Emergence of a Sharp Quantum Collective Mode in a One-Dimensional Fermi Polaron

Pavel E. Dolgirev ,1,*,‡ Yi-Fan Qu ,2,3,‡ Mikhail B. Zvonarev,4,5,6 Tao Shi,2,7,† and Eugene Demler1,81Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

2CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China

3School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China4Universite Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France

5St. Petersburg Department of V.A. Steklov Mathematical Institute of Russian Academy of Sciences,Fontanka 27, St. Petersburg, 191023, Russia

6Russian Quantum Center, Skolkovo, Moscow 143025, Russia7CAS Center for Excellence in Topological Quantum Computation,University of Chinese Academy of Sciences, Beijing 100049, China

8Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland

(Received 4 December 2020; revised 18 July 2021; accepted 12 August 2021; published 21 October 2021)

The Fermi-polaron problem of a mobile impurity interacting with fermionic medium emerges in variouscontexts, ranging from the foundations of Landau’s Fermi-liquid theory to electron-exciton interaction insemiconductors, to unusual properties of high-temperature superconductors. While classically the mediumprovides only a dissipative environment to the impurity, the quantum picture of polaronic dressing is moreintricate and arises from the interplay of few- and many-body aspects of the problem. The conventionalexpectation for the dynamics of Fermi polarons is that it is dissipative in character, and any excess energy israpidly emitted away from the impurity as particle-hole excitations. Here we report a strikingly differenttype of polaron dynamics in a one-dimensional system of the impurity interacting repulsively with thefermions. When the total momentum of the system equals the Fermi momentum, there emerges a sharpcollective mode corresponding to long-lived oscillations of the polaronic cloud surrounding the impurity.This mode can be observed experimentally with ultracold atoms using Ramsey interferometry and radio-frequency spectroscopy.

DOI: 10.1103/PhysRevX.11.041015 Subject Areas: Atomic and Molecular Physics,Condensed Matter Physics

I. INTRODUCTION

The problem of a polaron—a mobile particle interactingwith a host medium—has a long history dating back toLandau’s seminal work on an electron inducing localdistortion of a crystal lattice [1]. Polarons are ubiquitousin many-body systems, especially in solid-state [2,3] andatomic physics [4–6], and provide one of the key paradigmsof modern quantum theory. Recent experimental progressin cold atoms and ion-based quantum simulators bringsnew motivation for studying polaronic phenomena [7–19],since these platforms offer a high degree of isolation,

tunability of the interaction strength and dispersion, andcontrol of dimensionality [20]. These setups are particularlywell suited for accurate studies of far-from-equilibriumdynamics [10,17] since system parameters can be modifiedmuch faster than intrinsic timescales of the many-bodyHamiltonians. On the theoretical side, recent progress inunderstanding polarons has come from using such powerfultechniques as variational Ansätze [21–30], renormalization-group calculations [31,32], Monte Carlo simulations [33–35], diagrammatic techniques [36–41], exact Bethe Ansatz(BA) calculations for integrable models [42–47], andapproaches based on nonlinear Luttinger liquids [48].Analysis of equilibrium and dynamical properties of polar-ons has played an important role in developing new ideas andconcepts, and in testing theoretical methods and approaches.One of the surprising recent discoveries in the far-from-

equilibrium dynamics of Fermi polarons has been theprediction of the effect called quantum flutter [25,49]:When a repulsive mobile impurity with large momentum isinjected into a one-dimensional Fermi gas, it undergoeslong-lived oscillations of its velocity. This should be

*p_dolgirev@g.harvard.edu†tshi@itp.ac.cn‡P. E. D. and Y.-F. Q. contributed equally to this work.

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.

PHYSICAL REVIEW X 11, 041015 (2021)

2160-3308=21=11(4)=041015(11) 041015-1 Published by the American Physical Society

contrasted with the classical situation in which the impuritygradually slows down while transferring its momentum tothe host atoms. It has also been found that the quantumflutter frequency does not depend on the initial conditions.The robustness of these oscillations naturally suggests thatthey represent an incarnation of a fundamental yetunknown property of the polaronic system at equilibrium.In the present work, we investigate collective modes—

elementary excitations describing small deviations from anequilibrium state—in the system of a mobile impurityinteracting repulsively with a one-dimensional Fermi gas.Remarkably, we find the density of states of these modesdisplays a sharp peak when the total momentum of thesystem equals the Fermi momentum. This peak signals theemergence of a distinct collective excitation with a fre-quency ωkF, representing a “breathing mode” of a polaro-nic cloud surrounding the impurity. The frequency ωkFmatches the magnon-plasmon energy difference at theFermi momentum, as has been checked for an arbitraryrange of model parameters. As we demonstrate below,modern cold-atom techniques, including Ramsey interfer-ometry and radio-frequency (rf) spectroscopy, can be usedto detect this mode. Specifically, we find that the impurityabsorption spectra at the Fermi momentum exhibit adouble-peak structure, with the second peak correspond-ing to the frequency ωkF. Our study provides a naturalinterpretation of such a complex far-from-equilibriumphenomenon as recently discovered quantum flutter interms of basic equilibrium properties. In particular, weargue that flutter oscillations, as well as their robustness,represent nothing but the signatures of the collectivemode ωkF .

II. THEORETICAL FRAMEWORK

A Fermi-polaron model represents a nontrivial many-body problem with the Hamiltonian consisting of threeparts: H¼ Hfþ Himpþ Hint, where Hf ¼ P

kðk2=2mÞc†kckis the fermionic kinetic energy, Himp ¼

Pkðk2=2MÞd†kdk

is the kinetic energy of the impurity, and Hint ¼ðg=LÞPk;k0;q d

†kþqdkc

†k0−qck0 describes contact interaction

between the two species of particles. The Planck constant isset to ℏ ¼ 1 throughout the paper. Operator d†k (dk) creates(annihilates) the impurity with momentum k; operators c†kand ck represent the host gas. Throughout this work, weassume periodic boundary conditions with the system sizeL, so that k ¼ ð2π=LÞn with n being integer. The totalnumber of host-gas particles N is fixed via the chemicalpotential μ ¼ ðk2F=2mÞ, and kF ¼ ðπN=LÞ is the Fermimomentum. In our calculations, we set kF ¼ ðπ=2Þ,which implicitly defines the unit of length. The case ofa single impurity restricts the Hilbert space to states withP

k d†kdk ¼ 1. The dimensionless interaction strength

between the impurity and medium is γ ¼ ðπmg=kFÞ. We

use the following convention for the Fourier transform:cx ¼ ð1= ffiffiffiffi

Lp ÞPk e

ikxck. We choose a sufficiently large UVmomentum cutoff Λ ≫ kF in our numerical simulations.A challenge one encounters when solving a many-body

problem is that the Hilbert space grows exponentially withthe system size, limiting direct numerical simulations torelatively small systems. One approach to overcoming thisdifficulty is to employ a variational method, where a limitednumber of parameters is used to parametrize a class ofmany-body states. In this approach, the complexity ofcomputations typically grows polynomially with the sys-tem size, allowing for efficient numerical analysis.However, one needs to ensure that the variational wavefunction contains the right class of quantum states that canreliably capture the many-body correlations. More specifi-cally, a variational family of states is required to satisfy thefollowing criteria: (i) it contains a manageable number ofvariational parameters, (ii) it accurately predicts ground-state properties, (iii) it captures real-time dynamics includ-ing the spectrum of collective modes, and (iv) it can be usedto compute observables relevant for experiments.We employ recent developments of approaches based on

non-Gaussian states (NGS) to realize this program [29,50].In this work, we deal with zero-temperature situations, andfinite-temperature ensembles can be studied using theformalism developed in Ref. [51]. For the Fermi-polaronproblem, one first performs a unitary transformation to theimpurity reference frame [28,52], S ¼ expð−iximpPfÞ,where Pf ¼ P

k kc†kck is the total fermionic momentum

and ximp is the impurity position operator, and then invokesthe Hartree-Fock approximation. The unitary transforma-tion S plays a twofold role: (i) it provides sufficiententanglement between the impurity and the medium sothat the Hartree-Fock approximation becomes accurate and(ii) it takes advantage of the total momentum conservationand decouples the impurity from the rest of the system. Inthe impurity frame, the transformed Hamiltonian is para-metrized by the total momentum Q:

HQ ¼Xk;k0

c†k

�k2

2mδkk0 þ

gL

�ck0 þ

ðQ − PfÞ22M

: ð1Þ

Note that only the degrees of freedom of the host gas enterEq. (1). The first term is the fermionic kinetic energy, thesecond term describes scattering off the impurity, and thethird term corresponds to its recoil energy.

A. Equations of motion

The Hartree-Fock approximation can be convenientlycast into a Gaussian wave function [29], which we write as

jψðtÞi ¼ e−iθ expðic†ξcÞjFSi; ð2Þ

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where ξ ¼ ξ† and jFSi≡Qjkj≤kF c

†kj0i is the wave function

of the filled Fermi sea (j0i corresponds to the vacuumstate). The information about the state (2) is then encodedin θ and U ≡ eiξ. Let us define the covariance matrix asΓk;k0 ≡ hc†kck0 iGS ¼ U�Γ0UT , where Γ0 is the covariancematrix of the filled Fermi sea. We emphasize that eventhough we choose the state to be Gaussian in the impurityreference frame, it is non-Gaussian in the laboratory framedue to the unitary operator S.To find the (momentum-dependent) ground-state wave

function, we employ the imaginary-time dynamics [29]:

dτΓ ¼ 2ΓhΓ − fh;Γg; ð3Þ

where

E½Γ�≡ hHQi ¼Xk;k0

Γkk0

��ϵk þ

k2

2M−Q · kM

�δkk0 þ

gL

�

−Xk;k0

k · k0

2MjΓkk0 j2 þ

P2f

2Mþ Q2

2M; ð4Þ

hkk0 ≡ δE½Γ�δΓk0k

¼�ϵk þ

k2

2Mþ k · ðPf −QÞ

M

�δkk0

þ gL−k · k0

MΓkk0 : ð5Þ

Here ϵk ¼ ðk2=2mÞ − μ. Note that initially pure states, withΓ2 ¼ Γ, will remain pure under the imaginary-time evolu-tion: dτðΓ2 − ΓÞ ¼ 0. For these states, the total number offermions N ¼ P

k Γkk is conserved.The equations of motion for the real-time dynamics are

obtained from Dirac’s variational principle:

∂tθ ¼ E½Γ� − trhΓ; i∂tU ¼ h�U: ð6Þ

From Eqs. (6) one can derive an equation of motion solelyon the covariance matrix [29]:

dtΓ ¼ i½h;Γ�: ð7Þ

During the real-time dynamics, the state remains pure, andthe total number of fermions is also conserved.

B. Analysis of collective modes

Collective excitations represent low-energy small-ampli-tude fluctuations on top of an equilibrium state, in our caseon top of a ground state, with the covariance matrix ΓQ,previously computed via the imaginary-time dynamics. Toobtain their spectrum within the NGS approach, we writeΓ ¼ ΓQ þ δΓ, and, assuming that δΓ is small, we linearizethe real-time equation of motion (7):

dtδΓ ¼ i½hQ; δΓ� þiM

½trðPδΓÞP − PδΓP;ΓQ�; ð8Þ

where hQ ¼ h½ΓQ� and Pk;k0 ≡ kδk;k0 . Importantly, thesefluctuations δΓ are constrained to satisfy (i) HermiticityδΓ ¼ δΓ†, (ii) particle number conservation trδΓ ¼ 0, and(iii) purity fΓQ; δΓg ¼ δΓ. To see the implications of theseconditions, we switch to the basis where the matrix ΓQ isdiagonal:

ΓQ ¼�0 0

0 IN×N

�⇒ δΓ ¼

�0 K

K† 0

�; ð9Þ

where K is an arbitrary ðNSP − NÞ × N matrix that fullyparametrizes all possible physical fluctuations δΓ. NSP isthe total number of single-particle modes in the fermionicsystem, and it is determined by the system size L and by theUV cutoff Λ. We, therefore, conclude that the aboveconstraints reduce the total number of degrees of freedomin δΓ from 2N2

SP to only 2NðNSP − NÞ. Plugging Eq. (9)into Eq. (8), one obtains

dtK ¼ iðh11K − Kh22Þ þiM

P12trðP12K† þ P21KÞ

−iM

ðP11KP22 þ P12K†P12Þ; ð10Þ

where we used that in the chosen basis hQ ¼�

h11 0

0 h22

�,

since hQ and ΓQ commute (in principle, one can alwayschoose a basis where both hQ and ΓQ are diagonal). Theeigenenergies ωQ

i of Eq. (10) constitute the spectrum ofcollective excitations (here Q is the total momentum of thesystem, and i labels excitations). From ωQ

i and correspond-ing eigenvectors, one can compute standard linear-responsefunctions such as the density response function [53]. For abosonic system, similar fluctuation analysis has beenproven to be equivalent to the generalized random phaseapproximation [54] and successfully applied to reproducethe Goldstone zero-mode naturally without imposing theHugenholtz-Pines condition.Using the outlined theoretical framework, we obtained

our main results, which we discuss in the next section.

III. MAIN RESULTS AND DISCUSSION

A. Emergent collective mode and its origin

An important feature of the polaron system is momen-tum conservation, which after the Lee-Low-Pines (LLP)transformation and via the imaginary-time dynamics allowsus to compute not only the ground-state energy but ratherthe entire polaron energy-momentum dispersion. An exam-ple of such a calculation for the integrable case of equal

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masses M ¼ m is shown in Fig. 1(a), where we comparemomentum-dependent ground-state energies to the exactBethe Ansatz results [42,47]. The agreement is excellent,and it becomes even better for a larger total number ofparticles N and/or larger momentum cutoff Λ. Note that inthe thermodynamic limit N → ∞, this energy-momentumrelation is 2kF periodic, since at Q ¼ 2kF one can alwaysexcite a zero-energy particle-hole pair across the Fermisurface. At finiteN, this is no longer true, and to excite sucha pair costs energy proportional to 1=N, explaining thediscrepancy in Fig. 1(a) at large momenta Q ≃ 2kF. InAppendix A, we demonstrate that our approach alsoreproduces exact many-body correlation functions.Below, we also investigate a generic situation of not equalmasses, where no known exact solutions are available.Using the approach outlined in Sec. II B, we turn to

investigate the spectrum of collective modes on top ofmomentum-dependent ground states ΓQ. Figure 1(b) showsthe density of states (DOS) of these excitations, νω ¼P

i δðω − ωQi Þ. Most spectacularly, we discover a sharp

peak atQ ¼ kF [Fig. 1(c)], which signals the onset of a newdistinct collective mode. Our primary goal below is toelucidate its physical origin and investigate the feasibilityof experimental verification with ultracold atoms.We turn to discuss the physical mechanism behind the

emergence of this mode ωkF . We first identify which statesin the many-body spectrum determine the frequency ωkF.Let us define plasmon as the lowest-energy excitation of theFermi gas in the absence of impurity. Its dispersion EpðQÞhas a familiar inverse-parabolic shape, shown in Fig. 1(a)with dashed line. Magnon is the lowest-energy excitation ofthe entire interacting system. The magnon dispersionEmðQÞ—the polaron energy-momentum relation—is illus-trated in Fig. 1(a) with a solid line. In the presence of the

impurity, the plasmon still exists, but no longer representsthe lowest-energy excitation. Note that both magnon andplasmon group velocities evaluated atQ ¼ kF—at the samewave vector where the mode ωkF emerges—are zero,suggesting that these two states can form a correlatedlong-lived excitation, with frequency ωPMðkFÞ ¼ EpðkFÞ−EmðkFÞ. Interestingly, our variational calculations showthat

ωkF ¼ ωPMðkFÞ ð11Þ

for any impurity-gas mass ratios M=m and couplingstrengths γ, as illustrated in Fig. 1(e).Now we demonstrate that the collective excitation ωkF

represents oscillations of the polaronic cloud surroundingthe impurity. To that end, we take the initial many-bodywave function jψ labð0Þi ¼ jGSQi to be the ground state ofthe interacting Fermi polaron model with the total momen-tum Q ¼ kF, and then suddenly change the interactionstrength. In response to such a quench, we find that thefermionic density in the vicinity of the impurity,

G2ðx; tÞ ¼LN

ZL

0

dyhψ labðtÞjd†ydyc†xþycxþyjψ labðtÞi;

demonstrates damped oscillatory behavior, illustrated inFig. 2, with the frequency ωkF. These real-time dynamicalcorrelations are, in principle, accessible with ultracold-atom setups. One can see, however, that the amplitude ofthe signal shown in Fig. 2 is rather small because the systemis close to the linear-response regime. We find that theamplitude of oscillations remains small even for strongerquenches. To overcome this issue, below we suggest acomplementary experimental verification of our findings

FIG. 1. (a) Polaron energy-momentum relation—the magnon branch—for the case of equal massesM ¼ m and the plasmon branch ofthe host Fermi sea. Note the plasmon-magnon excitation energy ωPM at kF. Our variational NGS approach remarkably reproduces theexact BA result (solid line), adopted from Refs. [42,47] (here we usedΛ ¼ 10kF). (b) Density of states (DOS) of collective excitations asa function of frequency ω and total momentum Q. Here tF ¼ 1=EF is the Fermi time. Note that the spectral signal becomes particularlypronounced at Q ¼ kF. (c),(d) Cuts of the DOS at Q ¼ 0 and Q ¼ kF, respectively. Dashed line in (d) [shown also in (b)] indicates theemergence of a sharp mode. (e) The collective mode ωkF , as a function of the mass ratioM=m, matches the plasmon-magnon mode ωPM

from (a). Parameters used are γ ¼ 5, N ¼ 51, and Λ ¼ 5kF.

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by computing observables accessible with rf spectroscopyand Ramsey-type interferometry.

B. Cold-atom setups

Possible experimental setups for investigating the phys-ics of a mobile impurity coupled to a Fermi bath are shownin Figs. 3(a) and 3(b). We assume that the impurity has twohyperfine states: j↓i is decoupled from the Fermi sea,whereas j↑i strongly interacts with the host gas. We startfrom an initial many-body wave function prepared in thestate jFSi ⊗ j0;↓i. jQ;↓i labels the impurity state with thetotal momentum Q. To reach a given total momentumsector Q, we suggest the quenching protocol illustrated inFig. 3(a). The impurity is first accelerated—for example, byapplication of an external force as in Ref. [17]—such thatits momentum becomesQ. Then a rf pulse is used to couplethe two hyperfine states. Similar to the case of a staticimpurity discussed in Ref. [55], Ramsey interferometry canprobe the dynamical overlap function, which in our case is

written as SðtÞ ¼ hFSjeitHð0ÞQ e−itHQ jFSi, where Hð0Þ

Q is givenby Eq. (1) with g ¼ 0. The impurity absorption spectra areobtained as

Aω ¼ ð1=πÞReZ

∞

0

dteiωtSðtÞ.

(a)

(b)

FIG. 2. Linear-response dynamics in the impurity frame after asoft quench of the coupling strength γ: from γ ¼ 6 to γ ¼ 5. Thewave function at t ¼ 0 corresponds to the ground state at Q ¼ kFand γ ¼ 6. (a) Evolution of δG2ðx; tÞ ¼ G2ðx; tÞ − G2ðx; 0Þshowing oscillatory behavior of the fermionic density surround-ing the impurity. (b) Dynamics of δG2ðx; tÞ at x ¼ x0 [dashed linein (a)]. Notably, the Fourier transform of this signal (inset)matches the frequency ωkF extracted from Fig. 1(b).

FIG. 3. (a),(b) Possible cold-atom setups. The initial wave function corresponds to jFSi ⊗ j0;↓i, where the hyperfine state j↓i doesnot interact with the fermionic medium. (a) The impurity is first accelerated such that it acquires momentum Q; subsequent rf pulsedrives it into the hyperfine state j↑i strongly interacting with the host gas. (b) Alternatively, the two states j0;↓i and jQ;↑i can bedirectly coupled by a two-photon Raman process. (c)–(f) The dynamical overlap function SðtÞ and the impurity absorption spectra Aω

for Q ¼ 0 (c),(e) and for Q ¼ kF (d),(f). We shifted frequencies in (e) and (f) such that the zero value in both panels represents thecorresponding ground-state oscillations. For Q ¼ kF, the Ramsey contrast jSðtÞj demonstrates switching from initial rapid decay fortimes t ≲ 10tF to the lasting regime of slow dynamics. This behavior reflects inAω as it acquires a double-peak structure: The frequencyof the first peak is close to that of the ground-state oscillations, whereas the second peak corresponds to the collective mode ωkF ¼ ωPM.Parameters are the same as in Fig. 1, except N ¼ 251.

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Figure 3(b) shows an alternative experimental setup [56],where one employs the two-photon Bragg spectroscopy[57–59]. In this latter situation, the dynamical overlapfunction is modified by a nonessential phase factor.The overlap function SðtÞ is computed analytically:

SðtÞ ¼ e−ifθðtÞ−E½Γ0�tg detf1 − ½1 − UðtÞ�ΓT0g: ð12Þ

This expression is a generalization of the approach used forthe case of static impurity [55]. A numerical simulation ofEqs. (6) indicates that SðtÞ exhibits long-time revivals(roughly at t ≃ L=kF) associated with the finite system sizeL. Below we, therefore, choose a sufficiently large systemsuch that these revivals do not appear up to the largestsimulation times.For the case Q ¼ 0, shown in Figs. 3(c) and 3(e), the

Ramsey contrast jSðtÞj demonstrates a slow monotonicdecay; at long times it saturates around R0 ¼ jhGS0jFSij2,which equals R0 ≃ 0.6 for the parameters used in Fig. 3.Note, however, that in the thermodynamic limit, L → ∞,this quasiparticle residue R0 should vanish—an analog ofthe Anderson orthogonality catastrophe for the case of astatic impurity. This latter statement we explicitly verifynumerically in Appendix C, where we show that R0 decayswith the system size L as a power law. We note in passingthat this decay is much slower than in the case M ¼ ∞. Inthe frequency domain, Aω displays a sharp peak at thepolaron binding energy E0, as it should be.Figures 3(d) and 3(f) show the results for Q ¼ kF. We

find that SðtÞ demonstrates a qualitative change in itsdynamics roughly at t ≃ 10tF: The initial quick decay ofjSðtÞj, associated with the fact that the initial wave functionrepresents a far-from-equilibrium state for Q ¼ kF, turnsinto a much slower power-law decay at longer times. Thisbehavior resembles the non-Markovian spontaneous emis-sion of a two-level atom coupled to a nonflat photon bath[60], in which case the initial fast decay is associated withthe large DOS of the collective mode, cf. Fig. 1(b). For theimpurity absorption spectra Aω, we find it acquires adouble-peak structure. Importantly, the second broad peakcorresponds to the collective mode ωkF—the dashed line inFig. 3(f) denotes the discussed plasmon-magnon modeωPMðkFÞ. The position of the first peak is close to thefrequency of the ground state at Q ¼ kF. There are a fewreasons for the small mismatch between them. First, wefind that the overlap RkF ¼ jhGSkF jFSij2 is suppressed: itequals 4 × 10−3 for the parameters used in Fig. 3.Therefore, the first maximum in Aω is shifted towardhigher frequencies, where the overlap of the initial wavefunction and an excited state is more pronounced. Second,such a small value of RkF further indicates that the intrinsicdynamics is far from equilibrium. However, in an out-of-equilibrium setting, our method is not expected to be exact.Indeed, an explicit comparison in Appendix B of dynamics

in the NGS approach to that of the BA indicates that ourmethod displays a similar small discrepancy with the exactresult. Qualitatively, the method provides correct predic-tions even for nonequilibrium problems. Finally, the smallmismatch could potentially be reduced by increasing thefrequency resolution, which requires a simulation of aneven larger system and for longer times.

C. Quantum flutter and its robustness

Equipped with our main results outlined above, we turnto discuss the phenomenon of quantum flutter. While thequantum flutter embodies a strongly out-of-equilibriumcharacter, its phenomenology, surprisingly, shares a lot incommon with the discovered here equilibrium collectivemode ωkF . Indeed, essentially arbitrary quenching of thepolaronic system [25,49] results in the development at longtimes of long-lived oscillations of the polaronic cloudsurrounding the impurity. For the integrable situations, itwas shown that the frequency of those oscillations matchesthe plasmon-magnon energy difference at kF. The robust-ness of quantum flutter oscillations to quenching condi-tions, together with the results of our work, suggests thefollowing interpretation. The initial (strong) perturbation ofthe polaronic system results in exciting all kinds of possibleexcitations. Only the ones with small relative groupvelocities contribute to the impurity dynamics at longtimes, while the role of the rest of them is to carry energyand momentum away from the impurity. Although thesystem is out of equilibrium, near the impurity, and at longtimes, the dynamics should resemble the equilibriumsituation, such as in Fig. 2. This is (i) because both theplasmon and magnon group velocities are zero at kF and(ii) due to the enhanced density of states of the associatedcollective mode ωkF , cf. Fig. 1(b). This physical pictureexplains long-time flutter oscillations and their robustness.We turn to discuss new insights that our work brings to

the physics of quantum flutter. The plasmon and magnon atkF form a long-lived collective mode ωkF , although theplasmon lives in the continuum of excitations of theinteracting system. The sharp collective mode is respon-sible for a robust, long-lived oscillatory dynamics, inde-pendent of quenching conditions. More generally, weexpect the development of such a collective mode providedthere exist two branches of excitations, which are requiredto have zero group velocities simultaneously (as such, thecorresponding density of states are enhanced due to vanHove singularities). In contrast to existing theoreticalapproaches, our variational states enable an accuratecomputation of the magnon dispersion away from theintegrable points. We remark that the plasmon-magnonenergy difference at kF shown in Fig. 1(e) is consistent withthe flutter frequency for M ≥ m obtained from densitymatrix renormalization group simulations [25]. In theregime of light impurity, M ≲m, the two approaches start

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to deviate, which warrants further investigation. A probablereason for this is that it becomes notoriously difficult toextract the flutter frequency reliably in this regime (seeAppendix D).

IV. CONCLUSION AND OUTLOOK

The Fermi-polaron Hamiltonian represents an instanceof an interacting many-body system without a smallparameter. As such, to establish the validity of any con-clusions, it is crucial to verify that our chosen variationalstates are sufficient to capture both equilibrium and out-of-equilibrium properties. Importantly, it turns out that thecaseM ¼ m is Bethe Ansatz solvable, providing us with thenecessary ground for testing our approach. Specifically, inAppendix A, we demonstrate that our non-Gaussian statesreproduce both the ground-state energies and many-bodycorrelation functions, for both repulsive and attractiveinteractions. Benchmarking the out-of-equilibrium situa-tion is more challenging. In Appendix B, we show that thedeveloped here variational approach captures the essentialfeatures of the quantum flutter dynamics, although notperfectly. Our work, therefore, establishes the reliability ofthe non-Gaussian states in one spatial dimension, wherefluctuations are expected to be the strongest. Because ofefficient numerical implementation, these variational statesare promising to solve interacting many-body problems inhigher dimensions, where no such powerful numericalmethods as density matrix renormalization group exist,with Monte Carlo-based approaches being the only alter-native. They can be used for computing ground-stateproperties, including many-body correlations and collectivemodes, as well as out-of-equilibrium real-time dynamics.Once we established the reliability of the non-Gaussian

approach, we turned to investigate collective excitations ofthe polaronic system at equilibrium. Our main result is thattheir spectrum contains a sharp peak when the totalmomentum of the system equals kF, signaling the onsetof a new distinct collective mode. By analyzing polarondispersion for an arbitrary range of the mass ratios M=mand interaction strengths γ, we concluded that this moderepresents nothing but the plasmon-magnon excitation atkF. Our work suggests a close connection between the far-from-equilibrium Fermi-polaron dynamics and equilibriumcollective excitations. This relation explains the robustnessof the phenomenon of quantum flutter to changes in modelparameters and initial conditions. Theoretical predictionsmade in this paper can be tested with currently availableexperimental systems of ultracold atoms. Specifically, onecan search for the following features that should appearwhen the momentum of the impurity relative to the hostatoms reaches kF: (i) long-lived oscillations of the polar-onic cloud, (ii) abrupt change in the time evolution of theRamsey contrast, and (iii) development of the double-peakstructure in the impurity absorption spectra at the Fermimomentum. The frequency of oscillations can be tuned

experimentally by varying either the interaction strength γor the mass ratio M=m. From a broader perspective, ourwork is inspired by the recent developments in designingand studying controlled quantum systems using both solid-state and cold-atom platforms. In particular, as modernsemiconductor technologies are approaching the quantumdomain with the current feature size of a few nanometers,understanding far-from-equilibrium dynamics of interact-ing fermionic systems will be crucial for the design andoperation of future electronic devices.

ACKNOWLEDGMENTS

We thank I. Bloch, M. Zwierlein, Z. Yan, I. Cirac,D.Wild, F. Grusdt, K. Seetharam, O. Gamayun, and D. Selsfor fruitful discussions. P. E. D. and E. D. are supported bythe Harvard-MIT Center of Ultracold Atoms, ARO GrantNo. W911NF-20-1-0163, and NSF Grant No. OAC-1934714 and NSF EAGER-QAC-QSA Grant awardNo. 2222-206-2014111.T. S. is supported by the NSFC(Grant No. 11974363). The work of M. B. Z. is supportedby Grant No. ANR-16-CE91-0009-01 and CNRS GrantNo. PICS06738.

APPENDIX A: BENCHMARKING THEGROUND-STATE CORRELATIONS

WITH THE BETHE ANSATZ RESULTS

In the main text, we showed that for the case of repulsivecoupling γ > 0, the non-Gaussian variational approachreliably reproduces exact ground-state energies, availablefrom the BA for M ¼ m. Figure 4 demonstrates that ourmethod also agrees with the BA for the case of attractivecoupling γ < 0. Remarkably, even the kinklike feature atQ ¼ kF is reproduced. Below, we further compute variousmany-body correlation functions via the variationalapproach and compare them to the BA results.

FIG. 4. Polaron energy-momentum relation for the case ofattractive coupling γ < 0 (here we fix Λ ¼ 10kF). We find thatour variational calculation nicely reproduces the BA result,adopted from Refs. [43,47].

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We first compute the two-point correlation functionG2ðxÞ defined in the main text. This correlator describesthe probability density to find a bath particle separated fromthe impurity by the distance x. Within the Gaussianvariational approach, it is obtained as

G2ðxÞ ¼LNhc†xcxiGS: ðA1Þ

Figure 5 shows the comparison of our calculation (A1) tothe BA result adopted from Refs. [42,43], and we observean excellent agreement for both repulsive and attractivecouplings.Another interesting observable to calculate is the

momentum distribution function of the impurity in thelaboratory frame:

nkðQÞ ¼ hψ labjd†kdkjψ labi ¼ hψLLPjd†kþPfdkþPf

jψLLPi

¼Z

L

0

dxLhexp ðixðkþ Pf −QÞiGS: ðA2Þ

Using the formalism of Ref. [29], the latter expectationvalue can be computed analytically:

nk ¼Z

L

0

dxL

eiðk−QÞx det½ðeiKx − 1ÞΓQ þ 1�; ðA3Þ

where Kk;k0 ¼ kδk;k0 . This integral we evaluate numericallyonce the covariance matrix ΓQ has been computed using theimaginary-time evolution. In Fig. 6, we compare ourapproach with the BA calculation of Ref. [47]: Theagreement between the methods is quite striking. We,therefore, conclude that our non-Gaussian approach notonly predicts correctly ground-state energies but alsocaptures properties of wave function correlations.

APPENDIX B: BENCHMARKINGFAR-FROM-EQUILIBRIUM DYNAMICS

We now aim to test our real-time approach, encoded inEq. (7), and apply our formalism to the so-called quantumflutter outlined below. Importantly, recent exact BetheAnsatz [49] calculations and simulations with matrixproduct states [25] provide a necessary ground to bench-mark our variational method.

FIG. 5. Comparison of the two-point static correlation functionG2ðxÞ: solid (red) curves correspond to the non-Gaussianvariational approach; dashed (black) lines represent the BAanalytical results adopted from Refs. [42,43]. We consider bothrepulsion (left) and attraction (right). Parameters used are Λ ¼15kF and N ¼ 51.

FIG. 6. Comparison of the impurity momentum distributionfunction nkðQÞ: solid (red) curves correspond to simulations ofEq. (A3); the BA results (dashed black lines) are taken fromRef. [47]. Parameters are the same as in Fig. 5.

FIG. 7. Comparison of out-of-equilibrium correlations. (a) Evo-lution of the impurity momentum for three different initial con-ditions. Solid curves represent our variational simulations anddashed curves show the BA results adopted from Ref. [49]. Note(i) damped quantum flutter oscillations for intermediate times and(ii) saturation of the impurity momentum at t → ∞. The non-Gaussian approach well captures the initial dynamics and satu-ration, but the discrepancy at intermediate times is clear. Parametersused are γ ¼ 10, N ¼ 51, Λ ¼ 10kF. (b),(c) Momentum distribu-tion function of the host-gas particles at t ¼ 5tF (b) and t ¼ 25tF(c). These graphs demonstrate that dynamical NGS correlations arein good agreement with those of the BA [49]. Parameters used areγ ¼ 5, N ¼ 31, Λ ¼ 10kF, Pimpð0Þ ≈ 1.35kF.

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We choose the following initial state for the real-timedynamics:

jΨlabQ i ¼ jFSi ⊗ jQiimp: ðB1Þ

Note that the total momentum of the system is Q and,hence, in the comoving frame, we can stick to a singlemomentum sector.In Fig. 7(a), we plot the evolution of the impurity

momentum PimpðtÞ and compare our results with the BAcalculations of Ref. [49]. The dynamics exhibits three stages:(i) initial rapid decay during which the impurity redistributesitsmomentum to the host-gas particles; (ii) intermediate-timeoscillations called quantum flutter; and (iii) saturation to asteady statewith a nonzero impuritymomentum.We see thatourmethod captures the three stages correctly, though there isa clear discrepancy compared to the exact result at inter-mediate times. This mismatch indicates that our variationalwave function is too restrictive to reproduce full transientdynamics quantitatively. At the same time, the essentialphysics is well reproduced qualitatively. Furthermore,Figs. 7(b) and 7(c) show that the evolution of the fermionicmomentum distribution function computed with the NGSapproach matches well the BA results.

APPENDIX C: QUASIPARTICLE RESIDUE

Within the Gaussian-states framework, the quasiparticleresidue, defined as R ¼ jhFSjψGSij2 for Q ¼ 0, can beeasily computed:

R ¼ det½1þ 2ΓGSΓFS − ðΓGS þ ΓFSÞ�; ðC1Þ

where ΓFS is the covariance matrix of the filled Fermi sea.The residue R becomes suppressed as a power law of thesystem size L—see Fig. 8. For M ¼ ∞, this result isknown as the Anderson orthogonality catastrophe.Because of the nonvanishing recoil energy in Eq. (1),the suppression of the residue for the case of mobileimpurity M ¼ m is slower compared to the case ofinfinitely heavy impurity.

APPENDIX D: COLLECTIVE MODESFOR M ≠ m

Here, we provide more discussion of the properties of thesharp mode for M ≠ m. Figure 9 shows the DOS ofcollective excitations for M ¼ 0.5m [Figs. 9(a)–9(c)] andM ¼ 2m [Figs. 9(d)–9(f)]. The case of heavy impuritiesqualitatively resembles the M ¼ m result in Fig. 1(b): Weobserve the development of a sharp peak at Q ¼ kF, but itsrelative spectral weight decreases with increasing theimpurity mass—see also Fig. 10, which shows the DOSat kF as a function of the mass ratioM=m. The case of lightimpurities is different. As the impurity mass decreases, theentire magnon branch approaches that of the plasmon sothat their dispersions become nearly parallel to each otherin a wide vicinity of Q ¼ kF. This, in turn, results in (i) arather dispersive peak in the DOS of collective excitationsFIG. 8. Quasiparticle residue as a function of the system size L.

The density of fermions is kept constant. Parameters used areγ ¼ 5 and Λ ¼ 5kF.

FIG. 9. The DOS of collective excitations as a function of frequency ω and total momentum Q forM ¼ 0.5m (a)–(c) and forM ¼ 2m(d)–(f). ForM ¼ 0.5m, we observe a rather dispersive peak; atQ ¼ kF, the DOS is notably broad (c). This is because the magnon branchapproaches plasmon for small impurity masses so that their dispersions are parallel to each other over a wide range of momenta near kF.For M ¼ 2m, the peak at Q ¼ kF is sharp (f), but its spectral weight is relatively weak. Parameters are the same as in Fig. 1.

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[Fig. 9(a)] and (ii) a substantial broadening at kF [Fig. 9(c)].This latter broadening indicates that it is challenging toextract the quantum flutter frequency in dynamical simu-lations reliably.

[1] L. D. Landau, Electron Motion in Crystal Lattices, Phys. Z.Sowjetunion 3, 664 (1933).

[2] A. S. Alexandrov and J. T. Devreese, Advances in PolaronPhysics (Springer, New York, 2010), Vol. 159.

[3] J. T. Devreese and A. S. Alexandrov, Fröhlich Polaron andBipolaron: Recent Developments, Rep. Prog. Phys. 72,066501 (2009).

[4] I. Bloch, J. Dalibard, and S. Nascimbene, QuantumSimulations with Ultracold Quantum Gases, Nat. Phys.8, 267 (2012).

[5] P. Massignan, M. Zaccanti, and G. M. Bruun, Polarons,Dressed Molecules and Itinerant Ferromagnetism in Ultra-cold Fermi Gases, Rep. Prog. Phys. 77, 034401 (2014).

[6] R. Schmidt, M. Knap, D. A. Ivanov, J.-S. You, M. Cetina,and E. Demler, Universal Many-Body Response of HeavyImpurities Coupled to a Fermi Sea: A Review of RecentProgress, Rep. Prog. Phys. 81, 024401 (2018).

[7] S. Nascimbene, N. Navon, K. J. Jiang, L. Tarruell, M.Teichmann, J. McKeever, F. Chevy, and C. Salomon,Collective Oscillations of an Imbalanced Fermi Gas: AxialCompression Modes and Polaron Effective Mass, Phys.Rev. Lett. 103, 170402 (2009).

[8] A. Schirotzek, C.-H. Wu, A. Sommer, and M.W. Zwierlein,Observation of Fermi Polarons in a Tunable Fermi Liquidof Ultracold Atoms, Phys. Rev. Lett. 102, 230402 (2009).

[9] S. Palzer, C. Zipkes, C. Sias, and M. Köhl, QuantumTransport through a Tonks-Girardeau Gas, Phys. Rev.Lett. 103, 150601 (2009).

[10] J. Catani, G. Lamporesi, D. Naik, M. Gring, M. Inguscio, F.Minardi, A. Kantian, and T. Giamarchi, Quantum Dynamicsof Impurities in a One-Dimensional Bose Gas, Phys. Rev. A85, 023623 (2012).

[11] C. Kohstall, M. Zaccanti, M. Jag, A. Trenkwalder, P.Massignan, G. M. Bruun, F. Schreck, and R. Grimm,Metastability and Coherence of Repulsive Polarons in aStrongly Interacting Fermi Mixture, Nature (London) 485,615 (2012).

[12] V. M. Stojanović, T. Shi, C. Bruder, and J. I. Cirac,QuantumSimulation of Small-Polaron Formation with Trapped Ions,Phys. Rev. Lett. 109, 250501 (2012).

[13] T. Fukuhara, A. Kantian, M. Endres, M. Cheneau, P.Schauß, S. Hild, D. Bellem, U. Schollwöck, T. Giamarchi,C. Gross, I. Bloch, and S. Kuhr, Quantum Dynamics of aMobile Spin Impurity, Nat. Phys. 9, 235 (2013).

[14] M.-G. Hu, M. J. Van de Graaff, D. Kedar, J. P. Corson, E. A.Cornell, and D. S. Jin, Bose Polarons in the StronglyInteracting Regime, Phys. Rev. Lett. 117, 055301(2016).

[15] N. B. Jørgensen, L. Wacker, K. T. Skalmstang, M.M.Parish, J. Levinsen, R. S. Christensen, G. M. Bruun, andJ. J. Arlt, Observation of Attractive and Repulsive Polaronsin a Bose-Einstein Condensate, Phys. Rev. Lett. 117,055302 (2016).

[16] M. Cetina, M. Jag, R. S. Lous, I. Fritsche, J. T. M. Walraven,R. Grimm, J. Levinsen, M.M. Parish, R. Schmidt, M. Knapet al., Ultrafast Many-Body Interferometry of ImpuritiesCoupled to a Fermi Sea, Science 354, 96 (2016).

[17] F. Meinert, M. Knap, E. Kirilov, K. Jag-Lauber, M. B.Zvonarev, E. Demler, and H.-C. Nägerl, Bloch Oscillationsin the Absence of a Lattice, Science 356, 945 (2017).

[18] Z. Yan, P. B. Patel, B. Mukherjee, R. J. Fletcher, J. Struck,and M.W. Zwierlein, Boiling a Unitary Fermi Liquid, Phys.Rev. Lett. 122, 093401 (2019).

[19] Z. Z. Yan, Y. Ni, C. Robens, and M.W. Zwierlein, BosePolarons nearQuantumCriticality, Science 368, 190 (2020).

[20] I. Bloch, J. Dalibard, and W. Zwerger, Many-Body Physicswith Ultracold Gases, Rev. Mod. Phys. 80, 885 (2008).

[21] F. Chevy, Universal Phase Diagram of a Strongly Interact-ing Fermi Gas with Unbalanced Spin Populations, Phys.Rev. A 74, 063628 (2006).

[22] M.M. Parish and J. Levinsen, Highly Polarized FermiGases in Two Dimensions, Phys. Rev. A 87, 033616 (2013).

[23] S. Peotta, D. Rossini, M. Polini, F. Minardi, and R. Fazio,Quantum Breathing of an Impurity in a One-DimensionalBath of Interacting Bosons, Phys. Rev. Lett. 110, 015302(2013).

[24] F. Massel, A. Kantian, A. J. Daley, T. Giamarchi, and P.Törmä, Dynamics of an Impurity in a One-DimensionalLattice, New J. Phys. 15, 045018 (2013).

[25] M. Knap, C. J. M. Mathy, M. Ganahl, M. B. Zvonarev, andE. Demler, Quantum Flutter: Signatures and Robustness,Phys. Rev. Lett. 112, 015302 (2014).

[26] W. Li and S. Das Sarma, Variational Study of Polaronsin Bose-Einstein Condensates, Phys. Rev. A 90, 013618(2014).

[27] Y. E. Shchadilova, R. Schmidt, F. Grusdt, and E. Demler,Quantum Dynamics of Ultracold Bose Polarons, Phys. Rev.Lett. 117, 113002 (2016).

[28] B. Kain and H. Y. Ling, Hartree-Fock Treatment of FermiPolarons Using the Lee-Low-Pines Transformation, Phys.Rev. A 96, 033627 (2017).

[29] T. Shi, E. Demler, and J. I. Cirac, Variational Study ofFermionic and Bosonic Systems with Non-Gaussian States:Theory and Applications, Ann. Phys. (Amsterdam) 390, 245(2018).

[30] S. I. Mistakidis, G. C. Katsimiga, G. M. Koutentakis, and P.Schmelcher, Repulsive Fermi Polarons and Their Induced

FIG. 10. The DOS at Q ¼ kF as a function of frequency ω andmass ratio M=m. Parameters are the same as in Fig. 1.

DOLGIREV, QU, ZVONAREV, SHI, and DEMLER PHYS. REV. X 11, 041015 (2021)

041015-10

Interactions in Binary Mixtures of Ultracold Atoms, New J.Phys. 21, 043032 (2019).

[31] F. Grusdt, Y. E. Shchadilova, A. N. Rubtsov, and E. Demler,Renormalization Group Approach to the Fröhlich PolaronModel: Application to Impurity-BEC Problem, Sci. Rep. 5,12124 (2015).

[32] F. Grusdt, K. Seetharam, Y. Shchadilova, and E. Demler,Strong-Coupling Bose Polarons Out of Equilibrium:Dynamical Renormalization-Group Approach, Phys. Rev.A 97, 033612 (2018).

[33] N. Prokof’ev and B. Svistunov, Fermi-Polaron Problem:Diagrammatic Monte Carlo Method for Divergent Sign-Alternating Series, Phys. Rev. B 77, 020408(R) (2008).

[34] L. A. P. Ardila and S. Giorgini, Impurity in a Bose-EinsteinCondensate: Study of the Attractive and Repulsive BranchUsing Quantum Monte Carlo Methods, Phys. Rev. A 92,033612 (2015).

[35] F. Grusdt, G. E. Astrakharchik, and E. Demler, Bose Polar-ons in Ultracold Atoms in One Dimension: Beyond theFröhlich Paradigm, New J. Phys. 19, 103035 (2017).

[36] A. Rosch and T. Kopp, Heavy Particle in a d-DimensionalFermionic Bath: A Strong Coupling Approach, Phys. Rev.Lett. 75, 1988 (1995).

[37] R. Schmidt, T. Enss, V. Pietilä, and E. Demler, FermiPolarons in Two Dimensions, Phys. Rev. A 85, 021602(R)(2012).

[38] S. P. Rath and R. Schmidt, Field-Theoretical Study of theBose Polaron, Phys. Rev. A 88, 053632 (2013).

[39] E. Burovski, V. Cheianov, O. Gamayun, and O. Lychkovskiy,Momentum Relaxation of a Mobile Impurity in a One-Dimensional Quantum Gas, Phys. Rev. A 89, 041601(R)(2014).

[40] O. Gamayun, O. Lychkovskiy, and V. Cheianov, KineticTheory for a Mobile Impurity in a Degenerate Tonks-Girardeau Gas, Phys. Rev. E 90, 032132 (2014).

[41] O. Gamayun, Quantum Boltzmann Equation for a MobileImpurity in a Degenerate Tonks-Girardeau Gas, Phys. Rev.A 89, 063627 (2014).

[42] J. B. McGuire, Interacting Fermions in One Dimension. I.Repulsive Potential, J. Math. Phys. (N.Y.) 6, 432 (1965).

[43] J. B. McGuire, Interacting Fermions in One Dimension. II.Attractive Potential, J. Math. Phys. (N.Y.) 7, 123 (1966).

[44] C. N. Yang, Some Exact Results for the Many-Body Problemin one Dimension with Repulsive Delta-Function Interac-tion, Phys. Rev. Lett. 19, 1312 (1967).

[45] O. Gamayun, A. G. Pronko, and M. B. Zvonarev, Timeand Temperature-Dependent Correlation Function of anImpurity in One-Dimensional Fermi and Tonks–GirardeauGases as a Fredholm Determinant, New J. Phys. 18,045005 (2016).

[46] O. Gamayun, O. Lychkovskiy, E. Burovski, M. Malcomson,V. V. Cheianov, and M. B. Zvonarev, Impact of the InjectionProtocol on an Impurity’s Stationary State, Phys. Rev. Lett.120, 220605 (2018).

[47] O. Gamayun, O. Lychkovskiy, and M. B. Zvonarev, ZeroTemperature Momentum Distribution of an Impurityin a Polaron State of One-Dimensional Fermi andTonks-Girardeau Gases, SciPost Phys. 8, 53 (2020).

[48] A. Imambekov, T. L. Schmidt, and L. I. Glazman, One-Dimensional Quantum Liquids: Beyond the LuttingerLiquid Paradigm, Rev. Mod. Phys. 84, 1253 (2012).

[49] C. J. M. Mathy, M. B. Zvonarev, and E. Demler, QuantumFlutter of Supersonic Particles in One-Dimensional Quan-tum Liquids, Nat. Phys. 8, 881 (2012).

[50] L. Hackl, T. Guaita, T. Shi, J. Haegeman, E. Demler, and I.Cirac, Geometry of Variational Methods: Dynamics ofClosed Quantum Systems, SciPost Phys. 9, 048 (2020).

[51] T. Shi, E. Demler, and J. I. Cirac, Variational Approach forMany-Body Systems at Finite Temperature, Phys. Rev. Lett.125, 180602 (2020).

[52] T. D. Lee, F. E. Low, and D. Pines, The Motion ofSlow Electrons in a Polar Crystal, Phys. Rev. 90, 297(1953).

[53] A. Kamenev, Field Theory of Non-Equilibrium Systems(Cambridge University Press, Cambridge, England, 2011).

[54] T. Guaita, L. Hackl, T. Shi, C. Hubig, E. Demler, and J. I.Cirac, Gaussian Time-Dependent Variational Principle forthe Bose-Hubbard Model, Phys. Rev. B 100, 094529(2019).

[55] M. Knap, A. Shashi, Y. Nishida, A. Imambekov, D. A.Abanin, and E. Demler, Time-Dependent Impurity inUltracold Fermions: Orthogonality Catastrophe andBeyond, Phys. Rev. X 2, 041020 (2012).

[56] G. Ness, C. Shkedrov, Y. Florshaim, O. K. Diessel, J. vonMilczewski, R. Schmidt, and Y. Sagi, Observation of aSmooth Polaron-Molecule Transition in a DegenerateFermi Gas, Phys. Rev. X 10, 041019 (2020).

[57] J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn,D. E. Pritchard, and W. Ketterle, Bragg Spectroscopy of aBose-Einstein Condensate, Phys. Rev. Lett. 82, 4569(1999).

[58] D. M. Stamper-Kurn, A. P. Chikkatur, A. Görlitz, S. Inouye,S. Gupta, D. E. Pritchard, and W. Ketterle, Excitation ofPhonons in a Bose-Einstein Condensate by Light Scatter-ing, Phys. Rev. Lett. 83, 2876 (1999).

[59] R. Ozeri, N. Katz, J. Steinhauer, and N. Davidson,Colloquium: Bulk Bogoliubov Excitations in a Bose-Einstein Condensate, Rev. Mod. Phys. 77, 187 (2005).

[60] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg,Atom-Photon Interactions: Basic Processes and Applica-tions (Wiley-VCH, New York, 1998).

EMERGENCE OF A SHARP QUANTUM COLLECTIVE MODE IN A … PHYS. REV. X 11, 041015 (2021)

041015-11