ONE ELECTRON / CONTINUOUS STERN–GERLACH

Single-Electron Penning Trap & Spin Readout

How can the spin of a single trapped electron be read from the frequency of its motion? A weak magnetic bottle makes the trap's restoring force slightly stronger or weaker depending on the spin direction, so the electron's axial oscillation has one of two nearby frequencies. A spin flip shows up as a jump between them. Counting flips during a microwave scan finds the spin resonance; comparing it with the measured cyclotron frequency gives the size of the electron's magnetic moment.

Calibrate the two axial frequencies that go with the two spin states. Then scan a microwave resonance, reading the axial frequency before and after each drive, and combine the measured trap frequencies to find the size of the electron's magnetic moment.

Prepare, deflect, then infer the moment

How can the spin of a single trapped electron be read from the frequency of its motion?

Uniform magnetic field

B_0/\mathrm{T}

5

Axial baseline frequency

\nu_{z,0}/\mathrm{MHz}

60

Magnetic bottle coefficient

B_2/(\mathrm{T}\,\mathrm{m}^{-2})

180

Source frequency noise · calibrated from records

\sigma_\nu/\mathrm{Hz}

0.12

Source magnetic-moment magnitude · excluded from fit

|\mu_e|/\mu_B

1.00116

Trials per drive setting

N_{\rm trial}

80

Inspect retained trial · no acquisition change

i

0

Inspection keeps every record. Acquisition changes clear current data; a saved comparison remains in memory.

The trap is an ideal static quadrupole Penning trap. The electron stays in its orbital ground state, and the weak magnetic bottle is treated to first order. The frequency noise is synthetic and white, state preparation is ideal, and the resonance follows a phenomenological incoherent model. Cavity effects, relativistic corrections, thermal cyclotron excitation, drift, quantum measurement backaction and the systematic corrections of precision work are left out. The orbit is drawn enlarged and the animation is schematic. The Lab does not reproduce 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, all observations kept

Prepare, drive, then read out again.

Ready to collect simulated data.

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

The raw data are the frequency jumps.

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

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 using stated calibrations

Estimate only what the data can 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 with a 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 records

Check the binomial residuals.

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

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

\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 field holds the electron along the trap axis, and the static magnetic field keeps its radial motion stable.

\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}

The magnetic bottle adds a restoring force that depends on the spin direction, so the two spin states have different axial frequencies. The orbital magnetic moment stays 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}

The measured trap frequencies and the center of the fitted microwave resonance give the size of the moment. The electron's moment points opposite to its 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 Fan et al. · Electron magnetic moment CERN · Trap and image-current readout

Molecular-Beam Proton Magnetic Moment Quantum evidence collection

Start with the essentials

Focus question: How can the spin of a single trapped electron be read from the frequency of its motion?
One-sentence intuition: Keep what the simulation assumes about the source apart from what you estimate from the recorded data.

Core mathematical model

Frequencies of the static trap

\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 field holds the electron along the trap axis, and the static magnetic field keeps its radial motion stable.

Reading the spin from the axial frequency

\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}

The magnetic bottle adds a restoring force that depends on the spin direction, so the two spin states have different axial frequencies. The orbital magnetic moment stays fixed.

The moment from a 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}

The measured trap frequencies and the center of the fitted microwave resonance give the size of the moment. The electron's moment points opposite to its spin.

Common difficulties

A simulation already contains its law

Typical misconceptionGetting a source parameter back out of the data independently confirms the law.

Better mental modelThe synthetic data test a measurement and inference procedure. They are not historical observations, they give no confidence interval for the parameter, and they are not external scientific certification.