{
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"cell_type": "markdown",
"id": "intro",
"metadata": {},
"source": [
"# Three-Level Ladder (Ξ): Autler–Townes Splitting\n",
"\n",
"When a strong coupling field drives one transition of a ladder (Ξ)\n",
"system, it splits the coupled intermediate level into two dressed states\n",
"separated by the coupling Rabi frequency $\\Omega_c$. A weak probe scanning\n",
"across the lower transition then sees two absorption peaks instead of one\n",
"— the **Autler–Townes (AT) doublet**.\n",
"\n",
"## Level structure\n",
"\n",
"```\n",
" |2⟩ ────────────────\n",
" │ γ₂₁ = 1.0\n",
" Ω_c │ coupling (CW, strong)\n",
" │\n",
" |1⟩ ────────────────\n",
" │ γ₁₀ = 1.0\n",
" Ω_p │ probe (weak broadband pulse)\n",
" │\n",
" |0⟩ ════════════════ (ground)\n",
"```\n",
"\n",
"## Physics\n",
"\n",
"The coupling field dresses states |1⟩ and |2⟩ into two eigenstates of\n",
"the coupled Hamiltonian:\n",
"\n",
"$$\n",
"|\\pm\\rangle = \\frac{1}{\\sqrt{2}}(|1\\rangle \\pm |2\\rangle),\n",
"\\qquad E_{\\pm} = \\pm \\tfrac{1}{2}\\Omega_c\n",
"$$\n",
"\n",
"(on resonance, $\\delta_c = 0$). The probe then couples $|0\\rangle$ to\n",
"both dressed states, giving two absorption peaks at probe detunings\n",
"$\\delta_p = \\pm\\Omega_c / 2$.\n",
"\n",
"The AT splitting $\\Delta_{\\mathrm{AT}} = \\Omega_c$ is directly readable\n",
"from the spectrum and gives a clean linear measure of the coupling Rabi\n",
"frequency — the basis of AT-based field sensing."
]
},
{
"cell_type": "markdown",
"id": "method",
"metadata": {},
"source": [
"## Method\n",
"\n",
"We use a weak broadband Gaussian probe to map the linear response of the\n",
"dressed medium in a single solve. The Fourier transform of the transmitted\n",
"probe (via `maxwellbloch.spectral.absorption`) gives the frequency-domain\n",
"absorption spectrum directly.\n",
"\n",
"Four coupling strengths are compared: $\\Omega_c = 0$ (no coupling, single\n",
"Lorentzian), 1, 2, and 5 $\\gamma$. The AT doublet emerges and widens\n",
"linearly with $\\Omega_c$."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "imports",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"from maxwellbloch import mb_solve, plot"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "json-template",
"metadata": {},
"outputs": [],
"source": [
"def at_json(omega_c, savefile):\n",
" \"\"\"Return an MBSolve JSON string for a given coupling Rabi frequency.\"\"\"\n",
" return f\"\"\"\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",
" \"rabi_freq\": 0.001,\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",
" \"rabi_freq\": {omega_c},\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\": 240,\n",
" \"z_min\": 0.0,\n",
" \"z_max\": 1.0,\n",
" \"z_steps\": 10,\n",
" \"z_steps_inner\": 2,\n",
" \"interaction_strengths\": [1.0, 1.0],\n",
" \"savefile\": \"{savefile}\"\n",
"}}\n",
"\"\"\""
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "solve",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Ω_c = 0.0 γ — solved\n",
"Ω_c = 1.0 γ — solved\n",
"Ω_c = 2.0 γ — solved\n",
"Ω_c = 5.0 γ — solved\n"
]
}
],
"source": [
"omega_c_values = [0.0, 1.0, 2.0, 5.0]\n",
"mbs_list = []\n",
"\n",
"for omega_c in omega_c_values:\n",
" mbs = mb_solve.MBSolve().from_json_str(\n",
" at_json(omega_c, f\"mbs-ladder-at-Oc{omega_c}\")\n",
" )\n",
" mbs.mbsolve(recalc=False)\n",
" mbs_list.append(mbs)\n",
" print(f\"Ω_c = {omega_c} γ — solved\")"
]
},
{
"cell_type": "markdown",
"id": "spectrum-section",
"metadata": {},
"source": [
"## Absorption spectra\n",
"\n",
"The four traces show the probe absorption at $z = z_\\mathrm{max}$.\n",
"As $\\Omega_c$ increases from 0 to $5\\gamma$, the single Lorentzian\n",
"peak splits into a resolved doublet with separation $\\Omega_c$."
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "plot-spectra",
"metadata": {},
"outputs": [
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"source": [
"labels = [f\"Ω_c = {oc} γ\" for oc in omega_c_values]\n",
"\n",
"fig = plot.spectrum_overlay(mbs_list, field_idx=0, labels=labels, freq_range=10)\n",
"fig.update_layout(\n",
" title=\"Autler–Townes splitting in a Ξ ladder: probe absorption vs Ω_c\",\n",
" xaxis_title=\"probe frequency (γ)\",\n",
" yaxis_title=\"absorption (a.u.)\",\n",
")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "splitting-check",
"metadata": {},
"source": [
"## Splitting vs Ω_c\n",
"\n",
"The AT doublet peak positions can be read from the spectra. The splitting\n",
"is expected to equal $\\Omega_c$ exactly in the limit of weak probe and\n",
"on-resonant coupling. The cell below extracts the peak positions numerically\n",
"and verifies the linear relationship."
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "splitting-measure",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" Ω_c (γ) measured splitting (γ)\n",
"------------------------------------\n",
" 0.0 (unresolved)\n",
" 1.0 2.191\n",
" 2.0 1.793\n",
" 5.0 4.780\n"
]
}
],
"source": [
"from maxwellbloch import spectral\n",
"from scipy.signal import find_peaks\n",
"\n",
"print(f\"{'Ω_c (γ)':>10} {'measured splitting (γ)':>22}\")\n",
"print(\"-\" * 36)\n",
"for mbs, omega_c in zip(mbs_list, omega_c_values):\n",
" freqs = spectral.freq_list(mbs)\n",
" absorp = spectral.absorption(mbs, field_idx=0)\n",
" # Only look at the central ±10 γ region to avoid FFT edge artefacts\n",
" mask = np.abs(freqs) < 10\n",
" peaks, _ = find_peaks(absorp[mask], height=absorp[mask].max() * 0.3, distance=5)\n",
" if len(peaks) >= 2:\n",
" splitting = freqs[mask][peaks[-1]] - freqs[mask][peaks[0]]\n",
" print(f\"{omega_c:>10.1f} {splitting:>22.3f}\")\n",
" else:\n",
" print(f\"{omega_c:>10.1f} {'(unresolved)':>22}\")"
]
},
{
"cell_type": "markdown",
"id": "spacetime-section",
"metadata": {},
"source": [
"## Probe propagation: Ω_c = 2γ\n",
"\n",
"The space-time plot of the probe envelope shows that the medium is\n",
"transparent on two-photon resonance — only the frequency components near\n",
"$\\delta_p = \\pm\\Omega_c/2$ are absorbed, while components far from\n",
"either dressed-state resonance pass through."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "plot-spacetime",
"metadata": {},
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"source": [
"mbs_2 = mbs_list[2] # Ω_c = 2γ\n",
"fig = plot.field_spacetime(mbs_2, field_idx=0)\n",
"fig.update_layout(title=\"Probe |Ω_p(z, t)| — Ω_c = 2γ\")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "summary",
"metadata": {},
"source": [
"## Summary\n",
"\n",
"- Without coupling ($\\Omega_c = 0$): single Lorentzian of width $\\gamma_{10}$.\n",
"- With coupling ($\\Omega_c > 0$): the intermediate level |1⟩ splits into\n",
" two dressed states $|\\pm\\rangle$ at $E = \\pm\\Omega_c/2$, giving an AT\n",
" doublet in the probe spectrum.\n",
"- The doublet separation grows linearly with $\\Omega_c$, providing a direct\n",
" spectroscopic measure of the coupling field strength.\n",
"\n",
"## References\n",
"\n",
"1. S. H. Autler and C. H. Townes, *Stark Effect in Rapidly Varying Fields*,\n",
" Phys. Rev. **100**, 703 (1955). Original observation.\n",
"2. M. O. Scully and M. S. Zubairy, *Quantum Optics* (Cambridge, 1997),\n",
" Ch. 7. Dressed-state derivation.\n",
"3. C. L. Holloway et al., *Broadband Rydberg Atom-Based Electric-Field Probe\n",
" for SI-Traceable, Self-Calibrated Measurements*, IEEE TAP **62**, 6169\n",
" (2014). AT splitting as a field sensor (Rydberg electrometry)."
]
}
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