Nonlinear quantum metrology with moving matter-wave solitons

The estimation of some parameters with super-Heisenberg (SH) sensitivity, i.e. beyond Heisenberg limit, is one of the principal problems for current quantum metrology. We propose to use Bose-Einstein condensate quantum bright solitons for this purpose. We have shown that solitons, as quantum nonlinear structured field objects, allow SH phase estimation even with coherent probes in the framework of a nonlinear metrology approach. To achieve ultimate scaling in nonlinear phase estimation, which is 1/N3, we examine soliton phase shift occurring due to interaction of weakly coupled 1D solitons. We have shown that steady states of coupled solitons can be used for the formation of maximally path-entangled N00N-state providing minimal propagation error of solitons' relative distance, momentum, and atom-atom scattering length parameter.