Emergence of a Bose polaron in a small ring threaded by the Aharonov-Bohm flux

The model of a ring threaded by the Aharonov-Bohm flux underlies our understanding of a coupling between gauge potentials and matter. The typical formulation of the model is based upon a single particle picture, and should be extended when interactions with other particles become relevant. Here, we illustrate such an extension for a particle in an Aharonov-Bohm ring subject to interactions with a weakly interacting Bose gas. We show that the ground state of the system can be described using the Bose polaron concept -- a particle dressed by interactions with a bosonic environment. We connect the energy spectrum to the effective mass of the polaron, and demonstrate how to change currents in the system by tuning boson-particle interactions. Our results suggest the Aharonov-Bohm ring as a platform for studying coherence and few- to many-body crossover of quasi-particles that arise from an impurity immersed in a medium.


Introduction
In the idealized model of a ring threaded by the Aharonov-Bohm (AB) flux, a particle moves in a region with zero fields, and the presence of an electromagnetic potential manifests itself only in a minimal substitution −i∂/∂x → −i∂/∂x + Φ, where the position-independent parameter Φ determines the strength of the flux. This model provides insight into many physical phenomena. For example, it illustrates the significance of potentials in quantum mechanics [1], geometric phases [2], the Josephson effect and persistent currents [3,4]. Foundations of the AB physics are based upon a single-particle picture [5,6], which already has the power to explain some experiments qualitatively such as spectroscopy in semiconductor rings [7]. However, one-body studies do not take into account interactions with other particles, in particular, with the environment. Therefore, they should be extended for realistic systems. In this paper, we discuss such an extension assuming a one-dimensional bosonic environment.
Before we proceed, let us briefly review known fewbody physics in the AB ring. If all particles are identical, then the flux couples only to the total angular momentum of the system. It can change the global minimum of the energy leaving the internal [i.e., in relative coordinates] dynamics intact, see, e.g., [6,[8][9][10]. In short, there is no interplay between particle-particle interactions and the AB flux for identical particles. This conclusion holds true also for distinguishable [by spin or quasi-spin] particles with identical masses and AB fluxes. In this case the strength of the AB flux can however change the symmetry of the ground state, see, e.g., Pecci et al. [11]. For particles with non-identical charges and/or masses, such as electrons and holes [12,13], the internal structure of a one-dimensional system is coupled to the AB flux. At the two-body level, this coupling can modify the threshold for binding (which may preclude formation of excitons for weakly attractive potentials in one dimension [14]) or lead to formation of dark excitonic states [15]. Systems with more than two particles are less explored, to the best of our knowledge.
In this paper, we study one of the simplest twocomponent many-body models -a particle (impurity) coupled to the AB flux that interacts with a Bose gas. The system is motivated by recent cold-atom experiments on Bose polarons [16][17][18][19][20][21][22], and by theoretical and experimental progress in realizing ring-shape potentials and artificial gauge fields with neutral cold atoms. For reviews of these advances see [23][24][25]. Ring-shaped condensates with effective gauge potentials have so far not been engineered together with impurities. As we show below such a combination may lead to rich physics. Note that recent advances in engineering ring shaped potentials [26][27][28] and tunable gauge fields [29] suggest experiments with polaritons as other set-ups to test our results.
The focus of the paper is on 'dressing' the impurity -a typical question addressed in many-body physics -which determines properties of the system such as transport and 'magnetization'. As such, our results complement previous works that investigated small systems using fewbody methods and approaches.
One of the main findings of our work is that the system can be described using ideas developed for the onedimensional Bose-polaron problem [30][31][32][33][34][35][36][37][38][39]. This connection leads to a number of useful conclusions. First, previous studies of the Bose polaron contribute insight into properties of our system, and provide an intuitive interpretation of our results. This insight can be also useful for understanding numerical lattice simulations where electron-phonon interactions are taken into account, see, e.g., Monisha et al. [40]. Second, persistent currents can be an experimental measure of validity of the Bosepolaron concept in one dimension. In particular, they can be used to investigate phase coherence of the polaron across the AB ring -a necessary condition for the existence of persistent currents. Third, the AB ring provides a conceptual model for defining the effective mass in a finite-size system, allowing one to better understand a few-to many-body crossover of one-dimensional systems. In particular, our work paves the way for studying this crossover beyond the standard testbed -the groundstate energy [41].

Results and Discussion
System. We study a one-dimensional system of N bosons and a single impurity atom, see Fig. 1. The system is in a ring of length L, which corresponds to periodic boundary conditions. The position of the impurity (ith boson) is given by Ly (Lx i ); the mass of the impurity (a boson) is m (M ). We assume that only the impurity is coupled to the AB flux Φ/L. For neutral particles, Φ is not generated by a magnetic flux threading the ring. Instead, other techniques are used [23,42], e.g., stirring with a weak external potential with speed v, in which case Φ = mvL/h. Note that the more gen-eral case, which might be more suitable for experimental realization, where the artificial flux is coupled to both particle species can be easily incorporated in our model, see Suppl. Note 1 for the flux coupled to bosons.
The Hamiltonian in first quantization reads where h =h 2 2mL 2 (−i∂/∂y + Φ) 2 describes the impurity; for the bosons, we have The impurity-boson, V ib , and boson-boson, V bb , interactions are parameterized by delta-function potentials where c and g define the strength of interactions. For simplicity, we shall use the system of units in which h = M = 1. In the main part of the paper, a boson and the impurity have identical masses, m = M . [A mass imbalance does not change the main conclusions of our study, see Suppl. Note 2]. For a fixed value of N , dimensionless parameters that determine all physical properties are c/g and γ = gL/N . For our numerical simulations, we shall use γ = 0.2, which corresponds to a weakly-interacting Bose gas amenable to the mean-field treatment discussed below. We focus on c > 0 to avoid bound states [43][44][45] that are beyond the polaron physics. Note that the case with γ = 0.2 and N = 19 for Φ = 0 was considered in [46] providing us with a reference point to benchmark our numerical calculations.
In what follows, we shall use the Hamiltonian from Eq. (1) in our analysis. However, it is worthwhile noting that the parameter Φ can in principle be excluded from this Hamiltonian via a gauge transformation Ψ → e iΦy Ψ, where Ψ is the wave function. The effect of the flux is then incorporated in a 'twisted' boundary condition that demands that the wave function acquires a phase e i2πΦ after a full turn [47,48]. Such a condition implies that the energy spectrum must be a periodic function with period Φ/(2π) as shown in Fig. 1. Note that for general multi-component systems (e.g, strongly interacting Bose-Fermi mixtures) a smaller period of the ground-state energy is also possible, see, e.g., [10,11]. As we demonstrate below, this does not happen for an impurity in a weakly-interacting Bose gas whose low-energy spectrum resembles that of a single particle.
The Hamiltonian H with Φ = 0 corresponds to one of the most studied one-dimensional models [49][50][51][52]. Therefore, we can use the already known methods to tackle our problem with Φ ̸ = 0. We choose to work in the frame co-moving with the impurity (see below), where the mean-field approach (MFA) and flow equations (IM-SRG) provide powerful theoretical tools for our investigation (see Methods). These methods allow us to investigate the effect of the AB flux on the properties of the one-dimensional polaron problem beyond previous studies [53,54], which investigated relevant molecularcrystal models. In particular, we can define and study flux-independent properties of the Bose polaron (e.g., the effective mass) in a finite system.
Co-moving frame. The total momentum of the system is conserved since all interactions are translationinvariant. Therefore, we eliminate the impurity coordinate by writing the wave function as (cf. [55]) where is the Heaviside step function], and P/L is the total momentum. The transformation {y, x i } → {z i } can be seen as a coordinate-space analogue of the Lee-Low-Pines transformation [56]. The parameter P is quantized to fulfill the periodic boundary conditions: P = 2πn, where n is an integer. Note that the transformation to the co-moving frame has been already used to study the few-to many-body transition in the ground state of the Bose-polaron problem with Φ = 0 [35]. Here, we study this transition for nonvanishing P where a continuous parameter Φ provides a bridge between the discrete values of P . The Schrödinger equation in the co-moving frame reads as follows where E is the dimensionless energy of the system [to obtain the dimensionful energy one needs to multiply E byh 2 /(mL 2 )]. Note that Φ and P enter this equation together as a sum P = P + Φ. P is a continuous variable that can be seen as an effective total momentum that determines total currents in the system. This observation will be crucial for interpreting our results in terms of an effective one-body picture, see below. Note that Ψ * P solves the Schrödinger Eq. (4) with −P, which is a manifestation of time-reversal symmetry.
Energy spectrum. The energy spectrum for Φ = P = 0 and finite values of N was calculated in Volosniev et al. [35]. Therefore, in what follows we only calculate E(P, Φ) − E(P = 0, Φ = 0), where E(P, Φ) is the energy of the Hamiltonian for a given value of the total momentum, P , and the AB flux, Φ. E(P = 0, Φ = 0) approaches the ground-state energy of the Bose-polaron problem in the thermodynamic limit (N, L → ∞ with a fixed value of N/L), see also Suppl. Note 4. Due to the periodicity of the energy (cf. Fig. 1), it is enough to focus on fluxes −π ≤ Φ ≤ π. Furthermore, the energy spectrum is symmetric with respect to Φ → −Φ due to time-reversal symmetry. Therefore, in what follows we shall calculate the lowest-energy states for fixed values of P , so called Yrast states (cf. [6]), and currents only for 0 ≤ Φ ≤ π. This also fixes the values of the flux needed to observe our findings experimentally. ]. The data are obtained using the mean-field ansatz (solid curves, MFA) and the in-medium similarity renormalization group (crosses with errorbars, IM-SRG), black triangles show results from Yang et al. [46] for quantized momenta. The dotted curves show the energy of the non-interacting system.
Energy for Φ ̸ = 0. We illustrate the energies calculated with MFA and IM-SRG in Fig. 2. For P = 0 and |P |= 2π both methods agree reasonably well on the energy, demonstrating that the MFA is a useful analytical tool to describe the system. We observed a worse agreement for |P |≥ 2π. The failure of the mean-field approach is expected for high values of P as there are various ways to distribute momentum between the bosons and the impurity, see also Suppl. Note 3.
Let us give a few general remarks about Fig. 2. For Φ = ±π there is a level crossing between two Yrast states with P = 0 and |P |= 2π. It is a consequence of the rotational symmetry of the problem. If a defect is introduced into a system, then it will lead to an avoided crossing, see below where we discuss the role of defects. In Fig. 2, we also present the ground-state energy of a non-interacting impurity (c = 0), E = (P + Φ) 2 /2. We see that the solid curves are always below this value. The effect is more pronounced for stronger impurity-boson interactions -compare the left and right panels of Fig. 2. These features can be easily understood using the concept of a polaron and its effective mass.
Effective mass. For the thermodynamic limit with Φ = 0, one finds that the low-energy spectrum of the system is quadratic in the total momentum (see, e.g., [39,57] for one-dimensional Bose polarons): where we introduce an effective mass, m TD eff ; other definitions of the effective mass are discussed in Suppl. Note 5. Eq. (5) is a cornerstone of the polaron concept and effective one-body descriptions of mobile impurities. Note that for small systems, the limit in Eq. (5) should be re-defined since P is discrete.
The parameter Φ is continuous. The mean-field solution as well as time-reversal symmetry suggest that  dΦ 2 for Φ → 0 with P = 0 and |P |= 2π. The parameters of the system are N = 19, γ = 0.2. The green curve shows the known result for P → 0 in the thermodynamic limit (TD. limit) [39,57]. The data in both panels are obtained using the mean-field approach. E(Φ, P = 0) is proportional to Φ 2 . By analogy to the Bose-polaron problem, we can define the effective mass of an impurity in a small AB ring This expression connects our problem to the body of knowledge developed by solving polaron problems. The connection allows one to make predictions about the behavior of an impurity in the AB ring. For example, the effective mass is an increasing function of c. Therefore, one reduces the current associated with the impurity by increasing c, see the discussion below. In addition, Eq. (6) demonstrates that the AB ring can be a physical testbed for studying the few-to many-body crossover in one-dimensional Bose-polaron problems. We illustrate this crossover for the effective mass in Fig. 3. For weak interactions (c/g = 0.05), we see that the effective mass converges to the thermodynamic limit quickly. This is not the case for strong interactions (c/g = 5), meaning that many bosons are needed to screen the impurity for large values of c/g. Although, our analysis suggests that the effective mass converges somewhat slower than the energy towards the thermodynamic limit (see also Suppl. Note 4), the basic mechanism is the same: A high compressibility of a weakly-interacting Bose gas requires a large number of bosons to screen a strongly interacting impurity. Note that the number of bosons needed for screening heavily depends on the parameter γ. In particular, in the limit γ → ∞, the system fermionizes and the impurity is screened by a handful of particles [41,58,59]. This observation highlights the fact that the few-to many-body crossover should be studied separately for fermions and weakly-interacting bosons.
Finally, we note that the effective mass computed with Eq. (6) describes only the Yrast curve with P = 0 well. To illustrate this, we calculate the second derivative of the energy in the limit Φ → 0 using the MFA. For a noninteracting impurity, this derivative is given by 1/m for all values of P . For an interacting impurity, this is not the case. The second derivative for P = 0 is by definition given by 1/m eff . The right panel of Fig. 3 shows that the effective mass increases for stronger impurity-boson repulsion in agreement with our expectations. The figure also shows that for the |P |= 2π state additional effects come into play and change the second derivative. The physical picture behind these effects will become more clear below, when we consider currents. The difference between 'effective masses' defined for P = 0 and |P |= 2π illustrates a shortcoming of the use of the quasiparticle picture for a small AB ring. However, even then, the polaron picture explains the qualitative features of the spectrum well.
Currents. The AB flux in our system induces currents that can be defined via the continuity equations for the impurity and the bosons in the laboratory frame: where tL 2 is time, ρ I (ρ B ) is the probability density of the impurity (bosonic) cloud. The (local) probability cur-rents are defined as The rotational symmetry implies that j I , j B , ρ I and ρ B are position-independent, allowing us to work with the integral quantities, e.g., ρ I = ρ I dy/(2π), which is more convenient. For example, using these quantities, it is easy to show that j I + N j B = P. Therefore, the total current -the current that corresponds to the total density ρ I + N ρ B -is given by P = P + Φ. Note that even though the AB flux is coupled only to the impurity, boson-impurity interactions also generate a current of bosons. We illustrate these currents for c/g = 1 and c/g = 5 in Fig. 4 (other parameters are N = 19, γ = 0.2). The increase in the boson-impurity interaction leads to an increase in the bosonic current. This observation is most easily explained using the Bose-polaron picture.
Using the Hellmann-Feynman theorem, it is straightforward to show that which coincides with the standard definition of the current in a one-body problem, see, e.g., [6]. [Note that this expression provides an indirect way for measuring currents by studying the energy landscape of the problem with RF spectroscopy (cf. Scazza et al. [60]).] For the polaron picture with P = 0, this leads to j I = Φ/m eff connecting the current (and transport properties) of the impurity to its effective mass. The bosonic current in the same approximation is given by The bosonic current generated by the AB flux follows the impurity, and leads to renormalization of its mass. We conclude that in the polaron picture the currents depend linearly on Φ, with the slope fully determined by the effective mass. The region of validity of this picture is determined by the boson-impurity interaction, see the dashed lines in panels a) and c) of Fig. (4). For c/g = 1 we observe that the polaron approximation, in which the energy is related to the AB flux via Eq. (6), is accurate for |P|≲ 2π, but for stronger interactions, c/g = 5, it is appropriate merely for |P|≲ π. For even stronger interactions, Eq. (6) is accurate only in the limit P → 0. Our interpretation here is that the coherent propagation of the impurity is not possible for strong boson-impurity interaction and large fluxes. Indeed, for c → ∞ an impurity can exchange its position with a boson in a coherent manner only at timescales given by 1/c. Thus strong (fast) impurity currents excite bosons. This leads to a nonlinear increase of the currents with Φ, see also [46] and Suppl. Note 3. To quantify these effects, it is convenient to rely on the second derivative of the energy (effective mass), which is larger for |P |= 2π, see Fig. 3 (note that the bosonic current is related to the impurity current via N j B = P + Φ − j I , i.e., one can reach the same conclusion by considering N j B instead of j I ). The IM-SRG results show a somewhat stronger generation of bosonic currents than the MFA, but the qualitative picture stays the same.
In addition limits of validity of the polaron picture can be investigated by considering states with higher values of |P |. For example, Fig. 4 shows that for |P |= 4π, the current of the impurity (almost) does not depend on Φ. This current is critical in a sense that by changing the flux of the impurity one generates only the current of bosons. The value of the critical current, j cr I , is decreased by increasing c, in agreement with the mean-field studies [61,62].
The critical current can be seen as an analogue of the critical velocity of a classic impurity that moves in a superfluid (cf. Landau critical velocity). Using this analogy, we can understand why the mean-field approximation in the co-rotating frame does not provide the correct value of the critical current. The MFA does not describe accurately the excitations of the Bose gas when it is decoupled from the impurity. In particular, the MFA leads to an incorrect phononic dispersion relation and implies that the critical velocity can be larger than the speed of sound for small values of c/g [62], which is unphysical. Furthermore, it does not capture type-II excitations of the Lieb-Liniger gas [63], which define the lowest energy state for a given value of the momentum of the Bose gas, see [64] for tutorial. Note that our IM-SRG method also does not capture these states well -the flow equations diverge when the type-II excitations become relevant. For some additional details, see Suppl. Note 3.
Role of defects. The rotational symmetry of the problem makes the Yrast energy spectrum of Fig. 2 double degenerate at Φ = ±π. In realistic systems, the symmetry is typically broken due to the presence of defects, leading to avoided crossings in the energy spectrum (cf. Fig. 5). At the maxima of the avoided crossings one-body currents defined via ∂E/∂Φ vanish affecting transport properties of the system [5]. We also note that the simplest experimental realization of the AB flux in cold-atom setups can be achieved with a rotating weak link [65], which utilizes the equivalence between the Coriolis force in a non-inertial frame and the Lorentz force on a charged particle in a uniform magnetic field. The rotating link introduces a 'defect' potential into the problem whose effect can be studied using the methods discussed in this section.
Our two-component set-up offers unique possibilities to modify currents that are not present in a single-body AB physics. To illustrate this, we add to the Hamiltonian a small perturbation: where H is the original Hamiltonian from Eq. (1) and This additional term describes a short-range potential coupled exclusively to the Bose gas. The current of the impurity is sensitive to W only via the boson-impurity interaction, and therefore the avoided crossing should contain information about the boson-impurity correlation function.
As long as a is small (a → 0), we can assume that the defect has only a minor influence on our system unless the system is close to the degeneracy point, Φ = ±π. Close to these points, we calculate the dimensionless energy using degenerate state perturbation theory: ) is the energy of the Yrast state with P = 0 (|P |= 2π); Ψ 0 (Ψ 1 ) is the corresponding eigenstate. Within the MFA, the matrix elements read as where α = f * i (z)f j (z)dz, and subscripts determine the Yrast state, e.g., i = 0 corresponds to P = 0.
To provide insight into the avoided crossing, we focus on Φ = ±π. In this case f 1 = f 0 , and ⟨Ψ i |W |Ψ j ⟩ depends only on the density in the co-moving frame (or equivalently on the impurity-boson correlation function in the laboratory frame). This density, hence, the splitting of the energy levels, is sensitive to the values of c. For example, if c = 0, then the defect should destroy the rotational invariance of the Bose gas only. Indeed, in this case, |f | 2 is constant and ⟨Ψ i |W |Ψ j ⟩ = 0.
Panel a) of Fig. 5 illustrates the avoided crossing for a small value of a. Note that the energy of the system, E(P, Φ), increases for a > 0. This effect does not appear in the figure as we only show the energy difference. The interesting part is that in the presence of W the energies of the first and second Yrast state no longer cross. The splitting of the energies, ∆E = 2aLN I, is determined by the integral which can be estimated using the density in the thermodynamic limit at Φ = 0 [66]: Finally, we note an interference effect that appears if we place a second small perturbation into the system: In analogy to above, we can define a coupling integral as If d = 1/2, this matrix element vanishes and the energy levels cross again (within the lowest order of perturbation theory). This happens because the perturbations are placed opposite to one another, which effectively restores the rotational symmetry in this case. This effect can be seen also for more than two perturbations, as long as they are placed in a symmetric order on the ring (for example three defects in a form of an equilateral triangle).

Conclusions
To summarize, we studied an impurity coupled to the AB flux in a Bose gas. We argued that i) the system can be described using the ideas developed for the Bose polaron, ii) the AB ring can be a testbed for studies of the fewto many-body crossover in cold-atom polaron problems.
In particular, observation of persistent currents in the AB ring with an impurity can shed light onto coherence properties of the Bose polaron. Note that the 1D world has inherent phase fluctuations, which can be captured using the IM-SRG approach (see also Suppl. Note 3). These fluctuations should be necessarily taken into account when studying persistent currents of polarons.
Our investigation of currents shows that the AB ring can provide a platform for studying a few-body analogue of the critical velocity in a Bose-polaron problem. Furthermore, if we assume that the critical current, j cr I , does not depend on Φ, then (according to Eq. (10)) the energy of the system is E = E cr + j cr I Φ. This expression connects the bosonic current and the energy of the system motivating a study of few-body precursors of collective excitations in a Bose gas.
Finally, we note that ∂ 2 E/(∂Φ 2 ) relates to the inverse of the effective mass, which defines transport properties of a polaron. This relation bears some similarity to the Thouless conductance in a disordered medium [67]. It might be interesting to explore this connection further, in particular, for a weakly interacting light impurity that within the Born-Oppenheimer approximation experiences a disorder potential created by heavy bosons.

Methods
Mean-field approach. We use two methods to investigate the system. The first one is the mean-field approach (MFA) in relative coordinates. It assumes that all bosons occupy one state in the frame co-moving with the impurity, so that the total wave function of the system can be approximated as where f (z) is a normalized function determined by minimizing the Hamiltonian. The variational procedure leads to the Gross-Pitaevskii equation, which can be solved semi-analytically, see Cominotti et al. [68] and Suppl. Note 3. Flow equations. The MFA is a well-established approach by now whose accuracy for stationary impurity problems has been shown by comparing to numerical quantum Monte Carlo [35,69] and state-of-the-art RG methods [35,39,68,70]. The MFA in time-dependent problems was discussed in [71,72]. In spite of those previous tests of MFA, we still find it necessary to validate it for the problem at hand. To this end, we shall use flow equations in the form of the so-called in-medium similarity renormalization group method (IM-SRG). This is an ab initio method that has been employed in condensed matter and nuclear physics [73][74][75] (for applications for one-dimensional problems with impurities, see [35,45,70]). For convience of the reader, we provide a brief introduction into IM-SRG and further compare its results to the MFA in Suppl. Note 3.

Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code Availability
The code used for this study is available from the corresponding author upon reasonable request.
In the main part of the paper, we consider the AB flux coupled to the impurity. Here, we discuss a more general scenario in which the AB flux is coupled also to the bosons. The corresponding Hamiltonian reads: As in the main part, we analyze the Schrödinger equation in the frame comoving with the impurity Note the differences between this equation and the one where the AB flux coupled only to the impurity. In particular, one cannot introduce a single effective variable (cf. P) that parameterizes the energy of the system. Still, one can cast Eq. (S2) into the form discussed in the main part . Therefore, one can use the methods and results of the main part to study a system with the AB flux coupled to bosons.
We illustrate this in Fig. S1, which presents the energy for the parameters of Fig. 2 assuming that Φ I = 0. The energies at Φ B = 0 are not modified, but there is dramatic change for Φ B > 0. First, the term N Φ 2 B /2 leads to a faster change of the energy with the flux. Second, we see that there is no energy level crossing for the Yrast states with P = 0 and |P |= 2π. While the energy spectrum must still be periodic in Φ B with the period The data are obtained using the mean-field ansatz (solid curves) and the IM-SRG (crosses with errorbars), black triangles are the results of Ref. [2] for Φ = 0. The dotted curves show the energy of the non-interacting system. Φ B /2π, the energy is no longer parameterized by an effective total momentum P, which means that the states with 0 < Φ B < π are not connected to the states with P = −2π and Φ B > π.
Finally, we mention the special case of equal fluxes Φ I = Φ B . This scenario corresponds to a 'rotation' of all particles around the ring with the same 'velocity'. In this case, the energy depends on the AB flux only via a constant energy shift. The flux couples exclusively to the center-of-mass motion of the system, in agreement with previous studies, see, e.g., [1].

Supplementary Note 2: Mass imbalance
In the main part of the paper, we assume that the masses of the impurity and the bosons are equal. Here, we briefly discuss a mass-imbalanced system. In this case, the Schrödinger equation in the co-moving frame reads: The corresponding Gross-Pitaevskii equation is: S2) with κ = m/(m + 1) the reduced mass. (Recall that we use a system of units in whichh = M = 1.) Note that κ is the only parameter that contains information about mass imbalance. One can write the solution to the mean-field equation using the same approach (see Supplementary Note 3), allowing us to transfer the conclusions of the main part of the paper to the mass-imbalanced system.
In Fig. S2, we present the energies as well as the currents for a heavy (m = 5) and a light (m = 0.5) impurities. [Note that the definition of the impurity current should contain an additional factor 1/m in Eq. 8 for the mass-imbalanced case. In practice, to calculate j I , we use conservation of momentum, i.e., we calculate the bosonic current first, and then use that j I = P − N j B .] First of all, note that the qualitative behavior of the observables is similar to that in the equal-mass case. The main differ-ence is that a light (heavy) impurity generates stronger (weaker) bosonic currents: the kinetic energy of the impurity is large (small) and therefore it is energetically more (less) favorable to excite the Bose gas.
Finally, we remark that for a heavy impurity, the mean-field results are in agreement with IM-SRG over a large range of Φ and P . This might be connected to the two observations: First, like discussed above, a heavy impurity leads to a weak bosonic current. Second, the mean-field approximation neglects the mixed derivative terms which scale as 1/m, therefore, a higher impurity mass leads to a weaker effect of this simplification. For a light impurity, the mean-field approximation is accurate only for low values of P.

Supplementary Note 3: Methods
Here, we provide some technical details about the methods used to investigate the system. In particular, we discuss the semi-analytical mean-field ansatz, its validity, and the numerical IM-SRG method.

Mean-field ansatz
For the mean-field approximation, we use a product state in the co-moving framẽ where f minimizes the expectation value of the Hamiltonian. Note that this ansatz is conceptually different from the strong-coupling approach in which the product state is written in a laboratory frame, and which predicts self-localization of the impurity [3]. f is computed using the Gross-Pitaevskii equation: where µ is the chemical potential, P = P +Φ is the 'total' momentum, κ = m/(m + 1) is the reduced mass, and is the momentum of one boson. The impurity-boson interaction leads to the boundary condition: The derived Gross-Pitaevskii equation has been discussed before, see, e.g., Ref. [4]. For convenience of the reader, we summarize here the steps to find its solution semianalytically. To simplify the notation, we introduce the momentum We assume that which leads to the two coupled differential equations: whose solutions read: p . sn and dn are Jacobi elliptic functions, F (K) is the (complete) elliptic integral of first kind, E the elliptic integral of second kind, Π the elliptic integral of third kind and am is the Jacobi amplitude [5].
The parameters s 1 , s 2 , s 3 determine the solution depending on the particle number and interactions. The boundary condition due to the impurity interaction gives an expression for s min as a function of s 1 , s 2 , s 3 . Note that there exist three solutions. We take the physical solution that reproduces the behavior of the Bose polaron in the limit P → 0 discussed in Ref. [6].
To find the parameters s 1 , s 2 , s 3 , we solve a set of equations that demand that the state solves the Gross-Pitaevskii equation, it is normalized, and its phase is periodic: These equations are solved numerically with s 1 ≤ s 2 ≤ s 3 . (Other orderings produce solutions that do not reduce to the mean-field solution of Ref. [6] in the limit P → 0.) The chemical potential is given by: Having obtained the solution, we can calculate properties of the system, e.g. the energy of the system:

Flow Equation Approach (IM-SRG)
Flow equations. The flow equation approach (also called in-medium similarity renormalization group or IM-SRG) is a beyond-mean-field method that (block)diagonalizes the Hamiltonian in second quantization, via the so-called flow equation Here, s is the flow parameter, which formally plays a role of (imaginary) time. The generator of the flow η should be chosen such that the off-diagonal matrix elements vanish in the limit s → ∞ [7]. As we are only interested in the lowest energy state of the system for a given value of P, we normal order the Hamiltonian using a condensate as a reference state, see Ref. [8]. This leads to the normal-ordered Hamiltonian where we denote normal ordered operators with : O :.
The matrix elements f ij and Γ ijkl describe one-and twoparticle excitations from the reference state. For the generator, we use these are the matrix elements that need to vanish in order to decouple the lowest energy state from the excitations. Once the flow equation reaches a steady state, the lowest energy state state is decoupled.
To calculate observables other than the energy, we evolve the operator O together with the Hamiltonian, i.e., we solve the flow equation in addition to Eq. (S16). We can calculate different observables, e.g., the current of a single boson, where ϕ k is the one-body basis function in which we expand the Hamiltonian and a k (a † k ) are annihilation (creation) operators.
Reference state. The transformation governed by the flow equations can be understood as a mapping between the reference state and an eigenstate of the system. Since we are interested in a system of bosons, it is reasonable to use condensate as reference state. Our reference state is constructed iteratively: Starting from the ground state solution of the non-interacting Hamiltonian for P = 0, the density and phase are calculated with IM-SRG and used as a new reference state. This procedure is repeated until both, the phase and the density, do not change upon iteration. Then, P is increased by a small amount and the reference state is calculated again. We repeat this until the desired value of P is reached.
Note that other choices for the reference state are possible in principle, e.g., the mean-field solution discussed in Supplementary Note 3, see Ref. [9]. However, we observed that for the system under consideration the procedure outlined above allowed us to calculate properties of the system for a larger range of P.
Accuracy of IM-SRG. Higher order terms induced in evolution of Eq. (S16) make it impossible to find a numerically exact solution for the considered values of N , and should be truncated. In our truncation scheme, we truncate at the two-body level. Three-body operators that contain at least one a † 0 a 0 operator are also kept, however, a † 0 a 0 is treated as a c-number. This means that we work only with zero-, one-and two-body operators in Eq. (S16) which leads to a system of coupled, closed, non-linear differential equations, which we solve numerically [8][9][10].
We estimate the error due to the neglected pieces (called W ) using second order perturbation theory where Φ p is a state that contains three-body excitations and Φ ref is our reference state. We construct the Hamiltonian in second quantization using the eigenfunctions of the one-body Hamiltonian without momentum or flux (P = 0). Since we can only work with a finite Hilbert space, we solve the flow equations for different numbers of basis states (in our case n ∈ [11,13,15,17,19,21]). For the energy, we fit these values with to estimate the result in infinite Hilbert space, E(n → ∞); b 1 and b 2 are fitting parameters. For other observables, such a fit is not always possible. In such cases, we use instead the result for the largest Hilbert space (n = 21) and estimate its error from the largest deviation to smaller Hilbert spaces (n = 11,13,15,17,19).
Therefore, there are in total two contributions to our error bars: The truncation error from neglecting higher order terms in the flow equation and the truncation error due to a finite Hilbert space. We assume that both errors are not correlated, and add their absolute values for our final error estimation For a more detailed description of the method we refer to Ref. [8], where the flow equations and our estimate of the truncation error are introduced, see also Ref. [9] for information about calculation of observables and a detailed explanation of our estimate of error bars.
To further validate our numerical results, we benchmark them against Ref. [2]. Overall we observe a good agreement between IM-SRG and Ref. [2], see the figures in the main part of the paper. Only for c/g = 1 and P = 4π there are some small deviations. We interpret this deviation as follows: IM-SRG uses a condensate as a reference state, and fails to describe excited states of the Bose gas, which become important when the impurity current reaches its critical value. This argument is supported by the fact that the bosonic currents calculated with IM-SRG are always smaller than those in Ref. [2]. To improve the IM-SRG method, a reference state that includes boson-boson interactions could be used, like it is done in nuclear physics for fermionic systems, see e.g. Ref. [11,12]. Such references states are outside the scope of the present work.

Validity of mean-field ansatz
In the main part of the paper, we showed results for the energy (Fig. 2) and the currents (Fig. 4). For P ≤ 3π, we observed a good agreement between the MFA and IM-SRG, justifying the use of the MFA. For larger values of P, the MFA was less accurate, see, in particular the currents for P = 4π. Here, we use IM-SRG results to further test the validity of the MFA.
To this end, we calculate the densities, ρ(z) = |f (z)| 2 , and phases, θ(z), of the Bose gas for Φ = 0 and |P |= 0, 2π, 4π, see Fig. S3. In our IM-SRG calculations these quantities can be expressed as follows: For P = 0 and |P |= 2π, there is a good agreement between the two methods. For |P |= 4π, we start to see a difference, which is particularly noticeable in the phase. This is in agreement with our observation (see the main text) that the IM-SRG and mean-field results for the currents disagree when the value of P is large. Recall that the bosonic and impurity currents can be related to the gradient of the phase using Eq. (S3). We can go one step further with the beyond-mean-field IM-SRG method and estimate directly the off-diagonal coherence of the bosons by calculating a property called phase fluctuations given by the one-body density matrix (see, e.g., [13][14][15]): (S21) The quantity δΦ zz ′ vanishes for a condensate. It is therefore a direct measure of the validity of the mean-field approximation. A plot for the same parameters as before is also shown in Fig. S3. We see that for P = 0 and |P |= 2π phase fluctuations are nearly identical and that for |P |= 4π there is a considerable increase. Together with the behavior of the energies, currents, phases and densities, we come to the conclusion that the mean-field approximation works well only for small values of P and Φ such that P < 3π.
Let us provide some physical intuition into why the MFA results become less accurate when we increase P + Φ. To this end, we consider the case of vanishing impurity-boson interaction strength, c → 0. In this limit, the bosons are described by the well-known Lieb-Liniger gas [16,17]. There exist two different types of excitations in the Lieb-Liniger model: type-I and type-II excitations.
To illustrate this discussion, we plot the energy of the system with c = 0 (non-interacting impurity) as a function of P in Fig. S4. The blue (solid) curve represents the case when the bosons are in the ground state and all of the momentum is distributed to the impurity such that The orange (dashed) curve is obtained using the lowest energy state of the Bose gas with a quantized value of |P |= 2π. The rest of the total momentum is given to the impurity We use E bosons from the Yrast curve presented in Ref. [2]. For comparison in Fig. S4, we also show the results from Dashed lines between these symbols are added to guide the eye. 0 π 0.5 π 1 π 1.5 π 2 π 2.5 π 3 π 3.5 π 4 π P Imp. + LL ground. Imp. + LL excited Ref. [2] FIG. S4. Energy for a system of a non-interacting impurity (c → 0) in the Lieb-Liniger gas. The Lieb-Liniger gas is either in the ground (blue solid curve) or in the excited (orange dashed curve) states. The parameters of the system are N = 19, γ = 0.2, m = M = 1. The triangles show the results from Ref. [2] for these parameters.
Ref. [2] for c = 0. The point where the curves cross imply a strong momentum exchange if c ̸ = 0, which might be beyond IM-SRG and MFA. Note that this crossing can be observed directly only if Φ ̸ = 0.
We see that at small values of P, it is energetically favorable to have all 'vorticity' in the impurity. For some parameters P < 4π, however, it is more favorable to excite the Bose gas. This analysis is in agreement with the calculations of [2] for quantized values of P . It be-comes clear now why the mean-field results presented in the main part of the paper are accurate only for P = 0 and |P |= 2π: The MFA does not describe the excitations of the bosons for c = 0 well.
The IM-SRG method faces a similar problem since we use a condensate as a reference state. For example, our flow-equation calculations for c/g ≪ 1 diverge close to the point where it is energetically favorable to excite the bosons (e.g., P ≃ 3.8π in Fig. S4). In this way IM-SRG signals that a simple condensate description of the Bose gas in the co-rotating frame is no longer valid. It is worth noting that IM-SRG converges over a larger range of P if the impurity-boson interactions are stronger. In particular, IM-SRG is in agreement with Ref. [2] for |P |= 4π for strong impurity-boson interactions. This leads us to the question about the effect of the impurity-boson interactions on the excitations of the system, which determines the overlap with the reference state of the IM-SRG method, a condensate. Note that this question might be connected to the critical velocity of an impurity in a onedimensional Bose gas. We leave this question to future studies.

Supplementary Note 4: Convergence towards thermodynamic limit
As the main part of the paper argues, an AB ring allows us to study the convergence of the effective mass to the thermodynamic limit (N, L → ∞, N/L = ρ = const). Here, we discuss this few-to many-body crossover for the effective mass and the energy in more detail. Note that the energy has been discussed already in Ref. [6]. In particular, it was found there that the energy decays as E − E(c = 0) = ρ 2 π 2 /(2N κ) for a non-interacting Bose gas with P = Φ = 0. For an interacting Bose-gas, the energy converges to a finite value that is determined by the polaron energy.
Below, we first use the results of Ref. [6] to provide some further analytical insight. After that, we use a fitting procedure to find the convergence of the energy and the effective mass in our numerical simulations.

Energy
Reference [6] shows that the energy of a system with an impenetrable impurity (c → ∞) is E − E(c = 0) ρ 2 = 8K 4 (p)p + 2K 2 (p)κγN (N − 1)(p + 1) where K is the complete elliptic integral of first kind and p is a parameter determined from the equation: with the complete elliptic integral of second kind, E, [5]. In the limit of N → ∞, p → 1 and therefore E(p) → 1. This allows us to derive an analytic expression for K(p) in terms of γ and N , which can be inserted into Eq. (S1) producing (S3) In the limit of large N , the energy decays as 2/κN to the value 16γ/9κ, which corresponds to the boundary energy of the Bose gas (cf. Refs. [18,19] for an infinitely heavy impurity with κ = 1).
Let us now consider finite impurity-boson interactions. To this end, we fit the IM-SRG results with In Fig. S5, we show the energy for γ = 0.2 and c/g = 0.05, 1, 5. Fitting reveals that the parameter σ is in the range of σ = 1.065 − 1.08 suggesting that also for finite impurity-boson interactions the energy decays like 1/N . Small deviations from the 1/N behavior are probably due to relatively small values of N . We checked this statement by adjusting the number of particles.

Effective mass
It appears complicated to gain analytic insight into the few-to many-body crossover of the effective mass. Therefore, we fit the results from Fig. 3 with Eq. (S4). We find that contrary to the self-energy of the impurity the convergence seems to strongly depend on the strength of the impurity-boson interactions. For weak interactions (c/g = 0.05), we find the fastest convergence with σ = 1.07 ± 0.15. For c/g = 1, the effective mass convergences with σ = 0.99 ± 0.01. For strong interactions (c/g = 5), the convergence is significantly slower σ = 0.67±0.03. To understand this, note that for an impenetrable impurity (c → ∞), the impurity acts as a wall moving through the Bose gas. This means that the effective mass should be the mass of the whole system, i.e., the effective mass should actually increase with N in this case. Our results demonstrate that the effective mass is highly sensitive to the value of c/g, unlike the energy. In particular for strong impurity-boson interactions, one requires more bosons to observe convergence of the effective mass.

Supplementary Note 5: Definition of effective mass
For convenience of the reader, here we review and explain equations that define the effective mass in the thermodynamic limit with Φ = 0, cf. Refs. [20,21]. These equations also appear in the main part of the paper, although in a somewhat modified form.
The problem of a quantum impurity moving through a medium is simplified by mapping this many-body system to an effective one-body set-up -polaron. This quasiparticle is intuitively understood as the impurity dressed by the excitations of the medium, which lead to an effective mass higher than the bare mass.
In the polaron picture, the low-lying states of the system have (approximately) the energy where m eff is the effective mass and P is total momentum of the system. Similar to Eq. 10, we can write the velocity (probability current) of the impurity as v imp = P/m eff .
Sometimes, one employs this expression to define the probability current of the polaron as v pol = v imp .
Using Eq. (S2), we write the energy of the system as where m eff is the effective mass and v imp is the velocity of the impurity. Finally, we connect the effective mass to the momentum carried by the medium (in our case, the Bose gas). To this end, we use the fact that the total momentum is distributed between the impurity and the bosons P = P imp + P bos , and that (by definition) the impurity momentum is given by P imp = mv imp . With this, we derive m/m eff = 1 − P bos /P.
Eqs. (S1)-(S4) are equivalent and can be used to compute the effective mass in numerical simulations. We noticed in our IM-SRG calculations that the last equation is numerically the most stable and therefore it was used in our study.