Q007 · A bigger shift from fewer kept events

Weak Measurement & Conditional Pointer

To make the pointer shift bigger in a weak measurement, how many events must you give up? Most of them. Turning the final analyzer nearly orthogonal to the input makes the pointer shift of the selected events large, but only a small fraction of events get through. A fair cost count includes every trial: those that left by the other port and those that never clicked, not only the shifted ones.

Prepare a Gaussian beam, pull its two polarization components apart in a birefringent crystal, and choose an analyzer axis. Record both position sensors, compare the center of the selected spot with that of the whole beam, and look at 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 shifts two components slightly, and the analyzer changes how they interfere. Compare the shift, the beam width and the number of kept clicks together.

Choose a prediction, then acquire and inspect the record.

No events yet. Replay only illustrates the modes and adds no records.

Change one condition at a time

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

The model is one-dimensional paraxial optics with exactly shifted Gaussian amplitudes. The crystal's separation is defined in the imaged sensor plane. Both analyzer ports and the trials with no click count toward the total. The finite detector window, position noise and pixels act during acquisition. Classical coherent light shows the same effect, and a large shift alone does not mean better precision.

What the same clicks show

Simulated detector events

How many clicks the shift costs

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{—}

Nothing to estimate 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.

Clicks, centroids and standard errors 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 shifts the two polarization components sideways in opposite directions. The analyzer then combines their amplitudes into two output ports. When the analyzer is nearly crossed, destructive interference can leave a faint selected spot that has moved.

Why it mattersThe selected position distribution, its spread and the fraction of photons that survive all matter. An exactly crossed analyzer also shows where the usual weak-value formula stops being defined.

Start with the essentials

Focus question: What happens when the selection is exactly orthogonal to the preparation?
One-sentence intuition: At every coupling strength, use the full shifted waves. Count the other port and the trials with no click, and compute the selected centroid only from positions that were actually observed.

Core mathematical model

Normalized input pointer

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

The width is the standard deviation of the input intensity. All transverse positions, the separation and the detector widths are in millimeters, in a sensor plane imaged at unit magnification.

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

The preparation sets the polarization and relative phase. The crystal shifts the two components in opposite directions, and the orthogonal analyzer row gives the other port. The sampled distribution uses no small-coupling expansion.

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 values assume an ideal, infinitely large sensor and serve only as references. If the selected probability is zero, the centroid is undefined. A finite sensor or losses change the observed fraction, and clipping at the sensor edge can shift the observed centroid.

The two ports add up to the same total

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

The total profile does not depend on the analyzer basis. Its two Gaussian components are used only to draw samples; no hidden polarization path is recorded.

Where the weak-value formula applies

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 formula needs a nonzero denominator and a distortion that is small compared with the beam width. The Lab always uses the exact model. When preparation and selection are exactly orthogonal and the coupling is not zero, the selected profile can be symmetric with two lobes, a finite chance of selection and a centroid of zero.

Measured centroid and its cost in events

\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 detector coordinates rounded to pixels. With fewer than 20 selected detections, no standard error is shown. The standard errors are not exact confidence intervals, and the Lab establishes no advantage in fixed-resource estimation or in Fisher information.

Common difficulties

The spot never moves infinitely far

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

Better mental modelThe actual profile comes from the wave at finite coupling. With orthogonal preselection and postselection, the selected distribution can be symmetric instead.

A bigger shift does not give free precision

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

Better mental modelTo claim better precision you need a specific estimator, the same resources on both sides and a competing measurement. This Lab shows the shift and its cost but does not make that comparison.

Classical optics shows the same effect

Typical misconceptionThis centroid amplification alone proves a nonclassical light source.

Better mental modelClassical coherent polarized light produces the same interference. Reading the data as photon counts is the sampling interpretation this Lab states.

Run the experiment

01
Compare against the full beam
Acquire with Nearly crossed, then compare with Aligned analyzer at the same number of emitted trials.
What to observe: The selected centroid can move farther than either component was shifted, but far fewer events are selected.
02
Keep track of what the shift costs
Look at the counts in both ports, the number of no-click trials and the standard errors. Export all trials.
What to observe: A large shift does not by itself compare measurement precision. The surviving fraction, the width, nuisance parameters and the information thrown away all matter.
03
Test exact orthogonality
Choose Exactly crossed and acquire several batches. Then choose No coupling, crossed.
What to observe: With finite coupling, a faint, symmetric profile with two lobes remains. With zero coupling nothing is selected, so there is no centroid to estimate.
04
Go beyond the weak approximation
Try Strong separation, then imperfect sensors or a narrow window.
What to observe: The exact distribution is still well defined. The ideal centroid on an infinite sensor need not match the centroid of clipped, noisy pixel data.