  We discuss how standard $T_2$-based quantum sensing and noise spectroscopy
protocols often give rise to an inadvertent quench of the system or environment
being probed: there is an effective sudden change in the environmental
Hamiltonian at the start of the sensing protocol. These quenches are extremely
sensitive to the initial environmental state, and lead to observable changes in
the sensor qubit evolution. We show how these new features can be used to
directly access environmental response properties. This enables methods for
direct measurement of bath temperature, and methods to diagnose non-thermal
equilibrium states. We also discuss techniques that allow one to deliberately
control and modulate this quench physics, which enables reconstruction of the
bath spectral function. Extensions to non-Gaussian quantum baths are also
discussed, as is the direct applicability of our ideas to standard diamond
NV-center based quantum sensing platforms.
