Atomic optics / Excited-state coherence

Hanle Lifetime Spectrometer

How can a magnetic field measure how long an excited atom lives? From the width of the Hanle dip. A transverse magnetic field makes the excited atoms' dipoles precess, and once they turn faster than they decay, the fluorescence loses its polarization. Plotting polarization against field gives the dip: the shorter the excited-state lifetime, the wider it is. A dip that is not centered on zero applied field reveals a stray field.

Excite an ideal atom with linearly polarized light, sweep a transverse magnetic field, and count the side fluorescence in two polarizations. From those counts, recover both the excited-state lifetime and an unknown stray field.

The atom has an ideal transition from angular momentum zero up to one and back down to zero. Excitation is weak, radiative decay is exponential and the field is uniform. Two polarization channels with equal efficiency collect light from the same direction, and their polarization contrast is known from calibration. The model does not represent a specific isotope, collisions or radiation trapping, and it fits no real historical data.

The exciting light travels horizontally, and fluorescence is collected upward, parallel to the scanned field. A polarizing beam splitter sends light polarized parallel to the pump straight on to the upper receiver and the crossed polarization sideways to the other receiver.

Repeated preparations, raw photon counts

Count both polarizations.

At each applied field, the two polarization channels, which have equal efficiency, count at the same time. Their Poisson counts fluctuate independently; conditioning on the sum of the two removes the unknown brightness. Playback is sped up.

Ready. Predict, then acquire.

Recorded counts and the source expectation

Both channels are kept in one record.

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

Teal is the polarization parallel to the pump and gold is the crossed polarization. Dots are recorded rates, thin curves are source expectations, and gray marks saved counts.

Inference from a conditional count likelihood

Tell the lifetime apart from the field offset.

Inferred radiative lifetime

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Inferred zero-field position

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Inferred half-width

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Deviance per degree

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Recorded pairs

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

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 search for the lifetime and the zero-field position does not read the simulated source. The known factor of magnitude 1.5 and the contrast are taken as exact.

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.

Save, restore and reuse a run

Latest six raw observations
#	Applied field / microtesla	Parallel	Crossed

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}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 of two comes from squaring the dipole projection; there is no second Larmor frequency. The exciting light travels along one horizontal axis with its electric field along the other; the magnetic field and the fluorescence collection are vertical. The two polarization channels share all collected photons between them, so their total expected count does not depend on the field. Exponential radiative decay is the only loss of coherence. Geometry and optical frequency are schematic. Finite-aperture averaging, collisions, saturation, hyperfine structure, radiation trapping and detector imbalance are left out. A real experiment may measure an effective coherence time instead.

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

Read optical methods and their history Measure ground-state relaxation

Start with the essentials

Focus question: How can a magnetic field measure how long an excited atom lives?
One-sentence intuition: Keep what the simulation assumes separate from what you estimate from the recorded data.

Core mathematical model

Averaging over the precessing 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 depends on the square of the dipole projection. Averaging over the emission times, which follow an exponential distribution, gives a Lorentzian width in field.

Lifetime from the field width

\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}

The applied field is in teslas in this equation. The zero-field position is fitted together with the lifetime, and the magnetic factor is taken as calibrated.

Using both polarization channels

\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}

In the chosen viewing geometry, the mean total count stays constant. Conditioning on the total removes the brightness but keeps the polarization, which depends on the lifetime.

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.

Run the experiment

01
Find the zero-field peak
Scan the applied field through negative and positive values and compare the two channels.
What to observe: Their sum stays flat within the noise, and their difference peaks near the field that cancels the stray field.
02
Lengthen the lifetime
Save the scan and choose the Longer lifetime preset.
What to observe: With a longer lifetime, the atoms precess further even in a smaller field, so the polarization curve gets narrower.
03
Remove the polarization contrast
Set the calibrated contrast to zero and scan again.
What to observe: When the two channels are equal, they carry no information about the lifetime. A large total photon count alone is not enough.

Hanle Lifetime Spectrometer

Count both polarizations.

Both channels are kept in one record.

Tell the lifetime apart from the field offset.

Which lifetime does light reveal?