ATOMIC OPTICS / EXCITED COHERENCE

Hanle Lifetime Spectrometer

Excite an ideal atom with linearly polarized light. Scan a transverse magnetic field and count two polarizations of side fluorescence. Recover the excited-state lifetime and an unknown stray field.

Weak excitation of an ideal zero–one–zero angular-momentum transition, exponential radiative decay and uniform field. Two equal-efficiency polarization channels share one collection direction. The known polarization contrast is calibrated. No isotope, collisions, radiation trapping or real historical data are fitted.

Excitation travels horizontally; fluorescence is collected upward, parallel to the scanned field. A polarizing splitter sends pump-parallel light straight to the upper receiver and crossed light sideways to the other receiver.

REPEATED PREPARATIONS / RAW PHOTONS

Count both polarizations.

At each applied field, both equal-efficiency polarization channels count simultaneously. Poisson counts fluctuate independently; conditioning on their sum removes unknown brightness. Playback is accelerated.

Ready. Predict, then acquire.

RECORDED COUNTS / SOURCE EXPECTATION

Two channels, one retained record.

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

Teal: pump-parallel polarization. Gold: crossed polarization. Dots: recorded rates. Thin curves: source expectations. Gray: saved counts.

CONDITIONAL COUNT LIKELIHOOD / INFERENCE

Separate lifetime and field offset.

Inferred radiative lifetime

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Inferred zero-field position

\text{—}

Inferred half-width

\text{—}

Deviance per degree

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Recorded pairs

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$P_i=\frac{N_{\parallel,i}-N_{\perp,i}}{N_{\parallel,i}+N_{\perp,i}}$

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.

Lifetime and zero-field position are searched independently of the simulated source. Known factor magnitude 1.5 and contrast are assumed exact.

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

SAVE / RESTORE / REUSE

Latest six raw observations
#	Applied field / microtesla	Parallel	Crossed

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}P(B)&=\frac{C}{1+(2\omega_L\tau)^2}\\\omega_L&=1.5\frac{\mu_B}{\hbar}(B-B_0)\\\Delta B_{1/2}&=\frac{\hbar}{3\mu_B\tau}\end{aligned}

The factor two comes from a squared dipole projection, not a second Larmor frequency. Excitation travels along the first horizontal axis, its electric vector along the other horizontal axis; field and fluorescence collection are vertical. The two polarization channels partition all collected photons, so their total expectation is field independent. Exponential radiative decay is the only coherence loss. Geometry and optical frequency are schematic; no finite-aperture averaging, collisions, saturation, hyperfine structure, radiation trapping or detector imbalance. An actual experiment may measure an effective coherence time instead.

University of Washington · Hanle apparatus and derivation · Togawa et al. · experimental Hanle lifetime method · method references; this is not their apparatus or data.

Read optical methods and their history Measure ground-state relaxation

Start with the essentials

Focus question: A magnetic width becomes a lifetime.
One-sentence intuition: Keep source assumptions separate from estimates made from records.

Core mathematical model

Average a rotating dipole

\begin{aligned}P/C&=\int_0^\infty\frac{e^{-t/\tau}}{\tau}\cos(2\omega_Lt)\,dt\\&=\frac1{1+(2\omega_L\tau)^2}\end{aligned}

The polarization difference contains a squared dipole projection. Averaging over exponential emission ages creates a Lorentz field width.

A field width gives a lifetime

\begin{aligned}\omega_L&=g\mu_B(B-B_0)/\hbar,\quad g=1.5\\\Delta B_{1/2}&=\frac{\hbar}{2g\mu_B\tau}\end{aligned}

Applied field is in teslas in this equation. The source zero is fitted along with lifetime; the magnetic factor is calibrated.

Keep both polarizations

\begin{aligned}K_i&=N_{\parallel,i}+N_{\perp,i}\\N_{\parallel,i}\mid K_i&\sim\operatorname{Binomial}\!\left(K_i,\frac{1+P_i}{2}\right)\end{aligned}

The total mean stays constant in the chosen observation geometry. Conditioning removes brightness but retains the lifetime-dependent polarization.

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
Find a zero-field peak
Scan negative and positive applied fields and compare both channels.
What to observe: Their sum stays statistically flat while their difference peaks around the stray-field compensation point.
02
Lengthen the lifetime
Save the scan and select the longer-lifetime preset.
What to observe: A longer lifetime allows more precession at a smaller field, producing a narrower polarization curve.
03
Lose the polarization contrast
Set calibrated contrast to zero and scan again.
What to observe: Equal channels contain no lifetime information; a high total photon count alone is insufficient.