  We theoretically study the equilibrium spin current fluctuations and the
corresponding charge noise generated by inverse spin Hall effect (ISHE) in a
metal with strong spin-orbit coupling deposited on top of a quantum paramagnet.
It is shown that the charge noise power spectra measured along different
spatial axes can directly probe the different spin components of the boundary
dynamic spin correlations of the quantum paramagnet. We report the utility of
this ISHE-facilitated spin noise probe as a tool to unambiguously detect
topological phase transitions in an S=1/2 quantum spin ladder that hosts a
trivial ground state of singlet product states, but topologically-protected
fractional spin excitations localized at its ends. Our work demonstrates the
general usefulness of the ISHE-mediated spin noise spectroscopy for the
detection of topological phases in quantum paramagnets.
