Quantum-limited position measurements of a dark matter-wave soliton

We show that the position of a dark matter-wave soliton can be determined with a precision that scales with the atomic density as $n^{-3/4}$. This surpasses the standard shot-noise detection limit for independent particles, without use of squeezing and entanglement, and it suggests that interactions among particles may present new advantages in high-precision metrology. We also take into account quantum density fluctuations due to phonon and Goldstone modes and we show that they, somewhat unexpectedly, actually improve the resolution. This happens because the fluctuations depend on the soliton position and make a larger amount of information available.