---
title: "Zeeman Atomic-beam Slower Interactive Lab | Sandbox Physics"
description: "Keep a slowing atom within reach of the light. Which atoms can follow a changing resonance, and what does an exit fluorescence scan establish?"
canonical_url: "https://sandboxphysics.com/labs/zeeman-atomic-beam-slower/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/zeeman-atomic-beam-slower/"
last_modified: "2026-10-18"
---

DOPPLER / ZEEMAN / EXIT PROBE

# Zeeman Atomic-beam Slower

Send a selected sodium beam through a graded solenoid against a fixed-frequency laser. Predict capture, retain a field-free exit fluorescence scan, then recover a coarse velocity distribution from those counts and a declared probe response.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Synthetic selected Gaussian beam and steady-state closed two-level mean force. The effective transition slope is one Bohr magneton. Recoil diffusion, hyperfine pumping, transverse motion, collisions, magnetic forces and light depletion are omitted. Probe brightness is an assumed calibration; a dark atom is invisible. The coarse regularized inference is not a unique reconstruction, a temperature measurement or historical data. Geometry is enlarged and motion slowed.

PREPARE VIEW · slowing light, then a blanked-pump exit probe gate.

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 / ACQUIRE / ANALYZE / EXPLAIN

## The exit light leaves the record.

Selected source Radiation-pressure slowing Field-free probe scan Count-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 counts Count-only fitted response Saved observations

The horizontal coordinate is a declared frequency-to-speed calibration, 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 / 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

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Visible in-band mean speed

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Mean squared standardized residual

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

The fit takes only retained counts, exposures and the declared probe calibration. It uses fixed regularization and can represent both slowed and uncaptured groups. The reported fraction is conditional on probe-visible in-band atoms. Missing brightness or out-of-band flux cannot be recovered, and a low residual does not establish a unique distribution.

SEPARATE MODEL REFERENCE / NOT A DETECTOR RECORD

### A resonance condition still needs a force budget.

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

Designed resonance branch Integrated mean-source trajectory

Physical coil length

$\text{—}$

Entrance / exit field

$\text{—}$

Available / demanded acceleration

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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 / RELATED EXPERIMENTS

[Phillips & Metcalf · 1982](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.48.596) [NIST apparatus history](https://www.nist.gov/nist-and-nobel/bill-phillips/context-discovery-bill-phillips) [Read the historical beam evidence](https://sandboxphysics.com/stories/quantum-evidence/zeeman-slowing-neutral-atom-beam/) [Cooling and MOT](https://sandboxphysics.com/labs/laser-cooling-magneto-optical-trap/)

Physics tutorial

## Zeeman Atomic-beam Slower

**Background** Send a selected sodium beam through a graded solenoid against a fixed-frequency laser. Predict capture, retain a field-free exit fluorescence scan, then recover a coarse velocity distribution from those counts and a declared probe response.

**Why it matters** A fixed laser stops slowing an atom when its changing Doppler shift moves it out of resonance. A graded magnetic field can move the transition along with it, but the available force still limits how quickly that resonance may change.

Start with the essentials

- **Focus question**: Which source velocities can follow this field, and what can a fluorescence spectrum establish when some atoms are dark or outside the scan?
- **One-sentence intuition**: The field defines a possible following branch; integrating the actual radiation force determines whether an atom follows. A separate exit probe supplies counts. The inverse calculation may use those counts and an independent probe calibration, but may not borrow the hidden trajectory as its answer.

### Core mathematical model

#### Choose the frequency signs

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

The positive axis runs from oven to exit. Slowing light travels against it, so a moving atom sees a positive Doppler contribution. The chosen transition increases in frequency with positive field. All terms in this equation use angular frequencies. The effective moment is a declared closed-transition approximation.

#### A mean force with a finite ceiling

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

Each absorbed photon removes forward momentum. Symmetric spontaneous emission has zero mean axial recoil; its nonzero variance is omitted here. The natural linewidth is 9.8 megahertz. Finite intensity gives a lower ceiling than the fully saturated bound, so increasing the coil gradient cannot supply missing force.

#### A designed branch is 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 uses entrance speed six hundred and exit speed one hundred metres per second. Its laser detuning is minus one hundred fifty megahertz, and its stable offset is calibrated for saturation two and demand forty percent. Changing demand changes length; actual detuning and intensity remain independent controls. Ring density only illustrates this prescribed axial field; coil currents and fringe fields are not solved.

#### Audit an exiting trajectory

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

The solver integrates kinetic energy along distance and independently accumulates scattering and force work. These two ledgers audit valid exiting branches. A speed below half a metre per second is a declared numerical stopping event, not a simulated trapped atom; its unused final step is excluded from the conservation claim. Recoil diffusion is absent, so no temperature follows from the energy loss.

#### The probe maps frequency to velocity

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

Repeated identical exit subensembles are read in stroboscopic gates after blanking the slowing light. An ideal shield gives a field-free exit plane; the enlarged drawing does not model finite drift. The incoming probe has negative axial projection; fluorescence is collected transversely. Its assumed total linewidth is twenty-five megahertz, including natural and laser broadening. Gain, background, angle and linewidth are exact calibrations. Weak-probe mechanical back action is neglected.

#### Condition the inference on what can be seen

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

A fixed one hundred forty iterations of a nonnegative Poisson response fit regularize ninety-one velocity bins. Complete coverage is required. Summaries are withheld for background-only scans, large residuals or edge weight. The fraction concerns probe-visible in-band atoms, rather than incident capture efficiency. Unknown brightness and unseen velocities remain unidentifiable. The residual statistic is a mean square diagnostic, not a chi-squared confidence test.

### Common difficulties

#### Magnetic field is not the braking force

**Typical misconception** The solenoid directly removes the atoms' kinetic energy.

**Better mental model** In this model the axial braking momentum comes from photon absorption. The field keeps the transition near resonance; magnetic-gradient forces are omitted. With the laser off, all source speeds pass unchanged.

#### A narrow fitted peak is not a thermometer

**Typical misconception** The inferred width supplies the final three-dimensional temperature.

**Better mental model** This directed beam has a prescribed longitudinal distribution and no transverse or recoil diffusion model. Probe resolution and regularization also shape fitted widths. No equilibrium temperature is reported.

#### Model verification is not historical reproduction

**Typical misconception** Recovering a synthetic spectrum validates the 1982 experiment independently.

**Better mental model** Source, force and detector are invented teaching models informed by the primary method. The historical apparatus had multilevel atoms and pumping complications. Internal conservation and inverse-fit checks audit this simulation; no historical data were refitted.

### Run the experiment

1. 01

   #### Predict the two groups

   Select Following. Inspect overview, solenoid side and exit-probe detail. Predict whether the high-speed tail will slow, then acquire a full scan.

   **What to observe:** A slow group and an uncaptured fast group can coexist. The fluorescence spectrum and the coarse inference come from the retained probe counts, while the lower trajectory panel is a separate source-informed reference.

2. 02

   #### Break the force budget

   Save the scan, select Too steep and reacquire. Compare available and demanded acceleration, the velocity reference, raw peaks and the saved gray observations.

   **What to observe:** The prescribed branch still ends at the design speed, but the actual atoms do not have enough radiation pressure to stay on it. A changed field does not impose a chosen velocity on an atom.

3. 03

   #### Retain an interrupted measurement

   Start a new scan, stop partway, export raw CSV and save. Change a control, restore the saved run, and resume to complete the same seeded scan.

   **What to observe:** Every acquired count survives. Unscanned frequencies are missing data, not zeros. Inference waits for the full band; restored settings, seed and retained records reproduce the same continuation exactly.

4. 04

   #### Separate darkness from slowing

   Compare Laser off, Too fast and Dark beam. Reduce exposure and repeat a scan. Explain which conclusion each spectrum supports.

   **What to observe:** A displaced peak supports a changed visible velocity distribution within this calibrated model. Lost fluorescence alone does not prove slowing, loss or trapping. Longer exposure reduces count noise but cannot reveal a completely dark population.

Physical conclusion

Synthetic selected Gaussian beam and steady-state closed two-level mean force. The effective transition slope is one Bohr magneton. Recoil diffusion, hyperfine pumping, transverse motion, collisions, magnetic forces and light depletion are omitted. Probe brightness is an assumed calibration; a dark atom is invisible. The coarse regularized inference is not a unique reconstruction, a temperature measurement or historical data. Geometry is enlarged and motion slowed.

Quantum & fields · Starter path

## Start from a single measurement

Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)