{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "intro",
   "metadata": {},
   "source": [
    "# Λ-Type Three-Level: Adiabatons\n",
    "\n",
    "When two strong fields co-propagate through a three-level Λ medium and the\n",
    "coupling field precedes the probe (**counterintuitive ordering**), the atomic\n",
    "population is transferred from one ground state to the other without ever\n",
    "significantly occupying the excited state. The fields reshape themselves\n",
    "during propagation until they converge on self-consistent **adiabaton**\n",
    "solutions — pulses that maintain the adiabatic dark-state condition at every\n",
    "point in the medium.\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 dark eigenstate of the Λ Hamiltonian is\n",
    "\n",
    "$$\n",
    "|D\\rangle = \\cos\\theta\\,|0\\rangle - \\sin\\theta\\,|2\\rangle,\n",
    "\\qquad\n",
    "\\theta(z,t) = \\arctan\\!\\left(\\frac{\\Omega_p(z,t)}{\\Omega_c(z,t)}\\right)\n",
    "$$\n",
    "\n",
    "When the mixing angle $\\theta$ varies **slowly** compared to the adiabatic\n",
    "gap $\\sqrt{\\Omega_p^2 + \\Omega_c^2}$, atoms initially in $|D\\rangle$ stay\n",
    "there as the pulses pass. If $\\Omega_c \\gg \\Omega_p$ initially\n",
    "($\\theta \\approx 0$, all population in $|0\\rangle$) and $\\Omega_p \\gg \\Omega_c$\n",
    "finally ($\\theta \\approx \\pi/2$, all population in $|2\\rangle$), complete\n",
    "ground-state population transfer occurs with the excited state remaining\n",
    "**dark** throughout.\n",
    "\n",
    "During propagation, the Maxwell equations force the fields to reshape until\n",
    "both carry the same temporal envelope — the **adiabaton** shape —\n",
    "differing only in amplitude (Grobe, Hioe, Eberly, PRL 1994).\n",
    "\n",
    "## Key requirement: substantial pulse overlap\n",
    "\n",
    "For the dark state to be maintained the two fields must be present\n",
    "simultaneously. If the coupling and probe are separated by much more than\n",
    "their temporal width, the coupling is absorbed before the probe arrives\n",
    "and no dark state forms. Here the pulses are offset by only $\\pm 1\\,\\gamma^{-1}$\n",
    "with width $T = 2\\,\\gamma^{-1}$, giving ~85% normalised overlap.\n",
    "\n",
    "## Parameters\n",
    "\n",
    "| Quantity | Value | Notes |\n",
    "|---|---|---|\n",
    "| $\\Omega_p^{(0)}$ | $5\\,\\gamma$ | sech, centred at $t = +1\\,\\gamma^{-1}$ |\n",
    "| $\\Omega_c^{(0)}$ | $5\\,\\gamma$ | sech, centred at $t = -1\\,\\gamma^{-1}$ (arrives first) |\n",
    "| Pulse width | $T = 2\\,\\gamma^{-1}$ | sech half-width |\n",
    "| Pulse overlap | ~85% | normalised $\\int\\Omega_p\\Omega_c\\,dt$ |\n",
    "| OD | $\\sim 6$ | interaction\\_strengths × z\\_max × 2 |\n",
    "| Doppler | none | pure coherent effect |"
   ]
  },
  {
   "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": "coprop-section",
   "metadata": {},
   "source": [
    "## Intuitive ordering: probe first\n",
    "\n",
    "With the probe arriving before the coupling field, the atoms encounter\n",
    "the probe before any dark state is established. The probe drives\n",
    "population into the excited state and is strongly absorbed. The coupling\n",
    "field arrives later but by then the medium has already scattered.\n",
    "No adiabaton forms."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "solve-intuitive",
   "metadata": {},
   "outputs": [],
   "source": [
    "mb_solve_json_intuitive = \"\"\"\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\": 5.0,\n",
    "        \"rabi_freq_t_func\": \"sech\",\n",
    "        \"rabi_freq_t_args\": {\"ampl\": 5.0, \"centre\": -1.0, \"width\": 2.0}\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\": \"sech\",\n",
    "        \"rabi_freq_t_args\": {\"ampl\": 5.0, \"centre\": 1.0, \"width\": 2.0}\n",
    "      }\n",
    "    ]\n",
    "  },\n",
    "  \"t_min\": -8.0,\n",
    "  \"t_max\": 8.0,\n",
    "  \"t_steps\": 200,\n",
    "  \"z_min\": 0.0,\n",
    "  \"z_max\": 1.0,\n",
    "  \"z_steps\": 20,\n",
    "  \"z_steps_inner\": 2,\n",
    "  \"interaction_strengths\": [3.0, 3.0],\n",
    "  \"savefile\": \"mbs-lambda-adiabatons-intuitive\"\n",
    "}\n",
    "\"\"\"\n",
    "\n",
    "mbs_int = mb_solve.MBSolve().from_json_str(mb_solve_json_intuitive)\n",
    "mbs_int.mbsolve(recalc=False)\n",
    "print(\"Done\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "counter-section",
   "metadata": {},
   "source": [
    "## Counterintuitive ordering: coupling first (adiabaton)\n",
    "\n",
    "With the coupling field centred at $t = -1\\,\\gamma^{-1}$ and the probe\n",
    "at $t = +1\\,\\gamma^{-1}$, both pulses overlap substantially while the\n",
    "coupling still leads. This maintains $\\Omega_c \\geq \\Omega_p$ at early\n",
    "times ($\\theta \\approx 0$, dark state $\\approx |0\\rangle$) and\n",
    "$\\Omega_p \\geq \\Omega_c$ at late times ($\\theta \\approx \\pi/2$, dark\n",
    "state $\\approx |2\\rangle$). Population transfers adiabatically from\n",
    "$|0\\rangle$ to $|2\\rangle$ with the excited state remaining dark."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "solve-counter",
   "metadata": {},
   "outputs": [],
   "source": [
    "mb_solve_json_counter = \"\"\"\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\": 5.0,\n",
    "        \"rabi_freq_t_func\": \"sech\",\n",
    "        \"rabi_freq_t_args\": {\"ampl\": 5.0, \"centre\": 1.0, \"width\": 2.0}\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\": \"sech\",\n",
    "        \"rabi_freq_t_args\": {\"ampl\": 5.0, \"centre\": -1.0, \"width\": 2.0}\n",
    "      }\n",
    "    ]\n",
    "  },\n",
    "  \"t_min\": -8.0,\n",
    "  \"t_max\": 8.0,\n",
    "  \"t_steps\": 200,\n",
    "  \"z_min\": 0.0,\n",
    "  \"z_max\": 1.0,\n",
    "  \"z_steps\": 20,\n",
    "  \"z_steps_inner\": 2,\n",
    "  \"interaction_strengths\": [3.0, 3.0],\n",
    "  \"savefile\": \"mbs-lambda-adiabatons-counter\"\n",
    "}\n",
    "\"\"\"\n",
    "\n",
    "mbs_ctr = mb_solve.MBSolve().from_json_str(mb_solve_json_counter)\n",
    "mbs_ctr.mbsolve(recalc=False)\n",
    "print(\"Done\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "probe-field-section",
   "metadata": {},
   "source": [
    "## Probe field: intuitive vs counterintuitive\n",
    "\n",
    "In the intuitive case the probe is strongly absorbed as it enters an\n",
    "un-prepared medium. In the counterintuitive case both fields propagate\n",
    "with little attenuation because the atoms are kept dark throughout."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "plot-probe-int",
   "metadata": {},
   "outputs": [],
   "source": [
    "fig = plot.field_spacetime(mbs_int, field_idx=0)\n",
    "fig.update_layout(title=\"Probe |Ω_p(z, t)| — intuitive ordering (probe first)\")\n",
    "fig.show(renderer='notebook_connected')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "plot-probe-ctr",
   "metadata": {},
   "outputs": [],
   "source": [
    "fig = plot.field_spacetime(mbs_ctr, field_idx=0)\n",
    "fig.update_layout(title=\"Probe |Ω_p(z, t)| — counterintuitive ordering (adiabaton)\")\n",
    "fig.show(renderer='notebook_connected')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "coupling-field-section",
   "metadata": {},
   "source": [
    "## Coupling field: reshaping into the adiabaton\n",
    "\n",
    "Both fields reshape as they propagate. In the counterintuitive case they\n",
    "converge on the same temporal envelope — the adiabaton."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "plot-coupling-ctr",
   "metadata": {},
   "outputs": [],
   "source": [
    "fig = plot.field_spacetime(mbs_ctr, field_idx=1)\n",
    "fig.update_layout(title=\"Coupling |Ω_c(z, t)| — counterintuitive ordering (adiabaton)\")\n",
    "fig.show(renderer='notebook_connected')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "field-envelope-section",
   "metadata": {},
   "source": [
    "## Field envelopes at input and output\n",
    "\n",
    "Comparing $|\\Omega(t)|$ at the input face ($z = z_\\mathrm{min}$) and\n",
    "output face ($z = z_\\mathrm{max}$) shows the adiabaton reshaping: both\n",
    "fields exit with a similar envelope, shifted toward a common shape.\n",
    "In the counterintuitive case neither field is strongly absorbed — contrast\n",
    "with the intuitive ordering where the probe is greatly attenuated."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "plot-envelopes",
   "metadata": {},
   "outputs": [],
   "source": [
    "import plotly.graph_objects as go\n",
    "from maxwellbloch.plot import theme  # ensures template is registered\n",
    "\n",
    "t = mbs_ctr.tlist\n",
    "\n",
    "fig = go.Figure(layout=go.Layout(\n",
    "    title=\"Adiabaton: field envelopes at input and output\",\n",
    "    xaxis_title=\"time (γ⁻¹)\",\n",
    "    yaxis_title=\"Ω (γ)\",\n",
    "    template=\"maxwellbloch\",\n",
    "))\n",
    "fig.add_trace(go.Scatter(x=t, y=abs(mbs_ctr.Omegas_zt[0, 0, :]),  name=\"probe in\",     line=dict(dash=\"solid\")))\n",
    "fig.add_trace(go.Scatter(x=t, y=abs(mbs_ctr.Omegas_zt[0, -1, :]), name=\"probe out\",    line=dict(dash=\"dash\")))\n",
    "fig.add_trace(go.Scatter(x=t, y=abs(mbs_ctr.Omegas_zt[1, 0, :]),  name=\"coupling in\",  line=dict(dash=\"solid\")))\n",
    "fig.add_trace(go.Scatter(x=t, y=abs(mbs_ctr.Omegas_zt[1, -1, :]), name=\"coupling out\", line=dict(dash=\"dash\")))\n",
    "fig.show(renderer='notebook_connected')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "population-section",
   "metadata": {},
   "source": [
    "## Population dynamics\n",
    "\n",
    "In the counterintuitive (adiabaton) case the excited state $|1\\rangle$\n",
    "stays near zero throughout, while population is completely transferred\n",
    "from $|0\\rangle$ to $|2\\rangle$. Compare with the intuitive case where\n",
    "$\\rho_{11}$ peaks strongly and transfer is incomplete."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "plot-pop-ctr",
   "metadata": {},
   "outputs": [],
   "source": [
    "fig = plot.population(mbs_ctr, state_indices=[0, 1, 2], z_idx=0)\n",
    "fig.update_layout(title=\"Population at z = z_min — counterintuitive ordering (adiabaton)\")\n",
    "fig.show(renderer='notebook_connected')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "plot-pop-int",
   "metadata": {},
   "outputs": [],
   "source": [
    "fig = plot.population(mbs_int, state_indices=[0, 1, 2], z_idx=0)\n",
    "fig.update_layout(title=\"Population at z = z_min — intuitive ordering (no adiabaton)\")\n",
    "fig.show(renderer='notebook_connected')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "mixing-angle-section",
   "metadata": {},
   "source": [
    "## Mixing angle evolution\n",
    "\n",
    "The mixing angle $\\theta(z, t) = \\arctan(\\Omega_p / \\Omega_c)$ at the\n",
    "input face rotates smoothly from $0$ (coupling dominates, dark state\n",
    "$\\approx |0\\rangle$) to $\\pi/2$ (probe dominates, dark state\n",
    "$\\approx |2\\rangle$) as the pulse sequence passes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "plot-mixing-angle",
   "metadata": {},
   "outputs": [],
   "source": [
    "Omega_p_in = abs(mbs_ctr.Omegas_zt[0, 0, :])\n",
    "Omega_c_in = abs(mbs_ctr.Omegas_zt[1, 0, :])\n",
    "# Only compute where the fields are appreciable\n",
    "mask = (Omega_p_in + Omega_c_in) > 0.5\n",
    "theta = np.where(mask, np.arctan2(Omega_p_in, Omega_c_in), np.nan)\n",
    "\n",
    "fig = go.Figure(layout=go.Layout(\n",
    "    title=\"Mixing angle θ(t) at z = z_min\",\n",
    "    xaxis_title=\"time (γ⁻¹)\",\n",
    "    yaxis_title=\"θ (rad)\",\n",
    "    template=\"maxwellbloch\",\n",
    "))\n",
    "fig.add_trace(go.Scatter(x=t, y=theta, mode=\"lines\", name=\"θ = arctan(Ω_p / Ω_c)\"))\n",
    "fig.update_layout(yaxis=dict(tickvals=[0, np.pi/4, np.pi/2],\n",
    "                              ticktext=[\"0\", \"π/4\", \"π/2\"]))\n",
    "fig.show(renderer='notebook_connected')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "summary",
   "metadata": {},
   "source": [
    "## Summary\n",
    "\n",
    "- **Intuitive ordering (probe first):** the probe arrives before the dark\n",
    "  state is established → strong absorption, high $\\rho_{11}$, incomplete\n",
    "  population transfer.\n",
    "- **Counterintuitive ordering (coupling first, overlapping):** the\n",
    "  dark state adiabatically follows $\\theta$ from $0$ to $\\pi/2$,\n",
    "  transferring all population $|0\\rangle \\to |2\\rangle$ while\n",
    "  $\\rho_{11} \\approx 0$ throughout. Both fields reshape into the\n",
    "  **adiabaton** solution — neither is strongly absorbed.\n",
    "- The overlap of the two pulses is essential: if they are separated by\n",
    "  much more than the pulse width, the coupling is absorbed before the\n",
    "  probe arrives and the dark state never forms.\n",
    "\n",
    "Adiabatons are the propagating, adiabatic analog of STIRAP in\n",
    "single-atom physics: the medium self-consistently maintains the adiabatic\n",
    "condition as the pulses travel.\n",
    "\n",
    "## References\n",
    "\n",
    "1. R. Grobe, F. T. Hioe, J. H. Eberly, *Formation of Shape-Preserving\n",
    "   Pulses in a Nonlinear Adiabatically Integrable System*,\n",
    "   PRL **73**, 3183 (1994). Prediction of adiabatons.\n",
    "2. A. Rahman, J. H. Eberly, *Theory of adiabatons in coherent optical\n",
    "   pulse propagation*, PRA **58**, R805 (1998). Analytic structure.\n",
    "3. K. Bergmann, H. Theuer, B. W. Shore, *Coherent population transfer\n",
    "   among quantum states of atoms and molecules*, Rev. Mod. Phys. **70**,\n",
    "   1003 (1998). STIRAP review — the single-atom counterpart."
   ]
  }
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