Q007 · A larger shift / fewer retained events

Weak Measurement & Conditional Pointer

Prepare a Gaussian beam, separate its polarization components in a birefringent crystal and choose an analyzer axis. Record both position sensors, compare the selected centroid with the full beam and inspect the events that were left out.

Ideal selected fraction before detectors $\text{—}$

Exact ideal selected centroid $\text{—}$

FIRST, MAKE A PREDICTION

Does a larger selected shift mean a more precise measurement?

The crystal moves two components a little. The analyzer reshapes their interference. Compare the shift, beam width and number of retained clicks together.

Choose a prediction, then acquire and inspect the record.

No events acquired. Replay illustrates modes and adds no records.

CHANGE ONE THING

Compare the cases

Analyzer basis angle

\beta

-40^{\circ}

Polarization separation

d

0.1\,\mathrm{mm}

Input intensity width

\sigma

0.6\,\mathrm{mm}

Drag the orange analyzer axis; the slider gives keyboard access.

Preparation, coherence and detector settings

Prepared polarization

\alpha

45^{\circ}

Polarization relative phase

\varphi

0^{\circ}

Detection efficiency

\eta

1

Sensor half-window

W

3\,\mathrm{mm}

Coordinate noise width

\sigma_{\mathrm{read}}

0\,\mathrm{mm}

Emitted trials per batch

N

3000

Sampling seed

\mathrm{seed}

1991

Exact translated Gaussian amplitudes in a one-dimensional paraxial optical model. The crystal separation is defined in the imaged sensor plane. Both analyzer ports and no-click trials remain in the budget. Finite detector window, coordinate noise and pixelization act in acquisition. This effect also occurs with classical coherent light; a large shift alone establishes no precision advantage.

Evidence from the same raw record

SIMULATED DETECTOR EVENTS

A shift and its sample cost

Acquire events to begin.

Top: each observed group normalized to compare shape; gray is the ideal selected shape. Bottom: counts per emitted trial per millimeter to keep the sample cost visible; orange plus blue equals the white total, gray is the ideal total. Both gray references precede loss, window, noise and pixels.

Selected per emitted trial, 95% Wilson range

\text{—}

Selected centroid and empirical standard error

\text{—}

All-click centroid and empirical standard error

\text{—}

No inference until events are acquired.

Centroids use pixel records only. Standard errors use the sample variance and require at least 20 clicks in the respective group to be shown; they are not exact confidence intervals or a precision comparison. Finite-window selection can bias a centroid.

Budget and moments recomputed from pixel records
Group	Clicks	Centroid (mm)	Standard error (mm)

Latest 12 raw trials; CSV contains every trial
Trial	Analyzer port	Position (mm)

Physics tutorial

A large centroid is not a free measurement

BackgroundA birefringent crystal translates two polarization components in opposite transverse directions. The analyzer combines their amplitudes into two output ports. Near a crossed basis, destructive interference can leave a shifted but faint selected spot.

Why it mattersThe selected position distribution, its variance and its survival fraction all matter. An exactly crossed analyzer also exposes where the usual weak-value formula ceases to be defined.

Start with the essentials

Focus question: What happens at exactly orthogonal preparation and selection?
One-sentence intuition: Use the full translated waves at every coupling strength. Count the other port and the no-click trials, then compute the selected centroid only from observed positions.

Core mathematical model

Normalized input pointer

\psi(x)=(2\pi\sigma^2)^{-1/4}e^{-x^2/(4\sigma^2)}

Width is the standard deviation of input intensity. All transverse coordinates, separation and detector widths are in millimeters at a unit-magnification sensor plane.

Exact selected amplitude

f_+(x)=a\psi(x-d/2)+b e^{i\varphi}\psi(x+d/2),\quad a=\cos\alpha\cos\beta,\quad b=\sin\alpha\sin\beta

Preparation sets polarization and relative phase. The crystal gives opposite translations; the orthogonal analyzer row gives the other port. No small-coupling expansion enters the sampled distribution.

Probability and exact ideal centroid

P_+=a^2+b^2+2ab\cos\varphi\,e^{-d^2/(8\sigma^2)},\quad \langle x\rangle_+=\frac{d(a^2-b^2)}{2P_+}

These ideal infinite-sensor values are references only. If the selected probability vanishes, its centroid is undefined. A finite sensor or losses alter the observed fraction; clipping can alter the observed centroid.

Both ports preserve the total

|f_+(x)|^2+|f_-(x)|^2=\cos^2\alpha\,|\psi(x-d/2)|^2+\sin^2\alpha\,|\psi(x+d/2)|^2

The unconditional profile does not depend on the analyzer basis. Its two Gaussian components form a sampling proposal; no hidden polarization trajectory is recorded.

Limited weak-value expansion

A_w=\frac{a-b e^{i\varphi}}{a+b e^{i\varphi}},\quad \langle x\rangle_+\approx\frac d2\operatorname{Re}A_w

The denominator must be nonzero and the induced distortion small compared with the beam width. The lab always uses the exact model. At exact orthogonality with nonzero coupling the selected profile can be symmetric and two-lobed, with a finite success probability and zero centroid.

Recorded centroid and sampling cost

\widehat x_+=\frac1{N_+}\sum_{i\in +}x_i,\quad \mathrm{SE}(\widehat x_+)=\frac{s_+}{\sqrt{N_+}},\quad \widehat p_+=\frac{N_+}{N_{\mathrm{emit}}}

Positions are pixelized detector coordinates. Empirical standard errors are withheld below 20 selected detections. They are not exact confidence intervals; no fixed-resource estimation or Fisher-information advantage is established.

Common difficulties

No infinite spot

Typical misconceptionAn undefined weak value means the detector sees an infinite displacement.

Better mental modelThe finite-coupling wave determines the actual profile. Orthogonal preselection and postselection can give a symmetric selected distribution instead.

No free precision gain

Typical misconceptionA larger centroid alone proves better sensitivity per emitted photon.

Better mental modelA precision claim needs an explicit estimator, identical resource budget and a competing measurement. This lab shows the shift and cost without claiming that comparison.

Optical classical realization

Typical misconceptionThis centroid amplification alone proves a nonclassical light source.

Better mental modelCoherent classical polarization fields realize the same interference. Photon counting is the declared sampling interpretation.

Run the experiment

01
Compare against the full beam
Acquire with Nearly crossed. Then compare Aligned analyzer using the same emitted trial count.
What to observe: The selected centroid can be larger than either bare component translation, while the selected sample becomes much smaller.
02
Keep the cost visible
Inspect both-port counts, the no-click budget and the empirical standard errors. Export all trials.
What to observe: A large displacement is not a comparison of estimation precision; survival fraction, width, nuisance parameters and discarded information matter.
03
Test exact orthogonality
Choose Exactly crossed and acquire several batches. Then choose No coupling, crossed.
What to observe: Finite coupling leaves a weak, symmetric two-lobe profile. With zero coupling there is no selected sample and no centroid to estimate.
04
Break the weak approximation
Try Strong separation, then imperfect sensors or a narrow window.
What to observe: The exact distribution remains defined. The ideal infinite-sensor centroid need not equal the clipped noisy pixel centroid.