goerz/+2021-12_RabiCycling_README.md

## +2021-12_RabiCycling_README.md

      
    Raw
  

              +2021-12_RabiCycling_README.md
            
          
    We solve the Schrödinger equation for the dynamics of a two-level system analytically.

Rabi Cycling in the Two-Level-System — From Scratch
Rabi Cycling in the Two-Level-System — From Scratch (Variation for Pauli Matrices)

See also: Analytical Ramsey Scheme for a single TLS, which includes the solution of the TLS dynamics for arbitrary initial conditions.

  
## .gitignore
.ipynb_checkpoints

## AnalyticalRabiCycling.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "e8b420fc",
   "metadata": {},
   "source": [
    "# Rabi Cycling in the Two-Level-System — From Scratch "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6d870f6f",
   "metadata": {},
   "source": [
    "This notebook uses sympy to symbolically solve the Schrödinger equation for a two-level system, resulting in the exact analytic expressions for the complex amplitudes and the population dynamics of Rabi cycling.\n",
    "\n",
    "Having this written out in a computer algebra system allows to easily adapt the calculation to variations in the form of the two-level-system Hamiltonian, or different boundary conditions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "099e6f02",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:09.576682Z",
     "start_time": "2021-12-15T20:45:09.266346Z"
    }
   },
   "outputs": [],
   "source": [
    "from sympy import Function, symbols, sqrt, Eq, Abs, exp, cos, sin, Matrix, dsolve, solve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "392d26c5",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:09.579927Z",
     "start_time": "2021-12-15T20:45:09.577819Z"
    }
   },
   "outputs": [],
   "source": [
    "from sympy import I as 𝕚"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "22151c90",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:09.583750Z",
     "start_time": "2021-12-15T20:45:09.580729Z"
    }
   },
   "outputs": [],
   "source": [
    "g = Function('g')\n",
    "e = Function('e')\n",
    "t = symbols('t', positive=True)\n",
    "Δ = symbols('Delta', real=True)\n",
    "Ω0 = symbols('Ω_0', real=True)\n",
    "Ω = symbols('Omega', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b7ac0d90",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:09.716024Z",
     "start_time": "2021-12-15T20:45:09.585275Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & - \\frac{Ω_{0}}{2}\\\\- \\frac{Ω_{0}}{2} & \\Delta\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[     0, -Ω_0/2],\n",
       "[-Ω_0/2,  Delta]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ĥ = Matrix([\n",
    "    [  0  , -Ω0/2],\n",
    "    [-Ω0/2,    Δ ]\n",
    "])\n",
    "Ĥ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "4d51a2cf",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:09.721307Z",
     "start_time": "2021-12-15T20:45:09.717074Z"
    }
   },
   "outputs": [],
   "source": [
    "TDSE_system = [\n",
    "    g(t).diff(t) + 𝕚 * (Ĥ[0,0] * g(t) + Ĥ[0,1] * e(t)),\n",
    "    e(t).diff(t) + 𝕚 * (Ĥ[1,0]*  g(t) + Ĥ[1,1] * e(t)),\n",
    "]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "74fa6ba2",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.060209Z",
     "start_time": "2021-12-15T20:45:09.722256Z"
    }
   },
   "outputs": [],
   "source": [
    "sols_gen = dsolve(TDSE_system, [g(t), e(t)])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a501c1d6",
   "metadata": {},
   "source": [
    "Note that `dsolve` allows to specify boundary conditions, but the resulting expressions are hard to simply. We do better by solving for the integration constants \"manually\" later on."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "2718f04e",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.063009Z",
     "start_time": "2021-12-15T20:45:10.061017Z"
    }
   },
   "outputs": [],
   "source": [
    "effective_rabi_freq = {\n",
    "    Ω: sqrt(Δ**2 + Ω0**2)\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "75ee2c2a",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.068994Z",
     "start_time": "2021-12-15T20:45:10.064272Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\Omega = \\sqrt{\\Delta^{2} + Ω_{0}^{2}}$"
      ],
      "text/plain": [
       "Eq(Omega, sqrt(Delta**2 + Ω_0**2))"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(Ω, effective_rabi_freq[Ω])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "79b723c4",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.071767Z",
     "start_time": "2021-12-15T20:45:10.069861Z"
    }
   },
   "outputs": [],
   "source": [
    "find_Ω = {\n",
    "    effective_rabi_freq[Ω]: Ω\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "b05f056e",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.098172Z",
     "start_time": "2021-12-15T20:45:10.074005Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle g{\\left(t \\right)} = - \\frac{C_{1} Ω_{0} e^{- \\frac{t \\left(i \\Delta - i \\Omega\\right)}{2}}}{\\Delta - \\Omega} - \\frac{C_{2} Ω_{0} e^{- \\frac{t \\left(i \\Delta + i \\Omega\\right)}{2}}}{\\Delta + \\Omega}$"
      ],
      "text/plain": [
       "Eq(g(t), -C1*Ω_0*exp(-t*(I*Delta - I*Omega)/2)/(Delta - Omega) - C2*Ω_0*exp(-t*(I*Delta + I*Omega)/2)/(Delta + Omega))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sols_gen[0].subs(find_Ω)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "e8e0977b",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.129875Z",
     "start_time": "2021-12-15T20:45:10.099175Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle e{\\left(t \\right)} = C_{1} e^{- \\frac{i \\Delta t}{2}} e^{\\frac{i \\Omega t}{2}} + C_{2} e^{- \\frac{i \\Delta t}{2}} e^{- \\frac{i \\Omega t}{2}}$"
      ],
      "text/plain": [
       "Eq(e(t), C1*exp(-I*Delta*t/2)*exp(I*Omega*t/2) + C2*exp(-I*Delta*t/2)*exp(-I*Omega*t/2))"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sols_gen[1].subs(find_Ω).expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "394f4efd",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.133217Z",
     "start_time": "2021-12-15T20:45:10.130725Z"
    }
   },
   "outputs": [],
   "source": [
    "boundary_conditions = {\n",
    "    t: 0,\n",
    "    g(t): 1,\n",
    "    e(t) : 0,\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "23be389b",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.136309Z",
     "start_time": "2021-12-15T20:45:10.134102Z"
    }
   },
   "outputs": [],
   "source": [
    "find_Ω0sq = {\n",
    "    Ω**2 - Δ**2: Ω0**2\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "be349bd8",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.138733Z",
     "start_time": "2021-12-15T20:45:10.137171Z"
    }
   },
   "outputs": [],
   "source": [
    "C1, C2 = symbols(\"C1, C2\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "7d321d1e",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.176835Z",
     "start_time": "2021-12-15T20:45:10.139910Z"
    }
   },
   "outputs": [],
   "source": [
    "_integration_constants = solve(\n",
    "    [sol.subs(find_Ω).subs(boundary_conditions) for sol in sols_gen],\n",
    "    [C1, C2]\n",
    ")\n",
    "integration_constants = {\n",
    "    k: v.subs(find_Ω0sq)\n",
    "    for (k, v) in _integration_constants.items()\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "894eeee2",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.182662Z",
     "start_time": "2021-12-15T20:45:10.177878Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle C_{1} = \\frac{Ω_{0}}{2 \\Omega}$"
      ],
      "text/plain": [
       "Eq(C1, Ω_0/(2*Omega))"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(C1, integration_constants[C1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "851352b3",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.188218Z",
     "start_time": "2021-12-15T20:45:10.183516Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle C_{2} = - \\frac{Ω_{0}}{2 \\Omega}$"
      ],
      "text/plain": [
       "Eq(C2, -Ω_0/(2*Omega))"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(C2, integration_constants[C2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "339ee928",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.191145Z",
     "start_time": "2021-12-15T20:45:10.189072Z"
    }
   },
   "outputs": [],
   "source": [
    "global_phase = exp(-𝕚 * Δ * t / 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "21dc5c9c",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.194138Z",
     "start_time": "2021-12-15T20:45:10.191962Z"
    }
   },
   "outputs": [],
   "source": [
    "def remove_global_phase(eq, global_phase=global_phase):\n",
    "    return Eq(eq.lhs, eq.rhs * global_phase.conjugate())\n",
    "\n",
    "def restore_global_phase(eq, global_phase=global_phase):\n",
    "    return Eq(eq.lhs, eq.rhs * global_phase)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "43fb88d4",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.196779Z",
     "start_time": "2021-12-15T20:45:10.195038Z"
    }
   },
   "outputs": [],
   "source": [
    "def collect(eq, term):\n",
    "    return Eq(eq.lhs, eq.rhs.collect(term))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "78cc5307",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.199875Z",
     "start_time": "2021-12-15T20:45:10.197590Z"
    }
   },
   "outputs": [],
   "source": [
    "def simplify_sol_g(sol_g):\n",
    "    rhs = (\n",
    "        sol_g\n",
    "        .rhs\n",
    "        .rewrite(sin)\n",
    "        .expand()\n",
    "        .collect(𝕚)\n",
    "        .subs(effective_rabi_freq)\n",
    "        .simplify()\n",
    "        .subs(find_Ω)\n",
    "    )\n",
    "    return Eq(sol_g.lhs, rhs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "3a034ee0",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.602923Z",
     "start_time": "2021-12-15T20:45:10.200800Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle g{\\left(t \\right)} = \\left(\\frac{i \\Delta \\sin{\\left(\\frac{\\Omega t}{2} \\right)}}{\\Omega} + \\cos{\\left(\\frac{\\Omega t}{2} \\right)}\\right) e^{- \\frac{i \\Delta t}{2}}$"
      ],
      "text/plain": [
       "Eq(g(t), (I*Delta*sin(Omega*t/2)/Omega + cos(Omega*t/2))*exp(-I*Delta*t/2))"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol_g = restore_global_phase(\n",
    "    simplify_sol_g(\n",
    "        collect(\n",
    "            remove_global_phase(\n",
    "                sols_gen[0]\n",
    "                .subs(find_Ω)\n",
    "                .subs(integration_constants)\n",
    "            ).expand(),\n",
    "            exp(𝕚 * Ω * t / 2                                                                   )\n",
    "        )\n",
    "    )\n",
    ")\n",
    "sol_g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "143d31e9",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.660312Z",
     "start_time": "2021-12-15T20:45:10.603849Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle e{\\left(t \\right)} = \\frac{i Ω_{0} e^{- \\frac{i \\Delta t}{2}} \\sin{\\left(\\frac{\\Omega t}{2} \\right)}}{\\Omega}$"
      ],
      "text/plain": [
       "Eq(e(t), I*Ω_0*exp(-I*Delta*t/2)*sin(Omega*t/2)/Omega)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol_e = restore_global_phase(\n",
    "    remove_global_phase(\n",
    "        sols_gen[1].subs(find_Ω).subs(integration_constants)\n",
    "    ).expand().rewrite(sin).expand()\n",
    ")\n",
    "sol_e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "5480adc7",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-12-15T20:45:10.670174Z",
     "start_time": "2021-12-15T20:45:10.661098Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left|{e{\\left(t \\right)}}\\right|^{2} = \\frac{Ω_{0}^{2} \\sin^{2}{\\left(\\frac{\\Omega t}{2} \\right)}}{\\Omega^{2}}$"
      ],
      "text/plain": [
       "Eq(Abs(e(t))**2, Ω_0**2*sin(Omega*t/2)**2/Omega**2)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pop_e = (sol_e.rhs * sol_e.rhs.conjugate())\n",
    "Eq(Abs(e(t))**2, pop_e)"
   ]
  }
 ],
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## AnalyticalRabiCyclingPauli.ipynb

      
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	{
	"cells": [
	{
	"cell_type": "markdown",
	"id": "e8b420fc",
	"metadata": {},
	"source": [
	"# Rabi Cycling in the Two-Level-System — From Scratch "
	]
	},
	{
	"cell_type": "markdown",
	"id": "6d870f6f",
	"metadata": {},
	"source": [
	"This notebook uses sympy to symbolically solve the Schrödinger equation for a two-level system, resulting in the exact analytic expressions for the complex amplitudes and the population dynamics of Rabi cycling.\n",
	"\n",
	"Having this written out in a computer algebra system allows to easily adapt the calculation to variations in the form of the two-level-system Hamiltonian, or different boundary conditions."
	]
	},
	{
	"cell_type": "code",
	"execution_count": 1,
	"id": "099e6f02",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:09.576682Z",
	"start_time": "2021-12-15T20:45:09.266346Z"
	}
	},
	"outputs": [],
	"source": [
	"from sympy import Function, symbols, sqrt, Eq, Abs, exp, cos, sin, Matrix, dsolve, solve"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 2,
	"id": "392d26c5",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:09.579927Z",
	"start_time": "2021-12-15T20:45:09.577819Z"
	}
	},
	"outputs": [],
	"source": [
	"from sympy import I as 𝕚"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 3,
	"id": "22151c90",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:09.583750Z",
	"start_time": "2021-12-15T20:45:09.580729Z"
	}
	},
	"outputs": [],
	"source": [
	"g = Function('g')\n",
	"e = Function('e')\n",
	"t = symbols('t', positive=True)\n",
	"Δ = symbols('Delta', real=True)\n",
	"Ω0 = symbols('Ω_0', real=True)\n",
	"Ω = symbols('Omega', real=True)"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 4,
	"id": "b7ac0d90",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:09.716024Z",
	"start_time": "2021-12-15T20:45:09.585275Z"
	}
	},
	"outputs": [
	{
	"data": {
	"text/latex": [
	"$\\displaystyle \\left[\\begin{matrix}0 & - \\frac{Ω_{0}}{2}\\\\- \\frac{Ω_{0}}{2} & \\Delta\\end{matrix}\\right]$"
	],
	"text/plain": [
	"Matrix([\n",
	"[ 0, -Ω_0/2],\n",
	"[-Ω_0/2, Delta]])"
	]
	},
	"execution_count": 4,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"Ĥ = Matrix([\n",
	" [ 0 , -Ω0/2],\n",
	" [-Ω0/2, Δ ]\n",
	"])\n",
	"Ĥ"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 5,
	"id": "4d51a2cf",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:09.721307Z",
	"start_time": "2021-12-15T20:45:09.717074Z"
	}
	},
	"outputs": [],
	"source": [
	"TDSE_system = [\n",
	" g(t).diff(t) + 𝕚 * (Ĥ[0,0] * g(t) + Ĥ[0,1] * e(t)),\n",
	" e(t).diff(t) + 𝕚 * (Ĥ[1,0]* g(t) + Ĥ[1,1] * e(t)),\n",
	"]"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 6,
	"id": "74fa6ba2",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.060209Z",
	"start_time": "2021-12-15T20:45:09.722256Z"
	}
	},
	"outputs": [],
	"source": [
	"sols_gen = dsolve(TDSE_system, [g(t), e(t)])"
	]
	},
	{
	"cell_type": "markdown",
	"id": "a501c1d6",
	"metadata": {},
	"source": [
	"Note that `dsolve` allows to specify boundary conditions, but the resulting expressions are hard to simply. We do better by solving for the integration constants \"manually\" later on."
	]
	},
	{
	"cell_type": "code",
	"execution_count": 7,
	"id": "2718f04e",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.063009Z",
	"start_time": "2021-12-15T20:45:10.061017Z"
	}
	},
	"outputs": [],
	"source": [
	"effective_rabi_freq = {\n",
	" Ω: sqrt(Δ2 + Ω02)\n",
	"}"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 8,
	"id": "75ee2c2a",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.068994Z",
	"start_time": "2021-12-15T20:45:10.064272Z"
	}
	},
	"outputs": [
	{
	"data": {
	"text/latex": [
	"$\\displaystyle \\Omega = \\sqrt{\\Delta^{2} + Ω_{0}^{2}}$"
	],
	"text/plain": [
	"Eq(Omega, sqrt(Delta2 + Ω_02))"
	]
	},
	"execution_count": 8,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"Eq(Ω, effective_rabi_freq[Ω])"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 9,
	"id": "79b723c4",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.071767Z",
	"start_time": "2021-12-15T20:45:10.069861Z"
	}
	},
	"outputs": [],
	"source": [
	"find_Ω = {\n",
	" effective_rabi_freq[Ω]: Ω\n",
	"}"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 10,
	"id": "b05f056e",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.098172Z",
	"start_time": "2021-12-15T20:45:10.074005Z"
	}
	},
	"outputs": [
	{
	"data": {
	"text/latex": [
	"$\\displaystyle g{\\left(t \\right)} = - \\frac{C_{1} Ω_{0} e^{- \\frac{t \\left(i \\Delta - i \\Omega\\right)}{2}}}{\\Delta - \\Omega} - \\frac{C_{2} Ω_{0} e^{- \\frac{t \\left(i \\Delta + i \\Omega\\right)}{2}}}{\\Delta + \\Omega}$"
	],
	"text/plain": [
	"Eq(g(t), -C1Ω_0exp(-t(IDelta - IOmega)/2)/(Delta - Omega) - C2Ω_0exp(-t(IDelta + IOmega)/2)/(Delta + Omega))"
	]
	},
	"execution_count": 10,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"sols_gen[0].subs(find_Ω)"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 11,
	"id": "e8e0977b",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.129875Z",
	"start_time": "2021-12-15T20:45:10.099175Z"
	}
	},
	"outputs": [
	{
	"data": {
	"text/latex": [
	"$\\displaystyle e{\\left(t \\right)} = C_{1} e^{- \\frac{i \\Delta t}{2}} e^{\\frac{i \\Omega t}{2}} + C_{2} e^{- \\frac{i \\Delta t}{2}} e^{- \\frac{i \\Omega t}{2}}$"
	],
	"text/plain": [
	"Eq(e(t), C1exp(-IDeltat/2)exp(IOmegat/2) + C2exp(-IDeltat/2)exp(-IOmegat/2))"
	]
	},
	"execution_count": 11,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"sols_gen[1].subs(find_Ω).expand()"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 12,
	"id": "394f4efd",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.133217Z",
	"start_time": "2021-12-15T20:45:10.130725Z"
	}
	},
	"outputs": [],
	"source": [
	"boundary_conditions = {\n",
	" t: 0,\n",
	" g(t): 1,\n",
	" e(t) : 0,\n",
	"}"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 13,
	"id": "23be389b",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.136309Z",
	"start_time": "2021-12-15T20:45:10.134102Z"
	}
	},
	"outputs": [],
	"source": [
	"find_Ω0sq = {\n",
	" Ω2 - Δ2: Ω0**2\n",
	"}"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 14,
	"id": "be349bd8",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.138733Z",
	"start_time": "2021-12-15T20:45:10.137171Z"
	}
	},
	"outputs": [],
	"source": [
	"C1, C2 = symbols(\"C1, C2\")"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 15,
	"id": "7d321d1e",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.176835Z",
	"start_time": "2021-12-15T20:45:10.139910Z"
	}
	},
	"outputs": [],
	"source": [
	"_integration_constants = solve(\n",
	" [sol.subs(find_Ω).subs(boundary_conditions) for sol in sols_gen],\n",
	" [C1, C2]\n",
	")\n",
	"integration_constants = {\n",
	" k: v.subs(find_Ω0sq)\n",
	" for (k, v) in _integration_constants.items()\n",
	"}"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 16,
	"id": "894eeee2",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.182662Z",
	"start_time": "2021-12-15T20:45:10.177878Z"
	}
	},
	"outputs": [
	{
	"data": {
	"text/latex": [
	"$\\displaystyle C_{1} = \\frac{Ω_{0}}{2 \\Omega}$"
	],
	"text/plain": [
	"Eq(C1, Ω_0/(2*Omega))"
	]
	},
	"execution_count": 16,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"Eq(C1, integration_constants[C1])"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 17,
	"id": "851352b3",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.188218Z",
	"start_time": "2021-12-15T20:45:10.183516Z"
	}
	},
	"outputs": [
	{
	"data": {
	"text/latex": [
	"$\\displaystyle C_{2} = - \\frac{Ω_{0}}{2 \\Omega}$"
	],
	"text/plain": [
	"Eq(C2, -Ω_0/(2*Omega))"
	]
	},
	"execution_count": 17,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"Eq(C2, integration_constants[C2])"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 18,
	"id": "339ee928",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.191145Z",
	"start_time": "2021-12-15T20:45:10.189072Z"
	}
	},
	"outputs": [],
	"source": [
	"global_phase = exp(-𝕚 * Δ * t / 2)"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 19,
	"id": "21dc5c9c",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.194138Z",
	"start_time": "2021-12-15T20:45:10.191962Z"
	}
	},
	"outputs": [],
	"source": [
	"def remove_global_phase(eq, global_phase=global_phase):\n",
	" return Eq(eq.lhs, eq.rhs * global_phase.conjugate())\n",
	"\n",
	"def restore_global_phase(eq, global_phase=global_phase):\n",
	" return Eq(eq.lhs, eq.rhs * global_phase)"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 20,
	"id": "43fb88d4",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.196779Z",
	"start_time": "2021-12-15T20:45:10.195038Z"
	}
	},
	"outputs": [],
	"source": [
	"def collect(eq, term):\n",
	" return Eq(eq.lhs, eq.rhs.collect(term))"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 21,
	"id": "78cc5307",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.199875Z",
	"start_time": "2021-12-15T20:45:10.197590Z"
	}
	},
	"outputs": [],
	"source": [
	"def simplify_sol_g(sol_g):\n",
	" rhs = (\n",
	" sol_g\n",
	" .rhs\n",
	" .rewrite(sin)\n",
	" .expand()\n",
	" .collect(𝕚)\n",
	" .subs(effective_rabi_freq)\n",
	" .simplify()\n",
	" .subs(find_Ω)\n",
	" )\n",
	" return Eq(sol_g.lhs, rhs)"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 22,
	"id": "3a034ee0",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.602923Z",
	"start_time": "2021-12-15T20:45:10.200800Z"
	}
	},
	"outputs": [
	{
	"data": {
	"text/latex": [
	"$\\displaystyle g{\\left(t \\right)} = \\left(\\frac{i \\Delta \\sin{\\left(\\frac{\\Omega t}{2} \\right)}}{\\Omega} + \\cos{\\left(\\frac{\\Omega t}{2} \\right)}\\right) e^{- \\frac{i \\Delta t}{2}}$"
	],
	"text/plain": [
	"Eq(g(t), (IDeltasin(Omegat/2)/Omega + cos(Omegat/2))exp(-IDelta*t/2))"
	]
	},
	"execution_count": 22,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"sol_g = restore_global_phase(\n",
	" simplify_sol_g(\n",
	" collect(\n",
	" remove_global_phase(\n",
	" sols_gen[0]\n",
	" .subs(find_Ω)\n",
	" .subs(integration_constants)\n",
	" ).expand(),\n",
	" exp(𝕚 * Ω * t / 2 )\n",
	" )\n",
	" )\n",
	")\n",
	"sol_g"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 23,
	"id": "143d31e9",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.660312Z",
	"start_time": "2021-12-15T20:45:10.603849Z"
	}
	},
	"outputs": [
	{
	"data": {
	"text/latex": [
	"$\\displaystyle e{\\left(t \\right)} = \\frac{i Ω_{0} e^{- \\frac{i \\Delta t}{2}} \\sin{\\left(\\frac{\\Omega t}{2} \\right)}}{\\Omega}$"
	],
	"text/plain": [
	"Eq(e(t), IΩ_0exp(-IDeltat/2)sin(Omegat/2)/Omega)"
	]
	},
	"execution_count": 23,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"sol_e = restore_global_phase(\n",
	" remove_global_phase(\n",
	" sols_gen[1].subs(find_Ω).subs(integration_constants)\n",
	" ).expand().rewrite(sin).expand()\n",
	")\n",
	"sol_e"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 24,
	"id": "5480adc7",
	"metadata": {
	"ExecuteTime": {
	"end_time": "2021-12-15T20:45:10.670174Z",
	"start_time": "2021-12-15T20:45:10.661098Z"
	}
	},
	"outputs": [
	{
	"data": {
	"text/latex": [
	"$\\displaystyle \\left\|{e{\\left(t \\right)}}\\right\|^{2} = \\frac{Ω_{0}^{2} \\sin^{2}{\\left(\\frac{\\Omega t}{2} \\right)}}{\\Omega^{2}}$"
	],
	"text/plain": [
	"Eq(Abs(e(t))2, Ω_02sin(Omegat/2)2/Omega2)"
	]
	},
	"execution_count": 24,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"pop_e = (sol_e.rhs * sol_e.rhs.conjugate())\n",
	"Eq(Abs(e(t))**2, pop_e)"
	]
	}
	],
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