Quantum Love numbers

The response of a gravitating object to an external tidal field is encoded in its Love numbers, which identically vanish for classical black holes (BHs). Here we show, using standard time-independent quantum perturbation theory, that for a quantum BH, generically, the Love numbers are nonvanishing and negative, and that their magnitude depends on the lowest-lying levels of the quantum spectrum of the BH. We calculate the quadrupolar electric quantum Love number of slowly rotating BHs and show that it depends most strongly on the first excited level of the quantum BH. We then compare our results to the same Love number of exotic ultra compact objects and to that of classical compact stars and highlight their different parametric dependence. Finally, we discuss the detectability of the quadrupolar quantum Love number in future precision gravitational-wave observations and show that, under favorable circumstances, its magnitude is large enough to imprint an observable signature on the gravitational waves emitted during the inspiral phase of two moderately spinning BHs.