We present a theory of radio-frequency spectroscopy of impurities interacting with a quantum gas at finite temperature. By working in the canonical ensemble of a single impurity, we show that the impurity spectral response is directly connected to the finite-temperature equation of state (free energy) of the impurity. We consider two different response protocols: "injection’‘, where the impurity is introduced into the medium from an initially non-interacting state; and "ejection’’, where the impurity is ejected from an initially interacting state with the medium. We show that there is a simple mapping between injection and ejection spectra, which is connected to the detailed balance condition in thermal equilibrium. To illustrate the power of our approach, we specialize to the case of the Fermi polaron, corresponding to an impurity atom that is immersed in a non-interacting Fermi gas. For a mobile impurity with a mass equal to the fermion mass, we employ a finite-temperature variational approach to obtain the impurity spectral response. We find a striking non-monotonic dependence on temperature in the impurity free energy, the contact, and the radio-frequency spectra. For the case of an infinitely heavy Fermi polaron, we derive exact results for the finite-temperature free energy, thus generalizing Fumi’s theorem to arbitrary temperature. We also determine the exact dynamics of the contact after a quench of the impurity-fermion interactions. Finally, we show that the injection and ejection spectra obtained from the variational approach compare well with the exact spectra, thus demonstrating the accuracy of our approximate method.

中文翻译：

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量子气体中杂质的射频光谱理论

我们提出了一种在有限温度下与量子气体相互作用的杂质的射频光谱学理论。通过在单个杂质的规范集合中工作，我们表明杂质光谱响应直接与杂质的状态（自由能）有限温度方程式相关。我们考虑两种不同的响应协议：“注入”，其中杂质从最初的非相互作用状态引入介质；和“喷射”，其中杂质从与介质的初始相互作用状态被排放。我们表明，在注入光谱和喷射光谱之间有一个简单的映射，它与热平衡中的详细平衡条件有关。为了说明我们的方法的力量，我们专门研究费米极化子的情况，对应于浸没在非相互作用费米气体中的杂质原子。对于质量等于费米质量的活动杂质，我们采用有限温度变化方法来获得杂质光谱响应。我们在杂质自由能，接触和射频频谱中发现了对温度的非显着单调依赖性。对于无限重的费米极化子，我们得出了有限温度自由能的精确结果，从而将Fumi定理推广到任意温度。我们还确定了杂质-费米子相互作用淬灭后的确切接触动力学。最后，我们证明了从变分方法获得的进样和发射光谱与精确光谱有很好的比较，从而证明了我们近似方法的准确性。