Polaron Problems in Ultracold Atoms: Role of a Fermi Sea across Different Spatial Dimensions and Quantum Fluctuations of a Bose Medium
Hiroyuki Tajima, Junichi Takahashi, Simeon I. Mistakidis, Eiji Nakano, Kei Iida
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Polaron problems in ultracold atoms: Role of the medium acrossdifferent spatial dimensions
Hiroyuki Tajima , Junichi Takahashi , Simeon I. Mistakidis , Eiji Nakano , and Kei Iida Department of Mathematics and Physics,Kochi University, Kochi 780-8520, Japan Department of Electronic and Physical Systems,Waseda University, Tokyo, 169-8555, Japan Center for Optical Quantum Technologies,Department of Physics, University of Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany
Abstract
The notion of a polaron, originally introduced in the context of electrons in ionic lattices, helpsus to understand how a quantum impurity behaves when being immersed in and interacting with amany-body background. We discuss the impact of the impurities on the medium particles by con-sidering feedback effects from polarons that can be realized in ultracold quantum gas experimentsand in particular we exemplify the modifications of the medium either in the presence of Fermi orBose polarons. Regarding Fermi polarons we present a corresponding many-body diagrammaticapproach operating at finite temperatures and discuss how mediated two- and three-body interac-tions are implemented within this framework. Utilizing this approach, we analyze the behavior ofthe spectral function of Fermi polarons at finite temperature by varying impurity-medium inter-actions as well as spatial dimensions from three to one. Interestingly, we reveal that the spectralfunction of the medium atoms could be a useful quantity for analyzing the transition/crossoverfrom attractive polarons to molecules in three-dimensions. As for the Bose polaron, we showcasethe depletion of the background Bose-Einstein condensate in the vicinity of the impurity atom.Such spatial modulations would be important for future investigations regarding the quantificationof interpolaron correlations in Bose polaron problems.
PACS numbers: 67.85.-d, 03.75.Ss, 03.75.Hh, . INTRODUCTION The quantum many-body problem, which is one of the central issues of modern physics,is encountered in various research fields such as condensed matter and nuclear physics. Themajor obstacle that prevents their adequate description stems from the involvement of manydegrees-of-freedom as well as the participation of strong correlations. The polaron concept,which was originally proposed by S. I. Pekar and L. Landau [1, 2] to characterize electronproperties in crystals, provides a useful playground for understanding related nontrivialmany-body aspects of quantum matter and interactions. For instance, a key advantage ofthe polaron picture is that, under specific circumstances, it enables the reduction of a compli-cated many-body problem to an effective single-particle or a few-body one with renormalizedparameters. Ultracold atoms, owing to the excellent controllability of the parameters char-acterizing the system, are utilized to quantitatively determine polaron properties, as hasbeen recently demonstrated in a variety of relevant experimental efforts [3–19].Polarons basically appear in two different types, namely, Fermi and Bose polarons wherethe impurity atoms are immersed in a Fermi sea and a Bose-Einstein condensate (BEC)respectively. Both cases are experimentally realizable by employing a mixture of atomsresiding in different hyperfine states or using distinct isotopes. The impurity-medium in-teraction strength can be flexibly adjusted with the aid of Feshbach resonances [20], andas such strong interactions between the impurity and the majority atoms can be achieved.Due to this non-zero interaction, the impurities are subsequently dressed by the elemen-tary excitations of their background atoms, leading to a quasi-particle state that is calledthe polaron. In that light, the polaron and more generally the quasiparticle generation isinherently related to the build-up of strong entanglement among the impurities and theirbackground medium [21–23]. Moreover, since various situations such as mass-imbalanced [5],low-dimensional [6], and multi-orbital [15] ultracold settings can be realized, atomic polaronscan also be expected to be quantum simulators of quasiparticle states in nuclear physics [24–28].The single-particle character of polarons has been intensively investigated theoreticallyin the past few years by using different approaches [29–46] ranging from variational treat-ments [29–32] to diagrammatic Monte-Carlo simulations [38–43]. Interestingly, a multitudeof experimental observations regarding polaronic excitations have been well described based2n theoretical frameworks relying on the single-polaron ansatz [3, 4, 13]. However, it isstill a challenging problem and highly unexplored topic how many polaron systems behave,especially during their nonequilibrium dynamics. While the single-polaron analysis clarifiesthe mechanism of polaron formation via the dressing from the surrounding of the major-ity cloud, the many-polaron analysis is dedicated to the question of how polarons interactwith each other through the exchange of the excitations of their host. Therefore, the back-ground medium plays a crucial role in understanding many-polaron physics. In this sense,the concept of induced interpolaron interactions has attracted much attention [47–57]. Forinstance, in recent experiments, the sizable shift of the effective scattering length due tothe fermion-mediated interaction has been observed in Fermi polaron systems [16, 17]. Thecorresponding impact on the medium atoms due to the presence of strong impurity-bath cor-relations is under active investigation [51]. In the case of Bose polarons [7–12, 58–60], theinfluence of the impurities on their environment (BEC) is more pronounced when comparedto Fermi polarons due to the absence of the Pauli blocking effect. Characteristic examples,here, constitute the self-localization [61–65] and temporal orthogonality catastrophe [21]phenomena as well as complex tunneling [66–69] and emergent relaxation processes [56, 70].They originate from the presence of the impurity which causes its environment to signifi-cantly deform when the interaction between the subsystems is finite.In this work, we first provide a discussion on the role of the background atoms in many-polaron problems that are tractable in ultracold atom settings. Particularly, we presentdiagrammatic approaches to Fermi polaron systems and elaborate on how mediated two-and three-body interpolaron interactions are consistently taken into account within theseframeworks [51, 52]. Importantly, a comparison of the Fermi polaron excitation spectralfunction in three dimensions and at finite temperatures is performed among different vari-ants of the diagrammatic T -matrix approach. Namely, the usual T -matrix approach (TMA)which is based on the self-energy including the repeated particle-particle scattering processesconsisting of bare propagators [71, 72], the extended T -matrix approach (ETMA) where thebare propagator in the self-energy is partially replaced [73–75], and the self-consistent T -matrix approach where all the propagators in the self-energy consist of dressed ones [76, 77]are employed. We reveal how medium-induced interpolaron interactions are involved in theseapproaches and examine their effects in mass-balanced Fermi polaron settings realized, e.g.,in Li atomic mixtures. Subsequently, we discuss the polaron excitation spectrum in two and3ne spatial dimensions. The behavior of the spectral function of the host and the impuritiesat strong impurity-medium interactions is exemplified. Finally, the real-space Bogoliubovapproach to Bose polarons is reviewed. The latter helps us to unveil the condensate deforma-tion due to the presence of the impurity and appreciate the resultant quantum fluctuations[78]. We argue that the degree of the quantum depletion of the condensate decreases (in-creases) for repulsive (attractive) impurity-medium interactions, a result that is associatedwith the deformation of its density distribution. This is in contrast to homogeneous setupswhere the depletion increases independently of the sign of the interaction.This work is organized as follows. In Sec. II, we present the model Hamiltonian de-scribing ultracold Fermi polarons in three dimensions. For the Fermi polaron, we consideruniform systems and develop the concept of the diagrammatic T -matrix approximation.After explaining the ingredients of the diagrammatic approaches in some detail, we clar-ify how mediated two- and three-body interactions are incorporated in these approaches.The behavior of the resultant polaron spectral function at finite temperatures and impurityconcentrations in three- two- and one-dimensions is discussed. In Sec. III, we utilize thereal-space mean-field formulation for Bose polarons and expose the presence of quantumdepletion for the three-dimensional trapped Bose polaron at zero temperature. In Sec. IV,we summarize our results and provide future perspectives. For convenience, in what follows,we use k B = ~ = 1. II. FERMI POLARONSA. T -matrix approach to Fermi polaron problems Here we explain the concept of many-body diagrammatic approaches to Fermi polarons,namely, settings referring to the situation where fermionic impurity atoms are immersed ina uniform Fermi gas. Since such a two-component Fermi mixture mimics spin-1 / σ = B = ↑ and the impurity one by σ = I = ↓ . Note thatthese are standard conventions without loss of generality. The model Hamiltonian describingthis system reads H = X p ,σ ξ p ,σ c † p ,σ c p ,σ + g X p , p ′ , q c † p + q / , ↑ c †− p + q / , ↓ c − p ′ + q / , ↓ c p ′ + q / , ↑ , (1)4here ξ p ,σ = p / (2 m σ ) − µ σ is the kinetic energy minus the chemical potential µ σ , and m σ is the atomic mass of the σ component. Also, c p ,σ and c † p ,σ refer to the annihilation andcreation operators of a σ component fermion, respectively, possessing momentum p .We measure the effective coupling constant g of the contact-type interaction betweentwo different component fermions by using the low-energy scattering parameter, namely,the scattering length a . In 3D, it is known [79] that the coupling constant g and thescattering length a are related via m r πa = 1 g + m r Λ π , (2)with m − = m − ↑ + m − ↓ being the reduced mass. In this expression, the momentum cutoffΛ is introduced to avoid an ultraviolet divergence in the momentum summation of theLippmann-Schwinger equation expressed in the momentum space. This allows us to achievethe effective short-range interaction of finite range r e ∝ / Λ. Similarly, the relevant relationsin 2D and 1D read [80] a = 1Λ e − πm r g , and a = 1 m r g , (3)respectively, where g and g are the coupling constants in 2D and 1D.First, we introduce a thermal single-particle Green’s function [81] G σ ( p , iω n ) = 1 iω n − ξ p ,σ − Σ σ ( p , iω n ) , (4)where ω n = (2 n + 1) πT is the fermion Matsubara frequency introduced within the finite-temperature T formalism and n ∈ Z [81]. The effect of the impurity-medium interaction istaken into account in the self-energy Σ σ ( p , iω n ). The excitation spectrum A ↓ ( p , ω ) of a Fermipolaron can be obtained via the retarded Green’s function G R ↓ ( p , ω ) = G ↓ ( p , iω n → ω + iδ )(where δ is a positive infinitesimal) through analytic continuation [81]. In particular, it canbe shown that A ↓ ( p , ω ) = − π Im G R ↓ ( p , ω ) . (5)Experimentally, this quantity can be monitored by using a radio-frequency (rf) spectroscopyscheme where the atoms are transferred from their thermal equilibrium state to a specificspin state which interacts with the medium [82]. Indeed, the reverse rf response I r ( ω ) [13]5 + + + ... (a) (b) (c) s= + + + ...s (d)(e) FIG. 1: Feynman diagrams for (a) the T -matrix approach (TMA), (b) the extended T -matrixapproach (ETMA), and (c) the self-consistent T -matrix approach (SCTMA). Γ and Γ S are themany-body T -matrices, whose perturbative expansions are shown schematically in (d) and (e),consisting of bare and dressed propagators G σ and G σ , respectively. While in TMA, all the lines inthe self-energy (a) consist of G σ , they are replaced with G σ partially [upper loop of (b)] in ETMAand fully in SCTMA (c) [see also (e) where G σ is replaced by G σ compared to (d)], respectively. and the ejection one I e ( ω ) [14] are given by I r ( ω ) = 2 π Ω X p f ( ξ p , i ) A ↓ ( p , ω + ξ p , ↓ ) (6)and I e ( ω ) = 2 π Ω X p f ( ξ p , ↓ − ω ) A ↓ ( p , ξ p , ↓ − ω ) , (7)respectively. Here, ξ p , i represents the kinetic energy of the initial state in the reverse rfscheme. In Eqs.(6) and (7), Ω Rabi is the Rabi frequency.Importantly, the self-energy Σ ↑ ( p , iω n ) of the background plays an important role indescribing the mediated interpolaron interactions. This fact will be evinced below and it isachieved by expanding Σ ↑ ( p , iω n ) with respect to G σ and G σ . Also, the chemical potentials µ σ are kept fixed by imposing the particle number conservation condition obeying N σ = T X p ,iω n G σ ( p , iω n ) . (8)Moreover, in the remainder of this work, we define the impurity concentration as follows x = N ↓ N ↑ . (9)6dditionally, within the TMA [30, 50] the self- energy Σ σ ( p , iω n ) of the σ componentreads Σ σ ( p , iω n ) = T X q ,iν ℓ Γ( q , iν ℓ ) G − σ ( q − p , iν ℓ − iω n ) , (10)where Γ( q , iν ℓ ) is the many-body T -matrix, as diagrammatically shown in Fig. 1(a), withthe boson Matsubara frequency iν ℓ = 2 ℓπT ( ℓ ∈ Z ). Here, G σ ( p , iω n ) = ( iω n − ξ p ,σ ) − is the bare thermal single-particle Green’s function. Furthermore, by adopting a ladderapproximation illustrated in Fig. 1(d), the T -matrix Γ( q , iν ℓ ) is given byΓ( q , iν ℓ ) = g g Π( q , iν ℓ ) , (11)where Π( q , iν ℓ ) = T X p ,iω n G ↑ ( p + q , iω n + iν ℓ ) G ↓ ( − p , − iω n ) (12)is the lowest-order particle-particle bubble. The latter describes a virtual particle-particlescattering process associated with the impurity-medium interaction g which is replaced by g , g , and g in 3D, 2D, and 1D, respectively. Note that in Eq.(10) the impurity-impurityinteraction is not taken into account.The extended T -matrix approach (ETMA) [51] constitutes one of the better approxi-mations that allows us to take the induced polaron-polaron interactions into account in aself-consistent way. In this method, as depicted in Fig. 1(b) we include higher-order cor-relations by replacing the bare Green function G in Eq. (10) with the dressed one G σ .Namely Σ E σ ( p , iω n ) = T X q ,iν ℓ Γ( q , iν ℓ ) G − σ ( q − p , iν ℓ − iω n ) . (13)Importantly, the TMA and ETMA approaches are equivalent to each other in the single-polaron limit i.e. x →
0, where the self-energy of the fermionic medium Σ E ↑ [capturing thedifference between G ↑ and G ↑ in Eqs. (10) and (13), respectively] is negligible. Additionally,at zero temperature these two treatments coincide with a variational ansatz proposed by F.Chevy [29]. Recall that µ ↑ = E F and µ ↓ = E (a)P at T = 0 and x →
0, where E F = p / (2 m ↑ )denotes the Fermi energy of the majority component atoms while E (a)P corresponds to theattractive polaron energy. 7 Γ (a) (b) Γ Γ Γ FIG. 2: Feynman diagrams for induced (a) two- and (c) three-body interactions V (2 , amongpolarons. The arrows represent the direction of momentum and energy transfer in each propagator. Proceeding one step further, it is possible to construct the so-called self-consistent T -matrix approach (SCTMA) [52, 83, 84] which deploys the many-body T -matrix Γ S composedof dressed propagators as schematically shown in Fig. 1(e). In particular, the corresponding T -matrix is given by Γ S ( q , iν ℓ ) = g g Π S ( q , iν ℓ ) , (14)where Π S ( q , iν ℓ ) = T X p ,iω n G ↑ ( p + q , iω n + iν ℓ ) G ↓ ( − p , − iω n ) , (15)which describes a scattering process denoted by G ↑ and G ↓ , of the dressed medium atomswith the impurities and the dressed ones (polarons), respectively. This is in contrast toEq. (12) obtained in ETMA and consisting of G σ which represents the impurity-mediumscattering process of the bare atoms. Using this T -matrix, we can express the SCTMAself-energy Σ S σ [see also Fig. 1(c)] asΣ S σ ( p , iω n ) = T X q ,iν ℓ Γ S ( q , iν ℓ ) G − σ ( q − p , iν ℓ − iω n ) . (16)We note that within the ETMA, the impurity self-energy Σ E ↓ [Eq. (11)] can be rewrittenas Σ E ↓ ( p , iω n ) = T X q ,iν ℓ Γ( q , iν ℓ ) (cid:2) G ↑ ( q − p , iν ℓ − iω n )+ G ↑ ( q − p , iν ℓ − iω n )Σ ↑ ( q − p , iν ℓ − iω n ) G ↑ ( q − p , iν ℓ − iω n ) (cid:3) ≡ Σ ↓ ( p , iω n ) + δ Σ ↓ ( p , iω n ) , (17)8ith the higher-order correction δ Σ ↓ ( p , iω n ) beyond the TMA being δ Σ ↓ ( p , iω n ) = T X q , q ′ ,iν ℓ ,iν ℓ ′ Γ( q , iν ℓ )Γ( q ′ , iν ℓ ′ ) G ↑ ( q − p , iν ℓ − iω n ) G ↑ ( q − p , iν ℓ − iω n ) × G ↓ ( q ′ − q + p , iν ℓ ′ − iν ℓ + iω n ) ≡ T X p ′ ,iω n ′ V (2)eff ( p , iω n , p ′ , iω n ′ ; p , iω n , p ′ , iω n ′ ) G ↓ ( p ′ , iω n ′ ) . (18)In this expression, V (2)eff ( p , iω n , p , iω n ; p ′ , iω n ′ , p ′ , iω n ′ ) represents the induced impurity-impurity interaction [diagrammatically shown in Fig. 2(a)] with incoming and outgoingmomenta and frequencies { p i , iω n i } and { p ′ i , iω n ′ i } , respectively, where i = 1 ,
2. It reads V (2)eff ( p , iω n , p , iω n ; p ′ , iω n ′ , p ′ , iω n ′ ) = δ p + p , p ′ + p ′ δ n + n ,n ′ + n ′ × T X q ,iν ℓ Γ( q , iν ℓ )Γ( q + p − p ′ , iν ℓ + iω n − iω n ′ ) G ↑ ( q − p , iν ℓ − iω n ) × G ↑ ( q − p ′ , iν ℓ − iω n ′ ) . (19)Here, δ i,j is the Kronecker delta imposing the energy and momentum conservation in thetwo-body scattering.The self-energy Σ S ↓ of the impurities within the SCTMA involves a contribution of inducedthree-impurity correlations due to the dressed pair propagator Σ S ↓ . The latter can again bedecomposed as Σ S ↓ ( p , iω n ) ≡ Σ E ↓ ( p , iω n ) + δ Σ ′↓ ( p , iω n ) , (20)where δ Σ ′↓ ( p , iω n ) = T X q ,iν ℓ [Γ S ( q , iν ℓ ) − Γ( q , iν ℓ )] G ↑ ( q − p , iν ℓ − iω n )= T X q ,iν ℓ Γ S ( q , iν ℓ )Γ( q , iν ℓ )Φ( q , iν ℓ ) G ↑ ( q − p , iν ℓ − iω n ) . (21)Here we definedΦ( q , iν ℓ ) = Π S ( q , iν ℓ ) − Π( q , iν ℓ )= T X p ,iω n (cid:2) G ↑ ( p + q , iω n + iν ℓ ) G ↓ ( − p , − iω n ) − G ↑ ( p + q , iω n + iν ℓ ) G ↓ ( − p , − iω n ) (cid:3) ≃ T X p ,iω n (cid:2) G ↑ ( p + q , iω n + iν ℓ ) (cid:3) Σ S ↑ ( p + q , iω n + iν ℓ ) G ↓ ( − p , − iω n ) , (22)9hich represents the difference between the Π and Π S , namely, the medium-impurity andthe medium-polaron propagators. In the last line of Eq. (22), we assumed that G ↑ ≃ G ↑ and Σ S ↓ ≃
0. Thus, one can find a three-body correlation effect beyond the ETMA as shownin Fig. 2(b) and captured by δ Σ ′↓ ( p , iω n ) ≃ T X p ′ ,iω n ′ V (3)eff ( p , iω n , p ′ , iω n ′ , p ′′ , iω n ′′ ; p ′ , iω n ′ , p , iω n , p ′′ , iω n ′′ ) × G ↓ ( p ′ , iω n ′ ) G ↓ ( p ′′ , iω n ′′ ) , (23)where V (3)eff ( p , iω n , p , iω n , p , iω n ; p ′ , iω n ′ , p ′ , iω n ′ , p ′ , iω n ′ ) is the induced three-polaroninteraction term. Its explicit form reads V (3)eff ( p , iω n , p , iω n , p , iω n ; p ′ , iω n ′ , p ′ , iω n ′ , p ′ , iω n ′ ) = δ p + p + p , p ′ + p ′ + p ′ δ n + n + n ,n ′ + n ′ + n ′ × T X q ,iν ℓ Γ( q , iν ℓ )Γ( q + p − p ′ , iν ℓ + iω n − iω n ′ )Γ( q + p ′ − p , iν ℓ + iω n ′ − iω n ) × G ↑ ( q − p ′ , iν ℓ − iω n ′ ) G ↑ ( q − p , iν ℓ − iω n ) G ↑ ( q − p ′ + p − p ′ , iν ℓ − iω n ′ + iω n − iω n ′ ) . (24)From the above discussion it becomes evident how the medium-induced two-body and three-body interpolaron interactions are included in the ETMA and the SCTMA treatments.Recall that in the TMA the interpolaron interaction is not taken into account. Even so,observables such as thermodynamic quantities (e.g. particle number density) and spectralfunctions obtained via rf spectroscopy can in principle provide indications of the effect ofinterpolaron interactions through Σ σ ( p , iω n ). B. Spectral response of Fermi polarons
In the following, we shall present and discuss the behavior of the spectral function ofFermi polarons at temperatures close to zero up to of the order of the majority componentFermi temperature as well as for different spatial dimensions ranging from three to one. Forsimplicity, we consider a mass-balanced fermionic mixture i.e. m ↑ = m ↓ ≡ m . The latteris experimentally relevant for instance by considering two different hyperfine states, e.g., | F = 1 / , m F = +1 / i and | F = 3 / , m F = − / i of Li. In this notation, F and m F are the total angular momentum and its projection, respectively, of the specific hyperfinestate [13] at thermal equilibrium. 10 ( ω + μ σ )/ E F A ↑ p = , ω ) E F A (cid:2) p = , ω ) E F TMAETMASCTMA ((cid:0)(cid:1) (cid:3)(cid:4)(cid:5) T = 0.3 T F x = 0.1 (cid:6) p F a ) -1 = 0 ω + μ σ )/ E F FIG. 3: Zero-momentum spectral functions A σ ( p = , ω ) of (a) the majority (medium) and (b)the minority (impurities) fermions for varying energy ω at unitarity, ( p F a ) − = 0. We consider atemperature T = 0 . T F and an impurity concentration x = 0 . A ↑ ( p = , ω ) is almost the same among the three approaches, A ↓ ( p = , ω ) within the SCTMAexperiences a sizable difference compared to the response obtained in the TMA and the ETMAapproaches.
1. Three-dimensional case
The resultant spectral function A σ ( p = 0 , ω ) of the fermionic medium ( σ = ↑ ) and theimpurities ( σ = ↓ ) is depicted in Figure 3 as a function of the single-particle energy ω . Here,we consider a temperature T = 0 . T F , impurity concentration x = 0 .
1, and impurity-mediuminteraction at unitarity, i.e., ( p F a ) − = 0. The Fermi temperature is T F = p / (2 m ↑ ) and theFermi momentum p F . Evidently, the spectral function of the majority component [Fig. 3(a)]exhibits a peak around ω + µ ↑ = 0 in all three diagrammatic approaches introduced in Sec. II.The sharp peak around ω + µ ↑ = 0 corresponds to the spectrum of the bare medium atomsgiven by A ( p , ω ) = δ ( ω − ξ p , ↑ ) at p = 0. This indicates that the imprint of the impurity-medium interaction on the fermionic host is negligible for such small impurity concentrations x = 0 .
1; see also Fig. 6 and the discussion below. Indeed, the renormalization of µ ↑ (whichessentially evinces the backaction on the majority atoms from the impurities) in the ETMAat unitarity is proportional to x [51] and in particular µ ↑ E F = 1 − . x. (25)11t can be shown that in the weak-coupling limit, this shift is given by the Hartree correctionΣ H ↑ = πam N ↓ [81]. However, at the unitarity limit presented in Fig. 3, such a weak-couplingapproximation cannot be applied and therefore the factor 0 .
526 in Eq. (25) originates fromthe existence of strong correlations between the majority and the minority component atoms.The corresponding polaronic excitation spectrum is captured by A ↓ ( p = 0 , ω ) [Fig. 3(b)]having a dominant peak at ω + µ ↓ = − E (a)P where E (a)P is the attractive polaron energy.Notice here that since this peak is located at negative energies it indicates the formation ofan attractive Fermi polaron. This observation can be understood from the fact that in theabsence of impurity-medium interactions, the bare-particle pole, namely, the position of thepole of the bare retarded single-particle Green’s function G , R ↓ ( p = , ω ) = ( ω + iδ + µ ↓ ) − ,occurs at ω + µ ↓ = 0. Moreover, the attractive polaron energy E (a)P (being of course negative)is defined by the self-energy energy shift as E (a)P = Σ ↓ ( , E (a)P ). Thus, one can regard thedeviation of the position of the peak from ω + µ ↓ = 0 as the attractive polaron energy E (a)P ,since it is given by A ↓ ( p = 0 , ω ) ∼ δ ( ω + µ ↓ − E (a)P ). Recall that, in general, for finitetemperatures T and impurity concentrations x , µ ↓ = E (a)P holds in contrast to the single-polaron limit at T = 0 [51]. Additionally, a weak amplitude peak appears in A ↓ ( p = 0 , ω )at positive energies ω ≃ E F . It stems from the metastable upper branch of the impuritieswhere excited atoms repulsively interact with each other. This peak becomes sharper atpositive scattering lengths away from unitarity. Indeed, for positive scattering lengths, thequasi-particle excitation called a repulsive Fermi polaron emerges [22].Figure 4(a) presents the polaron spectral function A ↓ ( p = , ω ) with respect to theinteraction parameter ( p F a ) − obtained within the ETMA method at T = 0 . T F and x = O (10 − ). From the position of the poles of G R ↓ ( p = , ω ), one can extract two kinds ofpolaron energies, namely, E (a)P and E (r)P corresponding to the attractive and the repulsivepolaron energies, respectively. The interaction dependence of these energies is provided inFig. 4(b). E (r)P approaches the Hartree shift Σ H ↓ = πam N ↑ without the imaginary part of theself-energy (being responsible for the width of the spectra) and finally becomes zero [22].Indeed, the spectrum in Fig. 4(a) shows that the peak of the repulsive polaron at ω + µ ↓ > p F a ) − , indicating the vanishing imaginary part of theself-energy. On the other hand, E (a)P decreases with increasing ( p F a ) − as depicted by theposition of the low-energy peak (where ω + µ ↓ <
0) in Fig. 4(a). Eventually, the attractivepolaron undergoes the molecule transition as we discuss below. Another important issue12 ( p F a ) -1 E P / E F -4 -3 -2 -1 0 1 2 3 4-0.4 0 0.4 0.8 1.20 ω + μ ↓ )/ E F A ↓ p = 0, ω ) E F (cid:7)(cid:8)(cid:9)(cid:10)(cid:11) p F a ) -1 E (cid:12)(cid:13)(cid:14)(cid:15) E (cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22) (cid:23)(cid:24)(cid:25) FIG. 4: (a) Polaron spectral function A ↓ ( p = , ω ) for several coupling strengths ( p F a ) − . Thespectrum is calculated within the ETMA at temperature T = 0 . T F and impurity concentration x = O (10 − ) [51]. Panel (b) represents the attractive and repulsive polaron energies, namely, E (a)P and E (r)P , respectively, as a function of ( p F a ) − . The polaron energies have been extracted fromthe peak position of A ↓ ( p = , ω ), that is, the pole of G R ↓ ( p = , ω ). The experimental data ofRef. [13] are plotted in black circles for direct comparison with the theoretical predictions. here is that in the strong-coupling regime the attractive polaron undergoes the transition tothe molecular state with increasing impurity-bath attraction [85]. Although this transitionwas originally predicted to be of first-order, recent experimental and theoretical studiesshowed an underlying crossover behavior and coexistence between polaronic and molecularstates [18]. We note that in the case of finite impurity concentrations, a BEC of moleculescan appear at low temperatures; see also Eq. (26) and Eq. (27) below. It is also a factthat the interplay among a molecular BEC, thermally excited molecules, and polarons mayoccur at finite temperatures [86]. In the calculation of the attractive polaron energy E (a)P fordifferent coupling strengths [Fig. 4(b)], however, we do not encounter the molecular BECtransition identified by the Thouless criterion [87]1 + g Π( q = , iν ℓ = 0) = 0 . (26)In particular, in the strong-coupling limit, from Eq. (26) combined with the particle numberconservation [Eq. (8)] the BEC temperature T BEC of molecules satisfies [88] T BEC ≃ π (cid:18) x π ζ (3 / (cid:19) T F , (27)where ζ (3 / ≃ .
612 is the zeta function. Since we consider a small impurity concentration x = O (10 − ) here, T = 0 . T F is far above T BEC ∝ x .13ccording to the above-description, induced polaron-polaron interactions mediated bythe host atoms, which are taken into account within the ETMA and the SCTMA methodsas explicated in Sec. II, are weak in the present mass-balanced fermionic mixture. Thesefinite temperature findings are consistent with previous theoretical works [47–49] predictinga spectral shift of the polaron energy ∆ E = F E FG x with F = 0 . ∼ . T = 0 (where E FG is the ground-state energy of a non-interacting single-component Fermi gas at T = 0) aswell as the experimental observations of Ref. [4]. On the other hand, the presence of inducedpolaron-polaron interactions in the repulsive polaron scenario cannot be observed experi-mentally [13], a result that is further supported by recent studies based on diagrammaticapproaches [51].Furthermore, the spectral deviations between the TMA and the ETMA treatments repre-sent the effect of induced two-body interpolaron interactions in the attractive polaron case.However, in our case there is no sizable shift between the spectral lines predicted in these ap-proaches [Fig. 3(b)]. Indeed, the induced two-body energy is estimated to be of the order of O (10 − ) E FG at x = 0 .
1. The induced three-body interpolaron interaction, which is responsi-ble for the difference among the ETMA and the SCTMA results, exhibits a sizable effect onthe width of the polaron spectra. Although the SCTMA treatment tends to overestimate thepolaron energy, the observed full-width-at-half maximum (FWHM) of the rf spectrum givenby 2 . T /T F ) [14] can be well reproduced by this approach. The latter gives 2 . T /T F ) whereas the FWHM in ETMA is 1 . T /T F ) [52]. We should also note that the decay raterelated to the FWHM for repulsive polarons as extracted using TMA (and simultaneouslyETMA) agree quantitatively with the experimental result of Ref. [13]. For the attractivepolaron the quantitative agreement between the experiment and these diagrammatic ap-proaches is broken at high temperatures. For instance, the recent experiment of Ref. [14]showed that the transition from polarons to the Boltzmann gas occurs at T ≃ . T F [14],while the prediction of the diagrammatic approaches is above T F [52]. Besides the fact thatsuch polaron decay properties may be related to multi-polaron scattering events leading tomany-body dephasing [19], they are necessary for further detailed polaron investigationsat various temperatures and interaction strengths that facilitate the understanding of theunderlying physics of the observed polaron–to–Boltzmann-gas transition.The dependence of the polaron spectra A ↓ ( p , ω ) on the energy and the momentum of theimpurities is illustrated in Fig. 5 for T = 0 . T F , x = 0, and ( p F a ) − = 0. To infer the impact14 ( ω + μ ↓ ) / E F p / p F (cid:26) (cid:27) A ↓ (cid:28) p , ω ) E F (cid:29)(cid:30)(cid:31) ! MA " TMA T ’ )*+ T F x = 0.1 , p F a ) -1 = 0 FIG. 5: Polaron spectral function A ↓ ( p , ω ) as a function of the momentum p and the energy ω of the impurities at temperature T = 0 . T F , impurity concentration x = 0 .
1, and interaction( p F a ) − = 0. A ↓ ( p , ω ) is calculated within (a) the ETMA and (b) the SCTMA approaches. Thevertical dashed line marks the Fermi momentum p = p F of the medium. While the two approachespredict qualitatively similar spectra with a sharp peak at low momenta and broadening above p = p F , the SCTMA result (b) shows a relatively broadened peak at low momenta compared tothe ETMA one (a). of the multi-polaron correlations on the spectrum we explicitly compare A ↓ ( p , ω ) betweenthe ETMA and the SCTMA methods. As it can be seen, A ↓ ( p , ω ) exhibits a sharp peakwhich is associated with the attractive polaron state and shows an almost quadratic behaviorfor increasing momentum of the impurities. It is also apparent that the SCTMA spectrum[Fig. 5(b)] at low momenta is broadened when compared to the ETMA one [Fig. 5(a)] dueto the induced beyond-two-body interpolaron correlations, e.g. three-body ones. At smallimpurity momenta, the spectral peak of the attractive Fermi polaron within the presentmodel as described by Eq. (1), is generally given by A ↓ ( p , ω ) ≃ Z a δ (cid:18) ω + µ ↓ − p m ∗ a − E (a)P (cid:19) , (28)where Z a and m ∗ a are the quasiparticle residue [22] and the effective mass of the attractivepolaron, respectively. At unitarity it holds that Z a ≃ . m ∗ a ≃ . m , and E (a)P ≃ − . E F within the zero-temperature and single-polaron limits [30]. The behavior of these quantitieshas been intensively studied in current experiments [3, 4, 13] and an adequate agreementhas been reported using various theories. For instance, Chevy’s variational ansatz (beingequivalent to the TMA at T = 0 and x →
0) [29, 30] gives Z a = 0 . m ∗ a = 1 . m ,15nd E (a)P = − . E F . More recently, the functional renormalization group [35] predicts Z a = 0 .
796 and E (a)P = − . E F , while according to the recent diagrammatic Monte Carlomethod [43] E (a)P = − . E F . In this sense, nowadays, the corresponding values of thesequantities can be regarded as important benchmarks, especially for theoretical approaches.As we demonstrated previously (see Fig. 3), besides the fact that the spectral responsewithin the SCTMA method is broader compared to the one obtained in the ETMA, the twospectra feature a qualitatively similar behavior. Indeed, both approaches evince that thespectra beyond p = p F are strongly broadened. Recall that in this region of momenta theatoms of the majority component, which form the Fermi sphere, cannot follow the impurityatoms. This indicates that the dressed polaron state ceases to exist due to the phenomenonof the Cherenkov instability [89, 90], where the polaron moves faster than the speed of soundof the medium and consequently it becomes unstable against the spontaneous emission ofelementary excitations of the medium. Such a spectral broadening can also be observedin mesoscopic spin transport measurements [91] and may also be related to the underlyingpolaron-Boltzmann gas transition [14] since the contribution of high-momentum polaronscan be captured in rf spectroscopy due to the thermal broadening of the Fermi distributionfunction in Eq. (7) at high temperatures. Moreover, the momentum-resolved photoemissionspectra would reveal these effects across this transition.We remark that the medium spectral function A ↑ ( p , ω ) is also useful to reveal the prop-erties of strong-coupling polarons in the case of finite temperature and impurity concentra-tion. Figure 6 presents A ↑ ( p , ω ) for various impurity-medium couplings [( p F a ) − = − . .
4, 0 .
7, and 1 .
0] at T = 0 . T F and x = 0 .
1. At ( p F a ) − = − . p F a ) − = 0, A ↑ ( p = , ω ) features a single peak at ω + µ ↑ = 0. On the other hand, at intermediate cou-plings ( p F a ) − = 0 . p F a ) − = 0 .
7, besides a dominant spectral maximum a second peakappears around ω + µ ↑ = E F . The latter evinces the backaction from the repulsive polaronbecause the inset of Fig. 6 shows that the repulsive polaron is located around ω + µ ↑ ≃ E F .Moreover, at ( p F a ) − = 1, another peak emerges in the low-energy region ( ω + µ ↑ ≃ − E F ).This low-energy peak elucidates the emergence of two-body molecules with the binding en-ergy given by E b = 1 / ( ma ) due to the strong impurity-medium attraction. Concluding,the spectral function of the medium atoms can provide us with useful information for therecently observed smooth crossover from polarons to molecules [18].In the following, we shall elaborate on the behavior of the spectral function of lower16 -. /0 -2 -1 -5
12 56 -2 -1 0 1 2 -2 0 210 :; -2 A ↑ ( p = , ω ) E F A <= p = , ω ) E F ω + μ ↑ )/ E F > ω + μ ? )/ E F @ p F a ) -1 A BCDE FGH
IJK T L MNO T F x = 0.1 FIG. 6: Spectral function of the medium A ↑ ( p = , ω ) within the ETMA approach at zero mo-mentum of the impurity and for different impurity-medium couplings ( p F a ) − = − .
4, 0, 0 .
4, 0 . .
0. The temperature and the impurity concentration are given by T = 0 . T F and x = 0 . A ↓ ( p = , ω ). Whilethe sharp peak at ω + µ ↑ ≃ A ↑ ( p = 0 , ω ) is associated with the bare state, the small amplitudeside peaks at positive ( ω + µ ↑ ≃ E F ) and negative energies [ ω + µ ↓ ≃ − E F for the case with( p F a ) − = 1] originate from the backaction due to the impurities. dimensional Fermi polarons solely within the TMA approach. The latter provides an ade-quate description of the polaron formation in our case since the induced interpolaron inter-action [55, 56] is weak in the considered mass-balanced system.
2. Spectral response of Fermi polarons in two-dimensions
In two spatial dimensions, the attractive impurity-medium effective interaction g D < − / ( ma ) [92]. Simultaneously, the repulsive polaron branch appears at positive ener-gies [22] in addition to the attractive one located at negative energies. This phenomenologyis similar to the case of a positive impurity-bath scattering length in 3D [93]. To elaborateon the typical spectrum of 2D Fermi polarons below we employ a homogeneous Fermi mix-ture characterized by an impurity concentration x = 0 .
1, temperature T = 0 . T F , and atypically weak dimensionless coupling parameter ln( p F a ) = 0 . a is the 2D scat-17 QR S ( ω + μ σ ) / E F T A σ U p , ω ) E F VWXY Z[\ ority ( ↑ ) (a2) Minority ( ] ) T = 0.3 T F x = 0.1ln( p F a ) = 0.4 -2 ^_ ‘ p / p F a (b1) Majority ( ↑ ) (b2) Minority ( b ) T = 0.3 T F x = 0.3ln( p F a ) = 0 FIG. 7: Spectral function A σ ( p , ω ) of the Fermi (a1) medium and (a2) impurities in two-dimensionsfor different momenta and energies of the impurities. We consider a temperature T = 0 . T F ,impurity concentration x = 0 .
1, and dimensionless coupling parameter ln( p F a ) = 0 .
4. Thevertical dashed line indicates the Fermi momentum p = p F of the majority component atoms. Whilethe majority component (a) exhibits a sharp peak with quadratic dispersion ω + µ ↑ = p / (2 m ),the minority atoms (b) form the attractive polaron at negative energies ( ω + µ ↓ <
0) and abroadened peak associated with the repulsive impurity branch at positive energies ( ω + µ ↓ > T = 0 . T F , x = 0 . p F a ) = 0. Evidently, the feedback on the medium fromthe impurities is enhanced in the low-momentum region ( p ≃ tering length introduced in Eq. (3). The spectral response of both the fermionic background( A ↑ ( p , ω )) and the impurities ( A ↓ ( p , ω )) for varying momenta and energies of the impuritieswithin the TMA approach is depicted in Figure 7. We observe that the small impurityconcentration, i.e., x = 0 .
1, leads to the non-interacting dispersion of the spectrum of themajority component given by A ↑ ( p , ω ) ≃ δ ( ω − ξ p , ↑ ); see Fig. 7(a). In this case, therefore,the medium does not experience any backaction from the impurities. Importantly, one canindeed identify a sizable backaction on the medium in the case of a larger impurity concen-tration and smaller impurity-medium 2D scattering length as shown in Figs. 7(b1) and (b2)18here T = 0 . T F , x = 0 .
3, and ln( p F a ) = 0. Moreover, since the repulsive interactionin the excited branch of the impurities [ ω + µ ↓ ≃ E F ] is relatively strong, the impurityexcitation spectrum at positive energies ( ω + µ ↓ >
0) is largely broadened. We note that thestable repulsive polaron branch can be found in the case of small a . It becomes also evi-dent that the impurity spectrum in 2D is largely broadened beyond p = p F as compared tothe 3D spectral response [Fig. 5] . Simultaneously, the intensity of the metastable impurityexcitation in the repulsive branch becomes relatively strong in both the 2D and 3D cases.This result implies that fast-moving impurities do not dress the medium atoms and occupythe non-interacting excited states in such high-momentum regions.
3. Fermi polarons in one-dimension
In one spatial dimension the quasiparticle notion is somewhat more complicated as com-pared to the higher-dimensional case. Interestingly, various experiments are nowadays pos-sible to realize 1D ensembles and thus probe the properties of the emergent quasiparticles.Below, we provide spectral evidences of 1D Fermi polarons and in particular calculate therespective A σ ( p , ω ) [Fig. 8] for the background fermionic medium and the minority atomswithin the T -matrix approach including the Hartree correction. The system has an impu-rity concentration x = 0 . T = 0 . T F , and the 1D dimensionlesscoupling parameter for the impurity-medium attraction is ( p F a ) − = 0 .
28 in Figs. 8(a1)and (a2). For comparison, we also provide A σ ( p , ω ) in Figs. 8(b1) and (b2) for the repulsiveinteraction case ( p F a ) − = − .
55 with system parameters x = 0 .
264 and T = 0 . T F .We remark that the impurity-medium attraction is considered weak herein such that theinduced interpolaron interactions are negligible. In this sense, we do not expect significantdeviations when considering the ETMA or even the SCTMA approaches.It is also important to note here that in sharp contrast to higher spatial dimensions, thecoupling constant g D does not vanish when Λ → ∞ in the renormalization procedure; seeSec. II A. Thus, we take the Hartree shift Σ H σ = g D N − σ into account in the building blockof the self-energy diagrams [94]. This treatment is not necessary in the single-polaron limitsince Σ H ↑ → H ↓ → g D T P p ,iω n G ↑ ( p , iω n ) (which is included in the TMA self-energy)when x →
0. The non-vanishing coupling constant in 1D plays an important role in theemergence of induced interpolaron interactions as it has been recently demonstrated, e.g., in19 ( ω + μ σ ) / T c A σ d p , ω ) E F efgh ijk ority ( ↑ ) (a2) Minority ( l ) T m nopqr T F x s tuvwx ( p F a ) yz { |}~(cid:127) -50510 ( ω + μ σ ) / T p / p T (cid:128) (b1) Majority ( ↑ ) (b2) Minority ( (cid:129) ) T (cid:130) (cid:131)(cid:132)(cid:133)(cid:134)(cid:135) T F x (cid:136) (cid:137)(cid:138)(cid:139)(cid:140)(cid:141) ( p F a ) (cid:142)(cid:143) (cid:144) (cid:145)(cid:146)(cid:147)(cid:148)(cid:149)
30 1 2 3
FIG. 8: Spectral function A σ ( p , ω ) of the fermionic (a1) background and (a2) impurity atomsof concentration x = 0 .
326 with an attractive medium-impurity interaction for varying momentaand energies of the impurities in one-dimension. The system is at temperature T = 0 . T F and dimensionless coupling parameter ( p F a ) − = 0 . P T = √ mT is the momentum scaleassociated with the temperature T . The vertical dashed line marks the Fermi momentum p = p F of the background atoms. The majority component (a1) is largely broadened due to the backactionfrom the impurities in the low-momentum region ( p < ∼ p T ). On the other hand, the minoritycomponent (a2) exhibits a sharp peak in the low-momentum region below p = p F and it is broadenedabove p = p F . For comparison, we show the (b1) medium and (b2) impurity spectral functions inthe case of repulsive medium-impurity interaction characterized by ( p F a ) − = − .
55, where thetemperature and the impurity concentraion are given by T = 0 . T F and x = 0 . ω + µ ↓ ≃
0) is shifted upward, the tendencyof a spectral broadening is similar to the attractive case.
Refs. [57, 95, 96]. The polaronic excitation properties obtained within the TMA approachshow an excellent agreement with the results of the thermodynamic Bethe ansatz [97]. Thelatter provides an exact solution in 1D and in the single-polaron limit at T = 0 [92, 98].From these results, it is found that there is no transition but rather a crossover behavior20etween polarons and molecules. As it can be seen by inspecting Fig. 8(a1) the spectrum ofthe majority component is affected by the scattering with the impurities. This is attributedto the relatively large impurity concentration x considered here. In particular, A ↑ ( p , ω ) isbroadened at low momenta below p = p F . On the other hand, the spectral response of theimpurities in Fig. 8(a2) exhibits a sharp peak associated with the attractive polaron below p = p F and it becomes broadened above p = p F . Apparently, the curvature of the position ofthe polaron peak corresponding to the effective mass (curvature of the dispersion) is changedaround this value of the momentum. Similar broadening effects of sharp peaks can be foundeven in the case of repulsive impurity-medium interaction shown in Figs. 8 (b1) and (b2).However, the low-energy sharp peak (corresponding to the repulsive polaron) in the impurityspectrum [Fig. 8(b2)] is shifted to larger energies as a consequence of the repulsion with themedium. III. BOSE POLARONS
In this section, we shall discuss the Bogoliubov theory of trapped Bose polaron systemsin real space [78, 99, 100]. The reason for focusing on a real-space Bogoliubov theoryis to elaborate on the deformation of the BEC medium in the presence of an impurity.Indeed, the interaction between the impurity and the medium bosons lead to significantinhomogeneities of the density distribution of the background which cannot be describedwithin a simple Thomas-Fermi approximation. Such a modification of the boson distributioncauses, for instance, enhanced phonon emission [57, 70]. Moreover, in cold atom experimentsthe background bosons and the impurity are generally trapped. Considering the impact ofinhomogeneity that naturally arises in trapped systems, therefore, we treat the Bose polaronin real space without plane wave expansion because the momentum is not a good quantumnumber. Below, we review the description of a Bose polaron in trapped systems at zerotemperature using the Bogoliubov theory and elaborate on the ground state properties.In particular, we consider a 3D setting where a single atomic impurity is trapped in anexternal harmonic potential denoted by V I ( r ) and is embedded in a BEC medium that isalso trapped in an another harmonic potential V B ( r ) whose center coincides with that of V I ( r ). Hereafter, we use units in which ~ = 1. This system is described by the following21odel Hamiltonianˆ H = Z d d r ˆ ψ ( r ) † (cid:20) − ∇ m I + V I ( r ) (cid:21) ˆ ψ ( r ) + g IB Z d d r ˆ ψ ( r ) † ˆ φ † ( r ) ˆ φ ( r ) ˆ ψ † ( r ) ˆ ψ ( r )+ Z d d r ˆ φ ( r ) † (cid:20) − ∇ m B + V B ( r ) + g BB ˆ φ ( r ) † ˆ φ ( r ) (cid:21) ˆ φ ( r ) . (29)Here, ˆ φ and ˆ ψ are the field operators of the bosonic medium and the impurity, respectively. m I(B) is the mass of the impurity atom (the medium bosons) and µ is the chemical potentialof the medium bosons. The effective couplings g IB and g BB refer to the impurity-boson andboson-boson interaction strengths, respectively. A. Bogoliubov theory for Bose polaron problems
First, we calculate the expectation value of the Hamiltonian in terms of the single-impurity state | imp i = ˆa † imp | i imp in order to integrate out the impurity’s degree-of-freedomˆ H B = Z d d r ψ ∗ ( r ) (cid:20) − ∇ m I + V I ( r ) (cid:21) ψ ( r )+ Z d d r ˆ φ ( r ) † (cid:20) − ∇ m B + V B ( r ) + g IB | ψ ( r ) | + g BB ˆ φ ( r ) † ˆ φ ( r ) (cid:21) ˆ φ ( r ) , (30)where ˆ a imp denotes the annihilation operator of an impurity in the ground state; ψ ( r ) is thecorresponding wave function that can be determined self-consistently by Eq. (35). In thisway, we have obtained the effective Hamiltonian for the medium bosons, in which the bosonsexperience an effective potential constructed by the external trap and the density of theimpurity g IB | ψ ( r ) | . Since we have set the temperature to zero in the present study, we haveto assume that the medium bosons possess a condensed part, so-called the order parameteror the macroscopic wavefunction, when using perturbation theory. It is known [79, 101, 102]that when a BEC occurs, the vacuum expectation value of the field operator ˆ φ leads to a non-zero function which is used as an order parameter, i.e., h ˆ φ ( r ) i b = φ ( r ), where h· · · i b means b h | · · · | i b . The vacuum | i b is determined from the effective Hamiltonian (30) within theBogoliubov theory to the second order of fluctuations. This is equivalent to splitting theoperator as ˆ φ = φ + ˆ ϕ , where h ˆ ϕ i b = 0. Substituting this into the Hamiltonian of Eq. (30)and expressing it in terms of the different orders of ˆ ϕ , we can readily obtain the expansionˆ H B ≃ H (0) + H (1) + H (2) because the number of the non-condensed bosons is significantlysmaller than that of the condensed ones at zero temperature at weak couplings. In this22xpression, the individual contributions correspond to H (0) = Z d d r ψ ∗ (cid:20) − ∇ m I + V I (cid:21) ψ + Z d d r φ ∗ (cid:20) − ∇ m B + V B + g IB | ψ | + g BB | φ | − µ (cid:21) φ, (31) H (1) = Z d d r ˆ ϕ † (cid:20) − ∇ m B + V B + g IB | ψ | + g BB | φ | − µ (cid:21) φ + h . c ., (32) H (2) = 12 Z d d r (cid:16) ˆ ϕ † ˆ ϕ (cid:17) L MM ∗ L ∗ ˆ ϕ ˆ ϕ † , (33)where L ( r ) = − ∇ m B + V B ( r ) + g IB | ψ ( r ) | + 2 g BB | φ ( r ) | − µ , and M ( r ) = g BB φ ( r ) . Notethat we assume the weakly interacting limit of the medium to ensure the BEC dominatingcondition and thus g BB is adequately small such that the perturbation theory is valid. Also,in the above expansion we ignore the contributions stemming from the third- and fourth-order terms in the field operator assuming that they are negligible for the same reason.Subsequently, let us derive the corresponding equations of motion that describe theBose-polaron system. From the Heisenberg equation, the bosonic field operator ˆ ϕ satis-fies i∂ t h ˆ ϕ i b = h [ ˆ ϕ, ˆ H (1) + ˆ H (2) ] i b = 0 in the interaction picture. Accordingly, it is possible toretrieve the celebrated Gross-Pitaevskii equation describing the BEC background (cid:20) − ∇ m B + V B ( r ) + g IB | ψ ( r ) | + g BB | φ ( r ) | − µ (cid:21) φ ( r ) = 0 . (34)We remark that here, for simplicity, we consider the stationary case where the condensate istime-independent. Next, by following the variational principle for ψ namely δ hH B i b /δψ ∗ = 0,we arrive at the Schr¨odinger equation for the impurity wavefunction, namely, (cid:20) − ∇ m I + V I ( r ) + g IB | φ ( r ) | + g IB n ex ( r ) (cid:21) ψ ( r ) = 0 , (35)where n ex ( r ) = h ˆ ϕ † ( r ) ˆ ϕ ( r ) i b is the density of the non-condensed bosons in vacuum, theso-called quantum depletion .To evaluate this expectation value, we need the ground state | i b of the Hamiltonian thatcan be obtained by the diagonalization of Eq. (33), namely, H (2) = P n E n ˆ b † n ˆ b n is achievedusing the following field expansion ˆ ϕ ( r ) = P n h ˆ b n u n ( r ) + ˆ b † n v ∗ n ( r ) i . Here the complete set { u i , v i } satisfies the following system of linear equations being the so-called Bogoliubov-deGennes (BdG) equations [103, 104] L ( r ) M ( r ) −M ∗ ( r ) −L ( r ) u n ( r ) v n ( r ) = E n u n ( r ) v n ( r ) . (36)23e remark that the BdG equations are commonly used in mode analysis of condensates.In this context, the real eigenvalues constitute the spectrum, while the complex eigenvaluesunveil the dynamically unstable modes of the condensate [105, 106]. More precisely, ifcomplex eigenvalues exist then the Hamiltonian can not be expressed in the above-mentioneddiagonal form in terms of the annihilation/creation operators. Therefore, the dynamicallyunstable situation is beyond the scope of the present description. By using this expansion,we can calculate the vacuum expectation, e.g., n ex ( r ) = P n | v n ( r ) | . For the numericalcalculations, to be presented below, the total number of bosons N B is conserved, i.e., N B = N + N ex , with N = Z d d r | φ ( r ) | and N ex = Z d d r n ex ( r ) . (37)This condition is achieved by tuning the chemical potential µ of the bosonic medium. Noticethat N ex becomes non-zero due to thermal fluctuations at finite temperature, while in theultracold regime it can be finite due to the presence of quantum fluctuations, otherwisetermed quantum depletion [107]. We also remark that all of the above Eqs. (34)–(36)need to be solved simultaneously. The above-described treatment will be referred to in thefollowing as the real-space formulation of the Bose-polaron problem. B. Quantum depletion around a Bose polaron
Since N B is fixed [Eq. (37)], the number of condensed particles N changes due to theexistence of N ex . This is a quantum effect that occurs even at zero temperature, and itis called quantum depletion [102]. We need to clarify that the term quantum depletionrefers to the beyond mean-field corrections for the description of the bosonic ensemble. Inthe following, we shall investigate the effect of an impurity on the quantum depletion ofthe medium bosons at zero temperature. Indeed, the quantum depletion is a measurablequantum effect that is included in Eq. (35) and its quantification makes it possible to evaluatethe backaction of the impurity on the medium condensate.A commonly used external confinement in cold atom experiments is the harmonic poten-tial. As such, here, we consider that the traps of the impurity and the bosonic medium arespherically symmetric, namely, V B ( r ) = 12 m B ω r and V I ( r ) = 12 m I ω r . (38)24ccordingly, the order parameter of the BEC and the impuritys’ wave function have spheri-cally symmetric forms, and therefore the underlying BdG eigenfunctions are separable withthe help of spherical harmonics as φ ( r ) = φ ( r ) , ψ ( r ) = ψ ( r ) , u n r ℓm ( r ) v n r ℓm ( r ) = U n r ℓ ( r ) V n r ℓ ( r ) Y ℓm ( θ , θ ) , (39)where r = | r | . Here, ( n r , ℓ, m ) denote the radial, azimuthal, and magnetic quantum num-bers, respectively.As a further simplification, we consider the situation where ω I is sufficiently larger than ω B , namely, the impurity is more tightly confined than the medium bosons. As such, theorder parameter φ of the condensate changes much slower with respect to the spatial changeof the impurity’s wave function ψ . Since the impurity’s wave function is relatively narrowcompared to the condensate and the impurity-medium interaction is weak, the impurityessentially experiences to a good approximation an almost flat (homogeneous) environment.This also means that trap effects are not very pronounced in this case. In this sense, φ canbe regarded as being constant and the impurity’s wave function can be well approximatedby a Gaussian function i.e. ψ ( r ) ≃ (cid:16) πm I ω I (cid:17) − exp (cid:0) − m I ω I r (cid:1) . To experimentally realize sucha setting it is possible to consider a K Fermi impurity immersed in a Rb BEC, where m I /m B ≃ . N B = 10 and the ratioof the strength of the trapping potentials ω I /ω B = 10 with ω B = 20 × π Hz [11]. Moreover,for the boson-boson and impurity-boson interactions, we utilize the values 1 / ( a BB n / ) = 100and 1 / ( a IB n / ) = ± n B = N B . (cid:0) π d (cid:1) and d B = √ m B ω B . TABLE I: The number of depletion N ex and its deviation δN ex = 4 π R dr r δn ex ( r ) from the caseof zero impurity-medium interaction. It is evident that degree of depletion increases (decreases)for attractive (repulsive) interactions.1 / ( a IB n / ) ∞ +1 -1 N ex δN ex × − × − To reveal the backaction of the impurity on the bosonic environment we provide thecorresponding ground state density profiles of the condensed and the depleted part of the25
IG. 9: Radial profiles of (a) the order parameter ¯ φ ( r ) = φ ( r ; g IB = 0) / p N / π and (c) thedensity of depletion ¯ n ex ( r ) = n ex ( r ; g IB = 0) in the absence of an impurity. Differences of theradial profiles of (b) the order parameter δ Φ( r ) = ( φ ( r ; g IB ) − φ ( r ; g IB = 0)) / p N / π and (d) thedensity of depletion δn ex ( r ) = n ex ( r ; g IB ) − n ex ( r ; g IB = 0) in the presence of an impurity from theresult depicted in (a) and (c), respectively. bath in Figs. 9(a) and (c), respectively. In the case of g IB > g IB < g IB > g IB < IV. CONCLUSIONS
In this work, we have discussed the existence and behavior of Fermi and Bose polaronsthat can be realized in ultracold quantum gases focusing on their backaction on the back-ground medium. We have explicated three different diagrammatic approaches applicableto Fermi polarons in the homogeneous case. These include the TMA, the ETMA, and theSCTMA frameworks, where the ETMA considers induced two-body interpolaron interac-tions and the SCTMA includes two- and three-body ones. Importantly, we have explicitlyderived the mediated two- and three-body interpolaron correlation effects as captured withinthe different diagrammatic approaches. Although these induced interactions are weak in theconsidered mass-balanced Fermi polaron systems, our framework can be applied to vari-ous systems such as mass-imbalanced Fermi polaron systems. Using this strong-couplingapproach, we analyze the spectral response of the Fermi polaron in one- two- and three-spatial dimensions at finite temperature. It has been shown that the spectral function of theminority component exhibits a sharp polaron dispersion in the low-momentum region but itis broadened for higher momenta. Moreover, we argue that the spectral response reflects thecharacter of majority atoms forming a Fermi sphere while a strong interaction between themajority and the minority atoms induces a two-body bound state between a medium atomand an impurity atom. The presence of this two-body bound state becomes more importantin lower dimensions.Next, we present the mean-field treatment of trapped Bose polarons and analyze the roleof quantum depletion identified by the deformation of the background density within theframework of Bogoliubov theory of excitations. A systematic investigation of the latter en-ables us to deduce that the repulsive (attractive) impurity-medium interaction, giving rise torepulsive (attractive) Bose polarons, induces a decreasing (increasing) condensate depletioncaptured by the deformation of the distribution of the host. This effect is a consequence ofthe presence of the external confinement since for a homogeneous background the quantum27epletion increases independently of the sign of the impurity-medium interaction. Therefore,this result is considered a particular feature of the trapped system.Our investigation opens up the possibility for further studies on various polaron real-izations. In particular, the effect of finite temperatures and the impurity concentration onthe 2D Fermi polaron spectral response is expected to play a significant role close to theBerezinskii-Kosterlitz-Thouless transition of molecules [108]. Furthermore, the backaction ofthe impurities on the medium when considering dipolar interactions may affect the densitycollapse of the medium at strong impurity-medium attractions [109].
Acknowledgments
The authors thank K. Nishimura, T. Hata, K. Ochi, T. M. Doi, and S. Tsutsui for usefuldiscussion. S. I. M. gratefully acknowledges financial support in the framework of the Lenz-Ising Award of the University of Hamburg. This research was funded bu a Grant-in-Aidfor JSPS fellows (Grant No. 17J03975) and for Scientific Research from JSPS (Grants No.17K05445, No. 18K03501, No. 18H05406, No. 18H01211, and No. 19K14619). [1] Pekar, S. I. Local quantum states of electrons in an ideal ion crystal
Zh. Eksp. Teor. Fiz. , , 341.[2] Landau, L. D.; Pekar, S. I. Effective mass of a polaron Zh. Eksp. Teor. Fiz. , , 418.[3] Nascimb`ene, S.; Navon, N.; Jiang, K. J.; Tarruell, L.; Teichmann, M.; McKeever, J.; Chevy,F.; Salomon, C. Collective Oscillations of an Imbalanced Fermi Gas: Axial CompressionModes and Polaron Effective Mass Phys. Rev. Lett. , , 170402.[4] Schirotzek, A.; Wu, c.-H.; Sommer, A.; Zwierlein, M. W. Observation of Fermi Polarons ina Tunable Fermi Liquid of Ultracold Atoms Phys. Rev. Lett. , , 230402.[5] Kohstall, C.; Zaccanti, M.; Jag, M.; Trenkwalder, A.; Massignan, P.; Bruun, G. M.; Schreck,F.; Grimm, R. Metastability and Coherence of Repulsive Polarons in a Strongly InteractingFermi Mixture Nature , , 615.[6] Koschorreck, M.; Pertot, D.; Vogt, E.; Fr¨ohlich, B.; Feld, M.; K¨ohl, M. Attractive andrepulsive Fermi polarons in two dimensions Nature , , 619.
7] Catani, J.; Lamporesi, G.; Naik, D.; Gring, M.; Inguscio, M.; Minardi, F.; Kantian, A.;Giamarchi, T. Quantum Dynamics of Impurities in a One-Dimensional Bose Gas
Phys. Rev.A , 023623.[8] Scelle, R.; Rentrop, T.; Trautmann, A.; Schuster, T.; Oberthaler, M. K. Motional Coherenceof Fermions Immersed in a Bose Gas Phys. Rev. Lett. , 070401.[9] Hohmann, M.; Kindermann, F.; G¨anger, B.; Lausch, T.; Mayer, D.; Schmidt, F.; Widera, A.Neutral Impurities in a Bose-Einstein Condensate for Simulation of the Fr¨ohlich-Polaron
EPJQuantum Technol. , 23.[10] Jorgensen, N. B.; Wacker, L.; Skalmstang, K. T.; Parish, M. M.; Levinsen, J.; Christensen,R. S.; Bruun, G. M.; Arlt, J. J. Observation of Attractive and Repulsive Polarons in aBose-Einstein Condensate Phys. Rev. Lett. , 055302.[11] Hu, M.-G.; Van de Graaff, M. J.; Kedar, D.; Corson, J. P.; Cornell, E. A.; Jin, D. S. BosePolarons in the Strongly Interacting Regime
Phys. Rev. Lett. , 055301.[12] Rentrop, T.; Trautmann, A.; Olivares, F. A.; Jendrzejewski, F.; Komnik, A.;Oberthaler, M. K. Observation of the Phononic Lamb Shift with a Synthetic Vacuum
Phys.Rev. X , 041041.[13] Scazza, F.; Valtolina, G.; Massignan, P.; Recati, A.; Amico, A.; Burchianti, A.; Fort, C.;Inguscio, M.; Zaccanti, M.; Roati, G. Repulsive Fermi Polarons in a Resonant Mixture ofUltracold Li Atoms
Phys. Rev. Lett. , , 083602.[14] Yan, Z.; Patel, P. B.; Mukherjee, B.; Fletcher, R. J.; Struck, J.; Zwierlein, M. W. Boiling aUnitary Fermi Liquid Phys. Rev. Lett. , , 093401.[15] Oppong, N. D.; Riegger, L.; Bettermann, O.; H¨ofer, M.; Levinsen, J.; Parish, M. M.; Bloch,I.; F¨olling, S. Observation of Coherent Multiorbital Polarons in a Two-Dimensional FermiGas Phys. Rev. Lett. , 193604.[16] DeSalvo, B. J.; Patel, K.; Cai, G.; Chin, C. Observation of fermion-mediated interactionsbetween bosonic atoms
Nature , , 61.[17] Edri, H.; Raz, B.; Matzliah, N.; Davidson, N.; Ozeri, R. Observation of Spin-Spin Fermion-Mediated Interactions between Ultracold Bosons Phys. Rev. Lett. , , 163401.[18] Ness, G.; Shkedrov, C.; Florshaim, Y.; Diessel, O. K.; von Milczewski, J.; Schmidt, R.; Sagi,Y. Observation of a smooth polaron-molecule transition in a degenerate Fermi gas Phys. Rev.X , , 041019.
19] Adlong, H. S.; Liu, W. E.;, Scazza, F.; Zaccanti, M.; Oppong, N. D.; F¨olling, S.; Parish,M. M.; Levinsen, J. Quasiparticle Lifetime of the Repulsive Fermi Polaron
Phys. Rev. Lett. , , 133401.[20] Chin, C.; Grimm, R.; Julienne, P.; Tiesinga, E. Feshbach resonances in ultracold gases Rev.Mod. Phys. , 1225.[21] Mistakidis, S. I.; Katsimiga, G. C.; Koutentakis, G. M.; Busch, Th.; Schmelcher, P. QuenchDynamics and Orthogonality Catastrophe of Bose Polarons Phys. Rev. Lett. ,183001.[22] Massignan, P.; Zaccanti, M.; Bruun, G. M.
Rep. Prog. Phys. , , 034401.[23] Schmidt, R.; Knap, M.; Ivanov, D. A.; You, J.-S.; Cetina, M.; Demler, E. Universal many-body response of heavy impurities coupled to a Fermi sea: a review of recent progress Rep.Prog. Phys. , , 024401.[24] Kutschera, M.; W´ojcik, W. Proton impurity in the neutron matter: A nuclear polaron prob-lem Phys. Rev. C , , 1077.[25] Forbes, M. M.; Gezerlis, A.; Hebeler, K.;Lesinski, T.; Schwenk, A. Neutron polaron as aconstraint on nuclear density functionals Phys. Rev. C , , 041301(R).[26] Tajima, H.; Hatsuda, T.; van Wyk, P.; Ohashi, Y. Superfluid Phase Transitions and EffectsThermal Pairing Fluctuations in Asymmetric Nuclear Matter Sci. Rep. , , 18477.[27] Nakano, E.; Iida, K.; Horiuchi, W. Quasiparticle properties of a single α particle in coldneutron matter Phys. Rev. C , 055802.[28] Vidana, I. Fermi polaron in low-density spin-polarized neutron matter arXiv:2101.02941 [nucl-th] .[29] Chevy, F. Universal phase diagram of a strongly interacting Fermi gas with unbalanced spinpopulations
Phys. Rev. A , , 063628.[30] Combescot, R.; Recati, A.; Lobo, C.; Chevy, F. Normal State of Highly Polarized FermiGases: Simple Many-Body Approaches Phys. Rev. Lett. , , 180402.[31] Combescot, R.; Giraud, S. Normal State of Highly Polarized Fermi Gases: Full Many-BodyTreatment Phys. Rev. Lett. , , 050404.[32] Cui, X.; Zhai, H. Stability of a fully magnetized ferromagnetic state in repulsively interactingultracold Fermi gases Phys. Rev. A , , 041602(R).[33] Bruun, G. M.; Massignan, P. Decay of Polarons and Molecules in a Strongly Polarized Fermi as Phys. Rev. Lett. , , 020403.[34] Mathy, C. J. M.; Parish, M. M.; Huse, D. A. Trimers, Molecules, and Polarons in Mass-Imbalanced Atomic Fermi Gases Phys. Rev. Lett. . , 166404.[35] Schmidt, R.;Enss, T. Excitation spectra and rf response near the polaron-to-molecule tran-sition from the functional renormalization group Phys. Rev. A , , 063620.[36] Trefzger, C.; Castin, Y. Impurity in a Fermi sea on a narrow Feshbach resonance: A varia-tional study of the polaronic and dimeronic branches Phys. Rev. A , , 053612.[37] Baarsma, J. E.; Armaitis, J.; Duine, R. A.; Stoof, H. T. C. Polarons in extremely polarizedFermi gases: The strongly interacting Li- K mixture
Phys. Rev. A , , 033631.[38] Prokof’ev, N.; Svistunov, B. Fermi-polaron problem: Diagrammatic Monte Carlo method fordivergent sign-alternating series Phys. Rev. B , , 020408(R).[39] Prokof’ev, N. V. Svistunov, B. V. Bold diagrammatic Monte Carlo: A generic sign-problemtolerant technique for polaron models and possibly interacting many-body problems Phys.Rev. B , , 125101.[40] Vlietinck, J.; Ryckebusch J.;Van Houcke, K. Quasiparticle properties of an impurity in aFermi gas Phys. Rev. B , , 115133.[41] Kroiss, P.; Pollet, L. Diagrammatic Monte Carlo study of a mass-imbalanced Fermi-polaronsystem Phys. Rev. B , Phys. Rev. A , Phys. Rev. B , , 045134.[44] Kamikado, K.; Kanazawa, T.; Uchino, S. Mobile impurity in a Fermi sea from the functionalrenormalization group analytically continued to real time Phys. Rev. A , , 013612.[45] Liu, W. E.; Shi, Z.-Y..; Parish, M. M.;Levinsen, J. Theory of radio-frequency spectroscopyof impurities in quantum gases Phys. Rev. A , , 023304.[46] Liu, W. E.; Shi, Z.-Y..; Levinsen, J.; Parish, M. M. Radio-Frequency Response and Contactof Impurities in a Quantum Gas Phys. Rev. Lett. , , 065301.[47] Pilati, S.; Giorgini, S. Phase Separation in a Polarized Fermi Gas at Zero Temperature Phys.Rev. Lett. , , 030401.[48] Mora, C.; Chevy, F. Normal State of an Imbalanced Fermi Gas Phys. Rev. Lett. , , Phys. Rev. A , , 013605.[50] Hu, H.; Mulkerin, B. C.; Wang, J.; Liu, X.-J.; Attractive Fermi polarons at nonzero temper-atures with a finite impurity concentration Phys. Rev. A , , 013626.[51] Tajima, H.; Uchino, S. Many Fermi polarons at nonzero temperature New J. Phys. , , 073048.[52] Tajima, H.; Uchino, S. Thermal crossover, transition, and coexistence in Fermi polaronicspectroscopies Phys. Rev. A , , 063606.[53] Tajima, H.; Takahashi, J.; Nakano, E.; Iida, K. Collisional dynamics of polaronic cloudsimmersed in a Fermi sea Phys. Rev. A , , 051302(R).[54] Tajima, H.; Takahashi, J.; Nakano, E.; Iida, K. Extracting non-local inter-polaron interac-tions from collisional dynamics arXiv:2011.07911 .[55] Mistakidis, S. I.; Volosniev, A. G. and Schmelcher, P. Induced correlations between impuritiesin a one-dimensional quenched Bose gas. Phys. Rev. Research , 023154.[56] Mistakidis, S. I., Katsimiga, G. C., Koutentakis, G. M., Busch, T. and Schmelcher, P. Pump-probe spectroscopy of Bose polarons: Dynamical formation and coherence. Phys. Rev. Re-search , 033380.[57] Mukherjee, K.; Mistakidis, S. I.; Majumder, S.; Schmelcher P. Induced interactions andquench dynamics of bosonic impurities immersed in a Fermi sea arXiv:2007.02166v1 [cond-mat.quant-gas] , .[58] Compagno, E.; De Chiara, G.; Angelakis, D. G.; G. M. Palma, Tunable Polarons in Bose-Einstein Condensates Scientific Reports , 2355.[59] Nakano, E.; Yabu, H.; Iida, K. Bose-Einstein-Condensate Polaron in Harmonic Trap Poten-tials in the Weak-Coupling Regime: Lee-Low-Pines–Type Approach Phys. Rev. A ,023626.[60] Watanabe, K.; Nakano, E.; Yabu, H. Bose Polaron in Spherically Symmetric Trap Potentials:Ground States with Zero and Lower Angular Momenta Phys. Rev. A , 033624.[61] Cucchietti, F. M.; Timmermans, E. Strong-Coupling Polarons in Dilute Gas Bose-EinsteinCondensates Phys. Rev. Lett. , 210401.[62] Sacha, K.; Timmermans, E. Self-Localized Impurities Embedded in a One-Dimensional Bose- instein Condensate and Their Quantum Fluctuations Phys. Rev. A , 063604.[63] Kalas, R. M.; Blume, D. Interaction-Induced Localization of an Impurity in a Trapped Bose-Einstein Condensate Phys. Rev. A , 043608.[64] Bruderer, M.; Bao, W.; Jaksch, D.; Self-trapping of impurities in Bose-Einstein condensates:Strong attractive and repulsive coupling, EuroPhys. Lett. , 30004.[65] Boudjemˆaa, A. Self-localized state and solitons in a Bose-Einstein-condensate-impurity mix-ture at finite temperature Phys. Rev. A , 013628.[66] Cai, Z., Wang, L., Xie, X. C. and Wang, Y. Interaction-induced anomalous transport behaviorin one-dimensional optical lattices. Phys. Rev. A , 043602.[67] Theel, F., Keiler, K., Mistakidis, S. I. and Schmelcher, P. Many-body collisional dynamics ofimpurities injected into a double-well trapped Bose-Einstein condensate. arXiv: .[68] Keiler, K., Mistakidis, S. I. and Schmelcher, P. Doping a lattice-trapped bosonic species withimpurities: From ground state properties to correlated tunneling dynamics. New J. Phys. , , 083003.[69] Siegl, P., Mistakidis, S. I. and Schmelcher, P. Many-body expansion dynamics of a Bose-Fermimixture confined in an optical lattice. Phys. Rev. A , 053626.[70] Mistakidis, S. I., Grusdt, F., Koutentakis, G. M. and Schmelcher, P. Dissipative correlateddynamics of a moving impurity immersed in a Bose–Einstein condensate. New J. Phys. , 103026.[71] Strinati, G. C.; Pieri, P.; R¨opke, G.; Schuck, P.; Urban, M. The BCS-BEC crossover: Fromultra-cold Fermi gases to nuclear systems. Phys. Rep.
Prog. Part. Nucl. Phys.
Phys. Rev. A , 043622.[74] Tajima, H.; van Wyk, P.; Hanai, R.; Kagamihara, D.; Inotani, D.; Horikoshi, M.; Ohashi, Y.Strong-coupling corrections to ground-state properties of a superfluid Fermi gas. Phys. Rev.A , 043625.[75] Horikoshi, M.; Koashi, M.; Tajima, H.; Ohashi, Y.; Kuwata-Gonokami, M. Ground-StateThermodynamic Quantities of Homogeneous Spin-1/2 Fermions from the BCS Region to theUnitarity Limit. Phys. Rev. X , 041004.
76] Haussmann, R.; Crossover from BCS superconductivity to Bose-Einstein condensation: Aself-consistent theory.
Z. Phys. B , 291.[77] Haussmann, R.; Rantner, W.; Cerrito, S.; Zwerger, W. THermodynamics of the BCS-BECcrossover Phys. Rev. A , , 023610.[78] Takahashi, J.; Imai, R.; Nakano, E.; Iida, K. Bose Polaron in Spherical Trap Potentials:Spatial Structure and Quantum Depletion Phys. Rev. A , 023624.[79] Pethick, C. J.; Smith, H.
Bose-Einstein Condensation in Dilute Gases (Cambridge UniversityPress, Cambridge, 2008).[80] Morgan, S. A.; Lee, M. D.; Burnett, K. Off-shell T matrices in one, two and three dimensions Phys. Rev. A , 022706.[81] Fetter, A. L.; Walecka, J. D. Quantum Theory of Many-Particle Systems , Dover, New York, .[82] T¨orm¨a, P.; Spectroscopies–Theory, in
Quantum Gas Experiments: Exploring Many-BodyStates , T¨orm¨a, P., Sengstock, K. Eds.; Imperial College, London, .[83] Frank, B; Lang, J.; Zwerger, W. Universal phase diagram and scaling functions of imbalancedFermi gases
J. Exp. Theor. Phys. , , 812.[84] Pini, M.; Pieri, P.; Strinati, G. C. Fermi gas throughout the BCS-BEC crossover: Compara-tive study of t -matrix approaches with various degrees of self-consistency Phys. Rev. B , , 094502.[85] Punk, M.; Dumitrescu, P. T.; Zwerger, W. Polaron-to-molecule transition in a stronglyimbalanced Fermi gas Phys. Rev. A , , 053605.[86] Cui, X. Fermi polaron revisited: Polaron-molecule transition and coexistence Phys. Rev. A , , 061301(R).[87] Thouless, D. J. Perturbation theory in statistical mechanics and the theory of superconduc-tivity Ann. Phys. (NY) , , 553.[88] Liu, X.-J.; Hu, H. BCS-BEC crossover in an asymmetric two-component Fermi gas Europhys.Lett. , , 364.[89] Grudst, F.; Seetharam, K.; Shchadilova, Y. Demler, E. Strong-coupling Bose polarons out ofequilibrium: Dynamical renormalization group approach Phys. Rev. A , , 033612.[90] Nielsen, K.; Pe˜na Ardila, L. A.; Bruun, G. M.; Pohl, T. Critical slowdown of non-equilibriumpolaron dynamics New J. Phys. , , 043014.
91] Sekino, Y.; Tajima.; Uchino, S. Mesoscopic spin transport between strongly interacting Fermigases
Phys. Rev. Res. , , 023152.[92] Klawunn, M.; Recati, A. The Fermi-polaron in two dimensions: Importance of the two-bodybound state Phys. Rev. A , , 033607.[93] Schmidt, R.; Enss, T.; Pietil¨a, V.; Demler, E. Fermi polarons in two dimensions Phys. Rev.A , , 021602(R).[94] Tajima, H.; Tsutsui, S.; Doi, T. M. Low-dimensional fluctuations and pseudogap in Gaudin-Yang Fermi gas Phys. Rev. Res. , , , 033441.[95] Mistakidis, S. I.; Katsimiga, G. C.; Koutentakis, G. M.; Schmelcher, P. Repulsive Fermipolarons and their induced interactions in binary mixtures of ultracold atoms New J. Phys. , , 043032.[96] Kwasniok, J.; Mistakidis, S. I.; Schmelcher P. Correlated dynamics of fermionic impuritiesof an emsemble of fermions Phys. Rev. A , , 053619.[97] Guan, X.-W.; Batchelor, M. T.; Lee, C. Fermi gases in one dimension: From Bethe ansatzto experiments Rev. Mod. Phys. , , 1633.[98] Doggen, E. V. H.; Kinnunen, J. J. Energy and Contact of the One-Dimensional Fermi Polaronat Zero and Finite Temperature Phys. Rev. Lett. , , 025302.[99] Lampo, A.; Charalambous, C.; Garc´ıa-March, M. ´A.; Lewenstein, M. Non-Markovian PolaronDynamics in a Trapped Bose-Einstein Condensate Phys. Rev. A , 063630.[100] Mistakidis, S. I.; Volosniev, A. G.; Zinner, N. T.; Schmelcher, P. Effective Approach toImpurity Dynamics in One-Dimensional Trapped Bose Gases Phys. Rev. A , 013619.[101] Pitaevskii, L.; Stringari, S.
Bose-Einstein Condensation (Oxford, New York, 2003).[102] Dalfovo, F.; Giorgini, S.; Pitaevskii, L. P.; Stringari, S. Theory of Bose-Einstein Condensationin Trapped Gases
Rev. Mod. Phys. , 463.[103] Bogoliubov, N. N.; J. Phys. (Moscow) , 32.[104] de Gennes, P. G.; Superconductivity of Metals and Alloys (Benjamin, New York, 1966).[105] Katsimiga, G. C.; Mistakidis, S. I.; Bersano, T. M.; Ome, M. K. H.; Mossman, S. M.;Mukherjee, K.; Schmelcher, P.; Engels, P.; and Kevrekidis, P. G. Observation and Analysisof Multiple Dark-Antidark Solitons in Two-Component Bose-Einstein Condensates.
Phys.Rev. A , 023301.[106] Katsimiga, G. C.; Mistakidis, S. I.; Schmelcher, P.; and Kevrekidis, P. G. Phase Diagram, tability and Magnetic Properties of Nonlinear Excitations in Spinor Bose-Einstein Conden-sates. arXiv: Phys. Rev. A , , 033612.[108] Tempere, J.; Klimin, S. N.; Devreese, J. T. Effect of population imbalance on the Berezinskii-Kosterlitz-Thouless phase transition in a superfluid Fermi gas Phys. Rev. A , , 053637.[109] Nishimura, K.; Nakano, E.; Iida, K.; Tajima, H.; Miyakawa, T.; Yabu, H. The ground stateof polaron in an ultracold dipolar Fermi gas arXiv:2010.15558 ..