Quantum behavior of a heavy impurity strongly coupled to a Bose gas
Jesper Levinsen, Luis A. Peña Ardila, Shuhei M. Yoshida, Meera M. Parish
QQuantum behavior of a heavy impurity strongly coupled to a Bose gas
Jesper Levinsen,
1, 2
Luis A. Pe˜na Ardila, Shuhei M. Yoshida, and Meera M. Parish
1, 2 School of Physics and Astronomy, Monash University, Victoria 3800, Australia ARC Centre of Excellence in Future Low-Energy Electronics Technologies, Monash University, Victoria 3800, Australia Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, Germany Biometrics Research Laboratories, NEC Corporation, Kanagawa 211-8666, Japan (Dated: February 15, 2021)We investigate the problem of an infinitely heavy impurity interacting with a dilute Bose gas atzero temperature. When the impurity-boson interactions are short ranged, we show that boson-boson interactions induce a quantum blockade effect, where a single boson can effectively block orscreen the impurity potential. Since this behavior depends on the quantum granular nature of theBose gas, it cannot be captured within a standard classical-field description. Using a combinationof exact quantum Monte Carlo methods and a truncated basis approach, we show how the quantumcorrelations between bosons lead to universal few-body bound states and a logarithmically slowdependence of the polaron ground-state energy on the boson-boson scattering length. Moreover, weexpose the link between the polaron energy and the spatial structure of the quantum correlations,spanning the infrared to ultraviolet physics.
The scenario of an infinitely heavy impurity in a quan-tum medium is a fundamental problem in physics, withrelevance ranging from electron gases [1] to open quan-tum systems [2]. The behavior is well understood in thecase of an ideal Fermi medium [3, 4] where the prob-lem can be solved exactly. Here, Anderson famouslydemonstrated that any interaction with the impurityleads to the orthogonality catastrophe in the thermody-namic limit [5]. However, there is currently much debateover the nature of the ground state for a fixed impuritystrongly coupled to a dilute Bose gas, which is of immedi-ate importance to ongoing cold-atom experiments [6–12].The bosonic problem — termed the Bose polaron —appears straightforward at first glance, since there is thepossibility of describing the condensed ground state ofthe Bose gas as a classical field, e.g., in the form of acoherent state [13–17], or governed by an effective Gross-Pitaevskii equation [18–20]. Furthermore, when the Bosegas is non-interacting, the ground state corresponds toall bosons occupying the lowest single-particle state inthe system, making it even simpler than the fermioniccase [21]. However, this tendency of bosons to cluster alsomeans that, in the absence of boson-boson interactions,the Bose polaron ground-state energy diverges when theimpurity-boson interaction is attractive enough to sup-port a bound state [19, 22]. Thus, it is an importantand non-trivial question how this pathological behavioris cured by boson-boson interactions, and whether thedetails of the impurity-boson interaction play a key role.This is of particular interest in the case of short-rangeresonant impurity-boson interactions, where the scatter-ing length a → ±∞ and there is the prospect of universalphysics, independent of the microscopic details.In this Letter, we show that in order to describe theground state of the Bose polaron, it is crucial to go be-yond classical-field descriptions and include the quantum“granular” nature of the Bose gas. Specifically, once r . a B r > a B FIG. 1. Bosons (circles) in the presence of an attractive impu-rity potential. If the range of the potential r is comparableto or smaller than the boson-boson scattering length a B , thena single boson can block the potential (left). Conversely, if r > a B , as for a Rydberg [9] or ionic [31] impurity, thenmany bosons can interact with the potential at once (right). the boson-boson scattering length a B is comparable toor larger than the range r of the attractive impurity-boson potential, a single boson from the gas can effec-tively screen or block the impurity potential, as illus-trated in Fig. 1. For a sufficiently attractive impurity-boson potential with r →
0, we find that this quantumblocking effect leads to universal few-body bound statesinvolving the impurity, in agreement with Refs. [23, 24].Using exact quantum Monte Carlo (QMC) methods [25–27], we show that the polaron energy in the many-bodylimit exhibits a logarithmic dependence on a B in the uni-tary regime a → ±∞ . We further illustrate the impor-tance of quantum correlations between bosons by show-ing that the QMC results for the polaron ground-stateenergy are well captured by a truncated basis variationalapproach [28–30] across a range of interactions. Model.—
We consider the following Hamiltonian fora single infinitely heavy impurity in a Bose gas:ˆ H = (cid:88) k (cid:15) k b † k b k + (cid:88) kk (cid:48) q V ( q )2 b † k b † k (cid:48) b k (cid:48) + q b k − q + g (cid:88) kk (cid:48) b † k b k (cid:48) . (1) a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b The three terms correspond, respectively, to the kineticenergy of the bosons, the boson-boson interaction, andthe boson-impurity interaction, where we have set thesystem volume and (cid:126) to one. In this model, a bosonof mass m and momentum k is created by the opera-tor b † k , and we consider bosons with the quadratic dis-persion (cid:15) k = | k | / m ≡ k / m . Furthermore, we de-scribe their interaction using the short-range potential V ( q ), which results in a low-energy boson-boson scatter-ing length a B >
0. The interaction between the impurityand a boson is taken to be short-ranged and of strength g up to a momentum cutoff Λ. The bare parameters g andΛ can be related to the physical impurity-boson scatter-ing length a via m πa = g + (cid:80) Λ k (cid:15) k . In the following, wetake the zero-range limit r →
0, which requires Λ → ∞ .For the QMC calculations, we solve the problem in realspace, using a Bethe-Peierls boundary condition for theimpurity-boson interactions, and taking the boson-bosonpotential to be a hard-sphere potential, where the diam-eter of the sphere coincides with the s-wave scatteringlength a B (see Supplemental Material [32]). Few-body bound states.—
We first discuss the few-body physics of an infinitely heavy impurity interactingwith N B identical bosons, where we assume that a > N B = 1, we simply have the impurity-boson boundstate with energy − ε b = − / ma , while N B = 2 cor-responds to the minimal number of bosons where boson-boson correlations can emerge. In Fig. 2(a) we displaythe QMC results for the N B = 2 energy for a range of a B . We find that a trimer (2-boson) bound state only ex-ists when the scattering length a is above a critical value a ∗ (cid:39) a B set by the boson repulsion. Moreover, thetrimer energy remains close to − ε b (i.e., the result for N B = 1) for the plotted range of a B /a spanning severalorders of magnitude, and it only slowly approaches the re-sult for uncorrelated bosons, − ε b , as we take a B /a → N B = 3, since we seethat the tetramer (3-boson) bound state also only existswhen a > a ∗ , and the tetramer energy lies well above theuncorrelated result, − ε b . Therefore, we conclude thatboson repulsion dominates the few-body behavior.Indeed, we find that we can reproduce these few-bodystates when the bosons only block each other at the im-purity and are non-interacting otherwise. Such a sce-nario is achieved with a bosonic Anderson model [23, 24],where the impurity-boson interaction features an openand closed channel like in a realistic cold-atom scatteringprocess [33]. Here, the impurity is unavailable for inter-actions with other bosons once a boson enters the closed-channel state, thus mimicking the quantum blockade ef-fect in Fig. 1. We previously solved the N B = 2 prob-lem exactly analytically for this model and we obtainedthe critical scattering length a ∗ = 3 . | r eff | , where r eff is the (negative) effective range of the impurity-bosoninteractions [23, 24]. Moreover, we found that a ∗ cor- FIG. 2. (a) Trimer energy as a function of inverse scatteringlength obtained from QMC (black circles), bosons with at-tractive contact interaction (black dot dashed line), and theAnderson model (purple dashed line). The inset comparesthe tetramer energy in the QMC with those of the Ander-son model. (b) Few-body energy at a/a B = 75 as a functionof boson number calculated within the QMC (black circles).We also show the energy of uncorrelated bosons, E = − N B ε b (gray dashed line), and that of interacting bosons in an ef-fective potential that accounts for three-body correlations,Eq. (2), with U = 0 . ε b = 1 . a B /ma (blue dotted line).Data for the Anderson model is taken from Ref. [23]. responded to a multibody resonance beyond which all N B > a ∗ /a , and the behavior is determined by quantumblocking at the impurity.Such few-body universality also extends to models with zero-range boson-boson interactions. In this case, a fi-nite positive a B requires an underlying attractive po-tential V ( q ), which features Efimov physics as well asdeeply bound dimers [32]. Thus, the relevant few-bodystates with effective boson-boson repulsion are actuallymetastable excited states. Nonetheless, it is possible tosolve for the energy of the metastable trimer state [32]and we see that it agrees well with the results of the othermodels in Fig. 2(a). We also find the critical scatteringlength to be a ∗ = 20 . a B , which differs slightly from thatestimated from the QMC simulations for a hard-spherepotential, indicating that finite-range effects are relevantin the relationship between a ∗ and boson repulsion.Within QMC, we can extend our results to even larger N B complexes. Fixing a ∗ /a <
1, we observe in Fig. 2(b)that the energy strongly deviates from the uncorrelatedresult E = − N B ε b (dashed gray line) and appears tosaturate to a finite value with increasing N B . Moreover,this does not match the energy of interacting bosons ina potential, E = − N B ε b + U N B ( N B − /
2, for any in-teraction energy U . We expect this behavior to also holdfor a non-zero range r as long as we satisfy the blockingcondition r (cid:46) a B , illustrated in Fig. 1. This condition isequivalent to requiring that the boson interaction energy, ∼ a B /mr , exceeds the depth of the potential, ∼ /mr ,assuming that the potential is close to resonance and us-ing the fact that bosons within the potential interact overa volume set by r [34].We can understand the result of Fig. 2(b) by consider-ing instead N B − ∼ a and the energy ofinteracting bosons is E = − ε b − ( N B − ε T + U N B − N B − , (2)where ε T is the trimer binding energy. In Fig. 2(b) where a (cid:29) a B and ε T (cid:28) ε b , we see that the small- N B behavioris well captured by Eq. (2) using U ∼ a B /ma . Thisillustrates the importance of three-body correlations aswell as demonstrating the role of the potential range. Many-body limit.—
We now turn to the behavior ofan impurity in a Bose gas of finite density n . In the ab-sence of the impurity and in the limit of vanishing boson-boson interactions, the ground state is a Bose-Einsteincondensate (BEC): | Φ (cid:105) = e √ n ( b † − b ) | (cid:105) , where | (cid:105) is thevacuum state for bosons. Thus, we can replace operators b † and b in the Hamiltonian (1) by √ n . Introducing theimpurity and turning on interactions, the polaron groundstate can be written in the general form [29] | Ψ (cid:105) = (cid:16) α + (cid:88) k (cid:54) = α k b † k + 12 (cid:88) k , k (cid:54) = α k k b † k b † k . . . (cid:17) | Φ (cid:105) , (3)where the complex coefficients α j are associated with dif-ferent numbers of bosons excited out of the condensate.In principle, one could write the expansion in Eq. (3) interms of Bogoliubov excitations rather than bare bosonicexcitations [28, 35]. However, this only modifies the op-erators at low momenta k < √ πna B , and this is not ex-pected to affect the leading order behavior of the polaronenergy in the extremely dilute limit n / a B (cid:28) FIG. 3. Ground-state energy of the infinitely heavy Bose po-laron as a function of inverse impurity-boson scattering lengthat fixed n / a ∗ = 0 .
215 (purple dashed) and n / a ∗ = 0 . E = 2 πna/m , is depicted as a dotted line. Applying the Hamiltonian (1) to the state (3) andkeeping only the leading order boson-boson interactionterms in the limit n / a B (cid:28)
1, we obtain the ground-state polaron energy [32]: E = n (cid:34) m πa + (cid:88) k (cid:18) (cid:15) k + G k − (cid:15) k (cid:19)(cid:35) − . (4)Crucially, we find that it depends on the repulsive corre-lations between bosons via the positive function G k = g √ n (cid:32)(cid:88) k (cid:48) α kk (cid:48) /α k − (cid:88) k (cid:48) α k (cid:48) /α (cid:33) . (5)Note that the case of uncorrelated non-interacting bosonscorresponds to α kk (cid:48) = α k α k (cid:48) /α , which gives G k = 0,such that the polaron energy E = 2 πna/m , in agree-ment with previous work [19, 22]. Thus, the presence ofcorrelations is necessary to ensure that the ground-stateenergy remains finite in the unitarity limit 1 /a → n / a ∗ . For weakimpurity-boson attraction 1 /n / a (cid:28) −
1, we recover themean-field uncorrelated result E = 2 πna/m , which cor-responds to the leading order dependence of Eq. (4) on a . However, as anticipated, the energy becomes sensitiveto boson-boson correlations as we increase the interac-tions towards unitarity. This behavior is not just lim-ited to zero-range impurity-boson interactions since thesame result is obtained for a finite-range potential when r < a B [25]. Note that this behavior goes beyond thefew-body results discussed previously since the impurity-boson bound state is either absent (when a <
0) or largerthan the interparticle spacing ( n / a (cid:38) r eff (cid:39) − a B so that the value of a ∗ matches the one from the QMCsimulations. As shown in Fig. 3, we find that the trun-cated basis approach accurately reproduces the QMC re-sults across a wide range of n / a ∗ (up to two ordersof magnitude) when we include up to three excitationsonly. This suggests that the boson-boson repulsion sup-presses impurity-induced excitations of the condensate.We stress that this is a highly quantum effect that can-not be captured by a classical mean-field description [38].At unitarity 1 /a = 0, the polaron energy takes theuniversal form E = − f ( n / a B ) n / /m, (6)where f ( x ) is a dimensionless function. When a B → E → −∞ , while in the zero-density limit n →
0, we must have E → n / a B → f ( x ) → ∞ slower than ∼ /x . Indeed, ourQMC results reveal a logarithmically slow dependence f ( x ) ∼ − ln( x ), as shown in Fig. 4. This behavior isdifficult to fully capture within the truncated basis ap-proach [32] since it requires an increasingly larger numberof boson excitations as n / a B →
0. On the other hand,if we use a coherent-state ansatz [13] with an infinitenumber of excitations but only the approximate mean-field repulsion of the Bogoliubov Hamiltonian, then wehave f ( x ) = (cid:112) π/ x which drastically overestimates thechange in energy (see Fig. 4).Indeed, the polaron energy is intimately connected tothe spatial structure of the boson-boson correlations viathe function G k in Eq. (4), which can be viewed as an ef-fective interaction potential between two excited bosons.In the infrared limit k →
0, where the bosons are atlarge separation, we should recover the behavior of un-correlated bosons. Here, we expect that the differencein energy between one and two excited bosons is theirmean-field interaction with the condensate, 8 πa B n/m .This large-distance infrared behavior is correctly cap-tured by the coherent state ansatz [13], which howeverfails at shorter length scales since it predicts a constant G k = 8 πa B n/m for all k and a [32]. In reality, we ex-pect the blockade effect to dominate at short distancessuch that α kk (cid:48) →
0, and in this case one can show that G k → − E as k → ∞ [32]. This short-distance ultravio-let behavior is captured by a “Chevy-type” ansatz witha single boson excitation [35, 39, 40], but this ansatzdoes not describe the large-distance physics since it has G k = − E at all momenta. However, the momentum FIG. 4. Bose polaron ground-state energy in the unitarityregime of impurity-boson interactions, 1 /a = 0. The QMC re-sults (symbols) are consistent with a logarithmic dependenceof the form E QMC (cid:39) .
37 ln (cid:16) . n / a B (cid:17) n / /m (solid line).The dashed red line is the prediction of the coherent stateansatz within the Bogoliubov approximation [13]. dependence of G k can be well approximated within atruncated basis approach that includes more boson exci-tations [32], as considered in this work. In particular, ourresults indicate that quantum blocking at short distancesdominates the behavior of the polaron energy while theinfrared physics only provides a small correction. Conclusion.—
To conclude, we have shown that theground state of the Bose polaron exhibits strong quantumcorrelations between bosons when the impurity-boson po-tential is short-ranged. This is due to a quantum block-ade effect at the position of the impurity, which gives riseto universal few-body bound states and a logarithmicallyslow dependence of the polaron energy on boson-bosoninteractions in the unitarity limit 1 /a →
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SUPPLEMENTAL MATERIAL:“QUANTUM BEHAVIOR OF A HEAVY IMPURITY STRONGLY COUPLED TO A BOSE GAS”
Jesper Levinsen, , Luis A. Pe˜na Ardila, Shuhei M. Yoshida, and Meera M. Parish , School of Physics and Astronomy, Monash University, Victoria 3800, Australia ARC Centre of Excellence in Future Low-Energy Electronics Technologies, Monash University, Victoria 3800, Australia Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, Germany Biometrics Research Laboratories, NEC Corporation, Kanagawa 211-8666, Japan
QUANTUM MONTE CARLO
The Hamiltonian of a system of N indistinguishable bosons and a single static impurity can be written as H = − (cid:126) m N (cid:88) i =1 ∇ i + (cid:88) i We use a hard-sphere potential to model the repulsion between bosons. The potential reads, V BB ( r ) = (cid:26) + ∞ r ≤ a B r > a B (S2)where the radius of the potential coincides with the boson-boson scattering length. The Jastrow term for the atom-atom correlations is chosen by matching the solution of the two-body scattering problem, f B ( r ) = (cid:26) q ( r − a B ) /r ] r ≤ a B a B < r < L/ q by imposing continuity of the wave-function, in addition, in order to be consistent with periodic boundaryconditions, the first derivative of the Jastrow function must be zero at half of the box size L = V / , where V is thesize of the simulation box. Impurity-boson interactions The impurity-boson Jastrow wave function is written as, f I ( r ) = (cid:26) ψ BP ( r ) A + B (exp [ − αr ] + exp [ − α ( L − r )]) r ≤ ¯ R ¯ R < r < L/ , (S4)where ψ BP = 1 − a/r in absence of bound state [26] or ψ BP = exp ( − kr ) /r in terms of the energy of a two-bodybound-state in vacuum ε b = k m = − ma . The wavefunction and its first derivative are continuous at the matchingpoint ¯ R , where ¯ R = xL/ x ∈ [0 , x and α are optimised using a variational Monte Carlo method.Note that, the Jastrow wave function ψ BP displays a divergence at r = 0 and hence we resort to the contact boundaryBethe-Peierls (BP) condition f I ( r ) = ψ BP which allow us to use a zero-range potential instead of using a finite-rangeone. This condition has been heavily used in QMC for both Fermi [45, 46] and Bose systems [26, 47]. FIG. S1. Few-body spectrum for an impurity and one or two bosons. The bosons interact with attractive zero-range interactions,resulting in either a B < a B > BOSONS WITH SHORT-RANGE ATTRACTION When the bosons in the medium have zero-ranged attractive interactions, the Hamiltonian in Eq. (1) becomesˆ H = (cid:88) k (cid:15) k b † k b k + (cid:88) kk (cid:48) q g B b † k b † k (cid:48) b k (cid:48) + q b k − q + g (cid:88) kk (cid:48) b † k b k (cid:48) , (S5)with the constant g B < 0. In this case, the boson-boson scattering length a B is determined via m πa B = 1 g B + Λ B (cid:88) k (cid:15) k , (S6)with Λ B an ultraviolet cutoff. When a B > − /ma B . Two- and three-body problems We start by making the following observation: The model in Eq. (S5) features Efimov few-body physics for any short-range attraction between bosons, rendering the model ultraviolet divergent. As a consequence, the infinitelyheavy Bose polaron in an ideal Bose gas corresponds to a quantum critical point for Efimov physics, a point that hasnot previously been discussed in the literature.The various regimes of the few-body spectrum for the case of N B = 1 and N B = 2 are illustrated in Fig. S1. For a B < a < a > 0. Below this continuum we have the existence of Efimov trimers and the spectrumis unbounded without an additional short-range cutoff. When a B > 0, panel (b), for N B = 2 we additionally have theboson-boson bound state with binding energy 1 /ma B , leading to the horizontal line in the figure. Note that the Efimoveffect can be quite significant even for the infinitely heavy impurity, and it does not necessarily involve huge scalingfactors. For instance, in the case where the two scattering lengths are identical, a = a B , and they greatly exceed anyother length scale in the problem, the discrete scaling factor determining the Efimov spectrum is λ = 15 . a = a B → λ l a = λ l a B and energy E → Eλ − l where l is an integer.The scaling factor can thus be even smaller than the λ = 22 . a B = 0 [50].What may appear puzzling in Fig. S1 is the apparent absence of the state where two bosons both occupy theboson-impurity bound state for a > 0. Indeed, such a configuration trivially exists when the bosons are completelyuncorrelated. However, we find this state only in a tiny subset of the already very small region marked “universaltrimer” in panel (b). To see this, we solve explicitly the three-body problem consisting of the impurity and twobosons. Taking a general state | Ψ (cid:105) = 12 (cid:88) k k α k k b † k b † k | (cid:105) (S7)with α k k = α k k , we find that the Schr¨odinger equation ( E − ˆ H ) | Ψ (cid:105) leads to( E − (cid:15) k − (cid:15) k ) α k k = g B (cid:88) k (cid:48) k (cid:48) α k (cid:48) k (cid:48) δ k + k − k (cid:48) − k (cid:48) + g (cid:88) k (cid:48) α k (cid:48) k + g (cid:88) k (cid:48) α k k (cid:48) . (S8)Defining the quantities on the right hand side, η k ≡ g B (cid:80) k (cid:48) k (cid:48) α k (cid:48) k (cid:48) δ k − k (cid:48) − k (cid:48) and χ k ≡ g (cid:80) k (cid:48) α kk (cid:48) , we obtain T − IB ( E − (cid:15) k ) χ k = (cid:88) k (cid:48) (cid:18) η k (cid:48) E − (cid:15) k − (cid:15) k − k (cid:48) + χ k (cid:48) E − (cid:15) k − (cid:15) k (cid:48) (cid:19) , (S9a) T − BB ( E − (cid:15) k / η k = 2 (cid:88) k (cid:48) χ k (cid:48) E − (cid:15) k (cid:48) − (cid:15) k − k (cid:48) . (S9b)Here we have defined the vacuum impurity-boson and boson-boson scattering T matrices T − IB ( E ) = m π (1 /a − √− mE − i , (S10a) T − BB ( E ) = m π (1 /a B − √− mE − i , (S10b)respectively.Equation (S9) represents a set of coupled integral equations that can, for instance, be solved as an eigenvalueequation for χ k with eigenvalue 1 /a by replacing η k in Eq. (S9a) by its representation in Eq. (S9b). In order touniquely determine the spectrum of Efimov trimers, one must additionally supply a cutoff on the sums on the right-hand-side of the equation. However, our focus is on the universal trimer which exists for large values of the ratio a/a B when a B > 0, and which is independent of such a cutoff. This trimer is unusual, as it is sitting in a continuumof states corresponding to the relative motion of a two-boson bound state relative to the impurity, see Fig. S1(b).These continuum states sensitively depend on the choice of grid whereas the trimer state is independent of thischoice. Consequently, the trimer state is easily distinguishable from the continuum states in our numerical solutionof Eq. (S9), and we simply discard the continuum states from our consideration. BOSE POLARON GROUND STATE For the model in Eq. (1) with generic repulsive boson-boson interactions, the Hamiltonian for the Bose polaron isˆ H = (cid:88) k (cid:15) k b † k b k + g √ n (cid:88) k (cid:54) = ( b † k + b − k ) + g (cid:88) k , q (cid:54) = b † k b q + gn + (cid:88) kk (cid:48) q V ( q )2 b † k b † k (cid:48) b k (cid:48) + q b k − q . (S11)Here we have replaced the operator b by √ n , the square root of the condensate density, in the terms involving theimpurity. Since we will focus on the scenario of an extremely dilute Bose gas near the resonance of impurity-bosoninteractions where n / a B (cid:28) n / | a | (cid:38) 1, we do not perform the Bogoliubov prescription on the last term ofthe Hamiltonian. Instead, we use a complete description of the boson-boson interactions that is needed to render themodel ultraviolet convergent in this limit.We obtain the polaron energy E by applying the Hamiltonian to the general state in Eq. (3) of the main text, | Ψ (cid:105) = (cid:16) α + (cid:88) k (cid:54) = α k b † k + 12 (cid:88) k , k (cid:54) = α k k b † k b † k + 13! (cid:88) k , k , k (cid:54) = α k k k b † k b † k b † k + . . . (cid:17) | Φ (cid:105) , (S12)and then projecting onto different numbers of bosons excited out of the condensate. This yields the (infinite) set ofcoupled equations: Eα = gnα + g √ n (cid:88) k α k (S13a) Eα k = (cid:15) k α k + gnα k + g (cid:88) k (cid:48) α k (cid:48) + g √ nα + g √ n (cid:88) k (cid:48) α kk (cid:48) (S13b) Eα k k = ( (cid:15) k + (cid:15) k ) α k k + g (cid:88) k (cid:48) α k k (cid:48) + g (cid:88) k (cid:48) α k (cid:48) k + (cid:88) k (cid:48) k (cid:48) V ( k − k (cid:48) ) α k (cid:48) , k + k − k (cid:48) + g √ n ( α k + α k ) + . . . (S13c)Here we have only kept the leading order boson-boson interaction terms that survive in the dilute limit n / a B → V ( q ) term in Eq. (S13c), which is required to correctly describe the three-body physics.Let us now rescale everything by α such that we have ρ k = α k /α and ρ k k = α k k /α . Combining Eqs. (S13a)and (S13b) to replace the energy E then gives0 = (cid:15) k ρ k + g √ n + g (cid:88) k (cid:48) ρ k (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) √ nχ + g √ n (cid:32)(cid:88) k (cid:48) ρ kk (cid:48) − ρ k (cid:88) k (cid:48) ρ k (cid:48) (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) ρ k G k . (S14)Solving for ρ k then gives ρ k = − g + χ(cid:15) k + G k √ n and χ = (cid:34) g + (cid:88) k (cid:15) k + G k (cid:35) − , (S15)where we have taken the renormalization condition g → → ∞ , and we have assumed that the function G k does not diverge as k → ∞ . This finally yields the energy in Eq. (4), E = nχ = n (cid:34) m πa + (cid:88) k (cid:18) (cid:15) k + G k − (cid:15) k (cid:19)(cid:35) − . (S16)The function G k can be viewed as the potential energy difference between 1 and 2 boson excitations, thus providing ameasure of the quantum correlations between bosons. In the absence of interactions and correlations, we have G k = 0and we recover the non-interacting boson limit discussed below. For the case of complete boson blocking where ρ kk (cid:48) = 0, the behavior reduces to the Bose polaron version of the Chevy ansatz [35, 39]. We then have G k = − E according to Eq. (S13a) and the definition of G k . Coherent state ansatz In the limit of non-interacting bosons a B → 0, the many-body wave function is known exactly and corresponds tothe coherent state [13]: | Ψ coh (cid:105) = N e (cid:80) k β k b † k | Φ (cid:105) , (S17)where the normalization factor N = e − (cid:80) k | β k | , and | Φ (cid:105) is the BEC ground state in the absence of the impurity. Ifwe compare this with the general form of the static impurity wave function in Eq. (S12), we clearly have α = N , α k = N β k , α k k = N β k β k , α k k k = N β k β k β k , and so on. Thus, the bosons are completely uncorrelatedwithin the coherent state | Ψ coh (cid:105) .The Hamiltonian (S11) in the limit of an ideal Bose gas is simplyˆ H = (cid:88) k (cid:15) k b † k b k + g √ n (cid:88) k ( b † k + b − k ) + g (cid:88) kq b † k b q + gn. (S18)Applying this Hamiltonian to the state | Ψ coh (cid:105) givesˆ H | Ψ coh (cid:105) = (cid:34)(cid:88) k (cid:32) (cid:15) k β k + g √ n + g (cid:88) q β q (cid:33) b † k + g √ n (cid:88) k β k + gn (cid:35) | Ψ coh (cid:105) , (S19)where we have used the fact that b k | Ψ coh (cid:105) = β k | Ψ coh (cid:105) . Therefore, in order for the state | Ψ coh (cid:105) to be an eigenstate ofthe Hamiltonian, we require the term involving b † k to be zero, which gives the constraint: (cid:15) k β k + g √ n + g (cid:88) q β q = 0 . (S20)Solving for β k and using the renormalization condition then yields the simple expression β k = − πam √ n(cid:15) k , (S21)and the exact eigenenergy just corresponds to the mean-field energy without boson correlations: E coh = g √ n (cid:88) k β k + gn = 2 πam n. (S22)Here we again used the renormalization condition, taking g → → ∞ .The energy in Eq. (S22) only corresponds to the ground-state energy for a < 0, while for a > a → 0. We see that the energy is finitefor a < /a = 0 when a two-body bound state appears. Moreover, the energyis unbounded for a > 0, where an arbitrary number of bosons can bind to the impurity.Taking boson-boson interactions into account, the coherent-state ansatz is only approximate, with the bare bosonicoperators in Eq. (S17) replaced by Bogoliubov quasiparticle operators. A variational minimization then gives [13] β k = − πm a − − a − √ n(cid:15) k E / k , (S23)with Bogoliubov dispersion E k = (cid:112) (cid:15) k ( (cid:15) k + 8 πa B n/m ) ≡ (cid:112) (cid:15) k ( (cid:15) k + 1 /mξ ), and a − ≡ πm (cid:88) k (cid:18) (cid:15) k − (cid:15) k E k (cid:19) = √ ξ , (S24)where we have defined the coherence length of the BEC, ξ = 1 / √ πna B . Within the coherent-state approximation,the energy is [13] E coh = n (cid:34) m πa + (cid:88) k (cid:18) (cid:15) k + 1 /mξ − (cid:15) k (cid:19)(cid:35) − = 2 πm na − − √ /ξ . (S25)Comparing this to the energy in Eq. (4) (or Eq. (S16)), we see that the correlation function G k is replaced bythe constant 1 /mξ ≡ πa B n/m , which is simply the mean-field interaction between an excited boson and thecondensate. One can recover this from Eq. (S13) by including the lowest-order mean-field terms due to interactionswith the condensate and taking the bosons to be uncorrelated, α kk (cid:48) = α k α k (cid:48) /α . Residue In the ideal gas limit, the overlap with the non-interacting state, i.e., the residue, can be calculated within thecoherent state ansatz: Z ≡ |(cid:104) Φ | Ψ (cid:105)| = exp (cid:34) − (cid:88) k β k (cid:35) = exp (cid:34) − (cid:18) πa √ nm (cid:19) (cid:88) k (cid:15) k (cid:35) → . (S26)This is seen to vanish since the momentum sum is infrared divergent. This divergence is cured once there areinteractions between bosons and the coherence length ξ is finite. In this case, using the variational parameters inEq. (S23) and expanding to lowest order in the impurity-boson scattering length a then gives [13]ln Z (cid:39) − (cid:88) k β k (cid:39) − (cid:18) πam (cid:19) (cid:88) k n √ (cid:15) k ( (cid:15) k + 1 /mξ ) / (S27)= − √ a ξn, (S28) FIG. S2. The function G k that quantifies boson-boson correlations. We show the results calculated within the bosonic Andersonmodel using a variational state with three excitations. The lines correspond to those of Fig. 3 of the main text, namely wehave n / a ∗ = 0 . 215 (purple dashed) and n / a ∗ = 0 . which matches second-order perturbation theory [51].On the other hand, we note that the Gross-Pitaevskii approach in Ref. [19] does not yield the correct residue inthe weak-coupling limit n / | a | (cid:28) 1, since it appears to consider everything with respect to the non-interacting BECrather than the weakly interacting state. Thus, instead of − (cid:80) k β k like in Eq. (S27), it givesln Z (cid:39) − (cid:88) k (cid:15) k E k β k = −√ πa ξn. (S29) BOSONIC ANDERSON MODEL As an alternative manner of introducing correlations between the bosons, we also consider the following model:ˆ H = (cid:88) k (cid:15) k b † k b k + ν d † d + λ (cid:88) k ( d † b k + b † k d ) + U d † d † dd, (S30)where we assume that the operator d is bosonic and we take the limit U → + ∞ at the end of the calculation.While this model features no explicit interactions between the bosons, correlations are induced by the presence of theimpurity since the impurity-boson interaction changes a boson to the auxilliary state described by the operator d (thismodels the coupling of an open and a closed interaction channel in a Feshbach resonance [33]). The infinite repulsionbetween d states described by the last term in Eq. (S30) then ensures that the interaction channel, which dependson the presence of the impurity, is only available to one boson at a time. Due to the similarity with the Andersonimpurity model, we refer to Eq. (S30) as the “bosonic Anderson model”. The few-body physics of this model wasinvestigated in detail in Ref. [23, 24]. Truncated basis approach To investigate the ground state of a Bose polaron within the bosonic Anderson model, we apply a variationalprinciple using the truncated basis ansatz. In this approach, we approximate the polaron ground state as | Ψ (cid:105) = | ψ (cid:105) + | ψ (cid:105) + | ψ (cid:105) + | ψ (cid:105) , (S31)where the states with different numbers of excitations are: | ψ (cid:105) = α | Φ (cid:105)| ψ (cid:105) = (cid:88) k (cid:54) =0 α k b † k + γ d † | Φ (cid:105)| ψ (cid:105) = (cid:88) k , k (cid:54) =0 α k k b † k b † k + (cid:88) k (cid:54) =0 γ k d † b † k | Φ (cid:105)| ψ (cid:105) = (cid:32) (cid:88) k , k , k (cid:54) =0 α k k k b † k b † k b † k + 12 (cid:88) k , k (cid:54) =0 γ k k d † b † k b † k (cid:33) | Φ (cid:105) . (S32)We do not include states that have multiple bosons in the d state because such states are prohibited by the limit U →∞ , as explained above. The variational parameters are determined by the variational equation ∂ α ∗ ,γ ∗ (cid:104) Ψ | ( ˆ H − E ) | Ψ (cid:105) =0, which reads Eα = λ √ nγ , (S33a) Eα k = (cid:15) k α k + λγ + λ √ nγ k , (S33b) Eα k k = ( (cid:15) k + (cid:15) k ) α k k + λ ( γ k + γ k ) + λ √ nγ k k , (S33c) Eα k k k = ( (cid:15) k + (cid:15) k + (cid:15) k ) α k k k + λ ( γ k k + γ k k + γ k k ) , (S33d) Eγ = ν γ + λ √ nα + λ (cid:88) k (cid:54) =0 α k , (S33e) Eγ k = ( ν + (cid:15) k ) γ k + λ √ nα k + λ (cid:88) k (cid:48) α kk (cid:48) , (S33f) Eγ k k = ( ν + (cid:15) k + (cid:15) k ) γ k k + λ √ nα k k + λ (cid:88) k (cid:48) α k k k (cid:48) . (S33g)These coupled equations are ultraviolet divergent with fixed bare parameters ν and λ . To remedy this, we rewritethem using physical low-energy parameters. To this end, we first remove the α coefficients and obtain T − A ( E ) γ = nγ E + (cid:88) k √ nγ k E − (cid:15) k , (S34a) T − A ( E − (cid:15) k ) γ k = √ nγ E − (cid:15) k + nγ k E − (cid:15) k + (cid:88) k (cid:48) (cid:18) γ k (cid:48) E − (cid:15) k − (cid:15) k (cid:48) + √ nγ kk (cid:48) E − (cid:15) k − (cid:15) k (cid:48) (cid:19) , (S34b) T − A ( E − (cid:15) k − (cid:15) k ) γ k k = √ n ( γ k + γ k ) E − (cid:15) k − (cid:15) k + nγ k k E − (cid:15) k − (cid:15) k + (cid:88) k (cid:48) γ k k (cid:48) + γ k k (cid:48) E − (cid:15) k − (cid:15) k − (cid:15) k (cid:48) . (S34c)Here, we have defined the scattering T matrix in the Anderson model T − A ( E ) = 1 λ ( E − ν ) − (cid:88) k E − (cid:15) k . (S35)To eliminate the ultraviolet divergence of the momentum sum, we carry out the renormalization procedure with amomentum cutoff Λ and obtain T − A ( E ) = m π (1 /a − r eff mE − √− mE − i , (S36)where a ≡ ( ν π − πmλ ) − is the s -wave scattering length, and r eff ≡ − πm λ is the effective range. Note thatthis is identical to the vacuum impurity-boson scattering T matrix in the two-channel model [52]. The resultingequations (S34a–S34c), with T − A given in Eq. (S36), do not contain the bare parameters and are ultraviolet finite,and we solved them by discretization.In Fig. S2, we illustrate the quantum correlations captured by the truncated basis approach, as encoded in thefunction G k from Eq. (5). We focus on the unitarity limit 1 /a = 0 where the behavior only depends on the three-bodyparameter n / a ∗ , with a ∗ = 3 . | r eff | [23, 24]. We see that G k → − E as k → ∞ , which corresponds to bosonblocking at short distances. Our truncated basis approach also captures how the blocking effect decreases at largerdistances (smaller k ), but it tends to overestimate the repulsion in the infrared limit k →