Dressed.ipynb

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     "source": [
      "#Dressed NV Ground State\n",
      "\n",
      "A small section of [Phys. Rev. B 79, 041302 (2009)](doi://10.1103/PhysRevB.79.041302) by Lukin group describes a way to reduce the influence of NV couplings to the nitrogen nucleus on the NV triplet ground state by dressing these states using microwaves.\n",
      "\n",
      "doi:10.1103/PhysRevB.79.041302"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
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     "source": [
      "## The problem\n",
      "\n",
      " - The nucleus imparts a B field oriented in some random direction\n",
      " - stable on timescales of interest\n",
      " - drifts between acquisitions, leads to inhomogeneous broadening"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import division"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import matplotlib\n",
      "import pylab as p\n",
      "import numpy as n"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matplotlib.rcParams.update({'font.size':16, 'figure.figsize':(4,5)})\n",
      "matplotlib.rcParams.update({'figure.max_open_warning':100})"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sympy as s\n",
      "s.init_printing()"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "om, D, l = s.symbols('Omega Delta lambda')"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## NV Hamiltonian\n",
      "\n",
      "V system. Basis: $(0, +1, -1)$. No applied B field means no $\\pm1$ splitting. Single x-polarized microwave field couples them with rabi frequency $\\Omega$. In the rotating frame, the Hamiltonian is"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H = s.Matrix([[0, om/2, om/2], [om/2, D, 0], [om/2, 0, D]])\n",
      "H"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0 & \\frac{\\Omega}{2} & \\frac{\\Omega}{2}\\\\\\frac{\\Omega}{2} & \\Delta & 0\\\\\\frac{\\Omega}{2} & 0 & \\Delta\\end{matrix}\\right]$$"
       ],
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       "prompt_number": 6,
       "text": [
        "\u23a1   \u03a9  \u03a9\u23a4\n",
        "\u23a20  \u2500  \u2500\u23a5\n",
        "\u23a2   2  2\u23a5\n",
        "\u23a2       \u23a5\n",
        "\u23a2\u03a9      \u23a5\n",
        "\u23a2\u2500  \u0394  0\u23a5\n",
        "\u23a22      \u23a5\n",
        "\u23a2       \u23a5\n",
        "\u23a2\u03a9      \u23a5\n",
        "\u23a2\u2500  0  \u0394\u23a5\n",
        "\u23a32      \u23a6"
       ]
      }
     ],
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    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
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     },
     "source": [
      "Symbolically diagonalize this:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H.eigenvals()"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "outputs": [
      {
       "latex": [
        "$$\\begin{Bmatrix}\\Delta : 1, & \\frac{\\Delta}{2} - \\frac{1}{2} \\sqrt{\\Delta^{2} + 2 \\Omega^{2}} : 1, & \\frac{\\Delta}{2} + \\frac{1}{2} \\sqrt{\\Delta^{2} + 2 \\Omega^{2}} : 1\\end{Bmatrix}$$"
       ],
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       "prompt_number": 7,
       "text": [
        "\u23a7             ___________            ___________   \u23ab\n",
        "\u23aa            \u2571  2      2            \u2571  2      2    \u23aa\n",
        "\u23a8      \u0394   \u2572\u2571  \u0394  + 2\u22c5\u03a9       \u0394   \u2572\u2571  \u0394  + 2\u22c5\u03a9     \u23ac\n",
        "\u23aa\u0394: 1, \u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500: 1, \u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500: 1\u23aa\n",
        "\u23a9      2         2            2         2          \u23ad"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H.eigenvects()"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}\\begin{pmatrix}\\Delta, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}0\\\\-1\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}, & \\begin{pmatrix}\\frac{\\Delta}{2} - \\frac{1}{2} \\sqrt{\\Delta^{2} + 2 \\Omega^{2}}, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}- \\frac{\\Omega}{- \\Delta + \\sqrt{\\Delta^{2} + 2 \\Omega^{2}}} + \\frac{\\Omega \\left(2 \\Delta - 2 \\sqrt{\\Delta^{2} + 2 \\Omega^{2}}\\right)}{8 \\left(- \\frac{\\Delta}{2} + \\frac{1}{2} \\sqrt{\\Delta^{2} + 2 \\Omega^{2}}\\right)^{2}}\\\\- \\frac{2 \\Delta - 2 \\sqrt{\\Delta^{2} + 2 \\Omega^{2}}}{- 2 \\Delta + 2 \\sqrt{\\Delta^{2} + 2 \\Omega^{2}}}\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}, & \\begin{pmatrix}\\frac{\\Delta}{2} + \\frac{1}{2} \\sqrt{\\Delta^{2} + 2 \\Omega^{2}}, & 1, & \\begin{bmatrix}\\left[\\begin{matrix}- \\frac{\\Omega}{- \\Delta - \\sqrt{\\Delta^{2} + 2 \\Omega^{2}}} + \\frac{\\Omega \\left(2 \\Delta + 2 \\sqrt{\\Delta^{2} + 2 \\Omega^{2}}\\right)}{8 \\left(- \\frac{\\Delta}{2} - \\frac{1}{2} \\sqrt{\\Delta^{2} + 2 \\Omega^{2}}\\right)^{2}}\\\\- \\frac{2 \\Delta + 2 \\sqrt{\\Delta^{2} + 2 \\Omega^{2}}}{- 2 \\Delta - 2 \\sqrt{\\Delta^{2} + 2 \\Omega^{2}}}\\\\1\\end{matrix}\\right]\\end{bmatrix}\\end{pmatrix}\\end{bmatrix}$$"
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LANmUJw90RFI4J7lWSlKptxhHhUmISCo4SuHYEn9i9Cpr37+1lgEM52UwPUQs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\nSUVORK5CYII=\n",
       "prompt_number": 8,
       "text": [
        "\u23a1                \u239b       ___________     \u23a1\u23a1                               \u239b   \n",
        "\u23a2                \u239c      \u2571  2      2      \u23a2\u23a2                               \u239c   \n",
        "\u23a2                \u239c\u0394   \u2572\u2571  \u0394  + 2\u22c5\u03a9       \u23a2\u23a2             \u03a9               \u03a9\u22c5\u239d2\u22c5\u0394\n",
        "\u23a2\u239b\u0394, 1, \u23a1\u23a10 \u23a4\u23a4\u239e, \u239c\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 1, \u23a2\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\n",
        "\u23a2\u239c      \u23a2\u23a2  \u23a5\u23a5\u239f  \u239c2         2            \u23a2\u23a2    \u239b         ___________\u239e         \n",
        "\u23a2\u239c      \u23a2\u23a2-1\u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2    \u239c        \u2571  2      2 \u239f     \u239b   \n",
        "\u23a2\u239c      \u23a2\u23a2  \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2    \u239c  \u0394   \u2572\u2571  \u0394  + 2\u22c5\u03a9  \u239f     \u239c   \n",
        "\u23a2\u239d      \u23a3\u23a31 \u23a6\u23a6\u23a0  \u239c                       \u23a2\u23a2  2\u22c5\u239c- \u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f     \u239c  \u0394\n",
        "\u23a2                \u239c                       \u23a2\u23a2    \u239d  2         2       \u23a0   8\u22c5\u239c- \u2500\n",
        "\u23a2                \u239c                       \u23a2\u23a2                               \u239d  2\n",
        "\u23a2                \u239c                       \u23a2\u23a2                                   \n",
        "\u23a2                \u239c                       \u23a2\u23a2               \u239b           ________\n",
        "\u23a2                \u239c                       \u23a2\u23a2               \u239c          \u2571  2     \n",
        "\u23a2                \u239c                       \u23a2\u23a2              -\u239d2\u22c5\u0394 - 2\u22c5\u2572\u2571  \u0394  + 2\u22c5\n",
        "\u23a2                \u239c                       \u23a2\u23a2              \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u23a2                \u239c                       \u23a2\u23a2                 \u239b         ________\n",
        "\u23a2                \u239c                       \u23a2\u23a2                 \u239c        \u2571  2     \n",
        "\u23a2                \u239c                       \u23a2\u23a2                 \u239c  \u0394   \u2572\u2571  \u0394  + 2\u22c5\n",
        "\u23a2                \u239c                       \u23a2\u23a2               4\u22c5\u239c- \u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u23a2                \u239c                       \u23a2\u23a2                 \u239d  2         2    \n",
        "\u23a2                \u239c                       \u23a2\u23a2                                   \n",
        "\u23a3                \u239d                       \u23a3\u23a3                           1       \n",
        "\n",
        "        ___________\u239e\u23a4\u23a4\u239e  \u239b       ___________     \u23a1\u23a1                           \n",
        "       \u2571  2      2 \u239f\u23a5\u23a5\u239f  \u239c      \u2571  2      2      \u23a2\u23a2                           \n",
        " - 2\u22c5\u2572\u2571  \u0394  + 2\u22c5\u03a9  \u23a0\u23a5\u23a5\u239f  \u239c\u0394   \u2572\u2571  \u0394  + 2\u22c5\u03a9       \u23a2\u23a2             \u03a9             \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\u23a5\u239f, \u239c\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 1, \u23a2\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 \n",
        "                  2 \u23a5\u23a5\u239f  \u239c2         2            \u23a2\u23a2    \u239b         ___________\u239e \n",
        "      ___________\u239e  \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2    \u239c        \u2571  2      2 \u239f \n",
        "     \u2571  2      2 \u239f  \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2    \u239c  \u0394   \u2572\u2571  \u0394  + 2\u22c5\u03a9  \u239f \n",
        "   \u2572\u2571  \u0394  + 2\u22c5\u03a9  \u239f  \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2  2\u22c5\u239c- \u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f \n",
        " + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f  \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2    \u239d  2         2       \u23a0 \n",
        "         2       \u23a0  \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2                           \n",
        "                    \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2                           \n",
        "___\u239e                \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2               \u239b           \n",
        " 2 \u239f                \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2               \u239c          \u2571\n",
        "\u03a9  \u23a0                \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2              -\u239d2\u22c5\u0394 + 2\u22c5\u2572\u2571 \n",
        "\u2500\u2500\u2500\u2500\u2500               \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2              \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "___\u239e                \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2                 \u239b         \n",
        " 2 \u239f                \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2                 \u239c        \u2571\n",
        "\u03a9  \u239f                \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2                 \u239c  \u0394   \u2572\u2571 \n",
        "\u2500\u2500\u2500\u239f                \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2               4\u22c5\u239c- \u2500 - \u2500\u2500\u2500\n",
        "   \u23a0                \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2                 \u239d  2      \n",
        "                    \u23a5\u23a5\u239f  \u239c                       \u23a2\u23a2                           \n",
        "                    \u23a6\u23a6\u23a0  \u239d                       \u23a3\u23a3                           \n",
        "\n",
        "    \u239b           ___________\u239e\u23a4\u23a4\u239e\u23a4\n",
        "    \u239c          \u2571  2      2 \u239f\u23a5\u23a5\u239f\u23a5\n",
        "  \u03a9\u22c5\u239d2\u22c5\u0394 + 2\u22c5\u2572\u2571  \u0394  + 2\u22c5\u03a9  \u23a0\u23a5\u23a5\u239f\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\u23a5\u23a5\u239f\u23a5\n",
        "                          2 \u23a5\u23a5\u239f\u23a5\n",
        "    \u239b         ___________\u239e  \u23a5\u23a5\u239f\u23a5\n",
        "    \u239c        \u2571  2      2 \u239f  \u23a5\u23a5\u239f\u23a5\n",
        "    \u239c  \u0394   \u2572\u2571  \u0394  + 2\u22c5\u03a9  \u239f  \u23a5\u23a5\u239f\u23a5\n",
        "  8\u22c5\u239c- \u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f  \u23a5\u23a5\u239f\u23a5\n",
        "    \u239d  2         2       \u23a0  \u23a5\u23a5\u239f\u23a5\n",
        "                            \u23a5\u23a5\u239f\u23a5\n",
        "___________\u239e                \u23a5\u23a5\u239f\u23a5\n",
        "  2      2 \u239f                \u23a5\u23a5\u239f\u23a5\n",
        " \u0394  + 2\u22c5\u03a9  \u23a0                \u23a5\u23a5\u239f\u23a5\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500               \u23a5\u23a5\u239f\u23a5\n",
        "___________\u239e                \u23a5\u23a5\u239f\u23a5\n",
        "  2      2 \u239f                \u23a5\u23a5\u239f\u23a5\n",
        " \u0394  + 2\u22c5\u03a9  \u239f                \u23a5\u23a5\u239f\u23a5\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f                \u23a5\u23a5\u239f\u23a5\n",
        "   2       \u23a0                \u23a5\u23a5\u239f\u23a5\n",
        "                            \u23a5\u23a5\u239f\u23a5\n",
        "1                           \u23a6\u23a6\u23a0\u23a6"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "source": [
      "One of the eigenvectors is very simple. This tells us that\n",
      "the asymmetric linear combination of $\\pm 1$ states is an eigenstate."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "The asymmetric linear combination of $\\pm1$ states\n",
      "\n",
      "$\\frac{-\\left|+1\\right> + \\left|-1\\right>}{\\sqrt{2}}$\n",
      "\n",
      "is an eigenstate, with eigenvalue $\\Delta$. We can verify that:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "dark = s.Matrix([0,-1,1]) / s.sqrt(2)\n",
      "dark"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0\\\\- \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADsAAABXCAMAAACNzc2gAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
       "prompt_number": 9,
       "text": [
        "\u23a1   0   \u23a4\n",
        "\u23a2       \u23a5\n",
        "\u23a2   ___ \u23a5\n",
        "\u23a2-\u2572\u2571 2  \u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2   2   \u23a5\n",
        "\u23a2       \u23a5\n",
        "\u23a2   ___ \u23a5\n",
        "\u23a2 \u2572\u2571 2  \u23a5\n",
        "\u23a2 \u2500\u2500\u2500\u2500\u2500 \u23a5\n",
        "\u23a3   2   \u23a6"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H * dark"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0\\\\- \\frac{\\sqrt{2} \\Delta}{2}\\\\\\frac{\\sqrt{2} \\Delta}{2}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAABXCAMAAACQs/M7AAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nz/ufBx39+K91+JYoBoQL3Z8HWsMf3HgwTJri5+AAAAAASUVORK5CYII=\n",
       "prompt_number": 10,
       "text": [
        "\u23a1    0    \u23a4\n",
        "\u23a2         \u23a5\n",
        "\u23a2   ___   \u23a5\n",
        "\u23a2-\u2572\u2571 2 \u22c5\u0394 \u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2    2    \u23a5\n",
        "\u23a2         \u23a5\n",
        "\u23a2   ___   \u23a5\n",
        "\u23a2 \u2572\u2571 2 \u22c5\u0394 \u23a5\n",
        "\u23a2 \u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u23a5\n",
        "\u23a3    2    \u23a6"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We call this the dark state because the eigenvalue does not involve $\\Omega$, that is, it is decoupled from the microwave field."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "We can make this state one of our basis states by applying a rotation to the $\\pm 1$ states. This takes us from the $(0, +1, -1)$ basis to the $(0, b, d)$ basis."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "theta = s.symbols('theta')"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$P$ is the matrix that we apply to things in the $(0, +1, -1)$ basis to go to a representation in the $(0, b, d)$ basis."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "P = s.Matrix([[1, 0,0],\n",
      "              [0, s.cos(theta), -s.sin(theta)],\n",
      "              [0, s.sin(theta), s.cos(theta)]])\n",
      "P"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & \\cos{\\left (\\theta \\right )} & - \\sin{\\left (\\theta \\right )}\\\\0 & \\sin{\\left (\\theta \\right )} & \\cos{\\left (\\theta \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMgAAABLCAMAAAACojjaAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nO+IPKhkjfwH/PE7YsI/izwAAAABJRU5ErkJggg==\n",
       "prompt_number": 12,
       "text": [
        "\u23a11    0        0   \u23a4\n",
        "\u23a2                  \u23a5\n",
        "\u23a20  cos(\u03b8)  -sin(\u03b8)\u23a5\n",
        "\u23a2                  \u23a5\n",
        "\u23a30  sin(\u03b8)  cos(\u03b8) \u23a6"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "source": [
      "$P^{-1}$ goes the other way, and is just a rotation in the opposite direction"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s.simplify((P**-1))"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & \\cos{\\left (\\theta \\right )} & \\sin{\\left (\\theta \\right )}\\\\0 & - \\sin{\\left (\\theta \\right )} & \\cos{\\left (\\theta \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMgAAABLCAMAAAACojjaAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
       "prompt_number": 13,
       "text": [
        "\u23a11     0       0   \u23a4\n",
        "\u23a2                  \u23a5\n",
        "\u23a20  cos(\u03b8)   sin(\u03b8)\u23a5\n",
        "\u23a2                  \u23a5\n",
        "\u23a30  -sin(\u03b8)  cos(\u03b8)\u23a6"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For $\\theta = -\\pi / 4$, we make the dark state into the last column of our new basis."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "P.subs(theta, -s.pi/4) * dark"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0\\\\0\\\\1\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABoAAABLCAMAAABZRmeuAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRInN3SJm77tsdo1uFAAAAN5JREFUOBHtld0agyAIQFHJtczM8f7vOpHmdNmut6+4\nKOWA4h+AoiwainjRACgymGQoBAL3NTFSb+27FRpkjUPzsmvRnAa1fhHPBoWVlW7uoJiDHMhm1nhR\nRhNJrDWyZNh8Itx5LeRYp+QHtdcXZMW8NyDIXEMnDPCR5wq94DEv2fSWDJ43at02sY4wqV06i/72\n8lRFWq+i5sbp0Wmu6G28y8m3V/QxVm/5uqJN+vqdlzJLxts/2CVu+eQT2Wj0AUqmeKFSv4xeyesg\nh/0v+fCwZFsu0IiTLIe/uWQjwhPNCRmFIPG2DQAAAABJRU5ErkJggg==\n",
       "prompt_number": 14,
       "text": [
        "\u23a10\u23a4\n",
        "\u23a2 \u23a5\n",
        "\u23a20\u23a5\n",
        "\u23a2 \u23a5\n",
        "\u23a31\u23a6"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "source": [
      "The first column is obviously untouched by the rotation, so the middle column must be the bright state. In the old basis, this would be represented as:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(P**-1).subs(theta, -s.pi/4) * s.Matrix([0,1,0])"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0\\\\\\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACsAAABXCAMAAACqY0xIAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
       "prompt_number": 15,
       "text": [
        "\u23a1  0  \u23a4\n",
        "\u23a2     \u23a5\n",
        "\u23a2  ___\u23a5\n",
        "\u23a2\u2572\u2571 2 \u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2  2  \u23a5\n",
        "\u23a2     \u23a5\n",
        "\u23a2  ___\u23a5\n",
        "\u23a2\u2572\u2571 2 \u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a3  2  \u23a6"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "source": [
      "That is, the symmetric linear combination of those same two states."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Ok, let's transform the Hamiltonian to the $(0, b, d)$ basis:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Hbd = (P * H * P**-1).subs(theta, -s.pi/4)\n",
      "Hbd"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0 & \\frac{\\sqrt{2} \\Omega}{2} & 0\\\\\\frac{\\sqrt{2} \\Omega}{2} & \\Delta & 0\\\\0 & 0 & \\Delta\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJgAAABXCAMAAADPonxyAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\n/uufWXr755G2FTyTuhZkb839maVt1T/mKkKTzXY0bgAAAABJRU5ErkJggg==\n",
       "prompt_number": 16,
       "text": [
        "\u23a1           ___     \u23a4\n",
        "\u23a2         \u2572\u2571 2 \u22c5\u03a9   \u23a5\n",
        "\u23a2   0     \u2500\u2500\u2500\u2500\u2500\u2500\u2500  0\u23a5\n",
        "\u23a2            2      \u23a5\n",
        "\u23a2                   \u23a5\n",
        "\u23a2  ___              \u23a5\n",
        "\u23a2\u2572\u2571 2 \u22c5\u03a9            \u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500     \u0394     0\u23a5\n",
        "\u23a2   2               \u23a5\n",
        "\u23a2                   \u23a5\n",
        "\u23a3   0        0     \u0394\u23a6"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The last row or column has no off-diagonal elements, so it is now decoupled from the microwave field (dark state). The coupling to the bright state is stronger ($\\Omega / \\sqrt{2}$ instead of $\\Omega / 2$)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Eigenenergies with respect to applied mw power\n",
      "\n",
      "Plot the eigenenergies of our hamiltonian with respect to the applied microwave power. By setting $\\Delta=1$, we are effectively plotting these in units of the detuning."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Hbd.eigenvals()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{Bmatrix}\\Delta : 1, & \\frac{\\Delta}{2} - \\frac{1}{2} \\sqrt{\\Delta^{2} + 2 \\Omega^{2}} : 1, & \\frac{\\Delta}{2} + \\frac{1}{2} \\sqrt{\\Delta^{2} + 2 \\Omega^{2}} : 1\\end{Bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAfAAAAAbBAMAAABxdsAgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 17,
       "text": [
        "\u23a7             ___________            ___________   \u23ab\n",
        "\u23aa            \u2571  2      2            \u2571  2      2    \u23aa\n",
        "\u23a8      \u0394   \u2572\u2571  \u0394  + 2\u22c5\u03a9       \u0394   \u2572\u2571  \u0394  + 2\u22c5\u03a9     \u23ac\n",
        "\u23aa\u0394: 1, \u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500: 1, \u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500: 1\u23aa\n",
        "\u23a9      2         2            2         2          \u23ad"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def plot(function, with_respect_to, x, **kwargs):\n",
      "    \"\"\"Plot a sympy function having one variable\n",
      "    over a given range of that variable.\n",
      "    \"\"\"\n",
      "    # if this doesn't work, try setting dtype of x to complex128\n",
      "    f = s.lambdify(with_respect_to, function, 'numpy')\n",
      "    y = f(x) * n.ones_like(x)\n",
      "    # ones_like ensures that y has the dimensions of x\n",
      "    # (for constants, f sometimes returns a single value)\n",
      "    return p.plot(x, y, **kwargs)"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = n.arange(0, 3, .1)\n",
      "p.figure(figsize=(3.5,3.5))\n",
      "for ev in Hbd.eigenvals():\n",
      "    plot(ev.subs(D, 1), om, x)\n",
      "p.xlabel('$\\Omega / \\Delta$');\n",
      "p.ylabel('$E / \\Delta$');\n",
      "p.text(1.8,2.2, r'$\\left|e\\right>$', fontdict={'size':30})\n",
      "p.text(1.8,-.7, r'$\\left|g\\right>$', fontdict={'size':30})\n",
      "p.text(1.8,1.2, r'$\\left|d\\right>$', fontdict={'size':30});\n",
      "#p.savefig('dressed.pdf', bbox_inches='tight', transparent=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": [
        "<matplotlib.text.Text at 0x105c7ac90>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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jSUlJAHTs2JH7778/d/uPP/5I7969adKkCdu3b6eGiXV8V65cSWRkpHWvyUU5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\nuVcAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x105c11110>"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The flat line is obviously the dark state, which is unaffected by the microwave field. The states coupled by the microwaves shift apart as the power is increased. This is the ac Stark shift."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##Digression: real numbers with respect to detuning"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = n.arange(0, 5, .02)\n",
      "omegaval = 2\n",
      "for ev in Hbd.eigenvals():\n",
      "    plot(ev.subs(om, 0.1), D, x, ls='--')\n",
      "p.gca()._get_lines.set_color_cycle()\n",
      "for ev in Hbd.eigenvals():\n",
      "    plot(ev.subs(om, omegaval), D, x)\n",
      "p.xlabel('$\\Delta$ (MHz)');\n",
      "p.ylabel('$E$ (MHz)');\n",
      "p.text(0.1, 0.9, '$\\Omega=%.1f$ MHz' % omegaval, transform=p.gca().transAxes);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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+ALWACOFNWhowahRw4wbX2VzSyQA/du7mbXjuaQeIZYj0uwo3u6al36kGUQuIEB6EhXGB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\nGUeOHIGbm1uhGxNtbGzQvn17/PXXXzx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       "text": [
        "<matplotlib.figure.Figure at 0x10302d2d0>"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Eigenenergies with respect to Magnetic-field perturbations\n",
      "\n",
      "We are interested in what goes on when the NV center is subject to a magnetic field. This new term in the hamiltonian goes as $S_z$. In the old $0pm$ basis, this spin operator looks like"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Sz_0pm = s.Matrix([[0, 0, 0],\n",
      "                   [0, 1, 0],\n",
      "                   [0, 0, -1]])\n",
      "Sz_0pm"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & -1\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAABLCAMAAABEOFk1AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\ne87C1dr5y3GM4QvlOSqGyI1AqAAAAABJRU5ErkJggg==\n",
       "prompt_number": 21,
       "text": [
        "\u23a10  0  0 \u23a4\n",
        "\u23a2        \u23a5\n",
        "\u23a20  1  0 \u23a5\n",
        "\u23a2        \u23a5\n",
        "\u23a30  0  -1\u23a6"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "That is, it leads to a Zeeman splitting of the $\\pm1$ states. If we rotate it into our $(0, b, d)$ basis:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Sz = (P * Sz_0pm * P**-1).subs(theta, -s.pi/4)\n",
      "Sz"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & -1\\\\0 & -1 & 0\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHkAAABLCAMAAABjltjdAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 22,
       "text": [
        "\u23a10  0   0 \u23a4\n",
        "\u23a2         \u23a5\n",
        "\u23a20  0   -1\u23a5\n",
        "\u23a2         \u23a5\n",
        "\u23a30  -1  0 \u23a6"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "we see that it amounts to introducing couplings between the bright and dark state."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let $\\delta$ be the strength of this coupling in units of frequency (typically 1 MHz), so that\n",
      "\n",
      "$H_{\\mathrm{tot}} = H_{\\mathrm{NV}} + \\delta S_z$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "delta = s.symbols('delta')\n",
      "Htot = Hbd + delta * Sz\n",
      "Htot"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0 & \\frac{\\sqrt{2} \\Omega}{2} & 0\\\\\\frac{\\sqrt{2} \\Omega}{2} & \\Delta & - \\delta\\\\0 & - \\delta & \\Delta\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKEAAABXCAMAAABbRNWAAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 23,
       "text": [
        "\u23a1           ___      \u23a4\n",
        "\u23a2         \u2572\u2571 2 \u22c5\u03a9    \u23a5\n",
        "\u23a2   0     \u2500\u2500\u2500\u2500\u2500\u2500\u2500  0 \u23a5\n",
        "\u23a2            2       \u23a5\n",
        "\u23a2                    \u23a5\n",
        "\u23a2  ___               \u23a5\n",
        "\u23a2\u2572\u2571 2 \u22c5\u03a9             \u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500     \u0394     -\u03b4\u23a5\n",
        "\u23a2   2                \u23a5\n",
        "\u23a2                    \u23a5\n",
        "\u23a3   0       -\u03b4     \u0394 \u23a6"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This leads to coupling between the bright and dark states. Again, brute-force solve for the eigenvalues:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "evals = Htot.eigenvals()\n",
      "#heinously ugly expressions supressed"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "Now let's investigate how the eigenvalues scale with the magnetic field. I'll do this by setting $\\Delta = 1$ (as usual, using it as the unit) and then plotting the B field scaling for certain values of $\\Omega$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def delta_scaling(Omega, **kwargs):\n",
      "    x = n.arange(0, 1.2, .02, dtype='complex128')\n",
      "    for ev in evals:\n",
      "        plot(ev.subs({D: 1, om: Omega}), delta, x, **kwargs)\n",
      "    p.xlabel('$\\delta / \\Delta$');\n",
      "    p.ylabel('$E / \\Delta$');\n",
      "    p.title(\"$\\Omega = %.1f \\Delta$\" % Omega);\n",
      "    #p.ylim(-1,3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "-"
      }
     },
     "source": [
      "for zero driving, recover zeeman splitting:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "delta_scaling(0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "/Users/tkb/.virtualenvs/science/lib/python2.7/site-packages/numpy/core/numeric.py:460: ComplexWarning: Casting complex values to real discards the imaginary part\n",
        "  return array(a, dtype, copy=False, order=order)\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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tGHJ2wpAje+DIriaGnJ0w5MieGHb/xZCzE4YcOQLDjiFnNww5ciRXDjuGnJ0w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\nRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x105c67610>"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "turn microwaves on, widen anticrossing"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "delta_scaling(.1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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OGFvIKSnD7tuz38JeZI8AcQD6uPahsCN6g0JOR4w15JRkchn+uP0Hvj37LWwt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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x105c8e9d0>"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "for high driving, curves flatten"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "delta_scaling(1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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UlBXgeOpxHEk+gqjkKNia2CpHef5d/elEoUS7i6sVFRWIiorCwYMHYWhoiClT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\nCBG0/w/EiAZAaTA5YgAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x105c5fa50>"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "At the \"sweet spot\", the curvature of the lowest and middle states is ideally matched:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "delta_scaling(4.06)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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IegwqIpIeg4qIpBd1QVVeXo67774bPXv2RHx8PJKSknD99dfjr3/9q9alEVGY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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x105c67890>"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This flattening of the curves near $\\delta = 0$ lowers the susceptibility to nuclear spin fluctuations."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "###Interactive plot\n",
      "\n",
      "To visualize how this plot changes with raman driving, use the slider below:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from ipywidgets import StaticInteract, RangeWidget\n",
      "\n",
      "def delta_interact(Omega):\n",
      "    fig = p.figure()\n",
      "    delta_scaling(Omega)\n",
      "    p.ylim(-3, 4)\n",
      "    return fig\n",
      "\n",
      "StaticInteract(delta_interact, Omega=RangeWidget(0, 5, .1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "\n",
        "    <script type=\"text/javascript\">\n",
        "      var mergeNodes = function(a, b) {\n",
        "        return [].slice.call(a).concat([].slice.call(b));\n",
        "      }; // http://stackoverflow.com/questions/914783/javascript-nodelist/17262552#17262552\n",
        "      function interactUpdate(div){\n",
        "         var outputs = div.getElementsByTagName(\"div\");\n",
        "         //var controls = div.getElementsByTagName(\"input\");\n",
        "         var controls = mergeNodes(div.getElementsByTagName(\"input\"), div.getElementsByTagName(\"select\"));\n",
        "         function nameCompare(a,b) {\n",
        "            return a.getAttribute(\"name\").localeCompare(b.getAttribute(\"name\"));\n",
        "         }\n",
        "         controls.sort(nameCompare);\n",
        "\n",
        "         var value = \"\";\n",
        "         for(i=0; i<controls.length; i++){\n",
        "           if((controls[i].type == \"range\") || controls[i].checked){\n",
        "             value = value + controls[i].getAttribute(\"name\") + controls[i].value;\n",
        "           }\n",
        "           if(controls[i].type == \"select-one\"){\n",
        "             value = value + controls[i].getAttribute(\"name\") + controls[i][controls[i].selectedIndex].value;\n",
        "           }\n",
        "         }\n",
        "\n",
        "         for(i=0; i<outputs.length; i++){\n",
        "           var name = outputs[i].getAttribute(\"name\");\n",
        "           if(name == value){\n",
        "              outputs[i].style.display = 'block';\n",
        "           } else if(name != \"controls\"){\n",
        "              outputs[i].style.display = 'none';\n",
        "           }\n",
        "         }\n",
        "      }\n",
        "    </script>\n",
        "    \n",
        "    <div>\n",
        "      \n",
        "    <div name=\"Omega0\" style=\"display:block\">\n",
        "      <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASoAAAFlCAYAAABcNKWWAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n",
        "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3XtYVHX+B/D34ZZDCIF3uWgiCF4zUVDL0NVadd1Ns2Az\n",
        "Ny+76WZqJPWzvKGVJpVKlpLp/lbd1X6Zz2Zb7aaoQ1fBVrRVRAURBNc7ogI6At/fHywj4wzCzJyZ\n",
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        "    \n",
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        "\">\n",
        "    </div>\n",
        "    \n",
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        "    </div>\n",
        "    \n",
        "    <div name=\"Omega4\" style=\"display:none\">\n",
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