Reproduce-PRA-80-033846-2009-Zueco.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Reproduce: Qubit-oscillator dynamics in the dispersive regime"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Analytic theory beyond the rotating-wave approximation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "D. Zueco, G. M. Reuther, S. Kohler, and P. Hänggi, <a href=\"http://journals.aps.org/pra/abstract/10.1103/PhysRevA.80.033846\">Phys. Rev. A <b>80</b>, 033846 (2009).</a>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Reproduced by <b>Eunjong Kim</b> (<a href=\"mailto:ekim7206@gmail.com\">ekim7206@gmail.com</a>)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup Modules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympsi import *\n",
    "from sympsi.boson import BosonOp\n",
    "from sympsi.pauli import (SigmaX, SigmaY, SigmaZ,\n",
    "                          SigmaPlus, SigmaMinus)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Jaynes-Cummings Model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Jaynes-Cummings model describes the light-matter interaction as a single harmonic oscillator and a two-level system. Within dipole approximation, the model is expressed by the Hamiltonian"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "H = \\frac{\\hbar\\epsilon}{2}\\sigma_z + \\hbar\\omega a^\\dagger a + \\hbar g \\sigma_x (a + a^\\dagger)\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $\\hbar\\epsilon$ is the level splitting of the two-level system, henchforth <i>qubit</i>, $\\omega$ is the frequency of the EM field mode, and $g$ is the dipole interaction strength."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, $\\sigma_\\alpha$, $\\alpha=x, y, z$, are Pauli spin matrices\n",
    "\\begin{align}\n",
    "\\sigma_{x} = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix},\\quad\n",
    "\\sigma_{y} = \\begin{pmatrix} 0 & -i \\\\ i & 0 \\end{pmatrix}, \\quad\n",
    "\\sigma_{z} = \\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix},\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and $a^\\dagger$, $a$ are bosonic creation and annihilation operators of the EM field mode."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To represent the Jaynes-Cummings model in SymPy, we use the operator classes <tt>BosonOp</tt> and <tt>SigmaX</tt>, <tt>SigmaY</tt>, and <tt>SigmaZ</tt>, and use these to construct the Hamiltonian (we work in units where $\\hbar = 1$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "w, e, g, D, t, Hsym = symbols(\"omega, epsilon, g, Delta, t, H\")\n",
    "sx, sy, sz = SigmaX(), SigmaY(), SigmaZ()\n",
    "a = BosonOp(\"a\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASAAAAAlBAMAAAAQFym0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n8QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$H = \\frac{\\epsilon {\\sigma_z}}{2} + g {\\sigma_x} \\left({{a}^\\dagger} + {a}\\right) + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "    ε⋅False           ⎛ †    ⎞      †  \n",
       "H = ─────── + g⋅False⋅⎝a  + a⎠ + ω⋅a ⋅a\n",
       "       2                               "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H0 = w * Dagger(a) * a + e / 2 * sz\n",
    "Hint = g * sx * (a + Dagger(a))\n",
    "H = H0 + Hint\n",
    "Eq(Hsym, H)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This Hamiltonian cannot be diagonalized exactly, and is often simplified by a rotating-wave approximation(RWA)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We express the qubit-cavity interaction terms of the ladder operators $\\sigma_{\\pm} = \\frac{1}{2}(\\sigma_x \\pm i\\sigma_y)$. In the language of SymPy, these operators are written as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sp, sm = SigmaPlus(), SigmaMinus()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We move to a frame co-rotating with the qubit and oscillator frequency."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAG0AAAAZBAMAAADJZb2yAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\naIDaXoNC1b77X8ya6gj1WqW6/BOTj2F38mzglAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$U_{1} = e^{i \\omega t {{a}^\\dagger} {a}}$$"
      ],
      "text/plain": [
       "             †  \n",
       "      ⅈ⋅ω⋅t⋅a ⋅a\n",
       "U₁ = ℯ          "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U1 = exp(I * w * t * Dagger(a) * a); Eq(symbols(\"U_1\"), U1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGQAAAAZBAMAAAA1cJZ4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n4Uti8pGbw2EAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$U_{2} = e^{\\frac{i \\epsilon}{2} t {\\sigma_z}}$$"
      ],
      "text/plain": [
       "      ⅈ⋅ε⋅t⋅False\n",
       "      ───────────\n",
       "           2     \n",
       "U₂ = ℯ           "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U2 = exp(I * e * t * sz / 2); Eq(symbols(\"U_2\"), U2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQQAAAAlBAMAAABWoYoeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{\\epsilon {\\sigma_z}}{2} + g e^{i \\omega t} {\\sigma_x} {{a}^\\dagger} + g e^{- i \\omega t} {\\sigma_x} {a}$$"
      ],
      "text/plain": [
       "ε⋅False      ⅈ⋅ω⋅t        †      -ⅈ⋅ω⋅t        \n",
       "─────── + g⋅ℯ     ⋅False⋅a  + g⋅ℯ      ⋅False⋅a\n",
       "   2                                           "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H1 = hamiltonian_transformation(U1, H.expand()); H1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAncAAAAaBAMAAAAnEWf8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nhnKZMwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$g e^{i \\omega t} \\left(- \\sin{\\left (\\epsilon t \\right )} {\\sigma_y} + \\cos{\\left (\\epsilon t \\right )} {\\sigma_x}\\right) {{a}^\\dagger} + g e^{- i \\omega t} \\left(- \\sin{\\left (\\epsilon t \\right )} {\\sigma_y} + \\cos{\\left (\\epsilon t \\right )} {\\sigma_x}\\right) {a}$$"
      ],
      "text/plain": [
       "   ⅈ⋅ω⋅t                                     †      -ⅈ⋅ω⋅t                    \n",
       "g⋅ℯ     ⋅(-sin(ε⋅t)⋅False + cos(ε⋅t)⋅False)⋅a  + g⋅ℯ      ⋅(-sin(ε⋅t)⋅False + \n",
       "\n",
       "                 \n",
       "cos(ε⋅t)⋅False)⋅a"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H2 = hamiltonian_transformation(U2, H1.expand()); H2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAjoAAAAaBAMAAABMVevsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nIFdluTFPYsWcr5zug/wHqRPaprg7yW8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$g e^{- i \\epsilon t - i \\omega t} {\\sigma_-} {a} + g e^{- i \\epsilon t + i \\omega t} {\\sigma_-} {{a}^\\dagger} + g e^{i \\epsilon t - i \\omega t} {\\sigma_+} {a} + g e^{i \\epsilon t + i \\omega t} {\\sigma_+} {{a}^\\dagger}$$"
      ],
      "text/plain": [
       "   -ⅈ⋅ε⋅t - ⅈ⋅ω⋅t              -ⅈ⋅ε⋅t + ⅈ⋅ω⋅t        †      ⅈ⋅ε⋅t - ⅈ⋅ω⋅t     \n",
       "g⋅ℯ              ⋅False⋅a + g⋅ℯ              ⋅False⋅a  + g⋅ℯ             ⋅Fals\n",
       "\n",
       "         ⅈ⋅ε⋅t + ⅈ⋅ω⋅t        †\n",
       "e⋅a + g⋅ℯ             ⋅False⋅a "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H3 = pauli_represent_minus_plus(H2).rewrite((sin,cos), exp).expand()\n",
    "H3 = powsimp(H3); H3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Trick to simplify exponents\n",
    "def simplify_exp(e):\n",
    "    if isinstance(e, exp):\n",
    "        return exp(simplify(e.exp.expand()))\n",
    "    \n",
    "    if isinstance(e, (Add, Mul)):\n",
    "        return type(e)(*(simplify_exp(arg) for arg in e.args))\n",
    "    \n",
    "    return e"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We introduce the detuning $\\Delta = \\epsilon - \\omega$ and substitute into the expression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgEAAAAbBAMAAAAXGkGGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nGpToOtciwnvPvxr0qUG5mg465v4Py4GyeEXb00QAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$g e^{i \\Delta t} {\\sigma_+} {a} + g e^{i t \\left(\\epsilon + \\omega\\right)} {\\sigma_+} {{a}^\\dagger} + g e^{- i t \\left(\\epsilon + \\omega\\right)} {\\sigma_-} {a} + g e^{- i \\Delta t} {\\sigma_-} {{a}^\\dagger}$$"
      ],
      "text/plain": [
       "   ⅈ⋅Δ⋅t              ⅈ⋅t⋅(ε + ω)        †      -ⅈ⋅t⋅(ε + ω)              -ⅈ⋅Δ\n",
       "g⋅ℯ     ⋅False⋅a + g⋅ℯ           ⋅False⋅a  + g⋅ℯ            ⋅False⋅a + g⋅ℯ    \n",
       "\n",
       "⋅t        †\n",
       "  ⋅False⋅a "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H4 = simplify_exp(H3).subs(e - w, D); H4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the interaction picture, with respect to the uncoupled hamiltonian $H_0 = \\hbar\\epsilon\\sigma_z/2 + \\hbar\\omega a^\\dagger a$, the coupling operators $\\sigma_+ a, \\sigma_-a^\\dagger$ and $\\sigma_- a, \\sigma_+ a^\\dagger$ oscillate with the phase factors $\\exp[\\pm i(\\omega-\\epsilon)t]$ and $\\exp[\\pm i(\\omega+\\epsilon)t]$, respectively."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Operating at or near resonance, the cavity-qubit detuning is small, $|\\epsilon-\\omega|\\ll\\epsilon+\\omega$, so that the former operators oscillate slowly, whereas the latter exhibit fast <i>counter-rotating</i> oscillations. If in addition, the coupling is sufficiently weak $g\\ll \\min\\{\\epsilon,\\omega\\}$, one can separate time scales and replace the counter-rotating terms by their vanishing time average."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We drop the fast oscillating terms containing the factors $e^{\\pm i(\\omega+\\epsilon)t}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANUAAAAaBAMAAADS0ATRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/9QAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$g e^{i \\Delta t} {\\sigma_+} {a} + g e^{- i \\Delta t} {\\sigma_-} {{a}^\\dagger}$$"
      ],
      "text/plain": [
       "   ⅈ⋅Δ⋅t              -ⅈ⋅Δ⋅t        †\n",
       "g⋅ℯ     ⋅False⋅a + g⋅ℯ      ⋅False⋅a "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H5 = drop_terms_containing(H4, [exp( I * (w + e) * t),\n",
    "                                exp(-I * (w + e) * t)])\n",
    "H5 = drop_c_number_terms(H5.expand())\n",
    "\n",
    "H5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the Jaynes-Cummings model in the interaction picture. If we transform back to the Schrödinger picture, we have:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAW0AAAAaBAMAAACOfxDcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nq6PnvtJTdN7P7LzkMlT8B+MkN2gk6PJ8AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$g e^{i \\Delta t} e^{i \\omega t} {\\sigma_+} {a} + g e^{- i \\Delta t} e^{- i \\omega t} {\\sigma_-} {{a}^\\dagger} + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "   ⅈ⋅Δ⋅t  ⅈ⋅ω⋅t              -ⅈ⋅Δ⋅t  -ⅈ⋅ω⋅t        †      †  \n",
       "g⋅ℯ     ⋅ℯ     ⋅False⋅a + g⋅ℯ      ⋅ℯ      ⋅False⋅a  + ω⋅a ⋅a"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U3 = exp(-I * w * t * Dagger(a) * a)\n",
    "H6 = hamiltonian_transformation(U3, H5); H6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAfAAAAAlBAMAAABPBqb9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{\\epsilon {\\sigma_z}}{2} + g e^{i \\Delta t} e^{- i \\epsilon t} e^{i \\omega t} {\\sigma_+} {a} + g e^{- i \\Delta t} e^{i \\epsilon t} e^{- i \\omega t} {\\sigma_-} {{a}^\\dagger} + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "ε⋅False      ⅈ⋅Δ⋅t  -ⅈ⋅ε⋅t  ⅈ⋅ω⋅t              -ⅈ⋅Δ⋅t  ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t        † \n",
       "─────── + g⋅ℯ     ⋅ℯ      ⋅ℯ     ⋅False⋅a + g⋅ℯ      ⋅ℯ     ⋅ℯ      ⋅False⋅a  \n",
       "   2                                                                          \n",
       "\n",
       "     †  \n",
       "+ ω⋅a ⋅a\n",
       "        "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U4 = exp(-I * e * t * sz / 2)\n",
    "H7 = hamiltonian_transformation(U4, H6.expand()); H7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUoAAAAlBAMAAADW0r0NAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nZWu4fZVKw/8KaU7ZP7hUXqFLB4u2AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$H_{RWA} = \\frac{\\epsilon {\\sigma_z}}{2} + g {\\sigma_x} \\left({{a}^\\dagger} + {a}\\right) + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "        ε⋅False           ⎛ †    ⎞      †  \n",
       "H_RWA = ─────── + g⋅False⋅⎝a  + a⎠ + ω⋅a ⋅a\n",
       "           2                               "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H8 = simplify_exp(H7).subs(D, e - w)\n",
    "\n",
    "H8 = simplify_exp(powsimp(H8)).expand()\n",
    "\n",
    "H8 = drop_c_number_terms(H8)\n",
    "\n",
    "Hrwa = collect(H8, g)\n",
    "\n",
    "Eq(symbols(\"H_RWA\"), H)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the Jaynes-Cummings Hamiltonian."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dispersive Theory within RWA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The dispersive regime is characterized by a large detuning $\\Delta = \\epsilon - \\omega$ as compared to the qubit-oscillator coupling $g$. Thus, the factor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align} \\lambda\\equiv\\frac{g}{\\Delta} \\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "is a small parameter, while the RWA Hamiltonian is valid for $|\\epsilon - \\omega|\\ll \\epsilon+\\omega$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is easier if we separate the coupling term from the RWA Hamiltonian."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Defining the operators $X_{\\pm} = \\sigma_{-}a^\\dagger \\pm \\sigma_{+}a$,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKYAAAAZBAMAAABJOkjkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nereejTuTkd5tJRsyTLbd/wAzu3Fkz+5vsAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$X_{+} = {\\sigma_-} {{a}^\\dagger} + {\\sigma_+} {a}$$"
      ],
      "text/plain": [
       "            †          \n",
       "X₊ = False⋅a  + False⋅a"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Xp = sm * Dagger(a) + sp * a; Eq(symbols(\"X_+\"),Xp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKYAAAAZBAMAAABJOkjkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n145V56iZWIOFbEEuuE4AKFlq3JONh2AAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$X_{-} = {\\sigma_-} {{a}^\\dagger} - {\\sigma_+} {a}$$"
      ],
      "text/plain": [
       "            †          \n",
       "X₋ = False⋅a  - False⋅a"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Xm = sm * Dagger(a) - sp * a; Eq(symbols(\"X_-\"),Xm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The RWA Hamiltonian is expressed as\n",
    "\\begin{align} H_{RWA} = H_0 + \\hbar g X_+ \\end{align}\n",
    "where $H_0$ is the non-interacting Hamiltonian\n",
    "\\begin{align} H_0 \\equiv \\frac{\\hbar\\epsilon}{2} \\sigma_z +\\hbar\\omega a^\\dagger a\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAlBAMAAACuZCbAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9GCYDSJnGEH/K7j7/QPLN7FOAInGMwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$H_{0} = \\frac{\\epsilon {\\sigma_z}}{2} + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "     ε⋅False      †  \n",
       "H₀ = ─────── + ω⋅a ⋅a\n",
       "        2            "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H0 = e/2 * sz + w * Dagger(a) * a; Eq(symbols(\"H_0\"),H0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWcAAAAlBAMAAABscTVtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ntaol/gskm22+oZKz4QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$H_{RWA} = \\frac{\\epsilon {\\sigma_z}}{2} + g \\left({\\sigma_-} {{a}^\\dagger} + {\\sigma_+} {a}\\right) + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "        ε⋅False     ⎛       †          ⎞      †  \n",
       "H_RWA = ─────── + g⋅⎝False⋅a  + False⋅a⎠ + ω⋅a ⋅a\n",
       "           2                                     "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hrwa = H0 + g * Xp; Eq(symbols(\"H_RWA\"), Hrwa)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We apply the unitary transformation\\begin{align}\n",
    "D_{RWA} \\equiv e^{- \\lambda X_-} = e^{-\\lambda(\\sigma_- a^\\dagger -\\sigma_{+} a)}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "to obtain the dispersive Hamiltonian $H_{RWA, \\textrm{disp}} = D_{RWA} H_{RWA} D_{RWA}^\\dagger$ to the second order in $\\lambda$:\n",
    "\\begin{align}\n",
    "H_{RWA, \\textrm{disp}} = H_{RWA} + \\lambda [H_{RWA}, X_-] + \\frac{1}{2}\\lambda^2 [[H_{RWA},X_-],X_-]\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the language of SymPy,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANoAAAAaBAMAAAAj219cAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$D_{RWA} = e^{- \\lambda \\left({\\sigma_-} {{a}^\\dagger} - {\\sigma_+} {a}\\right)}$$"
      ],
      "text/plain": [
       "            ⎛       †          ⎞\n",
       "         -λ⋅⎝False⋅a  - False⋅a⎠\n",
       "D_RWA = ℯ                       "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l = symbols(\"lambda\", real = True)\n",
    "Drwa = exp(-l * Xm); Eq(symbols(\"D_RWA\"), Drwa)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The transformed Hamiltonian is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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xYSTEUlFkVLlmSSYAbH1XB5Lb6NUzanwC2AOjZ7D+5zqWe8+qO3Tepv8/ZDSkho+k8oE+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\n",
      "text/latex": [
       "$$- \\frac{\\epsilon \\lambda^{2}}{2} - \\epsilon \\lambda^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda^{2}}{2} {\\sigma_z} - \\epsilon \\lambda {{a}^\\dagger} {\\sigma_-} - \\epsilon \\lambda {a} {\\sigma_+} + \\frac{\\epsilon {\\sigma_z}}{2} + g \\lambda^{4} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{3 g}{4} \\lambda^{4} {{a}^\\dagger} {\\sigma_-} - g \\lambda^{4} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} + \\frac{5 g}{4} \\lambda^{4} {a} {\\sigma_+} - 2 g \\lambda^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 2 g \\lambda^{2} {{a}^\\dagger} {\\sigma_-} - 2 g \\lambda^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - 2 g \\lambda^{2} {a} {\\sigma_+} + g \\lambda + 2 g \\lambda {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda {\\sigma_z} + g {{a}^\\dagger} {\\sigma_-} + g {a} {\\sigma_+} + \\frac{\\lambda^{4} \\omega}{4} {{a}^\\dagger} {a} - \\frac{\\lambda^{3} \\omega}{2} {{a}^\\dagger} {\\sigma_-} - \\frac{\\lambda^{3} \\omega}{2} {a} {\\sigma_+} + \\frac{\\lambda^{2} \\omega}{2} + \\lambda^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} + \\frac{\\lambda^{2} \\omega}{2} {\\sigma_z} + \\lambda \\omega {{a}^\\dagger} {\\sigma_-} + \\lambda \\omega {a} {\\sigma_+} + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "     2                        2                                               \n",
       "  ε⋅λ       2  †           ε⋅λ ⋅False        †                       ε⋅False  \n",
       "- ──── - ε⋅λ ⋅a ⋅a⋅False - ────────── - ε⋅λ⋅a ⋅False - ε⋅λ⋅a⋅False + ─────── +\n",
       "   2                           2                                        2     \n",
       "\n",
       "                         4  †                  2                4             \n",
       "    4  †  2         3⋅g⋅λ ⋅a ⋅False      4 ⎛ †⎞            5⋅g⋅λ ⋅a⋅False     \n",
       " g⋅λ ⋅a ⋅a ⋅False - ─────────────── - g⋅λ ⋅⎝a ⎠ ⋅a⋅False + ────────────── - 2⋅\n",
       "                           4                                     4            \n",
       "\n",
       "                                                2                             \n",
       "   2  †  2              2  †              2 ⎛ †⎞                 2            \n",
       "g⋅λ ⋅a ⋅a ⋅False - 2⋅g⋅λ ⋅a ⋅False - 2⋅g⋅λ ⋅⎝a ⎠ ⋅a⋅False - 2⋅g⋅λ ⋅a⋅False + g\n",
       "                                                                              \n",
       "\n",
       "                                                              4    †      3   \n",
       "            †                          †                     λ ⋅ω⋅a ⋅a   λ ⋅ω⋅\n",
       "⋅λ + 2⋅g⋅λ⋅a ⋅a⋅False + g⋅λ⋅False + g⋅a ⋅False + g⋅a⋅False + ───────── - ─────\n",
       "                                                                 4            \n",
       "\n",
       " †          3              2                        2                         \n",
       "a ⋅False   λ ⋅ω⋅a⋅False   λ ⋅ω    2    †           λ ⋅ω⋅False        †        \n",
       "──────── - ──────────── + ──── + λ ⋅ω⋅a ⋅a⋅False + ────────── + λ⋅ω⋅a ⋅False +\n",
       " 2              2          2                           2                      \n",
       "\n",
       "                     \n",
       "                  †  \n",
       " λ⋅ω⋅a⋅False + ω⋅a ⋅a\n",
       "                     "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H1 = hamiltonian_transformation(Drwa, Hrwa.expand(), expansion_search=False, N=3).expand()\n",
    "\n",
    "H1 = qsimplify(H1)\n",
    "\n",
    "H1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We drop the terms of orders in lambda higher than 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$- \\frac{\\epsilon \\lambda^{2}}{2} - \\epsilon \\lambda^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda^{2}}{2} {\\sigma_z} - \\epsilon \\lambda {{a}^\\dagger} {\\sigma_-} - \\epsilon \\lambda {a} {\\sigma_+} + \\frac{\\epsilon {\\sigma_z}}{2} - 2 g \\lambda^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 2 g \\lambda^{2} {{a}^\\dagger} {\\sigma_-} - 2 g \\lambda^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - 2 g \\lambda^{2} {a} {\\sigma_+} + g \\lambda + 2 g \\lambda {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda {\\sigma_z} + g {{a}^\\dagger} {\\sigma_-} + g {a} {\\sigma_+} + \\frac{\\lambda^{2} \\omega}{2} + \\lambda^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} + \\frac{\\lambda^{2} \\omega}{2} {\\sigma_z} + \\lambda \\omega {{a}^\\dagger} {\\sigma_-} + \\lambda \\omega {a} {\\sigma_+} + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "     2                        2                                               \n",
       "  ε⋅λ       2  †           ε⋅λ ⋅False        †                       ε⋅False  \n",
       "- ──── - ε⋅λ ⋅a ⋅a⋅False - ────────── - ε⋅λ⋅a ⋅False - ε⋅λ⋅a⋅False + ─────── -\n",
       "   2                           2                                        2     \n",
       "\n",
       "                                                   2                          \n",
       "      2  †  2              2  †              2 ⎛ †⎞                 2         \n",
       " 2⋅g⋅λ ⋅a ⋅a ⋅False - 2⋅g⋅λ ⋅a ⋅False - 2⋅g⋅λ ⋅⎝a ⎠ ⋅a⋅False - 2⋅g⋅λ ⋅a⋅False \n",
       "                                                                              \n",
       "\n",
       "                                                                 2            \n",
       "               †                          †                     λ ⋅ω    2    †\n",
       "+ g⋅λ + 2⋅g⋅λ⋅a ⋅a⋅False + g⋅λ⋅False + g⋅a ⋅False + g⋅a⋅False + ──── + λ ⋅ω⋅a \n",
       "                                                                 2            \n",
       "\n",
       "            2                                              \n",
       "           λ ⋅ω⋅False        †                          †  \n",
       "⋅a⋅False + ────────── + λ⋅ω⋅a ⋅False + λ⋅ω⋅a⋅False + ω⋅a ⋅a\n",
       "               2                                           "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H2 = drop_terms_containing(H1.expand(), [l**3, l**4]); H2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and substitute $g/\\Delta$ in place of $\\lambda$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\frac{\\epsilon {\\sigma_z}}{2} + g {{a}^\\dagger} {\\sigma_-} + g {a} {\\sigma_+} + \\omega {{a}^\\dagger} {a} - \\frac{\\epsilon g}{\\Delta} {{a}^\\dagger} {\\sigma_-} - \\frac{\\epsilon g}{\\Delta} {a} {\\sigma_+} + \\frac{g^{2}}{\\Delta} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} + \\frac{g \\omega}{\\Delta} {{a}^\\dagger} {\\sigma_-} + \\frac{g \\omega}{\\Delta} {a} {\\sigma_+} - \\frac{\\epsilon g^{2}}{2 \\Delta^{2}} - \\frac{\\epsilon g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon g^{2} {\\sigma_z}}{2 \\Delta^{2}} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} {a} {\\sigma_+} + \\frac{g^{2} \\omega}{2 \\Delta^{2}} + \\frac{g^{2} \\omega}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega {\\sigma_z}}{2 \\Delta^{2}}$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                                                 †                        2   \n",
       "ε⋅False      †                        †     ε⋅g⋅a ⋅False   ε⋅g⋅a⋅False   g    \n",
       "─────── + g⋅a ⋅False + g⋅a⋅False + ω⋅a ⋅a - ──────────── - ─────────── + ── + \n",
       "   2                                             Δ              Δ        Δ    \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "   2  †            2              †                          2      2  †      \n",
       "2⋅g ⋅a ⋅a⋅False   g ⋅False   g⋅ω⋅a ⋅False   g⋅ω⋅a⋅False   ε⋅g    ε⋅g ⋅a ⋅a⋅Fal\n",
       "─────────────── + ──────── + ──────────── + ─────────── - ──── - ─────────────\n",
       "       Δ             Δ            Δ              Δ           2           2    \n",
       "                                                          2⋅Δ           Δ     \n",
       "\n",
       "                                                              2               \n",
       "        2            3  †  2            3  †            3 ⎛ †⎞               3\n",
       "se   ε⋅g ⋅False   2⋅g ⋅a ⋅a ⋅False   2⋅g ⋅a ⋅False   2⋅g ⋅⎝a ⎠ ⋅a⋅False   2⋅g \n",
       "── - ────────── - ──────────────── - ───────────── - ────────────────── - ────\n",
       "           2              2                 2                 2               \n",
       "        2⋅Δ              Δ                 Δ                 Δ                \n",
       "\n",
       "                                              \n",
       "            2      2    †            2        \n",
       "⋅a⋅False   g ⋅ω   g ⋅ω⋅a ⋅a⋅False   g ⋅ω⋅False\n",
       "──────── + ──── + ─────────────── + ──────────\n",
       "  2           2           2               2   \n",
       " Δ         2⋅Δ           Δ             2⋅Δ    "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H3 = H2.subs(l, g/D); H3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also drop c-number terms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\frac{\\epsilon {\\sigma_z}}{2} + g {{a}^\\dagger} {\\sigma_-} + g {a} {\\sigma_+} + \\omega {{a}^\\dagger} {a} - \\frac{\\epsilon g}{\\Delta} {{a}^\\dagger} {\\sigma_-} - \\frac{\\epsilon g}{\\Delta} {a} {\\sigma_+} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} + \\frac{g \\omega}{\\Delta} {{a}^\\dagger} {\\sigma_-} + \\frac{g \\omega}{\\Delta} {a} {\\sigma_+} - \\frac{\\epsilon g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon g^{2} {\\sigma_z}}{2 \\Delta^{2}} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} {a} {\\sigma_+} + \\frac{g^{2} \\omega}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega {\\sigma_z}}{2 \\Delta^{2}}$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                                                 †                          2 \n",
       "ε⋅False      †                        †     ε⋅g⋅a ⋅False   ε⋅g⋅a⋅False   2⋅g ⋅\n",
       "─────── + g⋅a ⋅False + g⋅a⋅False + ω⋅a ⋅a - ──────────── - ─────────── + ─────\n",
       "   2                                             Δ              Δ             \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       " †            2              †                          2  †              2   \n",
       "a ⋅a⋅False   g ⋅False   g⋅ω⋅a ⋅False   g⋅ω⋅a⋅False   ε⋅g ⋅a ⋅a⋅False   ε⋅g ⋅Fa\n",
       "────────── + ──────── + ──────────── + ─────────── - ─────────────── - ───────\n",
       "  Δ             Δ            Δ              Δ                2               2\n",
       "                                                            Δ             2⋅Δ \n",
       "\n",
       "                                                  2                           \n",
       "         3  †  2            3  †            3 ⎛ †⎞               3            \n",
       "lse   2⋅g ⋅a ⋅a ⋅False   2⋅g ⋅a ⋅False   2⋅g ⋅⎝a ⎠ ⋅a⋅False   2⋅g ⋅a⋅False   g\n",
       "─── - ──────────────── - ───────────── - ────────────────── - ──────────── + ─\n",
       "              2                 2                 2                 2         \n",
       "             Δ                 Δ                 Δ                 Δ          \n",
       "\n",
       "                           \n",
       "2    †            2        \n",
       " ⋅ω⋅a ⋅a⋅False   g ⋅ω⋅False\n",
       "────────────── + ──────────\n",
       "       2               2   \n",
       "      Δ             2⋅Δ    "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H4 = drop_c_number_terms(H3); H4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and collect the terms with $a^\\dagger a$ and $\\sigma_z$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$g {{a}^\\dagger} {\\sigma_-} + g {a} {\\sigma_+} + \\left(\\frac{\\epsilon}{2} + \\frac{g^{2}}{\\Delta} - \\frac{\\epsilon g^{2}}{2 \\Delta^{2}} + \\frac{g^{2} \\omega}{2 \\Delta^{2}}\\right) {\\sigma_z} + {{a}^\\dagger} {a} \\left(\\omega + \\frac{2 g^{2}}{\\Delta} {\\sigma_z} - \\frac{\\epsilon g^{2}}{\\Delta^{2}} {\\sigma_z} + \\frac{g^{2} \\omega}{\\Delta^{2}} {\\sigma_z}\\right) - \\frac{\\epsilon g}{\\Delta} {{a}^\\dagger} {\\sigma_-} - \\frac{\\epsilon g}{\\Delta} {a} {\\sigma_+} + \\frac{g \\omega}{\\Delta} {{a}^\\dagger} {\\sigma_-} + \\frac{g \\omega}{\\Delta} {a} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} {a} {\\sigma_+}$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                         ⎛     2      2    2  ⎞              ⎛       2        \n",
       "   †                     ⎜ε   g    ε⋅g    g ⋅ω⎟          †   ⎜    2⋅g ⋅False  \n",
       "g⋅a ⋅False + g⋅a⋅False + ⎜─ + ── - ──── + ────⎟⋅False + a ⋅a⋅⎜ω + ────────── -\n",
       "                         ⎜2   Δ       2      2⎟              ⎜        Δ       \n",
       "                         ⎝         2⋅Δ    2⋅Δ ⎠              ⎝                \n",
       "\n",
       "                                                                              \n",
       "    2          2        ⎞        †                            †               \n",
       " ε⋅g ⋅False   g ⋅ω⋅False⎟   ε⋅g⋅a ⋅False   ε⋅g⋅a⋅False   g⋅ω⋅a ⋅False   g⋅ω⋅a⋅\n",
       " ────────── + ──────────⎟ - ──────────── - ─────────── + ──────────── + ──────\n",
       "      2            2    ⎟        Δ              Δ             Δ              Δ\n",
       "     Δ            Δ     ⎠                                                     \n",
       "\n",
       "                                                    2                       \n",
       "           3  †  2            3  †            3 ⎛ †⎞               3        \n",
       "False   2⋅g ⋅a ⋅a ⋅False   2⋅g ⋅a ⋅False   2⋅g ⋅⎝a ⎠ ⋅a⋅False   2⋅g ⋅a⋅False\n",
       "───── - ──────────────── - ───────────── - ────────────────── - ────────────\n",
       "                2                 2                 2                 2     \n",
       "               Δ                 Δ                 Δ                 Δ      "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H5 = collect(H4, [Dagger(a)*a, sz]); H5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now move to a frame co-rotating with the qubit and oscillator frequencies:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\frac{\\epsilon {\\sigma_z}}{2} + g e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} + g e^{- i \\omega t} {a} {\\sigma_+} - \\frac{\\epsilon g}{\\Delta} e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} - \\frac{\\epsilon g}{\\Delta} e^{- i \\omega t} {a} {\\sigma_+} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} + \\frac{g \\omega}{\\Delta} e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} + \\frac{g \\omega}{\\Delta} e^{- i \\omega t} {a} {\\sigma_+} - \\frac{\\epsilon g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon g^{2} {\\sigma_z}}{2 \\Delta^{2}} - \\frac{2 g^{3}}{\\Delta^{2}} e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} e^{i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} e^{- i \\omega t} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} e^{- i \\omega t} {a} {\\sigma_+} + \\frac{g^{2} \\omega}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega {\\sigma_z}}{2 \\Delta^{2}}$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                                                       ⅈ⋅ω⋅t  †              -\n",
       "ε⋅False      ⅈ⋅ω⋅t  †            -ⅈ⋅ω⋅t           ε⋅g⋅ℯ     ⋅a ⋅False   ε⋅g⋅ℯ \n",
       "─────── + g⋅ℯ     ⋅a ⋅False + g⋅ℯ      ⋅a⋅False - ─────────────────── - ──────\n",
       "   2                                                       Δ                  \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "ⅈ⋅ω⋅t              2  †            2              ⅈ⋅ω⋅t  †              -ⅈ⋅ω⋅t\n",
       "     ⋅a⋅False   2⋅g ⋅a ⋅a⋅False   g ⋅False   g⋅ω⋅ℯ     ⋅a ⋅False   g⋅ω⋅ℯ      \n",
       "───────────── + ─────────────── + ──────── + ─────────────────── + ───────────\n",
       "   Δ                   Δ             Δ                Δ                     Δ \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "              2  †              2            3  ⅈ⋅ω⋅t  †            3  ⅈ⋅ω⋅t ⎛\n",
       "⋅a⋅False   ε⋅g ⋅a ⋅a⋅False   ε⋅g ⋅False   2⋅g ⋅ℯ     ⋅a ⋅False   2⋅g ⋅ℯ     ⋅⎝\n",
       "──────── - ─────────────── - ────────── - ──────────────────── - ─────────────\n",
       "                   2               2                2                         \n",
       "                  Δ             2⋅Δ                Δ                         Δ\n",
       "\n",
       "   2                                                                          \n",
       " †⎞               3  -ⅈ⋅ω⋅t  †  2            3  -ⅈ⋅ω⋅t            2    †      \n",
       "a ⎠ ⋅a⋅False   2⋅g ⋅ℯ      ⋅a ⋅a ⋅False   2⋅g ⋅ℯ      ⋅a⋅False   g ⋅ω⋅a ⋅a⋅Fal\n",
       "──────────── - ──────────────────────── - ──────────────────── + ─────────────\n",
       "2                          2                        2                    2    \n",
       "                          Δ                        Δ                    Δ     \n",
       "\n",
       "               \n",
       "      2        \n",
       "se   g ⋅ω⋅False\n",
       "── + ──────────\n",
       "           2   \n",
       "        2⋅Δ    "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H6 = hamiltonian_transformation(U1, H5.expand()); H6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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JXdC82vuXu+LvvTskCkzjrQ67md8txEe3BLo8PvC0fBjZ57VDC3hbche0BgA/ij3CLhFL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\n3lFHI4XqXNolWGcg7NXNQSK5/w9Gp5eBm0/U0AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$g e^{i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_+} + g e^{- i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} - \\frac{\\epsilon g}{\\Delta} e^{i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_+} - \\frac{\\epsilon g}{\\Delta} e^{- i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} + \\frac{g \\omega}{\\Delta} e^{i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_+} + \\frac{g \\omega}{\\Delta} e^{- i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} - \\frac{\\epsilon g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon g^{2} {\\sigma_z}}{2 \\Delta^{2}} - \\frac{2 g^{3}}{\\Delta^{2}} e^{i \\epsilon t} e^{- i \\omega t} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} e^{i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} e^{- i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} e^{- i \\epsilon t} e^{i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} + \\frac{g^{2} \\omega}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega {\\sigma_z}}{2 \\Delta^{2}}$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                                                            ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t     \n",
       "   ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t              -ⅈ⋅ε⋅t  ⅈ⋅ω⋅t  †         ε⋅g⋅ℯ     ⋅ℯ      ⋅a⋅Fa\n",
       "g⋅ℯ     ⋅ℯ      ⋅a⋅False + g⋅ℯ      ⋅ℯ     ⋅a ⋅False - ───────────────────────\n",
       "                                                                   Δ          \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "           -ⅈ⋅ε⋅t  ⅈ⋅ω⋅t  †            2  †            2              ⅈ⋅ε⋅t  -\n",
       "lse   ε⋅g⋅ℯ      ⋅ℯ     ⋅a ⋅False   2⋅g ⋅a ⋅a⋅False   g ⋅False   g⋅ω⋅ℯ     ⋅ℯ \n",
       "─── - ─────────────────────────── + ─────────────── + ──────── + ─────────────\n",
       "                   Δ                       Δ             Δ                   Δ\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "ⅈ⋅ω⋅t                -ⅈ⋅ε⋅t  ⅈ⋅ω⋅t  †            2  †              2          \n",
       "     ⋅a⋅False   g⋅ω⋅ℯ      ⋅ℯ     ⋅a ⋅False   ε⋅g ⋅a ⋅a⋅False   ε⋅g ⋅False   2\n",
       "───────────── + ─────────────────────────── - ─────────────── - ────────── - ─\n",
       "                             Δ                        2               2       \n",
       "                                                     Δ             2⋅Δ        \n",
       "\n",
       "                                                                              \n",
       "  3  ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t  †  2            3  ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t              3  -ⅈ⋅ε⋅t  ⅈ\n",
       "⋅g ⋅ℯ     ⋅ℯ      ⋅a ⋅a ⋅False   2⋅g ⋅ℯ     ⋅ℯ      ⋅a⋅False   2⋅g ⋅ℯ      ⋅ℯ \n",
       "────────────────────────────── - ─────────────────────────── - ───────────────\n",
       "               2                               2                             2\n",
       "              Δ                               Δ                             Δ \n",
       "\n",
       "                                        2                                     \n",
       "⋅ω⋅t  †            3  -ⅈ⋅ε⋅t  ⅈ⋅ω⋅t ⎛ †⎞             2    †            2      \n",
       "    ⋅a ⋅False   2⋅g ⋅ℯ      ⋅ℯ     ⋅⎝a ⎠ ⋅a⋅False   g ⋅ω⋅a ⋅a⋅False   g ⋅ω⋅Fal\n",
       "───────────── - ───────────────────────────────── + ─────────────── + ────────\n",
       "                                 2                          2               2 \n",
       "                                Δ                          Δ             2⋅Δ  \n",
       "\n",
       "  \n",
       "  \n",
       "se\n",
       "──\n",
       "  \n",
       "  "
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H7 = hamiltonian_transformation(U2, H6.expand()); H7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$g e^{i t \\left(- \\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_-} + g e^{i t \\left(\\epsilon - \\omega\\right)} {a} {\\sigma_+} - \\frac{\\epsilon g}{\\Delta} e^{i t \\left(- \\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_-} - \\frac{\\epsilon g}{\\Delta} e^{i t \\left(\\epsilon - \\omega\\right)} {a} {\\sigma_+} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} + \\frac{g \\omega}{\\Delta} e^{i t \\left(- \\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_-} + \\frac{g \\omega}{\\Delta} e^{i t \\left(\\epsilon - \\omega\\right)} {a} {\\sigma_+} - \\frac{\\epsilon g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon g^{2} {\\sigma_z}}{2 \\Delta^{2}} - \\frac{2 g^{3}}{\\Delta^{2}} e^{i t \\left(- \\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} e^{i t \\left(- \\epsilon + \\omega\\right)} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} e^{i t \\left(\\epsilon - \\omega\\right)} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} e^{i t \\left(\\epsilon - \\omega\\right)} {a} {\\sigma_+} + \\frac{g^{2} \\omega}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega {\\sigma_z}}{2 \\Delta^{2}}$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                                                         ⅈ⋅t⋅(-ε + ω)  †      \n",
       "   ⅈ⋅t⋅(-ε + ω)  †            ⅈ⋅t⋅(ε - ω)           ε⋅g⋅ℯ            ⋅a ⋅False\n",
       "g⋅ℯ            ⋅a ⋅False + g⋅ℯ           ⋅a⋅False - ──────────────────────────\n",
       "                                                                Δ             \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "        ⅈ⋅t⋅(ε - ω)              2  †            2              ⅈ⋅t⋅(-ε + ω)  \n",
       "   ε⋅g⋅ℯ           ⋅a⋅False   2⋅g ⋅a ⋅a⋅False   g ⋅False   g⋅ω⋅ℯ            ⋅a\n",
       " - ──────────────────────── + ─────────────── + ──────── + ───────────────────\n",
       "              Δ                      Δ             Δ                   Δ      \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "†              ⅈ⋅t⋅(ε - ω)              2  †              2            3  ⅈ⋅t⋅\n",
       " ⋅False   g⋅ω⋅ℯ           ⋅a⋅False   ε⋅g ⋅a ⋅a⋅False   ε⋅g ⋅False   2⋅g ⋅ℯ    \n",
       "─────── + ──────────────────────── - ─────────────── - ────────── - ──────────\n",
       "                     Δ                       2               2                \n",
       "                                            Δ             2⋅Δ                 \n",
       "\n",
       "                                           2                                  \n",
       "(-ε + ω)  †            3  ⅈ⋅t⋅(-ε + ω) ⎛ †⎞               3  ⅈ⋅t⋅(ε - ω)  †  2\n",
       "        ⋅a ⋅False   2⋅g ⋅ℯ            ⋅⎝a ⎠ ⋅a⋅False   2⋅g ⋅ℯ           ⋅a ⋅a \n",
       "───────────────── - ──────────────────────────────── - ───────────────────────\n",
       "    2                               2                                 2       \n",
       "   Δ                               Δ                                 Δ        \n",
       "\n",
       "                                                                 \n",
       "            3  ⅈ⋅t⋅(ε - ω)            2    †            2        \n",
       "⋅False   2⋅g ⋅ℯ           ⋅a⋅False   g ⋅ω⋅a ⋅a⋅False   g ⋅ω⋅False\n",
       "────── - ───────────────────────── + ─────────────── + ──────────\n",
       "                      2                      2               2   \n",
       "                     Δ                      Δ             2⋅Δ    "
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H7 = simplify_exp(powsimp(H7)); H7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the absolute value of the detuning $\\Delta = \\epsilon-\\omega$ is much larger than $g$ (equivalently, $|\\lambda\\ll 1|$), the exponential terms $e^{i(\\epsilon-\\omega)t}$, $e^{-i(\\epsilon-\\omega)t}$ are now considered fast oscillating and can be ignored."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAiMAAAAvBAMAAADDWdfQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nk45IAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} - \\frac{\\epsilon g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon g^{2} {\\sigma_z}}{2 \\Delta^{2}} + \\frac{g^{2} \\omega}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega {\\sigma_z}}{2 \\Delta^{2}}$$"
      ],
      "text/plain": [
       "   2  †            2            2  †              2          2    †           \n",
       "2⋅g ⋅a ⋅a⋅False   g ⋅False   ε⋅g ⋅a ⋅a⋅False   ε⋅g ⋅False   g ⋅ω⋅a ⋅a⋅False   \n",
       "─────────────── + ──────── - ─────────────── - ────────── + ─────────────── + \n",
       "       Δ             Δ               2               2              2         \n",
       "                                    Δ             2⋅Δ              Δ          \n",
       "\n",
       " 2        \n",
       "g ⋅ω⋅False\n",
       "──────────\n",
       "      2   \n",
       "   2⋅Δ    "
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H8 = drop_terms_containing(H7, [exp(I * (e - w) * t), exp(I * (- e + w) * t)])\n",
    "\n",
    "H8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgcAAAA1BAMAAAAnolWHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left(\\frac{g^{2}}{\\Delta} - \\frac{\\epsilon g^{2}}{2 \\Delta^{2}} + \\frac{g^{2} \\omega}{2 \\Delta^{2}}\\right) {\\sigma_z} + {{a}^\\dagger} {a} \\left(\\frac{2 g^{2}}{\\Delta} {\\sigma_z} - \\frac{\\epsilon g^{2}}{\\Delta^{2}} {\\sigma_z} + \\frac{g^{2} \\omega}{\\Delta^{2}} {\\sigma_z}\\right)$$"
      ],
      "text/plain": [
       "⎛ 2      2    2  ⎞              ⎛   2            2          2        ⎞\n",
       "⎜g    ε⋅g    g ⋅ω⎟          †   ⎜2⋅g ⋅False   ε⋅g ⋅False   g ⋅ω⋅False⎟\n",
       "⎜── - ──── + ────⎟⋅False + a ⋅a⋅⎜────────── - ────────── + ──────────⎟\n",
       "⎜Δ       2      2⎟              ⎜    Δ             2            2    ⎟\n",
       "⎝     2⋅Δ    2⋅Δ ⎠              ⎝                 Δ            Δ     ⎠"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H9 = qsimplify(H8)\n",
    "\n",
    "H9 = collect(H9, [Dagger(a)*a, sz])\n",
    "\n",
    "H9"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we move back to the lab frame:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAmwAAAAvBAMAAACs6Iv9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\np0p39ITpfnlPfzJ0ejD4PxQ7fA0+kdxjZxb0/QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\omega {{a}^\\dagger} {a} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} - \\frac{\\epsilon g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon g^{2} {\\sigma_z}}{2 \\Delta^{2}} + \\frac{g^{2} \\omega}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega {\\sigma_z}}{2 \\Delta^{2}}$$"
      ],
      "text/plain": [
       "            2  †            2            2  †              2          2    †  \n",
       "   †     2⋅g ⋅a ⋅a⋅False   g ⋅False   ε⋅g ⋅a ⋅a⋅False   ε⋅g ⋅False   g ⋅ω⋅a ⋅a\n",
       "ω⋅a ⋅a + ─────────────── + ──────── - ─────────────── - ────────── + ─────────\n",
       "                Δ             Δ               2               2              2\n",
       "                                             Δ             2⋅Δ              Δ \n",
       "\n",
       "          2        \n",
       "⋅False   g ⋅ω⋅False\n",
       "────── + ──────────\n",
       "               2   \n",
       "            2⋅Δ    "
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H10 = hamiltonian_transformation(U3, H9.expand()); H10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAqUAAAAvBAMAAAAoQq6WAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nuJ5RNLfMMPwP9YsJaoX+GBMAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{\\epsilon {\\sigma_z}}{2} + \\omega {{a}^\\dagger} {a} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} - \\frac{\\epsilon g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon g^{2} {\\sigma_z}}{2 \\Delta^{2}} + \\frac{g^{2} \\omega}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega {\\sigma_z}}{2 \\Delta^{2}}$$"
      ],
      "text/plain": [
       "                      2  †            2            2  †              2        \n",
       "ε⋅False      †     2⋅g ⋅a ⋅a⋅False   g ⋅False   ε⋅g ⋅a ⋅a⋅False   ε⋅g ⋅False  \n",
       "─────── + ω⋅a ⋅a + ─────────────── + ──────── - ─────────────── - ────────── +\n",
       "   2                      Δ             Δ               2               2     \n",
       "                                                       Δ             2⋅Δ      \n",
       "\n",
       "  2    †            2        \n",
       " g ⋅ω⋅a ⋅a⋅False   g ⋅ω⋅False\n",
       " ─────────────── + ──────────\n",
       "         2               2   \n",
       "        Δ             2⋅Δ    "
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H11 = hamiltonian_transformation(U4, H10.expand()); H11"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simplifying the expression, and collecting the terms of $\\sigma_z$ and $a^\\dagger a$, with substitution $\\Delta = \\epsilon-\\omega$, we get the expression for the Hamiltonian in the dispersive regime."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAaMAAAA1BAMAAAAe85ktAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nq3Ip7Bz/B8bfZG3ZMl4zAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$H_{RWAdisp} = \\left(\\frac{\\epsilon}{2} + \\frac{g^{2}}{2 \\Delta}\\right) {\\sigma_z} + {{a}^\\dagger} {a} \\left(\\omega + \\frac{g^{2} {\\sigma_z}}{\\Delta}\\right)$$"
      ],
      "text/plain": [
       "            ⎛      2⎞              ⎛     2      ⎞\n",
       "            ⎜ε    g ⎟          †   ⎜    g ⋅False⎟\n",
       "H_RWAdisp = ⎜─ + ───⎟⋅False + a ⋅a⋅⎜ω + ────────⎟\n",
       "            ⎝2   2⋅Δ⎠              ⎝       Δ    ⎠"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H12 = qsimplify(H11)\n",
    "\n",
    "H12 = collect(H12, [Dagger(a) * a, sz])\n",
    "\n",
    "Hrwadisp = H12.subs(w, e - D).expand().collect([Dagger(a)*a, sz]).subs(e - D, w)\n",
    "\n",
    "Eq(symbols(\"H_RWAdisp\"), Hrwadisp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The physical interpretation of this equation is that the oscillator frequency is shifted as\n",
    "\\begin{align}\n",
    "\\omega \\rightarrow \\omega\\pm \\frac{g^2}{\\Delta}\n",
    "\\end{align}\n",
    "where the sign depends on the state of the qubit."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dispersive Theory Beyond RWA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To treat the original Hamiltonian (not the RWA Hamiltonian) \\begin{align}\n",
    "H = \\frac{\\hbar\\epsilon}{2}\\sigma_z + \\hbar\\omega a^\\dagger a + \\hbar g \\sigma_x (a^\\dagger + a)\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASAAAAAlBAMAAAAQFym0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n8QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$H = \\frac{\\epsilon {\\sigma_z}}{2} + g {\\sigma_x} \\left({{a}^\\dagger} + {a}\\right) + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "    ε⋅False           ⎛ †    ⎞      †  \n",
       "H = ─────── + g⋅False⋅⎝a  + a⎠ + ω⋅a ⋅a\n",
       "       2                               "
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(Hsym, H)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "in the dispersive regime correctly. That is, we derive the expression valid in the full dispersive regime defined only by inequality\n",
    "\\begin{align}\n",
    "g\\ll |\\epsilon -\\omega|.\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Going beyond RWA, we have to keep the counter-rotating coupling terms\n",
    "\\begin{align}\n",
    "Y_{\\pm} =  \\sigma_- a  \\pm\\sigma_+ a^\\dagger,\n",
    "\\end{align}\n",
    "which are relevant if either of the relations $g\\ll \\min\\{\\epsilon,\\omega\\}$ or $|\\epsilon-\\omega|\\ll \\epsilon+\\omega$ is violated."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAZBAMAAABEJDijAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nmqRmrSL4dpqkZu0Lq0ZN0lV5/7QbSf4CE4F9DfCw+f0AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$Y_{+} = {\\sigma_-} {a} + {\\sigma_+} {{a}^\\dagger}$$"
      ],
      "text/plain": [
       "                      †\n",
       "Y₊ = False⋅a + False⋅a "
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Yp = sm * a + sp * Dagger(a)\n",
    "Ypsym = symbols(\"Y_+\")\n",
    "Eq(Ypsym,Yp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAZBAMAAABEJDijAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDbXWv/DvP6ejKj4Bn+hy8sMZdCcAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$Y_{-} = {\\sigma_-} {a} - {\\sigma_+} {{a}^\\dagger}$$"
      ],
      "text/plain": [
       "                      †\n",
       "Y₋ = False⋅a - False⋅a "
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ym = sm * a - sp * Dagger(a)\n",
    "Ymsym = symbols(\"Y_-\")\n",
    "Eq(Ymsym,Ym)  # different from the definition in the paper"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These operators satify the following commutation relations:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$[Y_+, Y_-] = \\sigma_z (2a^\\dagger a + 1) -1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKkAAAAbBAMAAAD1+bJiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$-1 + {\\sigma_z} \\left(1 + 2 {{a}^\\dagger} {a}\\right)$$"
      ],
      "text/plain": [
       "           ⎛       †  ⎞\n",
       "-1 + False⋅⎝1 + 2⋅a ⋅a⎠"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "comm = Commutator(Yp, Ym).expand(commutator=True).expand(commutator=True).expand(commutator=True).doit()\n",
    "comm = simplify(comm.subs(2*sm*sp, (1-sz)))\n",
    "collect(comm, [sz, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$[H_0, Y_-] = -(\\epsilon+\\omega) Y_+ $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANIAAAAbBAMAAAD7UMwNAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$- \\left(\\epsilon + \\omega\\right) \\left({{a}^\\dagger} {\\sigma_+} + {a} {\\sigma_-}\\right)$$"
      ],
      "text/plain": [
       "         ⎛ †                ⎞\n",
       "-(ε + ω)⋅⎝a ⋅False + a⋅False⎠"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "comm = Commutator(H0, Ym).expand(commutator=True).expand(commutator=True).expand(commutator=True).expand(commutator=True).doit()\n",
    "comm = qsimplify(comm)\n",
    "comm = collect(comm, [a*sm, Dagger(a)*sp])\n",
    "simplify(comm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$[Y_+, X_-] = \\sigma_z \\left\\{a^2 + (a^\\dagger)^2\\right\\} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJwAAAAnBAMAAAAImzzIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$${\\sigma_z} \\left(\\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right)$$"
      ],
      "text/plain": [
       "      ⎛    2     ⎞\n",
       "      ⎜⎛ †⎞     2⎟\n",
       "False⋅⎝⎝a ⎠  + a ⎠"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "comm = Commutator(Yp, Xm).expand(commutator=True).expand(commutator=True).expand(commutator=True).expand(commutator=True).doit()\n",
    "collect(comm, [sz])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$[Y_-, X_+] = -\\sigma_z \\left\\{a^2 + (a^\\dagger)^2\\right\\} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKsAAAAnBAMAAACCtKWJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- {\\sigma_z} \\left(\\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right)$$"
      ],
      "text/plain": [
       "       ⎛    2     ⎞\n",
       "       ⎜⎛ †⎞     2⎟\n",
       "-False⋅⎝⎝a ⎠  + a ⎠"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "comm = Commutator(Ym, Xp).expand(commutator=True).expand(commutator=True).expand(commutator=True).expand(commutator=True).doit()\n",
    "simplify(collect(comm, [sz]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We separate the qubit-oscillator coupling from the bare terms and rewrite the Hamiltonian as\n",
    "\\begin{align}\n",
    "H = H_0 + \\hbar g X_+ + \\hbar g Y_+\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAeMAAAAlBAMAAACDn5wWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$H = \\frac{\\epsilon {\\sigma_z}}{2} + g \\left({\\sigma_-} {{a}^\\dagger} + {\\sigma_+} {a}\\right) + g \\left({\\sigma_-} {a} + {\\sigma_+} {{a}^\\dagger}\\right) + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "    ε⋅False     ⎛       †          ⎞     ⎛                 †⎞      †  \n",
       "H = ─────── + g⋅⎝False⋅a  + False⋅a⎠ + g⋅⎝False⋅a + False⋅a ⎠ + ω⋅a ⋅a\n",
       "       2                                                              "
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H = H0 + g * Xp + g * Yp; Eq(symbols(\"H\"),H)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A unitary transformation corresponding to this Hamiltonian is achieved by the operator\n",
    "\\begin{align}\n",
    "D = e^{-\\lambda_1 X_- - \\lambda_2 Y_-}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we have introduced the parameters\n",
    "\\begin{align}\n",
    "\\lambda_1 \\equiv \\frac{g}{\\Delta},\\quad \\lambda_2  \\equiv \\frac{g}{\\epsilon+\\omega} = \\frac{g}{2\\epsilon -\\Delta},\n",
    "\\end{align}\n",
    "which obviously fulfills the relation $\\lambda_2<\\lambda_1$, since $\\epsilon$ and $\\omega$ are assumed to be positive."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATgAAAAXBAMAAABt3E4OAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n5Bdbj37Or78aAZH2rinGwRVG/gGeyu1yYpMhvwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$D = e^{- \\lambda_{1} \\left({\\sigma_-} {{a}^\\dagger} - {\\sigma_+} {a}\\right) - \\lambda_{2} \\left({\\sigma_-} {a} - {\\sigma_+} {{a}^\\dagger}\\right)}$$"
      ],
      "text/plain": [
       "          ⎛       †          ⎞      ⎛                 †⎞\n",
       "     - λ₁⋅⎝False⋅a  - False⋅a⎠ - λ₂⋅⎝False⋅a - False⋅a ⎠\n",
       "D = ℯ                                                   "
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l1, l2 = symbols(\"lambda1, lambda2\", real=True)\n",
    "Disp = exp(-l1 * Xm - l2 * Ym); Eq(symbols(\"D\"), Disp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define the dispersive Hamiltonian $H_{disp} = DHD^\\dagger$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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/caKlSKSrDuQJvSLCEF+Hx3FsDI/BeK1wnWYa5IF3x2DZTtGR8iD8nc78IbMr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\nAp1AJ9AJdAKdQCiB/wPoLaT4p9pMZAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\frac{\\epsilon \\lambda_{1}^{2}}{2} - \\epsilon \\lambda_{1}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{1}^{2}}{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} \\left({a}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{1} {{a}^\\dagger} {\\sigma_-} - \\epsilon \\lambda_{1} {a} {\\sigma_+} + \\frac{\\epsilon \\lambda_{2}^{2}}{2} - \\epsilon \\lambda_{2}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{2}^{2}}{2} {\\sigma_z} - \\epsilon \\lambda_{2} {{a}^\\dagger} {\\sigma_+} - \\epsilon \\lambda_{2} {a} {\\sigma_-} + \\frac{\\epsilon {\\sigma_z}}{2} - \\frac{g \\lambda_{1}^{4}}{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} + g \\lambda_{1}^{4} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{3 g}{4} \\lambda_{1}^{4} {{a}^\\dagger} {\\sigma_-} + \\frac{3 g}{4} \\lambda_{1}^{4} {{a}^\\dagger} {\\sigma_+} - g \\lambda_{1}^{4} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} + \\frac{g \\lambda_{1}^{4}}{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - \\frac{g \\lambda_{1}^{4}}{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - \\frac{g \\lambda_{1}^{4}}{4} {a} {\\sigma_-} + \\frac{5 g}{4} \\lambda_{1}^{4} {a} {\\sigma_+} + \\frac{g \\lambda_{1}^{4}}{2} \\left({a}\\right)^{3} {\\sigma_+} - \\frac{3 g}{2} \\lambda_{1}^{3} \\lambda_{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} + \\frac{3 g}{2} \\lambda_{1}^{3} \\lambda_{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{3 g}{2} \\lambda_{1}^{3} \\lambda_{2} {{a}^\\dagger} {\\sigma_-} + \\frac{3 g}{2} \\lambda_{1}^{3} \\lambda_{2} {{a}^\\dagger} {\\sigma_+} - \\frac{3 g}{2} \\lambda_{1}^{3} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} + \\frac{3 g}{2} \\lambda_{1}^{3} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - \\frac{g \\lambda_{2}}{2} \\lambda_{1}^{3} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} + \\frac{g \\lambda_{2}}{2} \\lambda_{1}^{3} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} - \\frac{3 g}{2} \\lambda_{1}^{3} \\lambda_{2} {a} {\\sigma_-} + \\frac{3 g}{2} \\lambda_{1}^{3} \\lambda_{2} {a} {\\sigma_+} - \\frac{g \\lambda_{2}}{2} \\lambda_{1}^{3} \\left({a}\\right)^{3} {\\sigma_-} + \\frac{g \\lambda_{2}}{2} \\lambda_{1}^{3} \\left({a}\\right)^{3} {\\sigma_+} + \\frac{g \\lambda_{1}^{3}}{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} - \\frac{g \\lambda_{1}^{3}}{2} \\left({a}\\right)^{2} {\\sigma_z} - \\frac{g \\lambda_{1}^{2}}{2} \\lambda_{2}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - \\frac{g \\lambda_{1}^{2}}{2} \\lambda_{2}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} + \\frac{g \\lambda_{1}^{2}}{2} \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} + \\frac{g \\lambda_{1}^{2}}{2} \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} + \\frac{g \\lambda_{1}^{2}}{2} \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} + \\frac{g \\lambda_{1}^{2}}{2} \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} - g \\lambda_{1}^{2} \\lambda_{2}^{2} {a} {\\sigma_-} - g \\lambda_{1}^{2} \\lambda_{2}^{2} {a} {\\sigma_+} - \\frac{g \\lambda_{1}^{2}}{2} \\lambda_{2}^{2} \\left({a}\\right)^{3} {\\sigma_-} - \\frac{g \\lambda_{1}^{2}}{2} \\lambda_{2}^{2} \\left({a}\\right)^{3} {\\sigma_+} - \\frac{g \\lambda_{2}}{2} \\lambda_{1}^{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + \\frac{g \\lambda_{2}}{2} \\lambda_{1}^{2} \\left({a}\\right)^{2} {\\sigma_z} - g \\lambda_{1}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - 2 g \\lambda_{1}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 2 g \\lambda_{1}^{2} {{a}^\\dagger} {\\sigma_-} - g \\lambda_{1}^{2} {{a}^\\dagger} {\\sigma_+} - 2 g \\lambda_{1}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - g \\lambda_{1}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - g \\lambda_{1}^{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - g \\lambda_{1}^{2} {a} {\\sigma_-} - 2 g \\lambda_{1}^{2} {a} {\\sigma_+} - g \\lambda_{1}^{2} \\left({a}\\right)^{3} {\\sigma_+} + \\frac{3 g}{2} \\lambda_{1} \\lambda_{2}^{3} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - \\frac{3 g}{2} \\lambda_{1} \\lambda_{2}^{3} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} + \\frac{3 g}{2} \\lambda_{1} \\lambda_{2}^{3} {{a}^\\dagger} {\\sigma_-} - \\frac{3 g}{2} \\lambda_{1} \\lambda_{2}^{3} {{a}^\\dagger} {\\sigma_+} + \\frac{3 g}{2} \\lambda_{1} \\lambda_{2}^{3} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - \\frac{3 g}{2} \\lambda_{1} \\lambda_{2}^{3} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} + \\frac{g \\lambda_{1}}{2} \\lambda_{2}^{3} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - \\frac{g \\lambda_{1}}{2} \\lambda_{2}^{3} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} + \\frac{3 g}{2} \\lambda_{1} \\lambda_{2}^{3} {a} {\\sigma_-} - \\frac{3 g}{2} \\lambda_{1} \\lambda_{2}^{3} {a} {\\sigma_+} + \\frac{g \\lambda_{1}}{2} \\lambda_{2}^{3} \\left({a}\\right)^{3} {\\sigma_-} - \\frac{g \\lambda_{1}}{2} \\lambda_{2}^{3} \\left({a}\\right)^{3} {\\sigma_+} - \\frac{g \\lambda_{1}}{2} \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + \\frac{g \\lambda_{1}}{2} \\lambda_{2}^{2} \\left({a}\\right)^{2} {\\sigma_z} - 3 g \\lambda_{1} \\lambda_{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} {{a}^\\dagger} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} {{a}^\\dagger} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - g \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - g \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} {a} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} {a} {\\sigma_+} - g \\lambda_{1} \\lambda_{2} \\left({a}\\right)^{3} {\\sigma_-} - g \\lambda_{1} \\lambda_{2} \\left({a}\\right)^{3} {\\sigma_+} + g \\lambda_{1} + 2 g \\lambda_{1} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{1} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + g \\lambda_{1} \\left({a}\\right)^{2} {\\sigma_z} + g \\lambda_{1} {\\sigma_z} + g \\lambda_{2}^{4} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - \\frac{g \\lambda_{2}^{4}}{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} + \\frac{3 g}{4} \\lambda_{2}^{4} {{a}^\\dagger} {\\sigma_-} - \\frac{3 g}{4} \\lambda_{2}^{4} {{a}^\\dagger} {\\sigma_+} + \\frac{g \\lambda_{2}^{4}}{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - g \\lambda_{2}^{4} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - \\frac{g \\lambda_{2}^{4}}{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} + \\frac{5 g}{4} \\lambda_{2}^{4} {a} {\\sigma_-} - \\frac{g \\lambda_{2}^{4}}{4} {a} {\\sigma_+} + \\frac{g \\lambda_{2}^{4}}{2} \\left({a}\\right)^{3} {\\sigma_-} + \\frac{g \\lambda_{2}^{3}}{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} - \\frac{g \\lambda_{2}^{3}}{2} \\left({a}\\right)^{2} {\\sigma_z} - 2 g \\lambda_{2}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - g \\lambda_{2}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - g \\lambda_{2}^{2} {{a}^\\dagger} {\\sigma_-} - 2 g \\lambda_{2}^{2} {{a}^\\dagger} {\\sigma_+} - g \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - 2 g \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - g \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} - 2 g \\lambda_{2}^{2} {a} {\\sigma_-} - g \\lambda_{2}^{2} {a} {\\sigma_+} - g \\lambda_{2}^{2} \\left({a}\\right)^{3} {\\sigma_-} - g \\lambda_{2} + 2 g \\lambda_{2} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + g \\lambda_{2} \\left({a}\\right)^{2} {\\sigma_z} + g \\lambda_{2} {\\sigma_z} + g {{a}^\\dagger} {\\sigma_-} + g {{a}^\\dagger} {\\sigma_+} + g {a} {\\sigma_-} + g {a} {\\sigma_+} + \\frac{\\lambda_{1}^{4} \\omega}{4} {{a}^\\dagger} {a} - \\frac{\\lambda_{1}^{3} \\omega}{2} {{a}^\\dagger} {\\sigma_-} - \\frac{\\lambda_{1}^{3} \\omega}{2} {a} {\\sigma_+} - \\frac{\\lambda_{1}^{2} \\omega}{2} \\lambda_{2}^{2} {{a}^\\dagger} {a} - \\frac{\\lambda_{2} \\omega}{2} \\lambda_{1}^{2} {{a}^\\dagger} {\\sigma_+} - \\frac{\\lambda_{2} \\omega}{2} \\lambda_{1}^{2} {a} {\\sigma_-} + \\frac{\\lambda_{1}^{2} \\omega}{2} + \\lambda_{1}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} + \\frac{\\lambda_{1}^{2} \\omega}{2} {\\sigma_z} + \\frac{\\lambda_{1} \\omega}{2} \\lambda_{2}^{2} {{a}^\\dagger} {\\sigma_-} + \\frac{\\lambda_{1} \\omega}{2} \\lambda_{2}^{2} {a} {\\sigma_+} + \\lambda_{1} \\omega {{a}^\\dagger} {\\sigma_-} + \\lambda_{1} \\omega {a} {\\sigma_+} + \\frac{\\lambda_{2}^{4} \\omega}{4} {{a}^\\dagger} {a} + \\frac{\\lambda_{2}^{3} \\omega}{2} {{a}^\\dagger} {\\sigma_+} + \\frac{\\lambda_{2}^{3} \\omega}{2} {a} {\\sigma_-} + \\frac{\\lambda_{2}^{2} \\omega}{2} - \\lambda_{2}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\lambda_{2}^{2} \\omega}{2} {\\sigma_z} - \\lambda_{2} \\omega {{a}^\\dagger} {\\sigma_+} - \\lambda_{2} \\omega {a} {\\sigma_-} + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "      2                          2                     2                      \n",
       "  ε⋅λ₁        2  †           ε⋅λ₁ ⋅False           ⎛ †⎞                   2   \n",
       "- ───── - ε⋅λ₁ ⋅a ⋅a⋅False - ─────────── - ε⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅False - ε⋅λ₁⋅λ₂⋅a ⋅Fa\n",
       "    2                             2                                           \n",
       "\n",
       "                                                                              \n",
       "                                         2                          2         \n",
       "            †                        ε⋅λ₂        2  †           ε⋅λ₂ ⋅False   \n",
       "lse - ε⋅λ₁⋅a ⋅False - ε⋅λ₁⋅a⋅False + ───── - ε⋅λ₂ ⋅a ⋅a⋅False - ─────────── - \n",
       "                                       2                             2        \n",
       "\n",
       "                                                                              \n",
       "                                             4  †  2                          \n",
       "      †                        ε⋅False   g⋅λ₁ ⋅a ⋅a ⋅False       4  †  2      \n",
       "ε⋅λ₂⋅a ⋅False - ε⋅λ₂⋅a⋅False + ─────── - ───────────────── + g⋅λ₁ ⋅a ⋅a ⋅False\n",
       "                                  2              2                            \n",
       "\n",
       "                                                                         2    \n",
       "         4  †               4  †                   2               4 ⎛ †⎞     \n",
       "   3⋅g⋅λ₁ ⋅a ⋅False   3⋅g⋅λ₁ ⋅a ⋅False       4 ⎛ †⎞            g⋅λ₁ ⋅⎝a ⎠ ⋅a⋅F\n",
       " - ──────────────── + ──────────────── - g⋅λ₁ ⋅⎝a ⎠ ⋅a⋅False + ───────────────\n",
       "          4                  4                                          2     \n",
       "\n",
       "                 3                                                            \n",
       "           4 ⎛ †⎞              4                 4               4  3         \n",
       "alse   g⋅λ₁ ⋅⎝a ⎠ ⋅False   g⋅λ₁ ⋅a⋅False   5⋅g⋅λ₁ ⋅a⋅False   g⋅λ₁ ⋅a ⋅False   \n",
       "──── - ───────────────── - ───────────── + ─────────────── + ────────────── - \n",
       "               2                 4                4                2          \n",
       "\n",
       "                                                                              \n",
       "      3     †  2               3     †  2               3     †               \n",
       "3⋅g⋅λ₁ ⋅λ₂⋅a ⋅a ⋅False   3⋅g⋅λ₁ ⋅λ₂⋅a ⋅a ⋅False   3⋅g⋅λ₁ ⋅λ₂⋅a ⋅False   3⋅g⋅λ₁\n",
       "────────────────────── + ────────────────────── - ─────────────────── + ──────\n",
       "          2                        2                       2                  \n",
       "\n",
       "                               2                          2                   \n",
       "3     †               3    ⎛ †⎞                  3    ⎛ †⎞                3   \n",
       " ⋅λ₂⋅a ⋅False   3⋅g⋅λ₁ ⋅λ₂⋅⎝a ⎠ ⋅a⋅False   3⋅g⋅λ₁ ⋅λ₂⋅⎝a ⎠ ⋅a⋅False   g⋅λ₁ ⋅λ₂\n",
       "───────────── - ──────────────────────── + ──────────────────────── - ────────\n",
       "   2                       2                          2                       \n",
       "\n",
       "     3                      3                                                 \n",
       " ⎛ †⎞              3    ⎛ †⎞                3                    3            \n",
       "⋅⎝a ⎠ ⋅False   g⋅λ₁ ⋅λ₂⋅⎝a ⎠ ⋅False   3⋅g⋅λ₁ ⋅λ₂⋅a⋅False   3⋅g⋅λ₁ ⋅λ₂⋅a⋅False \n",
       "──────────── + ──────────────────── - ────────────────── + ────────────────── \n",
       " 2                      2                     2                    2          \n",
       "\n",
       "                                                    2                         \n",
       "      3     3             3     3             3 ⎛ †⎞              3  2        \n",
       "  g⋅λ₁ ⋅λ₂⋅a ⋅False   g⋅λ₁ ⋅λ₂⋅a ⋅False   g⋅λ₁ ⋅⎝a ⎠ ⋅False   g⋅λ₁ ⋅a ⋅False  \n",
       "- ───────────────── + ───────────────── + ───────────────── - ────────────── -\n",
       "          2                   2                   2                 2         \n",
       "\n",
       "                                                               2              \n",
       "     2   2  †  2             2   2  †  2             2   2 ⎛ †⎞               \n",
       " g⋅λ₁ ⋅λ₂ ⋅a ⋅a ⋅False   g⋅λ₁ ⋅λ₂ ⋅a ⋅a ⋅False   g⋅λ₁ ⋅λ₂ ⋅⎝a ⎠ ⋅a⋅False   g⋅λ\n",
       " ───────────────────── - ───────────────────── + ─────────────────────── + ───\n",
       "           2                       2                        2                 \n",
       "\n",
       "           2                         3                       3                \n",
       " 2   2 ⎛ †⎞                2   2 ⎛ †⎞              2   2 ⎛ †⎞                 \n",
       "₁ ⋅λ₂ ⋅⎝a ⎠ ⋅a⋅False   g⋅λ₁ ⋅λ₂ ⋅⎝a ⎠ ⋅False   g⋅λ₁ ⋅λ₂ ⋅⎝a ⎠ ⋅False       2  \n",
       "──────────────────── + ───────────────────── + ───────────────────── - g⋅λ₁ ⋅λ\n",
       "        2                        2                       2                    \n",
       "\n",
       "                                                                              \n",
       "                                     2   2  3             2   2  3            \n",
       " 2               2   2           g⋅λ₁ ⋅λ₂ ⋅a ⋅False   g⋅λ₁ ⋅λ₂ ⋅a ⋅False   g⋅λ\n",
       "₂ ⋅a⋅False - g⋅λ₁ ⋅λ₂ ⋅a⋅False - ────────────────── - ────────────────── - ───\n",
       "                                         2                    2               \n",
       "\n",
       "          2                                                                   \n",
       " 2    ⎛ †⎞              2     2                                               \n",
       "₁ ⋅λ₂⋅⎝a ⎠ ⋅False   g⋅λ₁ ⋅λ₂⋅a ⋅False       2  †  2               2  †  2     \n",
       "───────────────── + ───────────────── - g⋅λ₁ ⋅a ⋅a ⋅False - 2⋅g⋅λ₁ ⋅a ⋅a ⋅Fals\n",
       "      2                     2                                                 \n",
       "\n",
       "                                                                              \n",
       "                                                    2                     2   \n",
       "          2  †             2  †               2 ⎛ †⎞                2 ⎛ †⎞    \n",
       "e - 2⋅g⋅λ₁ ⋅a ⋅False - g⋅λ₁ ⋅a ⋅False - 2⋅g⋅λ₁ ⋅⎝a ⎠ ⋅a⋅False - g⋅λ₁ ⋅⎝a ⎠ ⋅a⋅\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                  3                                                           \n",
       "            2 ⎛ †⎞              2                 2               2  3        \n",
       "False - g⋅λ₁ ⋅⎝a ⎠ ⋅False - g⋅λ₁ ⋅a⋅False - 2⋅g⋅λ₁ ⋅a⋅False - g⋅λ₁ ⋅a ⋅False +\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "          3  †  2                  3  †  2                  3  †              \n",
       " 3⋅g⋅λ₁⋅λ₂ ⋅a ⋅a ⋅False   3⋅g⋅λ₁⋅λ₂ ⋅a ⋅a ⋅False   3⋅g⋅λ₁⋅λ₂ ⋅a ⋅False   3⋅g⋅λ\n",
       " ────────────────────── - ────────────────────── + ─────────────────── - ─────\n",
       "           2                        2                       2                 \n",
       "\n",
       "                                2                          2                  \n",
       "    3  †                  3 ⎛ †⎞                     3 ⎛ †⎞                   \n",
       "₁⋅λ₂ ⋅a ⋅False   3⋅g⋅λ₁⋅λ₂ ⋅⎝a ⎠ ⋅a⋅False   3⋅g⋅λ₁⋅λ₂ ⋅⎝a ⎠ ⋅a⋅False   g⋅λ₁⋅λ₂\n",
       "────────────── + ──────────────────────── - ──────────────────────── + ───────\n",
       "    2                       2                          2                      \n",
       "\n",
       "      3                      3                                                \n",
       "3 ⎛ †⎞                 3 ⎛ †⎞                   3                    3        \n",
       " ⋅⎝a ⎠ ⋅False   g⋅λ₁⋅λ₂ ⋅⎝a ⎠ ⋅False   3⋅g⋅λ₁⋅λ₂ ⋅a⋅False   3⋅g⋅λ₁⋅λ₂ ⋅a⋅False\n",
       "───────────── - ──────────────────── + ────────────────── - ──────────────────\n",
       "  2                      2                     2                    2         \n",
       "\n",
       "                                                        2                     \n",
       "          3  3                3  3                2 ⎛ †⎞                 2  2 \n",
       "   g⋅λ₁⋅λ₂ ⋅a ⋅False   g⋅λ₁⋅λ₂ ⋅a ⋅False   g⋅λ₁⋅λ₂ ⋅⎝a ⎠ ⋅False   g⋅λ₁⋅λ₂ ⋅a ⋅\n",
       " + ───────────────── - ───────────────── - ──────────────────── + ────────────\n",
       "           2                   2                    2                     2   \n",
       "\n",
       "                                                                              \n",
       "                                                                              \n",
       "False              †  2                    †  2                    †          \n",
       "───── - 3⋅g⋅λ₁⋅λ₂⋅a ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅a ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅a ⋅False - 3\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                  2                         2                 \n",
       "          †                   ⎛ †⎞                      ⎛ †⎞                  \n",
       "⋅g⋅λ₁⋅λ₂⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅a⋅False - 3⋅g⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅a⋅False - g⋅λ₁⋅λ\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "      3                     3                                                 \n",
       "  ⎛ †⎞                  ⎛ †⎞                                                  \n",
       "₂⋅⎝a ⎠ ⋅False - g⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅False - 3⋅g⋅λ₁⋅λ₂⋅a⋅False - 3⋅g⋅λ₁⋅λ₂⋅a⋅False - \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                                          2   \n",
       "         3                  3                        †                ⎛ †⎞    \n",
       "g⋅λ₁⋅λ₂⋅a ⋅False - g⋅λ₁⋅λ₂⋅a ⋅False + g⋅λ₁ + 2⋅g⋅λ₁⋅a ⋅a⋅False + g⋅λ₁⋅⎝a ⎠ ⋅Fa\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                                           4  †  2            \n",
       "            2                          4  †  2         g⋅λ₂ ⋅a ⋅a ⋅False   3⋅g\n",
       "lse + g⋅λ₁⋅a ⋅False + g⋅λ₁⋅False + g⋅λ₂ ⋅a ⋅a ⋅False - ───────────────── + ───\n",
       "                                                               2              \n",
       "\n",
       "                                             2                                \n",
       "   4  †               4  †             4 ⎛ †⎞                      2          \n",
       "⋅λ₂ ⋅a ⋅False   3⋅g⋅λ₂ ⋅a ⋅False   g⋅λ₂ ⋅⎝a ⎠ ⋅a⋅False       4 ⎛ †⎞           \n",
       "───────────── - ──────────────── + ─────────────────── - g⋅λ₂ ⋅⎝a ⎠ ⋅a⋅False -\n",
       "    4                  4                    2                                 \n",
       "\n",
       "           3                                                                  \n",
       "     4 ⎛ †⎞                4               4               4  3             3 \n",
       " g⋅λ₂ ⋅⎝a ⎠ ⋅False   5⋅g⋅λ₂ ⋅a⋅False   g⋅λ₂ ⋅a⋅False   g⋅λ₂ ⋅a ⋅False   g⋅λ₂ ⋅\n",
       " ───────────────── + ─────────────── - ───────────── + ────────────── + ──────\n",
       "         2                  4                4               2                \n",
       "\n",
       "    2                                                                         \n",
       "⎛ †⎞              3  2                                                        \n",
       "⎝a ⎠ ⋅False   g⋅λ₂ ⋅a ⋅False         2  †  2             2  †  2             2\n",
       "─────────── - ────────────── - 2⋅g⋅λ₂ ⋅a ⋅a ⋅False - g⋅λ₂ ⋅a ⋅a ⋅False - g⋅λ₂ \n",
       "  2                 2                                                         \n",
       "\n",
       "                                                                              \n",
       "                                         2                       2            \n",
       "  †               2  †             2 ⎛ †⎞                  2 ⎛ †⎞             \n",
       "⋅a ⋅False - 2⋅g⋅λ₂ ⋅a ⋅False - g⋅λ₂ ⋅⎝a ⎠ ⋅a⋅False - 2⋅g⋅λ₂ ⋅⎝a ⎠ ⋅a⋅False - g\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "         3                                                                    \n",
       "   2 ⎛ †⎞                2               2               2  3                 \n",
       "⋅λ₂ ⋅⎝a ⎠ ⋅False - 2⋅g⋅λ₂ ⋅a⋅False - g⋅λ₂ ⋅a⋅False - g⋅λ₂ ⋅a ⋅False - g⋅λ₂ + 2\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                            2                                                 \n",
       "       †                ⎛ †⎞                2                         †       \n",
       "⋅g⋅λ₂⋅a ⋅a⋅False + g⋅λ₂⋅⎝a ⎠ ⋅False + g⋅λ₂⋅a ⋅False + g⋅λ₂⋅False + g⋅a ⋅False \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                                         4    †       3    †           3      \n",
       "     †                                 λ₁ ⋅ω⋅a ⋅a   λ₁ ⋅ω⋅a ⋅False   λ₁ ⋅ω⋅a⋅F\n",
       "+ g⋅a ⋅False + g⋅a⋅False + g⋅a⋅False + ────────── - ────────────── - ─────────\n",
       "                                           4              2                2  \n",
       "\n",
       "                                                                              \n",
       "         2   2    †       2       †           2                  2            \n",
       "alse   λ₁ ⋅λ₂ ⋅ω⋅a ⋅a   λ₁ ⋅λ₂⋅ω⋅a ⋅False   λ₁ ⋅λ₂⋅ω⋅a⋅False   λ₁ ⋅ω     2    \n",
       "──── - ────────────── - ───────────────── - ──────────────── + ───── + λ₁ ⋅ω⋅a\n",
       "             2                  2                  2             2            \n",
       "\n",
       "                                                                              \n",
       "              2                2    †              2                          \n",
       "†           λ₁ ⋅ω⋅False   λ₁⋅λ₂ ⋅ω⋅a ⋅False   λ₁⋅λ₂ ⋅ω⋅a⋅False         †      \n",
       " ⋅a⋅False + ─────────── + ───────────────── + ──────────────── + λ₁⋅ω⋅a ⋅False\n",
       "                 2                2                  2                        \n",
       "\n",
       "                                                                              \n",
       "                    4    †       3    †           3               2           \n",
       "                  λ₂ ⋅ω⋅a ⋅a   λ₂ ⋅ω⋅a ⋅False   λ₂ ⋅ω⋅a⋅False   λ₂ ⋅ω     2   \n",
       " + λ₁⋅ω⋅a⋅False + ────────── + ────────────── + ───────────── + ───── - λ₂ ⋅ω⋅\n",
       "                      4              2                2           2           \n",
       "\n",
       "                                                                \n",
       "               2                                                \n",
       " †           λ₂ ⋅ω⋅False         †                           †  \n",
       "a ⋅a⋅False - ─────────── - λ₂⋅ω⋅a ⋅False - λ₂⋅ω⋅a⋅False + ω⋅a ⋅a\n",
       "                  2                                             "
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H1 = hamiltonian_transformation(Disp, H.expand(), expansion_search=False, N=3).expand()\n",
    "\n",
    "H1 = qsimplify(H1).expand()\n",
    "\n",
    "H1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we only leave the terms with $\\lambda_1, \\lambda_2$ up to the second order."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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L6brQSpgXh6HUPw/5O2pzK6ZOT1s+o/9A7DVaVI/ECaBhGrDA/Mun8lqiyRGS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\nayEtUwiSaA7t9KiZGoTCTIa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fZkaL3TGK0b7nCbMCswKzArMCswKzArMCswJXswK9L4J7\n403ITW8KvfEmhFSbsrKnNndrBu1jL1Pqb9qtr1DLfiJ0zMxorftFMdr1PGFWYFZgVmBWYFZgVmBW\nYFbgalag90Vwb7wJuelNoTfehJCqU+64/oY3VA2aBzeec/2nTjVbX6GGHUXol5nRWneMYrTvecKs\nwKzArMCswKzArMCswKzA1axA74vg3ngTctObQm+8CSFVpzzrwoXvVA2aBw9duHDhqr/D7ChCv8w0\np9AbdozCQ87vswKzArMCswKzArMCswKzArMCDQr0vgjujdcQQmrSm0JvvJTv6OP/D8TAgeOUT6U3\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\frac{\\epsilon \\lambda_{1}^{2}}{2} - \\epsilon \\lambda_{1}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{1}^{2}}{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} \\left({a}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{1} {{a}^\\dagger} {\\sigma_-} - \\epsilon \\lambda_{1} {a} {\\sigma_+} + \\frac{\\epsilon \\lambda_{2}^{2}}{2} - \\epsilon \\lambda_{2}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{2}^{2}}{2} {\\sigma_z} - \\epsilon \\lambda_{2} {{a}^\\dagger} {\\sigma_+} - \\epsilon \\lambda_{2} {a} {\\sigma_-} + \\frac{\\epsilon {\\sigma_z}}{2} - g \\lambda_{1}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - 2 g \\lambda_{1}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 2 g \\lambda_{1}^{2} {{a}^\\dagger} {\\sigma_-} - g \\lambda_{1}^{2} {{a}^\\dagger} {\\sigma_+} - 2 g \\lambda_{1}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - g \\lambda_{1}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - g \\lambda_{1}^{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - g \\lambda_{1}^{2} {a} {\\sigma_-} - 2 g \\lambda_{1}^{2} {a} {\\sigma_+} - g \\lambda_{1}^{2} \\left({a}\\right)^{3} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} {{a}^\\dagger} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} {{a}^\\dagger} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - g \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - g \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} {a} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} {a} {\\sigma_+} - g \\lambda_{1} \\lambda_{2} \\left({a}\\right)^{3} {\\sigma_-} - g \\lambda_{1} \\lambda_{2} \\left({a}\\right)^{3} {\\sigma_+} + g \\lambda_{1} + 2 g \\lambda_{1} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{1} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + g \\lambda_{1} \\left({a}\\right)^{2} {\\sigma_z} + g \\lambda_{1} {\\sigma_z} - 2 g \\lambda_{2}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - g \\lambda_{2}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - g \\lambda_{2}^{2} {{a}^\\dagger} {\\sigma_-} - 2 g \\lambda_{2}^{2} {{a}^\\dagger} {\\sigma_+} - g \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - 2 g \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - g \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} - 2 g \\lambda_{2}^{2} {a} {\\sigma_-} - g \\lambda_{2}^{2} {a} {\\sigma_+} - g \\lambda_{2}^{2} \\left({a}\\right)^{3} {\\sigma_-} - g \\lambda_{2} + 2 g \\lambda_{2} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + g \\lambda_{2} \\left({a}\\right)^{2} {\\sigma_z} + g \\lambda_{2} {\\sigma_z} + g {{a}^\\dagger} {\\sigma_-} + g {{a}^\\dagger} {\\sigma_+} + g {a} {\\sigma_-} + g {a} {\\sigma_+} + \\frac{\\lambda_{1}^{2} \\omega}{2} + \\lambda_{1}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} + \\frac{\\lambda_{1}^{2} \\omega}{2} {\\sigma_z} + \\lambda_{1} \\omega {{a}^\\dagger} {\\sigma_-} + \\lambda_{1} \\omega {a} {\\sigma_+} + \\frac{\\lambda_{2}^{2} \\omega}{2} - \\lambda_{2}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\lambda_{2}^{2} \\omega}{2} {\\sigma_z} - \\lambda_{2} \\omega {{a}^\\dagger} {\\sigma_+} - \\lambda_{2} \\omega {a} {\\sigma_-} + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "      2                          2                     2                      \n",
       "  ε⋅λ₁        2  †           ε⋅λ₁ ⋅False           ⎛ †⎞                   2   \n",
       "- ───── - ε⋅λ₁ ⋅a ⋅a⋅False - ─────────── - ε⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅False - ε⋅λ₁⋅λ₂⋅a ⋅Fa\n",
       "    2                             2                                           \n",
       "\n",
       "                                         2                          2         \n",
       "            †                        ε⋅λ₂        2  †           ε⋅λ₂ ⋅False   \n",
       "lse - ε⋅λ₁⋅a ⋅False - ε⋅λ₁⋅a⋅False + ───── - ε⋅λ₂ ⋅a ⋅a⋅False - ─────────── - \n",
       "                                       2                             2        \n",
       "\n",
       "                                                                              \n",
       "      †                        ε⋅False       2  †  2               2  †  2    \n",
       "ε⋅λ₂⋅a ⋅False - ε⋅λ₂⋅a⋅False + ─────── - g⋅λ₁ ⋅a ⋅a ⋅False - 2⋅g⋅λ₁ ⋅a ⋅a ⋅Fal\n",
       "                                  2                                           \n",
       "\n",
       "                                                     2                     2  \n",
       "           2  †             2  †               2 ⎛ †⎞                2 ⎛ †⎞   \n",
       "se - 2⋅g⋅λ₁ ⋅a ⋅False - g⋅λ₁ ⋅a ⋅False - 2⋅g⋅λ₁ ⋅⎝a ⎠ ⋅a⋅False - g⋅λ₁ ⋅⎝a ⎠ ⋅a\n",
       "                                                                              \n",
       "\n",
       "                   3                                                          \n",
       "             2 ⎛ †⎞              2                 2               2  3       \n",
       "⋅False - g⋅λ₁ ⋅⎝a ⎠ ⋅False - g⋅λ₁ ⋅a⋅False - 2⋅g⋅λ₁ ⋅a⋅False - g⋅λ₁ ⋅a ⋅False \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "             †  2                    †  2                    †                \n",
       "- 3⋅g⋅λ₁⋅λ₂⋅a ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅a ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅a ⋅False - 3⋅g⋅λ₁⋅\n",
       "                                                                              \n",
       "\n",
       "                            2                         2                       \n",
       "    †                   ⎛ †⎞                      ⎛ †⎞                    ⎛ †⎞\n",
       "λ₂⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅a⋅False - 3⋅g⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅a⋅False - g⋅λ₁⋅λ₂⋅⎝a ⎠\n",
       "                                                                              \n",
       "\n",
       "3                     3                                                       \n",
       "                  ⎛ †⎞                                                        \n",
       " ⋅False - g⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅False - 3⋅g⋅λ₁⋅λ₂⋅a⋅False - 3⋅g⋅λ₁⋅λ₂⋅a⋅False - g⋅λ₁⋅λ\n",
       "                                                                              \n",
       "\n",
       "                                                                    2         \n",
       "   3                  3                        †                ⎛ †⎞          \n",
       "₂⋅a ⋅False - g⋅λ₁⋅λ₂⋅a ⋅False + g⋅λ₁ + 2⋅g⋅λ₁⋅a ⋅a⋅False + g⋅λ₁⋅⎝a ⎠ ⋅False + \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "      2                            2  †  2             2  †  2             2  \n",
       "g⋅λ₁⋅a ⋅False + g⋅λ₁⋅False - 2⋅g⋅λ₂ ⋅a ⋅a ⋅False - g⋅λ₂ ⋅a ⋅a ⋅False - g⋅λ₂ ⋅a\n",
       "                                                                              \n",
       "\n",
       "                                       2                       2              \n",
       "†               2  †             2 ⎛ †⎞                  2 ⎛ †⎞               \n",
       " ⋅False - 2⋅g⋅λ₂ ⋅a ⋅False - g⋅λ₂ ⋅⎝a ⎠ ⋅a⋅False - 2⋅g⋅λ₂ ⋅⎝a ⎠ ⋅a⋅False - g⋅λ\n",
       "                                                                              \n",
       "\n",
       "       3                                                                      \n",
       " 2 ⎛ †⎞                2               2               2  3                   \n",
       "₂ ⋅⎝a ⎠ ⋅False - 2⋅g⋅λ₂ ⋅a⋅False - g⋅λ₂ ⋅a⋅False - g⋅λ₂ ⋅a ⋅False - g⋅λ₂ + 2⋅g\n",
       "                                                                              \n",
       "\n",
       "                          2                                                   \n",
       "     †                ⎛ †⎞                2                         †         \n",
       "⋅λ₂⋅a ⋅a⋅False + g⋅λ₂⋅⎝a ⎠ ⋅False + g⋅λ₂⋅a ⋅False + g⋅λ₂⋅False + g⋅a ⋅False + \n",
       "                                                                              \n",
       "\n",
       "                                       2                          2           \n",
       "   †                                 λ₁ ⋅ω     2    †           λ₁ ⋅ω⋅False   \n",
       "g⋅a ⋅False + g⋅a⋅False + g⋅a⋅False + ───── + λ₁ ⋅ω⋅a ⋅a⋅False + ─────────── + \n",
       "                                       2                             2        \n",
       "\n",
       "                                 2                          2                 \n",
       "      †                        λ₂ ⋅ω     2    †           λ₂ ⋅ω⋅False         \n",
       "λ₁⋅ω⋅a ⋅False + λ₁⋅ω⋅a⋅False + ───── - λ₂ ⋅ω⋅a ⋅a⋅False - ─────────── - λ₂⋅ω⋅a\n",
       "                                 2                             2              \n",
       "\n",
       "                               \n",
       "†                           †  \n",
       " ⋅False - λ₂⋅ω⋅a⋅False + ω⋅a ⋅a\n",
       "                               "
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H2 = drop_terms_containing(H1.expand(), [l1**3, l1 * l2**2, l1**2 * l2, l2**3,\n",
    "                                         l1**4, l1**3 * l2, l1**2 * l2**2, l1 * l2**3,  l2**4]); H2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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VYnAABJrBia30BFu6Xnp6Ol9oj2sFHv3KVZsSwejKEtYsMHFiWmlA/gRgA8IVqHQqMciG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\nCixagSW9GL3xaP9/zZ9LBunDQE8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\epsilon \\lambda_{1}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{1}^{2}}{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} \\left({a}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{1} {{a}^\\dagger} {\\sigma_-} - \\epsilon \\lambda_{1} {a} {\\sigma_+} - \\epsilon \\lambda_{2}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{2}^{2}}{2} {\\sigma_z} - \\epsilon \\lambda_{2} {{a}^\\dagger} {\\sigma_+} - \\epsilon \\lambda_{2} {a} {\\sigma_-} + \\frac{\\epsilon {\\sigma_z}}{2} - g \\lambda_{1}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - 2 g \\lambda_{1}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 2 g \\lambda_{1}^{2} {{a}^\\dagger} {\\sigma_-} - g \\lambda_{1}^{2} {{a}^\\dagger} {\\sigma_+} - 2 g \\lambda_{1}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - g \\lambda_{1}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - g \\lambda_{1}^{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - g \\lambda_{1}^{2} {a} {\\sigma_-} - 2 g \\lambda_{1}^{2} {a} {\\sigma_+} - g \\lambda_{1}^{2} \\left({a}\\right)^{3} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} {{a}^\\dagger} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} {{a}^\\dagger} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - g \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - g \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} {a} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} {a} {\\sigma_+} - g \\lambda_{1} \\lambda_{2} \\left({a}\\right)^{3} {\\sigma_-} - g \\lambda_{1} \\lambda_{2} \\left({a}\\right)^{3} {\\sigma_+} + 2 g \\lambda_{1} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{1} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + g \\lambda_{1} \\left({a}\\right)^{2} {\\sigma_z} + g \\lambda_{1} {\\sigma_z} - 2 g \\lambda_{2}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - g \\lambda_{2}^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - g \\lambda_{2}^{2} {{a}^\\dagger} {\\sigma_-} - 2 g \\lambda_{2}^{2} {{a}^\\dagger} {\\sigma_+} - g \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - 2 g \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - g \\lambda_{2}^{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} - 2 g \\lambda_{2}^{2} {a} {\\sigma_-} - g \\lambda_{2}^{2} {a} {\\sigma_+} - g \\lambda_{2}^{2} \\left({a}\\right)^{3} {\\sigma_-} + 2 g \\lambda_{2} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + g \\lambda_{2} \\left({a}\\right)^{2} {\\sigma_z} + g \\lambda_{2} {\\sigma_z} + g {{a}^\\dagger} {\\sigma_-} + g {{a}^\\dagger} {\\sigma_+} + g {a} {\\sigma_-} + g {a} {\\sigma_+} + \\lambda_{1}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} + \\frac{\\lambda_{1}^{2} \\omega}{2} {\\sigma_z} + \\lambda_{1} \\omega {{a}^\\dagger} {\\sigma_-} + \\lambda_{1} \\omega {a} {\\sigma_+} - \\lambda_{2}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\lambda_{2}^{2} \\omega}{2} {\\sigma_z} - \\lambda_{2} \\omega {{a}^\\dagger} {\\sigma_+} - \\lambda_{2} \\omega {a} {\\sigma_-} + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "                         2                     2                              \n",
       "      2  †           ε⋅λ₁ ⋅False           ⎛ †⎞                   2           \n",
       "- ε⋅λ₁ ⋅a ⋅a⋅False - ─────────── - ε⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅False - ε⋅λ₁⋅λ₂⋅a ⋅False - ε⋅\n",
       "                          2                                                   \n",
       "\n",
       "                                                    2                         \n",
       "    †                            2  †           ε⋅λ₂ ⋅False         †         \n",
       "λ₁⋅a ⋅False - ε⋅λ₁⋅a⋅False - ε⋅λ₂ ⋅a ⋅a⋅False - ─────────── - ε⋅λ₂⋅a ⋅False - \n",
       "                                                     2                        \n",
       "\n",
       "                                                                              \n",
       "               ε⋅False       2  †  2               2  †  2               2  † \n",
       "ε⋅λ₂⋅a⋅False + ─────── - g⋅λ₁ ⋅a ⋅a ⋅False - 2⋅g⋅λ₁ ⋅a ⋅a ⋅False - 2⋅g⋅λ₁ ⋅a ⋅\n",
       "                  2                                                           \n",
       "\n",
       "                                     2                     2                  \n",
       "            2  †               2 ⎛ †⎞                2 ⎛ †⎞                2 ⎛\n",
       "False - g⋅λ₁ ⋅a ⋅False - 2⋅g⋅λ₁ ⋅⎝a ⎠ ⋅a⋅False - g⋅λ₁ ⋅⎝a ⎠ ⋅a⋅False - g⋅λ₁ ⋅⎝\n",
       "                                                                              \n",
       "\n",
       "   3                                                                          \n",
       " †⎞              2                 2               2  3                    †  \n",
       "a ⎠ ⋅False - g⋅λ₁ ⋅a⋅False - 2⋅g⋅λ₁ ⋅a⋅False - g⋅λ₁ ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅a ⋅a\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "2                    †  2                    †                    †           \n",
       " ⋅False - 3⋅g⋅λ₁⋅λ₂⋅a ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅a ⋅False - 3⋅\n",
       "                                                                              \n",
       "\n",
       "            2                         2                       3               \n",
       "        ⎛ †⎞                      ⎛ †⎞                    ⎛ †⎞                \n",
       "g⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅a⋅False - 3⋅g⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅a⋅False - g⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅False - g⋅λ₁⋅λ\n",
       "                                                                              \n",
       "\n",
       "      3                                                                       \n",
       "  ⎛ †⎞                                                           3            \n",
       "₂⋅⎝a ⎠ ⋅False - 3⋅g⋅λ₁⋅λ₂⋅a⋅False - 3⋅g⋅λ₁⋅λ₂⋅a⋅False - g⋅λ₁⋅λ₂⋅a ⋅False - g⋅λ\n",
       "                                                                              \n",
       "\n",
       "                                             2                                \n",
       "      3                 †                ⎛ †⎞                2                \n",
       "₁⋅λ₂⋅a ⋅False + 2⋅g⋅λ₁⋅a ⋅a⋅False + g⋅λ₁⋅⎝a ⎠ ⋅False + g⋅λ₁⋅a ⋅False + g⋅λ₁⋅Fa\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "            2  †  2             2  †  2             2  †               2  †   \n",
       "lse - 2⋅g⋅λ₂ ⋅a ⋅a ⋅False - g⋅λ₂ ⋅a ⋅a ⋅False - g⋅λ₂ ⋅a ⋅False - 2⋅g⋅λ₂ ⋅a ⋅Fa\n",
       "                                                                              \n",
       "\n",
       "                2                       2                     3               \n",
       "          2 ⎛ †⎞                  2 ⎛ †⎞                2 ⎛ †⎞                \n",
       "lse - g⋅λ₂ ⋅⎝a ⎠ ⋅a⋅False - 2⋅g⋅λ₂ ⋅⎝a ⎠ ⋅a⋅False - g⋅λ₂ ⋅⎝a ⎠ ⋅False - 2⋅g⋅λ₂\n",
       "                                                                              \n",
       "\n",
       "                                                                          2   \n",
       "2               2               2  3                 †                ⎛ †⎞    \n",
       " ⋅a⋅False - g⋅λ₂ ⋅a⋅False - g⋅λ₂ ⋅a ⋅False + 2⋅g⋅λ₂⋅a ⋅a⋅False + g⋅λ₂⋅⎝a ⎠ ⋅Fa\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "            2                         †            †                          \n",
       "lse + g⋅λ₂⋅a ⋅False + g⋅λ₂⋅False + g⋅a ⋅False + g⋅a ⋅False + g⋅a⋅False + g⋅a⋅F\n",
       "                                                                              \n",
       "\n",
       "                            2                                                 \n",
       "         2    †           λ₁ ⋅ω⋅False         †                          2    \n",
       "alse + λ₁ ⋅ω⋅a ⋅a⋅False + ─────────── + λ₁⋅ω⋅a ⋅False + λ₁⋅ω⋅a⋅False - λ₂ ⋅ω⋅a\n",
       "                               2                                              \n",
       "\n",
       "              2                                                \n",
       "†           λ₂ ⋅ω⋅False         †                           †  \n",
       " ⋅a⋅False - ─────────── - λ₂⋅ω⋅a ⋅False - λ₂⋅ω⋅a⋅False + ω⋅a ⋅a\n",
       "                 2                                             "
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H3 = drop_c_number_terms(H2)\n",
    "H4 = qsimplify(H3)\n",
    "H4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Move to a frame co-rotating with the qubit-oscillator system."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Qm1NKzjL/UR5nDZAh8u+Ns8JG0FAHYT4PY1VWHASRJAfHhVjEafBTmAkCwca4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\n/w9uXeSLk0FBJQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\epsilon \\lambda_{1}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{1}^{2}}{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} e^{2 i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} e^{- 2 i \\omega t} \\left({a}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{1} e^{i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_+} - \\epsilon \\lambda_{1} e^{- i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} - \\epsilon \\lambda_{2}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{2}^{2}}{2} {\\sigma_z} - \\epsilon \\lambda_{2} e^{i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_+} - \\epsilon \\lambda_{2} e^{- i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_-} - g \\lambda_{1}^{2} e^{i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_+} - g \\lambda_{1}^{2} e^{i \\epsilon t} e^{i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - 2 g \\lambda_{1}^{2} e^{i \\epsilon t} e^{- i \\omega t} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 2 g \\lambda_{1}^{2} e^{i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_+} - g \\lambda_{1}^{2} e^{i \\epsilon t} e^{- 3 i \\omega t} \\left({a}\\right)^{3} {\\sigma_+} - g \\lambda_{1}^{2} e^{- i \\epsilon t} e^{3 i \\omega t} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - 2 g \\lambda_{1}^{2} e^{- i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} - 2 g \\lambda_{1}^{2} e^{- i \\epsilon t} e^{i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - g \\lambda_{1}^{2} e^{- i \\epsilon t} e^{- i \\omega t} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - g \\lambda_{1}^{2} e^{- i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_-} - g \\lambda_{1} \\lambda_{2} e^{i \\epsilon t} e^{3 i \\omega t} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} e^{i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} e^{i \\epsilon t} e^{i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} e^{i \\epsilon t} e^{- i \\omega t} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} e^{i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_+} - g \\lambda_{1} \\lambda_{2} e^{i \\epsilon t} e^{- 3 i \\omega t} \\left({a}\\right)^{3} {\\sigma_+} - g \\lambda_{1} \\lambda_{2} e^{- i \\epsilon t} e^{3 i \\omega t} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} e^{- i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} e^{- i \\epsilon t} e^{i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} e^{- i \\epsilon t} e^{- i \\omega t} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} e^{- i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_-} - g \\lambda_{1} \\lambda_{2} e^{- i \\epsilon t} e^{- 3 i \\omega t} \\left({a}\\right)^{3} {\\sigma_-} + g \\lambda_{1} e^{2 i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + 2 g \\lambda_{1} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{1} {\\sigma_z} + g \\lambda_{1} e^{- 2 i \\omega t} \\left({a}\\right)^{2} {\\sigma_z} - g \\lambda_{2}^{2} e^{i \\epsilon t} e^{3 i \\omega t} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} - 2 g \\lambda_{2}^{2} e^{i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_+} - 2 g \\lambda_{2}^{2} e^{i \\epsilon t} e^{i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - g \\lambda_{2}^{2} e^{i \\epsilon t} e^{- i \\omega t} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - g \\lambda_{2}^{2} e^{i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_+} - g \\lambda_{2}^{2} e^{- i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} - g \\lambda_{2}^{2} e^{- i \\epsilon t} e^{i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - 2 g \\lambda_{2}^{2} e^{- i \\epsilon t} e^{- i \\omega t} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - 2 g \\lambda_{2}^{2} e^{- i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_-} - g \\lambda_{2}^{2} e^{- i \\epsilon t} e^{- 3 i \\omega t} \\left({a}\\right)^{3} {\\sigma_-} + g \\lambda_{2} e^{2 i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + 2 g \\lambda_{2} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{2} {\\sigma_z} + g \\lambda_{2} e^{- 2 i \\omega t} \\left({a}\\right)^{2} {\\sigma_z} + g e^{i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_+} + g e^{i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_+} + g e^{- i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} + g e^{- i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_-} + \\lambda_{1}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} + \\frac{\\lambda_{1}^{2} \\omega}{2} {\\sigma_z} + \\lambda_{1} \\omega e^{i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_+} + \\lambda_{1} \\omega e^{- i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_-} - \\lambda_{2}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\lambda_{2}^{2} \\omega}{2} {\\sigma_z} - \\lambda_{2} \\omega e^{i \\epsilon t} e^{i \\omega t} {{a}^\\dagger} {\\sigma_+} - \\lambda_{2} \\omega e^{- i \\epsilon t} e^{- i \\omega t} {a} {\\sigma_-}$$"
      ],
      "text/plain": [
       "                         2                              2                     \n",
       "      2  †           ε⋅λ₁ ⋅False            2⋅ⅈ⋅ω⋅t ⎛ †⎞                   -2⋅\n",
       "- ε⋅λ₁ ⋅a ⋅a⋅False - ─────────── - ε⋅λ₁⋅λ₂⋅ℯ       ⋅⎝a ⎠ ⋅False - ε⋅λ₁⋅λ₂⋅ℯ   \n",
       "                          2                                                   \n",
       "\n",
       "                                                                              \n",
       "ⅈ⋅ω⋅t  2               ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t                 -ⅈ⋅ε⋅t  ⅈ⋅ω⋅t  †         \n",
       "     ⋅a ⋅False - ε⋅λ₁⋅ℯ     ⋅ℯ      ⋅a⋅False - ε⋅λ₁⋅ℯ      ⋅ℯ     ⋅a ⋅False - \n",
       "                                                                              \n",
       "\n",
       "                       2                                                      \n",
       "    2  †           ε⋅λ₂ ⋅False         ⅈ⋅ε⋅t  ⅈ⋅ω⋅t  †               -ⅈ⋅ε⋅t  -\n",
       "ε⋅λ₂ ⋅a ⋅a⋅False - ─────────── - ε⋅λ₂⋅ℯ     ⋅ℯ     ⋅a ⋅False - ε⋅λ₂⋅ℯ      ⋅ℯ \n",
       "                        2                                                     \n",
       "\n",
       "                                                                       2      \n",
       "ⅈ⋅ω⋅t               2  ⅈ⋅ε⋅t  ⅈ⋅ω⋅t  †             2  ⅈ⋅ε⋅t  ⅈ⋅ω⋅t ⎛ †⎞       \n",
       "     ⋅a⋅False - g⋅λ₁ ⋅ℯ     ⋅ℯ     ⋅a ⋅False - g⋅λ₁ ⋅ℯ     ⋅ℯ     ⋅⎝a ⎠ ⋅a⋅Fal\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "           2  ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t  †  2               2  ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t              \n",
       "se - 2⋅g⋅λ₁ ⋅ℯ     ⋅ℯ      ⋅a ⋅a ⋅False - 2⋅g⋅λ₁ ⋅ℯ     ⋅ℯ      ⋅a⋅False - g⋅λ\n",
       "                                                                              \n",
       "\n",
       "                                                          3                   \n",
       " 2  ⅈ⋅ε⋅t  -3⋅ⅈ⋅ω⋅t  3             2  -ⅈ⋅ε⋅t  3⋅ⅈ⋅ω⋅t ⎛ †⎞                2  -\n",
       "₁ ⋅ℯ     ⋅ℯ        ⋅a ⋅False - g⋅λ₁ ⋅ℯ      ⋅ℯ       ⋅⎝a ⎠ ⋅False - 2⋅g⋅λ₁ ⋅ℯ \n",
       "                                                                              \n",
       "\n",
       "                                                   2                          \n",
       "ⅈ⋅ε⋅t  ⅈ⋅ω⋅t  †               2  -ⅈ⋅ε⋅t  ⅈ⋅ω⋅t ⎛ †⎞                2  -ⅈ⋅ε⋅t  \n",
       "     ⋅ℯ     ⋅a ⋅False - 2⋅g⋅λ₁ ⋅ℯ      ⋅ℯ     ⋅⎝a ⎠ ⋅a⋅False - g⋅λ₁ ⋅ℯ      ⋅ℯ\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "-ⅈ⋅ω⋅t  †  2             2  -ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t                    ⅈ⋅ε⋅t  3⋅ⅈ⋅ω⋅t ⎛\n",
       "      ⋅a ⋅a ⋅False - g⋅λ₁ ⋅ℯ      ⋅ℯ      ⋅a⋅False - g⋅λ₁⋅λ₂⋅ℯ     ⋅ℯ       ⋅⎝\n",
       "                                                                              \n",
       "\n",
       "   3                                                                        2 \n",
       " †⎞                     ⅈ⋅ε⋅t  ⅈ⋅ω⋅t  †                    ⅈ⋅ε⋅t  ⅈ⋅ω⋅t ⎛ †⎞  \n",
       "a ⎠ ⋅False - 3⋅g⋅λ₁⋅λ₂⋅ℯ     ⋅ℯ     ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅ℯ     ⋅ℯ     ⋅⎝a ⎠ ⋅\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                     ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t  †  2                    ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t     \n",
       "a⋅False - 3⋅g⋅λ₁⋅λ₂⋅ℯ     ⋅ℯ      ⋅a ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅ℯ     ⋅ℯ      ⋅a⋅Fa\n",
       "                                                                              \n",
       "\n",
       "                                                                       3      \n",
       "               ⅈ⋅ε⋅t  -3⋅ⅈ⋅ω⋅t  3                  -ⅈ⋅ε⋅t  3⋅ⅈ⋅ω⋅t ⎛ †⎞       \n",
       "lse - g⋅λ₁⋅λ₂⋅ℯ     ⋅ℯ        ⋅a ⋅False - g⋅λ₁⋅λ₂⋅ℯ      ⋅ℯ       ⋅⎝a ⎠ ⋅False\n",
       "                                                                              \n",
       "\n",
       "                                                                    2         \n",
       "              -ⅈ⋅ε⋅t  ⅈ⋅ω⋅t  †                    -ⅈ⋅ε⋅t  ⅈ⋅ω⋅t ⎛ †⎞          \n",
       " - 3⋅g⋅λ₁⋅λ₂⋅ℯ      ⋅ℯ     ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅ℯ      ⋅ℯ     ⋅⎝a ⎠ ⋅a⋅False \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "             -ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t  †  2                    -ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t           \n",
       "- 3⋅g⋅λ₁⋅λ₂⋅ℯ      ⋅ℯ      ⋅a ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅ℯ      ⋅ℯ      ⋅a⋅False - \n",
       "                                                                              \n",
       "\n",
       "                                                       2                      \n",
       "         -ⅈ⋅ε⋅t  -3⋅ⅈ⋅ω⋅t  3               2⋅ⅈ⋅ω⋅t ⎛ †⎞                  †    \n",
       "g⋅λ₁⋅λ₂⋅ℯ      ⋅ℯ        ⋅a ⋅False + g⋅λ₁⋅ℯ       ⋅⎝a ⎠ ⋅False + 2⋅g⋅λ₁⋅a ⋅a⋅F\n",
       "                                                                              \n",
       "\n",
       "                                                                        3     \n",
       "                          -2⋅ⅈ⋅ω⋅t  2             2  ⅈ⋅ε⋅t  3⋅ⅈ⋅ω⋅t ⎛ †⎞      \n",
       "alse + g⋅λ₁⋅False + g⋅λ₁⋅ℯ        ⋅a ⋅False - g⋅λ₂ ⋅ℯ     ⋅ℯ       ⋅⎝a ⎠ ⋅Fals\n",
       "                                                                              \n",
       "\n",
       "                                                               2              \n",
       "          2  ⅈ⋅ε⋅t  ⅈ⋅ω⋅t  †               2  ⅈ⋅ε⋅t  ⅈ⋅ω⋅t ⎛ †⎞               \n",
       "e - 2⋅g⋅λ₂ ⋅ℯ     ⋅ℯ     ⋅a ⋅False - 2⋅g⋅λ₂ ⋅ℯ     ⋅ℯ     ⋅⎝a ⎠ ⋅a⋅False - g⋅λ\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       " 2  ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t  †  2             2  ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t               2  -ⅈ⋅ε⋅t  \n",
       "₂ ⋅ℯ     ⋅ℯ      ⋅a ⋅a ⋅False - g⋅λ₂ ⋅ℯ     ⋅ℯ      ⋅a⋅False - g⋅λ₂ ⋅ℯ      ⋅ℯ\n",
       "                                                                              \n",
       "\n",
       "                                          2                                   \n",
       "ⅈ⋅ω⋅t  †             2  -ⅈ⋅ε⋅t  ⅈ⋅ω⋅t ⎛ †⎞                  2  -ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t \n",
       "     ⋅a ⋅False - g⋅λ₂ ⋅ℯ      ⋅ℯ     ⋅⎝a ⎠ ⋅a⋅False - 2⋅g⋅λ₂ ⋅ℯ      ⋅ℯ      ⋅\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       " †  2               2  -ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t               2  -ⅈ⋅ε⋅t  -3⋅ⅈ⋅ω⋅t  3    \n",
       "a ⋅a ⋅False - 2⋅g⋅λ₂ ⋅ℯ      ⋅ℯ      ⋅a⋅False - g⋅λ₂ ⋅ℯ      ⋅ℯ        ⋅a ⋅Fal\n",
       "                                                                              \n",
       "\n",
       "                       2                                                      \n",
       "           2⋅ⅈ⋅ω⋅t ⎛ †⎞                  †                              -2⋅ⅈ⋅ω\n",
       "se + g⋅λ₂⋅ℯ       ⋅⎝a ⎠ ⋅False + 2⋅g⋅λ₂⋅a ⋅a⋅False + g⋅λ₂⋅False + g⋅λ₂⋅ℯ      \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "⋅t  2            ⅈ⋅ε⋅t  ⅈ⋅ω⋅t  †            ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t              -ⅈ⋅ε⋅t \n",
       "  ⋅a ⋅False + g⋅ℯ     ⋅ℯ     ⋅a ⋅False + g⋅ℯ     ⋅ℯ      ⋅a⋅False + g⋅ℯ      ⋅\n",
       "                                                                              \n",
       "\n",
       "                                                                   2          \n",
       " ⅈ⋅ω⋅t  †            -ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t             2    †           λ₁ ⋅ω⋅False  \n",
       "ℯ     ⋅a ⋅False + g⋅ℯ      ⋅ℯ      ⋅a⋅False + λ₁ ⋅ω⋅a ⋅a⋅False + ─────────── +\n",
       "                                                                      2       \n",
       "\n",
       "                                                                              \n",
       "       ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t                 -ⅈ⋅ε⋅t  ⅈ⋅ω⋅t  †           2    †        \n",
       " λ₁⋅ω⋅ℯ     ⋅ℯ      ⋅a⋅False + λ₁⋅ω⋅ℯ      ⋅ℯ     ⋅a ⋅False - λ₂ ⋅ω⋅a ⋅a⋅False\n",
       "                                                                              \n",
       "\n",
       "     2                                                                     \n",
       "   λ₂ ⋅ω⋅False         ⅈ⋅ε⋅t  ⅈ⋅ω⋅t  †               -ⅈ⋅ε⋅t  -ⅈ⋅ω⋅t        \n",
       " - ─────────── - λ₂⋅ω⋅ℯ     ⋅ℯ     ⋅a ⋅False - λ₂⋅ω⋅ℯ      ⋅ℯ      ⋅a⋅False\n",
       "        2                                                                  "
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H5 = hamiltonian_transformation(U1, H4)\n",
    "H6 = hamiltonian_transformation(U2, H5)\n",
    "\n",
    "H6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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7NuCfDQN4yFvYvlOp1/vT3PYAaE/HsPbvG+tzOWVPbqrTd3tszQCSVIcdItdI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\nGbjZGDjmzzZO3Lhx46g+hvz/qbqiLgYx4MsAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\epsilon \\lambda_{1}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{1}^{2}}{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} e^{2 i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} e^{- 2 i \\omega t} \\left({a}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{1} e^{i t \\left(- \\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_-} - \\epsilon \\lambda_{1} e^{i t \\left(\\epsilon - \\omega\\right)} {a} {\\sigma_+} - \\epsilon \\lambda_{2}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{2}^{2}}{2} {\\sigma_z} - \\epsilon \\lambda_{2} e^{i t \\left(\\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_+} - \\epsilon \\lambda_{2} e^{- i t \\left(\\epsilon + \\omega\\right)} {a} {\\sigma_-} - 2 g \\lambda_{1}^{2} e^{i t \\left(- \\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_-} - 2 g \\lambda_{1}^{2} e^{i t \\left(- \\epsilon + \\omega\\right)} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - g \\lambda_{1}^{2} e^{i t \\left(- \\epsilon + 3 \\omega\\right)} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - g \\lambda_{1}^{2} e^{i t \\left(\\epsilon - 3 \\omega\\right)} \\left({a}\\right)^{3} {\\sigma_+} - 2 g \\lambda_{1}^{2} e^{i t \\left(\\epsilon - \\omega\\right)} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 2 g \\lambda_{1}^{2} e^{i t \\left(\\epsilon - \\omega\\right)} {a} {\\sigma_+} - g \\lambda_{1}^{2} e^{i t \\left(\\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_+} - g \\lambda_{1}^{2} e^{i t \\left(\\epsilon + \\omega\\right)} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - g \\lambda_{1}^{2} e^{- i t \\left(\\epsilon + \\omega\\right)} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - g \\lambda_{1}^{2} e^{- i t \\left(\\epsilon + \\omega\\right)} {a} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} e^{i t \\left(- \\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} e^{i t \\left(- \\epsilon + \\omega\\right)} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - g \\lambda_{1} \\lambda_{2} e^{i t \\left(- \\epsilon + 3 \\omega\\right)} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - g \\lambda_{1} \\lambda_{2} e^{i t \\left(\\epsilon - 3 \\omega\\right)} \\left({a}\\right)^{3} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} e^{i t \\left(\\epsilon - \\omega\\right)} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} e^{i t \\left(\\epsilon - \\omega\\right)} {a} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} e^{i t \\left(\\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_+} - 3 g \\lambda_{1} \\lambda_{2} e^{i t \\left(\\epsilon + \\omega\\right)} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - g \\lambda_{1} \\lambda_{2} e^{i t \\left(\\epsilon + 3 \\omega\\right)} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} - g \\lambda_{1} \\lambda_{2} e^{- i t \\left(\\epsilon + 3 \\omega\\right)} \\left({a}\\right)^{3} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} e^{- i t \\left(\\epsilon + \\omega\\right)} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - 3 g \\lambda_{1} \\lambda_{2} e^{- i t \\left(\\epsilon + \\omega\\right)} {a} {\\sigma_-} + g \\lambda_{1} e^{2 i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + 2 g \\lambda_{1} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{1} {\\sigma_z} + g \\lambda_{1} e^{- 2 i \\omega t} \\left({a}\\right)^{2} {\\sigma_z} - g \\lambda_{2}^{2} e^{i t \\left(- \\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_-} - g \\lambda_{2}^{2} e^{i t \\left(- \\epsilon + \\omega\\right)} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - g \\lambda_{2}^{2} e^{i t \\left(\\epsilon - \\omega\\right)} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - g \\lambda_{2}^{2} e^{i t \\left(\\epsilon - \\omega\\right)} {a} {\\sigma_+} - 2 g \\lambda_{2}^{2} e^{i t \\left(\\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_+} - 2 g \\lambda_{2}^{2} e^{i t \\left(\\epsilon + \\omega\\right)} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_+} - g \\lambda_{2}^{2} e^{i t \\left(\\epsilon + 3 \\omega\\right)} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_+} - g \\lambda_{2}^{2} e^{- i t \\left(\\epsilon + 3 \\omega\\right)} \\left({a}\\right)^{3} {\\sigma_-} - 2 g \\lambda_{2}^{2} e^{- i t \\left(\\epsilon + \\omega\\right)} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_-} - 2 g \\lambda_{2}^{2} e^{- i t \\left(\\epsilon + \\omega\\right)} {a} {\\sigma_-} + g \\lambda_{2} e^{2 i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + 2 g \\lambda_{2} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{2} {\\sigma_z} + g \\lambda_{2} e^{- 2 i \\omega t} \\left({a}\\right)^{2} {\\sigma_z} + g e^{i t \\left(- \\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_-} + g e^{i t \\left(\\epsilon - \\omega\\right)} {a} {\\sigma_+} + g e^{i t \\left(\\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_+} + g e^{- i t \\left(\\epsilon + \\omega\\right)} {a} {\\sigma_-} + \\lambda_{1}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} + \\frac{\\lambda_{1}^{2} \\omega}{2} {\\sigma_z} + \\lambda_{1} \\omega e^{i t \\left(- \\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_-} + \\lambda_{1} \\omega e^{i t \\left(\\epsilon - \\omega\\right)} {a} {\\sigma_+} - \\lambda_{2}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\lambda_{2}^{2} \\omega}{2} {\\sigma_z} - \\lambda_{2} \\omega e^{i t \\left(\\epsilon + \\omega\\right)} {{a}^\\dagger} {\\sigma_+} - \\lambda_{2} \\omega e^{- i t \\left(\\epsilon + \\omega\\right)} {a} {\\sigma_-}$$"
      ],
      "text/plain": [
       "                         2                              2                     \n",
       "      2  †           ε⋅λ₁ ⋅False            2⋅ⅈ⋅ω⋅t ⎛ †⎞                   -2⋅\n",
       "- ε⋅λ₁ ⋅a ⋅a⋅False - ─────────── - ε⋅λ₁⋅λ₂⋅ℯ       ⋅⎝a ⎠ ⋅False - ε⋅λ₁⋅λ₂⋅ℯ   \n",
       "                          2                                                   \n",
       "\n",
       "                                                                              \n",
       "ⅈ⋅ω⋅t  2               ⅈ⋅t⋅(-ε + ω)  †               ⅈ⋅t⋅(ε - ω)              \n",
       "     ⋅a ⋅False - ε⋅λ₁⋅ℯ            ⋅a ⋅False - ε⋅λ₁⋅ℯ           ⋅a⋅False - ε⋅λ\n",
       "                                                                              \n",
       "\n",
       "                    2                                                         \n",
       " 2  †           ε⋅λ₂ ⋅False         ⅈ⋅t⋅(ε + ω)  †               -ⅈ⋅t⋅(ε + ω) \n",
       "₂ ⋅a ⋅a⋅False - ─────────── - ε⋅λ₂⋅ℯ           ⋅a ⋅False - ε⋅λ₂⋅ℯ            ⋅\n",
       "                     2                                                        \n",
       "\n",
       "                                                                     2        \n",
       "                2  ⅈ⋅t⋅(-ε + ω)  †               2  ⅈ⋅t⋅(-ε + ω) ⎛ †⎞         \n",
       "a⋅False - 2⋅g⋅λ₁ ⋅ℯ            ⋅a ⋅False - 2⋅g⋅λ₁ ⋅ℯ            ⋅⎝a ⎠ ⋅a⋅False\n",
       "                                                                              \n",
       "\n",
       "                             3                                                \n",
       "       2  ⅈ⋅t⋅(-ε + 3⋅ω) ⎛ †⎞              2  ⅈ⋅t⋅(ε - 3⋅ω)  3               2\n",
       " - g⋅λ₁ ⋅ℯ              ⋅⎝a ⎠ ⋅False - g⋅λ₁ ⋅ℯ             ⋅a ⋅False - 2⋅g⋅λ₁ \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "  ⅈ⋅t⋅(ε - ω)  †  2               2  ⅈ⋅t⋅(ε - ω)               2  ⅈ⋅t⋅(ε + ω) \n",
       "⋅ℯ           ⋅a ⋅a ⋅False - 2⋅g⋅λ₁ ⋅ℯ           ⋅a⋅False - g⋅λ₁ ⋅ℯ           ⋅\n",
       "                                                                              \n",
       "\n",
       "                                  2                                           \n",
       " †             2  ⅈ⋅t⋅(ε + ω) ⎛ †⎞                2  -ⅈ⋅t⋅(ε + ω)  †  2       \n",
       "a ⋅False - g⋅λ₁ ⋅ℯ           ⋅⎝a ⎠ ⋅a⋅False - g⋅λ₁ ⋅ℯ            ⋅a ⋅a ⋅False \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "      2  -ⅈ⋅t⋅(ε + ω)                      ⅈ⋅t⋅(-ε + ω)  †                    \n",
       "- g⋅λ₁ ⋅ℯ            ⋅a⋅False - 3⋅g⋅λ₁⋅λ₂⋅ℯ            ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅ℯ\n",
       "                                                                              \n",
       "\n",
       "                 2                                       3                    \n",
       "ⅈ⋅t⋅(-ε + ω) ⎛ †⎞                     ⅈ⋅t⋅(-ε + 3⋅ω) ⎛ †⎞                   ⅈ⋅\n",
       "            ⋅⎝a ⎠ ⋅a⋅False - g⋅λ₁⋅λ₂⋅ℯ              ⋅⎝a ⎠ ⋅False - g⋅λ₁⋅λ₂⋅ℯ  \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "t⋅(ε - 3⋅ω)  3                    ⅈ⋅t⋅(ε - ω)  †  2                    ⅈ⋅t⋅(ε \n",
       "           ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅ℯ           ⋅a ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅ℯ       \n",
       "                                                                              \n",
       "\n",
       "                                                                            2 \n",
       "- ω)                      ⅈ⋅t⋅(ε + ω)  †                    ⅈ⋅t⋅(ε + ω) ⎛ †⎞  \n",
       "    ⋅a⋅False - 3⋅g⋅λ₁⋅λ₂⋅ℯ           ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅ℯ           ⋅⎝a ⎠ ⋅\n",
       "                                                                              \n",
       "\n",
       "                                     3                                        \n",
       "                   ⅈ⋅t⋅(ε + 3⋅ω) ⎛ †⎞                   -ⅈ⋅t⋅(ε + 3⋅ω)  3     \n",
       "a⋅False - g⋅λ₁⋅λ₂⋅ℯ             ⋅⎝a ⎠ ⋅False - g⋅λ₁⋅λ₂⋅ℯ              ⋅a ⋅Fals\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "               -ⅈ⋅t⋅(ε + ω)  †  2                    -ⅈ⋅t⋅(ε + ω)             \n",
       "e - 3⋅g⋅λ₁⋅λ₂⋅ℯ            ⋅a ⋅a ⋅False - 3⋅g⋅λ₁⋅λ₂⋅ℯ            ⋅a⋅False + g⋅\n",
       "                                                                              \n",
       "\n",
       "                2                                                             \n",
       "    2⋅ⅈ⋅ω⋅t ⎛ †⎞                  †                              -2⋅ⅈ⋅ω⋅t  2  \n",
       "λ₁⋅ℯ       ⋅⎝a ⎠ ⋅False + 2⋅g⋅λ₁⋅a ⋅a⋅False + g⋅λ₁⋅False + g⋅λ₁⋅ℯ        ⋅a ⋅F\n",
       "                                                                              \n",
       "\n",
       "                                                              2               \n",
       "           2  ⅈ⋅t⋅(-ε + ω)  †             2  ⅈ⋅t⋅(-ε + ω) ⎛ †⎞                \n",
       "alse - g⋅λ₂ ⋅ℯ            ⋅a ⋅False - g⋅λ₂ ⋅ℯ            ⋅⎝a ⎠ ⋅a⋅False - g⋅λ₂\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "2  ⅈ⋅t⋅(ε - ω)  †  2             2  ⅈ⋅t⋅(ε - ω)                 2  ⅈ⋅t⋅(ε + ω)\n",
       " ⋅ℯ           ⋅a ⋅a ⋅False - g⋅λ₂ ⋅ℯ           ⋅a⋅False - 2⋅g⋅λ₂ ⋅ℯ           \n",
       "                                                                              \n",
       "\n",
       "                                     2                                    3   \n",
       "  †               2  ⅈ⋅t⋅(ε + ω) ⎛ †⎞                2  ⅈ⋅t⋅(ε + 3⋅ω) ⎛ †⎞    \n",
       "⋅a ⋅False - 2⋅g⋅λ₂ ⋅ℯ           ⋅⎝a ⎠ ⋅a⋅False - g⋅λ₂ ⋅ℯ             ⋅⎝a ⎠ ⋅Fa\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "          2  -ⅈ⋅t⋅(ε + 3⋅ω)  3               2  -ⅈ⋅t⋅(ε + ω)  †  2            \n",
       "lse - g⋅λ₂ ⋅ℯ              ⋅a ⋅False - 2⋅g⋅λ₂ ⋅ℯ            ⋅a ⋅a ⋅False - 2⋅g\n",
       "                                                                              \n",
       "\n",
       "                                               2                              \n",
       "   2  -ⅈ⋅t⋅(ε + ω)                 2⋅ⅈ⋅ω⋅t ⎛ †⎞                  †            \n",
       "⋅λ₂ ⋅ℯ            ⋅a⋅False + g⋅λ₂⋅ℯ       ⋅⎝a ⎠ ⋅False + 2⋅g⋅λ₂⋅a ⋅a⋅False + g\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "                  -2⋅ⅈ⋅ω⋅t  2            ⅈ⋅t⋅(-ε + ω)  †            ⅈ⋅t⋅(ε - ω\n",
       "⋅λ₂⋅False + g⋅λ₂⋅ℯ        ⋅a ⋅False + g⋅ℯ            ⋅a ⋅False + g⋅ℯ          \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       ")              ⅈ⋅t⋅(ε + ω)  †            -ⅈ⋅t⋅(ε + ω)             2    †      \n",
       " ⋅a⋅False + g⋅ℯ           ⋅a ⋅False + g⋅ℯ            ⋅a⋅False + λ₁ ⋅ω⋅a ⋅a⋅Fal\n",
       "                                                                              \n",
       "\n",
       "       2                                                                      \n",
       "     λ₁ ⋅ω⋅False         ⅈ⋅t⋅(-ε + ω)  †               ⅈ⋅t⋅(ε - ω)            \n",
       "se + ─────────── + λ₁⋅ω⋅ℯ            ⋅a ⋅False + λ₁⋅ω⋅ℯ           ⋅a⋅False - λ\n",
       "          2                                                                   \n",
       "\n",
       "                    2                                                         \n",
       " 2    †           λ₂ ⋅ω⋅False         ⅈ⋅t⋅(ε + ω)  †               -ⅈ⋅t⋅(ε + ω\n",
       "₂ ⋅ω⋅a ⋅a⋅False - ─────────── - λ₂⋅ω⋅ℯ           ⋅a ⋅False - λ₂⋅ω⋅ℯ           \n",
       "                       2                                                      \n",
       "\n",
       "         \n",
       ")        \n",
       " ⋅a⋅False\n",
       "         "
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H6 = powsimp(H6)\n",
    "H7 = simplify_exp(H6)\n",
    "\n",
    "H7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since we have assumed that the detuning $\\Delta = \\epsilon-\\omega$ is much larger than the coupling frequency $g$, it is okay to ignore terms oscillating fast in the co-rotating system. i.e., $e^{\\pm i(\\epsilon-\\omega)t}, e^{\\pm i(\\epsilon+\\omega)t}, e^{\\pm i(\\epsilon+3\\omega)t}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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CRcLY077sTLTfIbd6S9AM0z2zKal/WR6Mq902CcmGcymyuAgopeNBb+O3Avkn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\nSZgrQlQUAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\epsilon \\lambda_{1}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{1}^{2}}{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} e^{2 i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} e^{- 2 i \\omega t} \\left({a}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{2}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{2}^{2}}{2} {\\sigma_z} - g \\lambda_{1}^{2} e^{i t \\left(- \\epsilon + 3 \\omega\\right)} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - g \\lambda_{1}^{2} e^{i t \\left(\\epsilon - 3 \\omega\\right)} \\left({a}\\right)^{3} {\\sigma_+} - g \\lambda_{1} \\lambda_{2} e^{i t \\left(- \\epsilon + 3 \\omega\\right)} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - g \\lambda_{1} \\lambda_{2} e^{i t \\left(\\epsilon - 3 \\omega\\right)} \\left({a}\\right)^{3} {\\sigma_+} + g \\lambda_{1} e^{2 i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + 2 g \\lambda_{1} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{1} {\\sigma_z} + g \\lambda_{1} e^{- 2 i \\omega t} \\left({a}\\right)^{2} {\\sigma_z} + g \\lambda_{2} e^{2 i \\omega t} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + 2 g \\lambda_{2} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{2} {\\sigma_z} + g \\lambda_{2} e^{- 2 i \\omega t} \\left({a}\\right)^{2} {\\sigma_z} + \\lambda_{1}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} + \\frac{\\lambda_{1}^{2} \\omega}{2} {\\sigma_z} - \\lambda_{2}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\lambda_{2}^{2} \\omega}{2} {\\sigma_z}$$"
      ],
      "text/plain": [
       "                         2                              2                     \n",
       "      2  †           ε⋅λ₁ ⋅False            2⋅ⅈ⋅ω⋅t ⎛ †⎞                   -2⋅\n",
       "- ε⋅λ₁ ⋅a ⋅a⋅False - ─────────── - ε⋅λ₁⋅λ₂⋅ℯ       ⋅⎝a ⎠ ⋅False - ε⋅λ₁⋅λ₂⋅ℯ   \n",
       "                          2                                                   \n",
       "\n",
       "                                        2                                   3 \n",
       "ⅈ⋅ω⋅t  2             2  †           ε⋅λ₂ ⋅False       2  ⅈ⋅t⋅(-ε + 3⋅ω) ⎛ †⎞  \n",
       "     ⋅a ⋅False - ε⋅λ₂ ⋅a ⋅a⋅False - ─────────── - g⋅λ₁ ⋅ℯ              ⋅⎝a ⎠ ⋅\n",
       "                                         2                                    \n",
       "\n",
       "                                                                    3         \n",
       "            2  ⅈ⋅t⋅(ε - 3⋅ω)  3                  ⅈ⋅t⋅(-ε + 3⋅ω) ⎛ †⎞          \n",
       "False - g⋅λ₁ ⋅ℯ             ⋅a ⋅False - g⋅λ₁⋅λ₂⋅ℯ              ⋅⎝a ⎠ ⋅False - \n",
       "                                                                              \n",
       "\n",
       "                                                    2                         \n",
       "         ⅈ⋅t⋅(ε - 3⋅ω)  3               2⋅ⅈ⋅ω⋅t ⎛ †⎞                  †       \n",
       "g⋅λ₁⋅λ₂⋅ℯ             ⋅a ⋅False + g⋅λ₁⋅ℯ       ⋅⎝a ⎠ ⋅False + 2⋅g⋅λ₁⋅a ⋅a⋅Fals\n",
       "                                                                              \n",
       "\n",
       "                                                             2                \n",
       "                       -2⋅ⅈ⋅ω⋅t  2               2⋅ⅈ⋅ω⋅t ⎛ †⎞                 \n",
       "e + g⋅λ₁⋅False + g⋅λ₁⋅ℯ        ⋅a ⋅False + g⋅λ₂⋅ℯ       ⋅⎝a ⎠ ⋅False + 2⋅g⋅λ₂⋅\n",
       "                                                                              \n",
       "\n",
       "                                                                         2    \n",
       " †                              -2⋅ⅈ⋅ω⋅t  2           2    †           λ₁ ⋅ω⋅F\n",
       "a ⋅a⋅False + g⋅λ₂⋅False + g⋅λ₂⋅ℯ        ⋅a ⋅False + λ₁ ⋅ω⋅a ⋅a⋅False + ───────\n",
       "                                                                            2 \n",
       "\n",
       "                            2        \n",
       "alse     2    †           λ₂ ⋅ω⋅False\n",
       "──── - λ₂ ⋅ω⋅a ⋅a⋅False - ───────────\n",
       "                               2     "
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H8 = drop_terms_containing(H7, [exp(I * (e-w)* t), exp(I * (-e+w)* t),\n",
    "                                exp(I * (e+w)* t), exp(-I * (e+w)* t),\n",
    "                                exp(I * (e+3*w)* t), exp(-I * (e+3*w)* t),\n",
    "                                ])\n",
    "H8 = drop_c_number_terms(H8)\n",
    "H8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The terms associated with the exponents $e^{\\pm2i\\omega t}, e^{\\pm i(\\epsilon-3\\omega)}$ are not to be neglected in this case, because we are only with the large detuning condition $g\\ll |\\epsilon-\\omega|$. (For instance, if $\\omega\\ll g$, $e^{\\pm 2i\\omega t}$ term is oscillating slowly and cannot be ignored. In addition, even if $\\omega\\gg g$, this term has no significant effect whether or not we include in the sum. So, it is necessary that we include this term in the summation.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We move back to the lab frame:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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JoeKRAD6iHcNr0fLAC8q+XmeERAfJTxItITm0mFoGktHAEL0W/0lxkWj4QH5L/IotHg6d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\n",
      "text/latex": [
       "$$- \\epsilon \\lambda_{1}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{1}^{2}}{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{1} \\lambda_{2} \\left({a}\\right)^{2} {\\sigma_z} - \\epsilon \\lambda_{2}^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon \\lambda_{2}^{2}}{2} {\\sigma_z} + \\frac{\\epsilon {\\sigma_z}}{2} - g \\lambda_{1}^{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - g \\lambda_{1}^{2} \\left({a}\\right)^{3} {\\sigma_+} - g \\lambda_{1} \\lambda_{2} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-} - g \\lambda_{1} \\lambda_{2} \\left({a}\\right)^{3} {\\sigma_+} + 2 g \\lambda_{1} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{1} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + g \\lambda_{1} \\left({a}\\right)^{2} {\\sigma_z} + g \\lambda_{1} {\\sigma_z} + 2 g \\lambda_{2} {{a}^\\dagger} {a} {\\sigma_z} + g \\lambda_{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z} + g \\lambda_{2} \\left({a}\\right)^{2} {\\sigma_z} + g \\lambda_{2} {\\sigma_z} + \\lambda_{1}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} + \\frac{\\lambda_{1}^{2} \\omega}{2} {\\sigma_z} - \\lambda_{2}^{2} \\omega {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\lambda_{2}^{2} \\omega}{2} {\\sigma_z} + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "                         2                     2                              \n",
       "      2  †           ε⋅λ₁ ⋅False           ⎛ †⎞                   2           \n",
       "- ε⋅λ₁ ⋅a ⋅a⋅False - ─────────── - ε⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅False - ε⋅λ₁⋅λ₂⋅a ⋅False - ε⋅\n",
       "                          2                                                   \n",
       "\n",
       "                     2                             3                          \n",
       "  2  †           ε⋅λ₂ ⋅False   ε⋅False       2 ⎛ †⎞              2  3         \n",
       "λ₂ ⋅a ⋅a⋅False - ─────────── + ─────── - g⋅λ₁ ⋅⎝a ⎠ ⋅False - g⋅λ₁ ⋅a ⋅False - \n",
       "                      2           2                                           \n",
       "\n",
       "            3                                                         2       \n",
       "        ⎛ †⎞                   3                 †                ⎛ †⎞        \n",
       "g⋅λ₁⋅λ₂⋅⎝a ⎠ ⋅False - g⋅λ₁⋅λ₂⋅a ⋅False + 2⋅g⋅λ₁⋅a ⋅a⋅False + g⋅λ₁⋅⎝a ⎠ ⋅False \n",
       "                                                                              \n",
       "\n",
       "                                                            2                 \n",
       "        2                              †                ⎛ †⎞                2 \n",
       "+ g⋅λ₁⋅a ⋅False + g⋅λ₁⋅False + 2⋅g⋅λ₂⋅a ⋅a⋅False + g⋅λ₂⋅⎝a ⎠ ⋅False + g⋅λ₂⋅a ⋅\n",
       "                                                                              \n",
       "\n",
       "                                          2                                2  \n",
       "                       2    †           λ₁ ⋅ω⋅False     2    †           λ₂ ⋅ω\n",
       "False + g⋅λ₂⋅False + λ₁ ⋅ω⋅a ⋅a⋅False + ─────────── - λ₂ ⋅ω⋅a ⋅a⋅False - ─────\n",
       "                                             2                                \n",
       "\n",
       "               \n",
       "⋅False      †  \n",
       "────── + ω⋅a ⋅a\n",
       "2              "
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H9  = hamiltonian_transformation(U3, H8)\n",
    "H10 = hamiltonian_transformation(U4, H9)\n",
    "\n",
    "H10 = qsimplify(H10.expand())\n",
    "\n",
    "H10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "D_ = Symbol(\"\\Delta'\", real=True) # D_ = e + w = 2 * e - D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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GHS9GqnU1Q5LYbZuOUxaaJWYkL+F/fV8vuldkyGiwK1tzjzXqrA4Aya1GeOcm\nwPZDG5DE6k2uKJWzi4T0KUB4W7C71WRd0f9+mlypLPzevep6brOSPWLrQ7ehdn5BWfhHxyPVqpoh\nCdyuSccpC80SnRjpfuuFe34aiu4VGSTR1NaHbkPVB8DJrUeQIxSoDmy/2gakY/UWV5TK2eWFdClA\neGPAetXkXTH/fto7U1V6zryAEL/46bn60G2oZfuKxJWhgRnSkJladJyy0CyR5AiFx0+c+F8ouldk\nECT7pj50G6o+AE5uPWLoSQe2X20D0rF6iytK5ezyQroUILwxYL1qUq78H9+yo5Tr10GpAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{\\epsilon {\\sigma_z}}{2} + \\omega {{a}^\\dagger} {a} + \\frac{2 g^{2}}{\\Delta'} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\left({{a}^\\dagger}\\right)^{2}}{\\Delta'} {\\sigma_z} + \\frac{g^{2} \\left({a}\\right)^{2}}{\\Delta'} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta'} - \\frac{\\epsilon g^{2}}{\\Delta'^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon g^{2} {\\sigma_z}}{2 \\Delta'^{2}} - \\frac{g^{2} \\omega}{\\Delta'^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{g^{2} \\omega {\\sigma_z}}{2 \\Delta'^{2}} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\left({{a}^\\dagger}\\right)^{2}}{\\Delta} {\\sigma_z} + \\frac{g^{2} \\left({a}\\right)^{2}}{\\Delta} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} - \\frac{\\epsilon g^{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z}}{\\Delta \\Delta'} - \\frac{\\epsilon g^{2} \\left({a}\\right)^{2} {\\sigma_z}}{\\Delta \\Delta'} - \\frac{g^{3} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-}}{\\Delta \\Delta'} - \\frac{g^{3} \\left({a}\\right)^{3} {\\sigma_+}}{\\Delta \\Delta'} - \\frac{\\epsilon g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon g^{2} {\\sigma_z}}{2 \\Delta^{2}} - \\frac{g^{3} \\left({{a}^\\dagger}\\right)^{3}}{\\Delta^{2}} {\\sigma_-} - \\frac{g^{3} \\left({a}\\right)^{3}}{\\Delta^{2}} {\\sigma_+} + \\frac{g^{2} \\omega}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega {\\sigma_z}}{2 \\Delta^{2}}$$"
      ],
      "text/plain": [
       "                                            2                                 \n",
       "                      2  †            2 ⎛ †⎞           2  2          2        \n",
       "ε⋅False      †     2⋅g ⋅a ⋅a⋅False   g ⋅⎝a ⎠ ⋅False   g ⋅a ⋅False   g ⋅False  \n",
       "─────── + ω⋅a ⋅a + ─────────────── + ────────────── + ─────────── + ──────── -\n",
       "   2                   \\Delta'          \\Delta'         \\Delta'     \\Delta'   \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "    2  †              2          2    †            2              2  †        \n",
       " ε⋅g ⋅a ⋅a⋅False   ε⋅g ⋅False   g ⋅ω⋅a ⋅a⋅False   g ⋅ω⋅False   2⋅g ⋅a ⋅a⋅False\n",
       " ─────────────── - ────────── - ─────────────── - ────────── + ───────────────\n",
       "            2               2              2               2          Δ       \n",
       "     \\Delta'       2⋅\\Delta'        \\Delta'       2⋅\\Delta'                   \n",
       "\n",
       "          2                                           2                       \n",
       "    2 ⎛ †⎞           2  2          2            2 ⎛ †⎞             2  2       \n",
       "   g ⋅⎝a ⎠ ⋅False   g ⋅a ⋅False   g ⋅False   ε⋅g ⋅⎝a ⎠ ⋅False   ε⋅g ⋅a ⋅False \n",
       " + ────────────── + ─────────── + ──────── - ──────────────── - ───────────── \n",
       "         Δ               Δ           Δ          Δ⋅\\Delta'         Δ⋅\\Delta'   \n",
       "                                                                              \n",
       "\n",
       "         3                                                             3      \n",
       "   3 ⎛ †⎞           3  3            2  †              2          3 ⎛ †⎞       \n",
       "  g ⋅⎝a ⎠ ⋅False   g ⋅a ⋅False   ε⋅g ⋅a ⋅a⋅False   ε⋅g ⋅False   g ⋅⎝a ⎠ ⋅False\n",
       "- ────────────── - ─────────── - ─────────────── - ────────── - ──────────────\n",
       "    Δ⋅\\Delta'       Δ⋅\\Delta'            2               2             2      \n",
       "                                        Δ             2⋅Δ             Δ       \n",
       "\n",
       "                                             \n",
       "    3  3          2    †            2        \n",
       "   g ⋅a ⋅False   g ⋅ω⋅a ⋅a⋅False   g ⋅ω⋅False\n",
       " - ─────────── + ─────────────── + ──────────\n",
       "         2               2               2   \n",
       "        Δ               Δ             2⋅Δ    "
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H10.subs({l1: g/D, l2: g/D_})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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GHS9GqnU1Q5LYbZuOUxaaJWYkL+F/fV8vuldkyGiwK1tzjzXqrA4Aya1GeOcm\nwPZDG5DE6k2uKJWzi4T0KUB4W7C71WRd0f9+mlypLPzevep6brOSPWLrQ7ehdn5BWfhHxyPVqpoh\nCdyuSccpC80SnRjpfuuFe34aiu4VGSTR1NaHbkPVB8DJrUeQIxSoDmy/2gakY/UWV5TK2eWFdClA\neGPAetXkXTH/fto7U1V6zryAEL/46bn60G2oZfuKxJWhgRnSkJladJyy0CyR5AiFx0+c+F8ouldk\nECT7pj50G6o+AE5uPWLoSQe2X20D0rF6iytK5ezyQroUILwxYL1qUq78H9+yo5Tr10GpAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{\\epsilon {\\sigma_z}}{2} + \\omega {{a}^\\dagger} {a} + \\frac{2 g^{2}}{\\Delta'} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\left({{a}^\\dagger}\\right)^{2}}{\\Delta'} {\\sigma_z} + \\frac{g^{2} \\left({a}\\right)^{2}}{\\Delta'} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta'} - \\frac{\\epsilon g^{2}}{\\Delta'^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon g^{2} {\\sigma_z}}{2 \\Delta'^{2}} - \\frac{g^{2} \\omega}{\\Delta'^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{g^{2} \\omega {\\sigma_z}}{2 \\Delta'^{2}} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\left({{a}^\\dagger}\\right)^{2}}{\\Delta} {\\sigma_z} + \\frac{g^{2} \\left({a}\\right)^{2}}{\\Delta} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} - \\frac{\\epsilon g^{2} \\left({{a}^\\dagger}\\right)^{2} {\\sigma_z}}{\\Delta \\Delta'} - \\frac{\\epsilon g^{2} \\left({a}\\right)^{2} {\\sigma_z}}{\\Delta \\Delta'} - \\frac{g^{3} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-}}{\\Delta \\Delta'} - \\frac{g^{3} \\left({a}\\right)^{3} {\\sigma_+}}{\\Delta \\Delta'} - \\frac{\\epsilon g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\epsilon g^{2} {\\sigma_z}}{2 \\Delta^{2}} - \\frac{g^{3} \\left({{a}^\\dagger}\\right)^{3}}{\\Delta^{2}} {\\sigma_-} - \\frac{g^{3} \\left({a}\\right)^{3}}{\\Delta^{2}} {\\sigma_+} + \\frac{g^{2} \\omega}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega {\\sigma_z}}{2 \\Delta^{2}}$$"
      ],
      "text/plain": [
       "                                            2                                 \n",
       "                      2  †            2 ⎛ †⎞           2  2          2        \n",
       "ε⋅False      †     2⋅g ⋅a ⋅a⋅False   g ⋅⎝a ⎠ ⋅False   g ⋅a ⋅False   g ⋅False  \n",
       "─────── + ω⋅a ⋅a + ─────────────── + ────────────── + ─────────── + ──────── -\n",
       "   2                   \\Delta'          \\Delta'         \\Delta'     \\Delta'   \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "    2  †              2          2    †            2              2  †        \n",
       " ε⋅g ⋅a ⋅a⋅False   ε⋅g ⋅False   g ⋅ω⋅a ⋅a⋅False   g ⋅ω⋅False   2⋅g ⋅a ⋅a⋅False\n",
       " ─────────────── - ────────── - ─────────────── - ────────── + ───────────────\n",
       "            2               2              2               2          Δ       \n",
       "     \\Delta'       2⋅\\Delta'        \\Delta'       2⋅\\Delta'                   \n",
       "\n",
       "          2                                           2                       \n",
       "    2 ⎛ †⎞           2  2          2            2 ⎛ †⎞             2  2       \n",
       "   g ⋅⎝a ⎠ ⋅False   g ⋅a ⋅False   g ⋅False   ε⋅g ⋅⎝a ⎠ ⋅False   ε⋅g ⋅a ⋅False \n",
       " + ────────────── + ─────────── + ──────── - ──────────────── - ───────────── \n",
       "         Δ               Δ           Δ          Δ⋅\\Delta'         Δ⋅\\Delta'   \n",
       "                                                                              \n",
       "\n",
       "         3                                                             3      \n",
       "   3 ⎛ †⎞           3  3            2  †              2          3 ⎛ †⎞       \n",
       "  g ⋅⎝a ⎠ ⋅False   g ⋅a ⋅False   ε⋅g ⋅a ⋅a⋅False   ε⋅g ⋅False   g ⋅⎝a ⎠ ⋅False\n",
       "- ────────────── - ─────────── - ─────────────── - ────────── - ──────────────\n",
       "    Δ⋅\\Delta'       Δ⋅\\Delta'            2               2             2      \n",
       "                                        Δ             2⋅Δ             Δ       \n",
       "\n",
       "                                             \n",
       "    3  3          2    †            2        \n",
       "   g ⋅a ⋅False   g ⋅ω⋅a ⋅a⋅False   g ⋅ω⋅False\n",
       " - ─────────── + ─────────────── + ──────────\n",
       "         2               2               2   \n",
       "        Δ               Δ             2⋅Δ    "
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H11 = H10.subs({l1: g/D, l2: g/D_})\n",
    "H11"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Me7T+68Ja+a0LeQ6slHCAUiqFUlyuaJItdcgYBqn7Vrs/96PY0Tt1c66ckT0bSEbzG3iP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\nTtNpBdbquo31sgos28XlxNGVK+nQ/z/pmbsTuG3usQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\epsilon {\\sigma_z}}{2} + \\omega {{a}^\\dagger} {a} + \\frac{g^{2} {\\sigma_z}}{\\Delta'} \\left(1 + 2 {{a}^\\dagger} {a} + \\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right) + \\frac{g^{2} {\\sigma_z}}{\\Delta'^{2}} \\left(- \\frac{\\epsilon}{2} - \\epsilon {{a}^\\dagger} {a} - \\frac{\\omega}{2} - \\omega {{a}^\\dagger} {a}\\right) + \\frac{g^{2} {\\sigma_z}}{\\Delta} \\left(1 + 2 {{a}^\\dagger} {a} + \\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right) + \\frac{\\epsilon g^{2} {\\sigma_z}}{\\Delta \\Delta'} \\left(- \\left({{a}^\\dagger}\\right)^{2} - \\left({a}\\right)^{2}\\right) - \\frac{g^{3} \\left({{a}^\\dagger}\\right)^{3} {\\sigma_-}}{\\Delta \\Delta'} - \\frac{g^{3} \\left({a}\\right)^{3} {\\sigma_+}}{\\Delta \\Delta'} - \\frac{g^{3} \\left({{a}^\\dagger}\\right)^{3}}{\\Delta^{2}} {\\sigma_-} - \\frac{g^{3} \\left({a}\\right)^{3}}{\\Delta^{2}} {\\sigma_+} + \\frac{g^{2} {\\sigma_z}}{\\Delta^{2}} \\left(- \\frac{\\epsilon}{2} - \\epsilon {{a}^\\dagger} {a} + \\frac{\\omega}{2} + \\omega {{a}^\\dagger} {a}\\right)$$"
      ],
      "text/plain": [
       "                            ⎛                 2     ⎞    2       ⎛  ε      †  \n",
       "                    2       ⎜       †     ⎛ †⎞     2⎟   g ⋅False⋅⎜- ─ - ε⋅a ⋅a\n",
       "ε⋅False      †     g ⋅False⋅⎝1 + 2⋅a ⋅a + ⎝a ⎠  + a ⎠            ⎝  2         \n",
       "─────── + ω⋅a ⋅a + ────────────────────────────────── + ──────────────────────\n",
       "   2                            \\Delta'                                      2\n",
       "                                                                      \\Delta' \n",
       "\n",
       "   ω      †  ⎞            ⎛                 2     ⎞              ⎛      2     \n",
       " - ─ - ω⋅a ⋅a⎟    2       ⎜       †     ⎛ †⎞     2⎟      2       ⎜  ⎛ †⎞     2\n",
       "   2         ⎠   g ⋅False⋅⎝1 + 2⋅a ⋅a + ⎝a ⎠  + a ⎠   ε⋅g ⋅False⋅⎝- ⎝a ⎠  - a \n",
       "────────────── + ────────────────────────────────── + ────────────────────────\n",
       "                                 Δ                            Δ⋅\\Delta'       \n",
       "                                                                              \n",
       "\n",
       "⎞          3                              3                        2       ⎛  \n",
       "⎟    3 ⎛ †⎞           3  3          3 ⎛ †⎞           3  3         g ⋅False⋅⎜- \n",
       "⎠   g ⋅⎝a ⎠ ⋅False   g ⋅a ⋅False   g ⋅⎝a ⎠ ⋅False   g ⋅a ⋅False            ⎝  \n",
       "─ - ────────────── - ─────────── - ────────────── - ─────────── + ────────────\n",
       "      Δ⋅\\Delta'       Δ⋅\\Delta'           2               2                   \n",
       "                                         Δ               Δ                    \n",
       "\n",
       "ε      †     ω      †  ⎞\n",
       "─ - ε⋅a ⋅a + ─ + ω⋅a ⋅a⎟\n",
       "2            2         ⎠\n",
       "────────────────────────\n",
       "      2                 \n",
       "     Δ                  "
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H12 = collect(H11, [g**2*sz*e/(D*D_), \n",
    "                    g**2*sz/D_, g**2*sz/D])\n",
    "\n",
    "H12"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The terms containing $a^3, (a^\\dagger)^3$ has a higher order of $\\lambda_1 = g/\\Delta, \\lambda_2 = g/(2\\epsilon-\\Delta)$, so that these can be ignored:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\frac{\\epsilon {\\sigma_z}}{2} + \\omega {{a}^\\dagger} {a} + \\frac{g^{2} {\\sigma_z}}{\\Delta'} \\left(1 + 2 {{a}^\\dagger} {a} + \\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right) + \\frac{g^{2} {\\sigma_z}}{\\Delta'^{2}} \\left(- \\frac{\\epsilon}{2} - \\epsilon {{a}^\\dagger} {a} - \\frac{\\omega}{2} - \\omega {{a}^\\dagger} {a}\\right) + \\frac{g^{2} {\\sigma_z}}{\\Delta} \\left(1 + 2 {{a}^\\dagger} {a} + \\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right) + \\frac{\\epsilon g^{2} {\\sigma_z}}{\\Delta \\Delta'} \\left(- \\left({{a}^\\dagger}\\right)^{2} - \\left({a}\\right)^{2}\\right) + \\frac{g^{2} {\\sigma_z}}{\\Delta^{2}} \\left(- \\frac{\\epsilon}{2} - \\epsilon {{a}^\\dagger} {a} + \\frac{\\omega}{2} + \\omega {{a}^\\dagger} {a}\\right)$$"
      ],
      "text/plain": [
       "                            ⎛                 2     ⎞    2       ⎛  ε      †  \n",
       "                    2       ⎜       †     ⎛ †⎞     2⎟   g ⋅False⋅⎜- ─ - ε⋅a ⋅a\n",
       "ε⋅False      †     g ⋅False⋅⎝1 + 2⋅a ⋅a + ⎝a ⎠  + a ⎠            ⎝  2         \n",
       "─────── + ω⋅a ⋅a + ────────────────────────────────── + ──────────────────────\n",
       "   2                            \\Delta'                                      2\n",
       "                                                                      \\Delta' \n",
       "\n",
       "   ω      †  ⎞            ⎛                 2     ⎞              ⎛      2     \n",
       " - ─ - ω⋅a ⋅a⎟    2       ⎜       †     ⎛ †⎞     2⎟      2       ⎜  ⎛ †⎞     2\n",
       "   2         ⎠   g ⋅False⋅⎝1 + 2⋅a ⋅a + ⎝a ⎠  + a ⎠   ε⋅g ⋅False⋅⎝- ⎝a ⎠  - a \n",
       "────────────── + ────────────────────────────────── + ────────────────────────\n",
       "                                 Δ                            Δ⋅\\Delta'       \n",
       "                                                                              \n",
       "\n",
       "⎞    2       ⎛  ε      †     ω      †  ⎞\n",
       "⎟   g ⋅False⋅⎜- ─ - ε⋅a ⋅a + ─ + ω⋅a ⋅a⎟\n",
       "⎠            ⎝  2            2         ⎠\n",
       "─ + ────────────────────────────────────\n",
       "                      2                 \n",
       "                     Δ                  "
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H13 = drop_terms_containing(H12, [a**3, Dagger(a)**3]); H13"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We use the familiar relation of fractions:\n",
    "\\begin{align}\n",
    "\\frac{\\epsilon}{\\Delta \\Delta' }\n",
    "=\\frac{\\epsilon}{\\Delta(2\\epsilon-\\Delta)}\n",
    "=\\frac{1}{2\\Delta} + \\frac{1}{2(2\\epsilon-\\Delta)}\n",
    "=\\frac{1}{2\\Delta} + \\frac{1}{2\\Delta'}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$\\frac{\\epsilon {\\sigma_z}}{2} + g^{2} \\left(\\frac{1}{2 \\Delta'} + \\frac{1}{2 \\Delta}\\right) {\\sigma_z} \\left(- \\left({{a}^\\dagger}\\right)^{2} - \\left({a}\\right)^{2}\\right) + \\omega {{a}^\\dagger} {a} + \\frac{g^{2} {\\sigma_z}}{\\Delta'} \\left(1 + 2 {{a}^\\dagger} {a} + \\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right) + \\frac{g^{2} {\\sigma_z}}{\\Delta'^{2}} \\left(- \\frac{\\epsilon}{2} - \\epsilon {{a}^\\dagger} {a} - \\frac{\\omega}{2} - \\omega {{a}^\\dagger} {a}\\right) + \\frac{g^{2} {\\sigma_z}}{\\Delta} \\left(1 + 2 {{a}^\\dagger} {a} + \\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right) + \\frac{g^{2} {\\sigma_z}}{\\Delta^{2}} \\left(- \\frac{\\epsilon}{2} - \\epsilon {{a}^\\dagger} {a} + \\frac{\\omega}{2} + \\omega {{a}^\\dagger} {a}\\right)$$"
      ],
      "text/plain": [
       "                                                                        ⎛     \n",
       "                                     ⎛      2     ⎞             2       ⎜     \n",
       "ε⋅False    2 ⎛    1        1 ⎞       ⎜  ⎛ †⎞     2⎟      †     g ⋅False⋅⎝1 + 2\n",
       "─────── + g ⋅⎜───────── + ───⎟⋅False⋅⎝- ⎝a ⎠  - a ⎠ + ω⋅a ⋅a + ───────────────\n",
       "   2         ⎝2⋅\\Delta'   2⋅Δ⎠                                              \\D\n",
       "                                                                              \n",
       "\n",
       "            2     ⎞    2       ⎛  ε      †     ω      †  ⎞            ⎛       \n",
       "  †     ⎛ †⎞     2⎟   g ⋅False⋅⎜- ─ - ε⋅a ⋅a - ─ - ω⋅a ⋅a⎟    2       ⎜       \n",
       "⋅a ⋅a + ⎝a ⎠  + a ⎠            ⎝  2            2         ⎠   g ⋅False⋅⎝1 + 2⋅a\n",
       "─────────────────── + ──────────────────────────────────── + ─────────────────\n",
       "elta'                                      2                                 Δ\n",
       "                                    \\Delta'                                   \n",
       "\n",
       "          2     ⎞    2       ⎛  ε      †     ω      †  ⎞\n",
       "†     ⎛ †⎞     2⎟   g ⋅False⋅⎜- ─ - ε⋅a ⋅a + ─ + ω⋅a ⋅a⎟\n",
       " ⋅a + ⎝a ⎠  + a ⎠            ⎝  2            2         ⎠\n",
       "───────────────── + ────────────────────────────────────\n",
       "                                      2                 \n",
       "                                     Δ                  "
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H14 = H13.subs(e / (D * D_), 1/(2*D) + 1/(2*D_))\n",
    "H14"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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qXnmabvI9CfO0XxHLK+Ka36AeAsYrr1vNN0ipG+pXFcE3sUknn6RuKPaqNk4TztEcBDxY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\n",
      "text/latex": [
       "$$\\frac{\\epsilon {\\sigma_z}}{2} + \\omega {{a}^\\dagger} {a} - \\frac{g^{2} {\\sigma_z}}{2 \\Delta'} \\left(1 + 2 {{a}^\\dagger} {a}\\right) + \\frac{g^{2} {\\sigma_z}}{\\Delta'} \\left(1 + 2 {{a}^\\dagger} {a} + \\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right) - \\frac{g^{2} {\\sigma_z}}{2 \\Delta} \\left(1 + 2 {{a}^\\dagger} {a}\\right) + \\frac{g^{2} {\\sigma_z}}{\\Delta} \\left(1 + 2 {{a}^\\dagger} {a} + \\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right) - \\frac{g^{2} {\\sigma_z}}{2 \\Delta \\Delta'} \\left(\\Delta + \\Delta'\\right) \\left(\\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right)$$"
      ],
      "text/plain": [
       "                                                    ⎛                 2     ⎞ \n",
       "                    2       ⎛       †  ⎞    2       ⎜       †     ⎛ †⎞     2⎟ \n",
       "ε⋅False      †     g ⋅False⋅⎝1 + 2⋅a ⋅a⎠   g ⋅False⋅⎝1 + 2⋅a ⋅a + ⎝a ⎠  + a ⎠ \n",
       "─────── + ω⋅a ⋅a - ───────────────────── + ────────────────────────────────── \n",
       "   2                     2⋅\\Delta'                      \\Delta'               \n",
       "\n",
       "                                   ⎛                 2     ⎞                  \n",
       "   2       ⎛       †  ⎞    2       ⎜       †     ⎛ †⎞     2⎟    2             \n",
       "  g ⋅False⋅⎝1 + 2⋅a ⋅a⎠   g ⋅False⋅⎝1 + 2⋅a ⋅a + ⎝a ⎠  + a ⎠   g ⋅(Δ + \\Delta'\n",
       "- ───────────────────── + ────────────────────────────────── - ───────────────\n",
       "           2⋅Δ                            Δ                                2⋅Δ\n",
       "\n",
       "        ⎛    2     ⎞\n",
       "        ⎜⎛ †⎞     2⎟\n",
       ")⋅False⋅⎝⎝a ⎠  + a ⎠\n",
       "────────────────────\n",
       "⋅\\Delta'            "
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H15 = Add(*[simplify(Mul(*[f.factor() for f in t.args])) for t in H14.args]).subs({e-w: D, e+w: D_})\n",
    "H15"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "for simplicity, let $A = a^\\dagger + a$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "A = Operator('A')  # a + Dagger(a)\n",
    "Asq = qsimplify(((a + Dagger(a))**2).expand())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUwAAAAyBAMAAADFH/9pAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$\\frac{\\epsilon {\\sigma_z}}{2} + \\omega {{a}^\\dagger} {a} + \\frac{g^{2} {\\sigma_z} \\left(A\\right)^{2}}{2 \\Delta'} + \\frac{g^{2} {\\sigma_z} \\left(A\\right)^{2}}{2 \\Delta}$$"
      ],
      "text/plain": [
       "                    2        2    2        2\n",
       "ε⋅False      †     g ⋅False⋅A    g ⋅False⋅A \n",
       "─────── + ω⋅a ⋅a + ─────────── + ───────────\n",
       "   2                2⋅\\Delta'        2⋅Δ    "
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H16 = H15.subs({Asq: A**2,\n",
    "                1 + 2 * Dagger(a) * a: A**2 - a**2 - Dagger(a)**2}).expand()\n",
    "H16"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$H_{disp} = \\frac{\\epsilon {\\sigma_z}}{2} + g^{2} \\left(\\frac{1}{- 2 \\Delta + 4 \\epsilon} + \\frac{1}{2 \\Delta}\\right) {\\sigma_z} \\left({{a}^\\dagger} + {a}\\right)^{2} + \\omega {{a}^\\dagger} {a}$$"
      ],
      "text/plain": [
       "                                                         2         \n",
       "         ε⋅False    2 ⎛     1          1 ⎞       ⎛ †    ⎞       †  \n",
       "H_disp = ─────── + g ⋅⎜──────────── + ───⎟⋅False⋅⎝a  + a⎠  + ω⋅a ⋅a\n",
       "            2         ⎝2⋅(-Δ + 2⋅ε)   2⋅Δ⎠                         "
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H17 = collect(H16, g**2*sz*A).subs({A: a + Dagger(a),\n",
    "                                    D_: 2 * e - D})\n",
    "\n",
    "Eq(Symbol(\"H_disp\"), H17)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the dispersive Hamiltonian obtained without regard to the RWA Hamiltonian in the beginning. The prefactor of the coupling has a contribution arising not only from $\\lambda_1 = g/\\Delta$, but also from $\\lambda_2=g/(2\\epsilon-\\Delta)$. Also, the coupling is proportional to $(a^\\dagger + a)^2$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This Hamiltonian is not diagonal in the eigenbasis of the uncoupled Hamiltonian $H_0$. Nevertheless, it is still possible to interpret this result as a qubit-state dependent frequency shift."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Interpreting $\\hbar\\omega a^\\dagger a$ as the Hamiltonian of a particle with unit mass in the potential $\\frac{1}{2}\\omega^2 x^2$, where $x=\\sqrt{\\frac{\\hbar}{2\\omega}}(a^\\dagger+a)$. The qubit-oscillator coupling modifies the potential curvature $\\omega^2$, such that the oscillator frequency undergoes a shift according to\n",
    "\\begin{align}\n",
    "\\omega \\ \\rightarrow \\ \\overline{\\omega} = \\omega\\sqrt{1\\pm \\frac{2g^2}{\\omega} \\left( \\frac{1}{\\Delta} + \\frac{1}{2\\epsilon-\\Delta}\\right)}.\n",
    "\\end{align}\n",
    "where the $\\pm$ sign depends on the qubit state."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the experimentally interesting parameter regime where $g<\\omega$, we can expand this expression to obtain\n",
    "\\begin{align}\n",
    "\\overline{\\omega} = \\omega \\pm g^2 \\left( \\frac{1}{\\Delta} + \\frac{1}{2\\epsilon-\\Delta}\\right).\n",
    "\\end{align}"
   ]
  }
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