Q051 · The price of precision

Spin Squeezing Metrology

Does squeezing the noise of a spin ensemble always give a more precise measurement? Only if the signal slope survives. Squeezing lowers the noise in one direction, but phase precision is that noise divided by how steeply the signal changes with phase. Twist too hard and the contrast collapses, even while one variance still looks small.

Twist the collective spin of a small group of atoms and rotate the direction you read it out. Then compare its phase estimates with those from independent atoms, using exactly the same number of atoms and shots.

Recorded shots $0$

Reference / current setting $\text{—}$

01 / Make a prediction

Rotate the readout to find the narrow side of the noise.

Drag the gold handle. Then collect both samples: does the narrower projection beat the independent-atom control?

Start with a comparison

Atoms per shot

N

20

Twisting strength

\mu=\chi t

0.12

Readout alignment

\alpha ({}^\circ)

67

Applied test phase

\phi (\mathrm{rad})

0.04

Acquisition & additional controls

Collective phase jitter

\sigma_\phi (\mathrm{rad})

0

Shots per acquisition

M

200

Sampling seed

s_0

6173

Each shot starts from a fresh preparation. The same seed repeats the same sample. Changing physical settings starts new records; save a sample first.

The model treats 4–40 ideal two-level atoms as one permutation-symmetric collective spin. It includes collective Gaussian phase jitter but leaves out atom loss and local decoherence. The noise ellipse shows second moments; it is not a joint measurement. Local sensitivity and the error of a finite sample are different quantities.

Prepare → twist → align → sense

\text{—}

Horizontal $J_y$ · Upward $J_z$

Project onto the gold line

Circle: independent atoms. Filled ellipse: twisted state. A narrow shadow helps only if enough signal remains.

Signal contrast

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Protocol and second-moment diagram; not an atom image or simultaneous measurement of two spin components.

Ready. No measurements yet.

02 / What was actually recorded

Both groups have the same atom number and the same number of shots.

Twisted · recordedIndependent · recordedDashed: model reference

Population difference along the readout axis; frequencies are counts divided by shots.

M=0

03 / Check the precision

A narrow distribution is only half the story.

Local model reference at zero phase

\text{—}

Repeated estimates, twenty shots each

\text{—}

Each estimate uses the recorded mean and a zero-phase slope known separately. The applied phase is used only to score the error. This local linear estimator can be biased away from zero phase. Incomplete groups stay in the raw CSV but are left out of the RMSE.

Keep the evidence

Latest twelve shots shown; CSV retains all outcomes and settings. Save up to six samples and restore their exact settings and records.

Model, limits & scientific sources

The model keeps a single collective-spin sector, with ideal rotations and ideal atom-number detection. After twisting, the state is aligned, picks up a phase with Gaussian collective jitter, and then gets a final readout rotation. The count distribution is computed without a Gaussian-state approximation. The plotted ellipse summarizes only the covariance. Both groups use the same atoms and shots; calibration cost and preparation time are not counted in that comparison.

Kitagawa & Ueda · 1993 Wineland et al. · 1994

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Start with the essentials

Focus question: Does squeezing the noise of a spin ensemble always give a more precise measurement?
One-sentence intuition: Precision is set by the noise divided by how steeply the signal changes with phase. Too much twisting can destroy the contrast, even when one variance looks small.

Core mathematical model

Twist a coherent spin

|\psi\rangle=e^{-i\mu J_z^2}|{+x}\rangle^{\otimes N}

The whole symmetric spin sector is kept. Each magnetic-number amplitude picks up a phase that grows with the square of that number; the amplitudes are not sampled as classical spin directions.

Local precision compared with independent atoms

\xi_R^2=\frac{N\operatorname{Var}(J_y)}{|\partial_\phi\langle J_y\rangle|_{\phi=0}^2}

The variance is evaluated at zero phase, after the state is aligned and has undergone collective phase diffusion. A value below one means the local precision beats the projection noise of ideal independent atoms; it does not guarantee that a finite data set does better.

Estimate, then score

\widehat\phi=\overline m/s_0,\qquad\mathrm{RMSE}=\sqrt{\frac1K\sum_{k=1}^K(\widehat\phi_k-\phi)^2}

The known slope at zero phase converts the sample mean into a phase. Twenty shots make one estimate, and both ensembles use the same number of atoms and of estimates. Only the scoring step uses the applied phase.

Common difficulties

What was actually measured?

Typical misconceptionEvery plotted quantity was measured.

Better mental modelThe histograms and phase estimates use sampled counts. The covariance ellipse and the local sensitivity are computed separately from the model.

Scope

Typical misconceptionA small ideal model proves an advantage in a real laboratory.

Better mental modelThe model treats 4–40 ideal two-level atoms as one permutation-symmetric collective spin. It includes collective Gaussian phase jitter but leaves out atom loss and local decoherence. The noise ellipse shows second moments; it is not a joint measurement. Local sensitivity and the error of a finite sample are different quantities.

Run the experiment

01
Predict
Start with independent atoms and rotate the readout axis.
What to observe: The transverse noise is circular, and the local precision does not change.
02
Run and record
Add modest twisting, align the narrow direction, and record both ensembles.
What to observe: Compare the local sensitivity with the actual errors from repeated finite groups of shots.
03
Break an assumption
Over-twist, add phase jitter, or increase the test phase away from zero.
What to observe: Contrast loss, classical noise and estimator bias can erase the apparent gain.