Radio-Frequency Response and Contact of Impurities in a Quantum Gas

We investigate the radio-frequency spectroscopy of impurities interacting with a quantum gas at finite temperature. In the limit of a single impurity, we show using Fermi’s golden rule that introducing (or injecting) an impurity into the medium is equivalent to ejecting an impurity that is initially interacting with the medium, since the “injection” and “ejection” spectral responses are simply related to each other by an exponential function of frequency. Thus, the full spectral information for the quantum impurity is contained in the injection spectral response, which can be determined using a range of theoretical methods, including variational approaches. We use this property to compute the finite-temperature equation of state and Tan contact of the Fermi polaron. Our results for the contact of a mobile impurity are in excellent agreement with recent experiments and we find that the finite-temperature behavior is qualitatively different compared to the case of infinite impurity mass.

Figure 1

Examples of impurity ejection (a) and injection (b) spectra observable in a quantum-gas experiment. An rf field of frequency $\omega$ drives a transition between the interacting ($\uparrow$) and noninteracting ($\downarrow$) impurity spin states (insets). We take the Fermi polaron at unitarity with mass $M=m$ and temperature $T=0.2T_F$.

Figure 2

Finite-temperature equation of state for the Fermi polaron. (a) Relative free energy $\mathrm{\Delta }F$ at unitarity for both equal-mass (blue solid) and infinite-mass cases [black dashed, exact result, Eq. (14); black squares, variational approach]. (b) Tan contact at unitarity for the equal-mass case obtained in this work (blue solid) and from experiment in Ref. [15] (red filled circles). (c) Tan contact for the infinite-mass case [black dashed, exact result, Eq. (13); black squares, variational] at interaction strengths $1/(k_{F}a)=-0.5$, 0, 0.3 (bottom to top).