Fast apparent oscillations of fundamental constants

Precision spectroscopy of atoms and molecules allows one to search for and to put stringent limits on the variation of fundamental constants. These experiments are typically interpreted in terms of variations of the fine structure constant $\alpha$ and the electron to proton mass ratio $\mu=m_e/m_p$. Atomic spectroscopy is usually less sensitive to other fundamental constants, unless the hyperfine structure of atomic levels is studied. However, the number of possible dimensionless constants increases when we allow for fast variations of the constants, where "fast" is determined by the time scale of the response of the studied species or experimental apparatus used. In this case, the relevant dimensionless quantity is, for example, the ratio $m_e/\langle m_e \rangle$ and $\langle m_e \rangle$ is the time average. In this sense, one may say that the experimental signal depends on the variation of dimensionful constants ($m_e$ in this example).