Lab 2 - Quantum States Solutions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lab 2 - Quantum States, SOLUTIONS\n",
    "Useful for working examples and problems with photon quantum states. You may notice some similarity to the Jones Calculus ;-)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from qutip import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These are the polarization states:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "H = Qobj([[1],[0]])\n",
    "V = Qobj([[0],[1]])\n",
    "P45 = Qobj([[1/np.sqrt(2)],[1/np.sqrt(2)]])\n",
    "M45 = Qobj([[1/np.sqrt(2)],[-1/np.sqrt(2)]])\n",
    "R = Qobj([[1/np.sqrt(2)],[-1j/np.sqrt(2)]])\n",
    "L = Qobj([[1/np.sqrt(2)],[1j/np.sqrt(2)]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0\\\\1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\n",
       "Qobj data =\n",
       "[[0.]\n",
       " [1.]]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\n",
       "Qobj data =\n",
       "[[0.]]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hbra*V"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "Hbra = H.dag()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Devices:\n",
    "\n",
    "HWP - Half-wave plate axis at $\\theta$ to the horizontal\n",
    "\n",
    "LP - Linear polarizer, axis at $\\theta$\n",
    "\n",
    "QWP - Quarter-wave plate, axis at $\\theta$\n",
    "\n",
    "Note, these are functions so you need to call them with a specific value of theta."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "def HWP(theta):\n",
    "    return Qobj([[np.cos(2*theta),np.sin(2*theta)],[np.sin(2*theta),-np.cos(2*theta)]]).tidyup()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "def LP(theta):\n",
    "    return Qobj([[np.cos(theta)**2,np.cos(theta)*np.sin(theta)],[np.sin(theta)*np.cos(theta),np.sin(theta)**2]]).tidyup()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "def QWP(theta):\n",
    "    return Qobj([[np.cos(theta)**2 + 1j*np.sin(theta)**2,\n",
    "                 (1-1j)*np.sin(theta)*np.cos(theta)],\n",
    "                [(1-1j)*np.sin(theta)*np.cos(theta),\n",
    "                 np.sin(theta)**2 + 1j*np.cos(theta)**2]]).tidyup()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\\begin{equation*}\\left(\\begin{array}{*{11}c}(0.500+0.500j) & (0.500-0.500j)\\\\(0.500-0.500j) & (0.500+0.500j)\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\n",
       "Qobj data =\n",
       "[[0.5+0.5j 0.5-0.5j]\n",
       " [0.5-0.5j 0.5+0.5j]]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QWP(np.pi/4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 1) Check that the $|H\\rangle$ state is normalized"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\n",
       "Qobj data =\n",
       "[[1.]]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H.dag()*H"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To show more information on an object, use the question mark after the function or object:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.sin?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 2) Converting from ket to bra:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}(1.0+1.0j)\\\\(2.0-1.0j)\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\n",
       "Qobj data =\n",
       "[[1.+1.j]\n",
       " [2.-1.j]]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi = Qobj([[1+1j],[2-1j]])\n",
    "psi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [2]], shape = (1, 2), type = bra\\begin{equation*}\\left(\\begin{array}{*{11}c}(1.0-1.0j) & (2.0+1.0j)\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [2]], shape = (1, 2), type = bra\n",
       "Qobj data =\n",
       "[[1.-1.j 2.+1.j]]"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi.dag()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}(1.0+1.0j)\\\\(2.0-1.0j)\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\n",
       "Qobj data =\n",
       "[[1.+1.j]\n",
       " [2.-1.j]]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi.dag().dag()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "the `.dag()` python method computes the \"daggar\" or the complex transpose."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1) Is `psi` normalized? If not, find the normalization constant and confirm that constant normalizes `psi`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\\begin{equation*}\\left(\\begin{array}{*{11}c}7.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\n",
       "Qobj data =\n",
       "[[7.]]"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi.dag()*psi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "psi_norm = psi*np.sqrt(1/7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\\begin{equation*}\\left(\\begin{array}{*{11}c}1.000\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\n",
       "Qobj data =\n",
       "[[1.]]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi_norm.dag() * psi_norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2) Verify that the $|V\\rangle$ state is normalized"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\n",
       "Qobj data =\n",
       "[[1.]]"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V.dag()*V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3) Verify that the $|H\\rangle$ and $|V\\rangle$ states are orthogonal. Repeat for the other pairs of states."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\n",
       "Qobj data =\n",
       "[[0.]]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H.dag()*V"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\n",
       "Qobj data =\n",
       "[[0.]]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L.dag()*R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\n",
       "Qobj data =\n",
       "[[0.]]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P45.dag()*M45"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4) Calculate the horizontal component $c_H$ of the state $\\psi = \\frac{1}{\\sqrt{5}}|H\\rangle + \\frac{2}{\\sqrt{5}}|V\\rangle$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.447\\\\0.894\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\n",
       "Qobj data =\n",
       "[[0.4472136 ]\n",
       " [0.89442719]]"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi = 1/np.sqrt(5)*H + 2/np.sqrt(5)*V\n",
    "psi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.447\\\\0.894\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\n",
       "Qobj data =\n",
       "[[0.4472136 ]\n",
       " [0.89442719]]"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi2 = Qobj([[1/np.sqrt(5)],[2/np.sqrt(5)]])\n",
    "psi2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\\begin{equation*}\\left(\\begin{array}{*{11}c}0.447\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\n",
       "Qobj data =\n",
       "[[0.4472136]]"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H.dag()*psi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5) Verify Eq. (3.18), $P(H||45\\rangle)=\\frac{1}{2},$ (which states \"The probability that a photon prepared in the +45 state will leave a PA_HV in the Horizontal state is one half.\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4999999999999999"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(H.dag()*P45).norm()**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.5+0.j]])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.conjugate(H.dag()*P45) * (H.dag()*P45)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6) Demonstrate that a half-wave plate at 45-degrees converts $|H\\rangle$ to $|V\\rangle$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0\\\\1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\n",
       "Qobj data =\n",
       "[[0.]\n",
       " [1.]]"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "HWP(np.radians(45)) * H"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "HWP(np.pi/4) * H == V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7) Re-create Figure 3.9 by plotting the probability P(+45) vs phase φ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1a1fb031d0>]"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot([1,2,3,2,3,4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [],
   "source": [
    "phi_list = np.linspace(0,8*np.pi,num=100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1a20429c10>]"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(phi,np.sin(phi))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [],
   "source": [
    "sin_list = [np.sin(p) for p in phi] # notes "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "add text "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1a205661d0>]"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(phi,sin_list)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [],
   "source": [
    "def psi(phi):\n",
    "    return 1/np.sqrt(2)*(H + np.exp(1j*phi)*V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [],
   "source": [
    "answer = [(M45.dag()*psi(phi)).norm()**2 for phi in phi_list]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'P(-45)')"
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(phi_list/np.pi,answer,\"-o\")\n",
    "plt.xlabel(\"$\\phi$\")\n",
    "plt.ylabel(\"P(-45)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernel_info": {
   "name": "python3"
  },
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.4"
  },
  "nteract": {
   "version": "0.15.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}