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We present a theory of radio-frequency spectroscopy of impurities interactingwith a quantum gas at finite temperature. By working in the canonical ensembleof a single impurity, we show that the impurity spectral response is directlyconnected to the finite-temperature equation of state (free energy) of theimpurity. We consider two different response protocols: "injection", where theimpurity is introduced into the medium from an initially non-interacting state;and "ejection", where the impurity is ejected from an initially interactingstate with the medium. We show that there is a simple mapping between injectionand ejection spectra, which is connected to the detailed balance condition inthermal equilibrium. To illustrate the power of our approach, we specialize tothe case of the Fermi polaron, corresponding to an impurity atom that isimmersed in a non-interacting Fermi gas. For a mobile impurity with a massequal to the fermion mass, we employ a finite-temperature variational approachto obtain the impurity spectral response. We find a striking non-monotonicdependence on temperature in the impurity free energy, the contact, and theradio-frequency spectra. For the case of an infinitely heavy Fermi polaron, wederive exact results for the finite-temperature free energy, thus generalizingFumi's theorem to arbitrary temperature. We also determine the exact dynamicsof the contact after a quench of the impurity-fermion interactions. Finally, weshow that the injection and ejection spectra obtained from the variationalapproach compare well with the exact spectra, thus demonstrating the accuracyof our approximate method.

Theory of radio-frequency spectroscopy of impurities in quantum gases