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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Lecture 2 - Symbolic quantum mechanics using SymPsi - Two-level systems"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Author: J. R. Johansson (robert@riken.jp), [http://jrjohansson.github.io](http://jrjohansson.github.io).\n",
      "\n",
      "Status: Preliminary (work in progress)\n",
      "\n",
      "This notebook is part of a series of IPython notebooks on symbolic quantum mechanics computations using \n",
      "[SymPy](http://sympy.org) and [SymPsi](http://www.github.com/jrjohansson/sympsi). SymPsi is an experimental fork and extension of the [`sympy.physics.quantum`](http://docs.sympy.org/dev/modules/physics/quantum/) module in SymPy. The latest version of this notebook is available at [http://github.com/jrjohansson/sympy-quantum-notebooks](http://github.com/jrjohansson/sympy-quantum-notebooks), and the other notebooks in this lecture series are also indexed at [http://jrjohansson.github.io](http://jrjohansson.github.com).\n",
      "\n",
      "Requirements: A recent version of SymPy and the latest development version of SymPsi is required to execute this notebook. Instructions for how to install SymPsi is available [here](http://www.github.com/jrjohansson/sympsi).\n",
      "\n",
      "Disclaimer: The SymPsi module is still under active development and may change in behavior without notice, and the intention is to move some of its features to [`sympy.physics.quantum`](http://docs.sympy.org/dev/modules/physics/quantum/) when they matured and have been tested. However, these notebooks will be kept up-to-date the latest versions of SymPy and SymPsi."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Setup modules"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import *\n",
      "init_printing()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympsi import *\n",
      "from sympsi.pauli import *"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Introduction: Hamiltonian and Pauli matrices"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "One of the simplest possible quantum system is the two-level system, such as a spin-$1/2$, or an atom or artificial-atom (nano-device) with (effectively) only two quantum states. \n",
      "\n",
      "The Hamiltonian for a two-level quantum system is conveniently written in terms of the [Pauli matrices](http://en.wikipedia.org/wiki/Pauli_matrices) $\\sigma_x$, $\\sigma_y$ and $\\sigma_z$, and their annihilation and creation operators $\\sigma_-$ and $\\sigma_+$. In terms of these operators we can write the Hamiltonian on the form:\n",
      "\n",
      "$$\n",
      "H = -\\frac{1}{2}\\Delta\\sigma_x - \\frac{1}{2}\\epsilon\\sigma_z \n",
      "$$\n",
      "\n",
      "where $\\epsilon$ is the bare energy splitting and $\\Delta$ is the tunneling rate between the two states of the two-level system.\n",
      "\n",
      "In SymPy we can represent this Hamiltonian as:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "theta, t = symbols(\"theta, t\")\n",
      "eps, Delta, Omega = symbols(\"epsilon, Delta, Omega\", positive=True)\n",
      "Hsym = symbols(\"H\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sx, sy, sz = SigmaX(), SigmaY(), SigmaZ()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H = -eps/2 * sz - Delta/2 * sx\n",
      "\n",
      "Eq(Hsym, H)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$H = - \\frac{\\Delta {\\sigma_x}}{2} - \\frac{\\epsilon {\\sigma_z}}{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKUAAAArBAMAAADrv1VBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n6VDSzXNt6f4uN17Aeenbw9UzZlnNj5l/AjvFdBsq62cAAAAASUVORK5CYII=\n",
       "prompt_number": 5,
       "text": [
        "      \u0394\u22c5False   \u03b5\u22c5False\n",
        "H = - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "         2         2   "
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Instantaneous eigenbasis"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Is often convenient to perform basis transformation that simplifies the Hamiltonian. For example, we can transform the Hamiltonian to the eigenbasis (where the Hamiltonian is diagonal, that is only containing a $\\sigma_z$ term) by applying the unitary tranformation:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "U = exp(I * theta/2 * sy); U"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$e^{\\frac{i \\theta}{2} {\\sigma_y}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAC0AAAAWBAMAAACmiQjHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABA0lEQVQoFWNggABeBgaew1A2MsXIwKDKLYAs\nAmOzbuBdAGPDaU5jBqZN+pjqOVMY+AJWwJUhGK8Y2BmUEVwYi7EggU/gIpAn0v//K0wQRG8W4FIK\nYGBgLI75imkLAwOLA0MXsvL/YPCBgT+A4SOyOAPDIj2gMQzyDJwFDJzKDA/gkg/YCoBsdgGg5zhT\nuDbAxS8wggzg2uwKJD2ZDwCdoGwCMgFkAAxcBIVIegCrAkhAFiwLYjEctmBgYO1kEFkAZDNuBImA\nAeuBSQwM3D3GB0E8kUAFsCCQ4FI2YGDg3wDh1v8vgImDaX4FFC6cwwEUB2rDAMwXGEQCMESBAiY2\nB7AIAwABCzC0C1YXkQAAAABJRU5ErkJggg==\n",
       "prompt_number": 6,
       "text": [
        " \u2148\u22c5\u03b8\u22c5False\n",
        " \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "     2    \n",
        "\u212f         "
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This unitary tranformation transforms the operators in the Hamiltonian according to these well-known relations:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hamiltonian_transformation(U, sx)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sin{\\left (\\theta \\right )} {\\sigma_z} + \\cos{\\left (\\theta \\right )} {\\sigma_x}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALUAAAAVBAMAAADyYbJ0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\neajrvKkuV1r9O/MHfynMtcAwv4AAAAAASUVORK5CYII=\n",
       "prompt_number": 7,
       "text": [
        "sin(\u03b8)\u22c5False + cos(\u03b8)\u22c5False"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hamiltonian_transformation(U, sz)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\sin{\\left (\\theta \\right )} {\\sigma_x} + \\cos{\\left (\\theta \\right )} {\\sigma_z}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMcAAAAVBAMAAAAA3efRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9GZaz28WJNKVmeffoAAAAABJRU5ErkJggg==\n",
       "prompt_number": 8,
       "text": [
        "-sin(\u03b8)\u22c5False + cos(\u03b8)\u22c5False"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "so the Hamiltonian after this transformation takes the form "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H1 = hamiltonian_transformation(U, H)\n",
      "\n",
      "H1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{\\Delta}{2} \\left(\\sin{\\left (\\theta \\right )} {\\sigma_z} + \\cos{\\left (\\theta \\right )} {\\sigma_x}\\right) - \\frac{\\epsilon}{2} \\left(- \\sin{\\left (\\theta \\right )} {\\sigma_x} + \\cos{\\left (\\theta \\right )} {\\sigma_z}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAfMAAAArBAMAAACeO3yOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\naJ45IpQWjidWq39FZD0K6f8B8hM3AQBeiRsAAAAASUVORK5CYII=\n",
       "prompt_number": 9,
       "text": [
        "  \u0394\u22c5(sin(\u03b8)\u22c5False + cos(\u03b8)\u22c5False)   \u03b5\u22c5(-sin(\u03b8)\u22c5False + cos(\u03b8)\u22c5False)\n",
        "- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                 2                                 2                "
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H2 = collect(H1.expand(), (sx, sz))\n",
      "\n",
      "H2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(- \\frac{\\Delta}{2} \\sin{\\left (\\theta \\right )} - \\frac{\\epsilon}{2} \\cos{\\left (\\theta \\right )}\\right) {\\sigma_z} + \\left(- \\frac{\\Delta}{2} \\cos{\\left (\\theta \\right )} + \\frac{\\epsilon}{2} \\sin{\\left (\\theta \\right )}\\right) {\\sigma_x}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAhIAAAAyBAMAAAD7IC+TAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nVgIJtZalzQoEOb+gHfxPlEqsZSz/AQWCt8Jeh0WqAAAAAElFTkSuQmCC\n",
       "prompt_number": 10,
       "text": [
        "\u239b  \u0394\u22c5sin(\u03b8)   \u03b5\u22c5cos(\u03b8)\u239e         \u239b  \u0394\u22c5cos(\u03b8)   \u03b5\u22c5sin(\u03b8)\u239e      \n",
        "\u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5False + \u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5False\n",
        "\u239d     2          2    \u23a0         \u239d     2          2    \u23a0      "
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In the eigenbasis we require the coefficient of $\\sigma_x$ to be zero, so we have the condition:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c, o = split_coeff_operator(H2.args[0])\n",
      "\n",
      "Eq(c, 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{\\Delta}{2} \\cos{\\left (\\theta \\right )} + \\frac{\\epsilon}{2} \\sin{\\left (\\theta \\right )} = 0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAOwAAAArBAMAAACJEHkrAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 11,
       "text": [
        "  \u0394\u22c5cos(\u03b8)   \u03b5\u22c5sin(\u03b8)    \n",
        "- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 = 0\n",
        "     2          2        "
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "with the solution"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Eq(tan(theta), Delta/eps)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\tan{\\left (\\theta \\right )} = \\frac{\\Delta}{\\epsilon}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGwAAAAsBAMAAAByEECSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrhZmvIapAx657+lgPeY/NdKG0nib8HAAAAAASUVORK5CYII=\n",
       "prompt_number": 12,
       "text": [
        "         \u0394\n",
        "tan(\u03b8) = \u2500\n",
        "         \u03b5"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Substituting this into the Hamiltonian results in "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H3 = simplify(H2.subs(theta, atan(Delta/eps)))\n",
      "\n",
      "H3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{{\\sigma_z}}{2} \\sqrt{\\Delta^{2} + \\epsilon^{2}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIUAAAAlBAMAAACe6DfhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
       "prompt_number": 13,
       "text": [
        "    _________       \n",
        "   \u2571  2    2        \n",
        "-\u2572\u2571  \u0394  + \u03b5  \u22c5False \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "         2          "
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now introduce $\\Omega = \\sqrt{\\Delta^2 + \\epsilon^2}$, which is the eigenenergies of the two-level system:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H3.subs(Delta, sqrt(Omega ** 2 - eps ** 2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{\\Omega {\\sigma_z}}{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAArBAMAAAAu4Q2VAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiJUdkSZZquJ\nu+94txTYAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABTklEQVQ4EWNgQADW9FQEB43FoRkQqYImBueu\nX8DAUJ8A56IwWD4BuUxfUcTgHO4/QCbPTzgfhcH8Dcjl+osiBufw/gIy2X8zMEi+//9/AlwYwmAH\nmcj9i4HFss02FE2OgfExUIT5AYNcAsN6dDkGhvUBDAz+CQyTGBj6BTBkmQoYGM4yMACtrseQY2VV\nYGAwZWX9wsCgzcDAaiTagKSEh2EjA+sEHsaPDCwPGRg4s44jyTFwLzvBwHaskuEaQzBID2MCsiTP\n/18MXP+/M0iWHwQJRyDLobFZG1gwnQxTE31sLYyJSRf4KWIKDryIkDEIAFPdfwzwYeBdR8gFrDPv\nOuBUI83A8wOn5EYGhnc4JYGpdz9Oc/UF8EgCjbTHnUAYOEB5ERdgvoBLBig+F48c+wY8kpkMrAG4\npFkmMLDjlAw+c3IzLo0M+v//f4ZJAgCAzFaEWcD9TAAAAABJRU5ErkJggg==\n",
       "prompt_number": 14,
       "text": [
        "-\u03a9\u22c5False \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "    2    "
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In summary, to reach this basis, we have transformed $\\sigma_x$ and $\\sigma_z$ as:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hamiltonian_transformation(U, sx)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sin{\\left (\\theta \\right )} {\\sigma_z} + \\cos{\\left (\\theta \\right )} {\\sigma_x}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALUAAAAVBAMAAADyYbJ0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMnZUzRC73UTviSKZ\nZqszMyTAAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAC0klEQVQ4Eb2US2gTURSG/2kzTcwkTSxCzcI2\nVAWJ2g5dKIriLKIL8TEroTa2oT5ACpJFEZ840JWCtFgCgpWGtKIoSlY+KNoYXPjYpC5cqNAIurRW\nayXWRzw3MzeTTExaN57FPeeec75/7pzcCfA/bQU97EAak//yzCqMcMkUksKA4/AW1HvN3EJRNUaY\nN2mbCvhUjwrFzC0ULZYJkNBZeGSwwGKdlj3fVmN4D/NXAPcgPGl0FGf1uL08lc9UY6SlBcgRAexJ\nNMuwyYUkDypoV2SElS3K3d9YNtzQ6icNZxIY0NCm5aPQSC5HCW5cO3RzFGI8oKGhfRXVShnxWS73\nUyf2AREcA3b4pR+UqUsDn6anR2gyCoQz/cfHuTB5Q1sagi/RmECPGIbdyqDx6cbehA6dVxFj2keB\nX5Sx0yT6gK9AbQQdMqb0Ln01tG0KtnlPAc3Lo3BaGXEI7hkDsn9bC6Z9CPhCqSUaMA9hFhBm8ITm\noxp9zBna3UmqUocndq6HnbCEcSlwzLFmMnGMJkraRwra4ixcg3ltepE2vYvWsWDwZDC4i6IBP71V\nlrTD+/eyyZK2ydANq+HnHoc7W6RNMxGz8BDtiEifgT2A1NepkQKZ9dwvEpjS8nMsMJjwwpYB4o9b\nvdgMfCzSZr/lZVwnIbcizkGIAq6tcdoyM+cNeZjO3x9DPfUXM+hW6bNGjfrermGTil42EGMmzgyQ\nkuhjgCuJg7inUSTKtDAztKWrcCZ8XuysjaIuQXcwU8TUydIJGih2U/+Fl6+1ptyHpty7Dd+ph30H\noXUqFWwaQi3XKMB9tjAztNGVugXp+R3NcXv9DUqXMGIqwGiwC2210zzxgAeSJuS7TW1e4b6coVsm\n8Krp3/KQDxmPRl8ZuYu8ZvHlzIQzwm6+xdgAmYlK3tHi923nYQVfznStia8ub2b/88x4v76rvi6a\neajrvKkuV1r9O/MHfynMtcAwv4AAAAAASUVORK5CYII=\n",
       "prompt_number": 15,
       "text": [
        "sin(\u03b8)\u22c5False + cos(\u03b8)\u22c5False"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hamiltonian_transformation(U, sz)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\sin{\\left (\\theta \\right )} {\\sigma_x} + \\cos{\\left (\\theta \\right )} {\\sigma_z}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMcAAAAVBAMAAAAA3efRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9GZaz28WJNKVmeffoAAAAABJRU5ErkJggg==\n",
       "prompt_number": 16,
       "text": [
        "-sin(\u03b8)\u22c5False + cos(\u03b8)\u22c5False"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "and chosen $\\theta = \\arctan(\\Delta/\\epsilon)$."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Versions"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%reload_ext version_information\n",
      "\n",
      "%version_information sympy, sympsi"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>3.4.1 (default, Sep 20 2014, 19:44:17) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)]</td></tr><tr><td>IPython</td><td>2.3.0</td></tr><tr><td>OS</td><td>Darwin 13.4.0 x86_64 i386 64bit</td></tr><tr><td>sympy</td><td>0.7.5-git</td></tr><tr><td>sympsi</td><td>0.1.0.dev-0c6e514</td></tr><tr><td colspan='2'>Thu Oct 09 15:23:41 2014 JST</td></tr></table>"
       ],
       "json": [
        "{\"Software versions\": [{\"module\": \"Python\", \"version\": \"3.4.1 (default, Sep 20 2014, 19:44:17) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)]\"}, {\"module\": \"IPython\", \"version\": \"2.3.0\"}, {\"module\": \"OS\", \"version\": \"Darwin 13.4.0 x86_64 i386 64bit\"}, {\"module\": \"sympy\", \"version\": \"0.7.5-git\"}, {\"module\": \"sympsi\", \"version\": \"0.1.0.dev-0c6e514\"}]}"
       ],
       "latex": [
        "\\begin{tabular}{|l|l|}\\hline\n",
        "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
        "Python & 3.4.1 (default, Sep 20 2014, 19:44:17) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)] \\\\ \\hline\n",
        "IPython & 2.3.0 \\\\ \\hline\n",
        "OS & Darwin 13.4.0 x86\\letterunderscore{}64 i386 64bit \\\\ \\hline\n",
        "sympy & 0.7.5-git \\\\ \\hline\n",
        "sympsi & 0.1.0.dev-0c6e514 \\\\ \\hline\n",
        "\\hline \\multicolumn{2}{|l|}{Thu Oct 09 15:23:41 2014 JST} \\\\ \\hline\n",
        "\\end{tabular}\n"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "Software versions\n",
        "Python 3.4.1 (default, Sep 20 2014, 19:44:17) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)]\n",
        "IPython 2.3.0\n",
        "OS Darwin 13.4.0 x86_64 i386 64bit\n",
        "sympy 0.7.5-git\n",
        "sympsi 0.1.0.dev-0c6e514\n",
        "Thu Oct 09 15:23:41 2014 JST"
       ]
      }
     ],
     "prompt_number": 17
    }
   ],
   "metadata": {}
  }
 ]
}