Q024 · Coupled spin levels in a magnetic field

Coupled Spin Spectroscopy

Why can a spectral line be missing even though both levels exist? Two levels are not enough; the microwave drive must also couple them. With the chosen polarization, the matrix element between some pairs of states is zero, and those transitions never appear even though both levels are there.

Use a magnetic field to split the levels of a coupled electron and nuclear spin. Turn the microwave drive, record a spectrum, and trace each allowed line back to the states it connects.

Lowest model gap $\text{—}$

Singlet contribution $\text{—}$

01 / Probe the atom at one frequency

Raise the field and watch the lines separate.

Scan with the static field off and save it. Then turn the field on and scan again. Rotate 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 covers 121 frequencies with 200 fresh preparations at 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 keeps the records. Save before changing preparation; presets load a fresh simulated scan.

The model is an illustrative pair of spin-one-half particles. Energies are in reference units, and the nuclear magnetic ratio is a chosen value, not calibrated to any isotope. The line counts come from an incoherent weak-response model with ideal state preparation and a yes-or-no readout.

02 / What did the readout record?

Find the probe frequencies that change the state.

Show model reference

$h\nu/A_0$ · Blue: this sample. Orange: most recently saved other sample. Dashed: current model. Drag the cursor to tune.

At the selected frequency

\text{—}

Changed / preparations

—

\text{—}

Uncertainty at this frequency

95% Wilson interval for independent yes-or-no readouts at this frequency, shown only after sampling. Each interval holds point by point; together they do not form a simultaneous band, and they do not cover optional stopping.

Latest 60 recorded outcomes

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

Recorded sample

Energy levels and charge sensitivity

03 / Where do the lines come from?

The levels move, and the drive decides which pairs of levels it connects.

$b$ · $E/A_0$ · 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{—}

Inspect the prepared state and pulse

Model inspection · adds no counts

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.

Saved comparisons

How saved samples work

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}

The readout uses an assumed incoherent Lorentzian response with a fixed one-percent false-change background and a twenty-percent contrast shared among the transitions. Width is the full width at half maximum. Strengths are normalized within the chosen preparation, so peak heights cannot compare absolute dipole moments across settings. All levels enter the normalization, including transitions outside the displayed band. This readout model does not solve coherent pulse dynamics and is not a fitted instrument calibration.

Start with the essentials

Focus question: Why can a spectral line be missing even though both levels exist?
One-sentence intuition: A missing peak does not have to mean a missing level; the drive's matrix element may be zero.

Core mathematical model

Coupled spin Hamiltonian

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

The spin-one-half operators are dimensionless, the coupling is positive and the nuclear ratio is illustrative. The static field points along the vertical axis.

Levels at zero field

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

At zero field there is one singlet and three degenerate triplet states, and the gap between them equals the hyperfine coupling.

Probe operator

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

Turning the drive changes the matrix elements but not the static energy levels. At zero field the choice of triplet basis is a convention, and the total line strength does not depend on it.

Common difficulties

Not every level gives a peak

Typical misconceptionEvery energy level must produce a visible peak.

Better mental modelThe prepared state, the probe operator, the line width, the frequency range and the finite number of counts all matter. Lines outside the plotted band still count 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 the low-contrast Lorentzian response are inputs set in the model. The separate pulse test uses a rotating-wave model; it does not infer pulse dynamics or hardware noise.

Run the experiment

01
Predict before scanning
Turn off the static field. With the triplet degenerate, how many distinct transition frequencies should remain?
What to observe: The dashed curve comes from the model, not from recorded data. Hide it while you predict.
02
Record and save
Scan, save the sample, change one preparation setting, then scan again.
What to observe: Every frequency gets its own fresh, independent trials. The orange points keep the most recently saved preparation.
03
Explain the missing line
With a nonzero field, switch between longitudinal and transverse drive and watch which transitions are allowed.
What to observe: A line's position is set by an energy difference, but the line shows up only if its matrix element is not zero.

Coupled Spin Spectroscopy

Find the probe frequencies that change the state.

The levels move, and the drive decides which pairs of levels it connects.

One state, two ways to describe it.