Λ-Type Three-Level: Coherent Population Trapping (CPT)

Author

Thomas Ogden

When two coherent fields simultaneously drive a three-level Λ system, the atoms can be optically pumped into a dark state — a coherent superposition of the two ground states that is completely decoupled from the excited state.

Level structure

         |1⟩ ──────────────── (excited)
        /    \
   Ω_p /      \ Ω_c
      /        \     γ₁₀ = γ₁₂ = 1.0
    |0⟩        |2⟩  (ground states)

Physics

The Hamiltonian in the rotating frame has a zero-eigenvalue eigenstate — the dark state — when the two-photon resonance condition holds (\(\delta_p = \delta_c\)):

\[ |D\rangle = \frac{\Omega_c\,|0\rangle - \Omega_p\,|2\rangle} {\sqrt{\Omega_p^2 + \Omega_c^2}} \]

Atoms pumped into \(|D\rangle\) stop scattering photons entirely. The probe absorption spectrum then shows a narrow transparency window centred on \(\delta_p = \delta_c\): the CPT (or EIT) dip.

The width of the dip is set by the ground-state decoherence rate \(\gamma_g\), which can be much narrower than the excited-state linewidth \(\gamma_{10}\). This makes CPT-based dark resonances extremely sharp — the basis for CPT atomic clocks and all-optical magnetometry.

import numpy as np
from clerq import mb_solve, plot

Without coupling: Doppler-broadened absorption

First we solve for the probe alone (coupling Rabi frequency = 0, i.e. no coupling field). The probe sees a Doppler-broadened Lorentzian — the baseline against which the CPT dip will be compared.

A thermal velocity distribution is included (thermal_std = 10 γ) to reproduce realistic lineshape broadening.

mb_solve_json_no_coupling = """
{
  "atom": {
    "num_states": 3,
    "decays": [
      {"channels": [[0, 1]], "rate": 1.0},
      {"channels": [[1, 2]], "rate": 1.0}
    ],
    "fields": [
      {
        "label": "probe",
        "coupled_levels": [[0, 1]],
        "detuning": 0.0,
        "detuning_positive": true,
        "rabi_freq": 1.0e-3,
        "rabi_freq_t_func": "gaussian",
        "rabi_freq_t_args": {"ampl": 1.0, "centre": 0.0, "fwhm": 1.5}
      },
      {
        "label": "coupling",
        "coupled_levels": [[1, 2]],
        "detuning": 0.0,
        "detuning_positive": false,
        "rabi_freq": 0.0,
        "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": 200,
  "z_min": 0.0,
  "z_max": 1.0,
  "z_steps": 10,
  "z_steps_inner": 2,
  "interaction_strengths": [2.0, 0.0],
  "velocity_classes": {
    "thermal_width": 10.0,
    "thermal_delta_min": -30.0,
    "thermal_delta_max":  30.0,
    "thermal_delta_steps": 12,
    "thermal_delta_inner_min": -3.0,
    "thermal_delta_inner_max":  3.0,
    "thermal_delta_inner_steps": 6
  },
  "savefile": "mbs-lambda-cpt-no-coupling"
}
"""

mbs_no_c = mb_solve.MBSolve().from_json_str(mb_solve_json_no_coupling)
mbs_no_c.mbsolve(recalc=False)
print("Done")

Done

With coupling: CPT dark resonance

Now we add a strong CW coupling field (\(\Omega_c = 5\gamma\)) that is switched on before the probe pulse arrives. The two-photon resonance condition (\(\delta_p = \delta_c = 0\)) is satisfied on resonance, so a sharp transparency window opens at \(\delta_p = 0\).

mb_solve_json_coupling = """
{
  "atom": {
    "num_states": 3,
    "decays": [
      {"channels": [[0, 1]], "rate": 1.0},
      {"channels": [[1, 2]], "rate": 1.0}
    ],
    "fields": [
      {
        "label": "probe",
        "coupled_levels": [[0, 1]],
        "detuning": 0.0,
        "detuning_positive": true,
        "rabi_freq": 1.0e-3,
        "rabi_freq_t_func": "gaussian",
        "rabi_freq_t_args": {"ampl": 1.0, "centre": 0.0, "fwhm": 1.5}
      },
      {
        "label": "coupling",
        "coupled_levels": [[1, 2]],
        "detuning": 0.0,
        "detuning_positive": false,
        "rabi_freq": 5.0,
        "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": 200,
  "z_min": 0.0,
  "z_max": 1.0,
  "z_steps": 10,
  "z_steps_inner": 2,
  "interaction_strengths": [2.0, 0.0],
  "velocity_classes": {
    "thermal_width": 10.0,
    "thermal_delta_min": -30.0,
    "thermal_delta_max":  30.0,
    "thermal_delta_steps": 12,
    "thermal_delta_inner_min": -3.0,
    "thermal_delta_inner_max":  3.0,
    "thermal_delta_inner_steps": 6
  },
  "savefile": "mbs-lambda-cpt-coupling"
}
"""

mbs_c = mb_solve.MBSolve().from_json_str(mb_solve_json_coupling)
mbs_c.mbsolve(recalc=False)
print("Done")

Done

Absorption spectra: the CPT dip

The two traces are overlaid below. The broad Doppler-broadened background is the same in both cases. With the coupling field on, a narrow transparency window — the dark resonance — appears at \(\delta_p = 0\).

fig = plot.spectrum_overlay(
    [mbs_no_c, mbs_c],
    field_idx=0,
    labels=["coupling off", "coupling on (Ω_c = 5γ)"],
    freq_range=8,
)
fig.update_layout(title="CPT dark resonance in a Λ system: probe absorption")
fig.show(renderer='notebook_connected')

Probe propagation with coupling

The space-time plot of the probe shows that the medium becomes transparent for the resonant frequency component: the probe pulse propagates with much less attenuation than in the bare two-level case.

fig = plot.field_spacetime(mbs_c, field_idx=0)
fig.update_layout(title="Probe |Ω_p(z, t)| — CPT (Ω_c = 5γ)")
fig.show(renderer='notebook_connected')

Dark-state population

Once the atoms are in the dark state they accumulate in the two ground states. The plot below shows \(\rho_{00}\) and \(\rho_{22}\) at the input face, illustrating the coherent redistribution of population driven by the coupling field.

fig = plot.population(mbs_c, state_indices=[0, 1, 2], z_idx=0)
fig.update_layout(title="Ground/excited populations at z = z_min (CPT, Ω_c = 5γ)")
fig.show(renderer='notebook_connected')

Summary

Without coupling: Doppler-broadened Lorentzian absorption of width \(\sim 2\gamma \sqrt{1 + (\sigma_D/\gamma)^2}\).
With coupling: a narrow CPT dip of width \(\sim \gamma_g \ll \gamma\) appears at the two-photon resonance. The dip depth approaches unity as the coupling strength increases.
Atoms accumulate in the dark state: \(\rho_{11} \to 0\), population is distributed between \(|0\rangle\) and \(|2\rangle\) with ratio set by \(\Omega_p^2 : \Omega_c^2\).

References

E. Arimondo and G. Orriols, Non-absorbing atomic coherences by coherent two-photon transitions in a three-level optical pumping, Lett. Nuovo Cimento 17, 333 (1976). First observation of CPT.
G. Alzetta, A. Gozzini, L. Moi, G. Orriols, An experimental method for the observation of r.f. transitions and laser beat resonances in oriented Na vapour, Nuovo Cimento B 36, 5 (1976).
E. Arimondo, Coherent Population Trapping in Laser Spectroscopy, Prog. Opt. 35, 257 (1996). Comprehensive review.