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arXiv:2002.01211 · Physical Review A

Theory of radio-frequency spectroscopy of impurities in quantum gases

Weizhe Edward Liu, Zhe-Yu Shi, Meera M. Parish, Jesper Levinsen

Published Aug 5, 2020• 38 citations
arXivHTMLPDF

Abstract

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 noninteracting 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 noninteracting 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 nonmonotonic 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 approximation method.

DOI
10.1103/physreva.102.023304

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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 noninteracting 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 noninteracting 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 nonmonotonic 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 approximation method.
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.
Recent experimental advances enabled the realization of mobile impurities immersed in a Bose-Einstein condensate (BEC) of ultracold atoms. Here, we consider impurities with two or more internal hyperfine states, and study their radio-frequency (rf) absorption spectra, which correspond to transitions between two different hyperfine states. We calculate rf spectra for the case when one of the hyperfine states involved interacts with the BEC, while the other state is noninteracting, by performing a nonperturbative resummation of the probabilities of exciting different numbers of phonon modes. In the presence of interactions, the impurity gets dressed by Bogoliubov excitations of the BEC, and forms a polaron. The rf signal contains a $\ensuremath{\delta}$-function peak centered at the energy of the polaron measured relative to the bare impurity transition frequency with a weight equal to the amount of bare impurity character in the polaron state. The rf spectrum also has a broad incoherent part arising from the background excitations of the BEC, with a characteristic power-law tail that appears as a consequence of the universal physics of contact interactions. We discuss both the direct rf measurement, in which the impurity is initially in an interacting state, and the inverse rf measurement, in which the impurity is initially in a noninteracting state. In the latter case, in order to calculate the rf spectrum, we solve the problem of polaron formation: a mobile impurity is suddenly introduced in a BEC, and dynamically gets dressed by Bogoliubov phonons. Our solution is based on a time-dependent variational ansatz of coherent states of Bogoliubov phonons, which becomes exact when the impurity is localized. Moreover, we show that such an ansatz compares well with a semiclassical estimate of the propagation amplitude of a mobile impurity in the BEC. Our technique can be extended to cases when both initial and final impurity states are interacting with the BEC.
We present a systematic study of a mobile impurity immersed in a three-dimensional Fermi sea of fermions at finite temperature, by using the standard non-self-consistent many-body TTT-matrix theory that is equivalent to a finite-temperature variational approach with the inclusion of one-particle-hole excitation. The impurity spectral function is determined in the real-frequency domain, avoiding any potential errors due to the numerical analytic continuation in previous TTT-matrix calculations and the small spectral broadening parameter used in variational calculations. In the weak-coupling limit, we find that the quasiparticle decay rate of both attractive and repulsive polarons does not increase significantly with increasing temperature, and therefore Fermi polarons may remain well defined far above Fermi degeneracy. In contrast, near the unitary limit with strong coupling, the decay rate of Fermi polarons rapidly increases and the quasiparticle picture breaks down close to the Fermi temperature. We analyze in detail the recent ejection and injection radio-frequency (rf) spectroscopy measurements, performed at Massachusetts Institute of Technology (MIT) and at the European Laboratory for Non-Linear Spectroscopy (LENS), respectively. We show that the momentum average of the spectral function, which is necessary to account for the observed rf spectroscopy, has a sizable contribution to the width of the quasiparticle peak in spectroscopy. As a result, the measured decay rate of Fermi polarons could be significantly larger than the calculated quasiparticle decay rate at zero momentum. By taking this crucial contribution into account, we find that there is a reasonable agreement between theory and experiment for the lifetime of Fermi polarons in the strong-coupling regime, as long as they remain well defined.
We present a general variational principle for the dynamics of impurity particles immersed in a quantum-mechanical medium. By working within the Heisenberg picture and constructing approximate time-dependent impurity operators, we can take the medium to be in any mixed state, such as a thermal state. Our variational method is consistent with all conservation laws and, in certain cases, it is equivalent to a finite-temperature Green's function approach. As a demonstration of our method, we consider the dynamics of heavy impurities that have suddenly been introduced into a Fermi gas at finite temperature. Using approximate time-dependent impurity operators involving only one particle-hole excitation of the Fermi sea, we find that we can successfully model the results of recent Ramsey interference experiments on ^{40}K atoms in a ^{6}Li Fermi gas. We also show that our approximation agrees well with the exact solution for the Ramsey response of a fixed impurity at finite temperature. Our approach paves the way for the investigation of impurities with dynamical degrees of freedom in arbitrary quantum-mechanical mediums.
This thesis presents experiments investigating the phase diagram of ultracold atomic Fermi gases using radio-frequency spectroscopy. The tunability of many experimental parameters including the temperature, the interparticle interaction strength and the relative population of different Fermions allows to access very different physical regimes. Radio-frequency spectroscopy has been developed into an ideal tool to probe correlations between particles in these different phases. In particular, radio-frequency spectroscopy of highly population imbalanced atomic Fermi systems gives access to the impurity problem: A single Fermion, or Boson, immersed in a sea of Fermions constitutes a polaron, which can be described by Landau's Fermi liquid theory. A critical interaction strength can be identified separating the regime of a fermionic polaron and a bosonic polaron. Radio-frequency spectroscopy of the polarized superfluid phase allows an accurate measure of the superfluid gap [Delta] and allows to identify the importance of Hartree Mean-field energies. Furthermore, it is shown how these different physical regimes are connected.
We discuss the interaction of a mobile quantum impurity with a Bose-Einstein condensate of atoms at finite temperature. To describe the resulting Bose polaron formation we develop a dynamical variational approach applicable to an initial thermal gas of Bogoliubov phonons. We study the polaron formation after switching on the interaction, e.g., by a radio-frequency (rf) pulse from a noninteracting to an interacting state. To treat also the strongly interacting regime, interaction terms beyond the Fröhlich model are taken into account. We calculate the real-time impurity Green's function and discuss its temperature dependence. Furthermore we determine the rf absorption spectrum and find good agreement with recent experimental observations. We predict temperature-induced shifts and a substantial broadening of spectral lines. The analysis of the real-time Green's function reveals a crossover to a linear temperature dependence of the thermal decay rate of Bose polarons as unitary interactions are approached.