SSelf-stabilized Bose polarons
Richard Schmidt
1, 2 and Tilman Enss Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Straße 1, 85748 Garching, Germany Munich Center for Quantum Science and Technology, Schellingstraße 4, 80799 Munich, Germany Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, 69120 Heidelberg, Germany (Dated: March 1, 2021)The mobile impurity in a Bose-Einstein condensate (BEC) is a paradigmatic many-body problem.For weak interaction between the impurity and the BEC, the impurity deforms the BEC only slightlyand it is well described within the Fr¨ohlich model and the Bogoliubov approximation. For stronglocal attraction this standard approach, however, fails to balance the local attraction with the weakrepulsion between the BEC particles and predicts an instability where an infinite number of bosonsis attracted toward the impurity. Here we present a solution of the Bose polaron problem beyond theBogoliubov approximation which includes the local repulsion between bosons and thereby stabilizesthe Bose polaron even near and beyond the scattering resonance. We show that the Bose polaronenergy remains bounded from below across the resonance and the size of the polaron dressingcloud stays finite. Our results demonstrate how the dressing cloud replaces the attractive impuritypotential with an effective many-body potential that excludes binding. We find that at resonance,including the effects of boson repulsion, the polaron energy depends universally on the effectiverange. Moreover, while the impurity contact is strongly peaked at positive scattering length, itremains always finite. Our solution highlights how Bose polarons are self-stabilized by repulsion,providing a mechanism to understand quench dynamics and nonequilibrium time evolution at strongcoupling.
I. INTRODUCTION
Impurities in a Bose-Einstein condensate (BEC) ex-hibit a multitude of fundamental physical phenomena:the formation of quasiparticles [1], Efimov bound states[2, 3], synthetic Lamb shifts [4, 5], Casimir interactionsinduced by a fluctuating medium [6, 7], and quantumcriticality [8]. Current experiments with ultracold atomicgases are investigating several of these effects reachingfar into the strong-coupling regime [4, 8–13]. For un-derstanding experimental observations it is thus vitalto develop a theoretical model that applies to impuritysystems at strong coupling and that can address bothground state and non-equilibrium phenomena. energy Bose-Einstein condensateimpurity potential effective potentialbound state x
FIG. 1. Illustration of the self-stabilized Bose polaron. Thestrong-coupling Bose polaron mimics a microtrap (red dashedline) with bound state (red bar) within the Bose-Einstein con-densate. If several bosons (blue dots) occupy the bound state,boson repulsion results in a shallower effective potential seenby the remaining bosons (blue solid line) that no longer ad-mits a bound state. The dressing cloud itself thus stabilizesthe Bose polaron.
At weak interactions between the impurity and theBEC, the impurity deforms the BEC only slightly andphysics is well described within the Fr¨ohlich model interms of long-wavelength phonon excitations [1, 14–24].For strong local attraction, instead, the Fr¨ohlich model isincomplete and needs to be amended by quadratic termsthat absorb and re-emit phonons [25]. Importantly, theseterm are also required to correctly capture the formationof bound states between the impurity and bath atomswhich is a crucial ingredient for the physics of poloransat strong coupling [26]. Variational wave functions basedon a single phonon excitation [27, 28] are able to describethe single occupation of such a bound state. However,the bosons that make up the BEC tend to bunch, and atstrong coupling it is energetically favorable to occupy thebound state multiple times leading to a gain of severaltimes the binding energy. This process, recently observedwith Rydberg atoms immersed in a BEC [29], is describedby a coherent state variational ansatz [25, 26, 30, 31] thatallows for an arbitrary number of excitations and stronglocal deformations of the condensate [32–34].Generally, the application of the variational principlefor the determination of the ground state is only viableif the Hamiltonian is bounded from below. Under thiscondition a stable solution can be found, and theoreti-cal approaches should rely only on such approximationsthat preserve stability of the underlying problem . Forthe strongly interacting Bose polaron, the resulting the-oretical challenge can be understood in the simple toy In this work we restrict ourselves to models of cold dilute gasesthat disregard deeply bound states, transitions to solid or liquidphases, as well as large cluster formation. a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b model illustrated in Fig. 1. Here a static attractive im-purity is represented by a local potential well of finiterange around the impurity. This well acts as a microtrapwithin the BEC. At strong coupling beyond a scatteringresonance [35], the potential well (red dashed line) is deepenough to admit a bound state with energy ε i = − ε B < all bosons would occupy the boundstate and the ground-state energy E = − N ε B is indeedunbounded from below in the thermodynamic limit: thewhole BEC is collapsed onto the impurity.Naturally, a local boson repulsion counteracts this pro-cess by balancing the impurity attraction [36] and thusproviding a lower bound to the ground-state energy.This mechanism can be understood in terms of a single-site Bose-Hubbard model [37, 38] with local repulsion U n i ( n i −
1) on the impurity site r i , that competes witha local attractive potential energy − ε B n i for occupationnumber n i . For U > n i (cid:39) ε B U , which is nonperturbative in the strengthof the interaction: the size of this polaron dressing cloudgrows for weaker repulsion. It is crucial to adequatelycapture this repulsive effect in the theoretical descriptionof Bose polarons.Previous variational approaches have included the bo-son repulsion only at the level of the Bogoliubov approx-imation. Here the interaction between the Bogoliubovquasiparticles is neglected, and thus bosons in the boundstate fail to generate the compensating pressure requiredto ensure the stability of the ground state. As a conse-quence this approach falsely predicts instead a dynami-cal instability in presence of boson repulsion toward infi-nite occupation of the bound state in the strong couplingregime [30, 31, 33]. This shows the crucial importanceof including the boson repulsion beyond the Bogoliubovapproximation.In this work, we present a stable variational approachto the Bose polaron problem at strong coupling. Ourapproach applies to arbitrary dimension and impurity-boson scattering lengths and it provides a basis for thestudy of dynamical properties of Bose polarons. Inthis way our work complements and extends previousapproaches using quantum Monte Carlo to determineground-state properties [39], studies of the role of bo-son repulsion in one-dimensional systems [40], or a re-cently developed nonlocal Gross-Pitaevskii (GP) theoryfor nonequilibrium dynamics [36]. Moreover, previousworks raised the question to which extent polaron prop-erties are universal in the short-range limit [41, 42]. Inthe following we study this question across the Fesh-bach resonance, and in particular in the regime wherea bound state is supported by the impurity-bath poten-tial, extending previous work that considered how theproperties of the Bose polaron depend on the range ofthe interactions, both for the finite-range boson repul-sion in the nonlocal GP theory [36] and for finite-rangeimpurity potentials [38, 43, 44].Specifically, we study the effect of local boson repulsion and finite-range attractive impurity potentials employinga coherent variational state that allows for large dress-ing clouds and strong local deformations of the BEC. InSec. II we introduce the stable Bose polaron model anddiscuss its solution within Gross-Pitaevskii theory. InSection III we minimize the resulting GP energy func-tional and obtain the condensate wave function aroundthe impurity. We find that even if the bare impuritypotential admits a bound state, the emerging effectivepotential does not, thus providing a simple mechanismfor the self-stabilization of Bose polarons. Section IVpresents our results for the polaron energy, the size ofthe polaron dressing cloud and the Tan contact acrossthe resonance. The universality of the polaron energy isdiscussed in Sec. V, and we show that the energy at uni-tarity depends universally on the effective range, as longas Efimov states can be neglected. Finally, in Sec. VI wecompare variational approaches to the Bose polaron anddiscuss which Hamiltonians and energy functionals canprovide rigorous bounds on the ground-state energy. II. MODEL
We consider a single impurity particle immersed in aninteracting Bose gas. The combined system is describedby the Hamiltonian H = ˆ p m I + (cid:88) i V IB ( ˆ x B ,i − ˆ x )+ (cid:88) i ˆ p ,i m B + (cid:88) i Inorder to appreciate the importance of the adequate inclu-sion of boson repulsion we briefly review approximationsto the model (1) that are frequently applied to the studyof the Bose polaron problem. In the formalism of secondquantization Eq. (1) readsˆ H = (cid:88) p p m I ˆ d † p ˆ d p + (cid:88) p p m B ˆ a † p ˆ a p + 1 V (cid:88) kk (cid:48) q V IB ( q ) ˆ d † k (cid:48) + q ˆ d k (cid:48) ˆ a † k − q ˆ a k + 12 V (cid:88) kk (cid:48) q V BB ( q )ˆ a † k (cid:48) + q ˆ a † k − q ˆ a k ˆ a k (cid:48) . (3)Here V is the system volume, ˆ d † p and ˆ a † p are the cre-ation operators of impurity and bosons, respectively, and V BB ( q ) and V IB ( q ) are the Fourier transforms of the re-spective interactions in Eq. (1). Next, the ladder opera-tors of bosons are shifted using a simple canonical coher-ent state transformation leading to ˆ a † p → ˆ a † p + δ p , √ N .This is then followed by the crucial Bogoliubov approx-imation : All terms beyond quadratic order bosonic oper-ators are neglected, leading to the truncated Hamiltonianˆ H (cid:48) = (cid:88) p p m I ˆ d † p ˆ d p + (cid:88) p p m B ˆ a † p ˆ a p + 1 V (cid:88) kk (cid:48) q V IB ( q ) ˆ d † k (cid:48) + q ˆ d k (cid:48) ˆ a † k − q ˆ a k (4)+ g BB N V + g BB N V (cid:88) q (cid:54) =0 (cid:16) a † q ˆ a q + ˆ a † q ˆ a †− q + ˆ a q ˆ a − q (cid:17) , where, for simplicity, we have chosen the example of aboson contact interaction of strength g BB . As a resultof the Bogoliubov approximation the bosonic part of themodel can be diagonalized using the standard Bogoliubovrotationˆ b p = u p ˆ a p + v ∗− p ˆ a †− p , ˆ b † p = u ∗ p ˆ a † p + v − p ˆ a − p . (5)While the Bogoliubov approximation thus allows to ob-tain a simple dispersion relation for bosonic quasipar-ticles, it, however, captures the effect of repulsion onlywithin the Bogoliubov coefficients u p and v p . They giverise, e.g., to the modified quasiparticle dispersion ω p = (cid:112) (cid:15) p ( (cid:15) p + 2 ng BB ) where, notably, the non-deformed , ho-mogenous boson density n appears. It turns out that thisapproximate account of boson repulsion is insufficient toself-stabilize the Bose polaron. Indeed, the neglect ofterms beyond quadratic order is responsible for the ap-parent instability of the truncated Hamiltonian (4) . In- The instability becomes physical for a non-interacting BEC [30,34]. stead, it is crucial to keep all terms beyond quadraticorder which allows the repulsive interactions to act as astabilizing counter term to the impurity attraction. In-cluding these terms allows then to expand the theory notsimply around the homogenous BEC but around a BECthat is already deformed due to the presence of the im-purity (for a discussion of the one-dimensional case seeRef. [40]). As discussed in the following, this in turnallows one to effectively map the strong coupling Bosepolaron problem onto a weakly interacting one. A. Gross-Pitaevskii functional Following this strategy we focus on a variational ap-proach to the ground state of the full Hamiltonian (2).Our ansatz is based on a coherent state, expressed in theproduct form [46]Ψ( x , . . . , x N ) = φ ( x ) · · · φ ( x N ) (6)where the condensate wave function φ ( x ) is normalizedto the condensate particle number (cid:82) d d x | φ ( x ) | = N .Both φ ( x ) and the ground-state energy are found by min-imizing the resulting variational Gross-Pitaevskii (GP)energy functional E [ φ ] = (cid:2) p − (cid:82) d d x ¯ φ ( − i ∇ ) φ (cid:3) m I + (cid:90) d d x (cid:104) |∇ φ | m red + V IB ( x ) | φ ( x ) | + g | φ ( x ) | − µ | φ ( x ) | (cid:105) . (7)Here µ is the chemical potential and we assume weak bo-son repulsion represented in three dimensions by a con-tact interaction of strength g = 4 πa BB /m B with bosonscattering length a BB > 0. Note that normal ordering ofthe impurity kinetic term in Eq. (2) contributes to theboson kinetic term with reduced mass m − = m − + m − [21, 30, 36].The energy functional Eq. (7) exhibits two impor-tant limiting cases: (i) for an infinitely heavy impu-rity m I → ∞ , the normal-ordered kinetic recoil term inthe first line of (7) vanishes and the standard GP en-ergy functional for bosons in a static external potentialis recovered; (ii) for a Bose polaron at rest ( p = 0),and a radially symmetric impurity potential V IB ( | x | ),the wave function φ ( x ) is spherically symmetric and therecoil term again vanishes —it only re-appears beyondthe product ansatz when boson correlations are included[36, 47].We find the condensate wave function φ ( x ) by mini-mizing the GP functional (7) in the thermodynamic limitsubject to the boundary conditions | φ ( x → | < ∞ , For weak boson interactions N ≈ N . | φ ( | x | → ∞ ) | = √ n in terms of the condensate den-sity n far away from the impurity. For a radially sym-metric impurity potential V IB ( | x | ) at rest ( p = 0), theground-state wave function is spherically symmetric andreal. In the solution of the GP functional the energy isuniversally expressed in units of the BEC bulk chemi-cal potential µ = gn and the distance from the impu-rity r = | x | is measured in units of the modified heal-ing length ξ = 1 / √ m red µ = 1 / (cid:112) π ( m red /m B ) a BB n ,which involves the reduced mass and is therefore largerthan the usual bulk healing length ξ = 1 / √ m B µ of theBEC without impurity.We express the energy functional for the three di-mensional case in terms of the radial function u ( r ) = rφ ( r ) / √ n that is scaled in relation to the unperturbedBEC expectation value √ n as E [ u ] µ = 4 πn (cid:90) ∞ dr (cid:104) ξ (cid:18) dudr (cid:19) − V IB ( r ) µ u ( r ) + ( u ( r ) − r ) r (cid:105) . (8)Here the boundary conditions for φ ( x ) translate to u (0) = 0 and u ( r → ∞ ) = r . Numerically, E [ u ] is mini-mized by global optimization on r ∈ [0 , L ] with L = 10 ξ and an r -grid spacing ∆ r = 0 . ξ much smaller than thepotential range.We thus obtain the scaling solution of the GP equation(GPE) in units of µ and ξ for a given dimensionless poten-tial shape ˜ V ( x = r/ξ ) = V IB ( r ) /µ . This scaling solutionis universal for arbitrary values of the condensate density n , boson scattering length a BB and mass ratio m red /m B as long as GP theory applies [36, 38], with these parame-ters entering only indirectly via µ and ξ . For comparisonwith experiment, and in order to visualize the effect ofboson repulsion, the universal solution can be rescaled toobtain the specific solution for desired values of the bo-son scattering length a BB and the mass ratio m red /m B in density units of energy E n = (cid:126) n / / m red and length n − / . B. Impurity potential For the impurity potential V IB ( r ) we consider two dif-ferent functional forms. This allows us to study the uni-versality of the Bose polaron by analyzing how quasipar-ticle properties depend on the potential shape and range.Specifically, we consider an attractive Gaussian potential, V gauss ( r ) = − V exp[ − ( r/R ) ] , (9)and an exponentially decaying potential, V expon ( r ) = − V exp[ − r/R ] , (10)both of depth V > R . The low-energy scattering properties of the impurityand boson are characterized by the impurity-boson scat-tering length a IB and the effective range r eff , which deter-mine the leading terms of the effective range expansion ofthe momentum-dependent scattering phase shift in threedimensions. They are found by numerically solving theSchr¨odinger equation for the scattering of a boson withthe impurity in the center-of-mass system via the poten-tial V IB ( r ). Equivalently, one may solve the first-ordernonlinear variable phase equation [48], which yields a (cid:48) ( r ) = 2 m red V IB ( r )[ r − a ( r )] ,r (cid:48) eff ( r ) = − m red V IB ( r ) r (cid:16) ra ( r ) − (cid:17) · (cid:16) r a ( r ) − r eff ( r ) r (cid:17) . (11)Here a ( r ) and r eff ( r ) obey the boundary conditions a (0) = 0, r eff (0) = 0, and account for the phase shiftaccumulated by the scattering wave function (for the gen-eralization of the variable phase equation to singular po-tentials see [49]). Correspondingly, the differential equa-tions (11) are integrated from r = 0 . . . ∞ which yieldsthe scattering length a IB = a ( r → ∞ ) and effective range r eff = r eff ( r → ∞ ). III. EFFECTIVE POTENTIAL First, we present results for the condensate profilearound the impurity that we find by minimizing theGross-Pitaevskii energy functional (7)–(8). For at-tractive impurity-boson interaction, the wave function φ ( r ) —which, in the comoving frame, directly yieldsthe impurity-boson density-density correlation function g (2)IB ( r ) = | φ ( r ) | /n — is enhanced near the impurity,as shown in Fig. 2(a). In this figure we have chosen apotential V IB ( r ) (dashed line in Fig. 2(b)) that is suffi-ciently deep to support a two-body bound state at energy − ε B < 0. Correspondingly, the potential is character-ized by a positive impurity-boson scattering length, inFig. 2(b) chosen as a IB ≈ ξ > V eff ( r ) = V IB ( r ) + g | φ ( r ) | − µ, (12)that results from by the competition of the bare attrac-tive potential V IB ( r ) (dashed line in Fig. 2(b)) and therepulsion created by the already existing polaron cloud.In essence, this effect can be understood as arising inan effective density-functional theory (DFT) for the BEC g ( ) I B ( r ) = | ( r ) | / n (a) condensate densityunperturbed BEC r n V ( r ) / (b) eff. potential V eff ( r ) = V IB ( r ) + g | ( r )| bare potential V IB ( r ) = V exp[ ( r / R ) ] FIG. 2. (a) Impurity-boson density-density correlation func-tion g (2)IB ( r ) as determined from the condensate wave function φ ( r ) as function of the distance r from the impurity (bluesolid). For an attractive impurity potential the wave func-tion is enhanced near the impurity as compared to that ofan unperturbed BEC φ ( r ) = √ n (orange dashed). (b) Thebare impurity potential (orange dashed) of Gaussian shape( V /µ = 5 . R/ξ = 0 . r eff = ξ and positive scattering length a IB = 4 ξ , and correspondinglyadmits a bound state. An extra bosonic test particle is, how-ever, subject to the effective potential (blue solid) that isweakened by the repulsion from the polaron cloud; while stillattractive the effective potential is characterized by an effec-tive, negative scattering length a IB,eff = − . ξ (and renor-malized ˜ r eff = − ξ ) which thus no longer supports a boundstate. Hence, additional bosonic quantum fluctuations leadonly to a weak, additional dressing of the impurity particle. particles in the presence of the impurity: the bosonsalready attracted to the impurity screen the attractivepotential and make it shallower, as shown by the blueline in Fig. 2(b). Moreover, we find that they createa small repulsive barrier at intermediate distance. Asa consequence, the effective potential no longer admitsa bound state and it is correspondingly characterizedby a negative effective impurity-boson scattering length a IB,eff ≈ − . ξ . Hence, the single-particle excitationspectrum for each additional boson is bounded from be-low: the dynamical instability is replaced by Bose po-larons self-stabilized by their dressing cloud . IV. BOSE POLARON ENERGY AND CONTACT The value of the Gross-Pitaevskii functional (7) eval-uated at the ground-state wave function determines thepolaron energy E = E [ φ ] relative to the homogeneousBEC. The energy is shown in Fig. 3(a) as a function ofthe dimensionless impurity-boson interaction ξ/a IB . Itis always negative for an attractive impurity potential.In Fig. 3(b) we present the energy for a mobile impu-rity of arbitrary mass in units of E n which depends on / a IB E / n (a) scaling solution a IB n E / E n (b) m red / m B ) a BB n = 1.08 ( m red / m B ) a BB n = 0.5mean field a IB n N c l o u d (c) a IB n r e ff n / FIG. 3. Bose polaron energy across an impurity-boson Fesh-bach resonance. (a) Scaling solution for the polaron energy E/µn ξ as function of the impurity-Bose interaction ξ/a IB for a Gaussian potential of fixed range R = ξ . (b) Polaronenergy E in density units E n = (cid:126) n / / m red in dependenceon the impurity-Bose interaction 1 /a IB n / for different BECgas parameters. The polaron binding energy is larger for weakboson repulsion 8 π ( m red /m B ) a BB n / = 0 . π ( m red /m B ) a BB n / = 1 . N cloud withinthe polaron dressing cloud. Inset: the effective range r eff forfixed potential range Rn / = 1 is smallest on the repulsiveside. the BEC density n . Specifically, we show results fortwo BEC gas parameters, which for equal mass of bosonand impurity, i.e., m red /m B = 1 / 2, correspond to values n a = (4 π ) − = 5 . × − ( n ξ = 1) at strongerboson repulsion and n a = (8 π ) − = 0 . × − ( n ξ = 2 . 8) at weaker boson repulsion. We find that,in absolute terms, the polaron binding energy is largerfor weaker boson repulsion where the BEC can be morestrongly deformed and thus acquire more attractive po-tential energy.At weak impurity-bath attraction 1 /a IB n / (cid:28) − E mf = 2 πa IB n m red = 4 πa IB n / E n (13)irrespective of a BB . Within mean-field theory the po-laron energy diverges to −∞ at unitarity 1 /a IB = 0.Variational approaches based on the trunctated Hamil-tonian Eq. (4) predict that the inclusion of Bogoliubovcorrections is not sufficient to heal this instability, butrather results in a shift of the instability to the repul-sive side of the Feshbach resonance [30]. In contrast, wefind that going beyond the Bogoliubov approximation byworking with the full Hamiltonian (1) allows the bosonrepulsion to stabilize the polaron at a finite ground-stateenergy that smoothly crosses over from the attractive tothe repulsive side of the Feshbach resonance.The deformation of the BEC is also reflected in thenumber of bosons participating in the formation of thepolaron dressing cloud N cloud = 4 πn (cid:90) ∞ dr [ u ( r ) − r ] . (14)As shown in Fig. 3(c), for a smaller gas parameter (orangedashed line) the impurity attracts a larger polaron cloudbecause the bosons are less repulsive. Naturally, thislarger dressing cloud corresponds to the larger polaronbinding energy found in Fig. 3(b).Our results in Fig. 3 are shown for a constant range R across the Feshbach resonance as applicable to exper-iments where the microscopic range of interactions cantypically not be tuned synchronously with the scatteringlength. Thus the effective range r eff varies in dependenceon a IB : The inset of Fig. 3(c) shows the effective rangethat is obtained from the scattering phase shift usingEq. (11). At constant potential range Rn / = 1, theeffective range r eff reaches a minimum on the repulsiveside of the resonance ( a IB > 0) and grows towards bothweak-coupling limits a IB → ± .The variation of the polaron energy with a IB definesthe impurity contact parameter [34, 36, 38, 43, 50] whichcharacterizes the impurity-boson correlations g (2)IB ( r ) atshort distances outside the impurity potential: C = 8 πm red (cid:126) ∂E∂ ( − /a IB ) = 4 πn / ∂ ( E/E n ) ∂ ( − /a IB n / ) . (15)The contact is shown in Fig. 4: we find that it reachesa maximal value on the repulsive side of the resonance.Similar to the energy, we find that the contact is largerfor smaller boson repulsion a BB (blue solid line) wherethe BEC is more strongly deformed and thus g (2)IB ( r ) isenhanced (see Fig. 2). At weak attractive interaction thecontact approaches the ground-state value of an impurityin an ideal BEC ( a BB = 0) [34] C mf = 16 π n a , (16)which agrees with the mean-field result prediction. a IB n C n / m red / m B ) a BB n = 1.08 ( m red / m B ) a BB n = 0.5mean field FIG. 4. Tan contact C of the Bose polaron for differentboson repulsion (BEC gas parameter) across the impurity-boson Feshbach resonance. The contact Cn − / obtainedfrom Eq. (15) reaches its maximum on the repulsive side ofthe resonance and increases for weaker boson repulsion. Atweak coupling it approaches the mean-field result (16) (greendotted). V. UNIVERSALITY Finally, we test the notion of universality of the Bosepolaron by studying different shapes and ranges R of theimpurity potential. Generally, our local Gross-Pitaevskiitheory is reliable as long as the potential range R is notso short that the assumption of a slowly varying potentialis violated [51]. To test universality in this regime, wespecifically compare the predictions following from theGaussian potential V gauss ( r ) in Eq. (9), and the expo-nential potential V expon ( r ) (10) for various ranges R .We find that when tuning the depth of the potentials toobtain equal scattering length a IB = −∞ at equal range R , the polaron energies are very different. Instead, if R is tuned to yield the same effective range r eff for bothpotentials, as shown in Fig. 5(a), remarkable agreementbetween the polaron energies is found. Indeed, as shownin Fig. 5(b), for both potentials, the polaron energy ap-proximately follows a linear scaling law for r eff (cid:38) . ξ , E (1 /a IB = 0) E n = − . − . r eff n / . (17)This shows that in the Bose polaron problem themomentum-dependent scattering phase shift is probed inthe regime where the effective range expansion is valid.Remarkably, the polaron energy remains sizeable evenfor short effective range. Indeed, our result generalizesa recent study which found a power-law scaling of theunitary polaron energy at ranges much shorter than thehealing length, E/E n ∼ ( r eff /ξ ) / for a BB (cid:46) r eff (cid:28) ξ in the case of a square-well potential [38]. In the regionfor r eff = 2 a BB this approach yields a polaron energy of E ≈ − E n as indicated by the star in Fig. 5(a), whichis consistent with the linear extrapolation of our result. r eff n R n / R Gauss = 0.69 r eff R expon = 0.28 r eff (a) r eff n E / E n E Gauss / E n r eff n E expon / E n r eff n (b) Gaussian potentialexponential potential r eff n E / E n (c) a IB = 1 r eff n E / E n (d) a IB = + 1 FIG. 5. Universality of the Bose polaron. (a) In order toyield the same effective range r eff , the Gaussian and exponen-tial potentials need to be tuned to different potential ranges R ; here shown for fixed unitary scattering length 1 /a IB = 0.(b) Polaron energy E as function of the effective range r eff atunitarity 1 /a IB = 0 for two different potential shapes. Theboson repulsion is set to 8 π ( m red /m B ) a BB n / = 1. The po-laron energy coincides for both potential shapes and increaseslinearly with r eff . For short range it approaches the valuefor r eff = 2 a BB from [38] (star). (c) The polaron energy atnegative scattering length 1 /a IB n / = − r eff (cid:38) ξ . (d) Polaron energy at positive scattering length1 /a IB n / = 1. For a deep Gaussian potential, several boundstates can appear and there is no unique polaron energy fora given r eff (see text). These findings can be further extended to even shorterranges r eff (cid:28) a BB by using a nonlocal generalization ofGross-Pitaevskii theory [36].Also on the attractive side of the resonance, seeFig. 5(c) for 1 /a IB n / = − 1, we find a universal effec-tive range dependence of the polaron energy, albeit not alinear one. Note that as the Gaussian potential becomesdeeper more bound states appear, each one associatedwith a scattering resonance in a IB . Clearly, for fixed a IB and effective range r eff each bound state yields a differentpolaron energy, so that there is no longer a unique map-ping from r eff to E . This is illustrated in Fig. 5(d) for therepulsive side, at 1 /a IB n / = 1: universality can at besthold in the vicinity of the first bound state, and onlyfor a limited interval of r eff values. Furthermore, sincethe Efimov effect modifies the polaron energy spectrum[2, 3, 41], universality can also depend on the three-bodyparameter [42, 52–54]. VI. DISCUSSION The variational principle provides a powerful tool tocompute both ground-state and dynamical properties ofquantum many-body systems. However, in order to makefull use of its predictive power it is essential to under-stand the limitations of approximations applied to theHamiltonian to be analyzed. In this regard the strong-coupling Bose polaron is a case in point. The full Bosepolaron Hamiltonian (1) —and equivalently Eq. (3)— isbounded from below for repulsive boson interaction, andhence variational wave functions give rigorous bounds onthe ground-state energy. For finite Bose repulsion, thepolaron energy remains finite for any Bose-impurity scat-tering length, including resonant interactions, and theground state represents a strong-coupling Bose polaronthat is self-stabilized by its own dressing cloud.In contrast, when applying the Bogoliubov approxima-tion to the full model (1) by truncating terms of higher-than quadratic order in the boson operators, the result-ing, truncated Hamiltonian ˆ H (cid:48) in Eq. (4) becomes un-bound from below. Crucially, this results in an instabil-ity of the Bose polaron problem that is solely an artefactof this approximation.Quite remarkably, the instability of the truncatedHamiltonian ˆ H (cid:48) becomes, however, only evident whenconsidering wavefunctions that account for more thantwo phonon excitations from the homogenous BEC. Forinstance, for a Chevy-type wavefunction [25, 27, 28] thatitself is truncated at the single excitation level, the termsbeyond quadratic order in the repulsive interactions havevanishing expectation value. Thus, incidentally, theChevy ansatz yields the same prediction when appliedto both the full and the truncated model. Thus due toits tremendous simplicity the Chevy ansatz becomes im-mune to the instability of approximate Hamiltonian ˆ H (cid:48) .However, while being a well-defined approach, the sim-ple Chevy ansatz misses the fact that at the weak Bosonrepulsion (as typically present in cold gases) the polaroncloud—even within the full model Eq. (1)—can containan exceedingly large number of bosons (Fig. 3(b)). Sucha large local deformation of the BEC is naturally cap-tured by the coherent state (6). Crucially, while account-ing for an arbitrary number of boson excitations, whenapplied to the full Hamiltonian (1), it still leads to abounded energy functional (7). Its solution shows thatthe smaller the boson repulsion and the wider the impu-rity potential, the larger the polaron cloud becomes. Thecoherent state approach is complementary to the Chevyansatz including its extensions to multi-boson excitations[3, 41, 55], and it becomes particularly accurate for softimpurity potentials where it is justified to ignore bosoniccorrelations. Remarkably, the case of extremely soft po-tentials is realized with Rydberg excitation immersed inBECs. In this case it was predicted that up to hundredsof atoms can be bound to the single impurity leading tothe creation of Rydberg polarons [56]. Since for Ryd-berg impurities the range of interactions R dramaticallyexceeds the interparticle distance, our local GP theoryapplies and provides a so far missing explanation as towhy the experimental observation of Rydberg polarons[29] is described exceptionally well by a coherent stateapproach [26].Recently also first steps to the understanding of thecomplementary, intermediate regime of short-range im-purity potentials —with yet large dressing clouds— hasbeen achieved by using an extension to nonlocal Gross-Pitaevskii theory [36]. In conjunction with our presentresult, these combined new approaches resolve a funda-mental shortcoming of the Bogoliubov approximation:while the formulation of the interacting Bose gas in termsof Bogoliubov quasiparticles is exact, the additional ap-proximation to neglect the residual interaction betweenphonons is not. In particular, the quadratic Bogoli-ubov mean-field Hamiltonian is unbounded from belowfor the strong-coupling Bose polaron, most obviously inthe regime where a two-body bound state appears on therepulsive side of the resonance [30, 31].As discussed above, the Chevy-type ansatz appliedto the truncated the Bogoliubov Hamiltonian [28] stillyields a finite polaron energy since it is of such low or-der in boson excitations that it is not sensitive to thetruncated part of the Hamiltonian. However, when thecoherent state ansatz or higher-order excitation exten-sions of the Chevy ansatz [41, 55] are applied to the trun-cated Hamiltonian (4) they lead to the aforementioned,spurious divergence of the ground-state energy and dy-namical instabilities in the nonequilibrium time evolution[30, 31, 33]. Instead the local extension to the truncatedBogoliubov approach studied in the present work (seealso [38, 42, 43]) as well as its nonlocal counterpart [36]provides a stable starting point for strong-coupling Bosepolaron dynamics, and we showed how it can find an ef-fective description in terms of a renormalization of theimpurity-boson potential (Fig. 2).Beyond our treatment of the two-particle impurity-boson correlations, three-body and higher-order corre-lations give rise to the Efimov effect and three-body re-combination. The Efimov effect can occur either betweenone impurity and two bosons [3, 41, 55] or between twoimpurities and one boson [2, 6]. These few-body effectscan be captured by Gaussian variational wave functions [47]. Alternatively, extensions of the Chevy ansatz totwo or more independent bosonic excitations [3] can beemployed. The latter approach was applied in the anal-ysis of the truncated model (4) and universal scaling de-pending on the three-body parameter was found [41, 55].It remains an interesting open question how universalthree-body physics carries over to the many-body casein a dense bosonic medium when the full HamiltonianEq. (1) is considered.Finally, in ultracold atomic gases the boson repulsionoriginates from an attractive van-der-Waals potential be-tween atoms. This results in the existence of deeplybound molecular states into which atoms can decay inthree-body recombination. These deeply bound statesare neither accounted for in variational approaches norin quantum Monte Carlo [39]. Up to now most experi-ments probe the metastable Bose polaron state of matteron transient time scales where the deeply bound states —arising both from the fundamentally attractive Bose-Boseand Bose-impurity potentials— can be ignored, and thusthe stable variational approach discussed in our work iswell applicable. However, as one starts to explore longertime scales or the build-up of more complex correlatedstates of impurities, understanding the impact of the fun-damental dissipative nature arising from deeply boundstates becomes essential and requires the development ofnew theoretical approaches to quantum impurity prob-lems. ACKNOWLEDGMENTS This work is supported by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation),project-ID 273811115 (SFB1225 ISOQUANT) and underGermany’s Excellence Strategy EXC2181/1-390900948(the Heidelberg STRUCTURES Excellence Cluster).R. 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