Classical optical realization of a joint weak measurement through optical beam shifts

Quantum weak measurements became extremely popular in classical optics to amplify small optical signals for fundamental interests and potential applications. Later, as a more general extension, joint weak measurement has been proposed to extract the weak value from a joint quantum measurement. However, unlike traditional weak measurements, the detection of a joint weak value in the realm of classical optics remains less explored. Here, using the polarization-dependent longitudinal and transverse optical beam shift as a platform, we experimentally realize a classical optical analog of the quantum joint weak measurement. Polarization states are cleverly pre- and postselected, and different single and joint canonical position-momentum observables of the beam are experimentally extracted and subsequently analyzed for successful detection of the complex joint weak value. We envision that this work will find usefulness for gaining fundamental insights on quantum measurements and to tackle analogous problems in optics.