Q024 · Field / polarization / spectrum

Coupled Spin Spectroscopy

Split a coupled electron–nuclear spin with a magnetic field. Turn the microwave drive, record a spectrum, and trace each allowed line back to its state.

Lowest model gap $\text{—}$

State / design reference $\text{—}$

01 / ASK THE ATOM WITH A FREQUENCY.

Raise the field. Watch the lines separate.

Scan with no static field. Save it. Then open the field and scan again. Turn the drive to see which transitions remain.

Effective spin pair · static field vertical · microwave drive rotates in the vertical plane · conceptual state readout

Probe frequency

h\nu/A_0

1.2

PrepareProbeChanged?

Ready. A scan records 121 frequencies, 200 fresh preparations each.

MAKE A COMPARISON

Static field

b

1

Prepared energy level

i

0

Drive angle to static field

\theta/{}^\circ

90

Level zero is the ground state; at zero field it is the singlet. Excited levels use the declared basis within degeneracies. Drive angle is measured from the static field axis.

Line width, coupling & sampling

Hyperfine coupling

A/A_0

1

Line width (FWHM)

h\Gamma/A_0

0.06

Sampling seed

s_0

24133

Tuning frequency keeps the records. Changing preparation starts a fresh sample; save first. Presets and camera views do not acquire data.

Illustrative pair of spin-one-half particles. Reference energy units and a declared nuclear magnetic ratio; no isotope calibration. Line counts use an incoherent weak-response model with ideal state preparation and binary readout.

02 / WHAT DID THE READOUT RECORD?

Build a spectrum, one bit at a time.

Show model reference

$h\nu/A_0$ · Blue: this sample. Orange: most recently saved other sample. Dashed: current model. Drag the cursor to tune; curves do not create records.

AT THE SELECTED FREQUENCY

\text{—}

Changed / preparations

—

\text{—}

95% Wilson interval for independent binary records at this frequency. No interval before sampling. These are pointwise, not simultaneous bands or optional-stopping guarantees.

Latest 60 recorded outcomes

Blue means changed; gray means unchanged. Every trial starts freshly prepared.

Recorded sample

03 / WHERE DO THE LINES COME FROM?

Levels move. The drive chooses the connections.

$b$ · $E/A_0$ · Model only. Drag horizontally; the matching slider also supports the keyboard.

Colored arrows: allowed transitions from the prepared level. Their thickness shows normalized strength. Gray ticks: other levels.

\text{—}

MODEL INSPECTION · NO COUNTS ADDED

One state, two ways to describe it.

Product order $\uparrow\uparrow,\ \uparrow\downarrow,\ \downarrow\uparrow,\ \downarrow\downarrow$

Coupled order $(F,m)=(0,0),(1,1),(1,0),(1,-1)$

These bars describe the same prepared eigenstate in two bases. They are probabilities, not an atom hopping between hidden configurations. Within a zero-field degenerate triplet the chosen eigenbasis is conventional; rotating that basis changes components but not total strength at the degenerate line.

04 / HOLD ON TO THE COMPARISON

Save a sample before changing the atom.

Up to eight frozen samples. Restore to inspect; acquiring again starts a new run. CSV includes each bit, actual probe frequency and preparation. Reusing the seed reproduces draws, not independent evidence.

Model, basis convergence & sources

Energy is in the reference unit; probe frequency is ordinary frequency multiplied by Planck’s constant, not angular frequency. Curves and level arrows are model predictions. Recorded bits are simulated independent preparation–probe–readout trials.

H/A_0=a\,\mathbf I\!\cdot\!\mathbf S+b(S_z-rI_z),\quad I=S=\tfrac12

O=S_z\cos\theta+S_x\sin\theta,\quad r=0.0015

The dimensionless spin operators use the product basis. Positive static field is vertical. The microwave magnetic axis lies in the horizontal–vertical plane; angle zero is longitudinal. A positive hyperfine coupling gives a singlet ground state at zero field. The nuclear ratio is illustrative. No isotope-specific constants, quadrupole term, Doppler broadening or optical pumping dynamics.

Steck · Hyperfine / Zeeman Hamiltonians and Breit–Rabi limits — operator conventions only; this pair is not rubidium.

\text{—}

p(\nu)=0.01+0.20\sum_{j\ne i}w_{ji}\left[1+4((\nu-\nu_{ji})/\Gamma)^2\right]^{-1}

w_{ji}=\frac{|\langle j|O|i\rangle|^2}{\sum_{k\ne i}|\langle k|O|i\rangle|^2}

Declared incoherent Lorentzian response: fixed one-percent false-change background and twenty-percent contrast budget. Width is FWHM. Strengths are normalized within the chosen preparation, so heights cannot compare absolute dipole moments across settings. All levels enter the normalization, including transitions outside the displayed band. This readout model is not a coherent pulse solver or a fitted instrument calibration.

Start with the essentials

Focus question: The levels exist. Why does a line disappear?
One-sentence intuition: A missing peak need not mean a missing energy level. The drive matrix element may vanish.

Core mathematical model

Coupled spin Hamiltonian

H/A_0=a\,\mathbf I\!\cdot\!\mathbf S+b(S_z-rI_z)

Dimensionless spin-one-half operators, positive coupling, and illustrative nuclear ratio. The static field defines the vertical axis.

Zero-field multiplets

E_{F=0}/A_0=-3a/4,\qquad E_{F=1}/A_0=a/4

One singlet and three degenerate triplet states. The zero-field splitting is the hyperfine coupling.

Probe operator

O=S_z\cos\theta+S_x\sin\theta

Turning the drive changes matrix elements without changing static energy levels. At zero field the choice of triplet basis is conventional; the total line strength is invariant.

Common difficulties

Peaks and states

Typical misconceptionEvery energy level must produce a visible peak.

Better mental modelPreparation, the probe operator, line width, frequency range and finite counts all matter. Lines outside the shown band remain in the strength normalization.

A model is not a calibrated instrument

Typical misconceptionA narrow simulated line establishes a long experimental lifetime.

Better mental modelThe line width and low-contrast Lorentzian response are declared inputs. The separate pulse test is a rotating-wave model, not inferred pulse dynamics or hardware noise.

Run the experiment

01
Predict before scanning
Choose no static field. How many distinct transition frequencies should survive the triplet degeneracy?
What to observe: The dashed curve is a model reference, not recorded data. Hide it for the prediction.
02
Record and save
Scan, save the sample, change one preparation setting, then scan again.
What to observe: Each frequency has fresh independent trials. Orange points retain the most recently saved preparation.
03
Explain the missing line
At finite field, switch between longitudinal and transverse drive while watching the allowed connections.
What to observe: Line positions are energy differences; visibility also requires a nonzero matrix element.