2017_04_03-QMTrajectory-CPT-MultichannelDecay.ipynb

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 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
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   "source": [
    "import numpy as np\n",
    "from scipy import linalg\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "import itertools"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# %load SpinMatrices.py\n",
    "\"\"\"\n",
    "Created on Wed Apr  5 15:51:46 2017\n",
    "\n",
    "@author: hilaryhurst\n",
    "\"\"\"\n",
    "\n",
    "import numpy as np\n",
    "\n",
    "def SpinMatrices(S):\n",
    "    \"\"\"\n",
    "    This function takes the arguement S which must be integer, \n",
    "    then returns an array of spin matrices Sx, Sy, and Sz based on the \n",
    "    size of S.\n",
    "    \"\"\"\n",
    "    \n",
    "    m=-1*np.arange(-S,S+1,1);\n",
    "    Sz = np.diag(np.diag(np.diag(m)))\n",
    "    SPlusDiag=[];\n",
    "    for n in m:\n",
    "        SPlusDiag.append(np.sqrt(S*(S+1)-n*(n+1)))\n",
    "    SPlus = np.diag(SPlusDiag[1:2*S+1],k=1)\n",
    "    SMinus= SPlus.T\n",
    "    \n",
    "    Sx = (SPlus+SMinus)/2\n",
    "    Sy = -1j*(SPlus-SMinus)/2\n",
    "    \n",
    "    return [Sx,Sy,Sz,SPlus,SMinus]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Quantum Trajectories Simulation of Multilevel Driven-Dissipative system: Different Unravelings of the Master Equation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model and Motivation \n",
    "\n",
    "### General Concepts\n",
    "\n",
    "The goal of this notebook is to use a simple model of a driven-dissipative system with multiple jump operators to understand how the quantum trajectories formalism applies to this case. In particular, multiple jump operators allows for the idea of different \"unravelings\" of the master equation to occur - wherein we can probe the system using differen sets of POVMs or Kraus operators. In this case, we will consider Coherent Population Trapping (CPT), i.e. population of a dark state, in a two-level system with 3-fold Zeeman degeneracy. The general master equation for a system with multiple Linblad jump operators $\\hat{L}_{\\mu}$ is given by \n",
    "$$\\frac{\\partial \\hat{\\rho}}{\\partial t} = -\\frac{i}{\\hbar}\\left[H_{eff},\\hat{\\rho}\\right]' + \\sum_{\\mu = 1}^{M}\\hat{L}_\\mu\\hat{\\rho}\\hat{L}_{\\mu}^\\dagger$$ with $\\hat{H}_{eff} = \\hat{H} - \\frac{i\\hbar}{2}\\sum_{\\mu}\\hat{L}_{\\mu}^\\dagger\\hat{L}_{\\mu}$ and the $\\left[~,~\\right]'$ means we take the Hermitian conjugate of the non-Hermitian Hamiltonian $H_{eff}$ in the commutator. The probability for each possible jump is given by $\\delta p_{\\mu} = \\langle\\psi|\\hat{L}_{\\mu}^\\dagger\\hat{L}_{\\mu}|\\psi\\rangle\\delta t$ with the total jump probability $\\delta p = \\sum_{\\mu}\\delta p_{\\mu}$. \n",
    "\n",
    "The MCWF simulation runs like this: \n",
    "1. Pick a random number $0<\\epsilon<1$.\n",
    "2. Evolve the wavefunction forward in time as $|\\psi(\\delta t)\\rangle = e^{-i\\hat{H}_{eff}\\delta t/\\hbar}|\\psi(0)\\rangle$. When $|\\psi(\\tau)|^2 = \\langle\\psi(0)|e^{\\frac{i}{\\hbar}(\\hat{H}_{eff}^\\dagger-\\hat{H}_{eff})\\tau}|\\psi(0)\\rangle <\\epsilon $, then perform a jump. This is statistically equivalent to saying that when $\\delta p >\\epsilon \\rightarrow$ jump.\n",
    "3. **Choose *which* jump with another random number $0<\\eta<1$. The $\\mu$-th jump is taken for $\\delta p_{\\mu}/\\delta p > \\eta$**. (not totally clear on this step).\n",
    "4. Renormalize, then again look at the decaying norm and jump agaion when the new WF and $\\epsilon$ are at $|\\psi(\\tau_2)|^2 = \\langle\\psi(\\tau)|e^{\\frac{i}{\\hbar}(\\hat{H}_{eff}^\\dagger-\\hat{H}_{eff})\\tau_2}|\\psi(\\tau)\\rangle <\\tilde{\\epsilon}$ Repeat for times $\\tau$ as long out as you want. \n",
    "5. Finally, at the end, $|\\psi(t)\\rangle$ needs to be renormalized at each intermediate time where a jump was not taken. \n",
    "\n",
    "### Specific Example: Spontaneous Emission w/ Zeeman Degeneracy\n",
    "\n",
    "Consider an atom with ground and excited state manifolds $J_{g,e}$ with total angular momentum $F_{e,g} = 1$ and projection along some quantization axis z $m_F = 0,\\pm 1$. States $|F_{e},m_{F_e}\\rangle$ and $|F_{g},m_{F_g}\\rangle$ are connected by electric dipole transitions with $m_{e}-m_{g} = q = 0,\\pm 1$ driven by photons of polarization $\\vec{e}_q$. The jump operators $\\hat{L}_q$ then correspond to emission of a photon with $q = 0$ ($\\pi$ transition) or $q = \\pm 1$ ($\\sigma_{\\pm}$ transitions). The jump operators are then given by \n",
    "$$\\hat{L}_q = \\sqrt{\\Gamma}\\sum_{m_e,m_g}\\langle F_{e},m_e|F_{g},m_g,q\\rangle |F_g,m_g\\rangle\\langle F_e,m_e| = \\sum_{m_e,m_g}\\sqrt{\\Gamma_q}~|F_g,m_g\\rangle\\langle F_e,m_e|$$ where $\\Gamma_q = \\Gamma|\\langle F_e,m_e|F_g,m_g,1_q\\rangle| = \\Gamma C^{J_e,J_g}_{m_e,m_g,1_q}$ is the spontaneous emission rate from $|F_e,m_e\\rangle \\rightarrow |F_g,m_g\\rangle$ by emitting a photon of polarization $q$, determined by the Clebsch-Gordon coefficents $C^{J_e,J_g}_{m_e,m_g,1_q}$."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Unraveling 1: Quantization Along $\\vec{k} = \\vec{z}$\n",
    "\n",
    "First we suppose that the atom is driven by linearly polarized light with the quantization axis parallel to the momentum axis of the incoming light, $\\vec{k}~||~\\vec{e}_z$ and polarization along $\\vec{e}_y$. Thus, there is only $\\vec{e}_{\\pm 1}$ ($\\sigma_{\\pm}$) light, not $\\vec{e}_{0}$ ($\\pi$) light, because a transverse wave cannot induce transitions along its direction of propagation. The polarization momentum is related to the axes we have chosen by $\\vec{e}_0 = \\vec{e}_z$ and $\\vec{e}_{\\pm} = \\mp(\\vec{e}_x\\pm i\\vec{e}_y)/\\sqrt{2}$.\n",
    "\n",
    "The Hamiltonian for the coherent drive is given by \n",
    "$$\\hat{H} = -\\Delta_{-1}|g,-1\\rangle\\langle g,-1|-(\\Delta_{-1}-\\Delta_{1})|g,1\\rangle\\langle g,1|-\\frac{\\hbar\\Omega}{2}\\left(\\hat{D}_y+\\hat{D}_y^\\dagger\\right)$$\n",
    "Where $\\Delta_{1,-1}$ are the detunings between the $|e,0\\rangle$ state and $|g,\\pm 1\\rangle$ states, which we will set to zero, and $\\Omega$ is the rabi freqeuncy. The raising and lowering operators $\\hat{D}_y^{(\\dagger)}$ are given by $$\\hat{D}^\\dagger_y = \\sum_{q = 0,\\pm 1}\\hat{D}^\\dagger_q(\\vec{e}_q\\cdot\\vec{e}_y) = -\\frac{i}{\\sqrt{2}}(\\hat{D}^\\dagger_1 + \\hat{D}^\\dagger_{-1})$$\n",
    "$$\\hat{D}_y = \\sum_{q = 0,\\pm 1}\\hat{D}_q(\\vec{e}*_q\\cdot\\vec{e}_y) = \\frac{i}{\\sqrt{2}}(\\hat{D}_1 + \\hat{D}_{-1})$$\n",
    "\n",
    "We want to study the evolutioon of the population of the dark state using $m_z$ as our quantization axis, given by $$|\\psi_D\\rangle = \\frac{|F_g,m_z=-1\\rangle + |F_g,m_z=+1\\rangle}{\\sqrt{2}},$$ which is uncoupled from the laser due to destructive interference."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
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   "outputs": [],
   "source": [
    "S = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "Sx, Sy, Sz, Sp, Sm = SpinMatrices(S)#Zeeman sublevel space\n",
    "taux = np.array([[0,1],[1,0]]);#Excited Manifold/Ground Manifold space\n",
    "tauM = np.array([[0,1],[0,0]]);#Excited Manifold/Ground Manifold Lowering\n",
    "tauP = np.array([[0,0],[1,0]]);#Excited Manifold/Ground Manifold Raising"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def Rot(theta0,axis0):\n",
    "    rot = linalg.expm(-1j*theta0*axis0);\n",
    "    return rot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "theta = -np.pi/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "rotM = Rot(theta,Sx);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#Introduce Parameters, set hbar = 1\n",
    "delta1 = 0; #detuning\n",
    "delta2 = 0;\n",
    "Gamma=1;\n",
    "Omega = 5;#units of Gamma \n",
    "dt = 0.005; #time step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "tarray = np.arange(0,20,dt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#Formulated in Terms of Spin Matrices - May be a more efficient way to scale this up.\n",
    "D02=np.kron(tauM,Sz)\n",
    "D12=np.kron(tauM,-1/np.sqrt(2)*Sm)#1/np.sqrt(2) from Clebsch-Gordon\n",
    "Dm12=np.kron(tauM,1/np.sqrt(2)*Sp)\n",
    "\n",
    "Jump0=np.dot(np.conj(D02).T,D02)\n",
    "Jump1=np.dot(np.conj(D12).T,D12)\n",
    "Jumpm1=np.dot(np.conj(Dm12).T,Dm12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "Dy = 1j/np.sqrt(2)*(Dm12+D12)#1/np.sqrt(2) from Polarization Vectors dot product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "H=-Omega/2*(np.conj(Dy).T+Dy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "Heff=H-1j*Gamma/2*(np.dot(np.conj(D02).T,D02)+np.dot(np.conj(D12).T,D12)+np.dot(np.conj(Dm12).T,Dm12))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "psiDark=np.zeros((1,6),dtype=np.complex).T\n",
    "psiDark[0,0]=1\n",
    "psiDark[2,0]=1\n",
    "norm=np.linalg.norm(psiDark)#normalize\n",
    "psiDark=psiDark/norm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#Initialize in |g,mz=-1>\n",
    "psiinit=np.zeros((1,6),dtype=np.complex).T\n",
    "psiinit[2,0]=1\n",
    "norm=np.linalg.norm(psiinit)#normalize\n",
    "psiinit=psiinit/norm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#|e,mz = 0> state, for comparison of excited state population\n",
    "psiup=np.zeros((1,6),dtype=np.complex).T\n",
    "psiup[4,0]=1\n",
    "norm=np.linalg.norm(psiup)#normalize\n",
    "psiup=psiup/norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Single Trajectory"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.64647587919214655"
      ]
     },
     "execution_count": 150,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "epsilon = np.random.rand(1)[0];\n",
    "epsilon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "wf=psiinit;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "wfarray=[];\n",
    "dparray=[];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "t0=0;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.809327267196\n",
      "0.0504049203959\n"
     ]
    }
   ],
   "source": [
    "for el in tarray:\n",
    "    #Step 1: look at the decaying norm of the WF. When it is less than epsilon, jump. \n",
    "    #DO NOT reassign wf until this happens\n",
    "    wftemp=np.dot(linalg.expm(-1j*Heff*(el-t0)),wf)\n",
    "    norm=np.linalg.norm(wftemp)\n",
    "    if norm**2<epsilon:\n",
    "        epsilon=np.random.rand(1)[0];\n",
    "        t0=el;\n",
    "        print(epsilon)\n",
    "        #Step 2: if decaying norm is less than epsilon, we jump and then renormalize. How we pick which jump to do?\n",
    "        #projection onto excited state manifold for current WF\n",
    "        pjump=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jump0+Jump1+Jumpm1,wftemp))[0,0])*dt\n",
    "        #individual jump probabilities\n",
    "        pjump0=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jump0,wftemp))[0,0])*dt\n",
    "        pjump1=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jump1,wftemp))[0,0])*dt\n",
    "        pjumpm1=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jumpm1,wftemp))[0,0])*dt\n",
    "        #ratio of each jump probability to the total jump probability\n",
    "        pm1r=pjumpm1/pjump\n",
    "        p1r=pjump1/pjump\n",
    "        p0r=pjump0/pjump\n",
    "        #print(pm1r,p1r,p0r) - pm1r and p1r are equal to 1/2, p0r is 0 because that transition is forbidden\n",
    "        #poor man's QM jumps for equal probability options.\n",
    "        eta = np.random.rand(1)[0];#generate a new random number to determine which qm jump to take\n",
    "        if eta<p1r:\n",
    "            wftemp=np.dot(D1,wftemp)\n",
    "            norm=np.linalg.norm(wftemp)\n",
    "            wftemp=wftemp/norm\n",
    "            wf=wftemp\n",
    "            norm=np.linalg.norm(wftemp)\n",
    "        else:\n",
    "            wftemp=np.dot(Dm1,wftemp)\n",
    "            norm=np.linalg.norm(wftemp)\n",
    "            wftemp=wftemp/norm\n",
    "            wf=wftemp\n",
    "            norm=np.linalg.norm(wftemp)\n",
    "    wfarray.append(wftemp)\n",
    "    dparray.append(norm**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
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    "collapsed": false
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   "outputs": [
    {
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1zrmIedA651zEPGidcy5iHrTOORcxD1rnnIuYB61zzkXMg9Y55yLmQeuccxHz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\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x116a750b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig1 = plt.figure(figsize = (1.5*3.38,1.5*2))  \n",
    "ax1 = fig1.add_subplot(111)\n",
    "ax1.plot(list(tarray),dparray,color='k',label='Jumps')\n",
    "ax1.set_xlim([0,10])\n",
    "ax1.set_ylim([0,1.05])\n",
    "ax1.set_xlabel(r'$t$',fontsize=12)\n",
    "ax1.set_ylabel(r'$P$ $(\\tau)$',fontsize=12)\n",
    "ax1.set_title(r'Decaying Norm for a Single Run')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#Go back and normalize all the wafefunctions\n",
    "for i in range(0,len(wfarray)):\n",
    "    norm=np.linalg.norm(wfarray[i])\n",
    "    wfarray[i]=wfarray[i]/norm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "pD=np.zeros(len(wfarray));#Dark state overlap\n",
    "pE=np.zeros(len(wfarray));#Excited State mz = 0 overlap\n",
    "wfa=np.array(wfarray);\n",
    "for i in range(0,len(wfarray)):\n",
    "    pD[i]=np.absolute(np.dot(np.conj(psiDark).T,wfarray[i]))[0,0]**2\n",
    "    pE[i]=np.absolute(np.dot(np.conj(psiup).T,wfarray[i]))[0,0]**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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EiRMhbxOTlpZG3759+f3vf8/zzz/P2bNn+Y//+I+gQbt161aWLFkS8HY99957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\nGBNjFrTGGBNj/wfrcHm0r2MAHwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11c1855f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig2 = plt.figure(figsize = (1.5*3.38,1.5*2))  \n",
    "ax2 = fig2.add_subplot(111)\n",
    "ax2.plot(list(tarray),pD,color='k',label='Dark State')\n",
    "ax2.plot(list(tarray),pE,color='b',label=r'$|$ $e,m_z=0\\rangle$')\n",
    "ax2.set_xlim([0,20])\n",
    "ax2.set_ylim([0,1.05])\n",
    "ax2.set_xlabel(r'$t$',fontsize=12)\n",
    "ax2.set_ylabel(r'$|P_i|^2$',fontsize=12)\n",
    "ax2.legend(loc='best',title=r'',fontsize=10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Physical Interpretation of trajectory:\n",
    "\n",
    "In this unraveling, we see that the system gets to the dark state via a rotation. That is, the evolution under $H_{eff}$  causes the system to evolve into the dark state, whereas quantum jumps from $|e,m_z=0\\rangle$ project the state onto one of the state $|g,m_z=\\pm 1\\rangle$, thereby decreasing the overlap with the dark state, as we can see in the above figure. Quantum jumps do not explain the evolution of the state into the dark state, rather they prevent it. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Many Trajectories"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "tarray = np.arange(0,20,dt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "nruns=100\n",
    "size=(nruns,)+(len(tarray),)+psiinit.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "wfarrayall = np.zeros(size,dtype=np.complex)\n",
    "dparrayall = np.zeros((nruns,)+(len(tarray),))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "for i in range(0,nruns):\n",
    "    wf=psiinit;\n",
    "    dp=np.linalg.norm(wf)**2\n",
    "    wfarraytemp = [];\n",
    "    dparraytemp = [];\n",
    "    epsilon=np.random.rand(1)[0];\n",
    "    t0=0;\n",
    "    for el in tarray:\n",
    "        #Step 1: look at the decaying norm of the WF. When it is less than epsilon, jump. \n",
    "        #DO NOT reassign wf until this happens\n",
    "        wftemp=np.dot(linalg.expm(-1j*Heff*(el-t0)),wf)\n",
    "        norm=np.linalg.norm(wftemp)\n",
    "        if norm**2<epsilon:\n",
    "            epsilon=np.random.rand(1)[0];\n",
    "            t0=el;\n",
    "            #print(epsilon) if you want to know how many jumps you have\n",
    "            #Step 2: if decaying norm is less than epsilon, we jump and then renormalize. How we pick which jump to do?\n",
    "            #projection onto excited state manifold for current WF\n",
    "            pjump=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jump0+Jump1+Jumpm1,wftemp))[0,0])*dt\n",
    "            #individual jump probabilities\n",
    "            pjump0=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jump0,wftemp))[0,0])*dt\n",
    "            pjump1=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jump1,wftemp))[0,0])*dt\n",
    "            pjumpm1=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jumpm1,wftemp))[0,0])*dt\n",
    "            #ratio of each jump probability to the total jump probability\n",
    "            pm1r=pjumpm1/pjump\n",
    "            p1r=pjump1/pjump\n",
    "            p0r=pjump0/pjump\n",
    "            #print(pm1r,p1r,p0r) - pm1r and p1r are equal to 1/2, p0r is 0 because that transition is forbidden\n",
    "            #poor man's QM jumps for equal probability options. Not clear how to generalize to more jump options\n",
    "            eta = np.random.rand(1)[0];#generate a new random number to determine which qm jump to take\n",
    "            if eta<p1r:\n",
    "                wftemp=np.dot(D1,wftemp)\n",
    "                norm=np.linalg.norm(wftemp)\n",
    "                wftemp=wftemp/norm\n",
    "                wf=wftemp\n",
    "                norm=np.linalg.norm(wftemp)\n",
    "            else:\n",
    "                wftemp=np.dot(Dm1,wftemp)\n",
    "                norm=np.linalg.norm(wftemp)\n",
    "                wftemp=wftemp/norm\n",
    "                wf=wftemp\n",
    "                norm=np.linalg.norm(wftemp)\n",
    "        wfarraytemp.append(wftemp)\n",
    "        dparraytemp.append(norm**2)\n",
    "    \n",
    "    #Go back and normalize all the wafefunctions\n",
    "    for j in range(0,len(wfarraytemp)):\n",
    "        norm=np.linalg.norm(wfarraytemp[j])\n",
    "        wfarraytemp[j]=wfarraytemp[j]/norm\n",
    "    \n",
    "    #assign to a big array \n",
    "    wfarrayall[i]=wfarraytemp\n",
    "    dparrayall[i]=dparraytemp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "pDAll=np.zeros((nruns,len(tarray)));#Dark state probability Array\n",
    "wfa2=np.array(wfarrayall)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "for j in range(0,nruns):\n",
    "    for i in range(0,len(tarray)):\n",
    "        pDAll[j,i]=np.absolute(np.dot(np.conj(psiDark).T,wfa2[j,i])[0,0])**2 #projection onto the dark state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "pDAvg = np.zeros(len(tarray));\n",
    "for i in range(0,len(tarray)):\n",
    "    pDAvg[i]=np.mean(pDAll.T[i])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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587ngggu46KKLcq1v1KgR8fHxREdHA5CRkcHq1at9ii1YE+yPP8Irr8Dnn8OYMbBzp3PB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\nxs8s0RpjjJ9ZojXGGD+zRGuMMX72/wG4vyOCvMYuWQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x12230da90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig3 = plt.figure(figsize = (1.5*3.38,1.5*2))  \n",
    "ax3 = fig3.add_subplot(111)\n",
    "#ax3.plot(list(tarray),pDAll[0],color='y',linestyle='-',)\n",
    "ax3.plot(list(tarray),pDAll[1],color='b',linestyle='-',label=r'Single Trajectory')\n",
    "ax3.plot(list(tarray),pDAll[20],color='g',linestyle='-')\n",
    "ax3.plot(list(tarray),pDAll[35],color='r',linestyle='-')\n",
    "ax3.plot(list(tarray),pDAvg,color='k',linestyle='--',label=r'Average')\n",
    "ax3.set_xlim([0,12])\n",
    "ax3.set_ylim([0,1.05])\n",
    "ax3.set_xlabel(r'$t$',fontsize=12)\n",
    "ax3.set_ylabel(r'$|P_D|^2$',fontsize=12)\n",
    "ax3.set_title(r'$\\mathrm{Multiple}$ $\\mathrm{Trajectories}$')\n",
    "ax3.legend(loc='lower right',title=r'',fontsize=10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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H6+4MGTKE3/zmN5x++ukBRiYi6UR9tAlYsmQJZWVlfP/73w87FBFJcxmbaB999FEmTJig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\nXPSU/TkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x116aa2630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig4 = plt.figure(figsize = (1.5*3.38,1.5*2))  \n",
    "ax4 = fig4.add_subplot(111)\n",
    "ax4.plot(list(tarray),pDAvg,color='k',label=r'Average')\n",
    "ax4.set_xlim([0,20])\n",
    "ax4.set_ylim([0,1.05])\n",
    "ax4.set_xlabel(r'$t$',fontsize=12)\n",
    "ax4.set_ylabel(r'$|P_D|^2$',fontsize=12)\n",
    "ax4.set_title(r'$\\mathrm{Average}$ $\\mathrm{Trajectory}$')\n",
    "ax4.legend(loc='lower right',title=r'',fontsize=10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Unraveling 2: Quantization Axis along $\\vec{e}_y$.\n",
    "\n",
    "In this case, we choose our quantization axis to be along the polarization of the incoming laser, $\\vec{e}_y = \\vec{e}_0$. Therefore, we only drive transitions between states with the same $m_y$ number. The raising and lowering operators $\\hat{D}_y^{(\\dagger)}$ in the Hamiltonian are now given by\n",
    "\n",
    "$$\\hat{D}^{(\\dagger)}_y = \\sum_{q = 0,\\pm 1}\\hat{D}_q(\\vec{e}_q\\cdot\\vec{e}_y) = \\hat{D}^{(\\dagger)}_0$$\n",
    "With the dark state given by \n",
    "$$|\\psi_{D}\\rangle^{(y)} = |g,m_y = 0\\rangle$$\n",
    "We initialize the state again in $|g,m_z=0\\rangle$, which in the $m_y$ basis is given by \n",
    "$$|g,m_z=0\\rangle = -\\frac{i}{\\sqrt{2}}|g,m_y = 0\\rangle+\\frac{1}{2}(|g,m_y = -1\\rangle - |g,m_y = +1\\rangle)$$\n",
    "where we have rotated the (coherent) Hamiltonian and the eigenstates to the new quantization basis via $\\hat{H}^{(z)}\\rightarrow U^{\\dagger}\\hat{H}^{(z)}U = \\hat{H}^{(y)}$ and $|\\psi\\rangle^{(z)}\\rightarrow U^{\\dagger}|\\psi\\rangle^{(z)} = |\\psi\\rangle^{(y)}$ with $U = exp(i S_x\\frac{\\pi}{2})$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#Initialize in |g,mz=-1>\n",
    "psiinit_my=np.dot(np.conj(rotM).T,psiinit[0:3])\n",
    "normmy=np.linalg.norm(psiinit_my)\n",
    "psiinit_my=psiinit_my/normmy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "psiinit_my=np.concatenate((psiinit_my,np.array([[0+0j],[0+0j],[0+0j]])))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "Dy = D02#Sy is the polarization axis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "H=-Omega/2*(np.conj(Dy).T+Dy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "Heff=H-1j*Gamma/2*(np.dot(np.conj(D02).T,D02)+np.dot(np.conj(D12).T,D12)+np.dot(np.conj(Dm12).T,Dm12))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Single Trajectory"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.62836968696087392"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "epsilon = np.random.rand(1)[0];\n",
    "epsilon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "wf=psiinit_my;\n",
    "dp=np.linalg.norm(wf)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "wfarray_my=[];\n",
    "dparray_my=[];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "t0=0;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.37165678763\n",
      "0.915000185288\n"
     ]
    }
   ],
   "source": [
    "for el in tarray:\n",
    "    #Step 1: look at the decaying norm of the WF. When it is less than epsilon, jump. \n",
    "    #DO NOT reassign wf until this happens\n",
    "    wftemp=np.dot(linalg.expm(-1j*Heff*(el-t0)),wf)\n",
    "    norm=np.linalg.norm(wftemp)\n",
    "    if norm**2<epsilon:\n",
    "        epsilon=np.random.rand(1)[0];\n",
    "        t0=el;\n",
    "        print(epsilon)\n",
    "        #Step 2: if decaying norm is less than epsilon, we jump and then renormalize. How we pick which jump to do?\n",
    "        #projection onto excited state manifold for current WF\n",
    "        pjump=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jump0+Jump1+Jumpm1,wftemp))[0,0])*dt\n",
    "        #individual jump probabilities\n",
    "        pjump0=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jump0,wftemp))[0,0])*dt\n",
    "        pjump1=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jump1,wftemp))[0,0])*dt\n",
    "        pjumpm1=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jumpm1,wftemp))[0,0])*dt\n",
    "        #ratio of each jump probability to the total jump probability\n",
    "        pm1r=pjumpm1/pjump\n",
    "        p1r=pjump1/pjump\n",
    "        p0r=pjump0/pjump\n",
    "        #print(pm1r,p1r,p0r) - pm1r and p1r are equal to 1/4, p0r = 1/2. Not clear how to rigorously do this for multiple jumps\n",
    "        eta = np.random.rand(1)[0];#generate a new random number to determine which qm jump to take\n",
    "        if eta<p0r:\n",
    "            wftemp=np.dot(D02,wftemp)\n",
    "            norm=np.linalg.norm(wftemp)\n",
    "            wftemp=wftemp/norm\n",
    "            wf=wftemp\n",
    "            norm=np.linalg.norm(wftemp)\n",
    "        else:\n",
    "            eta2=np.random.rand(1)[0];\n",
    "            if eta2<p1r:\n",
    "                wftemp=np.dot(D12,wftemp)\n",
    "                norm=np.linalg.norm(wftemp)\n",
    "                wftemp=wftemp/norm\n",
    "                wf=wftemp\n",
    "                norm=np.linalg.norm(wftemp)\n",
    "            else:\n",
    "                wftemp=np.dot(Dm12,wftemp)\n",
    "                norm=np.linalg.norm(wftemp)\n",
    "                wftemp=wftemp/norm\n",
    "                wf=wftemp\n",
    "                norm=np.linalg.norm(wftemp)\n",
    "    wfarray_my.append(wftemp)\n",
    "    dparray_my.append(norm**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#Go back and normalize all the wafefunctions\n",
    "for i in range(0,len(wfarray_my)):\n",
    "    norm_my=np.linalg.norm(wfarray_my[i])\n",
    "    wfarray_my[i]=wfarray_my[i]/norm_my"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "psiDark_my=np.zeros((1,6),dtype=np.complex).T\n",
    "psiDark_my[1,0]=1\n",
    "norm=np.linalg.norm(psiDark_my)#normalize\n",
    "psiDark_my=psiDark_my/norm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "pD_my=np.zeros(len(wfarray_my));#Dark state overlap\n",
    "wfa_my=np.array(wfarray_my);\n",
    "for i in range(0,len(wfarray)):\n",
    "    pD_my[i]=np.absolute(np.dot(np.conj(psiDark_my).T,wfa_my[i]))[0,0]**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ncv311/Pe974373IA92jNyqInQTsgneYQSD5dQDLVYeGsW7eOE044gYsvvrgwLQP3aM3K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\nWcYctGZmGft/wZZUnYkkBIUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x121b94cc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig5 = plt.figure(figsize = (1.5*3.38,1.5*2))  \n",
    "ax5 = fig5.add_subplot(111)\n",
    "ax5.plot(list(tarray),pD_my,color='k',label='Dark State')\n",
    "ax5.set_xlim([0,10])\n",
    "ax5.set_ylim([0,1.05])\n",
    "ax5.set_xlabel(r'$t$',fontsize=12)\n",
    "ax5.set_ylabel(r'$|P_D|^2$',fontsize=12)\n",
    "ax5.set_title(r'$\\mathrm{Single}$ $\\mathrm{Trajectory}$')\n",
    "#ax5.legend(loc='lower right',title=r'',fontsize=10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Physical Interpretation of trajectory:\n",
    "\n",
    "In this unraveling, we see that the system begins to rotate toward the dark state, but then a quantum jump occurs. If we see a photon of polarization $q=0$, as in the above trajectory, the system is projected onto either $|g,m_y=\\pm 1$, which is orthogonal to the dark state, thus the overlap with the dark state is zero. After some time, it is pumped to the $|e,m_y=\\pm 1$ state (which also has zero overlap with the dark state), but eventually we might se a photon of $q=\\pm 1$ at this point, we know the system is prjected into the dark state, where it stays. Thus, this trajectory is more discontinuous, and quantum jumps are what eventually propel us to the dark state, unlike the other example above. **Note** It is also possible that the first thing we see is a photon of polarization $q = \\pm 1$. In this case, we project directly onto the dark state and evolution stops."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### Many Trajectories"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "nruns=100\n",
    "size=(nruns,)+(len(tarray),)+psiinit_my.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "wfarrayall_my = np.zeros(size,dtype=np.complex)\n",
    "dparrayall_my = np.zeros((nruns,)+(len(tarray),))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "eps = 0.0931437374159\n",
      "eta = 0.602438531036\n",
      "eta2 = 0.98037978433\n",
      "eps = 0.00666314232247\n",
      "eta = 0.16522223715\n",
      "eps = 0.293795426668\n",
      "eta = 0.380985555589\n",
      "eps = 0.648562832422\n",
      "eta = 0.0768754941791\n",
      "eps = 0.438532850913\n",
      "eta = 0.549593235362\n",
      "eta2 = 0.7260000908\n",
      "eps = 0.406445518381\n",
      "eta = 0.449868425653\n",
      "eps = 0.375486208237\n",
      "eta = 0.877316471482\n",
      "eta2 = 0.586394035126\n",
      "eps = 0.467787056365\n",
      "eta = 0.411672307023\n",
      "eps = 0.493452330878\n",
      "eta = 0.75836573384\n",
      "eta2 = 0.571663300052\n",
      "eps = 0.932759253329\n",
      "eta = 0.358312117034\n",
      "eps = 0.744758867468\n",
      "eta = 0.539785863942\n",
      "eta2 = 0.693678840085\n",
      "eps = 0.67461305719\n",
      "eta = 0.206045997435\n",
      "eps = 0.303534655531\n",
      "eta = 0.381847741807\n",
      "eps = 0.402267561335\n",
      "eta = 0.782808227357\n",
      "eta2 = 0.843895646442\n",
      "eps = 0.505355407603\n",
      "eta = 0.0276242716857\n",
      "eps = 0.593351600832\n",
      "eta = 0.478528202941\n",
      "eps = 0.716051700968\n",
      "eta = 0.945053784704\n",
      "eta2 = 0.0110219168637\n",
      "eps = 0.771621464482\n",
      "eta = 0.331852400895\n",
      "eps = 0.240140893494\n",
      "eta = 0.741283782697\n",
      "eta2 = 0.299504534931\n",
      "eps = 0.912703074217\n",
      "eta = 0.530003864111\n",
      "eta2 = 0.305779784122\n",
      "eps = 0.409149329618\n",
      "eta = 0.894227694683\n",
      "eta2 = 0.86320860281\n",
      "eps = 0.669070600449\n",
      "eta = 0.0874065182212\n",
      "eps = 0.994379385285\n",
      "eta = 0.0287434985725\n",
      "eps = 0.736768772597\n",
      "eta = 0.18952272203\n",
      "eps = 0.80978017847\n",
      "eta = 0.842757733488\n",
      "eta2 = 0.591654564995\n",
      "eps = 0.301607776441\n",
      "eta = 0.372032912241\n",
      "eps = 0.00572475724949\n",
      "eta = 0.797188378995\n",
      "eta2 = 0.188711191054\n",
      "eps = 0.0105545431379\n",
      "eta = 0.274729919688\n",
      "eps = 0.18846612716\n",
      "eta = 0.815033668907\n",
      "eta2 = 0.292974220958\n",
      "eps = 0.802351896219\n",
      "eta = 0.856296597353\n",
      "eta2 = 0.768068699434\n",
      "eps = 0.829434899312\n",
      "eta = 0.660363086633\n",
      "eta2 = 0.806712663269\n",
      "eps = 0.271343009904\n",
      "eta = 0.225881427402\n",
      "eps = 0.153236998445\n",
      "eta = 0.595883240888\n",
      "eta2 = 0.648872658511\n",
      "eps = 0.703340545711\n",
      "eta = 0.325589173576\n",
      "eps = 0.408915626847\n",
      "eta = 0.0867915183059\n",
      "eps = 0.475327572171\n",
      "eta = 0.810104001924\n",
      "eta2 = 0.213978860832\n",
      "eps = 0.13476014875\n",
      "eta = 0.429895391494\n",
      "eps = 0.896401183033\n",
      "eta = 0.666616267907\n",
      "eta2 = 0.741463941404\n",
      "eps = 0.322138460934\n",
      "eta = 0.194241814112\n",
      "eps = 0.56328943902\n",
      "eta = 0.479019157358\n",
      "eps = 0.4388395518\n",
      "eta = 0.283162245952\n",
      "eps = 0.891453781152\n",
      "eta = 0.0166194883322\n",
      "eps = 0.720884746397\n",
      "eta = 0.271365903878\n",
      "eps = 0.46787388618\n",
      "eta = 0.445661681597\n",
      "eps = 0.412227343815\n",
      "eta = 0.287567590509\n",
      "eps = 0.144192056136\n",
      "eta = 0.259631993078\n",
      "eps = 0.396338062982\n",
      "eta = 0.0955376535938\n",
      "eps = 0.636144184353\n",
      "eta = 0.0581216433141\n",
      "eps = 0.282866983358\n",
      "eta = 0.0205974732755\n",
      "eps = 0.790308871429\n",
      "eta = 0.022220672698\n",
      "eps = 0.76870334188\n",
      "eta = 0.86693023697\n",
      "eta2 = 0.149647087481\n",
      "eps = 0.857756310544\n",
      "eta = 0.294773890474\n",
      "eps = 0.876283294963\n",
      "eta = 0.132558936909\n",
      "eps = 0.472985493314\n",
      "eta = 0.740050127337\n",
      "eta2 = 0.172316312391\n",
      "eps = 0.206053494397\n",
      "eta = 0.171910775118\n",
      "eps = 0.0603159589273\n",
      "eta = 0.142167151741\n",
      "eps = 0.69200072334\n",
      "eta = 0.576136765819\n",
      "eta2 = 0.38217337914\n",
      "eps = 0.138910204812\n",
      "eta = 0.230893433259\n",
      "eps = 0.106165737458\n",
      "eta = 0.339941089567\n",
      "eps = 0.295711338792\n",
      "eta = 0.00486051146521\n",
      "eps = 0.539090640139\n",
      "eta = 0.940155240479\n",
      "eta2 = 0.238235112445\n",
      "eps = 0.728679559833\n",
      "eta = 0.671620483625\n",
      "eta2 = 0.713730867245\n",
      "eps = 0.0531995555663\n",
      "eta = 0.791053061848\n",
      "eta2 = 0.0462212387362\n",
      "eps = 0.167964443515\n",
      "eta = 0.938482818262\n",
      "eta2 = 0.656977175508\n",
      "eps = 0.704872324325\n",
      "eta = 0.441343840499\n",
      "eps = 0.681944658124\n",
      "eta = 0.752251430815\n",
      "eta2 = 0.396892221058\n",
      "eps = 0.587637253092\n",
      "eta = 0.916802762932\n",
      "eta2 = 0.191464524107\n",
      "eps = 0.136729700685\n",
      "eta = 0.421969037829\n",
      "eps = 0.516557395894\n",
      "eta = 0.0429421269078\n",
      "eps = 0.483702521902\n",
      "eta = 0.922031156835\n",
      "eta2 = 0.0344529051761\n",
      "eps = 0.0244539232888\n",
      "eta = 0.0106401742678\n",
      "eps = 0.831938210857\n",
      "eta = 0.582808313399\n",
      "eta2 = 0.499575860366\n",
      "eps = 0.640361555223\n",
      "eta = 0.561457193544\n",
      "eta2 = 0.787272906229\n",
      "eps = 0.447808388919\n",
      "eta = 0.937993047194\n",
      "eta2 = 0.472757275396\n",
      "eps = 0.0810272981176\n",
      "eta = 0.265966144329\n",
      "eps = 0.175236812366\n",
      "eta = 0.467705640453\n",
      "eps = 0.381748496848\n",
      "eta = 0.464268993642\n",
      "eps = 0.700042182561\n",
      "eta = 0.457394942395\n",
      "eps = 0.0718505168391\n",
      "eta = 0.785358033007\n",
      "eta2 = 0.700055775128\n",
      "eps = 0.754582009651\n",
      "eta = 0.696611917248\n",
      "eta2 = 0.710309142234\n",
      "eps = 0.193384879719\n",
      "eta = 0.754030987864\n",
      "eta2 = 0.254471167387\n",
      "eps = 0.97354465814\n",
      "eta = 0.0690212847257\n",
      "eps = 0.885511939591\n",
      "eta = 0.523794013106\n",
      "eta2 = 0.618782182969\n",
      "eps = 0.386109543202\n",
      "eta = 0.999270334761\n",
      "eta2 = 0.855846968645\n",
      "eps = 0.403978499374\n",
      "eta = 0.125017037526\n",
      "eps = 0.420087029831\n",
      "eta = 0.292956409026\n",
      "eps = 0.79836040468\n",
      "eta = 0.930138222478\n",
      "eta2 = 0.547532310964\n",
      "eps = 0.124995945738\n",
      "eta = 0.492827352799\n",
      "eps = 0.574786342473\n",
      "eta = 0.788278962192\n",
      "eta2 = 0.853309009736\n",
      "eps = 0.436066978381\n",
      "eta = 0.457587936304\n",
      "eps = 0.664264533126\n",
      "eta = 0.0292382407321\n",
      "eps = 0.13644634889\n",
      "eta = 0.284537830168\n",
      "eps = 0.374497945909\n",
      "eta = 0.555222837418\n",
      "eta2 = 0.48058155898\n",
      "eps = 0.0335362020738\n",
      "eta = 0.840537661393\n",
      "eta2 = 0.782293296662\n",
      "eps = 0.185426435576\n",
      "eta = 0.641368715179\n",
      "eta2 = 0.100777189585\n",
      "eps = 0.534189749275\n",
      "eta = 0.500840440803\n",
      "eta2 = 0.38308309155\n",
      "eps = 0.755048258824\n",
      "eta = 0.706485207136\n",
      "eta2 = 0.507235523252\n",
      "eps = 0.650022514487\n",
      "eta = 0.0987251773225\n",
      "eps = 0.134820553748\n",
      "eta = 0.738104014822\n",
      "eta2 = 0.303423299591\n",
      "eps = 0.741750190145\n",
      "eta = 0.988531592415\n",
      "eta2 = 0.540279643982\n",
      "eps = 0.958978299274\n",
      "eta = 0.11948022782\n",
      "eps = 0.864257604834\n",
      "eta = 0.346106597967\n",
      "eps = 0.107472244467\n",
      "eta = 0.649703197514\n",
      "eta2 = 0.417700681722\n",
      "eps = 0.88216262627\n",
      "eta = 0.016715852186\n",
      "eps = 0.461388187904\n",
      "eta = 0.150592624601\n",
      "eps = 0.313713515249\n",
      "eta = 0.499524652396\n",
      "eps = 0.307483657414\n",
      "eta = 0.23330667102\n",
      "eps = 0.0338995656527\n",
      "eta = 0.180841875148\n",
      "eps = 0.312541297968\n",
      "eta = 0.837773292101\n",
      "eta2 = 0.570572824143\n",
      "eps = 0.0774593892396\n",
      "eta = 0.609869214564\n",
      "eta2 = 0.671985045844\n",
      "eps = 0.153584877998\n",
      "eta = 0.635486229901\n",
      "eta2 = 0.216520227993\n",
      "eps = 0.063608552038\n",
      "eta = 0.340486987237\n",
      "eps = 0.0248274865395\n",
      "eta = 0.631301739947\n",
      "eta2 = 0.126773097167\n",
      "eps = 0.131000457596\n",
      "eta = 0.886612469461\n",
      "eta2 = 0.25690584998\n",
      "eps = 0.617203595973\n",
      "eta = 0.986485543422\n",
      "eta2 = 0.632834674006\n",
      "eps = 0.919796679377\n",
      "eta = 0.488368930198\n",
      "eps = 0.740131999899\n",
      "eta = 0.961421444732\n",
      "eta2 = 0.169457172682\n",
      "eps = 0.558806379737\n",
      "eta = 0.829410745975\n",
      "eta2 = 0.379017607067\n"
     ]
    }
   ],
   "source": [
    "for i in range(0,nruns):\n",
    "    wf=psiinit_my;\n",
    "    wfarraytemp = [];\n",
    "    dparraytemp = [];\n",
    "    epsilon=np.random.rand(1)[0];\n",
    "    t0=0;\n",
    "    for el in tarray:\n",
    "        #Step 1: look at the decaying norm of the WF. When it is less than epsilon, jump. \n",
    "        #DO NOT reassign wf until this happens\n",
    "        wftemp=np.dot(linalg.expm(-1j*Heff*(el-t0)),wf)\n",
    "        norm=np.linalg.norm(wftemp)\n",
    "        if norm**2<epsilon:\n",
    "            epsilon=np.random.rand(1)[0];\n",
    "            t0=el;\n",
    "            print('eps = '+str(epsilon))\n",
    "            #Step 2: if decaying norm is less than epsilon, we jump and then renormalize. How we pick which jump to do?\n",
    "            #projection onto excited state manifold for current WF\n",
    "            pjump=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jump0+Jump1+Jumpm1,wftemp))[0,0])*dt\n",
    "            #individual jump probabilities\n",
    "            pjump0=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jump0,wftemp))[0,0])*dt\n",
    "            pjump1=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jump1,wftemp))[0,0])*dt\n",
    "            pjumpm1=np.absolute(np.dot(np.conj(wftemp).T,np.dot(Jumpm1,wftemp))[0,0])*dt\n",
    "            #ratio of each jump probability to the total jump probability\n",
    "            pm1r=pjumpm1/pjump\n",
    "            p1r=pjump1/pjump\n",
    "            p0r=pjump0/pjump\n",
    "            eta = np.random.rand(1)[0];#generate a new random number to determine which qm jump to take\n",
    "            print('eta = '+str(eta))\n",
    "            if eta<p0r:\n",
    "                wftemp=np.dot(D02,wftemp)\n",
    "                norm=np.linalg.norm(wftemp)\n",
    "                wftemp=wftemp/norm\n",
    "                wf=wftemp\n",
    "                norm=np.linalg.norm(wftemp)\n",
    "            else:\n",
    "                #now generate a NEW number if you rule out the highest probability jump.\n",
    "                eta2=np.random.rand(1)[0];\n",
    "                print('eta2 = '+str(eta2))\n",
    "                if eta2<p1r:\n",
    "                    wftemp=np.dot(D12,wftemp)\n",
    "                    norm=np.linalg.norm(wftemp)\n",
    "                    wftemp=wftemp/norm\n",
    "                    wf=wftemp\n",
    "                    norm=np.linalg.norm(wftemp)\n",
    "                else:\n",
    "                    wftemp=np.dot(Dm12,wftemp)\n",
    "                    norm=np.linalg.norm(wftemp)\n",
    "                    wftemp=wftemp/norm\n",
    "                    wf=wftemp\n",
    "                    norm=np.linalg.norm(wftemp)\n",
    "        wfarraytemp.append(wftemp)\n",
    "        dparraytemp.append(norm**2)\n",
    "    \n",
    "    #Go back and normalize all the wafefunctions\n",
    "    for j in range(0,len(wfarraytemp)):\n",
    "        norm=np.linalg.norm(wfarraytemp[j])\n",
    "        wfarraytemp[j]=wfarraytemp[j]/norm\n",
    "    \n",
    "    wfarrayall_my[i]=wfarraytemp\n",
    "    dparrayall_my[i]=dparraytemp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "pDAll_my=np.zeros((nruns,len(tarray)));#Dark state probability Array\n",
    "pEAll_my=np.zeros((nruns,len(tarray)));#|e,my=-1> probability Array\n",
    "pSAll_my=np.zeros((nruns,len(tarray)));#|g,my=-1> probability Array\n",
    "wfa2_my=np.array(wfarrayall_my)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "for j in range(0,nruns):\n",
    "    for i in range(0,len(tarray)):\n",
    "        pDAll_my[j,i]=np.absolute(np.dot(np.conj(psiDark_my).T,wfa2_my[j,i])[0,0])**2\n",
    "        pEAll_my[j,i]=np.absolute(wfa2_my[j,i,5,0])**2\n",
    "        pSAll_my[j,i]=np.absolute(wfa2_my[j,i,2,0])**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "pDAvg_my = np.zeros(len(tarray));\n",
    "for i in range(0,len(tarray)):\n",
    "    pDAvg_my[i]=np.mean(pDAll_my.T[i])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Q5OZpAwptdnaQmWICR7S+9gIKrZ89TbJotjjEsrDs37+fGjVqULVqVWbNmsXO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\nY0JrGIYRY0xoDcMwYowJrWEYRoz5fxUEAo8ud4ADAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e30b5c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig6 = plt.figure(figsize = (1.5*3.38,1.5*2))  \n",
    "ax6 = fig6.add_subplot(111)\n",
    "ax6.plot(list(tarray),pDAll_my[9],color='k',linestyle='-',label=r'$\\mathrm{Dark}$ $\\mathrm{state:}$ $|g,m_y = 0\\rangle$')\n",
    "ax6.plot(list(tarray),pEAll_my[9],color='r',linestyle='-',label=r'$|e,m_y = -1\\rangle$')\n",
    "ax6.plot(list(tarray),pSAll_my[9],color='b',linestyle='-',label=r'$|g,m_y = -1\\rangle$')\n",
    "ax6.set_xlim([0,10])\n",
    "ax6.set_ylim([0,1.05])\n",
    "ax6.set_xlabel(r'$t$',fontsize=12)\n",
    "ax6.set_ylabel(r'$|P_D|^2$',fontsize=12)\n",
    "ax6.set_title('Example evolution of one trajectory')\n",
    "ax6.legend(loc='lower right',title=r'',fontsize=10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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jebTMBO/dYU+HITZjjIkIZR74W1XzgOkhiMWEgB2mG+M+IXnCgqquCkU9JjJZ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\nMLNEa4wxYWaJ1hhjwswSrTHGhNn/A7Kvj1sHazmiAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x116b9b6a0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig7 = plt.figure(figsize = (1.5*3.38,1.5*2))  \n",
    "ax7 = fig7.add_subplot(111)\n",
    "ax7.plot(list(tarray),pDAll_my[0],color='y',linestyle='-',)\n",
    "ax7.plot(list(tarray),pDAll_my[9],color='b',linestyle='-',label=r'Single Trajectory')\n",
    "ax7.plot(list(tarray),pDAll_my[45],color='g',linestyle='-')\n",
    "#ax7.plot(list(tarray),pDAll_my[89],color='r',linestyle='-')\n",
    "ax7.plot(list(tarray),pDAvg_my,color='k',linestyle='--',label=r'Average')\n",
    "ax7.set_xlim([0,10])\n",
    "ax7.set_ylim([0,1.05])\n",
    "ax7.set_xlabel(r'$t$',fontsize=12)\n",
    "ax7.set_ylabel(r'$|P_D|^2$',fontsize=12)\n",
    "ax7.set_title(r'$\\mathrm{Multiple}$ $\\mathrm{Trajectories}$')\n",
    "ax7.legend(loc='lower right',title=r'',fontsize=10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ObIzVXWZ2fdT1SOOmQWWkyQpGrfqFu6+PuhZp3BS00mSZ2Tvu3iPqOqTxU9eB\nNElm1oPYzfYws7ERlyONnIJWmqoPgf1ByOZEXIs0cuo6EBEJmY5oRURCpqAVEQmZglZEJGQKWhGR\nkCloRURCpqAVEQmZglZEJGQKWhGRkP1/Nz6DE4TqL18AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11c18f630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig8 = plt.figure(figsize = (1.5*3.38,1.5*2))  \n",
    "ax8 = fig8.add_subplot(111)\n",
    "ax8.plot(list(tarray),pDAvg_my,color='k',label=r'Average')\n",
    "ax8.set_xlim([0,20])\n",
    "ax8.set_ylim([0,1.05])\n",
    "ax8.set_xlabel(r'$t$',fontsize=12)\n",
    "ax8.set_ylabel(r'$|P_D|^2$',fontsize=12)\n",
    "ax8.set_title(r'$\\mathrm{Average}$ $\\mathrm{Trajectory}$')\n",
    "ax8.legend(loc='lower right',title=r'',fontsize=10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Comparison Between Unravellings"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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0dPeFFWz/EvBWeaPFzMx1VSlSc4Unyw2sEollPgt/HmhO27lzJ40bNyY/P7/a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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e24d160>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig9 = plt.figure(figsize = (2*1.5*3.38,2*1.5*2))\n",
    "\n",
    "ax9a = fig9.add_subplot(221)\n",
    "ax9a.plot(list(tarray),pD,color='k',label='Dark State')\n",
    "ax9a.set_xlim([0,12])\n",
    "ax9a.set_ylim([0,1.05])\n",
    "#ax9a.set_xlabel(r'$t$',fontsize=12)\n",
    "ax9a.set_ylabel(r'$|P_D|^2$',fontsize=12)\n",
    "ax9a.set_title(r'$\\mathrm{Single}$ $\\mathrm{Trajectory}$')\n",
    "\n",
    "ax9b = fig9.add_subplot(222)\n",
    "ax9b.plot(list(tarray),pD_my,color='k',label='Dark State')\n",
    "ax9b.set_xlim([0,12])\n",
    "ax9b.set_ylim([0,1.05])\n",
    "#ax9b.set_xlabel(r'$t$',fontsize=12)\n",
    "#ax9b.set_ylabel(r'$|P_D|^2$',fontsize=12)\n",
    "ax9b.set_title(r'$\\mathrm{Single}$ $\\mathrm{Trajectory}$')\n",
    "\n",
    "ax9a = fig9.add_subplot(223)\n",
    "ax9a.plot(list(tarray),pDAvg,color='k',label=r'Average')\n",
    "ax9a.set_xlim([0,12])\n",
    "ax9a.set_ylim([0,1.05])\n",
    "ax9a.set_xlabel(r'$\\Gamma t$',fontsize=12)\n",
    "ax9a.set_ylabel(r'$|P_D|^2$',fontsize=12)\n",
    "ax9a.set_title(r'$\\mathrm{Average}$ $\\mathrm{Trajectory}$')\n",
    "#ax9a.legend(loc='lower right',title=r'',fontsize=10)\n",
    "\n",
    "ax9d = fig9.add_subplot(224)\n",
    "ax9d.plot(list(tarray),pDAvg_my,color='k',label=r'Average')\n",
    "ax9d.set_xlim([0,12])\n",
    "ax9d.set_ylim([0,1.05])\n",
    "ax9d.set_xlabel(r'$\\Gamma t$',fontsize=12)\n",
    "#ax9d.set_ylabel(r'$|P_D|^2$',fontsize=12)\n",
    "ax9d.set_title(r'$\\mathrm{Average}$ $\\mathrm{Trajectory}$')\n",
    "#ax9b.legend(loc='lower right',title=r'',fontsize=10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On average, both unravelings show similar behavior, however the physical interpretation of individual trajectories is very different. Additionally, we see that the second unraveling (where we choose the quantization axis to be parallel to the polarization axis) the average is much noisier because of the discontinuities in the trajectories. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [Root]",
   "language": "python",
   "name": "Python [Root]"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}