Weak Measurements

In textbook (projective) measurement, observing an eigenvalue collapses the state and strongly disturbs future dynamics. Weak measurements do the opposite: couple so gently that a single trial is noisy and barely informative, but many repetitions reveal a small, average shift. If you also post-select on a final state, the mean meter readout is proportional to a complex number called the weak value.

Weak measurements are useful in two ways: (i) metrology — amplifying tiny effects without saturating detectors; and (ii) foundational probes — mapping where probability currents flow in interferometers without destroying interference.

Consider system observable \(A\) and a meter with pointer position \(q\) and conjugate momentum \(p\). A brief interaction \(H_{\text{int}}=g\,\delta(t-t_0)\,A\otimes p\) generates the unitary \(U=\exp(-\tfrac{i}{\hbar}gA\otimes p)\). Prepare the system in \(|i\rangle\) and the meter in a broad Gaussian \(\phi(q)\). For weak coupling (small \(g\)), expand \(U\) to first order and post-select the system in \(|f\rangle\).

The conditioned meter state acquires a shift \[ \langle q\rangle_f \approx \langle q\rangle_0 + g\,\mathrm{Re}\,A_w,\qquad \langle p\rangle_f \approx \langle p\rangle_0 + 2g\,(\Delta p)^2\,\mathrm{Im}\,A_w, \] where the weak value is \[ A_w=\frac{\langle f|A|i\rangle}{\langle f|i\rangle}. \] Because the denominator can be small, \(A_w\) can lie outside the eigenvalue range — a signature of interference between pre- and post-selected amplitudes.

If \(\langle f|i\rangle\) is small, a tiny physical effect (encoded in \(g\)) produces a larger pointer shift \(g\,A_w\). This enables sensitive phase or angle measurements in optics and atomic systems. The trade-off: the successful post-selection rate drops as \(|\langle f|i\rangle|^2\), so total information is conserved when you account for all trials.

In practice, weak-value amplification shines when technical noise (detector saturation, \(1/f\) noise) dominates rather than pure shot noise.

Optical polarization: Treat \(A=\sigma_z\) on polarization; couple to a spatial degree of freedom via a birefringent wedge (pointer \(q=y\)). Pre-select linear polarization, weakly shear the beam, then post-select nearly orthogonal polarization. The centroid shift \(\Delta y\propto g\,\mathrm{Re}\,A_w\) measures minute birefringence.

Interferometers: In a Mach–Zehnder, choose almost-dark output as the post-selection. A tiny phase in one arm becomes a large displacement at the dark port due to \(A_w\) amplification.

Atomic ensembles: Off-resonant Faraday interactions couple collective spin to light’s polarization; weak probing reads spin components while minimally disturbing dynamics — useful for feedback and squeezing.

Weak values are not ordinary eigenvalues; they are interference-weighted conditionals between pre- and post-selection. They can fall outside spectra or even be complex. Physically, the real part moves the pointer mean; the imaginary part skews its momentum.

In interferometers, mapping local weak values of projectors reveals “where the probability current flows” without killing fringes — a way to visualize quantum processes that strong measurements would erase.

The coupling must be weak enough that higher-order terms in \(g\) are negligible and that the meter’s back-action does not noticeably disturb \(|i\rangle\) across a single trial. Finite pointer width, drift, mis-mode-match, and detector nonlinearities bound achievable amplification. Good practice reports both the post-selection probability and signal-to-noise budget.