Rb 87 D2: Cycling Transition, σ⁺ Optical Pumping

The D2 line of \(^{87}\)Rb connects the \(5S_{1/2}\) ground state to the \(5P_{3/2}\) excited state at 780 nm. With circularly polarised (\(\sigma^+\), \(q=+1\)) light tuned to the \(F=2 \to F'=3\) transition, the atoms are optically pumped into the \(m_F=+2\) cycling state — the sub-transition \(|F=2,\,m_F=+2\rangle \to |F'=3,\,m_{F'}=+3\rangle\) from which the only allowed decay is back to \(|F=2,\,m_F=+2\rangle\).

Level structure

5P₃/₂  F'=3  mF = -3 -2 -1  0 +1 +2 +3
               ─────────────────────────
                    σ⁺ (q=+1) drive
5S₁/₂  F=2   mF = -2 -1  0 +1 +2
               ─────────────────

The \(F=2 \to F'=3\) transition is closed under \(\sigma^+\): spontaneous decay from any \(m_{F'}\) can only reach \(m_F \leq m_{F'}\), so population cascades toward \(m_F=+2\) over many scattering cycles.

Hyperfine structure construction

The hyperfine.Atom1e helper computes the Clebsch-Gordan-weighted coupling and decay factors for the full \(m_F\) sublevel structure. This eliminates manual look-up of angular-momentum coefficients.

[1]:

import numpy as np
from maxwellbloch import hyperfine, mb_solve, plot

Build the hyperfine structure

[2]:

# D2 line: 5S₁/₂ (J=1/2) → 5P₃/₂ (J=3/2), I = 3/2 for ⁸⁷Rb
Rb87_5s12_F2 = hyperfine.LevelF(I=1.5, J=0.5, F=2)
Rb87_5p32_F3 = hyperfine.LevelF(I=1.5, J=1.5, F=3)

atom1e = hyperfine.Atom1e(element="Rb", isotope="87")
atom1e.add_F_level(Rb87_5s12_F2)   # index 0: F=2, 5 mF sublevels
atom1e.add_F_level(Rb87_5p32_F3)   # index 1: F'=3, 7 mF sublevels

NUM_STATES = atom1e.get_num_mF_levels()
print(f"Total mF sublevels: {NUM_STATES}")  # 5 + 7 = 12

Total mF sublevels: 12

[3]:

# σ⁺ field: couples F=2 → F'=3 with q=+1 (Δm_F = +1)
q = 1
FIELD_CHANNELS = atom1e.get_coupled_levels(F_level_idxs_a=(0,), F_level_idxs_b=(1,))
FIELD_FACTORS  = atom1e.get_clebsch_hf_factors(F_level_idxs_a=(0,), F_level_idxs_b=(1,), q=q)
DECAY_CHANNELS = atom1e.get_coupled_levels(F_level_idxs_a=(0,), F_level_idxs_b=(1,))
DECAY_FACTORS  = atom1e.get_decay_factors(F_level_idxs_a=(0,), F_level_idxs_b=(1,))
ENERGIES       = atom1e.get_energies()

# Initial state: uniform across F=2 (mF = -2, -1, 0, +1, +2)
INITIAL_STATE = [1.0/5.0]*5 + [0.0]*7

# Detuning: on resonance with F=2 → F'=3
DETUNING = 0.0

print(f"Field channels: {len(FIELD_CHANNELS)}")
print(f"sum(|d_i|²) field factors: {np.sum(FIELD_FACTORS**2):.4f}")
print(f"sum(|d_i|²) decay factors: {np.sum(DECAY_FACTORS**2):.4f}")

Field channels: 35
sum(|d_i|²) field factors: 1.1667
sum(|d_i|²) decay factors: 3.5000

Solve: weak σ⁺ probe, optical pumping

[4]:

mb_solve_json = """
{{
  "atom": {{
    "decays": [
      {{
        "channels": {decay_channels},
        "rate": 1.0,
        "factors": {decay_factors}
      }}
    ],
    "energies": {energies},
    "fields": [
      {{
        "coupled_levels": {field_channels},
        "factors": {field_factors},
        "detuning": {detuning},
        "detuning_positive": true,
        "label": "probe",
        "rabi_freq": 1e-3,
        "rabi_freq_t_args": {{
          "ampl": 1.0,
          "centre": 0.0,
          "fwhm": 1.0
        }},
        "rabi_freq_t_func": "gaussian"
      }}
    ],
    "num_states": {num_states},
    "initial_state": {initial_state}
  }},
  "t_min": -2.0,
  "t_max": 10.0,
  "t_steps": 100,
  "z_min": -0.2,
  "z_max": 1.2,
  "z_steps": 100,
  "interaction_strengths": [1.0e2],
  "savefile": "mbs-Rb87_5s12_5p32_F23_q1-weak-pulse-decay"
}}
""".format(
    num_states=NUM_STATES,
    energies=ENERGIES,
    initial_state=INITIAL_STATE,
    detuning=DETUNING,
    field_channels=FIELD_CHANNELS,
    field_factors=FIELD_FACTORS.tolist(),
    decay_channels=DECAY_CHANNELS,
    decay_factors=DECAY_FACTORS.tolist(),
)

mbs = mb_solve.MBSolve().from_json_str(mb_solve_json)
mbs.mbsolve(recalc=False)
print("Done")

Done

/home/docs/checkouts/readthedocs.org/user_builds/maxwellbloch/envs/v0.12.0/lib/python3.11/site-packages/maxwellbloch/mb_solve.py:344: UserWarning: Savefile was built with maxwellbloch==0.10.0, current version is 0.12.0.
  self.load_results()

Probe propagation

The space-time plot shows the weak probe pulse propagating through the Rb vapour. On resonance, the pulse is absorbed as it passes through the dense medium.

[5]:

fig = plot.field_spacetime(mbs, field_idx=0)
fig.update_layout(title="Probe |Ω(z, t)| — Rb 87 D2, F=2→F'=3, σ⁺")
fig.show(renderer='notebook_connected')

Optical pumping: mF sublevel populations

The plot below shows the 12 sublevel populations at the input face as a function of time. States 0–4 are the \(F=2\) ground sublevels (\(m_F = -2, -1, 0, +1, +2\)) and states 5–11 are the \(F'=3\) excited sublevels (\(m_{F'} = -3, \ldots, +3\)).

Starting from a uniform distribution across \(F=2\), \(\sigma^+\) excitation preferentially drives \(\Delta m_F = +1\) transitions. After several absorption/emission cycles, population accumulates in \(m_F = +2\) — the cycling state.

[6]:

fig = plot.population(mbs, state_indices=list(range(NUM_STATES)), z_idx=0)
fig.update_layout(title="mF sublevel populations at z = z_min — D2 σ⁺ optical pumping")
fig.show(renderer='notebook_connected')

Ground-state populations only

Zooming in on just the \(F=2\) ground sublevels makes the optical pumping dynamics clearer: \(m_F = +2\) grows at the expense of lower \(m_F\) states.

[7]:

fig = plot.population(mbs, state_indices=list(range(5)), z_idx=0)
fig.update_layout(
    title="F=2 ground sublevel populations at z = z_min — D2 σ⁺ optical pumping",
)
fig.show(renderer='notebook_connected')

Summary

The \(F=2 \to F'=3\) transition on the Rb 87 D2 line is a closed cycling transition under \(\sigma^+\) excitation: spontaneous emission from \(F'=3\) can only reach \(F=2\), not \(F=1\).
Starting from uniform \(m_F\) distribution, \(\sigma^+\) optical pumping drives population toward \(m_F=+2\) over multiple scattering cycles.
The \(|m_F=+2\rangle \to |m_{F'}=+3\rangle\) sub-transition is the true cycling transition: \(m_{F'}=+3\) can only decay to \(m_F=+2\) (the only allowed \(\Delta m_F \in \{-1, 0, +1\}\) channel from \(m_{F'}=+3\) to \(F=2\)), making atoms permanently trapped in the cycling loop.
The hyperfine.Atom1e module constructs the full \(m_F\) sublevel structure with correct Clebsch-Gordan-weighted coupling and decay factors, valid for any alkali \(nS_{1/2} \to nP_J\) transition.

References

1. 1. Steck, Rubidium 87 D Line Data (2001, revision 2.2.2, 2021). Comprehensive Rb 87 spectroscopic data including D2 hyperfine constants, transition dipole elements, and natural linewidth. Available at https://steck.us/alkalidata.
1. 1. Foot, Atomic Physics, Oxford University Press (2004). Chapter 7: laser cooling and optical pumping. Cycling transitions and the MOT.
1. 1. Metcalf and P. van der Straten, Laser Cooling and Trapping, Springer (1999). Optical pumping and rate equations, Chapter 4.