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It is well established that measurement-induced quantum back action (QBA) canbe eliminated in composite systems by engineering so-calledquantum-mechanics-free subspaces (QMFSs) of commuting variables, leading to atrajectory of a quantum system without quantum uncertainties. This situationcan be realized in a composite system that includes a negative-mass subsystem,which can be implemented by, e.g., a polarized spin ensemble or atwo-tone-driven optomechanical system. The realization of a trajectory withoutquantum uncertainties implies entanglement between the subsystems, and allowsfor measurements of motion, fields and forces with, in principle, unlimitedprecision. To date, these principles have been developed theoretically anddemonstrated experimentally for a number of composite systems. However, theutility of the concept has been limited by the dominating requirement of closeproximity of the resonance frequencies of the system of interest and thenegative-mass reference system, and by the need to embed the subsystems in anarrowband cavity, which could be problematic while at the same time achievinggood overcoupling. Here we propose a general approach which overcomes theselimitations by employing periodic modulation of the driving fields (e.g.,two-tone driving) in combination with coherent or measurement-based anti-noisepaths. This approach makes it possible to engineer a QMFS of two systems withvastly different spectra and with arbitrary signs of their masses, whiledispensing with the need to embed the subsystems in a sideband-resolvingcavity. We discuss the advantages of this novel approach for applications suchas QBA evasion in gravitational wave detection, force sensing, and entanglementgeneration between disparate systems.

Trajectories without quantum uncertainties in composite systems with disparate energy spectra