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Sandbox Physics

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 modelSingle-Electron Penning Trap & Spin Readout

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,νz−νz,0  (Hz)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.

(νd−νc,ref)/MHz,P^flip(\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.

ri=ki−nip^inip^i(1−p^i)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.

IndexPhaseDriveBeforeAfter
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.

ν±=νc±νc2−2νz22νc=eB02πme\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.

By=B0+B2y2Δνz=2∣μe∣B24π2meνz\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.

ν^c=ν+2+νz2+ν−2∣μ^e∣μB=νc,ref+Δνd^ν^c\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.

Pflip=12[1−exp⁡ ⁣(−2A1+(δ/w)2)]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 lectureFan et al. · Electron magnetic momentCERN · Trap and image-current readout

Molecular-Beam Proton Magnetic MomentQuantum evidence collection

Physics tutorial

Single-Electron Penning Trap & Spin Readout

BackgroundCalibrate 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 mattersA 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

ν±=νc±νc2−2νz22νc=eB02πme\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

By=B0+B2y2Δνz=2∣μe∣B24π2meνz\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

ν^c=ν+2+νz2+ν−2∣μ^e∣μB=νc,ref+Δνd^ν^c\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 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

  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.
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