  Central spin decoherence is useful for detecting many-body physics in
environments and moreover, the spin echo control can remove the effects of
static thermal fluctuations so that the quantum fluctuations are revealed. The
central spin decoherence approach, however, is feasible only in some special
configurations and often requires uniform coupling between the central spin and
individual spins in the baths, which are very challenging in experiments. Here,
by making analogue between central spin decoherence and depolarization of
photons, we propose a scheme of Faraday rotation echo spectroscopy (FRES) for
studying quantum fluctuations in interacting spin systems. The echo control of
the photon polarization is realized by flipping the polarization with a
birefringence crystal. The FRES, similar to spin echo in magnetic resonance
spectroscopy, can suppress the effects of the static magnetic fluctuations and
therefore reveal dynamical magnetic fluctuations. We apply the scheme to a
rare-earth compound LiHoF4 and calculate the echo signal, which is related to
the quantum fluctuations of the system. We observe enhanced signals at the
phase boundary. The FRES should be useful for studying quantum fluctuations in
a broad range of spin systems, including cold atoms, quantum dots, solid-state
impurities, and transparent magnetic materials.
