We propose to utilize the Bose–Einstein condensate (BEC) for precision phase sensitive measurement of position and momentum displacements. The controlled two-soliton dynamics can be turned into a quantum probe to measure these parameters with high precision in an experimentally verified system of bright solitary trains. A careful phase space analysis of the dynamics of the mesoscopic wave packet is carried out through Wigner phase space picture for finding out the parameter domain exhibiting sub-Planck structures, required for precision quantum metrology. We propose two experimentally feasible scenarios, one involving the overall phase and the other through the relative phase between the two-solitons when they are oscillating inside the trap. In both the cases, detailed analytical studies are carried out through the overlap functions which reveal the sensitivity issue. A careful analysis is also performed to find the parametric domain relevant for the use of BEC for weak value estimation.
Soliton dynamics for trapped Bose-Einstein condensate with higher-order interaction