  Studying the dynamics of quantum many-body systems is often constrained by
the limitations in probing relevant observables, especially in continuous
systems. A prominent example is two parallel 1D Bose gases, which simulate 1D
quantum field theories through the phase difference probed by interference.
Here we introduce a method to extract the total phase of the combined system
based on a general principle of reconstructing phase gradient from density
dynamics using continuity equation. This approach reveals the previously hidden
sector of the sum mode of the two 1D Bose gases. We validate our technique
numerically and demonstrate its effectiveness by analysing data from selected
experiments, showcasing how our method expands the scope and capabilities of
cold-atomic quantum simulators.
