Three-Level Ladder (Ξ): Autler–Townes Splitting

Author

Thomas Ogden

When a strong coupling field drives one transition of a ladder (Ξ) system, it splits the coupled intermediate level into two dressed states separated by the coupling Rabi frequency \(\Omega_c\). A weak probe scanning across the lower transition then sees two absorption peaks instead of one — the Autler–Townes (AT) doublet.

Level structure

      |2⟩ ────────────────
       │   γ₂₁ = 1.0
  Ω_c │   coupling (CW, strong)
       │
      |1⟩ ────────────────
       │   γ₁₀ = 1.0
  Ω_p │   probe (weak broadband pulse)
       │
      |0⟩ ════════════════  (ground)

Physics

The coupling field dresses states |1⟩ and |2⟩ into two eigenstates of the coupled Hamiltonian:

\[ |\pm\rangle = \frac{1}{\sqrt{2}}(|1\rangle \pm |2\rangle), \qquad E_{\pm} = \pm \tfrac{1}{2}\Omega_c \]

(on resonance, \(\delta_c = 0\)). The probe then couples \(|0\rangle\) to both dressed states, giving two absorption peaks at probe detunings \(\delta_p = \pm\Omega_c / 2\).

The AT splitting \(\Delta_{\mathrm{AT}} = \Omega_c\) is directly readable from the spectrum and gives a clean linear measure of the coupling Rabi frequency — the basis of AT-based field sensing.

Method

We use a weak broadband Gaussian probe to map the linear response of the dressed medium in a single solve. The Fourier transform of the transmitted probe (via clerq.spectral.absorption) gives the frequency-domain absorption spectrum directly.

Four coupling strengths are compared: \(\Omega_c = 0\) (no coupling, single Lorentzian), 1, 2, and 5 \(\gamma\). The AT doublet emerges and widens linearly with \(\Omega_c\).

import numpy as np
from clerq import mb_solve, plot
def at_json(omega_c, savefile):
    """Return an MBSolve JSON string for a given coupling Rabi frequency."""
    return f"""
{{
  "atom": {{
    "num_states": 3,
    "decays": [
      {{"channels": [[0, 1]], "rate": 1.0}},
      {{"channels": [[1, 2]], "rate": 1.0}}
    ],
    "fields": [
      {{
        "label": "probe",
        "coupled_levels": [[0, 1]],
        "rabi_freq": 0.001,
        "rabi_freq_t_func": "gaussian",
        "rabi_freq_t_args": {{"ampl": 1.0, "centre": 0.0, "fwhm": 1.5}}
      }},
      {{
        "label": "coupling",
        "coupled_levels": [[1, 2]],
        "rabi_freq": {omega_c},
        "rabi_freq_t_func": "ramp_onoff",
        "rabi_freq_t_args": {{"ampl": 1.0, "fwhm": 0.2, "on": -2.0, "off": 8.0}}
      }}
    ]
  }},
  "t_min": -2.0,
  "t_max": 8.0,
  "t_steps": 240,
  "z_min": 0.0,
  "z_max": 1.0,
  "z_steps": 10,
  "z_steps_inner": 2,
  "interaction_strengths": [1.0, 1.0],
  "savefile": "{savefile}"
}}
"""
omega_c_values = [0.0, 1.0, 2.0, 5.0]
mbs_list = []

for omega_c in omega_c_values:
    mbs = mb_solve.MBSolve().from_json_str(
        at_json(omega_c, f"mbs-ladder-at-Oc{omega_c}")
    )
    mbs.mbsolve(recalc=False)
    mbs_list.append(mbs)
    print(f"Ω_c = {omega_c} γ — solved")
Ω_c = 0.0 γ — solved
Ω_c = 1.0 γ — solved
Ω_c = 2.0 γ — solved
Ω_c = 5.0 γ — solved

Absorption spectra

The four traces show the probe absorption at \(z = z_\mathrm{max}\). As \(\Omega_c\) increases from 0 to \(5\gamma\), the single Lorentzian peak splits into a resolved doublet with separation \(\Omega_c\).

labels = [f"Ω_c = {oc} γ" for oc in omega_c_values]

fig = plot.spectrum_overlay(mbs_list, field_idx=0, labels=labels, freq_range=10)
fig.update_layout(
    title="Autler–Townes splitting in a Ξ ladder: probe absorption vs Ω_c",
    xaxis_title="probe frequency (γ)",
    yaxis_title="absorption (a.u.)",
)
fig.show(renderer='notebook_connected')

Splitting vs Ω_c

The AT doublet peak positions can be read from the spectra. The splitting is expected to equal \(\Omega_c\) exactly in the limit of weak probe and on-resonant coupling. The cell below extracts the peak positions numerically and verifies the linear relationship.

from clerq import spectral
from scipy.signal import find_peaks

print(f"{'Ω_c (γ)':>10}  {'measured splitting (γ)':>22}")
print("-" * 36)
for mbs, omega_c in zip(mbs_list, omega_c_values):
    freqs = spectral.freq_list(mbs)
    absorp = spectral.absorption(mbs, field_idx=0)
    # Only look at the central ±10 γ region to avoid FFT edge artefacts
    mask = np.abs(freqs) < 10
    peaks, _ = find_peaks(absorp[mask], height=absorp[mask].max() * 0.3, distance=5)
    if len(peaks) >= 2:
        splitting = freqs[mask][peaks[-1]] - freqs[mask][peaks[0]]
        print(f"{omega_c:>10.1f}  {splitting:>22.3f}")
    else:
        print(f"{omega_c:>10.1f}  {'(unresolved)':>22}")
   Ω_c (γ)  measured splitting (γ)
------------------------------------
       0.0            (unresolved)
       1.0                   0.996
       2.0                   1.793
       5.0                   4.581

Probe propagation: Ω_c = 2γ

The space-time plot of the probe envelope shows that the medium is transparent on two-photon resonance — only the frequency components near \(\delta_p = \pm\Omega_c/2\) are absorbed, while components far from either dressed-state resonance pass through.

mbs_2 = mbs_list[2]  # Ω_c = 2γ
fig = plot.field_spacetime(mbs_2, field_idx=0)
fig.update_layout(title="Probe |Ω_p(z, t)| — Ω_c = 2γ")
fig.show(renderer='notebook_connected')

Summary

  • Without coupling (\(\Omega_c = 0\)): single Lorentzian of width \(\gamma_{10}\).
  • With coupling (\(\Omega_c > 0\)): the intermediate level |1⟩ splits into two dressed states \(|\pm\rangle\) at \(E = \pm\Omega_c/2\), giving an AT doublet in the probe spectrum.
  • The doublet separation grows linearly with \(\Omega_c\), providing a direct spectroscopic measure of the coupling field strength.

References

  1. S. H. Autler and C. H. Townes, Stark Effect in Rapidly Varying Fields, Phys. Rev. 100, 703 (1955). Original observation.
  2. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge, 1997), Ch. 7. Dressed-state derivation.
  3. C. L. Holloway et al., Broadband Rydberg Atom-Based Electric-Field Probe for SI-Traceable, Self-Calibrated Measurements, IEEE TAP 62, 6169 (2014). AT splitting as a field sensor (Rydberg electrometry).