ATOMIC OPTICS / GROUND POPULATIONS

Optical Pumping and Relaxation

Redistribute three ground-state populations with polarized light, wait in darkness, then count transmitted probe photons and a blank reference. Infer relaxation from repeated preparation cycles.

Illustrative closed angular-momentum-one to zero transition. Incoherent circular components, equal spontaneous branches and isotropic ground relaxation; the excited state is adiabatically eliminated. It is neither a rubidium model nor coherent population trapping. Source populations are a diagnostic; estimates use only recorded photon counts.

Main arm: polarization mixer → cell → transmission counter. A calibrated equal-intensity bypass feeds the reference counter. The bias field follows the beam. Paths and cell size are schematic.

REPEATED PREPARATIONS / RAW PHOTONS

How long does preparation survive?

Each delay uses fresh, initially uniform atoms: pump, dark wait, then a brief weak probe. Exposure accumulates many identical cycles, with negligible probe backaction. Two independent Poisson counts per point; this is not continuous illumination during the wait.

Ready. Predict, then acquire.

RECORDED COUNTS / SOURCE EXPECTATION

Two channels, one retained record.

$t_d/\mathrm{s},\quad N/t_{\rm exp}\;(\mathrm{s^{-1}})$

Teal: transmitted probe. Gold: bypass reference. Dots: recorded rates. Thin curves: source expectations. Gray: saved counts.

CONDITIONAL COUNT LIKELIHOOD / INFERENCE

Infer the decay from the records.

Inferred ground relaxation

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Inferred preparation amplitude

\text{—}

Inferred relaxation time

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Deviance per degree

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Recorded pairs

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$T_i=N_i/N_{{\rm ref},i}$

Points use the recorded counts. Purple dashes use the conditional likelihood fit; fine teal curves are source expectations. The fit excludes all source parameters. The ratio is only for display; the fit uses integer counts.

The unknown prepared population and relaxation are fitted jointly. A flat-response improvement below sixteen is marked weak and withholds the relaxation readout; it is a heuristic diagnostic, not a significance threshold.

Boundary optima are flagged. Deviance and Pearson residuals are descriptive; no parameter interval or historical precision is claimed.

SOURCE DIAGNOSTIC / NOT RECONSTRUCTED

A dark subspace can contain two states.

m=-1

0.333

m=0

0.333

m=1

0.333

These source probabilities follow the selected dark delay. Pure positive helicity removes population only from the negative sublevel; the other two are dark. They are not inferred from one probe channel. The unpolarized preset is an incoherent circular mixture, not coherent linear-light dark-state trapping.

SAVE / RESTORE / REUSE

Latest six raw observations
#	Delay / s	Probe	Reference

CSV keeps raw counts, exposure, seed, units and all settings. Source settings are simulator metadata. One comparison remains in this page session.

MODEL / SCIENTIFIC BOUNDARY

Which lifetime does light reveal?

\begin{aligned}\dot p_m&=\frac{1}{3}\sum_j r_jp_j-r_mp_m\\&\quad+\gamma(1/3-p_m)\\r_{-1}&=R(1+h)/2,\quad r_0=0\\r_{+1}&=R(1-h)/2\end{aligned}

\begin{aligned}p_b(t_d)&=1/3+A e^{-\gamma t_d}\\T(t_d)&=\exp[-0.8p_b(t_d)]\end{aligned}

Three ground populations, one eliminated excited state, equal decay branches. The prepared ensemble is uniform before every cycle. A secular, incoherent rate description excludes Raman coherence. A homogeneous calibrated pump rate is prescribed; pump depletion is not solved. The weak probe obeys Beer–Lambert attenuation with exact calibrated depth 0.8 and negligible population change per pulse. No real isotope, other ground manifolds, diffusion, spin exchange or field-induced precession is included.

TeachSpin · Optical Pumping manual · preparation and relaxation method; our closed three-state model is a deliberate reduction, not its eight-state rubidium apparatus.

Read optical methods and their history Measure an excited-state lifetime

Start with the essentials

Focus question: Prepare with light. Read after the dark.
One-sentence intuition: Keep source assumptions separate from estimates made from records.

Core mathematical model

Conservative optical transfer

\dot p_m=\frac13\sum_jr_jp_j-r_mp_m+\gamma(1/3-p_m)

Excited-state decay returns equal probability to each of three ground sublevels. All transfers conserve total population.

Prepare, wait, probe

\begin{aligned}p_b(t_d)&=1/3+A e^{-\gamma t_d}\\T(t_d)&=e^{-d p_b(t_d)},\quad d=0.8\end{aligned}

The probe selects one bright ground sublevel. The amplitude is unknown to inference, including its sign.

Eliminate photon brightness

\begin{aligned}K_i&=N_i+N_{{\rm ref},i}\\N_i\mid K_i&\sim\operatorname{Binomial}\!\left(K_i,\frac{T_i}{1+T_i}\right)\end{aligned}

Independent Poisson transmission and reference counts share an unknown incident mean. Conditioning eliminates that nuisance mean at each delay.

Common difficulties

A simulation supplies its law

Typical misconceptionRecovering a source parameter independently confirms the law.

Better mental modelThese synthetic records test a measurement and inference workflow. They are not historical observations, a parameter confidence interval, or external scientific certification.

Run the experiment

01
Measure dark relaxation
Scan all waiting times after the default positive-helicity preparation.
What to observe: Transmission relaxes toward the uniform-population baseline. The fitted rate reads only photon records.
02
Reverse the probe
Save a run, select the opposite-probe preset, then scan again.
What to observe: The signal can approach the same baseline from the other side. A dark population is relative to a selected transition.
03
Remove information
Select no preparation, then acquire the same delay grid.
What to observe: A nearly flat trace cannot reliably identify a relaxation rate. More points do not replace preparation contrast.