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Sandbox Physics

Q051 · THE PRICE OF PRECISION

Spin Squeezing Metrology

Twist a small atomic ensemble and rotate the readout direction. Compare its phase estimates with independent atoms using exactly the same atom and shot budgets.

Interactive modelSpin Squeezing Metrology
Recorded shots00
Reference / current setting—\text{—}
PREPARE → TWIST → ALIGN → SENSE—\text{—}

Horizontal JyJ_y · Upward JzJ_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

Same atoms. Same number of shots.

Twisted · recordedIndependent · recordedDashed: model reference

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

M=0M=0

03 / CHECK THE PRECISION

A narrow distribution is only half the story.

Local model reference at zero phase

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Fresh estimates · twenty shots per estimate

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Each estimate uses the recorded mean and a separately known zero-phase slope. The applied phase is used only to score error. This local linear estimator can be biased away from zero. Incomplete groups remain in raw CSV but do not enter 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

Single collective-spin sector, ideal rotations and number detection. Twisting is followed by alignment, phase sensing with Gaussian collective jitter, and a final readout rotation. No Gaussian-state approximation is used for the count distribution. The plotted ellipse summarizes only the covariance. Calibration cost and preparation time are excluded from the equal atom-shot budget.

Kitagawa & Ueda · 1993Wineland et al. · 1994

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Physics tutorial

Spin Squeezing Metrology

BackgroundTwist a small atomic ensemble and rotate the readout direction. Compare its phase estimates with independent atoms using exactly the same atom and shot budgets.

Why it mattersChange one physical ingredient and keep a matched control.

Start with the essentials

Focus question
Less noise. But a better measurement?
One-sentence intuition
Precision depends on noise divided by signal slope. Over-twisting can destroy contrast even when one variance looks small.

Core mathematical model

Twist a coherent spin

∣ψ⟩=e−iμJz2∣+x⟩⊗N|\psi\rangle=e^{-i\mu J_z^2}|{+x}\rangle^{\otimes N}

The full symmetric spin sector is retained. Magnetic-number amplitudes acquire quadratic phase; they are not sampled as classical spin directions.

A local precision reference

ξR2=NVar⁡(Jy)∣∂ϕ⟨Jy⟩∣ϕ=02\xi_R^2=\frac{N\operatorname{Var}(J_y)}{|\partial_\phi\langle J_y\rangle|_{\phi=0}^2}

Variance is evaluated after alignment and collective phase diffusion at zero phase. Below one beats the ideal independent-atom projection-noise reference locally; it is not a finite-record guarantee.

Estimate, then score

ϕ^=m‾/s0,RMSE=1K∑k=1K(ϕ^k−ϕ)2\widehat\phi=\overline m/s_0,\qquad\mathrm{RMSE}=\sqrt{\frac1K\sum_{k=1}^K(\widehat\phi_k-\phi)^2}

A known zero-phase slope calibrates the sample mean. Twenty shots form one estimate; both ensembles use the same atoms and number of estimates. Scoring alone uses the applied phase.

Common difficulties

What comes from the records?

Typical misconceptionEvery plotted quantity was measured.

Better mental modelHistograms and phase estimates use sampled counts. The covariance ellipse and local sensitivity are separate model references.

Scope

Typical misconceptionA small ideal model certifies a laboratory advantage.

Better mental modelPermutation-symmetric spin model with 4–40 ideal two-level atoms. Collective Gaussian phase jitter is included; particle loss and local decoherence are omitted. The noise ellipse shows second moments, not a joint measurement. Local sensitivity and finite-sample error are different quantities.

Run the experiment

  1. 01

    Predict

    Start with independent atoms; rotate the readout axis.

    What to observe: The transverse noise is circular and the local precision is unchanged.
  2. 02

    Operate and record

    Add modest twisting, align the narrow direction, and record both ensembles.

    What to observe: Compare the local sensitivity reference with errors from repeated finite groups.
  3. 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.