  Weak measurement in tandem with real-time feedback control is a new route
toward engineering novel non-equilibrium quantum matter. Here we develop a
theoretical toolbox for quantum feedback control of multicomponent
Bose-Einstein condensates (BECs) using backaction-limited weak measurements in
conjunction with spatially resolved feedback. Feedback in the form of a
single-particle potential can introduce effective interactions that enter into
the stochastic equation governing system dynamics. The effective interactions
are tunable and can be made analogous to Feshbach resonances --
spin-independent and spin-dependent -- but without changing atomic scattering
parameters. Feedback cooling prevents runaway heating due to measurement
backaction and we present an analytical model to explain its effectiveness. We
showcase our toolbox by studying a two-component BEC using a stochastic
mean-field theory, where feedback induces a phase transition between easy-axis
ferromagnet and spin-disordered paramagnet phases. We present the steady-state
phase diagram as a function of intrinsic and effective spin-dependent
interaction strengths. Our result demonstrates that closed-loop quantum control
of Bose-Einstein condensates is a powerful new tool for quantum engineering in
cold-atom systems.
