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arXiv:2107.06102 · Physical Review A

Signatures of the orthogonality catastrophe in a coherently driven impurity

Haydn S. Adlong, Weizhe Edward Liu, Lincoln D. Turner, Meera M. Parish, Jesper Levinsen

Published Oct 14, 2021• 5 citations
arXivHTMLPDF

Abstract

We consider a fixed impurity immersed in a Fermi gas at arbitrary temperature. We take the impurity to have two internal spin states where the $\ensuremath{\uparrow}$ state is assumed to interact with the medium such that it exhibits the orthogonality catastrophe, whereas the $\ensuremath{\downarrow}$ state is a bare noninteracting particle. Introducing a Rabi coupling between the impurity states, therefore, allows us to investigate the coupling between a discrete spectral peak and the Fermi-edge singularity, i.e., between states with and without a quasiparticle residue. Combining an exact treatment of the uncoupled impurity Green's functions with a variational approach to treat the Rabi driven dynamics, we find that the system features Rabi oscillations whose frequency scales as a nontrivial power of the Rabi drive at low temperatures. This reflects the power law of the Fermi-edge singularity and, importantly, this behavior is qualitatively different from the case of a mobile impurity quasiparticle where the scaling is linear. We, therefore, argue that the scaling law serves as an experimentally implementable probe of the orthogonality catastrophe. We additionally simulate rf spectroscopy beyond linear response, finding a remarkable agreement with an experiment using heavy impurities [Kohstall et al., Nature 485, 615 (2012)], thus, demonstrating the power of our approach.

DOI
10.1103/physreva.104.043309

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arXiv:1104.2577•Physical Review A•Dec 22, 2011•123 citations
We consider a fixed impurity immersed in a Fermi gas at arbitrary temperature. We take the impurity to have two internal spin states where the \ensuremath↑\ensuremath{\uparrow}\ensuremath↑ state is assumed to interact with the medium such that it exhibits the orthogonality catastrophe, whereas the \ensuremath↓\ensuremath{\downarrow}\ensuremath↓ state is a bare noninteracting particle. Introducing a Rabi coupling between the impurity states, therefore, allows us to investigate the coupling between a discrete spectral peak and the Fermi-edge singularity, i.e., between states with and without a quasiparticle residue. Combining an exact treatment of the uncoupled impurity Green's functions with a variational approach to treat the Rabi driven dynamics, we find that the system features Rabi oscillations whose frequency scales as a nontrivial power of the Rabi drive at low temperatures. This reflects the power law of the Fermi-edge singularity and, importantly, this behavior is qualitatively different from the case of a mobile impurity quasiparticle where the scaling is linear. We, therefore, argue that the scaling law serves as an experimentally implementable probe of the orthogonality catastrophe. We additionally simulate rf spectroscopy beyond linear response, finding a remarkable agreement with an experiment using heavy impurities [Kohstall et al., Nature 485, 615 (2012)], thus, demonstrating the power of our approach.
The Anderson orthogonality catastrophe (AOC) is a paradigmatic many-body phenomenon in which a local perturbation induces a macroscopic response of a Fermi sea. We probe signatures of the AOC by coherently driving heavy Fermi polarons in an ultracold 6^66Li-133^{133}133Cs mixture. We observe a power-law dependence of the measured Rabi frequency on the drive strength, with exponents consistent with AOC predictions. Finite-temperature simulations quantitatively reproduce the observed scaling, indicating that AOC signatures persist beyond the idealized zero-temperature, infinite-mass limit. The damping of the Rabi oscillations provides access to polaron dephasing and reveals a nonmonotonic drive dependence, qualitatively consistent with current theories. Our results establish coherently driven impurities as a versatile probe of quantum many-body dynamics through local coherent control.
We investigate the behavior of a mobile spin-1/2 impurity atom immersed in a Fermi gas, where the interacting spin-$\ensuremath{\uparrow}$ and noninteracting spin-$\ensuremath{\downarrow}$ states of the impurity are Rabi coupled via an external field. This scenario resembles the classic problem of a two-state system interacting with a dissipative environment, but with an added dimension provided by the impurity momentum degree of freedom. In this case, the impurity can become ``dressed'' by excitations of the Fermi sea to form a Fermi polaron quasiparticle. For the steady-state system, where the impurity has thermalized with the medium, we derive exact thermodynamic relations that connect the impurity magnetization with quasiparticle properties such as the number of fermions in the dressing cloud. We show how the thermodynamic properties evolve with increasing Rabi coupling and we present exact analytical results in the limits of weak and strong Rabi coupling. For the dynamics of the Rabi-driven Fermi polaron, we formulate a theoretical approach based on correlation functions that respects conservation laws and allows the efficient calculation of Rabi oscillations for a range of timescales and impurity momenta beyond what has been achieved previously. Our results are in good agreement with recent experiments on the Rabi oscillations of the attractive polaron, and they reveal how the Rabi oscillations are influenced by the interplay between the polaron and its dressing cloud.
In this report we discuss the dynamical response of heavy quantum impurities immersed in a Fermi gas at zero and at finite temperature. Studying both the frequency and the time domain allows one to identify interaction regimes that are characterized by distinct many-body dynamics. From this theoretical study a picture emerges in which impurity dynamics is universal on essentially all time scales, and where the high-frequency few-body response is related to the long-time dynamics of the Anderson orthogonality catastrophe by Tan relations. Our theoretical description relies on different and complementary approaches: functional determinants give an exact numerical solution for time- and frequency-resolved responses, bosonization provides accurate analytical expressions at low temperatures, and the theory of Toeplitz determinants allows one to analytically predict response up to high temperatures. Using these approaches we predict the thermal decoherence rate of the fermionic system and prove that within the considered model the fastest rate of long-time decoherence is given by [Formula: see text]. We show that Feshbach resonances in cold atomic systems give access to new interaction regimes where quantum effects can prevail even in the thermal regime of many-body dynamics. The key signature of this phenomenon is a crossover between different exponential decay rates of the real-time Ramsey signal. It is shown that the physics of the orthogonality catastrophe is experimentally observable up to temperatures [Formula: see text] where it leaves its fingerprint in a power-law temperature dependence of thermal spectral weight and we review how this phenomenon is related to the physics of heavy ions in liquid [Formula: see text]He and the formation of Fermi polarons. The presented results are in excellent agreement with recent experiments on LiK mixtures, and we predict several new phenomena that can be tested using currently available experimental technology.
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.
In this thesis, we have explored various aspects of the dynamics of an impurity moving in more than one 1D bath. A recurring theme has been the investigation of the orthogonality catastrophe (OC) that follows the injection of the impurity in the system. This phenomenon has been studied by calculating the Green’s function of the impurity, which also described the time evolution and spectral features of the latter. The distinctive signature of the OC is that the Green’s function shows a power-law tail at long times. This function has been calculated using suitable perturbative expansions in the impurity-bath coupling, namely the Linked Cluster Expansion (LCE) and a time-dependent perturbation theory around a nontrivial dynamics. All of our results are nonperturbative in the inter-bath hopping. In the two-bath scenario we have performed a detailed asymptotic expansion of the LCE Green's function at long times, which turned out to be very accurate in comparison with its numerical evaluation. The expansion has allowed us to obtain the renormalisation of the dispersion of the impurity bands, as well as the exponent of the power-law decay and the lifetime of the odd mode. One of our main results is that the OC, leading to the breakdown of the quasiparticle picture, survives the inclusion of a second 1D bath and dominates the long-time behaviour of all the components of the Green's function. In particular, the exponent characterising the long-time behaviour of the Green's function is given by half of the average of the exponents of the individual baths and, notably, is the same for the intra-bath Green's functions and for that connecting the two baths. In the case of two asymmetric baths, the Green's function is nonuniversal, acquiring a high-frequency component at short times and exhibiting persistent oscillations at longer times. In real experiments, the temperature of the baths is always finite, so we have extended the LCE treatment to this scenario. The effect of the temperature is to suppress the Green's function, limiting the possibility to observe its coherent oscillations and the power-law tail. At sufficiently long times, we found analytically that this suppression is exponential in time, with a different decay constant for the even and odd modes. For low temperatures, the two decay constants are approximately coinciding, and are proportional to the temperature. Using a perturbation theory in the inter-band part of the interaction we have been able to reproduce the LCE Green’s function and the OC with a method which also allows us to access the evolution of the whole impurity-bath system. The advantage of this approach is that it provides an analytic expression for the time evolution of the state of the whole impurity-baths system. This has allowed us to compute the time evolution of observables beyond the impurity Green's function, including the often-neglected properties of the baths. Moreover, the numerical effort required by the approach is sufficiently low so that we have been able to treat the case in which the impurity is initialised in a wave packet of an (almost) arbitrary shape. On the impurity side, we have analysed the time evolution of its population within each bath, observing how the persistent oscillations of the free impurity are damped and slowed down by the interaction with the baths. The impurity momentum is subjected to damping, as well. We have observed that momentum is transferred to the baths in two steps: a short transient, connected to the bath relaxation and independent of the inter-bath hopping, and a much slower decay caused by the emission of phonons during the deexcitation of the odd mode. Finally, we have also examined the time evolution of the probability density of finding the impurity in a given position and bath, for various Gaussian wave packets. We have studied the time evolution of observables describing the dynamics of the baths, which are rarely discussed in the literature on mobile impurities, but are nonetheless accessible to experiments. We have looked at the number of excited phonons, which shows a slow logarithmic divergence in time, related to the OC, superimposed with the faster growth caused by the emission of phonons from the odd mode decay. The particle density of the baths shows a semiclassical behaviour, intuitively similar to that of a pond in which a stone has been thrown. After the impurity has been injected into one of the baths, a localised density depletion forms and follows its motion. At the same time, two wave fronts are generated and propagate away. Moreover, each time the impurity oscillates between the baths, a new pair of ripples is emitted. The emission of ripples can be suppressed by employing wider wave packets. We have also found that the bath momentum density displays a behaviour analogous to that of the density. When the initial impurity wave packet has a markedly non-Gaussian shape, we find complex interference phenomena, both in the ripples and within the central trough. We examined the inter-bath, equal-times connected density and momentum density correlations, which revealed a rich structure in real space. This structure is best understood by taking “slices” of the correlation function along the relative and centre-of-mass coordinate, which show both the light-cone propagation of correlations and the motion of the impurity. We have made the first steps towards a many-baths system, in which the impurity moves in a 1D or 2D lattice of 1D baths. We have obtained the time evolution of the state of the system with the perturbative technique developed before. The impurity Green's function has revealed qualitative differences from the two-baths setup. First of all, the Green's function shows a complex short-time behaviour, caused by interference effects between the various paths of propagation within the lattice of baths. More importantly, we have found that each band of the noninteracting impurity is characterised by its own OC exponent, which is proportional to the degeneracy of the band. We have then speculated that it may be possible to tune the OC exponent either by properly designing the lattice of baths, or by changing the degeneracies by means of external magnetic fields. We also shown that for generic lattices of baths, the OC exponents vanish in the limit of an infinite number of lattice sites. Lastly, we showed that in the case of two identical baths there exists a unitary transformation that diagonalises the impurity degrees of freedom, thus achieving a complete decoupling of the latter from the baths. On the basis of this transformation, we have sketched two variational approximations for the ground-state and dynamics of the system.
We investigate the behavior of a two-level atom coupled to a one-dimensional, ultracold Fermi gas. The sudden switching on of the scattering between the two entities leads to the loss of any coherence in the initial state of the impurity and we show that the exact dynamics of this process is strongly influenced by the effect of the orthogonality catastrophe within the gas. We highlight the relationship between the Loschmidt echo and the retarded Green's function---typically used to formulate the dynamical theory of the catastrophe---and demonstrate that the effect is reflected in the impurity dynamics. We show that the expected nonexponential decay of the spectral function can be observed using Ramsey interferometry on the two-level atom and comment on finite temperature effects.