{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Julia Package `QuantumAlgebra`: https://github.com/jfeist/QuantumAlgebra.jl\n",
    "```julia\n",
    "using Pkg\n",
    "Pkg.add(\"QuantumAlgebra\")\n",
    "```\n",
    "\n",
    "Allows to write (somewhat) arbitrary expressions of quantum operators and perform commutator algebra etc with them.\n",
    "\n",
    "Possible operators (for now): Bosonic ($a$, $a^\\dagger$) and two-level systems (Pauli matrices) $\\sigma_{x,y,z}$.\n",
    "\n",
    "Documentation at https://jfeist.github.io/QuantumAlgebra.jl/stable/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:20:57.405000+01:00",
     "start_time": "2019-10-29T19:20:57.127Z"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "using QuantumAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:20:58.647000+01:00",
     "start_time": "2019-10-29T19:20:57.860Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$a^\\dagger a$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpProd(adag{Tuple{}}(()), a{Tuple{}}(()))"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "adag()*a()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "All expressions are reduced to a \"canonical\" form (normal ordering + some extra rules)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:20:59.296000+01:00",
     "start_time": "2019-10-29T19:20:58.880Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1 + a^\\dagger a$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpSum(scal{Int64}(1), QuantumAlgebra.OpProd(adag{Tuple{}}(()), a{Tuple{}}(())))"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a()*adag()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:00.306000+01:00",
     "start_time": "2019-10-29T19:20:59.675Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1i \\sigma_{z}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpProd(scal{Complex{Rational{Int64}}}(0//1 + 1//1*im), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.z, ()))"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "σx()*σy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:00.595000+01:00",
     "start_time": "2019-10-29T19:21:00.379Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1i \\sigma_{x}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpProd(scal{Complex{Rational{Int64}}}(0//1 + 1//1*im), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.x, ()))"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "σy()*σz()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:01.366000+01:00",
     "start_time": "2019-10-29T19:21:01.066Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1$"
      ],
      "text/plain": [
       "scal{Int64}(1)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "σy()*σy()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Operators can have indices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:02.118000+01:00",
     "start_time": "2019-10-29T19:21:01.925Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1 + a_{k}^\\dagger a_{k}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpSum(scal{Int64}(1), QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:k,)), a{Tuple{Symbol}}((:k,))))"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a(:k)*adag(:k)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Symbolic indices can be equal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:02.410000+01:00",
     "start_time": "2019-10-29T19:21:02.408Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$δ_{jk} + a_{j}^\\dagger a_{k}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpSum(QuantumAlgebra.δ((:j, :k)), QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:j,)), a{Tuple{Symbol}}((:k,))))"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a(:k)*adag(:j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Sums**: Can represent analytic sums over indices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:04.052000+01:00",
     "start_time": "2019-10-29T19:21:04.050Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\sum_{i}a_{i}^\\dagger$"
      ],
      "text/plain": [
       "OpSumAnalytic(:i, adag{Tuple{Symbol}}((:i,)))"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s = ∑(:i,adag(:i))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:04.795000+01:00",
     "start_time": "2019-10-29T19:21:04.380Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\sum_{i}a_{i}^\\dagger a_{j}$"
      ],
      "text/plain": [
       "OpSumAnalytic(:i, QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:i,)), a{Tuple{Symbol}}((:j,))))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s*a(:j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:03:50.460000+01:00",
     "start_time": "2019-10-29T19:03:50.117Z"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Sum indices are assumed to run over **all** possible values.  \n",
    "Summing over $\\delta_{i,j}$ removes the sum."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:05.522000+01:00",
     "start_time": "2019-10-29T19:21:05.472Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1 + \\sum_{i}a_{i}^\\dagger a_{j}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpSum(scal{Int64}(1), OpSumAnalytic(:i, QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:i,)), a{Tuple{Symbol}}((:j,)))))"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a(:j)*s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:05.522000+01:00",
     "start_time": "2019-10-29T19:21:05.472Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1 + \\sum_{i}a_{i}^\\dagger a_{k}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpSum(scal{Int64}(1), OpSumAnalytic(:i, QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:i,)), a{Tuple{Symbol}}((:k,)))))"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a(:k)*s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$2 a_{k}^\\dagger + \\sum_{i}a_{i}^\\dagger a_{i}^\\dagger a_{k}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpSum(QuantumAlgebra.OpProd(scal{Int64}(2), adag{Tuple{Symbol}}((:k,))), OpSumAnalytic(:i, QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:i,)), QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:i,)), a{Tuple{Symbol}}((:k,))))))"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a(:k)*∑(:i,adag(:i)*adag(:i))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Multiple indices, nested sums, etc...**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$δ_{im} δ_{jn} + a_{mnμ}^\\dagger a_{ijμ}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpSum(QuantumAlgebra.OpProd(QuantumAlgebra.δ((:i, :m)), QuantumAlgebra.δ((:j, :n))), QuantumAlgebra.OpProd(adag{Tuple{Symbol,Symbol,Symbol}}((:m, :n, :μ)), a{Tuple{Symbol,Symbol,Symbol}}((:i, :j, :μ))))"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a(:i,:j,:μ) * adag(:m,:n,:μ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:07.553000+01:00",
     "start_time": "2019-10-29T19:21:06.847Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1 + \\sum_{i}\\sum_{j}a_{mnμ}^\\dagger a_{ijμ}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpSum(scal{Int64}(1), OpSumAnalytic(:i, OpSumAnalytic(:j, QuantumAlgebra.OpProd(adag{Tuple{Symbol,Symbol,Symbol}}((:m, :n, :μ)), a{Tuple{Symbol,Symbol,Symbol}}((:i, :j, :μ))))))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "∑(:j, ∑(:i,a(:i,:j,:μ))) * adag(:m,:n,:μ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:07.613000+01:00",
     "start_time": "2019-10-29T19:21:07.309Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\sum_{i}1 + \\sum_{i}\\sum_{k}a_{k}^\\dagger a_{i}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpSum(OpSumAnalytic(:i, scal{Int64}(1)), OpSumAnalytic(:i, OpSumAnalytic(:k, QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:k,)), a{Tuple{Symbol}}((:i,))))))"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "∑(:i,a(:i)) * ∑(:k,adag(:k))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Caution: implementation of analytic sums not fully finished (especially for nested sums)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Parameters**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:10.443000+01:00",
     "start_time": "2019-10-29T19:21:10.234Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\sum_{j}ω_{j} a_{j}^\\dagger$"
      ],
      "text/plain": [
       "OpSumAnalytic(:j, QuantumAlgebra.OpProd(param{Tuple{Symbol}}(:ω, 'n', (:j,)), adag{Tuple{Symbol}}((:j,))))"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s = ∑(:j,param(:ω,:j)*adag(:j))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "string-based input form:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:10.443000+01:00",
     "start_time": "2019-10-29T19:21:10.234Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\sum_{j}ω_{j} a_{j}^\\dagger$"
      ],
      "text/plain": [
       "OpSumAnalytic(:j, QuantumAlgebra.OpProd(param{Tuple{Symbol}}(:ω, 'r', (:j,)), adag{Tuple{Symbol}}((:j,))))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s = ∑(:j,Pr\"ω_j\"*adag(:j))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:11.223000+01:00",
     "start_time": "2019-10-29T19:21:10.940Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$ω_{i} + \\sum_{j}ω_{j} a_{j}^\\dagger a_{i}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpSum(param{Tuple{Symbol}}(:ω, 'r', (:i,)), OpSumAnalytic(:j, QuantumAlgebra.OpProd(param{Tuple{Symbol}}(:ω, 'r', (:j,)), QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:j,)), a{Tuple{Symbol}}((:i,))))))"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a(:i)*s"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Back to the original motivation:\n",
    "**Emitter + many-mode Hamiltonian**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:13.121000+01:00",
     "start_time": "2019-10-29T19:21:12.528Z"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$E(t) \\sigma_{x} + \\frac{1}{2} Ω \\sigma_{z} + \\sum_{i}ω_{i} a_{i}^\\dagger a_{i} + \\sum_{i}g_{i} a_{i}^\\dagger \\sigma_{x} + \\sum_{i}g_{i} a_{i} \\sigma_{x}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpSum(QuantumAlgebra.OpProd(param{Tuple{}}(Symbol(\"E(t)\"), 'r', ()), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.x, ())), QuantumAlgebra.OpSum(QuantumAlgebra.OpProd(scal{Rational{Int64}}(1//2), QuantumAlgebra.OpProd(param{Tuple{}}(:Ω, 'r', ()), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.z, ()))), QuantumAlgebra.OpSum(OpSumAnalytic(:i, QuantumAlgebra.OpProd(param{Tuple{Symbol}}(:ω, 'r', (:i,)), QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:i,)), a{Tuple{Symbol}}((:i,))))), QuantumAlgebra.OpSum(OpSumAnalytic(:i, QuantumAlgebra.OpProd(param{Tuple{Symbol}}(:g, 'r', (:i,)), QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:i,)), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.x, ())))), OpSumAnalytic(:i, QuantumAlgebra.OpProd(param{Tuple{Symbol}}(:g, 'r', (:i,)), QuantumAlgebra.OpProd(a{Tuple{Symbol}}((:i,)), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.x, ()))))))))"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H = 1//2*Pr\"Ω\"*σz() + Pr\"E(t)\"*σx() + ∑(:i,Pr\"ω_i\"*adag(:i)*a(:i)) + ∑(:i,Pr\"g_i\"*(adag(:i)+a(:i))*σx())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:13.925000+01:00",
     "start_time": "2019-10-29T19:21:13.249Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$2i E(t) \\sigma_{y} + 2i \\sum_{i}g_{i} a_{i}^\\dagger \\sigma_{y} + 2i \\sum_{i}g_{i} a_{i} \\sigma_{y}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpSum(QuantumAlgebra.OpProd(scal{Complex{Int64}}(0 + 2im), QuantumAlgebra.OpProd(param{Tuple{}}(Symbol(\"E(t)\"), 'r', ()), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.y, ()))), QuantumAlgebra.OpSum(QuantumAlgebra.OpProd(scal{Complex{Int64}}(0 + 2im), OpSumAnalytic(:i, QuantumAlgebra.OpProd(param{Tuple{Symbol}}(:g, 'r', (:i,)), QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:i,)), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.y, ()))))), QuantumAlgebra.OpProd(scal{Complex{Int64}}(0 + 2im), OpSumAnalytic(:i, QuantumAlgebra.OpProd(param{Tuple{Symbol}}(:g, 'r', (:i,)), QuantumAlgebra.OpProd(a{Tuple{Symbol}}((:i,)), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.y, ())))))))"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "comm(σz(),H)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-10-29T20:21:14.471000+01:00",
     "start_time": "2019-10-29T19:21:13.847Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1i g_{j} \\sigma_{y} + 2i E(t) a_{j}^\\dagger \\sigma_{y} + -1 ω_{j} a_{j}^\\dagger \\sigma_{z} + 2i \\sum_{i}g_{i} a_{i}^\\dagger a_{j}^\\dagger \\sigma_{y} + 2i \\sum_{i}g_{i} a_{j}^\\dagger a_{i} \\sigma_{y}$"
      ],
      "text/plain": [
       "QuantumAlgebra.OpSum(QuantumAlgebra.OpProd(scal{Complex{Rational{Int64}}}(0//1 + 1//1*im), QuantumAlgebra.OpProd(param{Tuple{Symbol}}(:g, 'r', (:j,)), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.y, ()))), QuantumAlgebra.OpSum(QuantumAlgebra.OpProd(scal{Complex{Int64}}(0 + 2im), QuantumAlgebra.OpProd(param{Tuple{}}(Symbol(\"E(t)\"), 'r', ()), QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:j,)), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.y, ())))), QuantumAlgebra.OpSum(QuantumAlgebra.OpProd(scal{Int64}(-1), QuantumAlgebra.OpProd(param{Tuple{Symbol}}(:ω, 'r', (:j,)), QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:j,)), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.z, ())))), QuantumAlgebra.OpSum(QuantumAlgebra.OpProd(scal{Complex{Int64}}(0 + 2im), OpSumAnalytic(:i, QuantumAlgebra.OpProd(param{Tuple{Symbol}}(:g, 'r', (:i,)), QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:i,)), QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:j,)), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.y, ())))))), QuantumAlgebra.OpProd(scal{Complex{Int64}}(0 + 2im), OpSumAnalytic(:i, QuantumAlgebra.OpProd(param{Tuple{Symbol}}(:g, 'r', (:i,)), QuantumAlgebra.OpProd(adag{Tuple{Symbol}}((:j,)), QuantumAlgebra.OpProd(a{Tuple{Symbol}}((:i,)), QuantumAlgebra.σ{Tuple{}}(QuantumAlgebra.y, ()))))))))))"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "comm(adag(:j)*σz(),H)"
   ]
  }
 ],
 "metadata": {
  "@webio": {
   "lastCommId": null,
   "lastKernelId": null
  },
  "hide_input": false,
  "kernelspec": {
   "display_name": "Julia 1.2.0",
   "language": "julia",
   "name": "julia-1.2"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.2.0"
  },
  "rise": {
   "scroll": true,
   "theme": "white"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  },
  "varInspector": {
   "cols": {
    "lenName": 16,
    "lenType": 16,
    "lenVar": 40
   },
   "kernels_config": {
    "python": {
     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
   ],
   "window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}