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The estimation of physical parameters with Heisenberg sensitivity and beyondis one of the crucial problems for current quantum metrology. However,unavoidable lossy effect is commonly believed to be the main obstacle whenapplying fragile quantum states. To utilize the lossy quantum metrology, weoffer an interferometric procedure for phase parameters estimation at theHeisenberg (up to 1/N) and super-Heisenberg (up to 1/N^3) scaling levels in theframework of the linear and nonlinear metrology approaches, respectively. Theheart of our setup is the novel soliton Josephson Junction (SJJ) systemproviding the formation of the quantum probe, i.e, the entangled Fock(N00N-like) state, beyond the superfluid-Mott insulator quantum phasetransition point. We illustrate that such states are close to the optimal oneseven with moderate losses. The enhancement of phase estimation accuracy remainsfeasible both for the linear and nonlinear metrologies with the SJJs, andallows further improvement for the current experiments performed with atomiccondensate solitons with a mesoscopic number of particles.

Enhanced nonlinear quantum metrology with weakly coupled solitons and particle losses