Counter-Propagating Fields

Many atomic physics experiments use fields that propagate in opposite directions through the medium. A common example is Rydberg EIT, where a near-infrared probe and a blue coupling field enter the vapour cell from opposite ends to reduce (or eliminate) the Doppler broadening of the two-photon resonance.

Why a counter-propagating geometry changes the Doppler shift

An atom moving with velocity \(v\) along the propagation axis sees each field at a Doppler-shifted frequency. For a field with wave-vector \(\mathbf{k}\):

  • Co-propagating (\(\mathbf{k} \parallel \hat{z}\)): the atom sees the field blue-shifted by \(kv\), so its effective one-photon detuning shifts by \(+kv\).

  • Counter-propagating (\(\mathbf{k} \antiparallel \hat{z}\)): the shift is \(-kv\).

In a two-photon (Ξ or Λ) scheme, the two-photon detuning is the sum of the individual detunings. When both fields propagate in the same direction, the two-photon detuning picks up \(+kv + kv = 2kv\), and the resonance is Doppler-broadened. When the fields counter-propagate with similar wavelengths, the shifts partially or fully cancel, giving a narrower (Doppler-reduced or Doppler-free) two-photon resonance.

How MaxwellBloch handles this

MaxwellBloch propagates all fields in the \(+z\) direction (it solves an initial-value problem marching from \(z_\mathrm{min}\) to \(z_\mathrm{max}\)). A genuinely counter-propagating field would require boundary data at \(z_\mathrm{max}\) and backward integration, turning the problem into a boundary-value problem.

However, when the counter-propagating field does not deplete significantly across the cell—a valid approximation for many coupling fields in EIT experiments—the forward-propagating solver is equivalent to the true counter-propagating geometry. The only physical difference is the sign of the Doppler shift each velocity class contributes to that field. MaxwellBloch exposes this sign via the counter_propagating field attribute.

Note on phrasing. This is not an ‘undepleted-coupling approximation’ in the sense of neglecting absorption. The solver computes absorption honestly for every velocity class. The approximation is purely geometric: it exploits the spatial symmetry that holds when the counter-propagating field’s intensity profile is essentially flat across the cell.

Basic usage: counter_propagating

Set "counter_propagating": true on any field in the JSON configuration. This tells MaxwellBloch to apply the Doppler shift with the opposite sign for that field, while leaving the forward-propagation geometry unchanged.

[1]:

mb_solve_json = """
{
  "atom": {
    "num_states": 3,
    "decays": [{"channels": [[0, 1], [1, 2]], "rate": 1.0}],
    "fields": [
      {
        "label": "probe",
        "coupled_levels": [[0, 1]],
        "rabi_freq": 1.0e-3,
        "rabi_freq_t_func": "gaussian",
        "rabi_freq_t_args": {"ampl": 1.0, "centre": 0.0, "fwhm": 1.0}
      },
      {
        "label": "coupling",
        "coupled_levels": [[1, 2]],
        "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},
        "counter_propagating": true
      }
    ]
  },
  "t_min": -2.0,
  "t_max": 8.0,
  "t_steps": 120,
  "z_min": 0.0,
  "z_max": 1.0,
  "z_steps": 20,
  "z_steps_inner": 2,
  "interaction_strengths": [1.0, 1.0],
  "velocity_classes": {
    "thermal_width": 5.0,
    "thermal_delta_min": -15.0,
    "thermal_delta_max": 15.0,
    "thermal_delta_steps": 10
  },
  "savefile": "counter-propagating-eit"
}
"""

[2]:

from maxwellbloch import mb_solve

mbs = mb_solve.MBSolve().from_json_str(mb_solve_json)
print(f"Coupling field counter_propagating: {mbs.atom.fields[1].counter_propagating}")
print(f"Coupling factor_doppler_shift:      {mbs.atom.fields[1].factor_doppler_shift}")

Coupling field counter_propagating: True
Coupling factor_doppler_shift:      -1.0

[3]:

Omegas_zt, states_zt = mbs.mbsolve()

/home/docs/checkouts/readthedocs.org/user_builds/maxwellbloch/envs/v0.11.0/lib/python3.11/site-packages/qutip/solver/solver_base.py:598: FutureWarning: e_ops will be keyword only from qutip 5.3 for all solver
  warnings.warn(
100%|██████████| 20/20 [00:12<00:00,  1.57z/s, Ω_max=31.4]

Saving MBSolve to counter-propagating-eit.qu

Depletion check

After solving, MaxwellBloch automatically checks whether the counter-propagating field depleted significantly across the cell. The check computes

\[\text{depletion} = 1 - \frac{\max_t |\Omega(z_\mathrm{max}, t)|}{\max_t |\Omega(z_\mathrm{min}, t)|}\]

  • < 1 % — check passes silently. The forward-propagation result is reliable.

  • 1 – 10 % — a UserWarning is emitted. Results may still be usable but caution is advised.

  • > 10 % — a CounterPropagatingDepletionError is raised. The approximation has broken down.

You can suppress the check with check_counter_prop_depletion=False if you know the depletion is acceptable for your purposes.

[4]:

import numpy as np

coupling_idx = 1  # second field
Omega_coupling = Omegas_zt[coupling_idx]
peak_in  = np.max(np.abs(Omega_coupling[0]))
peak_out = np.max(np.abs(Omega_coupling[-1]))
depletion = 1.0 - peak_out / peak_in
print(f"Coupling field depletion: {depletion:.2%}")

Coupling field depletion: -0.00%

Advanced: factor_doppler_shift

For schemes where the two counter-propagating fields have different wavelengths (e.g. a three-photon Doppler-free ladder where \(k_1 \neq k_2\)), the Doppler cancellation is only partial. Use factor_doppler_shift directly to set the fractional weight:

{
  "coupled_levels": [[2, 3]],
  "rabi_freq": 1.5,
  "factor_doppler_shift": -0.62
}

The sign determines the direction; the magnitude scales the Doppler shift relative to the thermal width unit. factor_doppler_shift = 1.0 (the default) gives standard co-propagating behaviour; -1.0 is equivalent to counter_propagating = true.

Note on ``detuning_positive``. The detuning_positive flag on a field is a separate and orthogonal concept. It controls the sign convention for the RWA Hamiltonian based on the level ordering (used when the second coupled level sits below the first in energy, e.g. the lower leg of a Λ system). It has nothing to do with the laser k-vector direction. Both flags can be set independently.

Pulse-area check

As a sanity check, verify that the probe pulse area at the input is correct.

[5]:

probe_area_in = np.trapezoid(np.abs(Omegas_zt[0, 0, :]), mbs.tlist) / np.pi
print(f"Probe area at z=z_min: {probe_area_in:.4f} π")

Probe area at z=z_min: 0.0021 π

References

  1. Mohapatra, Jackson, Adams — Coherent Optical Detection of Highly Excited Rydberg States Using Electromagnetically Induced Transparency, PRL 98, 113003 (2007). Example of the counter-propagating Rydberg EIT geometry this feature supports.

  2. Adams, Pritchard, Shaffer — Rydberg atom physics, J. Phys. B 53, 012002 (2020). Broader review including Doppler-free multi-photon spectroscopy.