Lab 2 - Quantum States.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lab 2 - Quantum States\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": "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": "markdown",
   "metadata": {},
   "source": [
    "## 2) Verify that the $|V\\rangle$ state is normalized"
   ]
  },
  {
   "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": "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": "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": "markdown",
   "metadata": {},
   "source": [
    "## 6) Demonstrate that a half-wave plate at 45-degrees converts $|H\\rangle$ to $|V\\rangle$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7) Re-create Figure 3.9 by plotting the probability P(+45) vs phase φ"
   ]
  }
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