Enhanced nonlinear quantum metrology with weakly coupled solitons in the presence of particle losses

The estimation of physical parameters with Heisenberg sensitivity and beyond is one of the crucial problems for current quantum metrology. Commonly, an unavoidable lossy effect is believed to be the main obstacle when applying fragile quantum states. To utilize the lossy quantum metrology, we offer an interferometric procedure for estimation of phase parameters at the Heisenberg (up to 1/N) and super-Heisenberg (up to 1/N-3) scaling levels in the framework of the linear and nonlinear metrology approaches, respectively. The heart of our setup is a soliton Josephson junction (SJJ) system, which provides the formation of the quantum probe, the entangled Fock (NOON-like) state, beyond the superfluid-Mott insulator quantum phase transition point. We illustrate that such states are close to the optimal ones even with moderate losses. The enhancement of phase estimation accuracy remains feasible for both the linear and nonlinear metrologies with the SJJs and allows further improvement for the current experiments performed with atomic condensate solitons with a mesoscopic number of particles.