{2}}}{2}\\\\0 & \\frac{\\Omega_{2} \\hbar}{2} & - \\delta_{1} \\hbar - \\delta_{2} \\hbar\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡          __                 ⎤\n",
       "⎢       h̅⋅Ω₁                 ⎥\n",
       "⎢  0    ─────         0       ⎥\n",
       "⎢         2                   ⎥\n",
       "⎢                             ⎥\n",
       "⎢                      __     ⎥\n",
       "⎢Ω₁⋅h̅              h̅⋅Ω₂     ⎥\n",
       "⎢─────  -δ₁⋅h̅      ─────     ⎥\n",
       "⎢  2                  2       ⎥\n",
       "⎢                             ⎥\n",
       "⎢       Ω₂⋅h̅                 ⎥\n",
       "⎢  0    ─────   -δ₁⋅h̅ - δ₂⋅h̅⎥\n",
       "⎣         2                   ⎦"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Omega1,Omega2=symbols(\"Omega1 Omega2\",real=True)\n",
    "Omega1,Omega2=symbols(\"Omega1 Omega2\")\n",
    "Htilde=Htilde.subs({E0[0]:Omega1*hbar/r[1][1,0]/e})\n",
    "Htilde=Htilde.subs({E0[1]:Omega2*hbar/r[1][2,1]/e})\n",
    "\n",
    "fprint(Htilde, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define the density matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI4AAABLCAMAAACRBk89AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nuIY/bHcNVicAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\rho_{11} & \\rho_{12} & \\rho_{13}\\\\\\rho_{21} & \\rho_{22} & \\rho_{23}\\\\\\rho_{31} & \\rho_{32} & \\rho_{33}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ρ₁₁  ρ₁₂  ρ₁₃⎤\n",
       "⎢             ⎥\n",
       "⎢ρ₂₁  ρ₂₂  ρ₂₃⎥\n",
       "⎢             ⎥\n",
       "⎣ρ₃₁  ρ₃₂  ρ₃₃⎦"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho=define_density_matrix(Ne)\n",
    "fprint( rho , print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The hamiltonian part of the equations is\n",
    "\\begin{equation}\n",
    "    \\dot{\\hat{\\rho}}=\\frac{i}{\\hbar}[\\hat{\\rho}, \\hat{\\tilde{H}}]\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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tIxAk2opy4A2+/vefmFnEN9LxP8/3lPcDj5IHw2IzCQrPYoUl3oVC/aMa3TEpJf1KGDwQ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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{i \\Omega_{1}}{2} \\rho_{12} - \\frac{i \\rho_{21}}{2} \\overline{\\Omega_{1}} & \\frac{i \\Omega_{2}}{2} \\rho_{13} - i \\delta_{1} \\rho_{12} + \\frac{i \\rho_{11}}{2} \\overline{\\Omega_{1}} - \\frac{i \\rho_{22}}{2} \\overline{\\Omega_{1}} & - i \\delta_{1} \\rho_{13} - i \\delta_{2} \\rho_{13} + \\frac{i \\rho_{12}}{2} \\overline{\\Omega_{2}} - \\frac{i \\rho_{23}}{2} \\overline{\\Omega_{1}}\\\\- \\frac{i \\Omega_{1}}{2} \\rho_{11} + \\frac{i \\Omega_{1}}{2} \\rho_{22} + i \\delta_{1} \\rho_{21} - \\frac{i \\rho_{31}}{2} \\overline{\\Omega_{2}} & - \\frac{i \\Omega_{1}}{2} \\rho_{12} + \\frac{i \\Omega_{2}}{2} \\rho_{23} + \\frac{i \\rho_{21}}{2} \\overline{\\Omega_{1}} - \\frac{i \\rho_{32}}{2} \\overline{\\Omega_{2}} & - \\frac{i \\Omega_{1}}{2} \\rho_{13} - i \\delta_{2} \\rho_{23} + \\frac{i \\rho_{22}}{2} \\overline{\\Omega_{2}} - \\frac{i \\rho_{33}}{2} \\overline{\\Omega_{2}}\\\\\\frac{i \\Omega_{1}}{2} \\rho_{32} - \\frac{i \\Omega_{2}}{2} \\rho_{21} + i \\delta_{1} \\rho_{31} + i \\delta_{2} \\rho_{31} & - \\frac{i \\Omega_{2}}{2} \\rho_{22} + \\frac{i \\Omega_{2}}{2} \\rho_{33} + i \\delta_{2} \\rho_{32} + \\frac{i \\rho_{31}}{2} \\overline{\\Omega_{1}} & - \\frac{i \\Omega_{2}}{2} \\rho_{23} + \\frac{i \\rho_{32}}{2} \\overline{\\Omega_{2}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                             __                                           __ \n",
       "⎢            ⅈ⋅Ω₁⋅ρ₁₂   ⅈ⋅ρ₂₁⋅Ω₁               ⅈ⋅Ω₂⋅ρ₁₃              ⅈ⋅ρ₁₁⋅Ω₁ \n",
       "⎢            ──────── - ────────               ──────── - ⅈ⋅δ₁⋅ρ₁₂ + ──────── \n",
       "⎢               2          2                      2                     2     \n",
       "⎢                                                                             \n",
       "⎢                                         __                                __\n",
       "⎢  ⅈ⋅Ω₁⋅ρ₁₁   ⅈ⋅Ω₁⋅ρ₂₂              ⅈ⋅ρ₃₁⋅Ω₂    ⅈ⋅Ω₁⋅ρ₁₂   ⅈ⋅Ω₂⋅ρ₂₃   ⅈ⋅ρ₂₁⋅Ω₁\n",
       "⎢- ──────── + ──────── + ⅈ⋅δ₁⋅ρ₂₁ - ────────  - ──────── + ──────── + ────────\n",
       "⎢     2          2                     2           2          2          2    \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢ ⅈ⋅Ω₁⋅ρ₃₂   ⅈ⋅Ω₂⋅ρ₂₁                           ⅈ⋅Ω₂⋅ρ₂₂   ⅈ⋅Ω₂⋅ρ₃₃           \n",
       "⎢ ──────── - ──────── + ⅈ⋅δ₁⋅ρ₃₁ + ⅈ⋅δ₂⋅ρ₃₁   - ──────── + ──────── + ⅈ⋅δ₂⋅ρ₃₂\n",
       "⎣    2          2                                  2          2               \n",
       "\n",
       "        __                                __         __ ⎤\n",
       "  ⅈ⋅ρ₂₂⋅Ω₁                          ⅈ⋅ρ₁₂⋅Ω₂   ⅈ⋅ρ₂₃⋅Ω₁ ⎥\n",
       "- ────────   -ⅈ⋅δ₁⋅ρ₁₃ - ⅈ⋅δ₂⋅ρ₁₃ + ──────── - ──────── ⎥\n",
       "     2                                 2          2     ⎥\n",
       "                                                        ⎥\n",
       "         __                                __         __⎥\n",
       "   ⅈ⋅ρ₃₂⋅Ω₂    ⅈ⋅Ω₁⋅ρ₁₃              ⅈ⋅ρ₂₂⋅Ω₂   ⅈ⋅ρ₃₃⋅Ω₂⎥\n",
       " - ────────  - ──────── - ⅈ⋅δ₂⋅ρ₂₃ + ──────── - ────────⎥\n",
       "      2           2                     2          2    ⎥\n",
       "                                                        ⎥\n",
       "         __                                __           ⎥\n",
       "   ⅈ⋅ρ₃₁⋅Ω₁               ⅈ⋅Ω₂⋅ρ₂₃   ⅈ⋅ρ₃₂⋅Ω₂           ⎥\n",
       " + ────────             - ──────── + ────────           ⎥\n",
       "      2                      2          2               ⎦"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hamiltonian_terms=(I/hbar*(rho*Htilde-Htilde*rho)).expand()\n",
    "fprint(hamiltonian_terms, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are two Lindblad operators, since there are two spontaneous decay channels."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAawAAABLCAMAAAAbIEMQAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nXgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\gamma_{21} \\rho_{22} & - \\frac{\\gamma_{21} \\rho_{12}}{2} & - \\frac{\\gamma_{32} \\rho_{13}}{2}\\\\- \\frac{\\gamma_{21} \\rho_{21}}{2} & - \\gamma_{21} \\rho_{22} + \\gamma_{32} \\rho_{33} & - \\frac{\\gamma_{21} \\rho_{23}}{2} - \\frac{\\gamma_{32} \\rho_{23}}{2}\\\\- \\frac{\\gamma_{32} \\rho_{31}}{2} & - \\frac{\\gamma_{21} \\rho_{32}}{2} - \\frac{\\gamma_{32} \\rho_{32}}{2} & - \\gamma_{32} \\rho_{33}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                -γ₂₁⋅ρ₁₂             -γ₃₂⋅ρ₁₃      ⎤\n",
       "⎢ γ₂₁⋅ρ₂₂        ─────────            ─────────     ⎥\n",
       "⎢                    2                    2         ⎥\n",
       "⎢                                                   ⎥\n",
       "⎢-γ₂₁⋅ρ₂₁                          γ₂₁⋅ρ₂₃   γ₃₂⋅ρ₂₃⎥\n",
       "⎢─────────  -γ₂₁⋅ρ₂₂ + γ₃₂⋅ρ₃₃   - ─────── - ───────⎥\n",
       "⎢    2                                2         2   ⎥\n",
       "⎢                                                   ⎥\n",
       "⎢-γ₃₂⋅ρ₃₁     γ₂₁⋅ρ₃₂   γ₃₂⋅ρ₃₂                     ⎥\n",
       "⎢─────────  - ─────── - ───────       -γ₃₂⋅ρ₃₃      ⎥\n",
       "⎣    2           2         2                        ⎦"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lindblad_terms =gamma[1,0]*lindblad_operator(ket(1,Ne)*bra(2,Ne),rho)\n",
    "lindblad_terms+=gamma[2,1]*lindblad_operator(ket(2,Ne)*bra(3,Ne),rho)\n",
    "\n",
    "fprint(lindblad_terms, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Optical Bloch Equations\n",
    "The Optical Bloch equations are thus."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "eqs=hamiltonian_terms + lindblad_terms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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OBdbrs5qeOnIBRORWYjTHr6AljwXvQEIaQKthphGt6b4D6i4ZLcyKqnPonZHuwnhopxh8\nRZ1I6TeU1G0uKGjJI8E7kJBy17UwU4nWdN8Mu0dOC7Oy6rx4fN+D+og+teFMLP0Oc1fSU5xGR9eS\nJ4L3olNrYaYRreq+NXqv26aFmfWkpqRmvXOTNvt1VnTrSQ6+ABpJv1nS0+ICXUseC97LTq2GWViO\nCDQpvdPFi3n/9YNSu8oHsxW19simYVZWnUPvjHQPti/pM9VTs/QbIvJmF+ha8okRgYQ0MrAWZqZi\nSjRqbL4sFiBlCL02JGHGs4By1bk5ThtNNtJeFN4NJ9VTC+k3JD2NLihoyaU8HUhIpTO2isnBVmLu\nlE/CjGcB5apz6J2R7sR4dLdh8OU7ktJvSHoaXaBrySPBO5CQLrAuJZo1Bdus4PUbkjALs4D2VJ2/\n3guix2w4s0FMrmvJNwnemWlGlIveLxeHGc8C2lF1vquT3mc4M+OGX0PU2hGHGc8C2k91PuPescVv\nNJypG/priDozkjALs4D2U53X3Tu49I2GM3VLOxF9lQrahJkiB46eN9UN3ql0hINGYO7knsZuX6Vi\nL8iB3191PkImPgKzcX//8WrJSRPW/g7V+QiZ+AhMePWXpWJCVJxdbkchzGbvySzvaUCLETLxEZj9\nTL/+mPW/J6wXL4Av7h1KYst89nUqdiXMfoPqfIRMfARm2NW304MXIQ1bF2d+7ueP43S9TeEd43g6\n/3WYruatcPf7j12yFNtrXegqdr+8v2/YR8Wehxk/b6ox3Lksl4nHzllDbwTm9OHeNPY4337EYrdr\n6Jk2ZxNBx4kejxOsuW98u3/boJroPqd9lUO6vdZTQcUeMKltLxV7Hmb8vKlGcd8yRSYunbOK3AhM\nin0zY4IS+sNMeLthw7/DgzDDO8bNw1D6GLnTje4apNvr/agqdgJzmGaqh6N/anlZQq2rPMzC86Za\ns93LMpm4cE4buXz+wwhMz+Xudlcbs7lafkl5N1XLhYSVO6mvNa+BlVTsCLN+KvYszPh5U43g3mWa\nTBzOaeSWhdkITHC5uZkI+Lotdedh/45xZ3X5tea1rkoqdniyn4o9CzN+3lQjuHeZJhOHcxq5ZWE2\nAjNw+TIavD6fqz0y4h3jzurjx+l0dGoqP/Zv8UZBxR5Omv1U7HmYhedNwSkNAulQ90UZRSYenGMo\nNFDOwqw/5uVEj4bNh94O9cymazRwtI3Tf/aSNciVXThBic7vHm8JM4eMtnh9OQ8/+qnY8zBLrWKB\ndFaibthNdS4ca277+alMKkezMQszreY2zHt4Pftl+qHxefRp4Rg1wBc78zt+fbkdmtnyeDua1FJu\n619fzu+YT1XsNZh6mQkzlgNrdYPkmuLtxx2YK8JnDeEl29g5QiVO95HsIj+cMpeGMNMx4QakDJmE\n7vXspmPbGdgPez8DdCTHgMOFAjLPZodFzbq8WWlLpddExc4BUMIqbp+VA7NAGpqt9hewF3tdVqCo\nuusAQdMNqkhdM13TXYfM3QB3lDFprGSVH1YDZTTXkkbgCBxZSD/9r8/woTuvGz9r0eJXooPpCjLx\nSTN+dGUP8yyQNicQo+yn802/8ewKytQk4Sm+GjymzFRzyunRTIAk2RgTbkBqSvFJMZ0oxEbU4WlG\n6IEGc2ScUAi4LE2IzX7NAKIN9ea2aqxiDwEQwbR8icNMacGSaxaXVf2xv+qcKfN+zSmnIaHYLjYx\nJtyAVFRKTpp0s/SDbspe7OnSjj04zHI85ioRd80nKnbN4kZ+WphFk81iybX7eeb7rLG3/tXCwEZA\nS8qgipSrVcJMAZWYFBD20MQpUBPMC908p5GUDbMfexsi0NDwQiHgGlOFbmNLUU0DIe729siVrjr8\niBCWi5YtWS3M5OIGseR6cuKytf5oIbSsjjZciCiDKlKGT0KCC8LyUmJThElX/H6KBFLUTDDp6Y+5\nyWWWOLi5e6qgoeKhEGiNqeaDxqZcTQX5eJzvJDuezvczZnymFjNCNaeFmZ1s5q9zI8m1vbdhut17\nbBZM4oFN2BRrukEVqaimzKZxpQqo6ga4o4z5YR4/mjjDXgINFQ+FjNeUU+g2tYsqqSCZij23OAIp\nf9HCzE428zazQJowoNla6Y8yi9Ul2nAhogyqSLmn5MjDBZMCGmHCDUi5aRnT1wENFQ+FjNeUU+g2\ntYsqtYHkFkcg5S9pmKWLGwjJNWvlV/qjzGJTSTpckJTDgTenXNd0J6ASE9I1pIJ8HZMqgoaGFwoF\nYGs2odvaLK43C6JYHCMUvyVhxpPN3OIGUnLNmq3lL2Av9t+hIBkuSMq0X2mqn/kgdd/m/8egEhNu\nQDqPJWp4Ghreco4CN6YrCpZk50BWWewIJGHGk83clAIpuYZmC2+1X2LBwLrpfHJJGVSRNtNIQCUm\n3IC0GZMebR4/nx9mSKjhhcIFgKia0MXmZeksyAqLwSAJszDZbO6JINrvnq4eLtSYDwGtdbitrAvd\nLiAlO+Iw48lmv2Vxg/XDhZJHaPsQ0Ep/G4u60O0CUjTEhJmQA3/Tg7fnga7Cf8niBhuGC0WX0OP2\n8/n802tOdbmfTiVd6HYBKRuUyIEx2ez4/O9p8wPbcq+bSnAPyoBsGC5EHCRmN9Coh3FfuvigC0jZ\nxvikGeq9s+p8hDZ8BGZw5r8Z5bWt1in2QcAbu2eENnwE5hu78KXUCkez2Yvbl5LMOxuhDR+BmTN/\nly199etzVilh9t6q8xHa8BGYwfGd9Oa98Vi//jgdaRge5OoT9Otu+3aZtWGehxk/CAiWvVNmhDY8\nx2Snr7a9r96c9rab49Ffv36bzBDJy9VpfQ7o1932zTJr68A8zPhBwGoPD2w4QhuuYAanrzelt968\nNx5ZZvXrdOPGvPOH5epev+63s85pvSu0o1l4ELAFdmDb7drwnFyGGZye1y1sKU3OWKk3L8FNK/F0\n1la/fri746R7dG8ngFn9OrabOUJbP9nRjB8EbIUe057cQCOJ+LXkWx2hYS6dL1GKi5V68xIcCR06\n7HXsGh9fNyv/cjJ2oV+nhVpMxa3eNRhZmL276ryDNtzYHX00TL92QFSv9qUYF+v05kW4aR2eSt3q\n182jfBO6XsYe9OvYPijM8CCAia2VRzNCx1wHbXjGRsHE2gGuboMH4rjYrDeP4UhKW9GvN7DLTLYb\njH4dWnPI2MsadB2jbas5mtXlwHZi7oIJGzuozsVh3dzwW0C25CQ43ZW3gMZxsVlvHsOR4MMZdfjK\n9est7HQ7rX7da829XN3Pzaf6mQZdx2jbauXAYjflrYTq3L/9PUi588p7bClow/GqekjlF1HTX3VO\nv7ii7j6Ki1a9edmhEdyU4wUm/Np0khwc3TotFSF55AWvVpLbWpvKNg15M2u4+hJqllvjre+xPLqh\njxdWCVpukI0VO+vU1wEUYLEHFBE73Qxo0JuD4yycWYEvxgMTcm1gp8OtM7n3LosvARIdsumM5dH8\n1nfMaO9NpgEvYSi+6mTNgdpJ5cvgAiTJ2jbsAQbLPRAffrySs643rzk0gUvxmAmzq8FlxieGzn7N\nAJZtiMNMacvyaFPoXombO1k03FN1npNtU+wI+nmWQRks90ASFy16c9NVyaEJXKpfZybMrgaXW/Xa\nLVqYlVXndm0b1ue8lmraGw9PQkmk5fZkia7XiIdq5YyCGb2YHGBzYdaqNy86NAmzHA9MNJNzdmWL\nZYlmvSxfn9fCrKg6x1vf15qxnqbSUgxPUBppuUGWNeKoVk4VzPTF5E7+k3sgjotGvTk4zsGZ1YsT\n/bo3SzU5hyubLEo060XxlqwWZnaymb+yjeXRxWP8Fgor2/LwJACoZJcopRXMWMQOsHxHxmFGK2g3\n6c2LDk3gcjzPRDU5ZxdcVMto1tfqLyjTwsxONvNUI3k03v6+0owFtBqq8vAkVFbJLlHsKJjixeTU\nD8ByD6RxEUj5DFqoHFHIjebgwKQRjoErOc36SvUlRWmYlVXndhhxtQvF515Z0mXHusmoi7XcTHax\nYifBlC8mZ8lT7oE5wTlaaByV56dzcDCrEa7Z56n1zQ3rFZMw48lmuercHFNP9pn+Ugl3ncKG0lgn\nLbXcILtcsRNjxi8mD5Kn5R5Q9ObgSKN5r41vdgXM0kxeAcf9JtZzwbZcEmY82SxXneOt71vk0dvI\npq3t2Z03Si03yC5W7CSYnQTiwWcax1DIpszlYFYnuNBdan0o2JhJwixMNuvwXHAjs/nmGCjN12yv\nMQKzvfe9aw6zPg4znmz2C1TnGJ703DcjMHvyG4s1zvo4zHiy2furzjE86en5EZg9+Y3FKlsfqaXX\nkEjCLEw2o8xOqvNWkzA8abF6BGZLv7+rTtmjTi29wZokzIAUPW/CxtekIwTgIzBf440/0kshzHZe\n3GCEAHwE5h8Jgm1mFGd3CNhCmNnnTaLai7MjBOAjMPu5pa8IvC+aU0vTLOsVr1P3HiqE2aj7J7P7\nZYQAfATmrCELKwQReNbOi8Cz7bUNfdGcWpqknO516rWOS2VKmO26uMEIAfgIzODPPmsX8EvMg97d\nvxcTIvBDugZBYJBnBFrSag2aV0sHObpnEq96cD2fbK85GbslDzN+3lRoMnLzCAH4CMzeaw1Yn1oR\nuNe7hzUF8BLzn2QNgtm94NCSVuvQnFo6aRsYEpPb15FevHMsv8MsDzN+3jRryYgKHQTgGa0RmAPW\nGpjMA+OgdzdPPGk8ZKZkGhE4JdEaBJmR2QaDlrRah0atjNIuYSJ0wg87oYKeeNPLg/RPHmbheZPe\nYPBWZ1IsKlcmMNRYZHNoRmCCQNe1BuQiA16FyyJwrDXQ7A2LFrdqQsv859TSeVv3QzBaC6tOJ5d8\nlSTxWZjx8yZ48qVpBwF4wU3H6UTHh/PhYmVjC0XlGSac0nOtAfcSc3qVpj2OuZ0YRODUoVyDAP2X\nU48WtWpCy2x1aum8LcLsGF6wfbf6K4VTFmb8vEmpPX7TrAB8nkLBTV/PJ73m6vR8GhFsLCpfgRma\ndFxrwL7EPFBzOxEi8AlrDTQzt2hJqya0zH/O1qxtWPXgK5wrH09/QA7u8Zk8zMLzplB1tXg+IGzJ\nxALwBi4FN0kOWzGraw1MDRwlmZC3IvBAzYaZHRCZCukaBKFRMWN/TTSP7UB/HrQNTfdf1pbHZj9+\naGZ+xDSK1D55mGW1zD00Oy8Iqua6AKb34gaRAJy5eP23wkV3U2RWARNCdai3uU2Cma01IGkwR+BB\nq854as6ezT01L6VnEXi8BoHaPt5o0eJWbWiJrR41a8ti/5M9QZh6Hy7guC4ozYdZWNwAquaBAhiw\nKqWBCzgglQ10N8kaUT5gwj6kslaMma01IGlkeLJQYr5rPra1geXlGy++/DSzgGMRvWsfh1nyeMpW\nYfE8/T5MdXM0nxNyN5CjKrnwPiEgvsZcwAGp609ZZoAKBEiSjTGh2kZaxqTbBPFaA5LGjL9ym00/\nCbHZr46b5sEtaEv9Z2ia/h7+AvPgB2nQPYCli5vqGhosnjeNjEh6oACGeak55gIOSGX1Zb9GxjQY\nTgTOKXBTTKfMCGsXCBo5nigE3Funqa0NZL/MpQbd2IvuyMh28dHMlUSzgCLxfIuQu9PiBnK0A8IR\nlyAmT8U4FTcpoBEm7EOKjqcUM1u7ACLwScULXANgU0ah29ROr9SKltoaoWkg9ADA3Tijg7q9F9N2\nNJOq80jaD5E0HdFc+EYMun5RBzSRkD9wSLmU3aSBSkzYh5QtSjC1tQY8DQ1vpb80ukxpaa4ZLbE1\n6kcFocGUfQB7pWtFe+HRFmZSdR6J58PJZPwEDjnagaExF3BAilrZkScUaKPKGLP1pKmsNeBp6HgZ\nR+ZUyWk+qFSfKWpGq4WZCvLxON/pluR0vp/dNOW2MLNO8VUT8fzVDvKGCWDYU9qAJuICDki5adlN\nCmiECVU9p0BNMbO1C0BDxUMh0BpThW5jS61aM1pqqwRTQe50JWCWcnhQas9y82GWqs5ZPL9ByC2J\nLsingy7mQjfy6Oh8owe6SAVszU1hCIX6jAn7kKKGTeuYgkaOJwojyLYvqQ/aWpVqNaHN2ToPMhtm\nPAvIqc6ltN8cMI3qvCyAKVm3cns66GIhf+CgcamvC5CACkzYh1SyrmMGOrR2Ig9agaNxlNjVfEK3\nWne+sAltxtYwJi53l4vo3ZWmuQFinxPwLCB3K0SK51cLuct8qiXpgEZygQgHaRVIFiagEhP2IZXN\nZvKBhoYXCmdAtOKErlZlwbYuaLMgiYjePNagD93jO0tSNCYAAACFSURBVNmPGcWFWUBedS7F8wvs\n6VE1G9D04JKC9sCUxnbGS+nKrpbnu6AtBnm46JJ0eRbQ7qpzZdAlma7LDwFdR6WlVV+6XdC6gPAs\noL1V55sGNKVdOAS01Nn27X3pdkHrAkIXRR8nuvdBV6f7qc7d7tkyoCnu4CGgxd42F/Sl2wVtA8j/\nAfCesRJaOIZuAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{d}{d t} \\rho_{11} & = & \\frac{i \\Omega_{1}}{2} \\rho_{12} + \\gamma_{21} \\rho_{22} - \\frac{i \\rho_{21}}{2} \\overline{\\Omega_{1}}\\\\\\frac{d}{d t} \\rho_{22} & = & - \\frac{i \\Omega_{1}}{2} \\rho_{12} + \\frac{i \\Omega_{2}}{2} \\rho_{23} - \\gamma_{21} \\rho_{22} + \\gamma_{32} \\rho_{33} + \\frac{i \\rho_{21}}{2} \\overline{\\Omega_{1}} - \\frac{i \\rho_{32}}{2} \\overline{\\Omega_{2}}\\\\\\frac{d}{d t} \\rho_{33} & = & - \\frac{i \\Omega_{2}}{2} \\rho_{23} - \\gamma_{32} \\rho_{33} + \\frac{i \\rho_{32}}{2} \\overline{\\Omega_{2}}\\\\\\frac{d}{d t} \\rho_{21} & = & - \\frac{i \\Omega_{1}}{2} \\rho_{11} + \\frac{i \\Omega_{1}}{2} \\rho_{22} + i \\delta_{1} \\rho_{21} - \\frac{\\gamma_{21} \\rho_{21}}{2} - \\frac{i \\rho_{31}}{2} \\overline{\\Omega_{2}}\\\\\\frac{d}{d t} \\rho_{31} & = & \\frac{i \\Omega_{1}}{2} \\rho_{32} - \\frac{i \\Omega_{2}}{2} \\rho_{21} + i \\delta_{1} \\rho_{31} + i \\delta_{2} \\rho_{31} - \\frac{\\gamma_{32} \\rho_{31}}{2}\\\\\\frac{d}{d t} \\rho_{32} & = & - \\frac{i \\Omega_{2}}{2} \\rho_{22} + \\frac{i \\Omega_{2}}{2} \\rho_{33} + i \\delta_{2} \\rho_{32} - \\frac{\\gamma_{21} \\rho_{32}}{2} - \\frac{\\gamma_{32} \\rho_{32}}{2} + \\frac{i \\rho_{31}}{2} \\overline{\\Omega_{1}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                                        __                 ⎤\n",
       "⎢d                            ⅈ⋅Ω₁⋅ρ₁₂             ⅈ⋅ρ₂₁⋅Ω₁                 ⎥\n",
       "⎢──(ρ₁₁)  =                   ──────── + γ₂₁⋅ρ₂₂ - ────────                 ⎥\n",
       "⎢dt                              2                    2                     ⎥\n",
       "⎢                                                                           ⎥\n",
       "⎢                                                              __         __⎥\n",
       "⎢d             ⅈ⋅Ω₁⋅ρ₁₂   ⅈ⋅Ω₂⋅ρ₂₃                       ⅈ⋅ρ₂₁⋅Ω₁   ⅈ⋅ρ₃₂⋅Ω₂⎥\n",
       "⎢──(ρ₂₂)  =  - ──────── + ──────── - γ₂₁⋅ρ₂₂ + γ₃₂⋅ρ₃₃ + ──────── - ────────⎥\n",
       "⎢dt               2          2                              2          2    ⎥\n",
       "⎢                                                                           ⎥\n",
       "⎢                                                         __                ⎥\n",
       "⎢d                             ⅈ⋅Ω₂⋅ρ₂₃             ⅈ⋅ρ₃₂⋅Ω₂                ⎥\n",
       "⎢──(ρ₃₃)  =                  - ──────── - γ₃₂⋅ρ₃₃ + ────────                ⎥\n",
       "⎢dt                               2                    2                    ⎥\n",
       "⎢                                                                           ⎥\n",
       "⎢                                                                    __     ⎥\n",
       "⎢d                  ⅈ⋅Ω₁⋅ρ₁₁   ⅈ⋅Ω₁⋅ρ₂₂              γ₂₁⋅ρ₂₁   ⅈ⋅ρ₃₁⋅Ω₂     ⎥\n",
       "⎢──(ρ₂₁)  =       - ──────── + ──────── + ⅈ⋅δ₁⋅ρ₂₁ - ─────── - ────────     ⎥\n",
       "⎢dt                    2          2                     2         2         ⎥\n",
       "⎢                                                                           ⎥\n",
       "⎢d                 ⅈ⋅Ω₁⋅ρ₃₂   ⅈ⋅Ω₂⋅ρ₂₁                         γ₃₂⋅ρ₃₁      ⎥\n",
       "⎢──(ρ₃₁)  =        ──────── - ──────── + ⅈ⋅δ₁⋅ρ₃₁ + ⅈ⋅δ₂⋅ρ₃₁ - ───────      ⎥\n",
       "⎢dt                   2          2                                2         ⎥\n",
       "⎢                                                                           ⎥\n",
       "⎢                                                                         __⎥\n",
       "⎢d             ⅈ⋅Ω₂⋅ρ₂₂   ⅈ⋅Ω₂⋅ρ₃₃              γ₂₁⋅ρ₃₂   γ₃₂⋅ρ₃₂   ⅈ⋅ρ₃₁⋅Ω₁⎥\n",
       "⎢──(ρ₃₂)  =  - ──────── + ──────── + ⅈ⋅δ₂⋅ρ₃₂ - ─────── - ─────── + ────────⎥\n",
       "⎣dt               2          2                     2         2         2    ⎦"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqsign=symbols(\"=\")\n",
    "eqs_list=[]\n",
    "for mu in range(0,Ne**2-1 -(Ne**2 - Ne)/2+1):\n",
    "    ii,jj,s=IJ(mu,Ne)\n",
    "    i=ii-1; j=jj-1\n",
    "    eqs_list+=[[Derivative(rho[i,j],t),eqsign,eqs[i,j]]]\n",
    "eqs_list=Matrix(eqs_list)\n",
    "fprint(eqs_list, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which is how most literature will show the equations. However, a more convenient way to express this equations is to explicitly asume a normalized and hermitian density matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAABLCAMAAABqUbriAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nrfmHMHWtnxf+jG3+Bzh1rf4D4nBRjYtm2ZAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\rho_{22} - \\rho_{33} + 1 & \\overline{\\rho_{21}} & \\overline{\\rho_{31}}\\\\\\rho_{21} & \\rho_{22} & \\overline{\\rho_{32}}\\\\\\rho_{31} & \\rho_{32} & \\rho_{33}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                ___  ___⎤\n",
       "⎢-ρ₂₂ - ρ₃₃ + 1  ρ₂₁  ρ₃₁⎥\n",
       "⎢                        ⎥\n",
       "⎢                     ___⎥\n",
       "⎢     ρ₂₁        ρ₂₂  ρ₃₂⎥\n",
       "⎢                        ⎥\n",
       "⎣     ρ₃₁        ρ₃₂  ρ₃₃⎦"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho=define_density_matrix(Ne,explicitly_hermitian=True,normalized=True)\n",
    "fprint( rho ,print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "hamiltonian_terms = (I/hbar*(rho*Htilde-Htilde*rho)).expand()\n",
    "lindblad_terms    =gamma[1,0]*lindblad_operator(ket(1,Ne)*bra(2,Ne),rho)\n",
    "lindblad_terms   +=gamma[2,1]*lindblad_operator(ket(2,Ne)*bra(3,Ne),rho)\n",
    "\n",
    "eqs=hamiltonian_terms + lindblad_terms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and only consider the equations for the populations $\\rho_{ii}$ for $i>1$ and the real and imaginary parts of the coherences below the diagonal."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the density matrix is represented as a vector whose components are the these independent components of the density matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADsAAADICAMAAABh/Wk2AAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n5hb/NG7QrLvc3P/MwdH4CU03X+QkR7WOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\rho_{22}\\\\\\rho_{33}\\\\\\Re{\\rho_{21}}\\\\\\Re{\\rho_{31}}\\\\\\Re{\\rho_{32}}\\\\\\Im{\\rho_{21}}\\\\\\Im{\\rho_{31}}\\\\\\Im{\\rho_{32}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  ρ₂₂  ⎤\n",
       "⎢       ⎥\n",
       "⎢  ρ₃₃  ⎥\n",
       "⎢       ⎥\n",
       "⎢re(ρ₂₁)⎥\n",
       "⎢       ⎥\n",
       "⎢re(ρ₃₁)⎥\n",
       "⎢       ⎥\n",
       "⎢re(ρ₃₂)⎥\n",
       "⎢       ⎥\n",
       "⎢im(ρ₂₁)⎥\n",
       "⎢       ⎥\n",
       "⎢im(ρ₃₁)⎥\n",
       "⎢       ⎥\n",
       "⎣im(ρ₃₂)⎦"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho_vect=define_rho_vector(rho,Ne)\n",
    "fprint(rho_vect, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then the equations can be re-written as linear combinations of these components plus an independent term.\n",
    "\\begin{equation}\n",
    "    \\dot{\\vec{\\rho}} = \\hat{A} \\vec{\\rho} - \\vec{b}\n",
    "\\end{equation}\n",
    "with $\\hat{A}$ a linear operator acting in this vector space and $\\vec{b}$ the vector of independent terms."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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5cppCaEQgBS7DBSBEXAlB9mpXaOdftfXiU4lcOe2v4pflFbgMF4DI\nmlkzYK921fiiMUcvPpXIldM0YoOY/wei/pfl5PTUSQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\left[\\begin{matrix}- \\gamma_{21} & \\gamma_{32} & \\Im{\\Omega_{1}} & 0 & - \\Im{\\Omega_{2}} & - \\Re{\\Omega_{1}} & 0 & \\Re{\\Omega_{2}}\\\\0 & - \\gamma_{32} & 0 & 0 & \\Im{\\Omega_{2}} & 0 & 0 & - \\Re{\\Omega_{2}}\\\\- \\Im{\\Omega_{1}} & - \\frac{\\Im{\\Omega_{1}}}{2} & - \\frac{\\gamma_{21}}{2} & - \\frac{\\Im{\\Omega_{2}}}{2} & 0 & - \\delta_{1} & \\frac{\\Re{\\Omega_{2}}}{2} & 0\\\\0 & 0 & \\frac{\\Im{\\Omega_{2}}}{2} & - \\frac{\\gamma_{32}}{2} & - \\frac{\\Im{\\Omega_{1}}}{2} & \\frac{\\Re{\\Omega_{2}}}{2} & - \\delta_{1} - \\delta_{2} & - \\frac{\\Re{\\Omega_{1}}}{2}\\\\\\frac{\\Im{\\Omega_{2}}}{2} & - \\frac{\\Im{\\Omega_{2}}}{2} & 0 & \\frac{\\Im{\\Omega_{1}}}{2} & - \\frac{\\gamma_{21}}{2} - \\frac{\\gamma_{32}}{2} & 0 & - \\frac{\\Re{\\Omega_{1}}}{2} & - \\delta_{2}\\\\\\Re{\\Omega_{1}} & \\frac{\\Re{\\Omega_{1}}}{2} & \\delta_{1} & - \\frac{\\Re{\\Omega_{2}}}{2} & 0 & - \\frac{\\gamma_{21}}{2} & - \\frac{\\Im{\\Omega_{2}}}{2} & 0\\\\0 & 0 & - \\frac{\\Re{\\Omega_{2}}}{2} & \\delta_{1} + \\delta_{2} & \\frac{\\Re{\\Omega_{1}}}{2} & \\frac{\\Im{\\Omega_{2}}}{2} & - \\frac{\\gamma_{32}}{2} & - \\frac{\\Im{\\Omega_{1}}}{2}\\\\- \\frac{\\Re{\\Omega_{2}}}{2} & \\frac{\\Re{\\Omega_{2}}}{2} & 0 & \\frac{\\Re{\\Omega_{1}}}{2} & \\delta_{2} & 0 & \\frac{\\Im{\\Omega_{1}}}{2} & - \\frac{\\gamma_{21}}{2} - \\frac{\\gamma_{32}}{2}\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\0\\\\- \\frac{\\Im{\\Omega_{1}}}{2}\\\\0\\\\0\\\\\\frac{\\Re{\\Omega_{1}}}{2}\\\\0\\\\0\\end{matrix}\\right]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡  -γ₂₁      γ₃₂      im(Ω₁)      0        -im(Ω₂)    -re(Ω₁)     0        re\n",
       "⎢⎢                                                                            \n",
       "⎢⎢   0        -γ₃₂       0         0        im(Ω₂)        0        0        -r\n",
       "⎢⎢                                                                            \n",
       "⎢⎢          -im(Ω₁)    -γ₂₁     -im(Ω₂)                          re(Ω₂)       \n",
       "⎢⎢-im(Ω₁)   ────────   ─────    ────────       0         -δ₁     ──────       \n",
       "⎢⎢             2         2         2                               2          \n",
       "⎢⎢                                                                            \n",
       "⎢⎢                     im(Ω₂)    -γ₃₂      -im(Ω₁)     re(Ω₂)              -re\n",
       "⎢⎢   0         0       ──────    ─────     ────────    ──────   -δ₁ - δ₂   ───\n",
       "⎢⎢                       2         2          2          2                    \n",
       "⎢⎢                                                                            \n",
       "⎢⎢ im(Ω₂)   -im(Ω₂)              im(Ω₁)     γ₂₁   γ₃₂           -re(Ω₁)       \n",
       "⎢⎢ ──────   ────────     0       ──────   - ─── - ───     0     ────────      \n",
       "⎢⎢   2         2                   2         2     2               2          \n",
       "⎢⎢                                                                            \n",
       "⎢⎢           re(Ω₁)             -re(Ω₂)                 -γ₂₁    -im(Ω₂)       \n",
       "⎢⎢ re(Ω₁)    ──────      δ₁     ────────       0        ─────   ────────      \n",
       "⎢⎢             2                   2                      2        2          \n",
       "⎢⎢                                                                            \n",
       "⎢⎢                    -re(Ω₂)               re(Ω₁)     im(Ω₂)    -γ₃₂      -im\n",
       "⎢⎢   0         0      ────────  δ₁ + δ₂     ──────     ──────    ─────     ───\n",
       "⎢⎢                       2                    2          2         2          \n",
       "⎢⎢                                                                            \n",
       "⎢⎢-re(Ω₂)    re(Ω₂)              re(Ω₁)                          im(Ω₁)     γ₂\n",
       "⎢⎢────────   ──────      0       ──────       δ₂          0      ──────   - ──\n",
       "⎣⎣   2         2                   2                               2         2\n",
       "\n",
       "(Ω₂)   ⎤, ⎡   0    ⎤⎤\n",
       "       ⎥  ⎢        ⎥⎥\n",
       "e(Ω₂)  ⎥  ⎢   0    ⎥⎥\n",
       "       ⎥  ⎢        ⎥⎥\n",
       "       ⎥  ⎢-im(Ω₁) ⎥⎥\n",
       " 0     ⎥  ⎢────────⎥⎥\n",
       "       ⎥  ⎢   2    ⎥⎥\n",
       "       ⎥  ⎢        ⎥⎥\n",
       "(Ω₁)   ⎥  ⎢   0    ⎥⎥\n",
       "─────  ⎥  ⎢        ⎥⎥\n",
       "2      ⎥  ⎢   0    ⎥⎥\n",
       "       ⎥  ⎢        ⎥⎥\n",
       "       ⎥  ⎢ re(Ω₁) ⎥⎥\n",
       "-δ₂    ⎥  ⎢ ────── ⎥⎥\n",
       "       ⎥  ⎢   2    ⎥⎥\n",
       "       ⎥  ⎢        ⎥⎥\n",
       "       ⎥  ⎢   0    ⎥⎥\n",
       " 0     ⎥  ⎢        ⎥⎥\n",
       "       ⎥  ⎣   0    ⎦⎥\n",
       "       ⎥            ⎥\n",
       "(Ω₁)   ⎥            ⎥\n",
       "─────  ⎥            ⎥\n",
       "2      ⎥            ⎥\n",
       "       ⎥            ⎥\n",
       "₁   γ₃₂⎥            ⎥\n",
       "─ - ───⎥            ⎥\n",
       "     2 ⎦            ⎦"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A,b=calculate_A_b(eqs,rho,Ne)\n",
    "fprint([A,b], print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Explicitly, this is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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XtDo6Ts3XqCTo/KPTTfOQSRA8CxTFYtGGWNqPdE+oEy0C4pBAWpviytz7+fpJfu1tkmps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\nDcYwNRP+wsspPEZZyc13Ckr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uJllnNXww0V6G7BdTZi3y7Gg6fqWG/IovCS09Sx3p7pgk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\nPXY0h81RhiHuqQNsTuTP4F8EfmYE/h+9AQVQ2I7foAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{d}{d t} \\rho_{22} & = & - \\gamma_{21} \\rho_{22} + \\gamma_{32} \\rho_{33} - \\Re{\\Omega_{1}} \\Im{\\rho_{21}} + \\Re{\\Omega_{2}} \\Im{\\rho_{32}} + \\Re{\\rho_{21}} \\Im{\\Omega_{1}} - \\Re{\\rho_{32}} \\Im{\\Omega_{2}}\\\\\\frac{d}{d t} \\rho_{33} & = & - \\gamma_{32} \\rho_{33} - \\Re{\\Omega_{2}} \\Im{\\rho_{32}} + \\Re{\\rho_{32}} \\Im{\\Omega_{2}}\\\\\\frac{d}{d t} \\Re{\\rho_{21}} & = & - \\delta_{1} \\Im{\\rho_{21}} - \\frac{\\gamma_{21} \\Re{\\rho_{21}}}{2} - \\rho_{22} \\Im{\\Omega_{1}} - \\frac{\\rho_{33} \\Im{\\Omega_{1}}}{2} + \\frac{\\Re{\\Omega_{2}} \\Im{\\rho_{31}}}{2} - \\frac{\\Re{\\rho_{31}} \\Im{\\Omega_{2}}}{2} + \\frac{\\Im{\\Omega_{1}}}{2}\\\\\\frac{d}{d t} \\Re{\\rho_{31}} & = & - \\frac{\\gamma_{32} \\Re{\\rho_{31}}}{2} + \\left(- \\delta_{1} - \\delta_{2}\\right) \\Im{\\rho_{31}} - \\frac{\\Re{\\Omega_{1}} \\Im{\\rho_{32}}}{2} + \\frac{\\Re{\\Omega_{2}} \\Im{\\rho_{21}}}{2} + \\frac{\\Re{\\rho_{21}} \\Im{\\Omega_{2}}}{2} - \\frac{\\Re{\\rho_{32}} \\Im{\\Omega_{1}}}{2}\\\\\\frac{d}{d t} \\Re{\\rho_{32}} & = & - \\delta_{2} \\Im{\\rho_{32}} + \\frac{\\rho_{22} \\Im{\\Omega_{2}}}{2} - \\frac{\\rho_{33} \\Im{\\Omega_{2}}}{2} + \\left(- \\frac{\\gamma_{21}}{2} - \\frac{\\gamma_{32}}{2}\\right) \\Re{\\rho_{32}} - \\frac{\\Re{\\Omega_{1}} \\Im{\\rho_{31}}}{2} + \\frac{\\Re{\\rho_{31}} \\Im{\\Omega_{1}}}{2}\\\\\\frac{d}{d t} \\Im{\\rho_{21}} & = & \\delta_{1} \\Re{\\rho_{21}} - \\frac{\\gamma_{21} \\Im{\\rho_{21}}}{2} + \\rho_{22} \\Re{\\Omega_{1}} + \\frac{\\rho_{33} \\Re{\\Omega_{1}}}{2} - \\frac{\\Re{\\Omega_{1}}}{2} - \\frac{\\Re{\\Omega_{2}} \\Re{\\rho_{31}}}{2} - \\frac{\\Im{\\Omega_{2}} \\Im{\\rho_{31}}}{2}\\\\\\frac{d}{d t} \\Im{\\rho_{31}} & = & - \\frac{\\gamma_{32} \\Im{\\rho_{31}}}{2} + \\left(\\delta_{1} + \\delta_{2}\\right) \\Re{\\rho_{31}} + \\frac{\\Re{\\Omega_{1}} \\Re{\\rho_{32}}}{2} - \\frac{\\Re{\\Omega_{2}} \\Re{\\rho_{21}}}{2} - \\frac{\\Im{\\Omega_{1}} \\Im{\\rho_{32}}}{2} + \\frac{\\Im{\\Omega_{2}} \\Im{\\rho_{21}}}{2}\\\\\\frac{d}{d t} \\Im{\\rho_{32}} & = & \\delta_{2} \\Re{\\rho_{32}} - \\frac{\\rho_{22} \\Re{\\Omega_{2}}}{2} + \\frac{\\rho_{33} \\Re{\\Omega_{2}}}{2} + \\left(- \\frac{\\gamma_{21}}{2} - \\frac{\\gamma_{32}}{2}\\right) \\Im{\\rho_{32}} + \\frac{\\Re{\\Omega_{1}} \\Re{\\rho_{31}}}{2} + \\frac{\\Im{\\Omega_{1}} \\Im{\\rho_{31}}}{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  d                                                                          \n",
       "⎢  ──(ρ₂₂)    =          -γ₂₁⋅ρ₂₂ + γ₃₂⋅ρ₃₃ - re(Ω₁)⋅im(ρ₂₁) + re(Ω₂)⋅im(ρ₃₂) \n",
       "⎢  dt                                                                         \n",
       "⎢                                                                             \n",
       "⎢  d                                                                          \n",
       "⎢  ──(ρ₃₃)    =                                -γ₃₂⋅ρ₃₃ - re(Ω₂)⋅im(ρ₃₂) + re(\n",
       "⎢  dt                                                                         \n",
       "⎢                                                                             \n",
       "⎢d                                 γ₂₁⋅re(ρ₂₁)                ρ₃₃⋅im(Ω₁)   re(\n",
       "⎢──(re(ρ₂₁))  =      -δ₁⋅im(ρ₂₁) - ─────────── - ρ₂₂⋅im(Ω₁) - ────────── + ───\n",
       "⎢dt                                     2                         2           \n",
       "⎢                                                                             \n",
       "⎢d                 γ₃₂⋅re(ρ₃₁)                        re(Ω₁)⋅im(ρ₃₂)   re(Ω₂)⋅\n",
       "⎢──(re(ρ₃₁))  =  - ─────────── + (-δ₁ - δ₂)⋅im(ρ₃₁) - ────────────── + ───────\n",
       "⎢dt                     2                                   2                2\n",
       "⎢                                                                             \n",
       "⎢d                                ρ₂₂⋅im(Ω₂)   ρ₃₃⋅im(Ω₂)   ⎛  γ₂₁   γ₃₂⎞     \n",
       "⎢──(re(ρ₃₂))  =     -δ₂⋅im(ρ₃₂) + ────────── - ────────── + ⎜- ─── - ───⎟⋅re(ρ\n",
       "⎢dt                                   2            2        ⎝   2     2 ⎠     \n",
       "⎢                                                                             \n",
       "⎢d                                γ₂₁⋅im(ρ₂₁)                ρ₃₃⋅re(Ω₁)   re(Ω\n",
       "⎢──(im(ρ₂₁))  =      δ₁⋅re(ρ₂₁) - ─────────── + ρ₂₂⋅re(Ω₁) + ────────── - ────\n",
       "⎢dt                                    2                         2          2 \n",
       "⎢                                                                             \n",
       "⎢d                 γ₃₂⋅im(ρ₃₁)                       re(Ω₁)⋅re(ρ₃₂)   re(Ω₂)⋅r\n",
       "⎢──(im(ρ₃₁))  =  - ─────────── + (δ₁ + δ₂)⋅re(ρ₃₁) + ────────────── - ────────\n",
       "⎢dt                     2                                  2                2 \n",
       "⎢                                                                             \n",
       "⎢d                                ρ₂₂⋅re(Ω₂)   ρ₃₃⋅re(Ω₂)   ⎛  γ₂₁   γ₃₂⎞     \n",
       "⎢──(im(ρ₃₂))  =      δ₂⋅re(ρ₃₂) - ────────── + ────────── + ⎜- ─── - ───⎟⋅im(ρ\n",
       "⎣dt                                   2            2        ⎝   2     2 ⎠     \n",
       "\n",
       "                                         ⎤\n",
       "+ re(ρ₂₁)⋅im(Ω₁) - re(ρ₃₂)⋅im(Ω₂)        ⎥\n",
       "                                         ⎥\n",
       "                                         ⎥\n",
       "                                         ⎥\n",
       "ρ₃₂)⋅im(Ω₂)                              ⎥\n",
       "                                         ⎥\n",
       "                                         ⎥\n",
       "Ω₂)⋅im(ρ₃₁)   re(ρ₃₁)⋅im(Ω₂)   im(Ω₁)    ⎥\n",
       "─────────── - ────────────── + ──────    ⎥\n",
       "   2                2            2       ⎥\n",
       "                                         ⎥\n",
       "im(ρ₂₁)   re(ρ₂₁)⋅im(Ω₂)   re(ρ₃₂)⋅im(Ω₁)⎥\n",
       "─────── + ────────────── - ──────────────⎥\n",
       "                2                2       ⎥\n",
       "                                         ⎥\n",
       "      re(Ω₁)⋅im(ρ₃₁)   re(ρ₃₁)⋅im(Ω₁)    ⎥\n",
       "₃₂) - ────────────── + ──────────────    ⎥\n",
       "            2                2           ⎥\n",
       "                                         ⎥\n",
       "₁)   re(Ω₂)⋅re(ρ₃₁)   im(Ω₂)⋅im(ρ₃₁)     ⎥\n",
       "── - ────────────── - ──────────────     ⎥\n",
       "           2                2            ⎥\n",
       "                                         ⎥\n",
       "e(ρ₂₁)   im(Ω₁)⋅im(ρ₃₂)   im(Ω₂)⋅im(ρ₂₁) ⎥\n",
       "────── - ────────────── + ────────────── ⎥\n",
       "               2                2        ⎥\n",
       "                                         ⎥\n",
       "      re(Ω₁)⋅re(ρ₃₁)   im(Ω₁)⋅im(ρ₃₁)    ⎥\n",
       "₃₂) + ────────────── + ──────────────    ⎥\n",
       "            2                2           ⎦"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqs_new=A*rho_vect - b\n",
    "\n",
    "eqs_list=[]\n",
    "for mu in range(0,Ne**2-1 ):\n",
    "    eqs_list+=[[Derivative(rho_vect[mu],t),eqsign,eqs_new[mu]]]\n",
    "eqs_list=Matrix(eqs_list)\n",
    "fprint(eqs_list, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which is the same as the equations in the previous form."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAADICAMAAADBYgHYAAAANlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABHL6OuAAAAEXRSTlMAMquZdlQQ\nQN0iRIlmze+7fEotVsoAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAFLSURBVFgJ7VjtkoMgDIwVqQqt\nl/d/2SNwfGyPdhyvnZtpwx9s1kCMm42FBo7jRGXMyUI08GjCmApCXn6fWKChWuuVB8iOzrt8H0KX\nsKidl+QJkL+K0V060BaDnNhGDLw4QiunWFvI8ii3r2x+eS3sxDakiVqvJXn1oAcLUgpj6oRB2yZ7\n+V7w6ZHH3iPTLIm6/iSxjTCYXXgX/fTKVmWgVzHLxcdDH0PRM5/Tm0eKfrUFqxRtswGFEivFmqQ8\niNASlGWXHoLf7tJTijbpVYoqRUMZ/VejVxXNGrZbvrJDnMWr5BAQgk6E0PG9qiy/l4rmTnTzLQqd\nCBs95FBV9LGKKkX//ndJKZpFDEovG9N8HFKKvpai+WwCVXSa9Vu0Mvg4e2tfrquFq2cs+F6NPifq\nptFDopCiAGmjj43+7gmnlfNMY9bKwnjCaQx9A2i9PGT7tyAIAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0\\\\0\\\\0\\\\0\\\\0\\\\0\\\\0\\\\0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0⎤\n",
       "⎢ ⎥\n",
       "⎢0⎥\n",
       "⎢ ⎥\n",
       "⎢0⎥\n",
       "⎢ ⎥\n",
       "⎢0⎥\n",
       "⎢ ⎥\n",
       "⎢0⎥\n",
       "⎢ ⎥\n",
       "⎢0⎥\n",
       "⎢ ⎥\n",
       "⎢0⎥\n",
       "⎢ ⎥\n",
       "⎣0⎦"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ss_comp={ rho[i,j]:re(rho[i,j])+I*im(rho[i,j]) for j in range(Ne) for i in range(Ne)}\n",
    "\n",
    "test= (eqs_new - Matrix([re(eqs[1,1]),re(eqs[2,2]),\n",
    "                          re(eqs[1,0]),re(eqs[2,0]),re(eqs[2,1]),\n",
    "                          im(eqs[1,0]),im(eqs[2,0]),im(eqs[2,1])])).expand()\n",
    "\n",
    "test=test.subs(ss_comp)\n",
    "fprint(test, print_ascii=print_ascii)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[1]  H.J. Metcalf and P. van der Straten. Laser Cooling and Trapping. Graduate Texts in Contempo-\n",
    "rary Physics. Springer New York, 2001.\n",
    "\n",
    "[2] Daniel Adam Steck. Quantum and Atom Optics. Oregon Center for Optics and Department of Physics, University of Oregon Copyright © 200"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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