Multiple polaron quasiparticles with dipolar fermions in a bilayer geometry

Over the last two decades, significant progress has been made in manipulating ultracold gases of dipolar atoms and molecules. The surge in experimental activity in this field is motivated by the expectation that the anisotropic and long-range nature of dipole-dipole interactions can lead to exotic states of matter Lahaye et al. (2009); Baranov et al. (2012); Chomaz et al. (2022). Significant progress has already been made with dipolar gases of highly magnetic atoms such as Cr Griesmaier et al. (2005); Stuhler et al. (2005), Dy Lu et al. (2011, 2012), and Er Aikawa et al. (2012, 2014a), where the achievement of quantum degeneracy has allowed the investigation of droplets, supersolids and other quantum phenomena Böttcher et al. (2020). However, in order to access the regime of strong dipole-dipole interactions, one requires other cold-atom platforms such as Rydberg atoms Löw et al. (2012) or heteronuclear molecules Moses et al. (2017); Shaffer et al. (2018).

Fermionic polar molecules in layered geometries are particularly promising for realizing quantum phases with strong and tunable dipolar interactions. By confining dipolar molecules to two-dimensional (2D) layers, inelastic losses are suppressed, while the sign and strength of the dipolar interactions can be precisely controlled Valtolina et al. (2020). Degenerate 2D Fermi gases have already been achieved with KRb Marco et al. (2019); Tobias et al. (2020) and NaK Schindewolf et al. (2022); Duda et al. (2023), while other molecules, including LiCs Deiglmayr et al. (2008); Repp et al. (2013) and NaLi Park et al. (2023), are also being explored. Moreover, recent experiments with ultracold KRb molecules have demonstrated the possibility to image and control multiple layers individually Tobias et al. (2022), thus expanding the range of scenarios that can be explored with 2D dipolar gases.

From a theoretical standpoint, dipole-dipole interactions are expected to generate ordered phases of fermions in layered geometries. Of particular interest is the configuration where all dipole moments are aligned perpendicularly to the confining planes, such that the system has rotational symmetry. In the case of a single layer, intralayer superfluid $p$-wave pairing can be driven by dressing polar molecules with a microwave field Cooper and Shlyapnikov (2009); Levinsen et al. (2011). Furthermore, density-wave instabilities with dipolar fermions have been proposed Bruun and Taylor (2008); Yamaguchi et al. (2010); Sun et al. (2010), including the spontaneous appearance of a stripe phase Parish and Marchetti (2012) and Wigner crystallization Matveeva and Giorgini (2012) at sufficiently high densities or strong interactions. In the case of bilayers, in addition to density-wave instabilities Marchetti and Parish (2013), new interlayer bound pairs can arise due to the attractive part of the dipolar interaction Klawunn et al. (2010a); Potter et al. (2010); Pikovski et al. (2010); Klawunn et al. (2010b). Such pairing and associated interlayer superfluidity has been studied in the case of both balanced Bruun and Taylor (2008); Pikovski et al. (2010); Mazloom and Abedinpour (2018) and imbalanced populations Mazloom and Abedinpour (2017). The latter includes the possibility of realizing a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) modulated pairing phase Lee et al. (2017). Most notably, the stability of the FFLO state can be enhanced by the long-range character of the interlayer dipolar interaction, where different partial waves contribute to the pairing order parameter.

In this work, we consider the limit of extreme population imbalance for dipolar fermions in a bilayer, as illustrated in Fig. 1. Specifically, we have an impurity problem, where a single dipolar particle in one layer interacts with a Fermi sea of identical dipolar fermions in a different layer. This so-called “Fermi-polaron” problem has previously been considered in Refs. Klawunn and Recati (2013); Matveeva and Giorgini (2013) for the case of dipoles in a bilayer geometry ¹¹1The “repulsive Fermi polaron” problem with dipolar fermions in a single layer geometry has been considered in Ref. Bombín et al. (2019)., and the general problem of an impurity in a 2D Fermi gas has been extensively studied in other 2D platforms such as ultracold atoms Zöllner et al. (2011); Parish (2011); Schmidt et al. (2012a); Ngampruetikorn et al. (2012); Parish and Levinsen (2013); Tajima et al. (2021) and doped semiconductors Sidler et al. (2016); Efimkin et al. (2021); Tiene et al. (2022); Huang et al. (2023). Our scenario can be readily realized with polar molecules; however, note that our setup is quite general and could in principle apply to other dipoles such as Rydberg atoms. We consider two possible solutions of this problem. In the first, we generalize the interlayer bound state between two dipoles to include the effects of an inert Fermi sea and how it blocks the occupation below the Fermi momentum. In the second, we consider the possibility of “polaron” quasiparticles, where the impurity is dressed by particle-hole excitations of the Fermi sea. Throughout, we compare our results with those previously obtained within a $T$-matrix formalism which focused on the limit of weak dipolar interactions Klawunn and Recati (2013).

By employing a variational Ansatz, we evaluate the polaron spectrum to reveal that it is characterized by a series of attractive polaron branches, where the number of branches increases when the distance between the two layers decreases or the dipole moment increases. We associate the appearance of these polaron branches to the series of two-body bound states that, similarly to the 2D hydrogen atom Yang et al. (1991), is characterized by a specific orbital angular momentum component and a principal quantum number. In the limit of vanishingly small density of fermions, the attractive polaron energies recover those of the dipole-dipole bound states. By evaluating the orbital angular momentum component of each polaron branch, we observe that the partial wave character of the branches evolve and interchange when we either increase the Fermi density or, at a fixed density, we increase the bilayer distance.

In contrast to the well-studied case of contact impurity-medium interactions Massignan et al. (2014); Scazza et al. (2022), a distinctive feature of finite-range dipole-dipole interactions is that the energy of the lowest energy polaron branch can either redshift, i.e., lower its energy, with increasing Fermi density, or blueshift. We find that this depends on the precise value of the dipolar strength, a quantity related to the specific value of the dipole moment and the layer separation. We explain this qualitative different behavior in terms of the contribution of hole scattering in the polaron formation, which we find is particularly important for the dipolar potential.

Further, we consider the possibility that the lowest dipole-dipole bound state spontaneously acquires a finite center of mass momentum when the density of the Fermi sea increases. Because of the long-range nature of the dipole-dipole interaction, this finite-momentum bound state mixes different orbital angular momentum components. We show that, while for small densities, $s$-wave pairing dominates close to the transition, for larger densities, the bound state acquires $p$- and $d$-wave components. These results agree with those found at finite but large imbalance in Ref. Lee et al. (2017).

The paper is organized as follows. In Sec. II we introduce the model of identical dipolar fermions in a bilayer geometry, and we discuss the relevant length and energy scales including their typical values in current experiments for either strongly magnetic atoms or heteronuclear molecules. Section III describes the properties of the two-body interlayer bound states, generalized to the case where a Fermi gas in one of the layers acts to block the occupation of states below the Fermi sea. In Sec. IV we describe the spectral properties of the Fermi polaron, while in Sec. V we show that the polaron spectral function can be probed analogously to radiofrequency spectroscopy by introducing an auxiliary layer to the system. Conclusions and perspectives are gathered in Sec. VI.

We investigate the configuration schematically represented in Fig. 1. A Fermi sea of dipoles (e.g., polar molecules) is confined in one layer, with index $\sigma=1$, where the dipole moments of the molecules are aligned perpendicularly to the plane by an external field. Additionally, a single dipolar molecule with the same perpendicular alignment occupies layer $\sigma=2$. The Hamiltonian describing the systems is (we set $\hbar=1$ and the system area $\mathcal{A}=1$):

where $\hat{c}_{{\bf k},\sigma}^{{\dagger}}$ ($\hat{c}_{{\bf k},\sigma}$) is the creation (annihilation) operator of a fermionic dipole with momentum ${\bf k}$ in layer $\sigma$. Dipoles in different layers have the same mass $m$ and their kinetic energy is $\epsilon_{\bf k}=k^{2}/2m$.

In writing Eq. (1) we have implicitly assumed that the intralayer correlations are weak, such that the gas in layer $1$ is in a Fermi liquid phase where it can be treated as approximately non-interacting. This neglects the possibility of density instabilities which can occur in a (perpendicularly alligned) 2D dipolar Fermi gas with large dipole moments or high densities Parish and Marchetti (2012).

On the other hand, the second term in the Hamiltonian (1) describes the dipolar interaction between a molecule in layer $1$ and one in layer $2$. The interlayer dipolar potentials in real and momentum space are, respectively, given by Pikovski et al. (2010); Li et al. (2010); Klawunn et al. (2010b):

where ${\mathbf{r}}$ is the planar separation. Here $d$ is the layer separation and $D^{2}$ is the dipolar interaction strength. Out of these variables it is profitable to introduce a dimensionless dipolar strength

where $E_{0}\equiv 1/md^{2}$ is the energy scale associated with the layer separation. We see that $U_{0}$ increases by either increasing the dipolar interaction strength or by moving the two layers closer to each other.

The interlayer dipolar potential is plotted in Fig. 2 in real and momentum space. As expected, in real space the potential is attractive at short distances $r\lesssim d$, where the dipoles are effectively arranged head-to-tail, while it is repulsive at large distances $r\gtrsim d$, where the dipoles are arranged side-by-side. Note that the interlayer potential has a vanishing zero-momentum contribution, i.e.,

This implies that the two-dipole bound state becomes very shallow when $U_{0}\to 0$ Klawunn et al. (2010b), as discussed further in Sec. III.

The dipolar interaction strength $D^{2}$, which has the dimensions of energy times volume, is related to either the permanent magnetic dipole moment $\mu_{m}$ of a magnetic atom or to the dipole moment $D_{e}$ of an atom or a molecule induced by an electric field Lahaye et al. (2009):

where $\mu_{0}$ is the vacuum permeability and $\epsilon_{0}$ the vacuum permittivity. Usually heteronuclear molecules with induced electric dipole moments display much stronger dipole-dipole interactions than atoms with a permanent magnetic moment. In order to quantify the dipolar interaction, it is convenient to introduce the dipolar length $a_{dd}$ Lahaye et al. (2009); Chomaz et al. (2022) ²²2Note that Refs. Lahaye et al. (2009); Chomaz et al. (2022) use a different definition of $C_{dd}$ but the same definition of $a_{dd}$: $C_{dd}^{\text{Lehaye}}=4\pi C_{dd}^{\text{Chomaz}}=4\pi D^{2}=12\pi\frac{a_{dd}}{m}$. :

For heteronuclear molecules, $a_{dd}$ can be up to 3 orders of magnitude larger than for magnetic atoms. Typical values of $a_{dd}$ for the highly magnetic atom ¹⁶⁴Dy and for the heteronuclear molecules KRb and LiCs are given in Table 1. Furthermore, in current experiments, typically $d\simeq 500$ nm; however very recently a new super-resolution technique which localizes and arranges dipolar molecules on a sub-$50$ nm scale has been implemented Du et al. (2023). The corresponding values of $U_{0}$ that can be achieved for both highly magnetic atoms and heteronuclear molecules are listed in Table 1. Throughout this work we consider the dipoles to be structureless fermions such that we do not make a distinction between magnetic atoms or heteronuclear molecules.

The density $n$ of dipoles in layer $1$ is related to the Fermi momentum $k_{F}$ by

From this, we can define a many-body dimensionless parameter $U_{F}$ analogous to the dipolar strength $U_{0}$:

This dimensionless interaction strength characterizes the extent of many-body correlations in the system. As discussed above, we neglect the intralayer interaction and assume an ideal Fermi gas in layer $1$. Strictly speaking, this requires us to consider sufficiently small values of $U_{F}$ such that there are no ordered phases. Specifically, in the case of perpendicular orientation of the dipole moments, it has been found that the translational symmetry is broken for $U_{F}\gtrsim 6$ towards the appearances of a stripe phase Parish and Marchetti (2012), while Wigner crystallization can occur for $U_{F}\gtrsim 25$ Matveeva and Giorgini (2012). Nevertheless, to comprehensively characterize bound-state and polaron properties in the presence of a Fermi gas, we must extend our investigation to higher density values. Therefore, to preserve the assumption of an ideal Fermi gas in layer $1$, we will implicitly assume that there is a small but finite temperature that induces the melting of the strongly correlated phases without strongly affecting the impurity physics.

Typically, the density of the 2D Fermi gas in experiments is $n\simeq 10^{8}$ cm^-2. This leads to values of $U_{F}$ listed in the fourth column of Table 1. We furthermore list typical values for the dimensionless parameters $E_{F}/E_{0}$ and $1/k_{F}d$ at different bilayer separation $d$ in Table 2.

Due to the attractive part of the dipolar interaction, two dipoles can form an interlayer bound state. In this section, we discuss the properties of this two-body bound (dimer) state, generalized to the case where the Fermi gas in layer $1$ is inert and acts to block the occupation below the Fermi momentum $k_{F}$. We thus consider a general two-body state with a center-of-mass momentum ${\bf Q}$ described by:

where the sum over the relative momenta ${\bf k}$ is restricted by Pauli blocking, $k>k_{F}$, while $|FS\rangle\equiv\prod_{{\bf q}<k_{F}}\hat{c}_{{\bf q},1}^{{\dagger}}|0\rangle$ describes the Fermi sea in layer 1, and $\eta_{{\bf k}}^{({\bf Q})}$ is the two-body wavefunction. The energies $E$ can then be found by solving the corresponding Schrödinger equation:

which can be readily solved by numerical diagonalization.

Before discussing the solution of Eq. (8), it is useful to classify the dimer states according to their orbital angular momentum component. First, if the impurity momentum $Q=0$, the system is rotationally symmetric, and angular momentum is a good quantum number. Furthermore, when $E_{F}=0$, the center of mass and relative motion decouple, and therefore the energy at finite $Q$ is simply related to the energy at $Q=0$ via $E^{(Q)}=E^{(Q=0)}+Q^{2}/4m$, allowing us to take advantage of the rotational symmetry at $Q=0$. The presence of the Fermi sea complicates matters because then the center of mass motion no longer decouples. Thus, for a dimer state where both ${\bf Q}$ and $E_{F}$ are finite, the system is no longer rotationally invariant and orbital angular momentum is not conserved.

To proceed, we expand the dimer wavefunction as a Fourier series over the orbital angular momentum basis $e^{i\ell\varphi}$:

where $\varphi$ is the angle between ${\bf k}$ and ${\bf Q}$. The dimer Schrödinger equation (8) now reads

Here, we have taken the continuum limit, $\sum_{{\bf k}^{\prime}}\to\int\frac{d{\bf k}^{\prime}}{(2\pi)^{2}}$ and decomposed the interlayer potential in the orbital angular momentum basis by using the fact that the potential is diagonal in angular momentum, i.e.,

where ${\bf k}=(k,\varphi)$ and ${\bf k}^{\prime}=(k^{\prime},\varphi^{\prime})$. Note that the potential $\tilde{V}(k,k^{\prime},\ell)$ is real.

As expected, Eq. (10) becomes diagonal in $\ell$ when $Q=0$ since, in this limit, the orbital angular momentum is a good quantum number. Note also that, because the potential $\tilde{V}(k,k^{\prime},\ell)$ is symmetric under the exchange $\ell\mapsto-\ell$, the eigenvectors for $\ell\neq 0$ can either be symmetric or antisymmetric solutions:

where $\tilde{\eta}_{k0}^{(+,Q)}=\tilde{\eta}_{k0}^{(Q)}$, while $\tilde{\eta}_{k0}^{(-,Q)}=0$. In terms of these, the Schrödinger equation reads ($\ell\geq 0$):

One can see that the Schrödinger equation (10) now becomes block diagonal using the symmetric and antisymmetric dimer wavefunctions, i.e., the equations for $\tilde{\eta}_{k0}^{(Q)}$ and $\tilde{\eta}_{k\ell}^{(+,Q)}$ decouple from those for $\tilde{\eta}_{k\ell}^{(-,Q)}$, the former $l=0$ states being in general the lowest energy solutions ³³3Note that, as mentioned previously, the Schrödinger equation becomes diagonal in $\ell$ when either $Q=0$ or $E_{F}=0$, such that symmetric and antisymmetric solutions become degenerate..

In the following two subsections, we first analyze the two-body limit, i.e., the limit $E_{F}\to 0$ where there is a single particle in each of layers $1$ and $2$, and we study the appearance of additional bound dimer states when the interlayer dipolar strength $U_{0}$ increases. Then, in Sec. III.2, we consider the effect of a finite density of fermions in layer $1$ and how this can lead to the dimer spontaneously acquiring a finite center of mass momentum.

We now analyze the spectral properties of the polaron formed by the dipolar impurity in layer $2$ which is dressed by particle-hole excitations of the Fermi sea of dipoles in layer $1$. To this end, we employ a variational ansatz Chevy (2006) describing a polaron with zero center-of-mass momentum $Q=0$ as the superposition between the bare impurity weighted by the variational parameter $\phi_{0}$ and a single particle-hole excitation, described by $\phi_{{\bf k}{\bf q}}$:

Here, we use a notation where ${\bf k}$ is the momentum of the particle states ($k>k_{F}$) and ${\bf q}$ of the hole states ($q<k_{F}$). The polaron state is normalized so that $1=|\phi_{0}|^{2}+\sum_{{\bf k}{\bf q}}|\phi_{{\bf k}{\bf q}}|^{2}$.

The variational ansatz in Eq. (19) has previously been successfully employed to describe the impurity problem in different contexts, including ultracold atoms Zöllner et al. (2011); Parish (2011); Schmidt et al. (2012a); Ngampruetikorn et al. (2012); Parish and Levinsen (2013) and doped semiconductors Sidler et al. (2016); Efimkin et al. (2021); Tiene et al. (2022); Huang et al. (2023). In the case of contact interactions, truncating the dressing of the Fermi sea to a single particle-hole excitation has been demonstrated to be an excellent approximation, with an almost exact cancellation of higher order contributions Combescot and Giraud (2008).

Previous work on the dipolar case Klawunn and Recati (2013) employed a $T$-matrix approach to evaluate the lowest attractive polaron branch energy, but the entire spectrum of excitations and the important role played by excited dimer states have not previously been considered. Furthermore, we find that there are important qualitative differences between the variational ansatz (19) and the $T$-matrix approach. This is unlike the case of a pure contact interaction, where the variational ansatz in Eq. (19) is completely equivalent to the $T$-matrix approach within a ladder approximation Combescot et al. (2007). This highlights the important role played by the longer-range parts of the dipolar interaction potential, despite the dipole-dipole interaction formally corresponding to a short-range interaction in the sense that one can define an asymptotic region and an associated 2D scattering length Baranov et al. (2011). We discuss in App. B the precise relation between the ansatz (19) and the $T$-matrix approach employed in Ref. Klawunn and Recati (2013).

By minimizing the expectation value $\langle P_{3}|(\hat{H}-E)|P_{3}\rangle$ with respect to the variational parameters $\phi_{0}$ and $\phi_{{\bf k}{\bf q}}$, the polaron spectral properties can be found by solving the eigenvalue problem

where $E_{{\bf k}{\bf q}}=\epsilon_{{\bf k}}-\epsilon_{{\bf q}}+\epsilon_{{\bf q}-{\bf k}}$ and where we have omitted the terms that are zero because $V_{\bf 0}=0$. The polaron spectral function $A(\omega)$ can be evaluated as usual from the impurity Green’s function $G(\omega)$ Mahan (2000):

Similarly to the dimer problem, it is profitable to project the eigenvalue problem (20) onto an eigenbasis of the orbital angular momentum. This simplifies the numerical solution considerably, because we will see that the polaron branches are characterized by a small number of partial waves $\ell$. Thus, we consider the following Fourier series of the dimer-hole wavefunctions:

The eigenvalue equations in (20) now read:

where we have employed Eq. (11). As for the dimer wavefunction, it is convenient to introduce symmetric and antisymmetric solutions for the exchange $\ell\mapsto-\ell$:

where $\tilde{\phi}_{kq0}^{(+)}=\tilde{\phi}_{kq0}$ and $\tilde{\phi}_{kq0}^{(-)}=0$. Restricting to $\ell\geq 0$, the eigenvalue equations become

We thus find that the equations for symmetric and antisymmetric dimer-hole wavefunctions decouple. Most importantly, the impurity wavefunction $\phi_{0}$ only couples to the symmetric dimer-hole wavefunction $\tilde{\phi}_{kq\ell}^{(+)}$. Thus, the antisymmetric solutions will lead to additional eigenvalues for the polaron problem but these will all have zero spectral weight. As such, we can halve the degrees of freedom in the problem by neglecting the antisymmetric sector.

By discretizing the eigenvalue equations on a Gauss-Legendre grid in $k\in(k_{F},\infty)$ and $q\in(0,k_{F})$ and including a finite number of values $\ell$, we numerically diagonalize Eq. (24) to find eigenvalues and eigenvectors and thus evaluate the polaron spectral function according to Eq. (21). We have checked that all our results are numerically converged with respect to the number of points in the momentum grids as well as in the number of $\ell$ values. It is interesting to note that when $\omega<0$ we need only very few, around 4, values of $\ell$ to obtain converged results in the considered parameter regime, while to describe the scattering states we find that $9$ values of $\ell$ are sufficient.

We now discuss how one can experimentally probe the spectral function. Our proposal is inspired by radiofrequency spectroscopy Punk and Zwerger (2007), where one can inject (eject) the impurity by driving transitions from (to) an auxiliary hyperfine state that, in the ideal case, does not interact with the medium. Similarly, we suggest to use an auxiliary layer ($\sigma=3$) that the impurity can tunnel into/from — see Fig. 15. We furthermore assume that interactions between the dipolar particles can occur only between layers $1$ and $2$, while tunneling can only occur between layers $2$ and $3$. This situation can, e.g., be achieved if a potential barrier is present between layers $1$ and $2$, while layer $3$ is further away from both layers. We also note that, in practice, a small 1-3 interaction can be taken into account by considering the initial state to be a weakly dressed attractive polaron, similarly to what is routinely done in the case of RF spectroscopy for contact interactions, see, e.g., Ref. Baym et al. (2007).

We thus have to extend the Hamiltonian (1) to include two additional terms, $\hat{H}_{3}$ and $\hat{H}_{t}$, that describe, respectively, the kinetic energy of the particles in layer $3$, and the tunneling between layers 2 and $3$:

Note that in the kinetic term $\hat{H}_{3}$ we have included a “detuning” energy $\Delta$, which represents the difference between the energy minima of the confining potentials in the $z$-direction.

The additional terms in the Hamiltonian exactly match those used in radiofrequency spectroscopy on impurities Liu et al. (2020), where $\Delta$ plays the role of the radiofrequency detuning, and $t$ the role of the Rabi coupling. Therefore, this auxiliary layer provides access to the spectral properties. To be specific, within linear response the tunneling rate can be evaluated using Fermi’s golden rule between an initial state consisting of one particle in layer $3$ plus a Fermi sea of dipoles in layer $1$, $\hat{c}_{{\bf 0},3}^{\dagger}|FS\rangle$, while the final state is the polaron state $|P_{3}\rangle$ (19). This gives

Thus, measuring the tunneling rate as a function of the energy difference $\Delta$ between the two lowest eigenenergies of the $z$-confining lattice potentials is equivalent to measuring the polaron spectral function $A(\omega)$ (21b) evaluated in this work.

In this work we have studied a gas of dipolar fermions in a bilayer geometry in the limit of extreme imbalance, i.e., a single dipole in one layer interacting with a Fermi sea of dipoles in a different layer. We analyze two different yet connected solutions of this problem. We first investigate the properties of the interlayer dimer bound state, generalized to include the Pauli blocking effect from the inert Fermi sea. We find a series of bound states, characterized by the orbital angular momentum and the principal quantum number, binding for increasing dipolar interaction strength or decreasing bilayer distance. For the dimer ground state, we determine the spontaneous emergence of a finite center of mass momentum when increasing $E_{F}$ above a threshold value. The finite momentum dimer corresponds to the large imbalance limit of the FFLO state studied in Ref. Lee et al. (2017). We find that this state has a mixed partial wave character, including $s$-, $p$-, and $d$-wave contributions.

The other solution we analyze is the many-body polaron state, where the presence of the impurity in one layer leads to particle-hole excitations of the Fermi sea in the other layer. We derive the polaron spectral properties by employing a single particle-hole variational ansatz, and we propose that the tunneling rate of the impurity from an additional auxiliary layer can be employed to experimentally access the spectrum. We find that the polaron spectrum is characterized by a series of attractive polaron branches which we trace back to the dimer bound states. At small $E_{F}$, the polaron energies and their orbital character recover those of the dimer states. However, both energies and orbital angular momentum components evolve and interchange with $E_{F}$. We find that the energy of the ground-state $1s$ polaron branch evolves with $E_{F}$ in a qualitative different way depending on the value of $U_{0}$. We explain this distinctive property of finite-range dipole-dipole interactions in terms of whether hole scattering is negligible or not in the polaron formation. We characterize the transfer of oscillator strength from the repulsive branch to the series of attractive branches in terms of their partial wave character and their distance in energy from the repulsive branch.

In our model, we neglect the repulsive interactions between particles in the Fermi sea. As such, we neglect the possibility of strong intralayer correlations that could lead, at very low temperatures, to the spontaneous appearance of density modulated phases such as stripes Parish and Marchetti (2012) and Wigner crystals Matveeva and Giorgini (2012), which are predicted to occur for either strong enough dipolar interactions or large enough Fermi densities. While beyond the scope of our study, an exciting perspective of our work is the generalization of the polaron formalism to include the possibility of the impurity interacting with such strongly correlated phases Matveeva and Giorgini (2013), which could potentially leave signatures in the polaron spectral response. Indeed, this would mirror very recent experiments on exciton polarons in doped 2D semiconductor monolayers which have probed strongly correlated states of 2D electron gases, such as Wigner crystals Smoleński et al. (2021); Shimazaki et al. (2021), fractional quantum Hall states in proximal graphene layers Popert et al. (2022) and correlated-Mott states of electrons in a moiré superlattice Schwartz et al. (2021).

Another interesting perspective of our work would involve considering a configuration where the alignment of the dipoles is tilted at a slight angle relative to the normal direction. In this case, the anisotropy induced by the dipole-dipole interaction results in a distorted Fermi surface Bruun and Taylor (2008); Yamaguchi et al. (2010), consequently influencing the properties of the Fermi polaron and giving rise to spatial anisotropies, as illustrated in a three-dimensional setup by Ref. Nishimura et al. (2021). Indeed, deformations of the Fermi surface have already been experimentally observed in a three-dimensional degenerate dipolar Fermi gas composed of Er atoms Aikawa et al. (2014b). Furthermore, a tilted configuration is expected to stabilize the FFLO state (see, e.g., Ref. Kawamura and Ohashi (2022)) since it breaks the continuous rotational symmetry and thus suppresses the pairing fluctuations that destroy FFLO long-range order Shimahara (1998).

The research data underpinning this publication can be accessed at Ref. Tiene et al. (2024).