2023-04-26_sympy_recombination.ipynb

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   "source": [
    "## Analytical Derivation of the Recombination"
   ]
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   "source": [
    "from sympy import *\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
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   "source": [
    "𝕚 = I"
   ]
  },
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   "source": [
    "Δϕ, Δp, Δθ, θ, p = symbols(\"Δϕ, Δp, ΔΘ, Theta, p\", positive=True)\n",
    "Ψ0 = symbols(\"Psi_0\", real=True)\n",
    "Ψa = symbols(\"Psi_+\")\n",
    "Ψb = symbols(\"Psi_-\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "726264e0-0126-444f-97ec-d99a00e73a76",
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   "source": [
    "The initial wave function is"
   ]
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   "source": [
    "Ψinit = Matrix([\n",
    "    [Ψ0],\n",
    "    [0]\n",
    "])"
   ]
  },
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\Psi_{0}\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[Psi_0],\n",
       "[    0]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ψinit"
   ]
  },
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   "source": [
    "We apply a π/2 pulse:"
   ]
  },
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\\\\\frac{\\sqrt{2} i}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  sqrt(2)/2, sqrt(2)*I/2],\n",
       "[sqrt(2)*I/2,   sqrt(2)/2]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U_split = Matrix(\n",
    "    [[1, 𝕚], [𝕚, 1]]\n",
    ") / sqrt(2)\n",
    "U_split"
   ]
  },
  {
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   "id": "51c06e1a-99ec-42b0-a03e-a923306a322a",
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   "source": [
    "and introduce a relative phase"
   ]
  },
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}e^{- \\frac{i Δϕ}{2}} & 0\\\\0 & e^{\\frac{i Δϕ}{2}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[exp(-I*Δϕ/2),           0],\n",
       "[           0, exp(I*Δϕ/2)]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Φ = Matrix(\n",
    "    [[exp(-𝕚*Δϕ/2), 0], [0, exp(+𝕚*Δϕ/2)]]\n",
    ")\n",
    "Φ"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4ef301b5-4831-45b3-94a2-13b42f9da6e6",
   "metadata": {},
   "source": [
    "to get the state before the recombination:"
   ]
  },
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   "cell_type": "code",
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   "id": "180d32e8-017d-417b-aaaa-356179f07c62",
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    "execution": {
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2} \\Psi_{0} e^{- \\frac{i Δϕ}{2}}}{2}\\\\\\frac{\\sqrt{2} i \\Psi_{0} e^{\\frac{i Δϕ}{2}}}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ sqrt(2)*Psi_0*exp(-I*Δϕ/2)/2],\n",
       "[sqrt(2)*I*Psi_0*exp(I*Δϕ/2)/2]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ψ = Φ @ (U_split @ Ψinit)\n",
    "Ψ"
   ]
  },
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   "cell_type": "markdown",
   "id": "34aaca2b-caab-448e-91b1-596818c21073",
   "metadata": {},
   "source": [
    "Note that this is normalized:"
   ]
  },
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   "outputs": [],
   "source": [
    "def inner(a, b):\n",
    "    return a.dot(b, hermitian=True, conjugate_convention=\"left\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "20db26ba-9dc2-4229-b41b-fa69272b6b09",
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   "outputs": [],
   "source": [
    "def norm(Ψ):\n",
    "    return sqrt(inner(Ψ, Ψ)).expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "3fb61f05-61ee-4341-aa4d-19c0b87044c5",
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left|{\\Psi_{0}}\\right|$"
      ],
      "text/plain": [
       "Abs(Psi_0)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm(Ψ)"
   ]
  },
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   "cell_type": "markdown",
   "id": "50c45d06-afc7-4e7c-a837-84a6ba49ae4b",
   "metadata": {},
   "source": [
    "We also allow the wave packet $\\Psi_0$ to evolve differently"
   ]
  },
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   "cell_type": "code",
   "execution_count": 12,
   "id": "acdd7334-491f-41af-b71c-a56ad056d992",
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2} \\Psi_{+} e^{- \\frac{i Δϕ}{2}}}{2}\\\\\\frac{\\sqrt{2} i \\Psi_{-} e^{\\frac{i Δϕ}{2}}}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ sqrt(2)*Psi_+*exp(-I*Δϕ/2)/2],\n",
       "[sqrt(2)*I*Psi_-*exp(I*Δϕ/2)/2]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ψ = Matrix([\n",
    "    [Ψ[0,0].subs({Ψ0: Ψa})],\n",
    "    [Ψ[1,0].subs({Ψ0: Ψb})],\n",
    "])\n",
    "Ψ"
   ]
  },
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   "cell_type": "markdown",
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   "metadata": {},
   "source": [
    "We then recombine by applying an inverse π/2 pulse:"
   ]
  },
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2} i}{2}\\\\- \\frac{\\sqrt{2} i}{2} & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[   sqrt(2)/2, -sqrt(2)*I/2],\n",
       "[-sqrt(2)*I/2,    sqrt(2)/2]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U_recomb = U_split.conjugate().transpose()\n",
    "U_recomb"
   ]
  },
  {
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   "execution_count": 14,
   "id": "e31bad72-4a0e-4c05-9550-06ca87b0971f",
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{\\Psi_{+} e^{- \\frac{i Δϕ}{2}}}{2} + \\frac{\\Psi_{-} e^{\\frac{i Δϕ}{2}}}{2}\\\\- \\frac{i \\Psi_{+} e^{- \\frac{i Δϕ}{2}}}{2} + \\frac{i \\Psi_{-} e^{\\frac{i Δϕ}{2}}}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[     Psi_+*exp(-I*Δϕ/2)/2 + Psi_-*exp(I*Δϕ/2)/2],\n",
       "[-I*Psi_+*exp(-I*Δϕ/2)/2 + I*Psi_-*exp(I*Δϕ/2)/2]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ψ_recomb = U_recomb @ Ψ\n",
    "Ψ_recomb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "93b4f7b9-b011-4159-aad3-fca42d324de4",
   "metadata": {
    "execution": {
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sqrt{\\frac{\\Psi_{+} \\overline{\\Psi_{+}}}{2} + \\frac{\\Psi_{-} \\overline{\\Psi_{-}}}{2}}$"
      ],
      "text/plain": [
       "sqrt(Psi_+*conjugate(Psi_+)/2 + Psi_-*conjugate(Psi_-)/2)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm(Ψ_recomb).expand()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b76a15f5-7036-4c35-ba78-c099cb86abd0",
   "metadata": {},
   "source": [
    "The complex amplitude on the \"right\"/\"+\" surface (index 1) is"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c605f5cb-185d-4acb-badb-1264564464bc",
   "metadata": {},
   "source": [
    "## Amplitude and population for \"+\" surface"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "b26248a5-e28d-483e-96e1-61b12975f8cf",
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\Psi_{+} e^{- \\frac{i Δϕ}{2}}}{2} + \\frac{\\Psi_{-} e^{\\frac{i Δϕ}{2}}}{2}$"
      ],
      "text/plain": [
       "Psi_+*exp(-I*Δϕ/2)/2 + Psi_-*exp(I*Δϕ/2)/2"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = Ψ_recomb[0,0]\n",
    "\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "Which corresponds to the population"
   ]
  },
  {
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\Psi_{+} \\overline{\\Psi_{+}}}{4} + \\frac{\\Psi_{+} e^{- i Δϕ} \\overline{\\Psi_{-}}}{4} + \\frac{\\Psi_{-} e^{i Δϕ} \\overline{\\Psi_{+}}}{4} + \\frac{\\Psi_{-} \\overline{\\Psi_{-}}}{4}$"
      ],
      "text/plain": [
       "Psi_+*conjugate(Psi_+)/4 + Psi_+*exp(-I*Δϕ)*conjugate(Psi_-)/4 + Psi_-*exp(I*Δϕ)*conjugate(Psi_+)/4 + Psi_-*conjugate(Psi_-)/4"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pop_a = (a * a.conjugate()).expand()\n",
    "pop_a"
   ]
  },
  {
   "cell_type": "code",
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     "iopub.status.idle": "2023-05-25T18:42:02.796596Z",
     "shell.execute_reply": "2023-05-25T18:42:02.796224Z",
     "shell.execute_reply.started": "2023-05-25T18:42:02.793689Z"
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "overlap_subs = {\n",
    "    Ψa * Ψa.conjugate(): 1,\n",
    "    Ψb * Ψb.conjugate(): 1,\n",
    "    Ψa * Ψb.conjugate(): symbols(\"⟨Ψ_{-}|Ψ_{+}⟩\"),\n",
    "    Ψa.conjugate() * Ψb: symbols(\"⟨Ψ_{-}|Ψ_{+}⟩\").conjugate()\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "4b16b2e8-c686-4883-87f5-817634f938bc",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-05-25T18:42:02.797335Z",
     "iopub.status.busy": "2023-05-25T18:42:02.797133Z",
     "iopub.status.idle": "2023-05-25T18:42:02.808539Z",
     "shell.execute_reply": "2023-05-25T18:42:02.808178Z",
     "shell.execute_reply.started": "2023-05-25T18:42:02.797320Z"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{⟨Ψ_{-}|Ψ_{+}⟩ e^{- i Δϕ}}{4} + \\frac{e^{i Δϕ} \\overline{⟨Ψ_{-}|Ψ_{+}⟩}}{4} + \\frac{1}{2}$"
      ],
      "text/plain": [
       "⟨Ψ_{-}|Ψ_{+}⟩*exp(-I*Δϕ)/4 + exp(I*Δϕ)*conjugate(⟨Ψ_{-}|Ψ_{+}⟩)/4 + 1/2"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pop_a.subs(overlap_subs)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "992b22cd-9509-4c45-bbc7-284bc63237a4",
   "metadata": {},
   "source": [
    "For the adiabatic case:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "5af45bfd-aa98-46f0-a9d9-427f954b9f3f",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-05-25T18:42:02.809268Z",
     "iopub.status.busy": "2023-05-25T18:42:02.809067Z",
     "iopub.status.idle": "2023-05-25T18:42:02.820608Z",
     "shell.execute_reply": "2023-05-25T18:42:02.820244Z",
     "shell.execute_reply.started": "2023-05-25T18:42:02.809254Z"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\cos{\\left(Δϕ \\right)}}{2} + \\frac{1}{2}$"
      ],
      "text/plain": [
       "cos(Δϕ)/2 + 1/2"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pop_a.subs({Ψa: Ψ0, Ψb:Ψ0}).subs({Ψ0**2:1}).rewrite(exp, cos)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "2121d4f1-c86a-44df-984d-ce3efcaf5646",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-05-25T18:42:02.821366Z",
     "iopub.status.busy": "2023-05-25T18:42:02.821128Z",
     "iopub.status.idle": "2023-05-25T18:42:02.827796Z",
     "shell.execute_reply": "2023-05-25T18:42:02.827431Z",
     "shell.execute_reply.started": "2023-05-25T18:42:02.821352Z"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\cos{\\left(Δϕ \\right)}}{2} + \\frac{1}{2}$"
      ],
      "text/plain": [
       "cos(Δϕ)/2 + 1/2"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cos(Δϕ/2)**2).rewrite(cos, exp).expand().rewrite(exp, cos)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "39611aac-266e-45d1-9d52-b1f9eb70e866",
   "metadata": {},
   "source": [
    "## Amplitude and population for \"-\" surface"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "395a1c7f-1d25-4544-8243-899d0efe14e3",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-05-25T18:42:02.829963Z",
     "iopub.status.busy": "2023-05-25T18:42:02.829752Z",
     "iopub.status.idle": "2023-05-25T18:42:02.834249Z",
     "shell.execute_reply": "2023-05-25T18:42:02.833883Z",
     "shell.execute_reply.started": "2023-05-25T18:42:02.829948Z"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{i \\Psi_{+} e^{- \\frac{i Δϕ}{2}}}{2} + \\frac{i \\Psi_{-} e^{\\frac{i Δϕ}{2}}}{2}$"
      ],
      "text/plain": [
       "-I*Psi_+*exp(-I*Δϕ/2)/2 + I*Psi_-*exp(I*Δϕ/2)/2"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = Ψ_recomb[1,0]\n",
    "\n",
    "b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bda7a9ba-ce3b-43a6-8aec-5ebd30d25f92",
   "metadata": {},
   "source": [
    "Which corresponds to the population"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "77d70cd6-ff8b-4fd3-a2ad-a327f66c6275",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-05-25T18:42:02.835171Z",
     "iopub.status.busy": "2023-05-25T18:42:02.834822Z",
     "iopub.status.idle": "2023-05-25T18:42:02.841684Z",
     "shell.execute_reply": "2023-05-25T18:42:02.841323Z",
     "shell.execute_reply.started": "2023-05-25T18:42:02.835155Z"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\Psi_{+} \\overline{\\Psi_{+}}}{4} - \\frac{\\Psi_{+} e^{- i Δϕ} \\overline{\\Psi_{-}}}{4} - \\frac{\\Psi_{-} e^{i Δϕ} \\overline{\\Psi_{+}}}{4} + \\frac{\\Psi_{-} \\overline{\\Psi_{-}}}{4}$"
      ],
      "text/plain": [
       "Psi_+*conjugate(Psi_+)/4 - Psi_+*exp(-I*Δϕ)*conjugate(Psi_-)/4 - Psi_-*exp(I*Δϕ)*conjugate(Psi_+)/4 + Psi_-*conjugate(Psi_-)/4"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pop_b = (b * b.conjugate()).expand()\n",
    "pop_b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "53886653-616a-4f50-ba1e-c3dd0b6aa7a2",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-05-25T18:42:02.842596Z",
     "iopub.status.busy": "2023-05-25T18:42:02.842202Z",
     "iopub.status.idle": "2023-05-25T18:42:02.851470Z",
     "shell.execute_reply": "2023-05-25T18:42:02.851102Z",
     "shell.execute_reply.started": "2023-05-25T18:42:02.842579Z"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{⟨Ψ_{-}|Ψ_{+}⟩ e^{- i Δϕ}}{4} - \\frac{e^{i Δϕ} \\overline{⟨Ψ_{-}|Ψ_{+}⟩}}{4} + \\frac{1}{2}$"
      ],
      "text/plain": [
       "-⟨Ψ_{-}|Ψ_{+}⟩*exp(-I*Δϕ)/4 - exp(I*Δϕ)*conjugate(⟨Ψ_{-}|Ψ_{+}⟩)/4 + 1/2"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pop_b.subs(overlap_subs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "6264ca63-5114-4031-b270-49490924fadc",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-05-25T18:42:02.852238Z",
     "iopub.status.busy": "2023-05-25T18:42:02.852033Z",
     "iopub.status.idle": "2023-05-25T18:42:02.857566Z",
     "shell.execute_reply": "2023-05-25T18:42:02.857201Z",
     "shell.execute_reply.started": "2023-05-25T18:42:02.852223Z"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{2} - \\frac{\\cos{\\left(Δϕ \\right)}}{2}$"
      ],
      "text/plain": [
       "1/2 - cos(Δϕ)/2"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pop_b.subs({Ψa: Ψ0, Ψb:Ψ0}).subs({Ψ0**2:1}).rewrite(exp, cos)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "0fa392d7-d29f-4356-af67-847aca0f544e",
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-05-25T18:42:02.858416Z",
     "iopub.status.busy": "2023-05-25T18:42:02.858119Z",
     "iopub.status.idle": "2023-05-25T18:42:02.868357Z",
     "shell.execute_reply": "2023-05-25T18:42:02.867993Z",
     "shell.execute_reply.started": "2023-05-25T18:42:02.858400Z"
    },
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{2} - \\frac{\\cos{\\left(Δϕ \\right)}}{2}$"
      ],
      "text/plain": [
       "1/2 - cos(Δϕ)/2"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sin(Δϕ/2)**2).rewrite(sin, exp).expand().rewrite(exp, cos)"
   ]
  }
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