{ "metadata": { "name": "", "signature": "sha256:197d4fa7f8628301ebfe6cb9c6dff312f80a010e6bd7d25e177589869adbba64" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Unitary transformations of pauli operators" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from sympy import *\n", "init_printing()\n", "\n", "from sympy.physics.quantum import *\n", "from sympy.physics.quantum.pauli import *" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "code", "collapsed": false, "input": [ "from sympy_quantum_utils import *" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "code", "collapsed": false, "input": [ "eps, delta, theta, Hsym, t = symbols(\"epsilon, delta, theta, H, t\")\n", "Omega = symbols(\"Omega\", positive=True)\n", "n = 6" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "code", "collapsed": false, "input": [ "sx, sy, sz = SigmaX(), SigmaY(), SigmaZ()" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 4 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Two-level system in its eigenbasis" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Consider the Hamiltonian for a two-level quantum system on the form:\n", "\n", "$$\n", "H = \\frac{1}{2}\\Delta\\sigma_x + \\frac{1}{2}\\epsilon\\sigma_z \n", "$$\n", "\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H = eps/2 * sz + delta/2 * sx\n", "\n", "Eq(Hsym, H)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = \\frac{\\delta {\\sigma_x}}{2} + \\frac{\\epsilon {\\sigma_z}}{2}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 21, "text": [ " \u03b4\u22c5O \u03b5\u22c5O\n", "H = \u2500\u2500\u2500 + \u2500\u2500\u2500\n", " 2 2 " ] } ], "prompt_number": 21 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Instantaneous eigenbasis" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can transform the Hamiltonian to the eigenbasis (where the Hamiltonian is diagonal, only containing a $\\sigma_z$ term) by applying the unitary tranformation:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(I * theta/2 * sy)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 6 }, { "cell_type": "code", "collapsed": false, "input": [ "hamiltonian_transformation_auto(U, sx)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\sin{\\left (\\theta \\right )} {\\sigma_z} + \\cos{\\left (\\theta \\right )} {\\sigma_x}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 7, "text": [ "sin(\u03b8)\u22c5O + cos(\u03b8)\u22c5O" ] } ], "prompt_number": 7 }, { "cell_type": "code", "collapsed": false, "input": [ "hamiltonian_transformation_auto(U, sz)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\sin{\\left (\\theta \\right )} {\\sigma_x} + \\cos{\\left (\\theta \\right )} {\\sigma_z}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 8, "text": [ "-sin(\u03b8)\u22c5O + cos(\u03b8)\u22c5O" ] } ], "prompt_number": 8 }, { "cell_type": "code", "collapsed": false, "input": [ "H1 = hamiltonian_transformation_auto(U, H)\n", "\n", "H1" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{\\delta}{2} \\left(\\sin{\\left (\\theta \\right )} {\\sigma_z} + \\cos{\\left (\\theta \\right )} {\\sigma_x}\\right) + \\frac{\\epsilon}{2} \\left(- \\sin{\\left (\\theta \\right )} {\\sigma_x} + \\cos{\\left (\\theta \\right )} {\\sigma_z}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 9, "text": [ "\u03b4\u22c5(sin(\u03b8)\u22c5O + cos(\u03b8)\u22c5O) \u03b5\u22c5(-sin(\u03b8)\u22c5O + cos(\u03b8)\u22c5O)\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\n", " 2 2 " ] } ], "prompt_number": 9 }, { "cell_type": "code", "collapsed": false, "input": [ "H4 = collect(H1, (sx, sz))\n", "\n", "H4" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{\\delta}{2} \\left(\\sin{\\left (\\theta \\right )} {\\sigma_z} + \\cos{\\left (\\theta \\right )} {\\sigma_x}\\right) + \\frac{\\epsilon}{2} \\left(- \\sin{\\left (\\theta \\right )} {\\sigma_x} + \\cos{\\left (\\theta \\right )} {\\sigma_z}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 20, "text": [ "\u03b4\u22c5(sin(\u03b8)\u22c5O + cos(\u03b8)\u22c5O) \u03b5\u22c5(-sin(\u03b8)\u22c5O + cos(\u03b8)\u22c5O)\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\n", " 2 2 " ] } ], "prompt_number": 20 }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the eigenbasis we require the coefficient of $\\sigma_x$ to be zero, so we have the condition:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "c, o = split_coeff_operator(H4.args[1])\n", "\n", "Eq(c, 0)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{\\delta}{2} \\sin{\\left (\\theta \\right )} + \\frac{\\epsilon}{2} \\cos{\\left (\\theta \\right )} = 0$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAANMAAAAsBAMAAAAN7ZAmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWakT3Oo/gd4HPlR0yQ0AAAAASUVORK5CYII=\n", "prompt_number": 13, "text": [ "\u03b4\u22c5sin(\u03b8) \u03b5\u22c5cos(\u03b8) \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 = 0\n", " 2 2 " ] } ], "prompt_number": 13 }, { "cell_type": "code", "collapsed": false, "input": [ "Eq(tan(theta), delta/eps)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\tan{\\left (\\theta \\right )} = \\frac{\\delta}{\\epsilon}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAGUAAAAtBAMAAABFWbj9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n", "prompt_number": 14, "text": [ " \u03b4\n", "tan(\u03b8) = \u2500\n", " \u03b5" ] } ], "prompt_number": 14 }, { "cell_type": "code", "collapsed": false, "input": [ "eq = Eq(theta, atan(delta/eps))\n", "\n", "eq" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\theta = \\operatorname{atan}{\\left (\\frac{\\delta}{\\epsilon} \\right )}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAHsAAAAyBAMAAACOvrnoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEGZEu1SZMu/ddqvN\niSKgCzzVAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADAElEQVRIDZ1WO2wTQRCdi7/n2D5TQEEKWwgQ\nP8kuUiBR2CCKFBQuKEJl0wUobKCDSE5ECAWNK0s06JSCAhRhUQAiQnGRiiBhUaQAobikDESEnyDM\nzO7Zvj07vmWKnTdz7+3O7e7tLYCeJXKSP6Gnk+wN8oGlKgRaMqPjEk1iH5n8DnBKRye50ykCTUAX\nbGvrjTsseU+tcVdbHiyzJN8hN81Ypylx7RCvk8jUrr7BYyU3/pAP/eXIfzMmBLOdd6wRnfmXmwXi\n4vxlqwSy4lUI+rJajmjhFtRYnrYx0rAZ5kYzUONx4y0NLVLvMz1dheMMYnpzF/nKKqsK5xgkf7Hz\n28jREnagICQL5IxDIhjZBucE5cMjSZ0iP7m/LMMRLlpQCJsdgMg8fFHSHAY8yXRZSWVxIc0MfFPS\nHB7wJPO2kqph4hiAmFHl2VElBigVlRSV04DIoNFDOwoX4Cnvtb601YTANsS2+1IETyzbcPLnSpHB\n7OKtJzYTKh12vcYqwPjvlZdbmEm8ZSsjDD2Ea1SVBGt2UmwvjzzaArMFVgs1fRa6B/kUyxlMOZOz\nmuojEYzWwWpDTZ2SyDIOhKMDgwsAYneuCvUuGRUM5hykizAj0t3WWOvgF4lyAc4r8i6P5TnRd+IB\nWxMf4lFWu55qgC1AV15RizfrEM0l6r3+GKXbULpZXYe2AF057VGX4buH7c+ebAE2b6SuQBs3OYKu\n3LPuOPPG1cuuHjFIPn988A2Mv+4wuL374uIPeiegLe4yXHcNK6lyzze0Z2cDP5k9Fa6HnsE835CL\nrgY4U27zLIX7sRIFt5SE3IZKdlgYk6eS8fGZoCwMYw7MO+fCabkCmge1/E0kndNF8zcBFS4quPMp\nI0B9YJFDk3mu2jq7TzAseyhz4AOzTGmLWwS6P+gxnvp4kzpBWxfOf7tI1MA8GOz1frAoKaVId2bJ\nJievWQR9WrzcRxRXxL7ESJjEy6hj8orohL78pR5Lv3a8FjmzDnC415N/9Mqh/teFHMK88bCPCV6E\nf3wuvRpKR39qAAAAAElFTkSuQmCC\n", "prompt_number": 15, "text": [ " \u239b\u03b4\u239e\n", "\u03b8 = atan\u239c\u2500\u239f\n", " \u239d\u03b5\u23a0" ] } ], "prompt_number": 15 }, { "cell_type": "code", "collapsed": false, "input": [ "H5 = simplify(H4.subs(theta, atan(delta/eps)))\n", "\n", "H5" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{\\epsilon {\\sigma_z}}{2} \\sqrt{\\frac{\\delta^{2}}{\\epsilon^{2}} + 1}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAHIAAAAyBAMAAAByq5IiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nbgjJGMP6ocDfkF+7FdILrRSB/w8qF+FuhahCXwAAAABJRU5ErkJggg==\n", "prompt_number": 16, "text": [ " ________ \n", " \u2571 2 \n", " \u2571 \u03b4 \n", "\u03b5\u22c5 \u2571 \u2500\u2500 + 1 \u22c5O\n", " \u2571 2 \n", " \u2572\u2571 \u03b5 \n", "\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": "markdown", "metadata": {}, "source": [ "Now introduce $\\Omega = \\sqrt{\\Delta^2 + \\epsilon^2}$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "powsimp(simplify(H5.subs(eps, sqrt(Omega ** 2-delta ** 2))), force=True)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{\\Omega {\\sigma_z}}{2}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAACUAAAArBAMAAAANulxAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmarid27\nEO8D7p43AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABNklEQVQoFbWSPUsDQRCGn028XC7xQgQJiM2J\njWVEEKxy+AdipVgIaSwEa0sVayH+AUGilSjGQgtJITa2sUkp6UX8Qo1iOCfGO9i9OlPMvu8zs1/s\nAmpyAiOGFvOFJYNVt6HkadB+F2u9aMz5Fpv50Fj2VWz6R2PDHbHJL2g9BUGlX0r2pjkd7Mv5xkgf\n4T6IyLa596j+I6jmYctjFcrNEFpFOAFZthQipRbgXKlPWJZ7Xtz5cjT2UZWM28V+hNTUoTQ7O0ck\narPcMC4duJ6kTNAhHbzRmjkVx2gv6aF8O9o8rIzVdkMZjcXNlUgPVgSxeB7shsbqhfVbg+DuceYb\n0GmSaxssV8eRZ9LC6saZNGS7WtefKR/EWSOOUtdxNhdHiQrh94uKGzAdmb5QV8drdYNZ8gQh+wXt\nVFVpO/xAUwAAAABJRU5ErkJggg==\n", "prompt_number": 17, "text": [ "\u03a9\u22c5O\n", "\u2500\u2500\u2500\n", " 2 " ] } ], "prompt_number": 17 }, { "cell_type": "markdown", "metadata": {}, "source": [ "So we have reached the eigenbasis by applying the unitary transformation" ] }, { "cell_type": "code", "collapsed": false, "input": [ "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{\\frac{i \\theta}{2} {\\sigma_y}}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAC0AAAAWBAMAAACmiQjHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABA0lEQVQoFWNggABeBgaew1A2MsXIwKDKLYAs\nAmOzbuBdAGPDaU5jBqZN+pjqOVMY+AJWwJUhGK8Y2BmUEVwYi7EggU/gIpAn0v//K0wQRG8W4FIK\nYGBgLI75imkLAwOLA0MXsvL/YPCBgT+A4SOyOAPDIj2gMQzyDJwFDJzKDA/gkg/YCoBsdgGg5zhT\nuDbAxS8wggzg2uwKJD2ZDwCdoGwCMgFkAAxcBIVIegCrAkhAFiwLYjEctmBgYO1kEFkAZDNuBImA\nAeuBSQwM3D3GB0E8kUAFsCCQ4FI2YGDg3wDh1v8vgImDaX4FFC6cwwEUB2rDAMwXGEQCMESBAiY2\nB7AIAwABCzC0C1YXkQAAAABJRU5ErkJggg==\n", "prompt_number": 18, "text": [ " \u2148\u22c5\u03b8\u22c5O\n", " \u2500\u2500\u2500\u2500\u2500\n", " 2 \n", "\u212f " ] } ], "prompt_number": 18 }, { "cell_type": "markdown", "metadata": {}, "source": [ "with" ] }, { "cell_type": "code", "collapsed": false, "input": [ "eq" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\theta = \\operatorname{atan}{\\left (\\frac{\\delta}{\\epsilon} \\right )}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 19, "text": [ " \u239b\u03b4\u239e\n", "\u03b8 = atan\u239c\u2500\u239f\n", " \u239d\u03b5\u23a0" ] } ], "prompt_number": 19 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Versions" ] }, { "cell_type": "code", "collapsed": false, "input": [ "%reload_ext version_information\n", "\n", "%version_information sympy" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "
SoftwareVersion
Python3.4.0 (default, Apr 11 2014, 13:05:11) [GCC 4.8.2]
IPython3.0.0-dev
OSposix [linux]
sympy0.7.4.1-git
Sat Jun 07 16:27:37 2014 JST
" ], "json": [ "{\"Software versions\":[{\"version\":\"3.4.0 (default, Apr 11 2014, 13:05:11) [GCC 4.8.2]\",\"module\":\"Python\"},{\"version\":\"3.0.0-dev\",\"module\":\"IPython\"},{\"version\":\"posix [linux]\",\"module\":\"OS\"},{\"version\":\"0.7.4.1-git\",\"module\":\"sympy\"}]}" ], "latex": [ "\\begin{tabular}{|l|l|}\\hline\n", "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n", "Python & 3.4.0 (default, Apr 11 2014, 13:05:11) [GCC 4.8.2] \\\\ \\hline\n", "IPython & 3.0.0-dev \\\\ \\hline\n", "OS & posix [linux] \\\\ \\hline\n", "sympy & 0.7.4.1-git \\\\ \\hline\n", "\\hline \\multicolumn{2}{|l|}{Sat Jun 07 16:27:37 2014 JST} \\\\ \\hline\n", "\\end{tabular}\n" ], "metadata": {}, "output_type": "pyout", "prompt_number": 16, "text": [ "Software versions\n", "Python 3.4.0 (default, Apr 11 2014, 13:05:11) [GCC 4.8.2]\n", "IPython 3.0.0-dev\n", "OS posix [linux]\n", "sympy 0.7.4.1-git\n", "\n", "Sat Jun 07 16:27:37 2014 JST" ] } ], "prompt_number": 16 } ], "metadata": {} } ] }