---
title: "Single-Electron Penning Trap & Spin Readout Interactive Lab | Sandbox Physics"
description: "Read a spin through a frequency jump. Calibrate two axial-frequency levels, retain before-and-after frequency pairs, scan a microwave resonance and combine measured trap frequencies to infer the electron magnetic-moment magnitude."
canonical_url: "https://sandboxphysics.com/labs/penning-electron-spin-readout/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/penning-electron-spin-readout/"
last_modified: "2026-10-18"
---

ONE ELECTRON / CONTINUOUS STERN–GERLACH

# Single-Electron Penning Trap & Spin Readout

Calibrate two axial-frequency levels, retain before-and-after frequency pairs, scan a microwave resonance and combine measured trap frequencies to infer the electron magnetic-moment magnitude.

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

Ideal static quadrupole Penning trap; fixed ground orbital state; first-order weak magnetic bottle. Synthetic white frequency noise, ideal state preparation and a phenomenological incoherent resonance. No cavity, relativistic, thermal cyclotron, drift, quantum measurement backaction or precision systematic corrections. Enlarged orbit and animation are schematic; this is not a recreation of a precision g-factor result.

Static confinement; the field axis is vertical. The electron orbit is greatly enlarged and slowed independently of real frequency ratios. The pickup senses image current; no electron hits a detector.

SYNTHETIC RECORDS / RETAINED OBSERVATIONS

### Prepare. Drive. Read again.

Ready to acquire synthetic records.

Sixty-four calibration records precede an interleaved 41-frequency scan. Every trial retains its measured before / after axial offset and the three measured trap frequencies. State preparation is ideal; acquisition is accelerated. The three trap-frequency channels are separate bottle-off calibrations. The fixed microwave reference is recorded in CSV.

RAW OBSERVATIONS

### Frequency jumps are the record.

$i,\quad \nu_z-\nu_{z,0}\;(\mathrm{Hz})$

First 240 drive trials, excluding calibration. Green: before; brown: after. Gray dashes show saved after-readings, clipped to the current plot range. Full records remain in CSV. The inspection slider selects one retained pair.

INFERENCE / DECLARED CALIBRATIONS

### Estimate only what the records resolve.

Inferred moment magnitude

$\text{—}$

Fitted spin resonance offset

$\text{—}$

Reconstructed cyclotron frequency

$\text{—}$

Calibrated axial separation

$\text{—}$

Calibrated frequency noise

$\text{—}$

Inspected before / after offset

$\text{—}$

CLASSIFIED TRIALS / BINOMIAL FIT

### Locate the microwave response.

$(\nu_d-\nu_{c,\rm ref})/\mathrm{MHz},\quad \widehat P_{\rm flip}$

A calibrated difference threshold classifies each frequency pair. A binomial likelihood fits center, width and drive strength. Error bars are a count scale, not a confidence interval. The source moment never enters the fit.

CHECK THE RECORD

### Keep the binomial discrepancies visible.

$r_i=\frac{k_i-n_i\widehat p_i}{\sqrt{n_i\widehat p_i(1-\widehat p_i)}}$

Residuals outside five count scales appear as red edge dots. This tests the chosen phenomenological response, not a fundamental magnetic law.

Inspect the last eight raw records

Drive offsets are in megahertz; before / after axial offsets are in hertz.

| Index | Phase | Drive | Before | After |
| --- | --- | --- | --- | --- |

Model, apparatus and primary evidence

Ideal static quadrupole Penning trap; fixed ground orbital state; first-order weak magnetic bottle. Synthetic white frequency noise, ideal state preparation and a phenomenological incoherent resonance. No cavity, relativistic, thermal cyclotron, drift, quantum measurement backaction or precision systematic corrections. Enlarged orbit and animation are schematic; this is not a recreation of a precision g-factor result.

$$
\begin{aligned}\nu_\pm&=\frac{\nu_c\pm\sqrt{\nu_c^2-2\nu_z^2}}2\\\nu_c&=\frac{eB_0}{2\pi m_e}\end{aligned}
$$

The electric quadrupole confines axially; the static magnetic field stabilizes radial motion.

$$
\begin{aligned}B_y&=B_0+B_2y^2\\\Delta\nu_z&=\frac{2|\mu_e|B_2}{4\pi^2m_e\nu_z}\end{aligned}
$$

A spin-dependent magnetic restoring force shifts the axial frequency. The orbital magnetic moment is fixed.

$$
\begin{aligned}\widehat\nu_c&=\sqrt{\nu_+^2+\nu_z^2+\nu_-^2}\\\frac{|\widehat\mu_e|}{\mu_B}&=\frac{\nu_{c,\rm ref}+\widehat{\Delta\nu_d}}{\widehat\nu_c}\end{aligned}
$$

Measured trap frequencies and the fitted microwave center determine the moment magnitude; the electron moment is antiparallel to spin.

Positive magnetic moment points upward and lowers the axial frequency in a positive bottle. The displayed prepared high-frequency state has negative moment; electron spin points oppositely. The constant orbital contribution is absorbed in the baseline. The microwave field is transverse to the static field. Frequency readout is an ideal calibrated image-current estimator, not a synthesized electrical waveform.

$$
P_{\rm flip}=\frac12\left[1-\exp\!\left(\frac{-2A}{1+(\delta/w)^2}\right)\right]
$$

The illustrative source uses drive strength 1.5 and a width of two megahertz plus 0.02 megahertz per unit bottle coefficient. This phenomenological broadening demonstrates a readout / linewidth tradeoff; it does not calculate the physical bottle lineshape. A symmetric readout error estimated from calibration corrects the binomial likelihood. Weak calibration or a fit boundary withholds the magnetic moment.

[Dehmelt · Nobel lecture](https://www.nobelprize.org/uploads/2018/06/dehmelt-lecture.pdf) [Fan et al. · Electron magnetic moment](https://arxiv.org/abs/2209.13084) [CERN · Trap and image-current readout](https://base.web.cern.ch/base-experiment)

[Molecular-Beam Proton Magnetic Moment](https://sandboxphysics.com/labs/molecular-beam-proton-moment/) [Quantum evidence collection](https://sandboxphysics.com/stories/quantum-evidence/)

Physics tutorial

## Single-Electron Penning Trap & Spin Readout

**Background** Calibrate two axial-frequency levels, retain before-and-after frequency pairs, scan a microwave resonance and combine measured trap frequencies to infer the electron magnetic-moment magnitude.

**Why it matters** A constant is inferred through a calibrated response and retained observations.

Start with the essentials

- **Focus question**: Read a spin through a frequency jump.
- **One-sentence intuition**: Keep source assumptions separate from estimates made from records.

### Core mathematical model

#### Static confinement

$$
\begin{aligned}\nu_\pm&=\frac{\nu_c\pm\sqrt{\nu_c^2-2\nu_z^2}}2\\\nu_c&=\frac{eB_0}{2\pi m_e}\end{aligned}
$$

The electric quadrupole confines axially; the static magnetic field stabilizes radial motion.

#### Spin readout

$$
\begin{aligned}B_y&=B_0+B_2y^2\\\Delta\nu_z&=\frac{2|\mu_e|B_2}{4\pi^2m_e\nu_z}\end{aligned}
$$

A spin-dependent magnetic restoring force shifts the axial frequency. The orbital magnetic moment is fixed.

#### Frequency ratio

$$
\begin{aligned}\widehat\nu_c&=\sqrt{\nu_+^2+\nu_z^2+\nu_-^2}\\\frac{|\widehat\mu_e|}{\mu_B}&=\frac{\nu_{c,\rm ref}+\widehat{\Delta\nu_d}}{\widehat\nu_c}\end{aligned}
$$

Measured trap frequencies and the fitted microwave center determine the moment magnitude; the electron moment is antiparallel to spin.

### Common difficulties

#### A simulation supplies its law

**Typical misconception** Recovering a source parameter independently confirms the law.

**Better mental model** These synthetic records test a measurement and inference workflow. They are not historical observations, a parameter confidence interval, or external scientific certification.

### Run the experiment

1. 01

   #### Calibrate the readout

   Acquire the resolved-jump preset. Inspect paired axial offsets.

   **What to observe:** Calibration uses both prepared magnetic-moment directions, without reading hidden trial spin labels.

2. 02

   #### Scan the resonance

   Compare inferred transitions with the retained frequency pairs. Move the inspection slider.

   **What to observe:** The resonance is fitted to classified records; source magnetic moment is excluded from the estimator.

3. 03

   #### Remove the bottle

   Select no readout coupling and acquire again.

   **What to observe:** The electron remains statically confined, but axial frequency no longer distinguishes spin.

Physical conclusion

Ideal static quadrupole Penning trap; fixed ground orbital state; first-order weak magnetic bottle. Synthetic white frequency noise, ideal state preparation and a phenomenological incoherent resonance. No cavity, relativistic, thermal cyclotron, drift, quantum measurement backaction or precision systematic corrections. Enlarged orbit and animation are schematic; this is not a recreation of a precision g-factor result.

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)