  It is well established that measurement-induced quantum back action (QBA) can
be eliminated in composite systems by engineering so-called
quantum-mechanics-free subspaces (QMFSs) of commuting variables, leading to a
trajectory of a quantum system without quantum uncertainties. This situation
can be realized in a composite system that includes a negative-mass subsystem,
which can be implemented by, e.g., a polarized spin ensemble or a
two-tone-driven optomechanical system. The realization of a trajectory without
quantum uncertainties implies entanglement between the subsystems, and allows
for measurements of motion, fields and forces with, in principle, unlimited
precision. To date, these principles have been developed theoretically and
demonstrated experimentally for a number of composite systems. However, the
utility of the concept has been limited by the dominating requirement of close
proximity of the resonance frequencies of the system of interest and the
negative-mass reference system, and by the need to embed the subsystems in a
narrowband cavity, which could be problematic while at the same time achieving
good overcoupling. Here we propose a general approach which overcomes these
limitations by employing periodic modulation of the driving fields (e.g.,
two-tone driving) in combination with coherent or measurement-based anti-noise
paths. This approach makes it possible to engineer a QMFS of two systems with
vastly different spectra and with arbitrary signs of their masses, while
dispensing with the need to embed the subsystems in a sideband-resolving
cavity. We discuss the advantages of this novel approach for applications such
as QBA evasion in gravitational wave detection, force sensing, and entanglement
generation between disparate systems.
