Reproduce-Nature-460-724-2009-Groblacher.ipynb

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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Reproduce: Gr\u00f6blacher et. al., Nature 460, 724 (2009)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Reproduced by: Eunjong Kim (<a href=\"mailto:ekim7206@gmail.com\">ekim7206@gmail.com</a>)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In this notebook, I find the expression for the normal mode splitting of optomechanical system, following the supplementary information of [<a href=\"http://dx.doi.org/10.1038/nature08171\">Gr\u00f6blacher et. al., Nature <b>460</b>, 724 (2009).</a>]"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Setup Modules"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import *\n",
      "init_printing(use_latex='png')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympsi.operator import Operator\n",
      "from sympsi.boson import BosonOp\n",
      "from sympsi.dagger import Dagger"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Constants\n",
      "d = symbols(\"Delta\", real=True)  # detuning\n",
      "hbar, g, O, wp, wm = symbols(\"hbar, g, Omega, omega_+, omega_-\", positive=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The linearized Hamiltonian for a driven cavity mode coupled via radiation pressure to a harmonically bound mirror is\n",
      "\n",
      "$$\n",
      "H = -\\frac{\\hbar\\Delta}{2} (X_c^2 + P_c^2 ) + \\frac{\\hbar\\Omega}{2} (X_m^2 + P_m^2 ) - \\hbar g X_c X_m\n",
      "$$\n",
      "\n",
      "where $\\Omega$ is the frequency of the mechanical resonator, $\\Delta\\equiv \\omega_L - (\\omega_c-\\delta_{rp})$ is the effective detuning between the driving field and the cavity frequency$.($\\delta_{rp}$ is the mean shift of the cavity frequency arising from radiation frequency.)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This can either be expressed in terms of the matrix equation with vector $\\vec{R}\\equiv [X_c,\\ P_c,\\ X_m,\\ P_m]^T$:\n",
      "\n",
      "$$\n",
      "H =\\frac{\\hbar}{2} \\vec{R}^T M \\vec{R}\n",
      "$$\n",
      "where $M$ is the matrix defined as"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M = Matrix([[-d, 0, -g, 0],\n",
      "            [0, -d, 0, 0],\n",
      "            [-g, 0, O, 0],\n",
      "            [0, 0, 0, O]])\n",
      "M"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
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       "prompt_number": 4,
       "text": [
        "\u23a1-\u0394  0   -g  0\u23a4\n",
        "\u23a2             \u23a5\n",
        "\u23a20   -\u0394  0   0\u23a5\n",
        "\u23a2             \u23a5\n",
        "\u23a2-g  0   \u03a9   0\u23a5\n",
        "\u23a2             \u23a5\n",
        "\u23a30   0   0   \u03a9\u23a6"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Operators\n",
      "Xc, Xm = Operator('X_c'), Operator('X_m')  # Position Quadratures\n",
      "Pc, Pm = Operator('P_c'), Operator('P_m')  # Momentum Quadratures\n",
      "ac, am = BosonOp('a_c'), BosonOp('a_m')    # Boson Operators"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The quadrature operators are related to the annihilation and creation operators by the following linear equation:\n",
      "$$ \\vec{R} = Q \\vec{a}$$\n",
      "where $\\vec{a} \\equiv [a_c,\\ a^\\dagger_c,\\ a_m,\\ a^\\dagger_m]^T$, and $Q$ is the coefficient matrix"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "R = Matrix([Xc, Pc, Xm, Pm])\n",
      "a = Matrix([ac, Dagger(ac), am, Dagger(am)])\n",
      "Q = Matrix([[1, 1, 0, 0],\n",
      "            [-I, I, 0, 0],\n",
      "            [0, 0, 1, 1],\n",
      "            [0, 0, -I, I]])/sqrt(2)\n",
      "Eq(R, MatMul(Q, a))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 6,
       "text": [
        "        \u23a1    ___       ___                     \u23a4       \n",
        "        \u23a2  \u2572\u2571 2      \u2572\u2571 2                      \u23a5       \n",
        "\u23a1X_c\u23a4 = \u23a2  \u2500\u2500\u2500\u2500\u2500     \u2500\u2500\u2500\u2500\u2500       0         0   \u23a5\u22c5\u23a1a_c \u23a4\n",
        "\u23a2   \u23a5   \u23a2    2         2                       \u23a5 \u23a2    \u23a5\n",
        "\u23a2P_c\u23a5   \u23a2                                      \u23a5 \u23a2   \u2020\u23a5\n",
        "\u23a2   \u23a5   \u23a2   ___       ___                      \u23a5 \u23a2a_c \u23a5\n",
        "\u23a2X_m\u23a5   \u23a2-\u2572\u2571 2 \u22c5\u2148   \u2572\u2571 2 \u22c5\u2148                    \u23a5 \u23a2    \u23a5\n",
        "\u23a2   \u23a5   \u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  \u2500\u2500\u2500\u2500\u2500\u2500\u2500      0         0   \u23a5 \u23a2a_m \u23a5\n",
        "\u23a3P_m\u23a6   \u23a2    2         2                       \u23a5 \u23a2    \u23a5\n",
        "        \u23a2                                      \u23a5 \u23a2   \u2020\u23a5\n",
        "        \u23a2                        ___       ___ \u23a5 \u23a3a_m \u23a6\n",
        "        \u23a2                      \u2572\u2571 2      \u2572\u2571 2  \u23a5       \n",
        "        \u23a2    0         0       \u2500\u2500\u2500\u2500\u2500     \u2500\u2500\u2500\u2500\u2500 \u23a5       \n",
        "        \u23a2                        2         2   \u23a5       \n",
        "        \u23a2                                      \u23a5       \n",
        "        \u23a2                       ___       ___  \u23a5       \n",
        "        \u23a2                    -\u2572\u2571 2 \u22c5\u2148   \u2572\u2571 2 \u22c5\u2148\u23a5       \n",
        "        \u23a2    0         0     \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5       \n",
        "        \u23a3                        2         2   \u23a6       "
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "i.e., $$X_{\\alpha} = \\frac{a_{\\alpha}+a^\\dagger_{\\alpha}}{\\sqrt{2}},\\quad P_{\\alpha} = -i\\frac{a_{\\alpha}-a^\\dagger_{\\alpha}}{\\sqrt{2}}\\quad (\\alpha = c, m)$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The quadratures satisfy the canonical commutation relations $$[X_\\alpha, P_\\beta] = i \\delta_{\\alpha, \\beta},\\quad [X_\\alpha, X_\\beta] = [P_\\alpha, P_\\beta] = 0.$$\n",
      "which are condensed into one simple equation\n",
      "$$\n",
      "[\\vec{R}_{i}, \\vec{R}_{j}] = i J_{ij}\n",
      "$$\n",
      "where $J$ is the matrix defined as"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Constraint representative of the canonical commutation relation\n",
      "J = Matrix([[0, 1, 0, 0],\n",
      "            [-1, 0, 0, 0],\n",
      "            [0, 0, 0, 1],\n",
      "            [0, 0, -1, 0]])\n",
      "J"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJgAAABkCAMAAABNTAlxAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nf5bYh/4itel/AVrXl9y0f0o//CK1rs0/v51EehgG0m0AAAAASUVORK5CYII=\n",
       "prompt_number": 7,
       "text": [
        "\u23a10   1  0   0\u23a4\n",
        "\u23a2            \u23a5\n",
        "\u23a2-1  0  0   0\u23a5\n",
        "\u23a2            \u23a5\n",
        "\u23a20   0  0   1\u23a5\n",
        "\u23a2            \u23a5\n",
        "\u23a30   0  -1  0\u23a6"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note that the matrix $J$ has simple properties:   $\\quad J^T = J^{-1} = -J, \\quad J^2=-I$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Our goal is to find a linear transformation $S$ that converts the original coordinates $\\vec{R}$ into the normal mode coordinates $\\vec{R}^{NM} = [X_+, P_+, X_-, P_-]^T = S\\vec{R}$, whose Hamiltonian representation is\n",
      "$$ H = \\frac{\\hbar\\omega_+}{2} \\left( X_+^2 + P_+^2 \\right) + \\frac{\\hbar \\omega_-}{2} \\left( X_-^2 + P_-^2\\right) $$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Operators in the normal mode coordinates\n",
      "X_p, X_m = Operator('X_+'), Operator('X_-')  # Position Quadratures\n",
      "P_p, P_m = Operator('P_+'), Operator('P_-')  # Momentum Quadratures\n",
      "a_p, a_m = BosonOp('a_+'), BosonOp('a_-')    # Boson Operators"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Rnm = Matrix([X_p, P_p, X_m, P_m])\n",
      "anm = Matrix([a_p, Dagger(a_p), a_m, Dagger(a_m)])\n",
      "Eq(Rnm, MatMul(Q, anm))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAYkAAAB0CAMAAABHeD78AAAAP1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADFBd4eAAAAFHRS\nTlMAMquZdlQQQO0wRGa774nNIt18bMlzHnYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAA4CSURBVHgB\n7V1ro6uqDrQv9zl92PZe//9vPQQFCUxQgtrdvZcfVgGZJEzqC6espvnUdu7dpomgDq3x+Mdirl3N\n0OrQNZ6/DnsYvvBHMfB72zTd6X4yH4qtDq1wKELew0DF/R/fcehPZ7NdpEAuZ5OIV9Mcn1KPXHsd\nOme5eN+Nhnnsi3G7AQ79IeuLEvA2mTj05k/xVocudjcLuH1vJm43M7rHtWlevflTutWhS70t6P/F\nmfDnpEuvuFDUoRcwW9rlCzPxGi4c0/Xjba4Xi7c69GI35R1XycTsTc4QV+ktAr5OXDt7CLR3N9ju\n5EoLPuvQg4P21J1P+UtYJhIJDTKxkFdztXQ3X3M3OWNcpbcIOBNN86THiJO7NtxKEmGAdWgaytMc\nlC3dKqg2CQ0zkb959P49sxJlvmdQAA6Dvawomb2ag+LlDgl6PruWfEHr0CbA24Oi7PyFhsU8WxHR\ngBiJAOyEDJQggENsOGPWfK2P41fy8LxcLvSMtnyrQzfN3T5qqm4TTJAiGhBTwqv5hkSZuN37/mG+\np+2z7+/goQw4lEgUA7n2V/fg/bBnSMkCbK9DN01vfV97MDjojzeKaECMSAA3OdbiTJhH3iHG2xF+\nU4FDaNc0yoE8Vc9yzk8duu3thenal9yxOdeNjAbEyAR4e0EhzcShp1PoS7iQAoeBNVaUAzmoWHDG\n69Cv3s47HoYPZ3Ppp4wGxMgEIHdpJhr6zr3c6SPGAIdxF1cvC8Shtv6UuVziWUYDYsoIAJm49KdW\nSoS9riwJ2fSBgQx3zvHfhRabGDfUl6Jtv3Y4GNRnJ3tEAfQWmWgePb5G0EiAQ4kImAmp837tw3Xi\nor1i27M2QANiIgJex+58eMOrL40eHBNNl7maAYcSiVEgUTf9iwkyVIF+20eZm2ayyzgW0YAYTsD1\nYe7c7/KMLcrEsbcPP4469zRs68Ch6xd/8kCivfoXE2SoBn22gzspn+xENCCGEdA+aO755J5oG8Yq\njQlk4vQab2Rpv9k8mCrAITWjjQUSd9C/mCBLVeg3zXY8Sp7rw+AlNCCGEXCyB+E028lYJQdpJszk\n2KEP+4Xl1TKhfjFhWalCt515w6ZNRCOh5zLxIBrb6eUZYxVmoqPD5h0+ejEMcGi5AX/YVwLsb7Qz\nDoOtOjSKp6oNEBMSMNyyXcxlYrxkM1bJc3xMnO356zbcYAyhMQxwKA0gDAT2mQ5VuHumsQ49Y7x8\nNyAmJKC192r3Z3MZZ9wYq+QuysTR3i6byZng1oJhgEMp6jAQ1KfoxURioA6dmKtuAMQwAp5mXuH2\nPjYjwfzqS97DTBzMrB+dgZurKRgQyS7M9qA/dOGnDTgcdqR/WSDp7tIXE9xCHZrbWqUGiGEEtMdz\nd3gd6ZyTskoRhJkQItrmmCh+McGiq0MzUytV5jIRu2Gs0s5PZULxYiIYSx06MLRe8WszoXgxEbBW\nhw4MrVf82kxMFOiExjrU5HX10h6ZYG8qgENpUOyCJXVqdEJjHUoMYoUdgJgsAYxV8k8GsogoSOAw\n6uGry8wyofE5ic9b4wUdittYtwaIWUaAC+PTmeBC48NC7YsO5ca8yefXZ4KmRMvn83SoTTLgjOoy\ncbRTGtbGh48JndBYh3KcbfP57ZnwbwloPu96Fl/acvZ0KG5j7ZouE930Cu9DxwQUGndtMNuFmNKh\nkKX12zSZuD/vp4fTXaWZML+O6e/H4/0Jf3UFHEqjytw6QKHxq728JVu2XYdKTEoa46QjbJDQgJgM\nAYPptulaf1CkmTATscPRf3/4XlNMwOG0k5eygUCh8X1OB6VD8bD2VCjPvo8K3liBTDhR1gUJCwoy\n8av/FXEQVJHQ2JycZkLXoQK3pihqjHk3oSai/5dqBbJfxcF+IHgFmTiPv7Y6h6+LXGAFmcgHAoTG\nl6efvXfu4k8dilkRNcasl1QR0YCYPAHWQz4TTgnCdQVjaMChFHQ+ECA0bo+k2chuOhQzKWqMWS+p\nIqIBMXkCrIeju17D2Y7xMvFCh8SKb4p0QmMdKuBV1hgHncSijM5nQlKdZZ/sxsvE9e3e87GwgEO2\nP6jMfCV0QmMdKohKVrYGncSijAbETASIqrPgV2ZkYEJQBOf+2HWn03k8UUyP4zY84FAKm5uVeu3d\nLnO5JBIZDYjxBMiqs+B0TAbaS3i7euePV9FbvvHhaknUUyDPt9+edHeEhcZzcmMdKo50R4Wy51VW\nnZ0n5olZT5mNuuePV1EmQOrjsbo6N+taP/75AYWyrDqbEpG+nzA/RGVkbZ0JndZYh7IDEzXGbNhS\nRUSDr6j7Ks6rzsgbGXAI6909TbhQNs6ETmusQw1DEjXGbsTZTxGdzQTdq+ZUZ+QyycSTXyZihRRw\nKEXOEix1Kn83QZZ0qDEGSWMshcjbJTQgxhMwqzojF1EmTvSTUzf7gxRSwCGPdKr5QKamtKTTGutQ\no3dJY5wGh1okNCDGEzCrOiNHUSZS3/qzUxvcGKR2gxad1liHCtyuW3y57+9k1mdiahpKEadD44aZ\nEAOJA9NpjXWo2Pdq9dwxETv5XTOh0xrrUDEn69X/gEzotMY61HrEJ5a2zwR/urDXlTCKjoTkbuN9\nl52ddFpjHSoMfO1ySSY4T2MkZGAZZQMAOJTGtMisTmusQ0mBrtIOiFlEgHf+6UzotMY6lB/0FoVV\nMpF9zRlFDV4SRj18tewr4WFfWgCZKOG1aYjZEspih52fZTUFLlcqMbs7/XVac4COiTEjKiOADJQg\ngEOJxRKzko2t2uu05ggNiCkj4C/NBNOaF6cboVfPxB5rnhWPfHUA15qXmofo1TOxy5pnpUNfvX+d\n1hyiN8jEamueld06rM52xmCd1hyjwU1l9XXiz17zjBLEtOaZjOFdGL3BMWGW09hwzTM3OEno6/bn\nP5VoqDXPewr25tFbZGKXNc+kpYiDkWeKOjTUmme88F0z6PlMSOqz0Q0ZiM9nO6x5Jgp9+eiFmhoN\nteaCk7Q5i57NhKg+Gx2hTERrnrGQgEO2P6jECQ52yUsRh53EsigTFhHjDqQ1n8NM+7NoQAwjIFaf\nTWbHEhn49c+/YXu85lm4r/n/P6yaq7BAoo6i0Dfqh6t6NNCaYxewNYcGmWC8xuqzxAExyylL1jxj\nGOCQ7Q8q3Gywg9b+slP0YLFP1k2oVKCB1lxwgppzaEAMIyBWnyX2yQBDpGueMQxwyPYHFWY2aDdF\nWV7K++FaDbpOa55BA2JCAhL1WTK0OBNgzTOGAQ7Z/qASBhI0U7GGyzp0ndY8gwbEhAQka55FjCQq\nG7TmGcMAh2x/UAkDCZqpuJVMOHLDq1jhzPvItTk0IIYREKvPEk9kwCHSNc8ud7c5KS1wmNgcG5xZ\ntH8jmTBytVMbIIYREKjPUlYpxjATS2IGDiVYTnkmCn0lY6y9Dl2zALMJQxBHlyjP2GBcZcNMHEi9\nKmyi0Ffoz5vr0DXyZnn5ZvBbcnZM8BGA2paZmJapTR1LQt+0J2qpQlfJm0VxNDhZfEcmJKEv4j1t\nq0JXyZvF/0r5tZlI2d21pU7ejNA/mdAlsE7ejNA/mVBlok7eDNE/mdBkok7ejNHKTEw/jScDJdd4\n4FDiosSsZGOT9jp5s4AGxCwh4K/ORJ28WUIrMzEtepYeE3ustLXJV32p0Tp5s4RWZSJc9CzNxD4r\nbS2l7Wv6qTIRLnoGMrHWSltLTpMfJtpPsJbEAeTJBq7KhNH5+bXFQCbcb+O3XWmrZOxb9UVK41lf\nAkiZiWmpLZCJfVbamh3xDh2M0niaWV26kjaTJ0+gLTKx00pbOzA948IojYN52YUraXN58gRSZmJa\n9Cw9JvZaaWuGph12G1ll+bwslCebYPOZEFVnueeJaKWtiBHgMOrhq7/7FZuUxsXzsliePJcJWXU2\nLXqWHhPRSlue2KHwB2XCKY3tzOrSlbQdyP4zvhAEiPFfxYzqjE6PE7MeYZuilbZcx6B/1CRVudmo\nl1JjPFqpQsdKYzuzOreSdgxqCBWCcpnIqM6mtU2SYyJeaStiEDiMevhqNhM6jbGzXYOOlcZ2ZnVu\nJe0YZO65jNonXH4bEOMJyKjOnFIDKArc04QbdPQJHEY9fNUH4lumglpjbE3UoRumNHYzq34tpSlI\nVmKgZkQFIECMI2BedUaeyIBDWM/xSlssHHiLEPXwVWbWtw4FtcbYwuvQTag0djOrsytphyD7z5Wu\nByPbmpbfzmYiWvMsImOoRplgK20hAHCIulFbLhN6jTFZrkOb3xR1zXG4UvqZ1fmVtCdQ41AhCBDj\nCZhVndGgokxQk9+QQgo49P2jgg8kajfVCo1xNdoYmJTGfmZ1fiXtCWR+6kObGUaw/DYgxhMwqzoj\nhnKZoP3xBhzGXVzdB+Iaps/P6WKHGDJK4ynIuJQHAWIyBMS2Tf3vzERGaQw4GpvyoC/NxG4K5TZe\nvlm3frOfPQeFMU9fmonm71Moy8dakMqS8xlIveQjZ7ZOY1yH9vFOM+K+aUFBQAFicgSkjsjAYTjm\n+JpAadf30C3dIbTkAqnTGNehXbjBjLhrWvApoWAmLGNzvJoZYcdse7Zb9P8NkqhuQ7ekXWrIZaKp\n0hhXoseAy2fECSihQCYW8mpumwqZlRiX2rOZqNIYN3XoMeDiGXGLk1AgExIxu7dnM7F7NNgh0hrj\nnmErQv1kImSovIy0xvNWEOonE/O8ZXpArXGm/7ALon4yMctbpoObEc90Absw6vfOxMKbODDaXZrc\njHiZsxTlb0LLDO3Xe/FN3H4hMU9ubps1zlYAauub0NmYvr2Dm9suG4cOtczHf3OaxxmAaw7PAAAA\nAElFTkSuQmCC\n",
       "prompt_number": 9,
       "text": [
        "       \u23a1    ___       ___                     \u23a4      \n",
        "       \u23a2  \u2572\u2571 2      \u2572\u2571 2                      \u23a5      \n",
        "\u23a1X\u208a\u23a4 = \u23a2  \u2500\u2500\u2500\u2500\u2500     \u2500\u2500\u2500\u2500\u2500       0         0   \u23a5\u22c5\u23a1a\u208a \u23a4\n",
        "\u23a2  \u23a5   \u23a2    2         2                       \u23a5 \u23a2   \u23a5\n",
        "\u23a2P\u208a\u23a5   \u23a2                                      \u23a5 \u23a2  \u2020\u23a5\n",
        "\u23a2  \u23a5   \u23a2   ___       ___                      \u23a5 \u23a2a\u208a \u23a5\n",
        "\u23a2X\u208b\u23a5   \u23a2-\u2572\u2571 2 \u22c5\u2148   \u2572\u2571 2 \u22c5\u2148                    \u23a5 \u23a2   \u23a5\n",
        "\u23a2  \u23a5   \u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  \u2500\u2500\u2500\u2500\u2500\u2500\u2500      0         0   \u23a5 \u23a2a\u208b \u23a5\n",
        "\u23a3P\u208b\u23a6   \u23a2    2         2                       \u23a5 \u23a2   \u23a5\n",
        "       \u23a2                                      \u23a5 \u23a2  \u2020\u23a5\n",
        "       \u23a2                        ___       ___ \u23a5 \u23a3a\u208b \u23a6\n",
        "       \u23a2                      \u2572\u2571 2      \u2572\u2571 2  \u23a5      \n",
        "       \u23a2    0         0       \u2500\u2500\u2500\u2500\u2500     \u2500\u2500\u2500\u2500\u2500 \u23a5      \n",
        "       \u23a2                        2         2   \u23a5      \n",
        "       \u23a2                                      \u23a5      \n",
        "       \u23a2                       ___       ___  \u23a5      \n",
        "       \u23a2                    -\u2572\u2571 2 \u22c5\u2148   \u2572\u2571 2 \u22c5\u2148\u23a5      \n",
        "       \u23a2    0         0     \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5      \n",
        "       \u23a3                        2         2   \u23a6      "
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This can either be expressed in terms of the matrix equation with vector $\\vec{R}^{NM}$:\n",
      "\n",
      "$$\n",
      "H = \\frac{\\hbar}{2} \\left(\\vec{R}^{NM}\\right)^T D \\vec{R}^{NM}\n",
      "= \\frac{\\hbar}{2} \\left(S\\vec{R}\\right)^T D \\left(S\\vec{R}\\right)\n",
      "= \\frac{\\hbar}{2} \\vec{R}^T \\left(S^T D S\\right)\\vec{R}\n",
      "$$\n",
      "where $D$ is the diagonal matrix defined as"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "D = diag(wp, wp, wm, wm)\n",
      "D"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALkAAABkCAMAAADt02GfAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADFBd4eAAAAFHRS\nTlMAMquZdlQQQO0wRLsi72aJ3c18bJS25Z0AAAAJcEhZcwAADsQAAA7EAZUrDhsAAARfSURBVHgB\n7Zxtm5owEEWjItv6hrb8/9/aEIgBPRNnWODRLX4oMVxuTsYssHepblOH19Z9yuvaAju3qXeFf+0/\nBdxdGtxt3ZBvPgY6gV4mIS93VbHTT98od6ifhvzmV1p5PaWC5FtGuUP9JOSXc0Na3fK8971GuWP9\nJOTHcFra1+UdLtswyh3rJyGvA/mhVp6djHLH+iH5savV6XzIVm24s6x3TcehLob9wjuj3An6Afkl\n8pZKiJbtVFdNY9NuBN7UbZQ7QT8gD6ULQ1TdBx+nksaFlmANytBllGvID4Hz2thfujNcXD4SRegv\n22LrV0v4iLRyv1pQ3695WKabUPig9Vgqcteu8736JzQMoZYL9n3yYBhuX0428muY4EV7VjTKHev7\n5Ee/RPzdly/1LV4OdTUvwpVop70SGeWO9X3y6lztzvt667dxXevI3bW5+p/VNy5GOdv3yctjfdv4\nGZ7Dgt/e/Ovc/HOJ8xC3ZeXvOtXgzihnfZ8cwJQ1hyNn71rJZy/x0wA/tubhFP803bfoeFHzt2Bk\niJWc6zJn71rzOavL3j+t5hjN8NTH9FrtUc81x2hmDCMfY7VHPZJzNMMUI3qt9qxHco5mRjDyIVZ7\n1iM5RzOMMaLXas96IheimWfGePc+a7CkSoo6Nm0gskywpMlbYlG15OlGcs5gyUIuRDNxYnG7ULCU\nS4q+6q9I0251yc9SwZKQFP2hv7ZwNDOcnXNLBUuKpOiOxtHMfXfXWCpYUiRFCU0V5SwWLL1OihK5\nKspZLFgalRSlyaha8dKkEn9TRNfQ8ZYruaZ2a83bKqX7AU3VvqeZtubfY7Ed/dnkZaH8s7etKDOr\nT/6+6X9+vmXm8qL9Z6/z59WC0QzOvO206jNWtAvtueYYzZBn12fVZ6xoF9ojOUcz5Nn2WfWyE+5h\neyTnaAZdQ6dVLzvhHrZHco5m0DV0WvWyE+5heyJXJ0XdOGp9vAmeJFkicm3eEguk1U+bLC1Jnu4k\nY7IU557dCoUhcmVSdB9OqYdk6W6Rawj2DfnXr9/DI3VJUTpGp4dkKVnkWmz/9xfccemSojSYTg/J\nUrLItdieVosQzcjmo5Ml2bK/h+2RnKOZvtlDe2yy9GAjvUV7JlclRb1xVPphsrQ/xtfrX2vQnsl7\nUG/bXMmX/2jWmq8111dgXS36Wk2lXGs+VSX1Pj+t5hjNZMph1BvlPDDXHKMZNgi9Rr1RzgMjOUcz\nbND0GvVGuTAuknM0Izj4bqPeKBfGRXKOZgQH323UG+XCuESuTn46T6NeL88mS0QuBBzC3J306Iyk\nV9vnk6V3Js8nS0QuRDNSDaWHfiS91v5FskTkwkM8Eon/CQ3VUf93MqX8RbKE5BzNyORGvVL+IllC\nco5mZHKjXimHZ5b6CEg+T1KUhsXkJ+3uWvDMUl/D5BjN9A97aBv1OvmLZInJH8De8u1KvvzHstZ8\nrbm+Au1q+dTvzCmbr6ApCtW3EuhrMqMyfGdOUbh/QjxHy03JGXsAAAAASUVORK5CYII=\n",
       "prompt_number": 10,
       "text": [
        "\u23a1\u03c9\u208a  0   0   0 \u23a4\n",
        "\u23a2              \u23a5\n",
        "\u23a20   \u03c9\u208a  0   0 \u23a5\n",
        "\u23a2              \u23a5\n",
        "\u23a20   0   \u03c9\u208b  0 \u23a5\n",
        "\u23a2              \u23a5\n",
        "\u23a30   0   0   \u03c9\u208b\u23a6"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "1) Comparing with the matrix representation of Hamiltonian in the original coordinates, it follows that $M = S^T D S$.\n",
      "\n",
      "(<b>NOTE</b>:  This transformation betweeen $M$ and $D$ is not in general a similarity transformation, so that $M$ and $D$ doesn't need not share a same set of eigenvalues.)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "2) In addition to this condition, the transformation matrix $S$ is subject to the cannonical commutation relations between quadratures of the normal mode coordinates, which gives another constraint on $S$ :\n",
      "\n",
      "\\begin{align}\n",
      "iJ_{ij} = \\left[\\vec{R}^{NM}_{i}, \\vec{R}^{NM}_{j}\\right] = \\left[\\sum_{k} S_{ik} \\vec{R}_{k}, \\sum_{l} S_{jl} \\vec{R}_{l}\\right] = \\sum_{k, l} S_{ik} \\left[ \\vec{R}_{k}, \\vec{R}_{l}\\right] (S^T)_{lj} = i \\sum_{k, l} S_{ik} J_{kl} (S^T)_{lj} = i(SJS^T)_{ij} \\quad\\Rightarrow\\quad J=SJS^T\n",
      "\\end{align}"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The latter property, which guarantees that the canonical commutation relations are conserved under the transformation, are often termed as a <i>symplectic transformation</i>."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The conditions 1) and 2) described above themselves doesn't say anything about the normal mode frequencies $\\omega_+,\\ \\omega_-$ or the transformation matrix $S$. But, combining two matrix equations, we can get a meaningful relation useful in getting normal mode frequencies."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "First, we can readily notice that the matrix $S$ should be invertible. This is due to the fact that $\\det{J} = 1$, so the determinant of the rhs of condition 2),\n",
      "\n",
      "$$\\det{(SJS^T)} = \\det{S} \\cdot\\det{J}\\cdot \\det{S^T} = \\det{J} (\\det{S})^2$$\n",
      "\n",
      "should also be 1. If $\\det{S} = 0$, this cannot be the case. It naturally follows that $|\\det{S}|=1$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Second, multiplying $J^{-1} S^{-1}$ on the left of the both sides of 2), we get\n",
      "\n",
      "$$\n",
      "J^{-1} S ^{-1}J = -J S ^{-1} J = (-J S^{-1}) SJS^T = -J^2 S^T = S^T.\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Finally, substituting into 1),\n",
      "\n",
      "$$\n",
      "M = S^T D S = (J^{-1} S^{-1} J)DS = J^{-1} S^{-1}(JD) S \\quad \\Rightarrow \\quad JM = S^{-1}(JD) S.\n",
      "$$\n",
      "\n",
      "we find that $JM$ and $JD$ are related by the similarity transformation. As a consequence, the matrices $JD$ and $JM$ share the same set of eigenvalues."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, let \n",
      "$$\n",
      "A^{-1} (JM) A = B^{-1} (JD) B = \\mathrm{diag}{(\\lambda_1, \\lambda_2, \\lambda_3, \\lambda_4)}.\n",
      "$$\n",
      "\n",
      "(with this definition, $S = BA^{-1}$)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Extracting the eigenvectors and eigenvalues for two matrices $JM$ and $JD$,"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# extracting eigenvectors and eigenvalues for J*M\n",
      "[(e1,_,[v1]), (e2,_,[v2]), (e3,_,[v3]), (e4,_,[v4])] = (J * M).eigenvects()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e1, e2, e3, e4"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 12,
       "text": [
        "\u239b       _______________________________________________        _______________\n",
        "\u239c      \u2571                ______________________________        \u2571               \n",
        "\u239c     \u2571     2    2     \u2571  4      2  2          2    4        \u2571     2    2     \n",
        "\u239c    \u2571     \u0394    \u03a9    \u2572\u2571  \u0394  - 2\u22c5\u0394 \u22c5\u03a9  - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9        \u2571     \u0394    \u03a9    \u2572\u2571\n",
        "\u239c-  \u2571    - \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 ,   \u2571    - \u2500\u2500 - \u2500\u2500 - \u2500\u2500\n",
        "\u239d \u2572\u2571       2    2                    2                   \u2572\u2571       2    2      \n",
        "\n",
        "________________________________         _____________________________________\n",
        " ______________________________         \u2571                _____________________\n",
        "\u2571  4      2  2          2    4         \u2571     2    2     \u2571  4      2  2        \n",
        "  \u0394  - 2\u22c5\u0394 \u22c5\u03a9  - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9         \u2571     \u0394    \u03a9    \u2572\u2571  \u0394  - 2\u22c5\u0394 \u22c5\u03a9  - 4\u22c5\u0394\u22c5\u03a9\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 , -  \u2571    - \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                    \u2572\u2571       2    2                    2       \n",
        "\n",
        "__________        _______________________________________________\u239e\n",
        "_________        \u2571                ______________________________ \u239f\n",
        "  2    4        \u2571     2    2     \u2571  4      2  2          2    4  \u239f\n",
        "\u22c5g  + \u03a9        \u2571     \u0394    \u03a9    \u2572\u2571  \u0394  - 2\u22c5\u0394 \u22c5\u03a9  - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9   \u239f\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 ,   \u2571    - \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 \u239f\n",
        "            \u2572\u2571       2    2                    2                 \u23a0"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# extracting eigenvectors and eigenvalues for J*D\n",
      "[(f1,_,[w1]), (f2,_,[w2]), (f3,_,[w3]), (f4,_,[w4])] = (J * D).eigenvects()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f1, f2, f3, f4"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQ4AAAAVBAMAAABF+R16AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 14,
       "text": [
        "(-\u2148\u22c5\u03c9\u208a, \u2148\u22c5\u03c9\u208a, -\u2148\u22c5\u03c9\u208b, \u2148\u22c5\u03c9\u208b)"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "there is a one-to-one correspondence between eigenvalues, which gives us the normal mode frequencies $\\omega_{\\pm}$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "wpsol = solve(e2-f2, wp)[0]\n",
      "Eq(wp, wpsol)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAgoAAAAvBAMAAABwEf/KAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n6XbX9cJud92eBXa76/YkALvddWQW/gUj0IT2ebk62wAAAABJRU5ErkJggg==\n",
       "prompt_number": 15,
       "text": [
        "                  _______________________________________________ \n",
        "                 \u2571                ______________________________  \n",
        "        ___     \u2571     2    2     \u2571  4      2  2          2    4   \n",
        "     -\u2572\u2571 2 \u22c5\u2148\u22c5\u2572\u2571   - \u0394  - \u03a9  - \u2572\u2571  \u0394  - 2\u22c5\u0394 \u22c5\u03a9  - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9    \n",
        "\u03c9\u208a = \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                              "
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "wmsol = solve(e4-f4, wm)[0]\n",
      "Eq(wm, wmsol)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAgoAAAAvBAMAAABwEf/KAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nOgvsdtfpJAC73S1IFv4G1juMuTc3SfQAAAAASUVORK5CYII=\n",
       "prompt_number": 16,
       "text": [
        "                  _______________________________________________ \n",
        "                 \u2571                ______________________________  \n",
        "        ___     \u2571     2    2     \u2571  4      2  2          2    4   \n",
        "     -\u2572\u2571 2 \u22c5\u2148\u22c5\u2572\u2571   - \u0394  - \u03a9  + \u2572\u2571  \u0394  - 2\u22c5\u0394 \u22c5\u03a9  - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9    \n",
        "\u03c9\u208b = \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                              "
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# simpler expression without an imaginary number\n",
      "wpsol = sqrt((d**2 + O**2 + sqrt((d**2 -O**2)**2 -4*d*O*g**2))/2)\n",
      "wmsol = sqrt((d**2 + O**2 - sqrt((d**2 -O**2)**2 -4*d*O*g**2))/2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Eq(wp, wpsol), Eq(wm, wmsol)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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u4sHy/MEvuX+cZbX+v6q8n66ZXQuLTZaifZhwfsHIsCidFSXequLRrHsPuYnXLaGrrJpd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\nTkSuQmCC\n",
       "prompt_number": 18,
       "text": [
        "\u239b            _________________________________________              __________\n",
        "\u239c           \u2571               _________________________              \u2571          \n",
        "\u239c          \u2571               \u2571                       2              \u2571           \n",
        "\u239c         \u2571    2    2     \u2571           2   \u239b 2    2\u239e              \u2571    2    2  \n",
        "\u239c        \u2571    \u0394    \u03a9    \u2572\u2571   - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u239d\u0394  - \u03a9 \u23a0             \u2571    \u0394    \u03a9   \n",
        "\u239c\u03c9\u208a =   \u2571     \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 , \u03c9\u208b =   \u2571     \u2500\u2500 + \u2500\u2500 -\n",
        "\u239d     \u2572\u2571      2    2                  2                      \u2572\u2571      2    2   \n",
        "\n",
        "_______________________________\u239e\n",
        "     _________________________ \u239f\n",
        "    \u2571                       2  \u239f\n",
        "   \u2571           2   \u239b 2    2\u239e   \u239f\n",
        " \u2572\u2571   - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u239d\u0394  - \u03a9 \u23a0   \u239f\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 \u239f\n",
        "               2               \u23a0"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, try to find the transformation matrix $S$. First construct matrices $A$ and $B$ from the eigenvectors multiplied by arbitrary constants (which will be determined later)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# arbitrary constants\n",
      "aa, bb, cc, dd = symbols(\"alpha, beta, gamma, delta\")\n",
      "x, y, z, w = symbols(\"x, y, z, w\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ee2, ff2 = simplify(sqrt(2)*e2), sqrt(2)*f2\n",
      "ee4, ff4 = simplify(sqrt(2)*e4), sqrt(2)*f4\n",
      "subs_dic = {ee2:ff2, ee2**2+2*d**2:ff2**2+2*d**2, ee4:ff4, ee4**2+2*d**2: ff4**2+2*d**2}\n",
      "u1, u2, u3, u4 = (v1.subs(subs_dic), v2.subs(subs_dic),\n",
      "                  v3.subs(subs_dic), v4.subs(subs_dic))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = (x*u1).row_join(y*u2).row_join(z*u3).row_join(w*u4)\n",
      "A = simplify(A)\n",
      "A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAgEAAAB+CAMAAACK0zJuAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nCl9E5gTBF8TbhKx8f2CpM7CIjif/B1jP/xcbiQKIAAAAAElFTkSuQmCC\n",
       "prompt_number": 21,
       "text": [
        "\u23a1 -\u2148\u22c5\u0394\u22c5\u03a9\u22c5g\u22c5x      \u2148\u22c5\u0394\u22c5\u03a9\u22c5g\u22c5y     -\u2148\u22c5\u0394\u22c5\u03a9\u22c5g\u22c5z      \u2148\u22c5\u0394\u22c5\u03a9\u22c5g\u22c5w  \u23a4\n",
        "\u23a2\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\u23a5\n",
        "\u23a2   \u239b 2     2\u239e     \u239b 2     2\u239e     \u239b 2     2\u239e     \u239b 2     2\u239e\u23a5\n",
        "\u23a2\u03c9\u208a\u22c5\u239d\u0394  - \u03c9\u208a \u23a0  \u03c9\u208a\u22c5\u239d\u0394  - \u03c9\u208a \u23a0  \u03c9\u208b\u22c5\u239d\u0394  - \u03c9\u208b \u23a0  \u03c9\u208b\u22c5\u239d\u0394  - \u03c9\u208b \u23a0\u23a5\n",
        "\u23a2                                                          \u23a5\n",
        "\u23a2   \u03a9\u22c5g\u22c5x          \u03a9\u22c5g\u22c5y          \u03a9\u22c5g\u22c5z          \u03a9\u22c5g\u22c5w     \u23a5\n",
        "\u23a2  \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   \u23a5\n",
        "\u23a2   2     2        2     2        2     2        2     2   \u23a5\n",
        "\u23a2  \u0394  - \u03c9\u208a        \u0394  - \u03c9\u208a        \u0394  - \u03c9\u208b        \u0394  - \u03c9\u208b    \u23a5\n",
        "\u23a2                                                          \u23a5\n",
        "\u23a2    \u2148\u22c5\u03a9\u22c5x         -\u2148\u22c5\u03a9\u22c5y          \u2148\u22c5\u03a9\u22c5z         -\u2148\u22c5\u03a9\u22c5w    \u23a5\n",
        "\u23a2    \u2500\u2500\u2500\u2500\u2500         \u2500\u2500\u2500\u2500\u2500\u2500\u2500         \u2500\u2500\u2500\u2500\u2500         \u2500\u2500\u2500\u2500\u2500\u2500\u2500   \u23a5\n",
        "\u23a2      \u03c9\u208a             \u03c9\u208a             \u03c9\u208b             \u03c9\u208b     \u23a5\n",
        "\u23a2                                                          \u23a5\n",
        "\u23a3      x              y              z              w      \u23a6"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B = (aa*w1).row_join(bb*w2).row_join(cc*w3).row_join(dd*w4)\n",
      "B"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALsAAABkCAMAAADpJrGiAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nR/IuzvwLAAAAAElFTkSuQmCC\n",
       "prompt_number": 22,
       "text": [
        "\u23a1\u2148\u22c5\u03b1  -\u2148\u22c5\u03b2   0    0  \u23a4\n",
        "\u23a2                    \u23a5\n",
        "\u23a2 \u03b1    \u03b2     0    0  \u23a5\n",
        "\u23a2                    \u23a5\n",
        "\u23a2 0    0    \u2148\u22c5\u03b3  -\u2148\u22c5\u03b4\u23a5\n",
        "\u23a2                    \u23a5\n",
        "\u23a3 0    0     \u03b3    \u03b4  \u23a6"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "and use the relation $S = B A^{-1}$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Ainv = A.inv()\n",
      "Ainv = simplify(Ainv)\n",
      "S = simplify(B * Ainv)\n",
      "\n",
      "S = Matrix([[simplify(S[i, j].factor()) for j in range(4)] for i in range(4)])\n",
      "S"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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kZma4QF63fB1up1Pym26iWiZoqdVyJGcWaKqlsmhmSIcV87GW2PybzTLLi/iZ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\n/j//GD7rrSwAhAAAAABJRU5ErkJggg==\n",
       "prompt_number": 23,
       "text": [
        "\u23a1-\u03c9\u208a\u22c5(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)\u22c5(\u03b1\u22c5y + \u03b2\u22c5x)   \u2148\u22c5(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u0394 \n",
        "\u23a2\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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u23a2          2\u22c5\u0394\u22c5\u03a9\u22c5g\u22c5x\u22c5y\u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)                       2\u22c5\u03a9\u22c5g\u22c5x\u22c5y\u22c5(\u03c9\u208a\n",
        "\u23a2                                                                             \n",
        "\u23a2\u2148\u22c5\u03c9\u208a\u22c5(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)\u22c5(\u03b1\u22c5y - \u03b2\u22c5x)   (\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u0394 -\n",
        "\u23a2\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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u23a2          2\u22c5\u0394\u22c5\u03a9\u22c5g\u22c5x\u22c5y\u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)                       2\u22c5\u03a9\u22c5g\u22c5x\u22c5y\u22c5(\u03c9\u208a\n",
        "\u23a2                                                                             \n",
        "\u23a2 \u03c9\u208b\u22c5(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)\u22c5(\u03b4\u22c5z + \u03b3\u22c5w)   \u2148\u22c5(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u0394 \n",
        "\u23a2 \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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u23a2          2\u22c5\u0394\u22c5\u03a9\u22c5g\u22c5w\u22c5z\u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)                       2\u22c5\u03a9\u22c5g\u22c5w\u22c5z\u22c5(\u03c9\u208a\n",
        "\u23a2                                                                             \n",
        "\u23a2\u2148\u22c5\u03c9\u208b\u22c5(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)\u22c5(\u03b4\u22c5z - \u03b3\u22c5w)  -(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u0394 -\n",
        "\u23a2\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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u23a3          2\u22c5\u0394\u22c5\u03a9\u22c5g\u22c5w\u22c5z\u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)                       2\u22c5\u03a9\u22c5g\u22c5w\u22c5z\u22c5(\u03c9\u208a\n",
        "\n",
        "- \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)\u22c5(\u03b1\u22c5y - \u03b2\u22c5x)  -\u03c9\u208a\u22c5(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u03b1\u22c5y + \u03b2\u22c5x)   -\u2148\u22c5(\u0394 - \u03c9\u208a)\u22c5(\u0394\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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        " - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)               2\u22c5\u03a9\u22c5x\u22c5y\u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)          2\u22c5x\u22c5y\u22c5(\u03c9\u208a \n",
        "                                                                              \n",
        " \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)\u22c5(\u03b1\u22c5y + \u03b2\u22c5x)   \u2148\u22c5\u03c9\u208a\u22c5(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u03b1\u22c5y - \u03b2\u22c5x)   -(\u0394 - \u03c9\u208a)\u22c5(\u0394 \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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        " - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)               2\u22c5\u03a9\u22c5x\u22c5y\u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)          2\u22c5x\u22c5y\u22c5(\u03c9\u208a \n",
        "                                                                              \n",
        "- \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)\u22c5(\u03b4\u22c5z - \u03b3\u22c5w)   \u03c9\u208b\u22c5(\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)\u22c5(\u03b4\u22c5z + \u03b3\u22c5w)   -\u2148\u22c5(\u0394 - \u03c9\u208b)\u22c5(\u0394\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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        " - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)               2\u22c5\u03a9\u22c5w\u22c5z\u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)          2\u22c5w\u22c5z\u22c5(\u03c9\u208a \n",
        "                                                                              \n",
        " \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)\u22c5(\u03b4\u22c5z + \u03b3\u22c5w)   \u2148\u22c5\u03c9\u208b\u22c5(\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)\u22c5(\u03b4\u22c5z - \u03b3\u22c5w)    (\u0394 - \u03c9\u208b)\u22c5(\u0394 \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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        " - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)               2\u22c5\u03a9\u22c5w\u22c5z\u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)          2\u22c5w\u22c5z\u22c5(\u03c9\u208a \n",
        "\n",
        " + \u03c9\u208a)\u22c5(\u03b1\u22c5y - \u03b2\u22c5x) \u23a4\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "- \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)    \u23a5\n",
        "                   \u23a5\n",
        "+ \u03c9\u208a)\u22c5(\u03b1\u22c5y + \u03b2\u22c5x)  \u23a5\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u23a5\n",
        "- \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)    \u23a5\n",
        "                   \u23a5\n",
        " + \u03c9\u208b)\u22c5(\u03b4\u22c5z - \u03b3\u22c5w) \u23a5\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "- \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)    \u23a5\n",
        "                   \u23a5\n",
        "+ \u03c9\u208b)\u22c5(\u03b4\u22c5z + \u03b3\u22c5w)  \u23a5\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  \u23a5\n",
        "- \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)    \u23a6"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The aritrary constants can be determined by the condition 2), $J = SJS^T$. Calculating $SJS^T$, we get "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "SJST = simplify(S * J * S.T)\n",
      "SJST = Matrix([[simplify(SJST[i,j].subs({wp:wpsol, wm:wmsol}).factor()) for j in range(4)] for i in range(4)])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The non-vanishing terms are (0, 1), (1, 0), (2, 3), (3, 2)-th elements and should be set to be equal to"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Eq(1, simplify(SJST[0, 1].subs(subs_dic).factor()))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 25,
       "text": [
        "                  _____________________________________________               \n",
        "                 \u2571              ______________________________  \u239b             \n",
        "      ___       \u2571   2    2     \u2571  4      2  2          2    4   \u239c 4      2  2 \n",
        "    \u2572\u2571 2 \u22c5\u03b1\u22c5\u03b2\u22c5\u2572\u2571   \u0394  + \u03a9  + \u2572\u2571  \u0394  - 2\u22c5\u0394 \u22c5\u03a9  - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9   \u22c5\u239d\u0394  - 2\u22c5\u0394 \u22c5\u03a9  \n",
        "1 = \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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                                             \u239b\n",
        "                                                                     4\u22c5\u03a9\u22c5x\u22c5y\u22c5\u239d\n",
        "\n",
        "                                                                              \n",
        "        ______________________________                         _______________\n",
        "   2   \u2571  4      2  2          2    4           2    4    2   \u2571  4      2  2  \n",
        "- \u0394 \u22c5\u2572\u2571  \u0394  - 2\u22c5\u0394 \u22c5\u03a9  - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9   - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9  + \u03a9 \u22c5\u2572\u2571  \u0394  - 2\u22c5\u0394 \u22c5\u03a9  -\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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        " 4      2  2          2    4\u239e                                                 \n",
        "\u0394  - 2\u22c5\u0394 \u22c5\u03a9  - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9 \u23a0                                                 \n",
        "\n",
        "                \n",
        "_______________\u239e\n",
        "        2    4 \u239f\n",
        " 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9  \u23a0\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                \n",
        "                "
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Eq(1, simplify(SJST[2, 3].subs(subs_dic).factor()))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 26,
       "text": [
        "                  _____________________________________________               \n",
        "                 \u2571              ______________________________  \u239b             \n",
        "      ___       \u2571   2    2     \u2571  4      2  2          2    4   \u239c 4      2  2 \n",
        "    \u2572\u2571 2 \u22c5\u03b4\u22c5\u03b3\u22c5\u2572\u2571   \u0394  + \u03a9  - \u2572\u2571  \u0394  - 2\u22c5\u0394 \u22c5\u03a9  - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9   \u22c5\u239d\u0394  - 2\u22c5\u0394 \u22c5\u03a9  \n",
        "1 = \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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                                             \u239b\n",
        "                                                                     4\u22c5\u03a9\u22c5w\u22c5z\u22c5\u239d\n",
        "\n",
        "                                                                              \n",
        "        ______________________________                         _______________\n",
        "   2   \u2571  4      2  2          2    4           2    4    2   \u2571  4      2  2  \n",
        "+ \u0394 \u22c5\u2572\u2571  \u0394  - 2\u22c5\u0394 \u22c5\u03a9  - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9   - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9  - \u03a9 \u22c5\u2572\u2571  \u0394  - 2\u22c5\u0394 \u22c5\u03a9  -\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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        " 4      2  2          2    4\u239e                                                 \n",
        "\u0394  - 2\u22c5\u0394 \u22c5\u03a9  - 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9 \u23a0                                                 \n",
        "\n",
        "                \n",
        "_______________\u239e\n",
        "        2    4 \u239f\n",
        " 4\u22c5\u0394\u22c5\u03a9\u22c5g  + \u03a9  \u23a0\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                \n",
        "                "
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "I will choose the arbitrary constants to be\n",
      "\n",
      "\\begin{align}\n",
      "x &= y = \\sqrt{\\frac{\\omega_+ (\\omega_+^2 - \\Delta^2)}{\\Omega (\\omega_+^2 - \\omega_-^2)}},\\\\\n",
      "z &= w = \\sqrt{\\frac{\\omega_- (\\Delta^2 - \\omega_-^2)}{\\Omega (\\omega_+^2 - \\omega_-^2)}},\\\\\n",
      "\\alpha&=\\beta=\\gamma=\\delta = 1\n",
      "\\end{align}"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "then, the transformation matrix $S$ becomes"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "S_ = S.subs({x: sqrt(wp * (wp**2 - d**2)/(O*(wp**2-wm**2))),\n",
      "        y: sqrt(wp * (wp**2 - d**2)/(O*(wp**2-wm**2))),\n",
      "        z: sqrt(wm * (d**2 - wm**2)/(O*(wp**2-wm**2))),\n",
      "        w: sqrt(wm * (d**2 - wm**2)/(O*(wp**2-wm**2))),\n",
      "        aa:1, bb:1, cc:1, dd:1})\n",
      "S_ = Matrix([[simplify(S_[i,j].factor()).factor() for j in range(4)] for i in range(4)])\n",
      "S_"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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MMbAxA6TpNKH/wX1P79ZK81RTip6/K61kuBymq4c3FsLaO/EYSJ/EwO4MkKbT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\nRp+Hc6pJ/QxsqnMSev2COEA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       "prompt_number": 27,
       "text": [
        "\u23a1         ____                                                                \n",
        "\u23a2      -\u2572\u2571 \u03c9\u208a \u22c5(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)                            \n",
        "\u23a2\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",
        "\u23a2               _____________________                                         \n",
        "\u23a2    ___       \u2571 -(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)                                           \n",
        "\u23a2\u0394\u22c5\u2572\u2571 \u03a9 \u22c5g\u22c5   \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)                     \n",
        "\u23a2            \u2571          2     2                                               \n",
        "\u23a2          \u2572\u2571         \u03c9\u208a  - \u03c9\u208b                                                \n",
        "\u23a2                                                                             \n",
        "\u23a2                                                                             \n",
        "\u23a2                                                                       (\u0394 - \u03c9\n",
        "\u23a2                           0                              \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u23a2                                                                             \n",
        "\u23a2                                                            ___     ____     \n",
        "\u23a2                                                          \u2572\u2571 \u03a9 \u22c5g\u22c5\u2572\u2571 \u03c9\u208a \u22c5   \u2571\n",
        "\u23a2                                                                           \u2571 \n",
        "\u23a2                                                                         \u2572\u2571  \n",
        "\u23a2                                                                             \n",
        "\u23a2         ____                                                                \n",
        "\u23a2       \u2572\u2571 \u03c9\u208b \u22c5(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)                            \n",
        "\u23a2 \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",
        "\u23a2                ___________________                                          \n",
        "\u23a2     ___       \u2571 (\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)                                           \n",
        "\u23a2 \u0394\u22c5\u2572\u2571 \u03a9 \u22c5g\u22c5   \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)                      \n",
        "\u23a2             \u2571         2     2                                               \n",
        "\u23a2           \u2572\u2571        \u03c9\u208a  - \u03c9\u208b                                                \n",
        "\u23a2                                                                             \n",
        "\u23a2                                                                             \n",
        "\u23a2                                                                      -(\u0394 - \u03c9\n",
        "\u23a2                           0                               \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u23a2                                                                             \n",
        "\u23a2                                                             ___     ____    \n",
        "\u23a2                                                           \u2572\u2571 \u03a9 \u22c5g\u22c5\u2572\u2571 \u03c9\u208b \u22c5   \n",
        "\u23a2                                                                            \u2571\n",
        "\u23a3                                                                          \u2572\u2571 \n",
        "\n",
        "                                                            ____              \n",
        "                                                         -\u2572\u2571 \u03c9\u208a \u22c5(\u0394 - \u03c9\u208a)\u22c5(\u0394 +\n",
        "           0                                \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",
        "                                                       _____________________  \n",
        "                                              ___     \u2571 -(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)    \n",
        "                                            \u2572\u2571 \u03a9 \u22c5   \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5(\n",
        "                                                    \u2571          2     2        \n",
        "                                                  \u2572\u2571         \u03c9\u208a  - \u03c9\u208b         \n",
        "                                                                              \n",
        "                                                                              \n",
        "\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)                                                 \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                           0        \n",
        " _____________________                                                        \n",
        "\u2571 -(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)                                                          \n",
        "  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)                                    \n",
        "         2     2                                                              \n",
        "       \u03c9\u208a  - \u03c9\u208b                                                               \n",
        "                                                                              \n",
        "                                                            ____              \n",
        "                                                          \u2572\u2571 \u03c9\u208b \u22c5(\u0394 - \u03c9\u208b)\u22c5(\u0394 +\n",
        "           0                                 \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",
        "                                                        ___________________   \n",
        "                                               ___     \u2571 (\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)    \n",
        "                                             \u2572\u2571 \u03a9 \u22c5   \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5(\u03c9\n",
        "                                                     \u2571         2     2        \n",
        "                                                   \u2572\u2571        \u03c9\u208a  - \u03c9\u208b         \n",
        "                                                                              \n",
        "                                                                              \n",
        "\u208a)\u22c5(\u0394 + \u03c9\u208a)\u22c5(\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)                                                 \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                            0        \n",
        "  ___________________                                                         \n",
        " \u2571 (\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)                                                          \n",
        "\u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)                                     \n",
        "         2     2                                                              \n",
        "       \u03c9\u208a  - \u03c9\u208b                                                               \n",
        "\n",
        "                                                                         \u23a4\n",
        " \u03c9\u208a)                                                                     \u23a5\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                            0                          \u23a5\n",
        "                                                                         \u23a5\n",
        "                                                                         \u23a5\n",
        "\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)                                                       \u23a5\n",
        "                                                                         \u23a5\n",
        "                                                                         \u23a5\n",
        "                                                                         \u23a5\n",
        "                                     ___                                 \u23a5\n",
        "                                  -\u2572\u2571 \u03a9 \u22c5(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)               \u23a5\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\u23a5\n",
        "                                _____________________                    \u23a5\n",
        "                      ____     \u2571 -(\u0394 - \u03c9\u208a)\u22c5(\u0394 + \u03c9\u208a)                      \u23a5\n",
        "                    \u2572\u2571 \u03c9\u208a \u22c5   \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)\u23a5\n",
        "                             \u2571          2     2                          \u23a5\n",
        "                           \u2572\u2571         \u03c9\u208a  - \u03c9\u208b                           \u23a5\n",
        "                                                                         \u23a5\n",
        "                                                                         \u23a5\n",
        " \u03c9\u208b)                                                                     \u23a5\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                             0                          \u23a5\n",
        "                                                                         \u23a5\n",
        "                                                                         \u23a5\n",
        "\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b)                                                        \u23a5\n",
        "                                                                         \u23a5\n",
        "                                                                         \u23a5\n",
        "                                                                         \u23a5\n",
        "                                     ___                                 \u23a5\n",
        "                                   \u2572\u2571 \u03a9 \u22c5(\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)               \u23a5\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 \u23a5\n",
        "                                 ___________________                     \u23a5\n",
        "                       ____     \u2571 (\u0394 - \u03c9\u208b)\u22c5(\u0394 + \u03c9\u208b)                      \u23a5\n",
        "                     \u2572\u2571 \u03c9\u208b \u22c5   \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5(\u03c9\u208a - \u03c9\u208b)\u22c5(\u03c9\u208a + \u03c9\u208b) \u23a5\n",
        "                              \u2571         2     2                          \u23a5\n",
        "                            \u2572\u2571        \u03c9\u208a  - \u03c9\u208b                           \u23a6"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's check if this matrix satisfies the two conditions we discussed above:"
     ]
    },
    {
     "cell_type": "heading",
     "level": 5,
     "metadata": {},
     "source": [
      "Check 1 : $M = S^T D S$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#check1\n",
      "STDS_ = simplify(S_.T * D * S_)\n",
      "STDS_ = Matrix([[simplify(STDS_[i,j].subs({wp:wpsol, wm:wmsol}).factor()) for j in range(4)] for i in range(4)])\n",
      "STDS_"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALsAAABkCAMAAADpJrGiAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 28,
       "text": [
        "\u23a1-\u0394  0   -g  0\u23a4\n",
        "\u23a2             \u23a5\n",
        "\u23a20   -\u0394  0   0\u23a5\n",
        "\u23a2             \u23a5\n",
        "\u23a2-g  0   \u03a9   0\u23a5\n",
        "\u23a2             \u23a5\n",
        "\u23a30   0   0   \u03a9\u23a6"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "heading",
     "level": 5,
     "metadata": {},
     "source": [
      "Check 2 : $J = S J S^T$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#check2\n",
      "SJST_ = simplify(S_ * J * S_.T)\n",
      "SJST_ = Matrix([[simplify(SJST_[i,j].subs({wp:wpsol, wm:wmsol}).factor()) for j in range(4)] for i in range(4)])\n",
      "SJST_"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJgAAABkCAMAAABNTAlxAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nf5bYh/4itel/AVrXl9y0f0o//CK1rs0/v51EehgG0m0AAAAASUVORK5CYII=\n",
       "prompt_number": 29,
       "text": [
        "\u23a10   1  0   0\u23a4\n",
        "\u23a2            \u23a5\n",
        "\u23a2-1  0  0   0\u23a5\n",
        "\u23a2            \u23a5\n",
        "\u23a20   0  0   1\u23a5\n",
        "\u23a2            \u23a5\n",
        "\u23a30   0  -1  0\u23a6"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Both conditions are fully satisfied with this transformation $S$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The transformation between bosonic operators in different frames is readily obtained:\n",
      "$$\n",
      "Q\\vec{a}^{NM} = S \\left(Q\\vec{a}\\right) \\quad \\Rightarrow \\quad \\vec{a}^{NM} = (Q^{-1} S Q) \\vec{a} = T\\vec{a}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "T = simplify((Q.inv() * S_ * Q))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lhs = anm[0]\n",
      "rhs = simplify((T * a)[0])\n",
      "Eq(lhs, rhs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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8dV6msfM0YU2ZOaK89fEw7KMIS2mi2MZmrI5VXPo7AlnPpLHXdXE663hJ4i2W8ta4cZm/\nxSN3AT9UWolrlMWlPyOwHWE9hr/1PiTOxxeaLMGUeeshj/82hCQnwGV0AdOjNb8DkiwufRmBdA1h\n0Y92XuReB+AH90qd89ajHfg0dmHBW2kocS3kcd1/EXgcElV+RuWdFw9V4SzEqPLWeLFuEZDT9D//\nBBLX/edKjIgjkMGJqpqM8LDPL7Iqbz0JtEUMV1xajkOJ6ziAfRqB9x+16sM6jY3PSbuz9C6m8tZj\n1eQuSG9GyVqAULq5Tx2KYb04g/9Vnk5jQ+Zl/vaj8tb5g49qwt7r2/xmE0pcx7EbkAg892wQaMHH\nd1EPAt8V4yfeYIz6nJvEKTSlAWNt/MqwC3g5VvJ4SlUkBqK1dsdhXn03/BJR4x9GqJQE+xBP6UBM\npRNk+bBTJqhjZYvebHSJp1SHYmCJ0RN9G3vJZmJ6ICNQ2UxvNnF5B0WgLHchv4Ncete78uS7PgL9\nFYD/AqYDxwjv46CUAAAAAElFTkSuQmCC\n",
       "prompt_number": 31,
       "text": [
        "           ____________                                                       \n",
        "          \u2571    2     2                                                        \n",
        "         \u2571  - \u0394  + \u03c9\u208a   \u239b    \u239b             \u2020               \u239e   \u239b 2     2\u239e \u239b   \n",
        "        \u2571   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u239d\u0394\u22c5g\u22c5\u239d(-\u03a9 + \u03c9\u208a)\u22c5a_m  + (\u03a9 + \u03c9\u208a)\u22c5a_m\u23a0 + \u239d\u0394  - \u03c9\u208b \u23a0\u22c5\u239d(-\u0394\n",
        "       \u2571      2     2                                                         \n",
        "     \u2572\u2571     \u03c9\u208a  - \u03c9\u208b                                                          \n",
        "a\u208a = \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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                     ___     ____             \n",
        "                                               2\u22c5\u0394\u22c5\u2572\u2571 \u03a9 \u22c5g\u22c5\u2572\u2571 \u03c9\u208a              \n",
        "\n",
        "                            \n",
        "                            \n",
        "                         \u2020\u239e\u239e\n",
        " + \u03c9\u208a)\u22c5a_c + (\u0394 + \u03c9\u208a)\u22c5a_c \u23a0\u23a0\n",
        "                            \n",
        "                            \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\n",
        "                            \n",
        "                            "
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lhs = anm[1]\n",
      "rhs = simplify((T * a)[1])\n",
      "Eq(lhs, rhs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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VmJCJlNVRSqM3LoeGdJ7IQjWwY6sTEHUT/E4qBiYkAhQSUgA/PohfioekuSycanvXvuWY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\nRK5CYII=\n",
       "prompt_number": 32,
       "text": [
        "            ____________                                                      \n",
        "           \u2571    2     2                                                       \n",
        "          \u2571  - \u0394  + \u03c9\u208a   \u239b    \u239b                            \u2020\u239e   \u239b 2     2\u239e \u239b  \n",
        "         \u2571   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u239d\u0394\u22c5g\u22c5\u239d(-\u03a9 + \u03c9\u208a)\u22c5a_m + (\u03a9 + \u03c9\u208a)\u22c5a_m \u23a0 + \u239d\u0394  - \u03c9\u208b \u23a0\u22c5\u239d(-\n",
        "        \u2571      2     2                                                        \n",
        "  \u2020   \u2572\u2571     \u03c9\u208a  - \u03c9\u208b                                                         \n",
        "a\u208a  = \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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                      ___     ____            \n",
        "                                                2\u22c5\u0394\u22c5\u2572\u2571 \u03a9 \u22c5g\u22c5\u2572\u2571 \u03c9\u208a             \n",
        "\n",
        "                             \n",
        "                             \n",
        "           \u2020               \u239e\u239e\n",
        "\u0394 + \u03c9\u208a)\u22c5a_c  + (\u0394 + \u03c9\u208a)\u22c5a_c\u23a0\u23a0\n",
        "                             \n",
        "                             \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\n",
        "                             \n",
        "                             "
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lhs = anm[2]\n",
      "rhs = simplify((T * a)[2])\n",
      "Eq(lhs, rhs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 33,
       "text": [
        "           ___________                                                        \n",
        "          \u2571   2     2                                                         \n",
        "         \u2571   \u0394  - \u03c9\u208b   \u239b    \u239b             \u2020               \u239e   \u239b 2     2\u239e \u239b    \n",
        "        \u2571   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u239d\u0394\u22c5g\u22c5\u239d(-\u03a9 + \u03c9\u208b)\u22c5a_m  + (\u03a9 + \u03c9\u208b)\u22c5a_m\u23a0 + \u239d\u0394  - \u03c9\u208a \u23a0\u22c5\u239d(-\u0394 \n",
        "       \u2571      2     2                                                         \n",
        "     \u2572\u2571     \u03c9\u208a  - \u03c9\u208b                                                          \n",
        "a\u208b = \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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                    ___     ____              \n",
        "                                              2\u22c5\u0394\u22c5\u2572\u2571 \u03a9 \u22c5g\u22c5\u2572\u2571 \u03c9\u208b               \n",
        "\n",
        "                           \n",
        "                           \n",
        "                        \u2020\u239e\u239e\n",
        "+ \u03c9\u208b)\u22c5a_c + (\u0394 + \u03c9\u208b)\u22c5a_c \u23a0\u23a0\n",
        "                           \n",
        "                           \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\n",
        "                           \n",
        "                           "
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lhs = anm[3]\n",
      "rhs = simplify((T * a)[3])\n",
      "Eq(lhs, rhs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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YrvG9IMgNYXud5/VKMHtZ8PbqSmNL/PlZH0cKfsKtPh0XS5TXX1cLQWCAD6tBvn4q5T5s\nGZBG1f/A69H1EmzQN9jTcDm7G4dQGlvmysnAJd5seH+rkXaRI5DBOys+OyXpBptpYFXDKd2DLaWx\nZa4cOXgU7sPayoaTMCojG4F0U+TaNuJpe0oH57SCsyfT2CpXjuBxNdXGxiSOiYnK6EZgM0J7Gj/r\n2hC7GJ950gRVprHHZK4cJYkWnsyoG7yAp/FRWewIJEuIgD60l8V5G4AfDMYrnMZWuXJUX9KVDUdh\nVEY2As9ArMirPb6nPl7kP11QaWyVK0f0p9HbSVc2fGRdioBhihY3CBTjHImNRdxJ3uaHXpXGFrly\nvADTasqbDY/CN8IR+MAxKw+DZzEzPjOzvo0LqJ1peuBTaWyRK5erKW82fIS9iqBtX8B/J7hevPnF\nG2zqLX6zotLYnCtXqylvNjwKXogi8OrLbrDZtrWacqsiLkQR2IbzaIrMlcvVlJFHVNgiYO0SNLly\nuZoKmy8R3h4RULlyuZrq0SsShzUCcjUVVvgR7l4R8K6mevWL5OGKgHs1FS7sEdreEXCtpnp3izT7\nKgL/A8/ghZK+h5vTAAAAAElFTkSuQmCC\n",
       "prompt_number": 34,
       "text": [
        "            ___________                                                       \n",
        "           \u2571   2     2                                                        \n",
        "          \u2571   \u0394  - \u03c9\u208b   \u239b    \u239b                            \u2020\u239e   \u239b 2     2\u239e \u239b   \n",
        "         \u2571   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u239d\u0394\u22c5g\u22c5\u239d(-\u03a9 + \u03c9\u208b)\u22c5a_m + (\u03a9 + \u03c9\u208b)\u22c5a_m \u23a0 + \u239d\u0394  - \u03c9\u208a \u23a0\u22c5\u239d(-\u0394\n",
        "        \u2571      2     2                                                        \n",
        "  \u2020   \u2572\u2571     \u03c9\u208a  - \u03c9\u208b                                                         \n",
        "a\u208b  = \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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                     ___     ____             \n",
        "                                               2\u22c5\u0394\u22c5\u2572\u2571 \u03a9 \u22c5g\u22c5\u2572\u2571 \u03c9\u208b              \n",
        "\n",
        "                            \n",
        "                            \n",
        "          \u2020               \u239e\u239e\n",
        " + \u03c9\u208b)\u22c5a_c  + (\u0394 + \u03c9\u208b)\u22c5a_c\u23a0\u23a0\n",
        "                            \n",
        "                            \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\n",
        "                            \n",
        "                            "
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Special Case: Red-detuned side ($\\Delta = -\\Omega$)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here, I will consider the special case of the optomechanical system, where the driving field is tuned to the upper sideband ($\\Delta = -\\Omega$). The transformation matrix $S$ becomes"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "S__ = simplify(S_.subs({d: -O, wp: wpsol.subs(d, -O), wm: wmsol.subs(d, -O)}))\n",
      "lhs = Rnm\n",
      "rhs = MatMul(S__, R)\n",
      "Eq(lhs, rhs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 35,
       "text": [
        "       \u23a1   ___ 4 _______                    ___ 4 _______             \u23a4      \n",
        "       \u23a2-\u2572\u2571 2 \u22c5\u2572\u2571 \u03a9 + g                   \u2572\u2571 2 \u22c5\u2572\u2571 \u03a9 + g              \u23a5      \n",
        "\u23a1X\u208a\u23a4 = \u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500        0        \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500       0     \u23a5\u22c5\u23a1X_c\u23a4\n",
        "\u23a2  \u23a5   \u23a2       4 ___                            4 ___                 \u23a5 \u23a2   \u23a5\n",
        "\u23a2P\u208a\u23a5   \u23a2     2\u22c5\u2572\u2571 \u03a9                           2\u22c5\u2572\u2571 \u03a9                  \u23a5 \u23a2P_c\u23a5\n",
        "\u23a2  \u23a5   \u23a2                                                              \u23a5 \u23a2   \u23a5\n",
        "\u23a2X\u208b\u23a5   \u23a2                      ___ 4 ___                      ___ 4 ___\u23a5 \u23a2X_m\u23a5\n",
        "\u23a2  \u23a5   \u23a2                   -\u2572\u2571 2 \u22c5\u2572\u2571 \u03a9                     \u2572\u2571 2 \u22c5\u2572\u2571 \u03a9 \u23a5 \u23a2   \u23a5\n",
        "\u23a3P\u208b\u23a6   \u23a2        0          \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500         0         \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5 \u23a3P_m\u23a6\n",
        "       \u23a2                      4 _______                      4 _______\u23a5      \n",
        "       \u23a2                    2\u22c5\u2572\u2571 \u03a9 + g                     2\u22c5\u2572\u2571 \u03a9 + g \u23a5      \n",
        "       \u23a2                                                              \u23a5      \n",
        "       \u23a2   ___ 4 _______                    ___ 4 _______             \u23a5      \n",
        "       \u23a2 \u2572\u2571 2 \u22c5\u2572\u2571 \u03a9 - g                   \u2572\u2571 2 \u22c5\u2572\u2571 \u03a9 - g              \u23a5      \n",
        "       \u23a2 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500         0        \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500       0     \u23a5      \n",
        "       \u23a2       4 ___                            4 ___                 \u23a5      \n",
        "       \u23a2     2\u22c5\u2572\u2571 \u03a9                           2\u22c5\u2572\u2571 \u03a9                  \u23a5      \n",
        "       \u23a2                                                              \u23a5      \n",
        "       \u23a2                      ___ 4 ___                      ___ 4 ___\u23a5      \n",
        "       \u23a2                    \u2572\u2571 2 \u22c5\u2572\u2571 \u03a9                     \u2572\u2571 2 \u22c5\u2572\u2571 \u03a9 \u23a5      \n",
        "       \u23a2        0           \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500          0         \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5      \n",
        "       \u23a2                      4 _______                      4 _______\u23a5      \n",
        "       \u23a3                    2\u22c5\u2572\u2571 \u03a9 - g                     2\u22c5\u2572\u2571 \u03a9 - g \u23a6      "
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "so that\n",
      "\n",
      "\n",
      "\\begin{align}\n",
      "X_{\\pm} &= \\sqrt[4]{\\frac{\\Omega\\pm g}{\\Omega}} \\frac{\\mp X_c + X_p}{\\sqrt{2}},\\\\\n",
      "P_{\\pm} &= \\sqrt[4]{\\frac{\\Omega}{(\\Omega\\pm g)}} \\frac{\\mp P_c + P_p}{\\sqrt{2}}.\n",
      "\\end{align}"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "It can be easily shown that this transformation is in consistent with the original Hamiltonian."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H = hbar * wp/2 *(X_p**2 + P_p**2) + hbar * wm/2 * (X_m**2 + P_m**2)\n",
      "H"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAa8AAAArCAMAAAD44/dJAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\nxHGTATbktv+JPqOsp2Eyz/k/ARpCqt+YOdsAAAAASUVORK5CYII=\n",
       "prompt_number": 36,
       "text": [
        "      \u239b  2     2\u239e         \u239b  2     2\u239e\n",
        "h\u0305\u22c5\u03c9\u208a\u22c5\u239dP\u208a  + X\u208a \u23a0   h\u0305\u22c5\u03c9\u208b\u22c5\u239dP\u208b  + X\u208b \u23a0\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\n",
        "        2                   2        "
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "By substituting into the normal mode Hamiltonian,"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H_ = H.subs({wp: wpsol.subs(d, -O), wm: wmsol.subs(d, -O),\n",
      "             X_p: (rhs.doit())[0],\n",
      "             P_p: (rhs.doit())[1],\n",
      "             X_m: (rhs.doit())[2],\n",
      "             P_m: (rhs.doit())[3],\n",
      "             })\n",
      "simplify(H_.expand().factor()).subs({sqrt(((O+g)**2).expand()): O+g,\n",
      "                                     Xm * Xc: Xc * Xm,\n",
      "                                     Pm * Pc: Pc * Pm}).expand()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAhAAAAArBAMAAADbDKz9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n6ejbgGtCm4H9D09fxCKroxxwAAAAAElFTkSuQmCC\n",
       "prompt_number": 37,
       "text": [
        "        2           2           2           2               \n",
        "\u03a9\u22c5h\u0305\u22c5P_c    \u03a9\u22c5h\u0305\u22c5P_m    \u03a9\u22c5h\u0305\u22c5X_c    \u03a9\u22c5h\u0305\u22c5X_m                \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 - g\u22c5h\u0305\u22c5X_c\u22c5X_m\n",
        "    2           2           2           2                   "
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Version Information"
     ]
    },
    {
     "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>posix [darwin]</td></tr><tr><td>sympy</td><td>0.7.5-git</td></tr><tr><td>sympsi</td><td>0.1.0.dev-11eaf6c</td></tr><tr><td colspan='2'>Fri Oct 24 15:05:38 2014 JST</td></tr></table>"
       ],
       "json": [
        "{\"Software versions\": [{\"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\": \"Python\"}, {\"version\": \"2.3.0\", \"module\": \"IPython\"}, {\"version\": \"posix [darwin]\", \"module\": \"OS\"}, {\"version\": \"0.7.5-git\", \"module\": \"sympy\"}, {\"version\": \"0.1.0.dev-11eaf6c\", \"module\": \"sympsi\"}]}"
       ],
       "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 & posix [darwin] \\\\ \\hline\n",
        "sympy & 0.7.5-git \\\\ \\hline\n",
        "sympsi & 0.1.0.dev-11eaf6c \\\\ \\hline\n",
        "\\hline \\multicolumn{2}{|l|}{Fri Oct 24 15:05:38 2014 JST} \\\\ \\hline\n",
        "\\end{tabular}\n"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 1,
       "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 posix [darwin]\n",
        "sympy 0.7.5-git\n",
        "sympsi 0.1.0.dev-11eaf6c\n",
        "<tr><td colspan='2'>Fri Oct 24 15:05:38 2014 JST</td></tr>"
       ]
      }
     ],
     "prompt_number": 1
    }
   ],
   "metadata": {}
  }
 ]
}