Measure-invariant parameter paths for quantum control of many-body systems​​​​​

Selective gap engineering without globally shifting the many-body spectra is a nontrivial control task. Here, we introduce measure-invariant spectral transformations (MIST): parameter-space paths that preserve the coarse spectral scale while reshaping the spectrum. In an operator–coefficient decomposition, MIST constrains spectral moments (i.e., fixing the centroid and width of the spectrum) so the smoothed DOS slope is unchanged even if the gaps and matrix elements are reconfigured. We construct moment-locked circular paths and demonstrate them on the Bose–Hubbard trimer as an example. Sector-resolved paths keep the spectral invariants yet produce sizable, targeted changes in low-lying gaps. Thermodynamically, we observe energy- and entropy-driven spontaneity previously seen in geometric settings, now generalized beyond geometry. For control, MIST enhances selectivity of a chosen transition over all others and suppresses leakage under π-pulses; for a fixed leakage target it enables shorter pulses, and at fixed pulse time it lowers leakage. The protocol uses native ultracold-atom knobs and is directly portable to superconducting circuits (local detunings, tunable couplers, and interaction control), providing a broadly applicable route to spectral-gap engineering.