Atomic optics / Ground-state populations

Optical Pumping and Relaxation

How can light measure how fast atoms relax in the dark? Pump them with polarized light, then read them with a weak probe after different waits in the dark. Pumping puts the atoms into particular ground states, and relaxation slowly evens the populations out. How much probe light gets through depends on the populations; comparing it with a beam that bypasses the cell removes the source brightness. Readings at several waits give the relaxation rate.

Use polarized light to move atoms among three ground states, wait in the dark, then count the probe photons that get through, along with a blank reference. Repeat the preparation many times to work out how fast the atoms relax.

The atom has an illustrative closed transition from a ground level with angular momentum one to an excited level with angular momentum zero. The two circular components of the light act incoherently, spontaneous decay is equally likely to reach each ground state, and ground-state relaxation is isotropic. The excited state is adiabatically eliminated. This is not a model of rubidium, and it does not include coherent population trapping. The simulated populations are shown only as a diagnostic; the estimates use only the recorded photon counts.

In the main arm, light passes through the polarization mixer and the cell to the transmission counter. A calibrated bypass of equal intensity feeds the reference counter. The bias field points along the beam. Light paths and cell size are schematic.

Repeated preparations, raw photon counts

How long does the prepared population last?

Every delay starts with fresh atoms whose populations are uniform. Each cycle pumps them, waits in the dark, then applies a brief weak probe. One exposure adds up many identical cycles, and the probe backaction is negligible. Each point gives two independent Poisson counts; the atoms are not lit during the wait.

Ready. Predict, then acquire.

Recorded counts and the source expectation

Both channels are kept in one record.

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

Teal is the transmitted probe and gold is the bypass reference. Dots are recorded rates, thin curves are source expectations, and gray marks saved counts.

Inference from a conditional count likelihood

Infer the decay from the records.

Inferred ground relaxation

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Inferred preparation amplitude

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Inferred relaxation time

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Deviance per degree

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Recorded pairs

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

The points come from the recorded counts. The purple dashed curve is the conditional-likelihood fit, and the fine teal curves are source expectations. The fit uses none of the source parameters. The ratio is only for display; the fit works with integer counts.

The unknown prepared population and the relaxation are fitted together. If the deviance improves on a flat response by less than sixteen, the result is marked weak and the relaxation readout is withheld. This cutoff is a rule of thumb, not a significance threshold.

A best fit that lands on a search boundary is flagged. The deviance and Pearson residuals only describe the fit; the Lab gives no parameter interval and claims no historical precision.

Source diagnostic, not a reconstruction

A dark subspace can contain two states.

m=-1

0.333

m=0

0.333

m=1

0.333

These source probabilities change with the chosen dark delay. Light of pure positive helicity removes population only from the negative sublevel; the other two sublevels are dark. The probabilities are not inferred from the single probe channel. The unpolarized preset is an incoherent mixture of circular polarizations, not coherent dark-state trapping by linear light.

Save, restore and reuse a run

Latest six raw observations
#	Delay / s	Probe	Reference

The CSV keeps raw counts, exposure, seed, units and every setting. The source settings are simulator metadata. One comparison is kept for as long as this page stays open.

The model and its scientific limits

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}

The model has three ground-state populations and one excited state that is eliminated from the equations, with equal decay branches. The ensemble is uniform before every cycle. The rate description is secular and incoherent, so Raman coherence is left out. The pump rate is prescribed, uniform and calibrated; depletion of the pump light is not computed. The weak probe follows Beer–Lambert attenuation with a calibrated depth of exactly 0.8, and each pulse changes the populations negligibly. No real isotope, other ground manifolds, diffusion, spin exchange or field-induced precession is included.

TeachSpin · Optical Pumping manual · source of the preparation and relaxation method. Our closed three-state model is a deliberate simplification, not the manual’s eight-state rubidium apparatus.

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

Start with the essentials

Focus question: How can light measure how fast atoms relax in the dark?
One-sentence intuition: Keep what the simulation assumes separate from what you estimate from the recorded data.

Core mathematical model

Rate equations for optical pumping

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

The excited state decays back to each of the three ground sublevels with equal probability. Every transfer keeps the total population fixed.

Probe transmission after a waiting time

\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 light addresses one ground sublevel, the bright one. The fit does not know the amplitude, not even its sign.

Removing the unknown light intensity

\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}

The transmitted and reference counts are independent Poisson counts with the same unknown incident mean. Conditioning on their sum removes that unwanted mean at each delay.

Common difficulties

The simulation already contains the law

Typical misconceptionIf I recover the parameter the simulation used, I have independently confirmed the law.

Better mental modelThese simulated records test the method of measuring and inferring. They are not historical observations, they are not a confidence interval for the parameter, and they do not count as outside scientific certification.