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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Dysypatywne oscylacje neutrin\n",
      "==============================\n",
      "\n",
      "Plan\n",
      "----\n",
      "\n",
      "1. Fenomenologiczny model dynamiki oscylacji neutrin w materii\n",
      "2. Numeryczna analiza oscylacji\n",
      "3. Wizualizacja prawdopodobie\u0144stwa oscylacji neutrin\n",
      "4. Zadania\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Oscylacje neutrin to postawowe zjawisko, w kt\u00f3rym mo\u017cemy obserwow\u0107 szerg interesuj\u0105cych w\u0142\u0105sno\u015bci tych tajemniczych cz\u0105stek. \n",
      "Zak\u0142adamy, \u017ce mo\u017cliwy jest fenomenolobicxzny opis zaczepni\u0119ty z prac F.Benattiego i R, Floreanniniego:\n",
      "\n",
      "$ H=H_0+ H_A$\n",
      "\n",
      "gdzie cz\u0119\u015b\u0107 $H_0$  opisuje oscylacje sowobodne, za\u015b $H_A$ oddzia\u0142ywanie z materi\u0105. Za\u0142o\u017cymy, \u017ce mo\u017cemy si\u0119 ograniczy\u0107 do rozwa\u017cania neutrin mionowych i elektronowych, w\u00f3wczas: \n",
      "\n",
      "$H_0=\\left(\\begin{array}{cc}E-\\omega_0-\\omega_3 & \\omega_1-i\\omega_2 \\\\\n",
      "\\omega_1 +i\\omega_2 & E+\\omega_0+\\omega_3\\end{array}\\right)$\n",
      "\n",
      "gdzie $E$ to \u015brednia energia neutrin, $\\omega_0\\sim \\Delta m^2$ opisuje r\u00f3\u017cnic\u0119 mas r\u00f3\u017cnych typ\u00f3w neutrin. Pozosta\u0119 parametry \n",
      "opisuj\u0105 oddzia\u0142ywanie z otoczeniem, ale dla uproszczenia za\u0142o\u017cymy ich znikanie.\n",
      "\n",
      "Cz\u0119\u015b\u0107 hamiltonianu opisuj\u0105ca oddzia\u0142ywanie z materi\u0105:\n",
      "\n",
      "$H_A= \\frac{A}{2}\\left(\\begin{array}{cc} 1+\\cos(2\\theta) & e^{-i\\phi}\\sin(2\\theta)\\\\\n",
      "e^{i\\phi}\\sin(2\\theta) &  1-\\cos(2\\theta)\\end{array} \\right) $\n",
      "\n",
      "zale\u017cy od $A$  reprezentujacego efektywn\u0105 \"gesto\u015b\u0107 materii\" o\u015brodka, w kt\u00f3rym neutrina propaguj\u0105, za\u015b $\\theta$ to tzw. k\u0105t mieszania, parametr znany z coraz liczniejszych eksperyment\u00f3w.  \n",
      "\n",
      "Pocz\u0105tkowo zak\u0142adamy neutrina elektronowe w stanie: \n",
      "\n",
      "$\\rho_{\\nu_e}=\\left(\\begin{array}{cc} \\cos^2(\\theta) & e^{-i\\phi}\\cos(\\theta)\\sin(\\theta) \\\\\n",
      "e^{i\\phi}\\cos(\\theta)\\sin(\\theta)  &  \\sin^2(\\theta)\\end{array}\\right)  $\n",
      "\n",
      "odpowiadaj\u0105cy im stan neutrin mionowych otrzymujemy z zale\u017cno\u015bci:\n",
      "\n",
      "$\\rho_{\\nu_\\mu}=1-\\rho_{\\nu_e}$\n",
      "\n",
      "Paramert $\\phi$ jest niezerowy w przpadku, gdy neutrina mia\u0142yby by\u0107 cz\u0105stkami typu Majorany."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from qutip import *\n",
      "from pylab import *\n",
      "%pylab inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "WARNING: pylab import has clobbered these variables: ['power', 'linalg', 'draw_if_interactive', 'random', 'save', 'load', 'info', 'fft']\n",
        "`%pylab --no-import-all` prevents importing * from pylab and numpy\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Wprowad\u017amy oznaczenia:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "up=basis(2,0)\n",
      "dn=basis(2,1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "rpp=up*up.dag()\n",
      "rpm=up*dn.dag()\n",
      "rmp=rpm.trans()\n",
      "rmm=dn*dn.dag()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Okre\u015blmy parametry oscylacji:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "E=1.0\n",
      "w0=1.0\n",
      "w3=1.0\n",
      "w2=0.0\n",
      "w1=0.0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Hamiltonian:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H0=(E-w0-w3)*rpp+(E+w0+w3)*rmm+(w1-1j*w2)*rpm+(w1+1j*w2)*rpm"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Dalsze parametry:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A=1.0\n",
      "theta=pi/3.0\n",
      "phi=0.0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "HA=A/2.0*((1.0+cos(2.0*theta))*rpp+(1.0-cos(2.0*theta))*rmm+exp(-1j*phi)*sin(2.0*theta)*rpm+exp(1j*phi)*sin(2.0*theta)*rmp) "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "i ca\u0142y hamiltonian:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H=H0+HA"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "a tak\u017ce stan pocz\u0105tkowy:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "psi0=cos(theta)**2.0*rpp+sin(theta)**2.0*rmm+cos(theta)*sin(theta)*(exp(-1j*phi)*rpm+exp(1j*phi)*rmp)\n",
      "psi_mu=qeye(2)-psi0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Nie wykluczajmy r\u00f3wnie\u017c obecno\u015bci dekoherencji wynikaj\u0105cej z wp\u0142ywu otoczenia:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "D=sqrt(0.1)*sigmaz()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "tlist=linspace(0,5.0,10.0)\n",
      "outlist=zeros(len(tlist))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "obliczmy dynamik\u0119 neutrin:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "qq=mesolve(H, psi0, tlist, [D], [])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Charakterystyk\u0105 badan\u0105 b\u0119dzie prawdopodobie\u0144stwo oscylacji:\n",
      "\n",
      "$P_{\\nu_{e}\\rightarrow \\nu_{\\mu}}=\\mbox{Tr}(\\rho_{\\nu_{e}}(t)\\rho_{\\nu_{\\mu}})$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "for i1 in range(len(tlist)):\n",
      "    rho=qq.states[i1]\n",
      "    outlist[i1]=abs((rho*psi_mu).tr())"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Przedstawmy wizualizacj\u0119 otrzymanego wyniku: "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(tlist,outlist,'-',label='', linewidth=4)\n",
      "xlabel('t',fontsize=20)\n",
      "ylabel('P(nu->mu)',fontsize=20)\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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XwPjxzo2R2E72aTmMDAYDcnNzsW/fPuzatUv04ank7rQ2ql4d6NNHuI8G1slv/XphggB4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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x2ceec10>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Zadania\n",
      "-------\n",
      "1. Przedstawi\u0107 wp\u0142yw $\\phi$ na wyniki $P$\n",
      "2. Okre\u015bli\u0107 wp\u0142yw $\\omega_{1,2}$ na wyniki $P$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    }
   ],
   "metadata": {}
  }
 ]
}