Hybrid measurement scheme for generating non-Gaussian spin states

We present a protocol for generating nonclassical states of atomic spin ensembles through the backaction induced by a hybrid measurement of light that is entangled with atoms, combining both homodyne and single-photon detection. In phase I of the protocol we create a spin-squeezed state by measuring the light's polarization rotation due to the Faraday effect in a balanced polarimeter, equivalent to a homodyne measurement. In phase II we send a second probe beam through the sample and detect single photons scattered into the signal mode. Before doing so, we rotate the uncertainty bubble to increase the projection fluctuations of the measured spin component. This increases the coupling strength between the atoms and photons and thus the rate of scattering of single photons into the signal mode. In the ideal case, the result is a squeezed Dicke state, with substantial quantum advantage for sensing spin rotations. We benchmark the protocol's utility in the presence of inevitable decoherence due to optical pumping using the Fisher information as a measure of quantum advantage. We show that in the presence of decoherence, the quantum Fisher information associated with the non-Gaussian mixed state we prepare is substantially larger than the classical Fisher information obtained from the standard measurement of spin rotations. We deduce a measurement basis that is close to optimal for achieving the quantum Cram\'er-Rao bound in the presence of decoherence.