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We consider the effective dynamics obtained by double-passing a far-detunedlaser probe through a large atomic spin system. The net result of theatom-field interaction is a type of coherent positive feedback that amplifiesthe values of selected spin observables. An effective equation of motion forthe atomic system is presented, and an approximate 2-parameter model of thedynamics is developed that should provide a viable approach to modeling eventhe extremely large spin systems, with N>>1 atoms, encountered under typicallaboratory conditions.  Combining the nonlinear dynamics that result from the positive feedback withcontinuous observation of the atomic spin offers an improvement in quantumparameter estimation. We explore the possibility of reaching the Heisenberguncertainty scaling in atomic magnetometry without the need for any appreciablespin-squeezing by analyzing our system via the quantum Cramer-Rao inequality.Finally, we develop a realistic quantum parameter estimator for atomicmagnetometry that is based on a two-parameter family of Gaussian states andinvestigate the performance of this estimator through numerical simulations. Indoing so, we identify several issues, such as numerical convergence and thereduction of estimator bias, that must be addressed when incorporating ourparameter estimation methods into an actual laboratory setting.

Amplified Quantum Dynamics and Enhanced Parameter Sensitivity via Coherent Feedback in Collective Atomic Spin Systems