We consider a large class of Ramsey interferometry protocols which are enhanced by squeezing and un-squeezing operations before and after a phase signal is imprinted on the collective spin of $N$ particles. We report an analytical optimization for any given particle number and strengths of (un-)squeezing. These results can be applied even when experimentally relevant decoherence processes during the squeezing and un-squeezing interactions are included. Noise between the two interactions is however not considered in this work. This provides a generalized characterization of squeezing echo protocols, recovering a number of known quantum metrological protocols as local sensitivity maxima, thereby proving their optimality. We discover a single new protocol. Its sensitivity enhancement relies on a double inversion of squeezing. In the general class of echo protocols, the newly found over-un-twisting protocol is singled out due to its Heisenberg scaling even at strong collective dephasing.
Many theoretical proposals were developed, which suggest the use of entangled probes in order to improve upon the quantum projection noise limit e.g. by squeezing quantum fluctuations. However, the experimental implementation of these schemes faces major hurdles due to the often extremely demanding requirements on the manipulation and measurement of the particles and due to increased susceptibility to noise and decoherence of highly entangled states. Notably, some of the best metrological measurements have been achieved in schemes where well-controlled squeezing interactions were used several times, before and after imprinting a signal. The application of such an ‘echo’ was essential to encode the signal in a robust and readily measurable quantity and thus allow an improvement already at small particle numbers and with imperfect control capabilities.
In our work we give for the first time a theory of generalized squeezing echoes in Ramsey interferometry. We show how to perform an analytical optimization of the geometrical parameters involved in the problem. Our methods allow us to attain a complete overview of all possible echo protocols within this large variational class, and we recover many prominent known proposals as local sensitivity maxima. In some cases our treatment proves optimality of these protocols which could not be claimed so far. Most importantly, we are able to show that there is exactly one more, so-far unknown protocol. Remarkably, this new protocol achieves a sensitivity at the Heisenberg limit and turns out to be very robust with respect to the experimentally most relevant decoherence processes.
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