{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "intro",
   "metadata": {},
   "source": [
    "# Λ-Type Three-Level: Coherent Population Trapping (CPT)\n",
    "\n",
    "When two coherent fields simultaneously drive a three-level Λ system, the\n",
    "atoms can be optically pumped into a **dark state** — a coherent\n",
    "superposition of the two ground states that is completely decoupled from\n",
    "the excited state.\n",
    "\n",
    "## Level structure\n",
    "\n",
    "```\n",
    "         |1⟩ ──────────────── (excited)\n",
    "        /    \\\n",
    "   Ω_p /      \\ Ω_c\n",
    "      /        \\     γ₁₀ = γ₁₂ = 1.0\n",
    "    |0⟩        |2⟩  (ground states)\n",
    "```\n",
    "\n",
    "## Physics\n",
    "\n",
    "The Hamiltonian in the rotating frame has a zero-eigenvalue eigenstate —\n",
    "the **dark state** — when the two-photon resonance condition holds\n",
    "($\\delta_p = \\delta_c$):\n",
    "\n",
    "$$\n",
    "|D\\rangle = \\frac{\\Omega_c\\,|0\\rangle - \\Omega_p\\,|2\\rangle}\n",
    "                  {\\sqrt{\\Omega_p^2 + \\Omega_c^2}}\n",
    "$$\n",
    "\n",
    "Atoms pumped into $|D\\rangle$ stop scattering photons entirely. The probe\n",
    "absorption spectrum then shows a **narrow transparency window** centred on\n",
    "$\\delta_p = \\delta_c$: the CPT (or EIT) dip.\n",
    "\n",
    "The width of the dip is set by the ground-state decoherence rate $\\gamma_g$,\n",
    "which can be much narrower than the excited-state linewidth $\\gamma_{10}$.\n",
    "This makes CPT-based dark resonances extremely sharp — the basis for\n",
    "CPT atomic clocks and all-optical magnetometry."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "imports",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from maxwellbloch import mb_solve, plot"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "no-coupling-section",
   "metadata": {},
   "source": [
    "## Without coupling: Doppler-broadened absorption\n",
    "\n",
    "First we solve for the probe alone (coupling Rabi frequency = 0, i.e.\n",
    "no coupling field). The probe sees a Doppler-broadened Lorentzian — the\n",
    "baseline against which the CPT dip will be compared.\n",
    "\n",
    "A thermal velocity distribution is included\n",
    "(`thermal_std = 10 γ`) to reproduce realistic lineshape broadening."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "solve-no-coupling",
   "metadata": {},
   "outputs": [],
   "source": [
    "mb_solve_json_no_coupling = \"\"\"\n",
    "{\n",
    "  \"atom\": {\n",
    "    \"num_states\": 3,\n",
    "    \"decays\": [\n",
    "      {\"channels\": [[0, 1]], \"rate\": 1.0},\n",
    "      {\"channels\": [[1, 2]], \"rate\": 1.0}\n",
    "    ],\n",
    "    \"fields\": [\n",
    "      {\n",
    "        \"label\": \"probe\",\n",
    "        \"coupled_levels\": [[0, 1]],\n",
    "        \"detuning\": 0.0,\n",
    "        \"detuning_positive\": true,\n",
    "        \"rabi_freq\": 1.0e-3,\n",
    "        \"rabi_freq_t_func\": \"gaussian\",\n",
    "        \"rabi_freq_t_args\": {\"ampl\": 1.0, \"centre\": 0.0, \"fwhm\": 1.5}\n",
    "      },\n",
    "      {\n",
    "        \"label\": \"coupling\",\n",
    "        \"coupled_levels\": [[1, 2]],\n",
    "        \"detuning\": 0.0,\n",
    "        \"detuning_positive\": false,\n",
    "        \"rabi_freq\": 0.0,\n",
    "        \"rabi_freq_t_func\": \"ramp_onoff\",\n",
    "        \"rabi_freq_t_args\": {\"ampl\": 1.0, \"fwhm\": 0.2, \"on\": -2.0, \"off\": 8.0}\n",
    "      }\n",
    "    ]\n",
    "  },\n",
    "  \"t_min\": -2.0,\n",
    "  \"t_max\": 8.0,\n",
    "  \"t_steps\": 200,\n",
    "  \"z_min\": 0.0,\n",
    "  \"z_max\": 1.0,\n",
    "  \"z_steps\": 10,\n",
    "  \"z_steps_inner\": 2,\n",
    "  \"interaction_strengths\": [2.0, 0.0],\n",
    "  \"velocity_classes\": {\n",
    "    \"thermal_width\": 10.0,\n",
    "    \"thermal_delta_min\": -30.0,\n",
    "    \"thermal_delta_max\":  30.0,\n",
    "    \"thermal_delta_steps\": 12,\n",
    "    \"thermal_delta_inner_min\": -3.0,\n",
    "    \"thermal_delta_inner_max\":  3.0,\n",
    "    \"thermal_delta_inner_steps\": 6\n",
    "  },\n",
    "  \"savefile\": \"mbs-lambda-cpt-no-coupling\"\n",
    "}\n",
    "\"\"\"\n",
    "\n",
    "mbs_no_c = mb_solve.MBSolve().from_json_str(mb_solve_json_no_coupling)\n",
    "mbs_no_c.mbsolve(recalc=False)\n",
    "print(\"Done\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "coupling-section",
   "metadata": {},
   "source": [
    "## With coupling: CPT dark resonance\n",
    "\n",
    "Now we add a strong CW coupling field ($\\Omega_c = 5\\gamma$) that is\n",
    "switched on before the probe pulse arrives. The two-photon resonance\n",
    "condition ($\\delta_p = \\delta_c = 0$) is satisfied on resonance, so a\n",
    "sharp transparency window opens at $\\delta_p = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "solve-coupling",
   "metadata": {},
   "outputs": [],
   "source": [
    "mb_solve_json_coupling = \"\"\"\n",
    "{\n",
    "  \"atom\": {\n",
    "    \"num_states\": 3,\n",
    "    \"decays\": [\n",
    "      {\"channels\": [[0, 1]], \"rate\": 1.0},\n",
    "      {\"channels\": [[1, 2]], \"rate\": 1.0}\n",
    "    ],\n",
    "    \"fields\": [\n",
    "      {\n",
    "        \"label\": \"probe\",\n",
    "        \"coupled_levels\": [[0, 1]],\n",
    "        \"detuning\": 0.0,\n",
    "        \"detuning_positive\": true,\n",
    "        \"rabi_freq\": 1.0e-3,\n",
    "        \"rabi_freq_t_func\": \"gaussian\",\n",
    "        \"rabi_freq_t_args\": {\"ampl\": 1.0, \"centre\": 0.0, \"fwhm\": 1.5}\n",
    "      },\n",
    "      {\n",
    "        \"label\": \"coupling\",\n",
    "        \"coupled_levels\": [[1, 2]],\n",
    "        \"detuning\": 0.0,\n",
    "        \"detuning_positive\": false,\n",
    "        \"rabi_freq\": 5.0,\n",
    "        \"rabi_freq_t_func\": \"ramp_onoff\",\n",
    "        \"rabi_freq_t_args\": {\"ampl\": 1.0, \"fwhm\": 0.2, \"on\": -2.0, \"off\": 8.0}\n",
    "      }\n",
    "    ]\n",
    "  },\n",
    "  \"t_min\": -2.0,\n",
    "  \"t_max\": 8.0,\n",
    "  \"t_steps\": 200,\n",
    "  \"z_min\": 0.0,\n",
    "  \"z_max\": 1.0,\n",
    "  \"z_steps\": 10,\n",
    "  \"z_steps_inner\": 2,\n",
    "  \"interaction_strengths\": [2.0, 0.0],\n",
    "  \"velocity_classes\": {\n",
    "    \"thermal_width\": 10.0,\n",
    "    \"thermal_delta_min\": -30.0,\n",
    "    \"thermal_delta_max\":  30.0,\n",
    "    \"thermal_delta_steps\": 12,\n",
    "    \"thermal_delta_inner_min\": -3.0,\n",
    "    \"thermal_delta_inner_max\":  3.0,\n",
    "    \"thermal_delta_inner_steps\": 6\n",
    "  },\n",
    "  \"savefile\": \"mbs-lambda-cpt-coupling\"\n",
    "}\n",
    "\"\"\"\n",
    "\n",
    "mbs_c = mb_solve.MBSolve().from_json_str(mb_solve_json_coupling)\n",
    "mbs_c.mbsolve(recalc=False)\n",
    "print(\"Done\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "spectrum-section",
   "metadata": {},
   "source": [
    "## Absorption spectra: the CPT dip\n",
    "\n",
    "The two traces are overlaid below. The broad Doppler-broadened background\n",
    "is the same in both cases. With the coupling field on, a narrow\n",
    "transparency window — the dark resonance — appears at $\\delta_p = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "plot-spectra",
   "metadata": {},
   "outputs": [],
   "source": "fig = plot.spectrum_overlay(\n    [mbs_no_c, mbs_c],\n    field_idx=0,\n    labels=[\"coupling off\", \"coupling on (Ω_c = 5γ)\"],\n    freq_range=8,\n)\nfig.update_layout(title=\"CPT dark resonance in a Λ system: probe absorption\")\nfig.show(renderer='notebook_connected')"
  },
  {
   "cell_type": "markdown",
   "id": "spacetime-section",
   "metadata": {},
   "source": [
    "## Probe propagation with coupling\n",
    "\n",
    "The space-time plot of the probe shows that the medium becomes\n",
    "transparent for the resonant frequency component: the probe pulse\n",
    "propagates with much less attenuation than in the bare two-level case."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "plot-spacetime",
   "metadata": {},
   "outputs": [],
   "source": [
    "fig = plot.field_spacetime(mbs_c, field_idx=0)\n",
    "fig.update_layout(title=\"Probe |Ω_p(z, t)| — CPT (Ω_c = 5γ)\")\n",
    "fig.show(renderer='notebook_connected')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dark-state-section",
   "metadata": {},
   "source": [
    "## Dark-state population\n",
    "\n",
    "Once the atoms are in the dark state they accumulate in the two ground\n",
    "states. The plot below shows $\\rho_{00}$ and $\\rho_{22}$ at the input\n",
    "face, illustrating the coherent redistribution of population driven by\n",
    "the coupling field."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "plot-population",
   "metadata": {},
   "outputs": [],
   "source": [
    "fig = plot.population(mbs_c, state_indices=[0, 1, 2], z_idx=0)\n",
    "fig.update_layout(title=\"Ground/excited populations at z = z_min (CPT, Ω_c = 5γ)\")\n",
    "fig.show(renderer='notebook_connected')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "summary",
   "metadata": {},
   "source": [
    "## Summary\n",
    "\n",
    "- Without coupling: Doppler-broadened Lorentzian absorption of width\n",
    "  $\\sim 2\\gamma \\sqrt{1 + (\\sigma_D/\\gamma)^2}$.\n",
    "- With coupling: a narrow CPT dip of width $\\sim \\gamma_g \\ll \\gamma$\n",
    "  appears at the two-photon resonance. The dip depth approaches unity\n",
    "  as the coupling strength increases.\n",
    "- Atoms accumulate in the dark state: $\\rho_{11} \\to 0$, population is\n",
    "  distributed between $|0\\rangle$ and $|2\\rangle$ with ratio set by\n",
    "  $\\Omega_p^2 : \\Omega_c^2$.\n",
    "\n",
    "## References\n",
    "\n",
    "1. E. Arimondo and G. Orriols, *Non-absorbing atomic coherences by\n",
    "   coherent two-photon transitions in a three-level optical pumping*,\n",
    "   Lett. Nuovo Cimento **17**, 333 (1976). First observation of CPT.\n",
    "2. G. Alzetta, A. Gozzini, L. Moi, G. Orriols, *An experimental method\n",
    "   for the observation of r.f. transitions and laser beat resonances in\n",
    "   oriented Na vapour*, Nuovo Cimento B **36**, 5 (1976).\n",
    "3. E. Arimondo, *Coherent Population Trapping in Laser Spectroscopy*,\n",
    "   Prog. Opt. **35**, 257 (1996). Comprehensive review."
   ]
  }
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