This work aims to help lay the foundation for new technologies that use quantum physics to out-perform conventional non-quantum devices, a situation referred to as "quantum supremacy". The first goal of this project is to borrow an element of conventional engineering known as the feedback loop (familiar from, e. g., the cruise control in a car), apply it to a manifestly quantum object (here, a collection of atoms trapped by a laser beam), and show that "quantum feedback" can greatly improve the sensitivity, speed, and reliability of magnetic sensors. From a basic science perspective, feedback control involves measurement and corrective action (e. g., measuring the speed of a car and increasing/decreasing engine power to keep the speed of the car reliably constant). In the quantum realm, this interplay between measurement and correction is fundamentally limited by the quantum uncertainty principle, according to which measurement also causes disturbance, which then limits the accuracy of the correction. The second goal of this project is to study this measurement/disturbance tradeoff in quantum feedback when controlling a physical system (again, atoms in a laser trap) subject to chaos, a scenario that is challenging even in conventional engineering. A recurrent theme in quantum metrology has been the use of continuous quantum measurement and feedback to construct adaptive measurement strategies that outperform their open-loop equivalents. One good example is the pioneering work of Wiseman on adaptive measurement strategies for homodyne phase estimation, along with related strategies for coherent state discrimination. In the early 2000's, Mabuchi and co-workers developed the theoretical framework for closed-loop strategies that use optical dispersive quantum nondemolition (QND) measurements and real-time feedback to create spin squeezing and perform magnetometry below the standard quantum limit (SQL), but were unable to implement these ideas in practice. The main objective of this project is to show that quantum feedback and closed-loop control of collective atomic spins is an experimentally viable idea with applications in quantum metrology and quantum simulation. Over the past few years the Principal Investigator has systematically solved many of the problems that plagued the early work, allowing significant deterministic squeezing of the collective angular momentum of an atomic ensemble, as well as a proof-of-principle demonstration of closed-loop control and magnetometry below the SQL. Building on these accomplishments, the goals of this project are (i) Further development of closed-loop magnetometry, and (ii) Exploration of a novel type of quantum nonlinear dynamics where the evolution of a quantum system is conditioned on the outcome of a time-continuous QND measurement. The latter provides an attractive platform for fundamental studies of the information gain/disturbance tradeoffs involved in real-time quantum feedback. It also provides access to new regimes of quantum simulation, and an opportunity to study long-standing issues related to quantum-classical correspondence in chaotic systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.