Doppler shift, Zeeman shift and exit probe

Zeeman Atomic-beam Slower

Which atoms does a Zeeman slower actually slow down, and what can the glow of atoms at its exit prove? Atoms that enter slow enough to meet the resonance and then keep up with it. Light pushes each atom back while the tapered magnetic field shifts its transition to track the falling Doppler shift, but no faster than the light can decelerate the atom. Too-fast atoms, or a too-steep taper, get left behind. An exit probe sees only atoms that still glow, so fading light alone does not prove slowing.

Send a velocity-selected sodium beam through a solenoid with a graded field, head-on into a fixed-frequency laser. Predict which atoms will be captured and slowed, record a fluorescence scan in the field-free region at the exit, and then work out a coarse velocity distribution using only those counts and the stated probe response.

A fixed laser frequency meets a changing resonance

Give the light enough distance.

Solenoid field scale

b

1

Design deceleration fraction

\eta

0.4

Slowing-laser detuning

\Delta/(2\pi\,\mathrm{MHz})

-150

Slowing-laser saturation

s

2

Selected source mean speed

\overline v_{\rm in}/(\mathrm{m\,s^{-1}})

550

Selected source speed spread

\sigma_{\rm in}/(\mathrm{m\,s^{-1}})

55

Probe-visible fraction · source model

q

1

Probe exposure per frequency

T/\mathrm{ms}

8

The deceleration setting sets the coil length and keeps the designed entrance and exit speeds. Changing the field scale or the laser detuning can break the compensation. The source has a prescribed Gaussian distribution of selected speeds; it is not derived from an oven temperature.

Settings changes clear the current scan. The saved comparison keeps exact settings, seed and counts. Range controls support arrow keys.

The source is an invented velocity-selected beam with a Gaussian speed distribution, and the atoms feel the steady-state mean force of a closed two-level transition. The transition's effective magnetic moment is one Bohr magneton. Recoil diffusion, hyperfine pumping, transverse motion, collisions, magnetic forces and light depletion are left out. Probe brightness is an assumed calibration, and an atom in a dark state gives no signal. The coarse, regularized velocity fit is not a unique reconstruction, does not measure a temperature and is not historical data. The drawing enlarges the geometry and slows the motion.

Preparation view · slowing light first, then the exit probe with the slowing light switched off.

Atoms travel from oven through collimators and graded coils to the shielded exit probe. Slowing light runs against the atoms; fluorescence goes sideways to a photon counter. Geometry enlarged, time slowed.

Predict first, then measure, analyze and explain

The record comes from light at the exit.

Selected sourceRadiation-pressure slowingField-free probe scanCount-only inference

At each frequency, repeated identically prepared exit subensembles are probed stroboscopically with slowing light blanked. The weak probe crosses the ideal field-free exit plane at forty-five degrees; its frequency selects longitudinal speed. A transverse collector counts fluorescence, not atoms. Exposure sums repeated read gates. The drawn exit spacing is magnified: no finite downstream drift is modeled. Every scan retains ninety-five Poisson count observations.

Ready. Predict which group reaches the exit, then acquire.

With the workbench focused: Space acquires or stops; R resets. Native controls keep their keyboard behavior. Leaving the workbench or hiding the page stops acquisition and retains counts.

No saved comparison.

Retained fluorescence · Poisson counts

Two velocity groups can leave two peaks.

$v_{\rm probe}/(\mathrm{m\,s^{-1}}),\quad C\;[\mathrm{counts}]$

Retained countsCount-only fitted responseSaved observations

The horizontal axis converts probe frequency to speed with an assumed calibration; it is not a measured atom trajectory. The probe has an assumed Lorentzian total linewidth of twenty-five megahertz, including natural and laser broadening. Counts and raw detunings remain in the CSV.

Nonnegative response fit in ten-metre-per-second bins

Infer a distribution, then inspect its residual.

$v/(\mathrm{m\,s^{-1}}),\quad \widehat w_j$

$v_{\rm probe}/(\mathrm{m\,s^{-1}}),\quad r_i=(C_i-\widehat C_i)/\sqrt{\widehat C_i}$

Retained probe frequencies

\text{—}

Calibrated visible in-band weight

\text{—}

Visible in-band fraction below threshold

\text{—}

Visible in-band mean speed

\text{—}

Mean squared standardized residual

\text{—}

$v_{\rm threshold}=200\,\mathrm{m\,s^{-1}},\qquad 0\leq v_{\rm fitted}\leq900\,\mathrm{m\,s^{-1}}$

The fit uses only the kept counts, the exposures and the assumed probe calibration. Its regularization is fixed, and it can represent both slowed and uncaptured groups. The reported fraction counts only atoms that the probe can see and that fall within the band. Dark atoms and out-of-band flux cannot be recovered, and a low residual does not prove the distribution is unique.

A separate model reference, not a detector record

Staying on resonance still needs enough force.

$z/L,\quad v/(\mathrm{m\,s^{-1}})$

Designed resonance branchTrajectory integrated from the mean source speed

Physical coil length

\text{—}

Entrance / exit field

\text{—}

Available / demanded acceleration

\text{—}

Mean-source trajectory exit speed

\text{—}

This panel uses source parameters and force integration. It is deliberately separate from the count-only inverse calculation. A large demand shortens the coil; finite laser intensity may then be unable to keep atoms on its branch. No final temperature or trapped-atom count is inferred.

Inspect the latest twelve raw observations

Scan index	$\Delta_p/(2\pi\,\mathrm{MHz})$	$T/\mathrm{ms}$	Counts

Primary evidence and related experiments

Phillips & Metcalf · 1982 NIST apparatus history Read the historical beam evidence Cooling and MOT

Physics tutorial

Zeeman Atomic-beam Slower

BackgroundSend a velocity-selected sodium beam through a solenoid with a graded field, head-on into a fixed-frequency laser. Predict which atoms will be captured and slowed, record a fluorescence scan in the field-free region at the exit, and then work out a coarse velocity distribution using only those counts and the stated probe response.

Why it mattersAs an atom slows, its Doppler shift changes, and a fixed-frequency laser soon falls out of resonance and stops slowing it. A graded magnetic field can shift the transition to keep up, but the light can only push so hard, and that limits how quickly the resonance may change along the beam.

Start with the essentials

Focus question: Which starting speeds can keep up with this field, and what can a fluorescence spectrum tell you when some atoms are dark or outside the scan?
One-sentence intuition: The field only sets the path an atom could follow; whether it does depends on the radiation force it actually feels along the way. The velocity estimate uses the counts from a separate exit probe and the probe's independent calibration, never the hidden simulated trajectories.

Core mathematical model

Detuning and its sign conventions

\begin{aligned}\delta(z,v)&=\Delta+kv-\mu_{\rm eff}B(z)/\hbar\\k&=2\pi/\lambda,\quad \mu_{\rm eff}=\mu_B\end{aligned}

The positive axis points from the oven to the exit. The slowing light travels the other way, so a moving atom sees a positive Doppler term. The chosen transition rises in frequency as the field increases. Every term is an angular frequency. The effective moment is an assumed value that treats the transition as closed.

The radiation force and its upper limit

\begin{aligned}R&=\frac{\Gamma}{2}\frac{s}{1+s+(2\delta/\Gamma)^2}\\F_z&=-\hbar kR,\quad a_{\max}=\frac{\hbar k\Gamma}{2m}\\|a|&\leq a_{\max}\frac{s}{1+s}\end{aligned}

Each absorbed photon takes away some forward momentum. Spontaneous emission is symmetric, so on average it gives no kick along the axis; the random spread of those kicks is left out here. The natural linewidth is 9.8 megahertz. At finite intensity the force tops out below the fully saturated limit, and a steeper coil gradient cannot make up the missing force.

The designed velocity profile is only a reference

\begin{aligned}v_d(z)&=\sqrt{v_i^2-2\eta a_{\max}z}\\L&=\frac{v_i^2-v_f^2}{2\eta a_{\max}}\\B(z)&=b\frac{\hbar}{\mu_B}\left(\Delta_d+kv_d-\delta_d\right)\end{aligned}

The design takes atoms from six hundred to one hundred metres per second. It assumes a laser detuning of minus one hundred fifty megahertz, and its stable offset is set for a saturation parameter of two and a deceleration of forty percent of the maximum. Asking for a different fraction changes the coil length; the actual detuning and intensity stay independent. The spacing of the drawn rings only illustrates this prescribed axial field; coil currents and fringe fields are not solved.

Momentum and energy checks for atoms that exit

\begin{aligned}m(v_{\rm in}-v_{\rm out})&=\hbar k\int R\,dt\\\frac{m}{2}(v_{\rm out}^2-v_{\rm in}^2)&=\int F_z\,dz\end{aligned}

The solver integrates kinetic energy along the beam and separately adds up the scattered photons and the work done by the force. For every atom that exits, both sums must match its change in momentum and energy. An atom that drops below half a metre per second is simply stopped by the solver; it is not a simulated trapped atom, and its unused last step is left out of the conservation check. Because recoil diffusion is not modeled, the energy loss gives no temperature.

The probe maps frequency to velocity

\begin{aligned}\delta_p&=\Delta_p+kv\cos\theta,\quad\theta=45^\circ\\K_{ij}&=\left[1+\left(\frac{2k\cos\theta(v_j-u_i)}{\Gamma_p}\right)^2\right]^{-1}\\\widehat C_i&=T_i\left[\beta+g\sum_jK_{ij}\widehat w_j\right]\end{aligned}

The slowing light is switched off during each stroboscopic read gate, and the probe reads many identically prepared groups of exiting atoms. An ideal shield makes the exit plane field-free; the drawing enlarges the gap, but no drift beyond the shield is modeled. The probe comes in with a component against the atoms' motion, and the fluorescence is collected from the side. Its total linewidth is assumed to be twenty-five megahertz, including natural and laser broadening. Gain, background, angle and linewidth are treated as exactly known. The weak probe's mechanical back action on the atoms is neglected.

The estimate covers only atoms the probe can see

\begin{aligned}C_i&\sim\operatorname{Poisson}(\widehat C_i)\\f_{\rm slow,vis}&=\frac{\sum_{v_j<200\,\mathrm{m\,s^{-1}}}\widehat w_j}{\sum_j\widehat w_j},\quad \widehat w_j\geq0\end{aligned}

A nonnegative Poisson response fit over ninety-one velocity bins runs for a fixed one hundred forty iterations, and that fixed count is what regularizes it. The scan must be complete. No summary is reported when the scan shows only background, the residuals are large or too much weight sits in the edge bins. The fraction refers to atoms the probe can see within the scanned band, not to how many incoming atoms were captured. An unknown brightness and velocities outside the scan cannot be determined. The residual statistic is a mean-square diagnostic, not a chi-squared confidence test.

Common difficulties

The light does the braking, not the magnet

Typical misconceptionThe magnetic field in the solenoid is what slows the atoms down.

Better mental modelIn this model the atoms lose forward momentum by absorbing photons. The field only keeps the transition close to resonance, and magnetic-gradient forces are left out. With the laser off, every atom passes through at its starting speed.

A narrow peak does not give a temperature

Typical misconceptionThe width of the fitted peak gives the atoms' final three-dimensional temperature.

Better mental modelThe beam has a prescribed speed distribution along its axis, and the model includes no transverse motion or recoil diffusion. The probe resolution and the regularization also change the fitted width. The Lab reports no equilibrium temperature.

Checking the model does not reproduce the original experiment

Typical misconceptionIf the fit recovers the simulated spectrum, that independently confirms the 1982 experiment.

Better mental modelThe source, force and detector here are invented teaching models based on the original method. The historical apparatus had to cope with multilevel atoms and optical pumping. The conservation and inverse-fit checks test only this simulation; no historical data were refitted.

Run the experiment

01
Predict the two groups
Select Following and look at the overview, the solenoid side view and the exit-probe detail. Predict whether the fast tail of the beam will slow down, then record a full scan.
What to observe: A slowed group and a fast group that was never captured can appear together. The fluorescence spectrum and the coarse velocity estimate come only from the recorded probe counts. The trajectory panel below is a separate reference that uses the source settings.
02
Make the field too steep
Save the scan, select Too steep and record again. Compare the deceleration the light can supply with the deceleration the field requires, then look at the velocity reference, the raw peaks and the saved scan shown in gray.
What to observe: The designed profile still ends at the design speed, but the radiation pressure is too weak to keep the atoms on it. Changing the field cannot force an atom to a chosen speed.
03
Keep an interrupted scan
Start a new scan, stop it partway, export the raw CSV and save. Change a control, restore the saved run and resume it to finish the same seeded scan.
What to observe: Every count already recorded is kept. Frequencies not yet scanned are missing data, not zeros. The fit waits until the whole band is scanned, and the restored settings, seed and saved records continue the scan exactly as before.
04
Tell a dark beam from a slowed one
Compare Laser off, Too fast and Dark beam. Shorten the exposure and scan again. Decide what each spectrum lets you conclude.
What to observe: Within this calibrated model, a shifted peak shows that the velocities of the visible atoms changed. Missing fluorescence on its own does not prove slowing, loss or trapping. A longer exposure lowers the counting noise but cannot reveal atoms that are completely dark.

Zeeman Atomic-beam Slower

The record comes from light at the exit.

Two velocity groups can leave two peaks.

Infer a distribution, then inspect its residual.

Staying on resonance still needs enough force.

Zeeman Atomic-beam Slower

Core mathematical model

Detuning and its sign conventions

The radiation force and its upper limit

The designed velocity profile is only a reference

Momentum and energy checks for atoms that exit

The probe maps frequency to velocity

The estimate covers only atoms the probe can see

Common difficulties

The light does the braking, not the magnet

A narrow peak does not give a temperature

Checking the model does not reproduce the original experiment

Run the experiment

Predict the two groups

Make the field too steep

Keep an interrupted scan

Tell a dark beam from a slowed one

Related experiments and articles

Clocks, traps and single-quantum control