{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "qubit-decay-title",
   "metadata": {},
   "source": [
    "# Qubit Decay with `mesolve`\n",
    "\n",
    "This tutorial builds the smallest open quantum system in OpenQuantumSim: a two-level atom with spontaneous emission. It compares the excited-state population against the analytic result $P_e(t)=e^{-\\gamma t}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "qubit-decay-imports",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import openquantumsim as oqs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "qubit-decay-system",
   "metadata": {},
   "outputs": [],
   "source": [
    "gamma = 0.35\n",
    "atom = oqs.SpinSpace(0.5, label=\"atom\")\n",
    "\n",
    "H = 0.0 * oqs.sigmaz(atom)\n",
    "collapse = np.sqrt(gamma) * oqs.sigmam(atom)\n",
    "excited = oqs.basis(atom, \"up\")\n",
    "rho0 = oqs.ket2dm(excited)\n",
    "P_excited = oqs.Operator(oqs.ket2dm(excited), atom, \"P_excited\")\n",
    "times = np.linspace(0.0, 6.0, 61)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "qubit-decay-backend-note",
   "metadata": {},
   "source": [
    "`mesolve` uses the Julia backend. Run `python setup_julia.py` once before executing this notebook locally."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "qubit-decay-run",
   "metadata": {},
   "outputs": [],
   "source": [
    "result = oqs.mesolve(\n",
    "    H,\n",
    "    rho0,\n",
    "    times,\n",
    "    c_ops=[collapse],\n",
    "    e_ops=[P_excited],\n",
    "    options=oqs.Options(rtol=1e-9, atol=1e-11),\n",
    ")\n",
    "\n",
    "analytic = np.exp(-gamma * times)\n",
    "np.max(np.abs(result.expect[0].real - analytic))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "qubit-decay-plot",
   "metadata": {},
   "outputs": [],
   "source": [
    "ax = oqs.expect_plot(times, result.expect[0], label=\"OpenQuantumSim\")\n",
    "ax.plot(times, analytic, \"--\", label=\"analytic\")\n",
    "ax.legend()"
   ]
  }
 ],
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