Anyonic molecules in atomic fractional quantum Hall liquids: a quantitative probe of fractional charge and anyonic statistics

We study the quantum dynamics of massive impurities embedded in a strongly interacting two-dimensional atomic gas driven into the fractional quantum Hall (FQH) regime under the effect of a synthetic magnetic field. For suitable values of the atom-impurity interaction strength, each impurity can capture one or more quasi-hole excitations of the FQH liquid, forming a bound molecular state with novel physical properties. An effective Hamiltonian for such molecules is derived within the Born-Oppenheimer approximation, which provides renormalized values for the effective mass, charge and statistics of such anyonic molecules by combining the finite mass of the impurity and the fractional charge and anyonic statistics of the quasi-holes. The anyonic statistics is shown to provide a long-range Aharonov-Bohm-like interaction between molecules. The resulting relative phase of the direct and exchange scattering channels can be thus extracted from the angular position of the interference fringes in the scattering cross section of a pair of colliding molecules. Different configurations providing direct and quantitative insight on the fractional charge and the anyonic statistics of quasi-hole excitations in FQH liquids are highlighted for both cold atoms and photonic systems.

We study the quantum dynamics of massive impurities embedded in a strongly interacting twodimensional atomic gas driven into the fractional quantum Hall (FQH) regime under the effect of a synthetic magnetic field. For suitable values of the atom-impurity interaction strength, each impurity can capture one or more quasi-hole excitations of the FQH liquid, forming a bound molecular state with novel physical properties. An effective Hamiltonian for such molecules is derived within the Born-Oppenheimer approximation, which provides renormalized values for the effective mass, charge and statistics of such anyonic molecules by combining the finite mass of the impurity and the fractional charge and anyonic statistics of the quasi-holes. The anyonic statistics is shown to provide a long-range Aharonov-Bohm-like interaction between molecules. The resulting relative phase of the direct and exchange scattering channels can be thus extracted from the angular position of the interference fringes in the scattering cross section of a pair of colliding molecules. Different configurations providing direct and quantitative insight on the fractional charge and the anyonic statistics of quasi-hole excitations in FQH liquids are highlighted for both cold atoms and photonic systems.

I. INTRODUCTION
The discovery of the fractional quantum Hall (FQH) effect in two-dimensional (2D) electron gases under a strong transverse magnetic field [1][2][3] is a cornerstone of modern physics. Not only did it pave the way towards the study of topological phases of matter [4] but also changed the paradigm of the boson-fermion dicotomy when the possibility of observing quasi-particles with fractional statistics (and fractional charge) in 2D systems was proposed, the so-called anyons [5][6][7][8][9][10]. Such exotic quasi-particles have been predicted to arise as emergent excitations of FQH fluids with different properties depending on the fluid density and the applied magnetic field [2]. In addition to their intrinsic interest as exotic quantum mechanical objects, in the recent years they have started attracting a lot of attention also for the crucial role that they are expected to play in the development of fault-tolerant quantum computers [11]. Even though shot-noise experiments confirmed the existence of fractionally charged quasi-particles in 2D electron gases in the FQH state [12], a clear signature of fractional statistics remains elusive [13,14].
In parallel to these studies in the electronic context, impressive developments in the experimental study of ultracold atomic gases [15] opened the door to the exploration of topological phases of matter using these highly controllable quantum systems [16]. Several protocols have been investigated to drive a 2D gas of ultracold atoms into the FQH regime. Conceptually, the most straightforward one relies on the Coriolis force experienced by neutral atoms set into rotation, which formally recovers the Lorentz force felt by charged particles in a * a.munozdelasheras@unitn.it magnetic field. Alternative strategies to induce effective Lorentz forces on atoms involve the application of suitable optical and magnetic fields to the atoms, so to associate a Berry phase to the particle motion and generate a synthetic magnetic field [17,18]. For growing values of the angular speed or of the synthetic magnetic field strength, the atoms are expected to turn into a stronglycorrelated FQH liquid state at sufficiently low temperatures [2,3,19]. Pioneering experimental investigations in this direction were reported in [20].
In the last decade, a new platform has emerged as a promising candidate to study many-body physics, including strongly correlated FQH liquids. Starting from microcavity polaritons in semiconductor microstructures, assemblies of photons in nonlinear optical devices are presently under active study as the so-called quantum fluids of light [21,22]. In addition to the effective mass (typically induced by the spatial confinement) and the binary interactions (mediated by the optical nonlinearity of the medium), recent developments have demonstrated synthetic magnetic fields and realized various topological models for light [23]. The first experimental study of the interplay of strong photon-photon interactions with a synthetic magnetic field was reported for a three-site system in [24]. The realization of a two-particle Laughlin state was presented in [25] using the giant nonlinearity of Rydberg polaritons and the synthetic magnetic field of a twisted optical cavity [26].
Simultaneously to these exciting experimental advances, theorists have started investigating new strategies to probe in an unambiguous way the anyonic nature of the excitations of quantum Hall fluids. A Ramsey-like interferometry scheme to detect the many-body braiding phase arising upon exchange of two anyons was proposed for a cold atom cloud in [27]. A related proposal exploiting the peculiarities of driven-dissipative photonic systems was presented in [28]. Spectroscopical conse-arXiv:2004.02477v1 [cond-mat.quant-gas] 6 Apr 2020 quences of the Haldane exclusion statistics were pointed out in [29] and soon translated to the photonic context in [30]. A subtle quantitative relation between the density profile of quasi-holes (QHs) and their anyonic statistics was theoretically put forward in [31,32] and numerically confirmed for discrete lattice geometries in [33]. Finally, random unitary techniques to measure the manybody Chern number were investigated in [34].
In analogy with polarons arising from the many-body dressing of an impurity immersed in a cloud of quantum degenerate atoms [35,36], a series of works [37][38][39] have anticipated the possibility of using impurity particles immersed in a FQH liquid to capture quasi-hole excitations (that is, flux tubes) and thus generate new anyonic molecules that inherit the fractional statistics of the quasi-hole. Observable consequences of the fractional statistics were pointed out in the fractional angular momentum of the impurities and, correspondingly, in their correlation functions [37]. An interferometric scheme to measure fractional charges by binding a mobile impurity to quasi-particles was proposed in [40]. Alternative models where heavy particles may acquire fractional statistics by interacting with phonons in the presence of strong magnetic fields and/or fast rotation were proposed in [41,42]. The transport properties of impurities embedded in a Fermi gas in a (integer) Chern-insulating state were recently studied in [43].
In the present paper, we take inspiration from the aforementioned theoretical works and from the highly developed experimental techniques that are available to address and manipulate single atoms in large atomic gases to theoretically illustrate how such anyonic molecules are a very promising tool to observe fractional statistics and shine new light on the microscopic physics of FQH fluids. In particular, we investigate the quantum mechanical motion of a few impurities inside an FQH fluid. Capitalizing on previous works, we provide a rigorous derivation of the effective molecular Hamiltonian based on a Born-Oppenheimer (BO) approximation [44,45] where the positions of the impurities play the role of the slow degrees of freedom and the surrounding fluid provides the fast ones. Whereas bare quasi-holes typically do not support motional degrees of freedom [46], the molecule is found to display a fully fledged spatial dynamics, with a mass determined by the impurity mass supplemented by a non-trivial correction due to the quasi-hole inertia.
Binding to the QH also modifies the effective charge of the impurity by including the Berry phase [47] that this latter accumulates during its motion in space. In the single impurity case the molecule then behaves as a free charged particle describing cyclotron orbits. Detailed information on the renormalized mass and on the fractional charge of the molecule can be extracted from the orbit radius.
In the two impurity case, the fractional statistics of the QHs results in a long-range Aharonov-Bohm-like interaction between the molecules. The consequences of this long-range topological interaction are illustrated in the simplest scattering process where two such objects are made to collide. For both hard-disk and dipolar interaction potentials, we calculate the differential scattering cross section for indistinguishable impurities, finding that for large relative momenta it features alternate maxima and minima due to the interference of direct and exchange scattering channels: analogously to textbook twoslit experiments, the interference pattern rigidly shifts when the statistical phase that the anyonic molecules acquire upon exchange is varied and it fully disappears when the exchange channel is suppressed by considering distinguishable impurities. An experiment of this kind would therefore allow to confirm the existence of particles beyond the traditional boson-fermion classification and to quantitatively measure the statistics of the QHs in a direct way.
The structure of the article is the following. In Sec. II we review the system Hamiltonian and in Sec. III we develop the rigorous Born-Oppenheimer framework that we employ to study the quantum dynamics of the anyonic molecules: In Subsec. III B we establish the single particle parameters of the anyonic molecule and in Subsec. III C we recover the interaction Hamiltonian between molecules. The theory of two-body scattering is presented in Sec. IV, where we summarize our predictions for the angular dependence of the differential scattering cross-section and we highlight the qualitative impact of the fractional statistics. Conclusions are finally drawn in Sec. V.

II. THE PHYSICAL SYSTEM AND THE MODEL
We consider a system of quantum particles confined to the two-dimensional x-y plane and formed by a small number N of mobile impurities of mass M immersed in a large bath of n N atoms of mass m in a FQH state. For simplicity, in what follows the former will be indicated as impurities, while the latter will be indicated as atoms. A transverse and spatially uniform synthetic magnetic field B = B u z is applied to the whole system (where u z is the unit vector in the z direction), and we consider that the impurities and the atoms possess effective (synthetic) charges Q and q, respectively. In the particular case in which the magnetic field is generated by rotating the trap around the z axis, the value of these quantities is set by the atomic masses and the rotation frequency of the trap ω rot [18,48], while the corresponding centrifugal potential can be compensated by a harmonic trap potential of frequency ω hc = ω rot , as done in [49]. Even though our discussion will be focused on atomic systems, all our conclusions directly extend to any other platform where quantum particles are made to experience a synthetic gauge field and strong interparticle interactions, for instance photons in twisted cavity set-ups where Landau levels [26] and Laughlin states [25] have been recently observed. In = 1 units, the system Hamiltonian then reads where We denote by r j and −i∇ rj the position and canonical momentum of the j-th atom, while R j and −i∇ Rj represent those of the j-th impurity. A(r j ) = B(−y j /2, x j /2, 0) and A(R j ) = B(−Y j /2, X j /2, 0) are the vector potentials corresponding to the synthetic magnetic field (B = ∇ × A) at the positions of atoms and impurities, respectively.
The strength of the contact binary interaction between atoms is quantified by the g aa parameter, whereas v ia and v ii denote the impurity-atom and impurity-impurity interaction potentials, respectively. At low temperatures and for sufficiently strong repulsive atom-atom interactions g aa , when the rotation frequency (or in general the synthetic magnetic field) is large enough, the number of vortices n v becomes comparable with the number n of atoms and the atomic gas enters the so-called FQH regime described by a rational value of the filling fraction ν = n/n v [2,3,48]. This incompressible state is characterized by excitations with fractional charge and statistics (quasi-holes and quasi-particles).
As it was first anticipated in [37][38][39] a repulsive interaction potential v ia between the impurities and the atoms leads to the pinning of quasi-hole excitations at the impurities' positions. As a result, quasi-holes adiabatically follow the motion of the impurity forming composite objects that can be regarded as anyonic molecules. By looking at the density pattern of quasi-hole excitations shown in [50,51], we anticipate that the number of quasi-holes pinned by each impurity can be controlled via the strength of v ia : a stronger and/or longer-ranged interaction will provide space for more quasi-holes. Even though the generalization to many quasi-holes is straightforward, in this work we will focus on the case of a single quasi-hole per impurity.
Finally, the impurities are assumed to be separated enough in space to study the few-body physics of a few molecules. In what follows we will consider impurityimpurity potentials v ii of a far larger range than the atom-atom and impurity-atom interactions. In particular we will focus on interaction potentials with hard-disk or dipolar spatial shapes.

III. THE BORN-OPPENHEIMER APPROXIMATION
Several authors have theoretically addressed the quantum mechanics of mobile impurities immersed in FQH fluids and have written effective Hamiltonians for the motion of the resulting charge-flux-tube complexes [37][38][39][40][41][42][43]. Most such treatments were however based on heuristic models of the binding mechanism: while this was sufficient to get an accurate answer for the synthetic charge and the fractional statistics, it did not provide a quantitative prediction for the mass of the anyonic molecule: this is in fact determined by the bare mass of the impurity, supplemented by a correction due to the inertia of the FQH quasi-holes.
To fill this gap, in this Section we will summarize a rigorous approach to this problem. The reader that is already familiar with such effective Hamiltonians and is not interested in the technical details and in the quantitative value of the parameters can jump to the experimental remarks in the final Subsec. III B 3 and then move on to the scattering theory in Sec. IV.

A. General framework
Our theoretical description is based on a Born-Oppenheimer formalism in which we treat the impurities' positions as the slowly-varying degrees of freedom, while those of the surrounding atoms play the role of the fast ones [44]. For each position of the impurities, the atoms are assumed to be in their many-body ground state, which contains quasi-holes at the impurities' positions to minimize the repulsive interaction energy. Given the spatial coincidence of the impurity and the quasi-hole, in the following the positions of the resulting molecules will be indicated with the same variables R i . While our approach is known to be exact for fixed impurities, it extends to moving impurities as long as their kinetic energy is smaller than the gap between the quasi-hole state and the first excited state.
Under this approximation, the total wave function can be factorized as where the wave function χ({R i }, t) describes the quantum motion of the impurities and the atomic wave function ϕ that includes the kinetic and interaction energy of the atoms and the interaction potential between atoms and impurities.
In our specific FQH case, the atomic wave function can be written in terms of the magnetic length l B = 1/ √ qB and the complex in-plane coordinates z = x − iy of the atoms as a many-quasi-hole wave function of the form The positions of the quasi-holes are parameterically fixed by the (complex) positions Z = X − iY of the impurities, while the last factor φ L is the well-known Laughlin wave function of the FQH state [52], In Eq. (9), the normalization constant N is chosen to ensure the partial normalization condition Provided that the impurity lives in the bulk of the atomic cloud far from its edges, the energy BO of the Born-Oppenheimer ground state is independent of the impurity position R and can be safely neglected.
As we will discuss in full detail in the following subsections, the dynamics of the anyonic molecules will be governed by an effective Hamiltonian acting on the molecule wave function χ(R), that combines the properties of impurities and quasiholes.
Within this picture, each molecule then features a mass M -only approximately equal to the one of the impurities, see Sec. III B 1-and a total charge Q = Q − νq resulting from the sum of the bare charge Q of the impurity and the one −νq of the quasi-hole -see Sec. III B 2. These values are of course only accurate as long as the impurities are located in a region of constant density of the atomic cloud, that is in the bulk of an incompressible FQH phase. Under this condition, both the BO energy resulting from the interaction with the atoms (0) BO and the scalar potential arising in the BO approximation give spatially constant energy shifts that can be safely neglected.
In addition to these single-particle properties, the molecules inherit the interaction potential V ii ({R j }) between the impurities and experience a Berry connection A stat,j ({R k }) that now depends on the position of all molecules and encodes their quantum statistics. For the Abelian FQH states under investigation here, we will see in Sec. III C that this can be summarized by a single statistical parameter α = α i + ν, which indicates that the phase picked upon exchange of two molecules is exp(iπα). The intrinsic statistics of the impurities is encoded in the α i parameter, bosons (fermions) corresponding to α i = 0 (α i = 1). The statistical parameter of the quasi-hole is instead fixed by the filling fraction ν of the FQH fluid. Of course, if more N qh > 1 quasi-holes are pinned to the same impurity, the statistical parameter would have the quadratic dependence α = α i + N 2 qh ν [2]. Finally, note that we are restricting our attention to anyonic molecules that are separated enough in space for their internal structure not to be distorted by the interactions with the neighboring molecules. This is expected to be an accurate approximation if the inter-impurity distance is much larger than the range of the atom-impurity potential and the internal size of the quasi-hole -typically of the order of the magnetic length B [2]. Under this approximation, the values of the renormalized mass and of the synthetic charge that we obtain for single impurities directly translate to the many-impurity case.

B. Single Impurity
In this section we will investigate the parameters in the effective Hamiltonian (12) that control the single-particle physics of the molecules, namely the renormalized mass M and charge Q. A simple experimental configuration to extract these values will also be proposed at the end of this subsection.

Mass renormalization
A crucial, yet often disregarded feature of the BO approximation is the renormalization of the effective mass of the slow degrees of freedom. In molecular physics such a renormalization affects the effective mass of the nuclei dressed by the electrons and is essential to guarantee consistency of the description [45,47]. In our case it concerns the change of the effective mass of the impurity when this is dressed by the quasi-hole excitation in the surrounding FQH fluid. As far as we know, this feature was always overlooked in previous literature, even though it may give a quantitatively significant bias to observable quantities such as the effective magnetic length considered in [37].
In order to obtain a quantitative estimate for the effective mass M, we generalize the molecular physics approach of Ref. [45] by including the synthetic magnetic field in the formalism. As it is discussed in detail in Appendix A, one needs to include the first perturbative correction to the BO adiabatic approximation, which amounts to taking into account the distortion of the quasi-hole profile due to the motion of the impurity. To this purpose we expand ϕ R (r, t) ϕ R (r, t), where the BO wave function ϕ (0) R (r) was obtained as a ground state of the Hamiltonian (8) in the presence of a single impurity at R and ϕ (1) R (r, t) is the first order correction. Note that the BO wave function ϕ (0) R (r) only depends here on the coordinate differences r − R.
Following the theory of Ref. [45], the mass tensor of the molecule is then given at first order by where M is the 2 × 2 unity matrix multiplied by the bare impurity mass and the correction is such that the corresponding kinetic energy recovers the increase in the BO energy due to the motion of the impurity. This latter induces in fact a correction ϕ R (r, t) to the atomic wave function, which is obtained at the lowest perturbative level in the impurity speed v = (v X , v Y ) by applying the inverse of the fast Hamiltonian to the gradient of the atomic wave function with respect to the in-plane coordinates of the impurity α = {X, Y }.
In physical terms, this correction is such to recover the evolution of the BO wave function ϕ R (r, t) due to the spatial displacement of the impurity under the action of the fast BO Hamiltonian H BO . In our case, the correction to the mass tensor is diagonal and its magnitude ∆M is dominated by the lowest excited state of H BO , which corresponds to a chiral ∆L = −1 oscillation of the quasi-hole around the impurity [53]. The result for ∆M is then on the order of where ∆ω −1 is the frequency of the quasi-hole oscillation mode described above and T is a numerical factor quantifying the many-body matrix element of the perturbation. Its calculation requires a detailed study of the many-body wave function of the FQH state and goes beyond this work. On the other hand, in Appendix B, we show how the excitation energy ∆ω −1 grows for stronger impurity-atom interaction potentials V ia which reinforce the internal rigidity of the composite quasi-hole-impurity. Quantitatively, the excitation energy ∆ω −1 is at most on the order of a fraction of the bulk many-body gap above the fractional quantum Hall state, namely a fraction of the atom-atom interaction energy scale V 0 = g aa /2l 2 B [51]. Since this latter is a fraction of the cyclotron energy in the magnetic field, the relative mass correction ∆M/M will be equal to a (possibly quite large) numerical factor times the (small) ratio m/M of the atomic and the impurity masses. To reduce the mass correction and make the BO approximation more accurate, it is then natural to choose heavy impurities in a FQH fluid of light atoms or to reinforce the atomic interactions g aa energy V 0 using a Feshbach resonance [54].

Synthetic charge
As in the previous subsection, we consider the simplest situation in which a single quasi-hole is bound to the single impurity present in the system at a position R. In this case the effective molecule Hamiltonian takes the form where is the Berry connection related to the quasi-hole motion across the FQH fluid, which enters the equation above in the form of an effective vector potential. The effective scalar potential is instead equal to Exception made for a constant energy shift, the scalar potential can be safely neglected as long as the impurity lives in the bulk of the (incompressible) FQH fluid where the fluid density is -to a high precision-constant.
To explicitly calculate the Berry connection A(R), we make use of the plasma analogy [2,52]. To this purpose, the atomic wave function ϕ (0) R needs to satisfy the partial normalization condition in Eq. (11), i.e.
with |Z the unnormalized single quasi-hole state (9) and its normalization. Use of the plasma analogy relies on identifying the exponential in Eq. (21) with the Boltzmann factor of a 2D classical plasma at a ficticious temperature T given by β = 1/T = 2ν and requiring its neutrality. The free energy of the plasma can be considered independent of the position of the impurities as long as its Debye screening length is sufficiently short compared with the inter-impurity distance, which is obviously guaranteed in the single impurity case. The normalization constant then takes the form where C is an irrelevant constant for the calculation of the effective potentials. For a single impurity, the Berry connection gets a contribution from the braiding of the quasi-hole through the atomic cloud The expression gets physically transparent if one notes that A(R) = −νqA(R), so that the effective singlemolecule Hamiltonian can be recast in the compact form in terms of the effective charge that results from the bare charge Q of the impurity and the fractional charge −νq of the quasi-hole that is bound to it.

Experimental remarks
Since the Hamiltonian (24) describes a free particle in a magnetic field, we can envisage a simple experiment to measure the fractional charge of the molecule based on its motion in a cyclotron orbit, as sketched in Fig. 1. Once the molecule receives a momentum kick p (e.g. by applying a time-dependent force to the impurity), it starts describing a cyclotron orbit. For a given value of the momentum kick, the molecule mass M can be directly obtained from the actual speed via p = Mv. Since only the product of charge and magnetic field actually matters for synthetic fields, the interesting parameter to characterize the fractional charge is the ratio Q/Q. Using the formula for the cyclotron radius, it is straightforward to extract Q/Q by comparing the cyclotron radius for a molecule immersed in the FQH fluid with the one of the bare impurity in the absence of the surrounding FQH fluid. The relation (25) allows then to relate the observed charge Q to the fractional charge of the quasi-holes in the FQH fluid and, thus, to the filling parameter ν. If the synthetic magnetic field is generated by rotating the system, its calibration is made even simpler by the fact that the product qB (QB) is determined by the rotation speed Ω via qB = mΩ (QB = M Ω) [18,19]. Taking advantage of the different nature of the impurity particle as compared to the atoms forming the FQH fluid, reconstruction of the trajectory of the anyonic molecule can be done by imaging the position of the impurity at different evolution times after a deterministic preparation at a given location and a deterministic kick, e.g. via an external potential. In this context, a heavy mass of the impurity gives the further advantage of allowing for a more accurate definition of its position and velocity against Heisenberg indetermination principle.
On the other hand, forming the bound impurity-quasihole state may be itself a non-trivial task since quasiholes are associated to a global rotation of the FQH fluid. In [33,51], it was shown that the quasi-hole state naturally forms as the ground state in the presence of the impurity potential provided the atomic cloud is able to exchange angular momentum with the external world. Alternatively, a quasi-hole can be created by inserting a localized flux through the cloud, and then introducing the impurity particle at its location [55,56]. Finally, a speculative strategy yet to be fully explored may consist of inserting the impurity into the FQH fluid through its edge: provided the impurity's motion is slow enough, one can reasonably expect that it will be energetically favourable for the impurity to capture a quasi-hole from the edge and bring it to the bulk of the FQH cloud.
Besides these technical difficulties, we anticipate that our proposed experiment will have great conceptual advantages over the shot noise measurements of electronic currents that were first used to detect charge fractionalization [12]. These experiments involve in fact complex mechanisms for charge transport and charge injection/extraction into/from the edge of the electron gas. On the other hand, we foresee that our proposed experiment has the potential to provide a direct and unambiguous characterization of the fractional charge of the quasi-hole excitations in the bulk of a fractional quantum Hall fluid.

C. Two impurities
After completing in the previous Section the calculation of the single-particle parameters M and Q, we now move on to the many-particle case. The two molecule case is already of particular interest as it allows to obtain information about the fractional statistics of the anyonic molecules. In the following we will focus on this case and we will leave the complexities of the three-and moreparticle cases [10] to future investigations.
As already stated, we assume that the two impurities are located in the bulk of the FQH cloud, far apart from the edges and they are well separated by a distance much larger than the magnetic length. The effective molecule Hamiltonian is now given by where the Berry connection now contains two terms, The former term is the extra magnetic field B q = ν/ 2 B due to the bound quasi-hole, as discussed in the previous section. The latter term depends on the relative position of the two impurities, with R rel = (X rel , Y rel ) = R 1 − R 2 the relative position between the two molecules: each impurity experiences the vector potential corresponding to ν quanta of magnetic flux spatially localized on the other impurity. Since ∇ × A stat,j = 0, there is no magnetic field involved in the interaction between spatially separated impurities and the effect can be viewed as an Aharonov-Bohm-like interaction [57]. Since the impurities are assumed to be located in the bulk of the FQH cloud, the effective scalar potential only gives a global energy shift that can be safely neglected.
Grouping the extra magnetic field B q with the one felt by the bare impurity as done in the previous section, we can write the Hamiltonian in the compact form in terms of the effective charge Q = Q − νq of each molecule. According to this Hamiltonian, the molecules interact via the interaction potential V ii between the bare impurities and via the Aharanov-Bohm interaction encoded by the vector potential A stat,j . Differently from the usual case of an externally applied vector potential, now the interaction vector potential A stat,j depends on the relative position R 1 − R 2 between the two molecules. Given the translational invariance of the configuration, we can separate the center of mass and the relative motion of the two molecules. Assuming a central impurityimpurity interaction V ii (R i − R j ) = V ii (R rel ), we define the reduced and the total mass as usual as the relative and center of mass position the corresponding momenta and vector potentials to be included in the center of mass and relative Hamiltonians The center of mass Hamiltonian (37) describes a free particle motion of total mass 2M and charge 2Q. On the other hand, the relative Hamiltonian (38) contains the uniform magnetic field experienced by the reduced charge Q/2 plus a non-trivial vector potential corresponding to ν quanta of magnetic flux localized at R rel = 0.
As it was discussed in the seminal works [5][6][7][8][9][10], the presence of this latter vector potential is the key feature that encodes the fractional statistics of the anyonic molecules. In the following of this work, we will study the effect of this vector potential onto the scattering cross section of two molecules. This is a measurable quantity that can serve as a probe of the statistical parameter of the molecules. In the case of a single quasi-hole pinned to each impurity, this is given by the sum α = α i + ν of the contribution α i = 0 (α i = 1) of the bare bosonic (fermionic) impurities plus the fractional one ν of the quasi-holes. Fig. 1, two indistinguishable anyonic molecules (green circles) formed by the binding of the same number of quasiholes to a pair of identical impurities in the bulk of a FQH fluid (blue region) are considered. The two molecules are given momentum kicks against each other (P1 and P2, respectively). Due to their indistinguishability, two scattering channels contribute to the differential scattering cross section at an angle φ, see Eq. (56): the two channels are labelled as direct (red, solid trajectories) and exchange (yellow, dashed ones) and involve a relative phase determined by the anyonic statistics. As one can guess from textbook two-slit interference, information about the statistics can be extracted from the global position of the interference fringe pattern. This is illustrated in the next figures.

IV. SCATTERING OF ANYONIC MOLECULES AND FRACTIONAL STATISTICS
In the previous Section, we have summarized the conceptual framework to study the quantum mechanical motion and the interactions of anyonic molecules. Based on this complete and flexible framework, we can now attack the core subject of this work, namely the observable consequences of the fractional statistics. As a simplest and most exciting example, we consider the differential cross section for the scattering of two anyonic molecules and, in particular, we will highlight a simple relation between the angular position of its maxima and minima and the fractional statistics.
To simplify our discussion, from now on we assume that the process underlying the synthetic magnetic field is designed in a way to have a vanishing effective charge Q = 0 of the molecules. Even though this condition is not naturally fulfilled in the laboratory reference frame, viewing the physical process from a rotating reference frame [17,18] with an angular frequency Ω around the z axis such that QA(R) + MΩ × R = 0 allows to compensate for the non-zero Q. This condition is beneficial to have rectilinear trajectories in the asymptotic states of the scattering molecules. The only vector potential remaining in Eq. (29) will then be the Aharonov-Bohm interaction A rel , which simplifies enormously the study of the scattering process.
A scheme of the proposed experimental strategy can be found in Fig. 2. If one prepares a pair of identical molecules inside the bulk of the FQH droplet, each one composed of the same kind of impurity and a binded quasi-hole excitation, and then makes them collide, e.g. by using a suitable potential, the angular dependence of the differential scattering cross section will show a pattern of maxima and minima whose angular dependence can be directly related with the fractional statistics of the molecules, as we will show in Sec. IV D.

A. General scattering theory
In order to study the two-molecule scattering we focus on the relative Hamiltonian (38) in 2D cylindrical coordinates and we consider the time-independent Schrödinger equation where k 2 = 2µE is related to the energy E of the scattering process. For the sake of notational simplicity we use the shorthand r, µ in place of R rel , M rel . Eq. (39) represents the scattering of a particle of mass µ by a flux tube of radius r 0 → 0 giving a vector potential qA rel = νu φ /r that incorporates the fractional statistics (u φ is a unit vector in the φ coordinate). For a short-range potential (i.e. such that rV ii (r) → 0 when r → ∞) the solution far from the origin can be written as the sum of an incoming plane wave [58] and an outgoing cylindrical wave [59,60] where f (k, φ) is the scattering amplitude. We will solve Eq. (39) using the method of partial waves. Given the cylindrical symmetry of the problem, we can look for factorized solutions ψ(r, φ) = e imφ u m,ν (r)/ √ r of angular momentum m with a radial function satisfying In contrast to usual scattering problems, for any noninteger value of ν the centrifugal barrier is present here for all values of m. This guarantees that the wave function vanishes for r = 0 when the two particles overlap [10].
In the limit of r → ∞, the solution of Eq. (41) can be written in the asymptotic form where the phase shifts have been defined w.r.t. the free case without quasi-holes attached to the impurities, i.e. V ii = 0 and ν = 0. As usual, the scattering amplitude in Eq. (40) can be related to the phase shifts δ m,ν (k) of this asymptotic expansion: using the fact that the cylindrical harmonics are a complete basis and replacing all cylindrical Bessel functions J m (kr) with their asymptotic form at r → ∞, we can write Eq. (40) as: from which it is easy to obtain an expression of the scattering amplitude in terms of the phase shifts, where a m,ν (k) = 2i πk e iδm,ν (k) sin δ m,ν (k) .

B. A general result for short-range potentials
The general solution of Eq. (41) in the free case with ν = 0 has the form that tends to in the r → ∞ limit. For a non-vanishing and non-integer ν the free solution changes to which approaches in the r → ∞ limit. For any short-range potential V ii , the total phase shift in the cosine cos(kr + ∆) in the asymptotic limit r → ∞ can be referred to the fully free case with V ii = 0 and ν = 0 [as done for δ m,ν (k) in Eq. (42)] or to the noninteracting case V ii = 0 with ν = 0. These two choices give respectively, where ∆ V m,ν is the phase shift exclusively due to the intermolecular potential V ii . Combining these equations we obtain where the total phase shift δ m,ν (k) is decomposed as the sum of the phase shift due to the topological flux attached to the impurities plus the one ∆ V m,ν (k) due to the interaction potential. In the non-interacting V ii = 0 case, this yields the same result as calculated in the original work by Aharonov and Bohm [57].
In the general case, assuming 0 < ν < 1, we can combine Eqs. (46) and (53) and decompose the scattering amplitude as The terms on the first two lines are geometric series that can be analytically summed up to m = ∞. They give the Aharonov-Bohm contribution to the scattering amplitude [57] and carry all information on the particle statistics. The terms on the third and fourth line summarize instead the contribution f V (k, φ) of the interaction potential V ii to the scattering amplitude. These terms must be evaluated by numerically summing the series.
This decomposition is of crucial technical importance as it enables to isolate the Aharonov-Bohm contribution f AB that can be analytically computed, and restrict the numerical calculation to the potential contribution f V only, for which convergence on the high angularmomentum side is straightforward. The differential cross section is finally obtained by adding the two terms and summing over exchange processes via Eq. (56).
However, this decomposition is more than just a mathematical trick, since it tells us about the different physical nature of the two contributions to the scattering amplitude. The statistical part of the scattering amplitude f AB originates from a vector potential A rel that extends to infinity. As a result, it affects all angular momentum components. Its divergent behaviour for φ → 0 can be physically related to the step-like jump of the geometric phase that is accumulated along paths passing on either side of r = 0. On the other hand, for a short-range interaction potential, the particles only see each other up to a certain distance, and therefore one only needs to sum up to a finite number of partial waves to achieve convergence in f V .

C. Indistinguishable impurities
The scattering process is most interesting when the impurities are indistinguishable particles. In this case, the differential scattering cross section involves a sum over exchange processes according to and may thus allow for interesting interference features in the angular dependence. As usual, the ± signs here correspond to bosonic and fermionic impurities, respectively. Indistinguishability guarantees that the cross section has the same value for φ and φ + π.
Repeating the same calculation leading to (54) in the 1 < ν < 2 case and noting that ∆ V m,ν is a function of m − ν only, one can show that holds for any 0 < ν < 1. It is then immediate to deduce that the scattering cross sections in the bosonic and fermionic cases are related by As a remarkable consequence, one can summarize the statistics into a single statistical parameter α, defined as α = ν for bosonic impurities and as α = 1 + ν for fermionic ones. A similar reasoning leads to the interesting relation from which one extracts the symmetry relation that translates into the compact form D. Numerical results for the differential scattering cross section The key feature of the topological contribution (55) to the differential scattering amplitude is the peak in the scattering cross section at small angles, proportional to 1/φ 2 . The height of the peak depends on ν, but its shape does not. No additional feature appears when performing the indistinguishability sum (56).
The situation gets much more interesting when the interaction potential V ii is included. This introduces a more complex angular dependence of the scattering amplitude f V (k, φ) and clear features in the differential scattering cross section. Keeping an eye on possible experimental realizations of this work we choose a dipolar form of the repulsive interaction potential [61], V ii = b/r 3 . A dipolar length can then be defined as a = 2µb. For this potential Eq. (41) can only be solved numerically, so we have calculated the differential scattering cross section dσ/dφ employing Numerov's method [62], which gives a global error of order O(h 4 ), being h = r i+1 − r i the numerical step size in the r coordinate. As an additional check, we have also calculated the differential scattering cross section for the case of a hard-wall potential of radius a, for which we can benchmark our predictions against the semi-analytical results available in Ref. [63].
For fixed values of µ, ν and ka the numerics allows us to calculate the phase shift δ m,ν (k) between the wave function of the molecules interacting according to V ii in the presence of the vector potential A rel and compare it with the result in Eq. (49) for V ii = 0. Using the result for ∆ V m,ν , we can then sum partial waves in Eq. (54) until convergence is reached. Summing the resulting f V to the analytically computed f AB and plugging the outcome into Eq. (56), we finally obtain the differential cross section dσ/dφ in the different cases.
Figs. 3 and 4 show the differential scattering cross section as function of the relative angle φ for a hard-disk and a dipolar interaction between bosonic impurities, respectively. The filling fraction of the FQH bath is fixed in both cases to ν = 0.5, while the different solid curves represent different values of the relative incident momentum ka for indistinguishable impurities. The qualitative behavior of dσ/dφ is identical for the two potentials: for small momenta (ka 1) the only effect of the fractional statistics beyond the peaks around φ = 0, π is the slight breaking of the φ ↔ π − φ symmetry (or, equivalently, of the φ ↔ −φ symmetry), so that the minimum of the curve is displaced to angles larger than φ = π/2, as was first found in Ref. [63]. For larger momenta ka 1, the peaks around φ = 0, π persist, and marked oscillations appear in the angular dependence because of interference effects, with a strong suppression of the differential scattering cross section occurring at some particular angles φ min . This oscillating behavior is reached for smaller ka in the case of hard-disk interactions. For ka = 5 (the largest value of the momentum considered here) several periods of oscillations are clearly visible. For dipolar interactions, a larger momentum is required to develop a comparable oscillating pattern. For instance for ka = 5, the curve only shows a single oscillation period between φ = 0 and π, with a strongly reduced contrast, even weaker than the one for ka = 1 in the hard-disk case. An oscillating behaviour is however recovered for far larger momenta, for instance the curve for ka = 50 features four welldeveloped minima.
Figs. 3 and 4 also show, as dashed curves, the angular dependence of the differential scattering cross section for distinguishable impurities. In this case, only one of the two scattering channels contributes to the scattering cross section, and therefore the oscillating behavior disappears due to the absence of the sum over direct and exchange pro- cesses. While the peak at φ = 0 due to the Aharanov-Bohm contribution (55) is still present, the one at φ = π is no longer present since the forward and backward directions are no longer equivalent.
As a next step, it is interesting to compare the two cases of bosonic and fermionic impurities (from now on, we will always consider the indistinguishable case).  tion with no peaks at φ = 0, π. For bosons at intermediate values of 0 < ν < 1, the peaks at φ → 0, π appear and, more interestingly, the oscillation pattern at large ka features a global shift towards smaller angles for growing ν, with again a smooth recovery to the fermionic case when ν → 1.
The linearity of this shift as a function of ν is illustrated in Fig. 7 for both choices of interaction potential. In order to have a good contrast in the oscillations, a relatively large ka is chosen. The deviations that are visible for the dipolar case disappear when even larger values of ka are chosen. Of course, the minimum recovers the usual location at φ min = π/2 of standard ν = 0 fermions for α = 1. The smaller slope of the hard-disk case is a consequence of the faster angular periodicity of the oscillations visible when comparing the panels in Figs. 6 b) and d).
This simple dependence on ν of the angular interference pattern shown in Fig. 7 is a key conclusion of our study. On the one hand, it provides a quantitatively accurate way to extract the fractional statistics of the quasi-holes in the FQH cloud just by detecting the oscillations in the angular dependence of the differential cross section and measuring the position of the minimum (φ min ). If such experiment is carried out using bosonic impurities, one can measure φ min in the presence and absence (i.e. ν = 0) of the FQH liquid, and then interpolate for the value of ν in the former case using the fact that bosons in a FQH liquid with ν = 1 have a minimum at an angle φ min = π/2 and the linear dependence on α = ν shown in Fig. 7. A qualitative signature of the fractional statistics for 0 < ν < 1 is offered by the asymmetry of the differential cross section for φ ↔ π − φ (or, equivalently, for φ ↔ −φ or φ ↔ 2π − φ), that indicates a preferential chirality in the scattering process.
On the other hand, it suggests an intuitive understanding of the underlying physical mechanism: the oscillations can be interpreted as an interference pattern for the two scattering channels contributing to the scattering in a given direction, say at an angle φ. In one channel, each particle is deflected by an angle φ during the scattering process. In the other channel, each particle is deflected by π + φ. Because of indistinguishability, the two processes have to be summed up with a relative phase α resulting from the sum of the intrinsic statistics α i = 0, 1 of the bosonic/fermionic impurities and of the fractional statistics ν of the attached quasi-holes. As it happens in generic interference experiments, e.g. two-slit interference, a phase-shift on one of the two arms results in a rigid shift of the fringe pattern. This intuitive interpretation is further confirmed by the complete disappearance of the fringe pattern when distinguishable impurities are considered.
In practice, a scattering experiment will begin with the simultaneous generation of a pair of anyonic molecules at different and controlled spatial locations. This can be done following one of the schemes discussed in Sec. III B 3. The two molecules will then have to be pushed against each other at a controlled speed with suitable potentials. The angular dependence of their differential scattering cross section will be finally extracted by repeating the experiment many times and collecting the statistical distribution of the trajectories of the scattering products. As we have mentioned at the beginning of Sec. IV, the analysis of the scattering experiment could be made simpler if the system parameters were chosen in such a way to give a vanishing effective charge Q for the anyonic molecules.

V. CONCLUSIONS
In this work we have shown how the quantum dynamics of impurity atoms immersed in a two-dimensional fractional quantum Hall (FQH) fluid of ultracold atoms may reveal crucial information about the fractional charge and statistics of the FQH quasi-hole excitations. Even though the discussion was carried out with a special attention to realizations in ultracold atomic gas platforms, equally promising candidates for experimental observation of the fractional charge and statistics are offered by FQH fluids of photons [23,25] or hybrid electronic-optical systems [64,65].
We considered impurities that repulsively interact with the atoms of the FQH fluid. In this case, for suitable parameters the impurities can form bound states with quasi-hole excitations, the so-called anyonic molecules. A rigorous Born-Oppenheimer [44,45] framework was set up to derive the effective charge and statistics of the anyonic molecules. Quite remarkably, this same formalism provides a quantitative prediction for the effective mass of the molecules, which combines the bare impurity mass with a correction due to the quasi-hole inertia.
As a main result of our work we proposed and characterized specific configurations where the fractional charge and statistics can be experimentally highlighted with state-of-the-art technology. If a single anyonic molecule is prepared inside the FQH fluid with some initial momentum, the values of the renormalized mass and of the fractional charge can be extracted from the experimentally accessible cyclotron orbit that it describes as a free charged particle in a magnetic field. This provides direct and unambiguous information on the fractional charge of FQH quasi-holes.
In the case of two anyonic molecules, the fractional statistics of the quasi-holes provides a long-range Aharonov-Bohm-like interaction between the molecules. For large values of the relative incident momentum, the differential cross section for the considered hard-disk and dipolar potentials displays a clear oscillatory pattern due to the interference of direct and exchange processes: the fractional statistical phase that the quasi-hole acquires upon exchange is then directly observable as a rigid shift of the angular interference pattern in the differential cross section.
As future perspectives, we envision to extend our approach to the case of impurities binding with different numbers of quasi-holes, leading to molecules with different anyonic statistics, and to the case of a larger number of molecules forming few-body complexes with a richer structure of eigenstates determined by the interplay of the inter-impurity interaction and the fractional statistics [10]. An even more intriguing development will be to extend our treatment to more subtle FQH fluids supporting non-Abelian excitations [2] and explore the consequences of the topological degeneracy on the quantum dynamics of the molecules [38,66].
In this first Appendix, we show how the approach of Ref. [45] can be straightforwardly extended to include synthetic magnetic fields.
We start by writing the full action functional where A(R j ) is the magnetic field (not present in the original formulation of Ref. [45]) evaluated at the position of the impurity j. Requiring and using Eqs. (7) and (11) we obtain the expressions to be satisfied by the factorized wave functions where is the impurity-bath coupling operator, is the time-dependent energy surface.
We see that the corrected expressions in Eqs. (A4)-(A8) do not include any first order modification to the renormalized molecule mass calculated in Ref. [45], which is given by Eqs. (13)- (15). In this second Appendix, we give quantitative evidence of the binding of impurities to FQH quasi-holes in the presence of a repulsive interaction between the former and the atoms.
To this purpose, we calculate the energy difference between the ground state displaying a quasi-hole located at the impurity's position (taken as the origin, i.e. R = 0) and the lowest energy oscillation state with the quasihole orbiting around the impurity with ∆L = −1. For simplicity, we employ a hard-disk interaction potential between the impurity and the atoms of the FQH fluid of the form where v ia > 0, a is the hard-disk radius and n is the number of atoms. By means of exact diagonalization, we first identify the states of interest using the Jack polynomials approach adopted in Ref. [51]. We then compute the expectation values of the atom-impurity potentials for different values of the hard-disk radius a. We have checked that the density profiles of the relevant many-body states in the vicinity of the impurity (i.e. for distances ≤ a) do not depend on the total number n of atoms in the FQH state.
The energies of the ground state and of the lowest excited state for several values of a are shown in Fig. 8 as red squares and blue circles, respectively. While the ground state is almost insensitive to the presence of the impurity as long as the radius a of the interaction potential is well smaller than the magnetic length, the density in the excited state always has a significant overlap with the impurity. As a result, as the hard-disk radius or the potential strength are increased, the energy gap between the two states also increases, thus reinforcing the rigidity of the impurity-quasihole molecule and reducing the Born-Oppenheimer mass correction ∆M .