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Polaron problems in ultracold atoms: Role of the medium across different spatial dimensions

The notion of a polaron, originally introduced in the context of electrons in ionic lattices, helps us to understand how a quantum impurity behaves when being immersed in and interacting with a many-body background. We discuss the impact of the impurities on the medium particles by considering feedback effects from polarons that can be realized in ultracold quantum gas experiments and in particular we exemplify the modifications of the medium either in the presence of Fermi or Bose polarons. Regarding Fermi polarons we present a corresponding many-body diagrammatic approach operating at finite temperatures and discuss how mediated two- and three-body interactions are implemented within this framework. Utilizing this approach, we analyze the behavior of the spectral function of Fermi polarons at finite temperature by varying impurity-medium interactions as well as spatial dimensions from three to one. Interestingly, we reveal that the spectral function of the medium atoms could be a useful quantity for analyzing the transition/crossover from attractive polarons to molecules in three-dimensions. As for the Bose polaron, we showcase the 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 quantification of interpolaron correlations in Bose polaron problems.


I. 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. The major obstacle that prevents their adequate description stems from the involvement of many degrees-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 electron properties in crystals, provides a useful playground for understanding related nontrivial many-body aspects of quantum matter and interactions. For instance, a key advantage of the polaron picture is that, under specific circumstances, it enables the reduction of a complicated many-body problem to an effective single-particle or a few-body one with renormalized parameters. Ultracold atoms, owing to the excellent controllability of the parameters characterizing the system, are utilized to quantitatively determine polaron properties, as has been recently demonstrated in a variety of relevant experimental efforts [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].
Polarons basically appear in two different types, namely, Fermi and Bose polarons where the 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 atoms residing in different hyperfine states or using distinct isotopes. The impurity-medium interaction strength can be flexibly adjusted with the aid of Feshbach resonances [20], and as 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 elementary excitations of their background atoms, leading to a quasi-particle state that is called the polaron. In that light, the polaron and more generally the quasiparticle generation is inherently related to the build-up of strong entanglement among the impurities and their background medium [21][22][23]. Moreover, since various situations such as mass-imbalanced [5], low-dimensional [6], and multi-orbital [15] ultracold settings can be realized, atomic polarons can also be expected to be quantum simulators of quasiparticle states in nuclear physics [24][25][26][27][28].
They originate from the presence of the impurity which causes its environment to significantly 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 manypolaron problems that are tractable in ultracold atom settings. Particularly, we present diagrammatic approaches to Fermi polaron systems and elaborate on how mediated twoand three-body interpolaron interactions are consistently taken into account within these frameworks [51,52]. Importantly, a comparison of the Fermi polaron excitation spectral function in three dimensions and at finite temperatures is performed among different variants 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 processes consisting of bare propagators [71,72], the extended T -matrix approach (ETMA) where the bare propagator in the self-energy is partially replaced [73][74][75], and the self-consistent Tmatrix 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 these approaches and examine their effects in mass-balanced Fermi polaron settings realized, e.g., in 6 Li atomic mixtures. Subsequently, we discuss the polaron excitation spectrum in two and one spatial dimensions. The behavior of the spectral function of the host and the impurities at strong impurity-medium interactions is exemplified. Finally, the real-space Bogoliubov approach to Bose polarons is reviewed. The latter helps us to unveil the condensate deformation 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 (increases) for repulsive (attractive) impurity-medium interactions, a result that is associated with the deformation of its density distribution. This is in contrast to homogeneous setups where the depletion increases independently of the sign of the interaction.
This work is organized as follows. In Sec. II, we present the model Hamiltonian describing ultracold Fermi polarons in three dimensions. For the Fermi polaron, we consider uniform systems and develop the concept of the diagrammatic T -matrix approximation.
After explaining the ingredients of the diagrammatic approaches in some detail, we clarify how mediated two-and three-body interactions are incorporated in these approaches.
The behavior of the resultant polaron spectral function at finite temperatures and impurity concentrations in three-two-and one-dimensions is discussed. In Sec. III, we utilize the real-space mean-field formulation for Bose polarons and expose the presence of quantum depletion 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.

A. 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 in a uniform Fermi gas. Since such a two-component Fermi mixture mimics spin-1/2 electrons, we denote the bath component as σ = B =↑ and the impurity one by σ = I =↓. Note that these are standard conventions without loss of generality. The model Hamiltonian describing where ξ p,σ = p 2 /(2m 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 and creation operators of a σ component fermion, respectively, possessing momentum p.
We measure the effective coupling constant g of the contact-type interaction between two 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 3D and the scattering length a are related via with m −1 r = m −1 ↑ + m −1 ↓ being the reduced mass. In this expression, the momentum cutoff Λ is introduced to avoid an ultraviolet divergence in the momentum summation of the Lippmann-Schwinger equation expressed in the momentum space. This allows us to achieve the effective short-range interaction of finite range r e ∝ 1/Λ. Similarly, the relevant relations in 2D and 1D read [80] respectively, where g 2D and g 1D are the coupling constants in 2D and 1D.
First, we introduce a thermal single-particle Green's function [81] where ω n = (2n + 1)πT is the fermion Matsubara frequency introduced within the finitetemperature T formalism and n ∈ Z [81]. The effect of the impurity-medium interaction is taken into account in the self-energy Σ σ (p, iω n ). The excitation spectrum A ↓ (p, ω) of a Fermi polaron 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 can be shown that Experimentally, this quantity can be monitored by using a radio-frequency (rf) spectroscopy scheme where the atoms are transferred from their thermal equilibrium state to a specific spin state which interacts with the medium [82]. Indeed, the reverse rf response I r (ω) [13] = + + + ... and the ejection one I e (ω) [14] are given by and respectively. Here, ξ p,i represents the kinetic energy of the initial state in the reverse rf scheme. 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 in describing the mediated interpolaron interactions. This fact will be evinced below and it is achieved by expanding Σ ↑ (p, iω n ) with respect to G σ and G 0 σ . Also, the chemical potentials µ σ are kept fixed by imposing the particle number conservation condition obeying Moreover, in the remainder of this work, we define the impurity concentration as follows Additionally, within the TMA [30,50] the self-energy Σ σ (p, iω n ) of the σ component where Γ(q, iν ℓ ) is the many-body T -matrix, as diagrammatically shown in Fig. 1(a), with the boson Matsubara frequency iν ℓ = 2ℓπT (ℓ ∈ Z). Here, G 0 σ (p, iω n ) = (iω n − ξ p,σ ) −1 is the bare thermal single-particle Green's function. Furthermore, by adopting a ladder approximation illustrated in Fig. 1(d), the T -matrix Γ(q, iν ℓ ) is given by where is the lowest-order particle-particle bubble. The latter describes a virtual particle-particle scattering process associated with the impurity-medium interaction g which is replaced by g 3D , g 2D , and g 1D in 3D, 2D, and 1D, respectively. Note that in Eq.(10) the impurity-impurity interaction is not taken into account.
The extended T-matrix approach (ETMA) [51] constitutes one of the better approximations that allows us to take the induced polaron-polaron interactions into account in a self-consistent way. In this method, as depicted in Fig. 1(b) we include higher-order correlations by replacing the bare Green function G 0 in Eq. (10) with the dressed one G σ . Namely Importantly, the TMA and ETMA approaches are equivalent to each other in the singlepolaron limit i.e. x → 0, where the self-energy of the fermionic medium Σ E ↑ [capturing the difference between G 0 ↑ 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  Proceeding one step further, it is possible to construct the so-called self-consistent Tmatrix approach (SCTMA) [52,83,84] which deploys the many-body T -matrix Γ S composed of dressed propagators as schematically shown in Fig. 1(e). In particular, the corresponding T -matrix is given by where which describes a scattering process denoted by G ↑ and G ↓ , of the dressed medium atoms with the impurities and the dressed ones (polarons), respectively. This is in contrast to Eq. (12) obtained in ETMA and consisting of G 0 σ which represents the impurity-medium scattering process of the bare atoms. Using this T -matrix, we can express the SCTMA We note that within the ETMA, the impurity self-energy Σ E ↓ [Eq. (11)] can be rewritten as with the higher-order correction δΣ ↓ (p, iω n ) beyond the TMA being In this expression, V represents the induced impurityimpurity interaction [diagrammatically shown in Fig. 2(a)] with incoming and outgoing momenta and frequencies Here, δ i,j is the Kronecker delta imposing the energy and momentum conservation in the two-body scattering.
The self-energy Σ S ↓ of the impurities within the SCTMA involves a contribution of induced three-impurity correlations due to the dressed pair propagator Σ S ↓ . The latter can again be decomposed as where Here we defined which represents the difference between the Π and Π S , namely, the medium-impurity and the medium-polaron propagators. In the last line of Eq. (22), we assumed that G ↑ ≃ G 0 ↑ and Σ S ↓ ≃ 0. Thus, one can find a three-body correlation effect beyond the ETMA as shown in Fig. 2(b) and captured by where V is the induced three-polaron interaction term. Its explicit form reads From the above discussion it becomes evident how the medium-induced two-body and threebody 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 spectral functions obtained via rf spectroscopy can in principle provide indications of the effect of interpolaron 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 of Fermi polarons at temperatures close to zero up to of the order of the majority component Fermi temperature as well as for different spatial dimensions ranging from three to one. For simplicity, we consider a mass-balanced fermionic mixture i.e. m ↑ = m ↓ ≡ m. The latter is experimentally relevant for instance by considering two different hyperfine states, e.g., |F = 1/2, m F = +1/2 and |F = 3/2, m F = −3/2 of 6 Li. In this notation, F and m F are the total angular momentum and its projection, respectively, of the specific hyperfine state [13] at thermal equilibrium.

Three-dimensional case
The resultant spectral function A σ (p = 0, ω) of the fermionic medium (σ =↑) and the impurities (σ =↓) is depicted in Figure 3 as a function of the single-particle energy ω. Here, we consider a temperature T = 0.3T F , impurity concentration x = 0.1, and impurity-medium interaction at unitarity, i.e., (p F a) −1 = 0. The Fermi temperature is T F = p 2 F /(2m ↑ ) and the Fermi 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 atoms given by A(p, ω) = δ(ω − ξ p,↑ ) at p = 0. This indicates that the imprint of the impuritymedium interaction on the fermionic host is negligible for such small impurity concentrations It can be shown that in the weak-coupling limit, this shift is given by the Hartree correction [81]. However, at the unitarity limit presented in Fig. 3, such a weak-coupling approximation cannot be applied and therefore the factor 0.526 in Eq. (25) originates from the 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 P is the attractive polaron energy. Notice here that since this peak is located at negative energies it indicates the formation of an attractive Fermi polaron. This observation can be understood from the fact that in the absence of impurity-medium interactions, the bare-particle pole, namely, the position of the pole of the bare retarded single-particle Green's function G 0,R ↓ (p = 0, ω) = (ω + iδ + µ ↓ ) −1 , occurs at ω +µ ↓ = 0. Moreover, the attractive polaron energy E . Recall that, in general, for finite temperatures T and impurity concentrations x, µ ↓ = E (a) P holds in contrast to the singlepolaron 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 impurities where excited atoms repulsively interact with each other. This peak becomes sharper at positive scattering lengths away from unitarity. Indeed, for positive scattering lengths, the quasi-particle excitation called a repulsive Fermi polaron emerges [22]. P corresponding to the attractive and the repulsive polaron energies, respectively. The interaction dependence of these energies is provided in Fig. 4(b). E (r) P approaches the Hartree shift Σ H ↓ = 4πa m N ↑ without the imaginary part of the self-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 ↓ > 0 becomes sharper when increasing (p F a) −1 , indicating the vanishing imaginary part of the self-energy. On the other hand, E (a) P decreases with increasing (p F a) −1 as depicted by the position of the low-energy peak (where ω + µ ↓ < 0) in Fig. 4(a). Eventually, the attractive polaron undergoes the molecule transition as we discuss below. Another important issue  and E (r) P , respectively, as a function of (p F a) −1 . The polaron energies have been extracted from the peak position of A ↓ (p = 0, ω), that is, the pole of G R ↓ (p = 0, ω). The experimental data of Ref. [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 to the molecular state with increasing impurity-bath attraction [85]. Although this transition was originally predicted to be of first-order, recent experimental and theoretical studies showed an underlying crossover behavior and coexistence between polaronic and molecular states [18]. We note that in the case of finite impurity concentrations, a BEC of molecules can appear at low temperatures; see also Eq. (26) and Eq. (27) below. It is also a fact that the interplay among a molecular BEC, thermally excited molecules, and polarons may occur at finite temperatures [86]. In the calculation of the attractive polaron energy E (a) P for different coupling strengths [ Fig. 4(b)], however, we do not encounter the molecular BEC transition identified by the Thouless criterion [87] 1 + gΠ(q = 0, iν ℓ = 0) = 0.
In particular, in the strong-coupling limit, from Eq. (26) combined with the particle number conservation [Eq. (8)] the BEC temperature T BEC of molecules satisfies [88] T BEC ≃ 2π x where ζ(3/2) ≃ 2.612 is the zeta function. Since we consider a small impurity concentration whereas the FWHM in ETMA is 1.61(T /T F ) 2 [52]. We should also note that the decay rate related to the FWHM for repulsive polarons as extracted using TMA (and simultaneously ETMA) agree quantitatively with the experimental result of Ref. [13]. For the attractive polaron the quantitative agreement between the experiment and these diagrammatic approaches 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 ≃ 0.75T F [14], while the prediction of the diagrammatic approaches is above T F [52]. Besides the fact that such polaron decay properties may be related to multi-polaron scattering events leading to many-body dephasing [19], they are necessary for further detailed polaron investigations at various temperatures and interaction strengths that facilitate the understanding of the underlying 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 the impurities is illustrated in Fig. 5 for T = 0.2T F , x = 0, and (p F a) −1 = 0. To infer the impact of the multi-polaron correlations on the spectrum we explicitly compare A ↓ (p, ω) between the ETMA and the SCTMA methods. As it can be seen, A ↓ (p, ω) exhibits a sharp peak which is associated with the attractive polaron state and shows an almost quadratic behavior for increasing momentum of the impurities. It is also apparent that the SCTMA spectrum to the induced beyond-two-body interpolaron correlations, e.g. three-body ones. At small impurity momenta, the spectral peak of the attractive Fermi polaron within the present model as described by Eq. (1), is generally given by where Z a and m * a are the quasiparticle residue [22] and the effective mass of the attractive polaron, respectively. At unitarity it holds that Z a ≃ 0.8, m * a ≃ 1.2m, and E (a) P ≃ −0.6E F within the zero-temperature and single-polaron limits [30]. The behavior of these quantities has been intensively studied in current experiments [3,4,13] and an adequate agreement has been reported using various theories. For instance, Chevy's variational ansatz (being equivalent to the TMA at T = 0 and x → 0) [ As we demonstrated previously (see Fig. 3), besides the fact that the spectral response within the SCTMA method is broader compared to the one obtained in the ETMA, the two spectra feature a qualitatively similar behavior. Indeed, both approaches evince that the spectra beyond p = p F are strongly broadened. Recall that in this region of momenta the atoms of the majority component, which form the Fermi sphere, cannot follow the impurity atoms. This indicates that the dressed polaron state ceases to exist due to the phenomenon of the Cherenkov instability [89,90], where the polaron moves faster than the speed of sound of the medium and consequently it becomes unstable against the spontaneous emission of elementary excitations of the medium. Such a spectral broadening can also be observed in mesoscopic spin transport measurements [91] and may also be related to the underlying polaron-Boltzmann gas transition [14] since the contribution of high-momentum polarons can be captured in rf spectroscopy due to the thermal broadening of the Fermi distribution function in Eq. (7) at high temperatures. Moreover, the momentum-resolved photoemission spectra would reveal these effects across this transition.
We remark that the medium spectral function A ↑ (p, ω) is also useful to reveal the properties of strong-coupling polarons in the case of finite temperature and impurity concentration. Figure 6 Fig. 6 shows that the repulsive polaron is located around ω + µ ↑ ≃ E F . Moreover, at (p F a) −1 = 1, another peak emerges in the low-energy region (ω + µ ↑ ≃ −3E F ).
This low-energy peak elucidates the emergence of two-body molecules with the binding energy given by E b = 1/(ma 2 ) due to the strong impurity-medium attraction. Concluding, the spectral function of the medium atoms can provide us with useful information for the recently observed smooth crossover from polarons to molecules [18].
In the following, we shall elaborate on the behavior of the spectral function of lower  dimensional Fermi polarons solely within the TMA approach. The latter provides an adequate description of the polaron formation in our case since the induced interpolaron interaction [55,56] is weak in the considered mass-balanced system.

Spectral response of Fermi polarons in two-dimensions
In two spatial dimensions, the attractive impurity-medium effective interaction g 2D < 0 is always accompanied by the existence of a two-body bound state whose energy scales as −1/(ma 2 2D ) [92]. Simultaneously, the repulsive polaron branch appears at positive energies [22] in addition to the attractive one located at negative energies. This phenomenology is similar to the case of a positive impurity-bath scattering length in 3D [93]. To elaborate on the typical spectrum of 2D Fermi polarons below we employ a homogeneous Fermi mixture characterized by an impurity concentration x = 0.1, temperature T = 0.3T F , and a typically weak dimensionless coupling parameter ln(p F a 2D ) = 0.4 where a 2D is the 2D scat- where T = 0.3T F , x = 0.3, and ln(p F a 2D ) = 0. Moreover, since the repulsive interaction in the excited branch of the impurities [ω + µ ↓ ≃ E F ] is relatively strong, the impurity excitation spectrum at positive energies (ω + µ ↓ > 0) is largely broadened. We note that the stable repulsive polaron branch can be found in the case of small a 2D . It becomes also evident that the impurity spectrum in 2D is largely broadened beyond p = p F as compared to the 3D spectral response [Fig. 5] . Simultaneously, the intensity of the metastable impurity excitation 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 occupy the non-interacting excited states in such high-momentum regions.

Fermi polarons in one-dimension
In one spatial dimension the quasiparticle notion is somewhat more complicated as com- We remark that the impurity-medium attraction is considered weak herein such that the induced interpolaron interactions are negligible. In this sense, we do not expect significant deviations when considering the ETMA or even the SCTMA approaches.
It is also important to note here that in sharp contrast to higher spatial dimensions, the coupling constant g 1D does not vanish when Λ → ∞ in the renormalization procedure; see Sec. II A. Thus, we take the Hartree shift Σ H σ = g 1D N −σ into account in the building block of the self-energy diagrams [94]. This treatment is not necessary in the single-polaron limit since Σ H ↑ → 0 and Σ H ↓ → g 1D T p,iωn G 0 ↑ (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 the emergence of induced interpolaron interactions as it has been recently demonstrated, e.g., in Refs. [57,95,96]. The polaronic excitation properties obtained within the TMA approach show an excellent agreement with the results of the thermodynamic Bethe ansatz [97]. The latter 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 behavior between polarons and molecules. As it can be seen by inspecting Fig. 8(a1) the spectrum of the majority component is affected by the scattering with the impurities. This is attributed to the relatively large impurity concentration x considered here. In particular, A ↑ (p, ω) is broadened at low momenta below p = p F . On the other hand, the spectral response of the impurities 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 of the polaron peak corresponding to the effective mass (curvature of the dispersion) is changed around this value of the momentum. Similar broadening effects of sharp peaks can be found even 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 impurity spectrum [ Fig. 8(b2)] is shifted to larger energies as a consequence of the repulsion with the medium.

III. BOSE POLARONS
In this section, we shall discuss the Bogoliubov theory of trapped Bose polaron systems in real space [78,99,100]. The reason for focusing on a real-space Bogoliubov theory is 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 significant inhomogeneities of the density distribution of the background which cannot be described within a simple Thomas-Fermi approximation. Such a modification of the boson distribution causes, for instance, enhanced phonon emission [57,70]. Moreover, in cold atom experiments the background bosons and the impurity are generally trapped. Considering the impact of inhomogeneity that naturally arises in trapped systems, therefore, we treat the Bose polaron in real space without plane wave expansion because the momentum is not a good quantum number. Below, we review the description of a Bose polaron in trapped systems at zero temperature 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 an external harmonic potential denoted by V I (r) and is embedded in a BEC medium that is also 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 following model Hamiltonian Here,φ andψ 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 potential of the medium bosons. The effective couplings g IB and g BB refer to the impurity-boson and boson-boson interaction strengths, respectively.

A. Bogoliubov theory for Bose polaron problems
First, we calculate the expectation value of the Hamiltonian in terms of the singleimpurity state |imp =â † imp |0 imp in order to integrate out the impurity's degree-of-freedom whereâ imp denotes the annihilation operator of an impurity in the ground state; ψ(r) is the corresponding wave function that can be determined self-consistently by Eq. (35). In this way, we have obtained the effective Hamiltonian for the medium bosons, in which the bosons experience an effective potential constructed by the external trap and the density of the impurity g IB |ψ(r)| 2 . Since we have set the temperature to zero in the present study, we have to assume that the medium bosons possess a condensed part, so-called the order parameter or 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 operatorφ leads to a nonzero function which is used as an order parameter, i.e., φ (r) b = φ(r), where · · · b means b 0| · · · |0 b . The vacuum |0 b is determined from the effective Hamiltonian (30) within the Bogoliubov theory to the second order of fluctuations. This is equivalent to splitting the operator asφ = φ +φ, where φ b = 0. Substituting this into the Hamiltonian of Eq. (30) and expressing it in terms of the different orders ofφ, we can readily obtain the expansion H B ≃ H (0) + H (1) + H (2) because the number of the non-condensed bosons is significantly smaller than that of the condensed ones at zero temperature at weak couplings. In this expression, the individual contributions correspond to where L(r) = − ∇ 2 2m B + V B (r) + g IB |ψ(r)| 2 + 2g BB |φ(r)| 2 − µ, and M(r) = g BB φ 2 (r). Note that we assume the weakly interacting limit of the medium to ensure the BEC dominating condition 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 fourthorder 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 the Bose-polaron system. From the Heisenberg equation, the bosonic field operatorφ satis- (2) ] b = 0 in the interaction picture. Accordingly, it is possible to retrieve the celebrated Gross-Pitaevskii equation describing the BEC background We remark that here, for simplicity, we consider the stationary case where the condensate is time-independent. Next, by following the variational principle for ψ namely δ H B b /δψ * = 0, we arrive at the Schrödinger equation for the impurity wavefunction, namely, where n ex (r) = φ † (r)φ(r) b is the density of the non-condensed bosons in vacuum, the so-called quantum depletion.
To evaluate this expectation value, we need the ground state |0 b of the Hamiltonian that can be obtained by the diagonalization of Eq. (33), namely, H (2) = n E nb † nb n is achieved using the following field expansionφ(r) = n b n u n (r) +b † n v * n (r) . Here the complete set {u i , v i } satisfies the following system of linear equations being the so-called Bogoliubov-de Gennes (BdG) equations [103,104] We 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 eigenvalues unveil the dynamically unstable modes of the condensate [105,106]. More precisely, if complex eigenvalues exist then the Hamiltonian can not be expressed in the above-mentioned diagonal form in terms of the annihilation/creation operators. Therefore, the dynamically unstable situation is beyond the scope of the present description. By using this expansion, we can calculate the vacuum expectation, e.g., n ex (r) = n |v n (r)| 2 . For the numerical calculations, to be presented below, the total number of bosons N B is conserved, i.e., This condition is achieved by tuning the chemical potential µ of the bosonic medium. Notice that N ex becomes non-zero due to thermal fluctuations at finite temperature, while in the ultracold regime it can be finite due to the presence of quantum fluctuations, otherwise termed 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 the following 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 0 changes due to the existence of N ex . This is a quantum effect that occurs even at zero temperature, and it is called quantum depletion [102]. We need to clarify that the term quantum depletion refers to the beyond mean-field corrections for the description of the bosonic ensemble. In the following, we shall investigate the effect of an impurity on the quantum depletion of the medium bosons at zero temperature. Indeed, the quantum depletion is a measurable quantum effect that is included in Eq. (35) and its quantification makes it possible to evaluate the backaction of the impurity on the medium condensate.
A commonly used external confinement in cold atom experiments is the harmonic potential. As such, here, we consider that the traps of the impurity and the bosonic medium are spherically symmetric, namely, Accordingly, the order parameter of the BEC and the impuritys' wave function have spherically symmetric forms, and therefore the underlying BdG eigenfunctions are separable with the help of spherical harmonics as φ(r) = φ(r), ψ(r) = ψ(r), where r = |r|. Here, (n r , ℓ, m) denote the radial, azimuthal, and magnetic quantum numbers, 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, the order parameter φ of the condensate changes much slower with respect to the spatial change of the impurity's wave function ψ. Since the impurity's wave function is relatively narrow compared to the condensate and the impurity-medium interaction is weak, the impurity essentially 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 can be regarded as being constant and the impurity's wave function can be well approximated by a Gaussian function i.e. ψ(r) ≃ π m I ω I  To reveal the backaction of the impurity on the bosonic environment we provide the corresponding ground state density profiles of the condensed and the depleted part of the bath in Figs. 9(a) and (c), respectively. In the case of g IB > 0 ( g IB < 0), the condensate experiences an additional potential hump (dip) at the location of the impurity and eventually it seems to be slightly repelled from (pulled towards) the impurity as shown in Fig. 9 (b), where the deformation of the radial profile of the condensate from the case of zero impuritymedium interactions is provided. Moreover, in order to appreciate the role of the quantum depletion of the BEC environment we illustrate its depletion density in the absence and in the presence of the impurity in Figs. 9 (b) and (d), respectively. Apparently, the degree of the quantum depletion decreases (increases) [Figs. 9 (d)] for g IB > 0 (g IB < 0), a phenomenon that is accompanied by the deformation of the condensate density. The effect of the impurity on the quantum depletion of the condensate is summarized in the Table I. Inspecting the latter we can deduce that the quantum depletion decreases (increases) when the interaction is repulsive (attractive). This is a non-trivial result caused by the presence of the trap since in a uniform system the depletion always increases irrespectively of whether the interaction is positive or negative.

IV. CONCLUSIONS
In this work, we have discussed the existence and behavior of Fermi and Bose polarons that can be realized in ultracold quantum gases focusing on their backaction on the background medium. We have explicated three different diagrammatic approaches applicable to Fermi polarons in the homogeneous case. These include the TMA, the ETMA, and the SCTMA frameworks, where the ETMA considers induced two-body interpolaron interactions and the SCTMA includes two-and three-body ones. Importantly, we have explicitly derived the mediated two-and three-body interpolaron correlation effects as captured within the different diagrammatic approaches. Although these induced interactions are weak in the considered mass-balanced Fermi polaron systems, our framework can be applied to various systems such as mass-imbalanced Fermi polaron systems. Using this strong-coupling approach, we analyze the spectral response of the Fermi polaron in one-two-and threespatial dimensions at finite temperature. It has been shown that the spectral function of the minority component exhibits a sharp polaron dispersion in the low-momentum region but it is broadened for higher momenta. Moreover, we argue that the spectral response reflects the character of majority atoms forming a Fermi sphere while a strong interaction between the majority and the minority atoms induces a two-body bound state between a medium atom and an impurity atom. The presence of this two-body bound state becomes more important in lower dimensions.
Next, we present the mean-field treatment of trapped Bose polarons and analyze the role of quantum depletion identified by the deformation of the background density within the framework of Bogoliubov theory of excitations. A systematic investigation of the latter enables us to deduce that the repulsive (attractive) impurity-medium interaction, giving rise to repulsive (attractive) Bose polarons, induces a decreasing (increasing) condensate depletion captured by the deformation of the distribution of the host. This effect is a consequence of the presence of the external confinement since for a homogeneous background the quantum depletion 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 realizations. In particular, the effect of finite temperatures and the impurity concentration on the 2D Fermi polaron spectral response is expected to play a significant role close to the Berezinskii-Kosterlitz-Thouless transition of molecules [108]. Furthermore, the backaction of the impurities on the medium when considering dipolar interactions may affect the density collapse of the medium at strong impurity-medium attractions [109].