Colloquium: Understanding quantum weak values: Basics and applications

Since its introduction 25 years ago, the quantum weak value has gradually transitioned from a theoretical curiosity to a practical laboratory tool. While its utility is apparent in the recent explosion of weak value experiments, its interpretation has historically been a subject of confusion. Here a pragmatic introduction to the weak value in terms of measurable quantities is presented, along with an explanation for how it can be determined in the laboratory. Further, its application to three distinct experimental techniques is reviewed. First, as a large interaction parameter it can amplify small signals above technical background noise. Second, as a measurable complex value it enables novel techniques for direct quantum state and geometric phase determination. Third, as a conditioned average of generalized observable eigenvalues it provides a measurable window into nonclassical features of quantum mechanics. In this selective review, a single experimental configuration to discuss and clarify each of these applications is used.

Figure 1

An experiment for illustrating how one can measure weak values. (a) A Gaussian beam from a single mode fiber is collimated by a lens and prepared in an initial polarization state by a quarter-wave plate (QWP) and half-wave plate (HWP). A polarizer postselects the beam on a final polarization state. A CCD then measures the position-dependent beam intensity. (b) A birefringent crystal is inserted between the wave plates and polarizer to displace different polarizations by a small amount. A lens images the transverse position on the output face of the crystal onto the CCD in order to measure the real part of the polarization weak value as a linear shift in the postselected intensity. (c) The lens is changed to imaging the far field of the crystal face onto the CCD as the transverse momentum in order to determine the imaginary part of the polarization weak value (details in the text).

Figure 2

A band around the equator of the Poincaré sphere showing the initial polarization $|i⟩$ (dot, back of sphere) from Eq. (8) and postselection polarization $|f⟩$ (dot, front of sphere) from Eq. (9). We also indicate the small angles that make $|f⟩$ almost orthogonal to $|i⟩$.

Figure 3

(a) Comparisons between perturbed profiles (solid, for various values of beam displacement $\epsilon$) and a fixed unperturbed profile (dashed, corresponding to $P$). Note that both curves represent postselected measurements. (b) The exact ratio of the two curves (solid) is compared to the first order approximation (dashed). When $\epsilon$ is sufficiently small, the first order approximation adequately models the quantity $P_{\epsilon }/P$ over most of the profile.

Figure 4

(a) Real (dashed) and imaginary (solid) parts of the polarization weak value $S_w=⟨f|\widehat{S}|i⟩/⟨f|i⟩$, with initial state $|i⟩$ given in Eq. (8) and shown in Fig. 2, and final state $|f⟩$ that depends on a varying angle $\theta$. The eigenvalue bounds of $\pm 1$ are shown as dotted lines for reference, while the dots indicate the final state chosen in Eq. (9). (b) The postselection probability $P(\theta )=|⟨f|i⟩{|}^2$ as a function of $\theta$, showing how a large weak value corresponds to a small detection probability. The inset shows the small probability region enlarged for clarity, while the dots similarly indicate the final state in Eq. (9).

Figure 5

Conditioned average (22) of generalized polarization eigenvalues $x/\epsilon$ for various values of the crystal length $\epsilon$, using the beam profile illustrated in Fig. 2. For large $\epsilon$ the average is a classical conditioned average constrained to the eigenvalue range (dotted lines). For sufficiently small $\epsilon$, however, the conditioned average (solid lines) approximates the real part (dashed lines) of the polarization weak value in Fig. 4.