Repeated weak measurements: watching quantum correlations evolve

Experimental access to many-body quantum systems is often limited by measurement backaction, and key dynamical properties are typically obtained by perturbing a system and measuring its response. Here we replace this active paradigm with a minimally invasive protocol based on a pair of weak quantum measurements that leverages measurement backaction as a strength. By correlating time-separated measurements with the first detecting fluctuations -- of any sort -- and the second tracking their time evolution, our method directly measures dynamical correlation functions without external perturbation. We demonstrate this technique in an atomic Bose-Einstein condensate using phase-contrast imaging to obtain the two-time density-density correlation function known as the Van Hove function and, through its Fourier transform, the dynamical structure factor. Due to the role of spatial correlations in scattering, these quantities underpin neutron and X-ray scattering and atomic Bragg spectroscopy. This approach is broadly applicable, providing access to correlation functions between any pair of observables amenable to weak measurement, thereby going beyond the capabilities of conventional strong measurements. We further isolate the role of quantum backaction through Aharonov's post-selection-based quantum weak values.