Phonon Fluctuations in a Bose-Einstein Condensate Subject to Repeated Weak Measurements

Weak measurements are minimally destructive measurement techniques that present new opportunities for understanding the system-reservoir dynamics of many-body systems. Weak measurements yield a controlled reservoir and consequently allow time-resolved study of the system evolution.~We experimentally study the phonon fluctuations that result from the quantum backaction induced by repeated weak measurements in quasi-one dimensional Bose-Einstein condensates (BECs). We use partial transfer absorption imaging (PTAI) technique to obtain minimally destructive and multiple successive \textit{in situ} images of the same BEC.~We describe the reciprocal space analysis of the density profiles and present the resultant space-time separated density-density correlation functions.~Further we discuss feedback control protocols for future applications of Hamiltonian engineering using weak measurements and feedback.~