Universality of an Impurity in a Bose-Einstein Condensate

PHYSICAL REVIEW X 8, 011024 (2018)

Shuhei M. Yoshida,1,2 Shimpei Endo,2 Jesper Levinsen,2 and Meera M. Parish2 1Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan 2School of Physics and Astronomy, Monash University, Victoria 3800, Australia

(Received 13 October 2017; revised manuscript received 12 December 2017; published 13 February 2018)

We consider the ground-state properties of an impurity particle (“polaron”) resonantly interacting with a Bose-Einstein condensate (BEC). Focusing on the equal-mass system, we use a variational wave function for the polaron that goes beyond previous work and includes up to three Bogoliubov excitations of the BEC, thus allowing us to capture both Efimov trimers and associated tetramers. We find that the length scale associated with Efimov trimers (i.e., the three-body parameter) can strongly affect the polaron’s behavior, even at densities where there are no well-defined Efimov states. However, by comparing our results with recent quantum Monte Carlo calculations, we argue that the polaron energy is a universal function of the Efimov three-body parameter for sufficiently low boson densities. We further support this conclusion by showing that the energies of the deepest bound Efimov trimers and tetramers at unitarity are universally related to one another, regardless of the microscopic model. On the other hand, we find that the quasiparticle residue and effective mass sensitively depend on the coherence length ξ of the BEC, with the residue tending to zero as ξ diverges, in a manner akin to the orthogonality catastrophe.

DOI: 10.1103/PhysRevX.8.011024 Subject Areas: Atomic and Molecular Physics, Condensed Matter Physics, Quantum Physics

I. INTRODUCTION question whether the unitary Bose system can also display universal behavior that has relevance to a range of different Universality is a powerful concept in physics, allowing physical systems. one to construct physical descriptions of systems that are In contrast to the Fermi case, bosonic systems at unitarity independent of the precise microscopic details or energy always support few-body Efimov bound states [21–26], scales. A prime example is the dilute two-component Fermi whose energy and size depend on short-distance parameters gas at unitarity [1], where the scattering length a → ∞ such as the interaction range r. Thus, there is an additional and the interparticle spacing greatly exceed the range of the interaction length scale that can impact the behavior of the interactions. Here, the large separation of length scales unitary Bose system and potentially make it sensitive to means that the unitary Fermi gas is independent of an microscopic details. Indeed, the deepest bound trimers and interaction length scale, thus resulting in universal proper- larger few-body clusters typically cannot be universally ties that describe systems ranging from atomic gases at related to each other without invoking a specific model for microkelvin temperatures to the inner crust of neutron stars. the short-range interactions [24,25]. Indeed, highly controlled cold-atom experiments have To investigate the universality of a Bose system at successfully determined its universal equation of state unitarity, we consider an impurity immersed in a Bose- [2–4], pairing correlations [5–12], and transport properties Einstein condensate (BEC), where the boson-impurity [13]. More recently, cold-atom experiments have started to interactions can be tuned to unitarity while the rest of probe Bose gases in the unitary regime [14–20].Even the system remains weakly interacting. This so-called though the strongly interacting Bose gas is expected to be “Bose polaron” has received much theoretical attention unstable, there are regimes where the system appears to be [27–41] and has very recently been observed in cold-atom sufficiently long-lived to observe well-defined static and experiments [15,16]. Moreover, it has been extended to dynamical properties [14,16]. However, it remains an open impurities with more complex internal structure [42–44], and it has potential relevance to polaron problems in solid- state systems—for instance, the limit of weak impurity- Published by the American Physical Society under the terms of boson interactions can be directly mapped to the Fröhlich the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to model [27,45]. For stronger interactions, two recent works the author(s) and the published article’s title, journal citation, investigated the possibility of coupling the polaron to and DOI. Efimov trimers [32,40]. However, there has yet to be a

2160-3308=18=8(1)=011024(16) 011024-1 Published by the American Physical Society YOSHIDA, ENDO, LEVINSEN, and PARISH PHYS. REV. X 8, 011024 (2018)

Four atoms continuum function for the ground-state Bose polaron which recovers both three- and four-body equations in the limit of zero density. Strikingly, we find that the polaron properties at j j -0.05 Two atoms+dimer continuum unitarity sensitively depend on the Efimov scale a− ,even in the regime where ja−j far exceeds the interparticle -0.10 Atom + trimer continuum spacing. However, we show that the ground-state polaron 1=3 -0.15 energy is a universal function of n ja−j that is model independent in the regime n1=3r ≪ 1. We corroborate this -0.20 finding by comparing our results with recent quantum -0.25 Monte Carlo (QMC) calculations [33,36]. In the nonuniversal, high-density limit n1=3r ≫ 1,we -0.30 consider the case of a narrow Feshbach resonance, and we -0.35 derive perturbative expressions for the polaron energy and -0.2 -0.1 0 0.1 0.2 contact at unitarity. This allows us to demonstrate, for the first time, that the Bose polaron at unitarity can be FIG. 1. Few-body energy spectrum calculated from Eqs. (7)– thermodynamically stable even in the limit of vanishing (9). The ground (first excited) trimer is shown as a red solid curve, boson-boson interactions. On the other hand, we find that the ð1Þ which first appears at a negative scattering length a− ða− Þ and quasiparticle residue and effective mass sensitively depend then dissociates into the atom-dimer continuum (delimited by the on the BEC coherence length. For the case of an ideal BEC, ð1Þ black solid line) at a positive scattering length a ða Þ, where a the effective mass converges to a finite value as we increase is outside the plotted region. We only show the ground and first the number of excitations of the condensate, while the excited trimers, but there exists an infinite series of higher excited residue vanishes in a manner reminiscent of the orthogon- trimers with an accumulation point at unitarity, 1=a ¼ 0. We also ality catastrophe for a static impurity in a Fermi gas [53]. find two tetramer states (blue dashed lines) tied to the ground Efimov trimer. The excited tetramer is very weakly bound, but it persists at unitarity and disappears into the atom-trimer con- II. MODEL AND VARIATIONAL APPROACH ≃ 9 103j j tinuum at the positive scattering length a × r0 . We consider an impurity atom immersed in a weakly interacting homogeneous Bose-Einstein condensate at zero systematic study of the effect of Efimov physics on the temperature. The interactions in the medium are charac- Bose polaron at unitarity. terized by the boson-boson scattering length aB, and we 3 ≪ 1 The Bose polaron is a promising system to search for thus require n0aB , with n0 the condensate density universal behavior since the Efimov trimers consisting of (which essentially equals the total density n in this regime). the impurity and two bosons are orders of magnitude larger This allows us to treat both the ground state and the than the short-distance scale r (provided the impurity mass excitations of the BEC within Bogoliubov theory [54]. is not too small) [46,47]. If we parametrize the size of the Note that we always implicitly assume aB > 0 to ensure the deepest Efimov trimer by ja−j, where a− is the scattering stability of the condensate. length at which the trimer crosses the three-atom con- The presence of the impurity in the medium adds two tinuum (see Fig. 1), then the hierarchy of length scales at length scales associated with the impurity-boson interaction. −1=3 unitarity is r ≪ n ≪ ja−j for the typical densities n in The first of these is the scattering length a between the experiment [15,16]. This suggests that there is a regime impurity and a boson from the medium. We disregard this where Efimov physics is irrelevant, such that the ground- length scale since we focus on the strong-coupling unitary state properties of the polaron only depend on the density, regime close to a Feshbach resonance, where jaj greatly like in the case of an impurity resonantly interacting with a exceeds all other length scales; i.e., we generally consider Fermi gas [48–52]. 1=a ¼ 0. Here, we assume short-range contact interactions, In this paper, we address this question using impurities a scenario that is well realized in current cold-atom that have a mass equal to that of the bosons—a situation experiments. that has already been realized in the 39K atomic gas The second additional length scale is the so-called three- experiments in Ref. [16]. We first investigate the few-body body parameter, which characterizes the size of the smallest limit and determine the Efimov trimer and associated (ground-state) Efimov trimer consisting of two bosons and tetramer states that involve the impurity. For vanishing the impurity. To demonstrate the model independence of boson-boson interactions, we show that the ratios of the our results, we fix this Efimov length scale in two differ- binding energies for the deepest bound states are universal, ent ways: unlike the case of three and four resonantly interacting (i) r0 model—We introduce an effective range r0, identical bosons. We then include this Efimov physics in which directly relates the three-body parameter to the many-body system by constructing a variational wave the two-body interaction [55].

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Λ — → 0 1 (ii) model We take r0 and instead apply an ð Þ¼− ð Þ Λ f k −1 1 2 : 3 explicit ultraviolet cutoff to the momenta involved a − 2 r0k þ ik in Efimov physics, which is equivalent to including a three-body repulsion [56]. Carrying out the renormalization procedure with high- As we explicitly show in Sec. III, for an impurity of the momentum cutoff k0, one obtains [58,59] same mass as the medium atoms—the scenario considered here—the precise manner of regularizing Efimov physics is 2 1 8π ¼ mg ¼ − ð Þ unimportant, owing to the large separation of scales a 2 ;r0 2 2 ; 4 4π g mk0 − ν m g between the short-range physics and Efimov physics. 2π2 0 Given the above considerations, we model the system which relates the physical low-energy observables—the using a two-channel description of the Feshbach resonance ≤ 0— [57], which corresponds to the Hamiltonian (setting ℏ and scattering length a and effective range r0 to the bare the volume to 1): parameters of the model. In all calculations, we take the limit k0, ν0 → ∞ for a given set of values for the X observables. We emphasize that our two-channel model ˆ ¼ ½ β† β þ ϵ † þðϵd þ ν Þ † H Ek k k kckck k 0 dkdk describes the exact impurity-boson interaction without any k X X approximations arising from the presence of the conden- pffiffiffiffiffi † † þ g n0 ðdkck þ H:c:Þþg ðdqcq−kbk þ H:c:Þ: sate. It thus goes beyond the Fröhlich model, which ignores k k;q terms where both incoming and outgoing bosons in a ð1Þ scattering process are Bogoliubov excitations. Such terms have been demonstrated to be important beyond the weak- † † coupling regime [31]. In our model, we recover the single- Here, bk and ck are the creation operators of a boson and ¼ 0 channel model (where r0 Pand the interaction part of the the impurity, respectively, with momentum k and single- † † 0 b 0 c 0 cq−kbk ϵ ¼ k2 2 Hamiltonian has the form k;k ;q k q−k ) by tak- particle energy k = m, where the mass m of a boson → ∞ and of the impurity are taken to be equal. ing g while keeping a fixed. In writing the Hamiltonian (1), we already applied the To explore the ground-state properties of the unitary Bogoliubov approximation for the single-particle excita- Bose polaron, we apply the variational principle using a † wave function consisting of terms jψ i with an increasing tions of the weakly interacting condensate: The operator β N k number N of Bogoliubov modes excited from the con- creates a Bogoliubov excitation and is related to the bare † densate by the impurity: boson operator bk by the transformation jΨi¼jψ 0iþjψ 1iþjψ 2iþjψ 3i: ð5Þ † † bk ¼ ukβk − vkβ−k: ð2Þ Explicitly, we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The coherence factors are uk ¼ ½ðϵk þ μÞ=Ek þ 1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jψ i¼α †jΦi ¼ ½ðϵ þμÞ −1 2 0 0c0 ; and vk pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik =Ek = , the Bogoliubov dispersion X ¼ ϵ ðϵ þ 2μÞ † † † is Ek k k , and the boson chemical potential is jψ 1i¼ αkc−kβk þ γ0d0 jΦi; 2 μ ¼ 4πn0aB=m ≡ 1=ð2mξ Þ, where ξ is the coherence k X X length of the condensate. Note that Eq. (1) is defined with 1 † † † † † jψ 2i¼ αk k c β β þ γkd β jΦi; respect to the energy of the BEC in the absence of the 2 1 2 −k1−k2 k1 k2 −k k impurity. k1k2 k X The interaction between a boson and the impurity is 1 † † † † jψ 3i¼ αk k k c β β β described by the terms in the second line of Eq. (1), where 6 1 2 3 −k1−k2−k3 k1 k2 k3 k1k2k3 the first of these arises from explicitly extracting the X condensate k ¼ 0 contribution from the second term. For 1 † † † þ γk k d β β jΦi; ð6Þ clarity, we write the interaction part of the Hamiltonian in 2 1 2 −k1−k2 k1 k2 k1k2 terms of the original boson operator, but note that we always use Eq. (2) to relate this to a Bogoliubov excitation. with jΦi the ground-state wave function of the weakly Within the two-channel model, a boson and the impurity interacting BEC. Including up to three excitations allows us † interact by forming a closed-channel dimer created by dk, to investigate the effect of both Efimov trimers and d with ϵk ¼ ϵk=2. The interchannel coupling g and the bare tetramers on the unitary Bose polaron. This goes beyond detuning ν0 are chosen such that they reproduce the previous works on the Bose polaron: The effect of dressing two-body scattering amplitude in vacuum at relative the impurity by a single excitation was investigated in momentum k: Ref. [28] using a T-matrix approach and in Ref. [29] with a

011024-3 YOSHIDA, ENDO, LEVINSEN, and PARISH PHYS. REV. X 8, 011024 (2018) variational approach. Furthermore, two of us previously However, this approximation is reasonable when the used two-excitation dressing to investigate the relationship Efimov states are insensitive to aB, i.e., when the micro- between polaronic and Efimov physics [32], as well as to scopic length scale determining the three-body parameter successfully compare with the experimentally obtained greatly exceeds aB. polaron spectral function [16] (see also Ref. [60]). In Fig. 1, we show the spectrum of bound states obtained To investigate the properties of the ground-state Bose by solving Eqs. (7)–(9) within the r0 model. For a positive polaron, we determine the variational parameters by the scattering length, a single two-body bound dimer exists, ˆ stationary condition ∂α;γ hΨjðH − EÞjΨi¼0, where the with energy energy E can be viewed as the Lagrange multiplier that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 1 − 2 − 1Þ2 ensures normalization. This yields a set of coupled equa- ε ¼ − r0=a ð Þ α B 2 ; 11 tions, from which we can eliminate the coefficients and mr0 obtain coupled integral equations for the γ’s only. The resulting equations are given in Appendix A. In the r0 as found by solving Eq. (7). The three-body problem model, we keep the effective range r0 in the two-body described by Eq. (8) is distinctly different: For strong interaction, while in the Λ model, we set r0 ¼ 0 and apply interactions, the impurity and two bosons can form an the cutoff Λ to the momenta in the γ coefficients. infinite series of three-body bound states, even in the absence of the two-body bound state. Remarkably, these III. UNIVERSAL FEW-BODY EFIMOV STATES so-called Efimov trimers [21,62] satisfy a discrete scaling symmetry whereby the energy E of one trimer can be Before proceeding to the many-body polaron physics, it mapped onto the next via the discrete transformation — −2l l is important to first discuss the few-body limit since as we E → λ E and a → λ a, with l an integer and λ0 a scaling — 0 0 explicitly demonstrate in Sec. IV this plays a crucial role parameter that depends on the particular three-body prob- in understanding the polaron properties. To determine the lem (for reviews, see Refs. [22,24–26]). In our case of two few-body spectrum, we use wave functions of the form (6) noninteracting identical bosons with the same mass as the → 0 jΦi in the limit n , such that now corresponds to the third particle, the scaling factor is found from a transcen- jψ i Fock vacuum. Here, the state with N excitations is π=s0 N dental equation to be λ0 ¼ e ≃ 1986.1 [62]. Because of identified as a few-body state containing N bosons and the this large scaling factor, we only show the two lowest trimer impurity. Consequently, the variational approach is exact states in the figure (indeed, for realistic experimental and the sectors of different particle numbers completely parameters, the size of the first excited Efimov trimer is decouple. Ignoring the trivial equation arising from the of order 1 cm). single-particle problem, the resulting few-body equations Below the ground-state Efimov trimer, we predict the take the form existence of two tetramer states, even at unitarity (see Fig. 1). This mirrors previous results for four identical T −1ðE; 0Þ¼0; ð Þ 0 7 bosons, where it was found, both theoretically [63–66] and X γ experimentally [67], that there are two lower-lying tet- −1 q T 0 ðE − ϵk; kÞγk ¼ ; ð8Þ ramers associated with the ground-state trimer. The present E − ϵ q kq scenario more closely resembles studies of a light atom Xγ þγ resonantly interacting with three identical heavy atoms −1 k1q qk2 T ðE−ϵk −ϵk ;k1 þk2Þγk k ¼ ; ð9Þ [47,68,69]. Our findings are generally consistent with the 0 1 2 1 2 −ϵ q E k1k2q results of these previous works, although no excited tetramer was reported for small mass imbalance [70]. which may be obtained as the zero-density limit of the Note that, within our theory, we do not require an additional polaron equations in Appendix A. Here, we have defined length scale to fix the positions of the tetramers relative to ϵ ≡ ϵ þ ϵ þ ϵ ϵ ≡ ϵ þ ϵ þ ϵ þ k1k2 k1 k2 k1þk2 and k1k2q k1 k2 q the Efimov trimers [65]. ϵ T k1þk2þq, while 0 is the vacuum two-body T matrix, A crucial point is that the few-body physics described here is essentially universal, in the sense that the entire 2 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m r0 m2 spectrum is independent of the details of the short-range T −1ðE;kÞ¼ − ðE−ϵd Þ− ϵd −E−i0: ð10Þ 0 4πa 8π k 4π k physics. This universality originates from the large separation of scales between the short-distance physics, charac- Equation (7) gives the condition for a two-body bound terized here by the effective range r0, and the typical size of state, while the solutions of Eqs. (8) and (9) yield, the ground-state Efimov trimer. The latter may be conven- respectively, the trimer spectrum and associated tetramers iently characterized by the “critical” scattering length a− at [61]. Note that boson-boson interactions are neglected in which the trimer crosses into the three-atom continuum. In this limit since they are only included via a density- our model, we have a−=r0 ≃ 2467 [72] (see Table I for a dependent mean-field shift in the boson chemical potential. summary of the length scales and energy scales associated

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TABLE I. Comparison of few-body data for the r0 and the Λ models. While the overall length scale set by a− depends on the details of the models, the dimensionless ratios characterizing the few-body spectra agree to an accuracy of 1% or less. These quantities include the scattering lengths at which the few-body bound states cross into the continuum, as well as their energies at unitarity, and are illustrated in Fig. 1. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j j 2 j j ð1Þ ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ a− ma− E− a− =a− E−=E− a−=a4B;− E4B;−=E− a−=a4B;− E4B;−=E− r0 model 2467jr0j 9.913 1991 1986.126 2.810 9.35 1.060 1.0030(3) Λ model 1354Λ−1 9.950 1987 1986.127 2.814 9.43 1.061 1.0036(1) with Efimov physics). This large ratio ensures that the four-body parameter to fix the tetramer relative to the energy of the ground-state Efimov trimer at unitarity is far trimer. This is consistent with theoretical results for smaller than that required to probe the short-range physics. identical bosons [63,64,66] and for heteronuclear systems Indeed, we see that the ratio of critical scattering lengths for [47,68,69]. ð1Þ the excited- and ground-state trimers, i.e., a− =a− ≃ 1991, We end this section by contrasting the few-body uni- is very close to the predicted scaling factor of 1986.1. versality found here with the so-called van der Waals Likewise, at unitarity, the ratio of the ground-state trimer universality of Efimov physics [77–81]. The latter refers to ð1Þ 2 energy to that of the excited state is E−=E− ≃ ð1986.1Þ ; the phenomenon where the Efimov states of various atomic i.e., it matches the universal prediction to 5 significant digits. species are universally related to their van der Waals length, ≃ −ð9 1 1 5Þ This is in remarkable contrast to the situation for three e.g., a− . . rvdW. This behavior, however, is identical bosons, where deviations from the universal rather different from that predicted by universal zero-range scaling factor of 22.7 are usually on the order of Efimov theory, in particular, for the ground-state trimer. ð1Þ 10%–20% because of finite-range corrections [22,24,73]. Indeed, the ratio a− =a− ≃ 20.7 found in van der Waals To test the model independence of the few-body physics, universal theory [81] significantly deviates from the we also consider the Λ model, where r0 ¼ 0 and we apply universal ratio 22.7, in contrast to the scenario considered an explicit cutoff Λ to the momentum sums in the three- and here. four-body equations (8) and (9). Here, we find that after fixing the three-body parameter, the spectrum is essentially indistinguishable from that in Fig. 1, in the sense that the IV. POLARON ENERGY AT UNITARITY differences are within the thickness of the lines [74]. We now turn to the many-body system and investigate The agreement between the two models is also evident the zero-temperature equation of state for the polaron at in the summary of length scales and energy scales in unitarity. Since we wish to properly describe Efimov Table I. Thus, we conclude that our description of the few- physics within our models, we focus on the regime where body spectrum can be accurately characterized by universal the boson-boson scattering length aB is always much Efimov theory. smaller than all other length scales. To test the accuracy Further evidence of the universality of our few-body of our variational approach outlined in Eqs. (5) and (6),we results is found when comparing the ratio of our ground- consider wave functions with up to one, two, or three state tetramer and trimer energies with recent QMC results Bogoliubov excitations, corresponding to jψ 0iþjψ 1i, ð1Þ ≃ 9 7 jψ iþjψ iþjψ i jΨi [33]. There, this ratio was reported to be E4B;−=E− . , 0 1 2 , and , respectively. For each model, ð1Þ −1 we convert the short-distance scale (e.g., r0 or Λ ) while we find E −=E− ≃ 9.35 and 9.43, respectively, for 4B; 1=3 into the dimensionless three-body parameter n ja−j in the r0 and Λ models (Table I). Importantly, the QMC calculation used a completely different model involving a order to expose the role of universal Efimov physics. The hard-sphere boson-boson interaction, characterized by the numerical results are displayed in Fig. 2, and, in the following, we analyze the different regimes defined by scattering length aB, and attractive square-well interactions between the impurity and the bosons. The range of the particle density. square well was taken to be much smaller than aB, and consequently, the energies of the ground-state Efimov A. Low-density limit trimer and associated tetramers are set by a . The univer- 1=3 B In the limit of vanishing density n ja−j → 0, the ground- sality of the spectrum thus means that we can relate the state polaron energy should reduce to that of the deepest three-body parameter in the QMC calculation to the boson- bound few-body cluster. If the polaron variational wave boson scattering length [76]: function only includes one Bogoliubov excitation, the ðQMCÞ 4 ðQMCÞ largest few-body state it can describe is the dimer, which a− ≃ −2.09ð36Þ × 10 a : ð12Þ B has zero binding energy at unitarity. Thus, as shown in Fig. 2, The agreement between the results from the different we obtain an energy that simply tends to zero with models also reinforces the point that we do not need a decreasing density, i.e., E ¼ −ð4πnÞ2=3=m ≃ −5.4n2=3=m

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that the pentamer has a closed-shell structure, where any additional bosons must occupy higher-energy orbitals, similar to what has been argued for mass-imbalanced fermions [86,87]. Such an effective repulsion naturally emerges in both the r0 and the Λ models: In the former, this arises from the fact that only one boson at a time can occupy the closed-channel dimer, while in the latter, it is due to the three-body cutoff being equivalent to a three- body repulsion [56]. From the above considerations, it is reasonable to conclude that the low-density limit is universal. Moreover, since there is a maximum size to the deepest bound Efimov cluster, we expect the polaron energy to have 2=3 FIG. 2. Energy equation of state for the Bose polaron at unitarity a well-defined thermodynamic limit, where mE=n is 1=a ¼ 0, obtained using the variational wave function with one finite for nonzero density. (gray lines), two (red lines), and three (blue lines) Bogoliubov excitations for vanishing boson-boson interactions a → 0.We B B. High-density limit show the results of the r0 model (dashed lines) and the Λ model (solid lines). In the low-density limit, the dotted straight lines To further support the claim that the polaron energy is correspond to the Efimov trimer and associated tetramer energies, finite at unitarity, we consider the opposite limit of high ¼−9 91 j j2 ð1Þ ¼−92 7 j j2 → ∞ with E− . =m a− and E4B;− . =m a− , respec- density, n . Here, the interparticle spacing becomes tively. The black dotted line in the high-density regime shows smaller than the range of the interactions, and thus, the the result of a perturbative expansion, Eq. (14). The gray shaded system will depend on the microscopic details of the model. region depicts the low-density few-body dominated regime, while However, in the case of the r0 model, it allows us to the blue shaded region corresponds to the high-density regime perform a controlled perturbative expansion in the inter- −1 −1=3 jr0j, Λ >n , where Efimov physics is suppressed and the 1=3 channel coupling g (or, equivalently, 1=n jr0j) at unitarity system is sensitive to the microscopic model. [88]. Physically, this corresponds to the limit of a narrow Feshbach resonance, a scenario that has already been [28,29]. Note that this is independent of the microscopic successfully investigated for an impurity in a Fermi sea, – model we use since the interparticle spacing greatly exceeds both theoretically [89 91] and experimentally [92,93]. any length scale of the impurity-boson interactions. Our starting point is the self-energy of the unitary Bose → 0 → 0 For polaron wave functions with up to two (three) polaron, which, in the limit g and aB , consists Bogoliubov excitations, the low-density limit recovers of the lowest-order diagrams shown in Fig. 3. Such the deepest bound universal trimer (tetramer) state dis- diagrams are included in the self-consistent T-matrix cussed in Sec. III. Here, the density drops out of the approach to the Bose polaron [28]. For an impurity with 2 momentum p and frequency ω, the self-energy in this limit problem, and we have polaron energy E ¼ −η=ðmja−j Þ, where η is a universal, model-independent constant that is explicitly depends on the Efimov cluster size (see Fig. 2). We also expect there to be larger bound clusters beyond the tetramer, similar to the case of identical bosons where (a) (b) larger clusters associated with the Efimov effect have been predicted [82–84] and, indeed, experimentally observed [85]. For the impurity case, recent QMC calculations (c) already demonstrate the existence of a pentamer with η ≃ 300 [33]. However, since there is a high-energy cutoff in the problem (e.g., set by Λ or r0), the system cannot support FIG. 3. Lowest-order diagrams for the unitary ground-state 1=3 arbitrarily deeply bound Efimov clusters, and the deepest polaron in the high-density limit n jr0j ≫ 1, where (a) and (b) bound state should be universal as long as η ≪ 106. Indeed, give, respectively, the first- and second-order terms for the as suggested by the QMC calculations [33], it is possible polaron energy in a weakly interacting BEC. Here, the dashed lines correspond to particles emitted or absorbed from the that there is no bound cluster larger than the pentamer for condensate, the blue solid line is the boson propagator, the wavy the equal-mass impurity case. First, the pairwise attraction lines represent closed-channel molecule propagators, and the between the impurity and the bosons scales with N rather filled circles correspond to the interchannel coupling g. The black than NðN − 1Þ, unlike the case for N identical bosons [25], double line, defined in (c), represents the dressed impurity to so larger bound states will be less favored. Second, an lowest order, where the black single line is the bare impurity effective short-range repulsion between bosons could mean propagator.

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2 4 ≃−4 6 2=3 Σðp ωÞ ≃ n0g þ n0g density, i.e., Epol . n =m [48]. Remarkably, for the ; d d 2 ω − ϵp ðω − ϵpÞ Bose polaron, we see in Fig. 2 that the energy strongly 1=3 X 1 1 depends on n ja−j at intermediate densities. This suggests × 2 þ ; ð13Þ that there exist resonant three-body interactions, such that ω − ϵ − ϵ − n0g 2ϵk k kþp k ω−ϵd −ϵ kþp k there are strong three-body correlations in the system even when the trimer binding energy is comparatively small. where the first and second terms correspond to the Even though the energy of the Bose polaron cannot be diagrams in Figs. 3(a) and 3(b), respectively. The assigned a universal value at unitarity, we argue that it is, in 1=3 ground-state polaron energy is then determined by taking fact, a universal function of n ja−j away from the high- the zero-momentum pole of the impurity propagator, which density regime. We have previously argued that the low- gives E ¼ Σð0;EÞ. Note that we must include self-energy density limit of the polaron energy universally depends on 1=3 insertions in the impurity propagator even at lowest order n ja−j because of the universal behavior of the few-body (see Fig. 3) since we cannot simply take ω ¼ 0 at the pole states demonstrated in Sec. III. In Fig. 2, we clearly see that when aB → 0. This is in contrast to previous perturbative this universality extends to the intermediate-density regime. treatments of the weakly interacting Bose polaron which Moreover, even for the higher densities shown in Fig. 4,we focused on aB > 0 [31]. see that different microscopic models of the Bose polaron In the regime of weak boson-boson interactions, where can essentially be collapsed onto the same curve when the 3 ≪ j j ≪ 1 1=3 naB aB= r0 , we obtain the ground-state polaron ground-state energy is plotted versus n ja−j. Note that the energy small difference between the curves of the Λ and the r0 1=3j j sffiffiffiffiffiffiffiffi rffiffiffi models is due to n r0 corrections to the two-body 1 8π 1 3 8π 1=4 scattering properties. Such a deviation from universality ≃ − n þ n ð Þ E 5 : 14 would also be present in the unitary Fermi system at these m jr0j m 7 jr0j densities [89–91]. Crucially, both the results for the Λ and the r0 models This demonstrates that the energy is well defined and are consistent with the QMC calculation from Ref. [36] bounded from below in this limit, even when the boson- when Eq. (12) is used to plot the QMC data in terms of boson interactions are vanishingly small. Such behavior the Efimov three-body parameter. This demonstrates arises from the fact that only one boson at a time can scatter that the polaron energy can be universally described in into the closed channel, thus producing an effective boson- 1=3j j boson repulsion that restricts the density of bosons that can terms of n0 a− , regardless of the microscopic details. cluster around the impurity. As shown in Fig. 2, the Furthermore, it suggests that our variational approach with perturbative expression correctly reproduces the energy three Bogoliubov excitations captures the dominant corre- for the r0 model in the high-density limit. Note that the one- lations in the full many-body problem, and that the Efimov excitation wave function only captures the leading-order term in Fig. 3(a), while the wave functions with two or three excitations correctly describe the next order correction 1=3 in 1=n jr0j. In the case of the Λ model, the high-density limit corresponds to n1=3=Λ → ∞, which is equivalent to taking the cutoff Λ to zero. Therefore, within our variational approach, only two-body correlations survive, and we obtain the equations for the wave function with one excitation (see Appendix A). For vanishing boson-boson interactions, this yields E ¼ −ð4πnÞ2=3=m, which is the polaron energy of the one-excitation wave function across all densities, as shown in Fig. 2.

C. Many-body universal regime ≪ −1=3 ≪ j j FIG. 4. Comparison of different models for the unitary Bose In the intermediate regime r n a− (where r ≪ −1=3 ≪ j j j j Λ−1 polaron in the regime r n a− , where r is the range of the can represent r0 or ), the interparticle distance is well interactions. We show the results for the polaron energy from the Λ separated from all length scales associated with the inter- model (solid line) and the r0 model (dashed line), which include up actions. If the medium were fermionic rather than bosonic, to three excitations of the condensate and which take aB → 0. The then the polaron energy at unitarity, in the limit r → 0, would QMC results of Ref. [36] are shown as the gray dots. The recent 1=3 be a universal value that only depends on the medium Aarhus experiment [16] is estimated to have n ja−j≃100.

011024-7 YOSHIDA, ENDO, LEVINSEN, and PARISH PHYS. REV. X 8, 011024 (2018) three-body parameter plays a more important role in the small. We find that there are clear differences between the polaron energy than the coherence length of the condensate results depending on the number of Bogoliubov excitations (see Appendix B). We emphasize that our comparison in used in the variational approach, and also between the Fig. 4 constitutes a complete reinterpretation of the QMC different models. Within the r0 model, the residue takes the results of Ref. [36] since that work did not consider the value 1=2 in the high-density regime where the polaron possibility that the variation of the polaron energy with wave function is an equal superposition of the impurity and boson-boson interaction was due to Efimov physics. the closed-channel dimer. However, while the residue within the one-excitation approximation is seen to increase 1 2 2 3 V. CHARACTERIZATION OF THE POLARON from = to = with increasing density, we observe a rapid decrease of Z towards the low-density regime when Having shown that the polaron energy is a universal including several excitations. On the other hand, for the 1=3 function of n ja−j for sufficiently low densities, we now Λ model, the residue is 2=3 if we consider only one turn to the other quasiparticle properties that characterize Bogoliubov excitation. By increasing the number of the polaron: the residue, effective mass, and contact. We Bogoliubov excitations, it is suppressed not only in the generally focus on the r0 model, which is the more physical few-body dominated regime but also in the high-density model and which allows a perturbative analysis in the high- limit, where the residue takes the values 1=3 and 1=9, density limit. respectively, when two and three excitations are taken into account. Thus, unlike the energy, the polaron residue does A. Residue not appear to converge to a universal function of the three- The residue Z is the squared overlap of the polaron wave body parameter. † The origin of this behavior can be understood within function with the noninteracting state c0jΦi. Making perturbation theory for the r0 model in the high-density reference to the wave function in Eq. (6), this is limit. Considering again the diagrams in Fig. 3 and the ¼jα j2 ð Þ explicit expression for the self-energy in Eq. (13), we find, Z 0 ; 15 at small g, where we assume hΨjΨi¼1 for normalization (see ∂Σð0 ωÞ −1 ; Appendix C). The residue can be accessed directly in Z ¼ 1 − ∂ω ω¼ experiment; indeed, in the case of the Fermi polaron, it has rffiffiffi E 3 2 been measured using both radio-frequency spectroscopy ≃2 − þ L ð Þ 3 1=4 ; 16 [94] and Rabi oscillations [92]. 7 ð8πnjr0j Þ jr0j In Fig. 5 we show our results for the polaron residue at unitarity as a function of the three-body parameter, where where L is a low-energy length scale that provides an the boson-boson interactions are taken to be vanishingly infrared cutoff of low momenta. In the thermodynamic limit, this is set by the coherence length of the BEC; i.e., we have L ∝ ξ. Thus, we find that the residue vanishes when the coherence length ξ → ∞, and consequently, the polaron wave function is orthogonal to the noninteracting impurity state in the case of an ideal BEC. This feature cannot properly be captured within our variational approach since this requires an infinite number of Bogoliubov excitations, hence the observed lack of convergence in Fig. 5. On the other hand, when aB ≠ 0, we find that the wave functions with two and three excitations both give the same leading- order correction to Z: rffiffiffiffiffiffiffi 1 a Z − ∝ B ; ð17Þ 2 jr0j

1=3 1=3 in the limit n a ≪ 1 and n jr0j → ∞ [95]. Thus, in Z 1=a ¼ 0 B FIG. 5. Polaron residue at unitarity as a function of this case, the residue is finite and initially increases the dimensionless three-body parameter. We show the results for with decreasing jr0j. Note, however, that the residue the Λ model (solid line) and the r0 model (dashed line) for always vanishes in the low-density regime, as shown in aB → 0. In order of decreasing residue, the results are obtained for wave functions including up to one, two, or three excitations Fig. 5, since the polaron evolves into a few-body bound of the condensate. The grey and blue shaded regions are the same state which has zero overlap with the noninteracting as in Fig. 2, i.e., the low- and high-density regimes, respectively. impurity state.

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The simultaneous convergence of the energy and sup- consider the high-density limit of the r0 model and use pression of the residue in the ideal BEC arises from the fact the self-energy in Eq. (13), we obtain that the impurity affects the condensate at arbitrarily large rffiffiffi distances and thus excites an infinite number of low-energy 3 3 1 m ≃ − þ α L ð Þ modes in the condensate. This result is akin to the Z 3 1=4 ; 19 m 2 7 ð8πnjr0j Þ jr0j orthogonality catastrophe for a static impurity in a Fermi gas. Indeed, in that case, the orthogonality of the interacting where α is a constant that must be determined from a proper and noninteracting impurity wave functions is also apparent finite-aB perturbative analysis. Thus, we see that there is a in perturbation theory, already at lowest order beyond term proportional to L that cancels the corresponding term mean-field theory [96]. We also note that the vanishing in the residue in Eq. (16), implying that the effective mass is residue of the Bose polaron is not confined to the strongly finite. interacting impurity at unitarity: Even for weak impurity- Indeed, in the low-density few-body regime, we already boson interactions, the residue of the attractive polaron was have the situation where the residue vanishes, while the 1 ξ found within perturbation theory to approach zero as = effective mass remains finite with m ¼ðN þ 1Þm, where → 0 when aB [31]. N is the number of bosons bound to the impurity. This behavior is captured by both r0 and Λ models, as shown in B. Effective mass Fig. 6, where the wave function with two Bogoliubov The interactions between the impurity and the bosonic excitations has m=m → 3 as n → 0. Likewise, we expect medium also modify the mobility of the impurity and give the wave function with three excitations to recover the mass rise to an effective mass m. This can be derived from the of the tetramer in this limit. For higher densities, we find impurity’s self-energy as follows: that the results of our variational approach become sensitive to the specific model used, as well as to the number of ∂2Σðq Þ excitations of the ideal BEC considered. Here, in the many- m ¼ 1 þ ;E ð Þ Z m 2 : 18 body limit, one requires an infinite number of low-energy m ∂q q¼0 excitations to correctly describe the effective mass for a → 0 B , similarly to the case of the residue. Note that this gives m ¼ m for the noninteracting impurity 0 1=3 ≪ 1 For a BEC with

Thus, the effective mass, in this case, initially decreases with decreasing density before increasing towards the mass of the deepest bound few-body cluster.

C. Contact Another quantity of interest is the unitary Tan contact, which is defined from the polaron energy as [97,98]

∂ 2 2 E C ¼ m g : ð21Þ ∂ν ν¼0

FIG. 6. Inverse effective mass of the Bose polaron at unitarity, ν ≡ − 2 4π 1=a ¼ 0, for a ¼ 0. We calculate it with the Λ model (solid Here, we have used the physical detuning g m= a, B ν lines) and the r0 model (dashed lines), taking into account one which can be related to the bare detuning 0 using Eq. (4). (gray lines) and two (red lines) Bogoliubov excitations. The grey Physically, the contact gives a measure of the density of and blue shaded regions are the same as in Fig. 2. pairs at short distances. For the two-channel Hamiltonian in

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1=3 3 For lower densities n ja−j ≲ 10 , the contact becomes insensitive to the underlying model and thus appears to be a universal function of the dimensionless three-body parameter (see Fig. 7). In the limit of vanishing density n → 0,we recover the contact of the deepest bound few-body cluster, where Cja−j is a universal number. For the Efimov trimer, we have Cja−j ≃ 171.4 in the r0 model and Cja−j ≃ 172.2 in the Λ model, while for the associated tetramer, we obtain Cja−j ≃ 535.5 (r0 model) and 542.6 (Λ model).

VI. CONCLUSIONS AND OUTLOOK Using a variational approach, we have investigated the ground state of the Bose polaron with unitarity limited FIG. 7. Contact of the Bose polaron at unitarity, 1=a ¼ 0, for impurity-boson interactions. For sufficiently low boson ¼ 0 Λ aB . The solid and dashed curves correspond to the and r0 densities, we have demonstrated that the polaron equation models, respectively. We show the results obtained using wave of state is a universal function of the three-body parameter functions with up to one (gray line), two (red line), and three (blue line) excitations. The grey and blue shading indicates the associated with Efimov physics. This is a highly nontrivial same regions as in Fig. 2. result since the regime of universality extends to densities at which the binding energies of the few-body states are several orders of magnitude smaller than the polaron Eq. (1), one can show that the contact is proportional to the energy. Our findings are corroborated by the fact that we population of closed-channel dimers [98]: observe the same behavior for two different microscopic X models and by the fact that our calculations agree well with ¼ 2 2 hΨj † jΨi ð Þ C m g dkdk : 22 recent QMC results [33,36] when they are appropriately k reinterpreted in terms of the three-body parameter. We have also demonstrated that the few-body spectrum of Efimov Within our variational ansatz, this corresponds to trimers and associated tetramers is model independent, X X further highlighting the universality of our results. 2 2 2 2 1 2 C ¼ m g jγ0j þ jγkj þ jγk k j : ð23Þ 2 1 2 In the nonuniversal, high-density limit, we have per- k k1k2 formed a controlled perturbative analysis of the r0 model, and we have derived new analytic expressions for the → ∞ Note that the limit g is well defined, and thus this unitary Bose polaron at a narrow Feshbach resonance. Our Λ expression also captures the single-channel model. results demonstrate that even when we take the ideal Bose Λ Figure 7 displays the contact at unitarity for both and gas limit, the polaron energy and contact can remain well r0 models in the case of vanishing boson-boson inter- defined thermodynamically. This is a consequence of the actions. Similarly to the polaron energy, we find that the fact that only one boson at a time can occupy the closed- contact in the high-density limit is well converged with channel dimer state, which thus generates an effective respect to the number of Bogoliubov excitations. For the r0 repulsion between bosons that acts to stabilize the system. model, we can evaluate the contact perturbatively for large We expect a similar scenario for the Λ model since the 1=3 n jr0j using the self-energy in Eq. (13). Here, we simply high-momentum cutoff in the three- and four-body terms of d d shift the closed-channel dispersion from ϵp to ϵp þ ν and the variational equations is equivalent to an explicit three- then determine the ground-state energy from the equation body repulsion [56]. However, in general, it is not clear that E ¼ Σð0;EÞ as a function of detuning ν. Applying Eq. (21) a well-defined thermodynamic limit exists for treatments of then yields the high-density expression the Bose polaron that focus on single-channel Hamiltonians rffiffiffi and neglect Efimov physics [39,99]. 8π 1 29 3 1 ≃ − ð Þ While the polaron equation of state at unitarity is C 3 1=4 : 24 jr0j 2 28 7 ð8πnjr0j Þ relatively insensitive to boson-boson interactions, we find that the quasiparticle residue and effective mass strongly We find that this perturbative result is correctly reproduced depend on the coherence length of the BEC. In particular, by both two- and three-excitation approximations in the the residue vanishes as ξ → ∞, a feature that resembles the high-density limit. The leading-order term is independent orthogonality catastrophe [53]. Thus, an interesting remain- of density since the closed-channel fraction tends to 1=2 at ing question is how the residue tends to zero with the unitarity as jr0j → ∞ (or, equivalently, as g → 0); see number of excitations of the condensate included in the Eq. (16). wave function. A related question is as follows: What finite

011024-10 UNIVERSALITY OF AN IMPURITY IN A BOSE- … PHYS. REV. X 8, 011024 (2018) result does the effective mass converge towards as ξ → ∞, Joint Research Co-operation Scheme. This work was and is it a universal function of the three-body parameter? performed in part at the Aspen Center for Physics, which We emphasize that the (two-channel) r0 model faithfully is supported by the National Science Foundation Grant describes the physics of an actual Feshbach resonance in an No. PHY-1607611. ultracold atomic gas [57], and thus we expect our results to accurately describe current and future cold atom experiments. Furthermore, we expect the universal behavior APPENDIX A: COUPLED INTEGRAL presented here to also exist for heavy impurities, a scenario EQUATIONS that can be realized by choosing atomic species with a large To obtain the integral equations for the r0 model, we first mass ratio or by pinning the impurity in an optical lattice. In h∂Ψjð ˆ − ÞjΨi¼0 α the particular case of a static impurity, there are no Efimov take the derivative H E with respect to 0, α α α γ γ γ few-body states, and therefore, it would be of great interest k, k1k2 , k1k2k3 , 0, k, and k1k2 for our variational wave to understand how the system behaves in the limit function in Eq. (5). This yields the following linear 1=3 n jr0j ≪ 1 of the two-channel model. equations: Our results may be probed in experiments similar to pffiffiffiffiffi X those carried out recently at JILA [15] and in Aarhus [16]. Eα0 ¼ g n0γ0 − g vkγk; ðA1Þ In particular, our results directly apply to the latter experi- k ment, which used 39K as both the impurity and host atoms. pffiffiffiffiffi X 1=3 ðE − ϵk − EkÞαk ¼ gukγ0 þ g n0γk − g γkqvq; ðA2Þ Furthermore, we estimate that n ja−j ≃ 100 in that q experiment, which indicates that it is already in a regime where the nontrivial dependence on the interparticle spac- pffiffiffiffiffi ðE − Ek k Þαk k ¼ gðγk uk þ γk uk Þþg n0γk k ; ing is strongly pronounced. Indeed, in a recent JILA 1 2 1 2 1 2 2 1 1 2 experiment [19], the density of the Bose gas was changed ðA3Þ by 2 orders of magnitude, and thus an experimental investigation of the density dependence of the unitary ðE − Ek k k Þαk k k ¼ gðγk k uk þ γk k uk þ γk k uk Þ; Bose polaron appears to be well within reach. 1 2 3 1 2 3 1 2 3 2 3 1 1 3 2 ðA4Þ ACKNOWLEDGMENTS pffiffiffiffiffi X ðE − ν0Þγ0 ¼ g n0α0 þ g ukαk; ðA5Þ We gratefully acknowledge fruitful discussions with Luis k A. Peña Ardila, Jan J. Arlt, Georg M. Bruun, Rasmus S. pffiffiffiffiffi X Christensen, Victor Gurarie, Nils B. Jørgensen, Richard d ðE − ϵk − ν0 − EkÞγk ¼ g n0αk − gvkα0 þ g uk0 αkk0 ; Schmidt, Zhe-Yu Shi, and Bing Zhu. We also thank Luis A. k0 Peña Ardila for the QMC data of Refs. [33,36] and for ð Þ helpful comments on the manuscript. S. M. Y. acknowl- A6 edges support from the Japan Society for the Promotion of d ðE − ϵ − ν0 − Ek − Ek Þγk k Science through Program for Leading Graduate Schools k1þk2 1 1 1 2 X (ALPS) and Grant-in-Aid for JSPS Fellows (KAKENHI pffiffiffiffiffi ¼ g n0αk k − gðαk vk þ αk uk Þþg αk k k uk ; Grant No. JP16J06706). J. L. and M. M. P. acknowledge 1 2 1 2 2 1 1 2 3 3 k3 financial support from the Australian Research Council via Discovery Project No. DP160102739. J. L. and S. E. are ðA7Þ supported by the Australian Research Council through ¼ þ þ ϵ ¼ þ Future Fellowship FT160100244 and DECRA Fellowship where Ek1k2 Ek1 Ek2 k1þk2 and Ek1k2k3 Ek1 þ þ ϵ DE180100781, respectively. J. L. and M. M. P. acknowl- Ek2 Ek3 k1þk2þk3 . Using the first four equations to edge funding from the Universities Australia-Germany remove the α coefficients, we obtain

X X pffiffiffiffiffi γ γ ukvqγkq T −1ð 0Þγ ¼ n0 γ þ uk k − vk k − ð Þ E; 0 0 n0 − ϵ − − ϵ − ; A8 E k E k Ek E kq E k Ek

X pffiffiffiffiffi ukuqγq vkvqγq T −1ð − kÞγ ¼ uk − vk γ þ n0 γ þ þ E Ek; k n0 − ϵ − 0 − ϵ − k − E k Ek E E k Ek q E Ekq E X pffiffiffiffiffi uqγkq vqγkq þ − ð Þ n0 − − ϵ − ; A9 q E Ekq E k Ek

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pffiffiffiffiffi γ γ γ −1 n0 uk1 k2 vk1 k2 uk1 vk2 0 T ðE − Ek − Ek ; k1 þ k2Þγk k ¼ γk k þ n0 − − 1 2 1 2 − 1 2 − − ϵ − − ϵ − E Ek1k2 E Ek1k2 E k2 Ek2 E k1 Ek1 X uk uqγk q vk vqγk q þ 1 2 þ 1 2 þðk ↔ k Þ ð Þ − − ϵ − 1 2 : A10 q E Ek1k2q E k2 Ek2

Here, the two-body T matrix T ðE; kÞ in the BEC medium is X 2 m mr0 uq 1 T −1ð kÞ¼ − ð − ϵd Þ − þ E; 4π 8π E k − − ϵ 2ϵ a q E Eq kþq q X 2 1 uq ¼ T −1ð kÞþ − ð Þ 0 E; − ϵ − ϵ − − ϵ ; A11 q E q kþq E Eq kþq

where the vacuum T matrix T 0ðE; kÞ was defined in the Λ model are shown in Fig. 8. We see that the polaron Eq. (10). energy is almost unaffected in the low-density, few-body Λ The coupled integral equations for the model have the dominated regime, while aB becomes more important with same form except that we take r0 → 0 with a fixed in the T increasing density. However, throughout the range plotted, matrix and impose a UV cutoff Λ on the integrals on the i.e., the universal many-body regime, the relative change in – 1=3 right-hand side of Eqs. (A8) (A10). The UV cutoff is energy arising from the change in n ja−j clearly greatly necessary because otherwise the coupled integral equations 1=3 exceeds that from the increase in n aB. We furthermore are ill posed and the three-body parameter cannot be see that both sets of results are consistent with our fixed [56]. Note that the coupled integral equations with reinterpretation of the QMC results in terms of a three- r0 → 0 also follow from the single-channel model, body parameter. which has a density-density coupling of the form In the inset of Fig. 8, we furthermore illustrate the P † † k;k0;qbk0 cq−k0 cq−kbk but no terms involving the closed- convergence of the polaron energy calculated within both channel molecules. In the latter case, we again need to interaction models as a function of (inverse) number of introduce a finite cutoff Λ by hand to have a fixed three-body parameter. In this paper, we show results obtained from variational wave functions, which include 1, 2, or 3 excitations of the condensate; see Eq. (6). These approximations correspond to solving Eq. (A8) with γk ¼ γkq ¼ 0, Eqs. (A8) and (A9) with γkq ¼ 0, and all of Eqs. (A8)–(A10), respectively.

APPENDIX B: EFFECT OF FINITE COHERENCE LENGTH AND NUMBER OF BOGOLIUBOV EXCITATIONS In Sec. IV C, we compared the results of our variational approach with those obtained in quantum Monte Carlo calculations by Peña Ardila and Giorgini [36], and we found very good agreement across more than an order of magnitude in interparticle spacing (see Fig. 2). In this comparison, we highlighted the fact that the dependence of FIG. 8. Polaron energy calculated within the Λ model as a 1=3j j ¼ 0 ¼ a− function of n a− for aB (solid line) and for aB the polaron energy on the three-body parameter is much 4 2.1 × 10 ja−j (dashed line). The latter choice of a ensures that stronger than the dependence on aB, and to emphasize this, B we take the same ratio of a− and aB as in the QMC calculation we took aB ¼ 0. However, as we have argued in Sec. III,in (gray dots) [see Eq. (12)], and the corresponding scale for aB is the QMC calculation the ratio between a− and a was kept B shown on the top x axis. Inset: Polaron energy at unitarity as a ðQMCÞ ≃ −2 09ð36Þ 104 ðQMCÞ fixed at a value of a− . × aB as function of inverse number N of Bogoliubov excitations for identified in Eq. (12). Therefore, we now consider the aB ¼ 0. We present the results of the Λ model (solid) and r0 1=3 1=3 residual effect of keeping a−=aB fixed in the same manner model (open) for n ja−j¼100 (blue circles) and n ja−j¼ as in the QMC. Our results for the polaron energy within 500 (red triangles).

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Bogoliubov excitations included in our variational Xˆ½EðPÞ; Pjγi¼0: ðC3Þ approach. In the many-body dominated regime, at 1=3 n ja−j¼500, we see that the convergence appears to jγi γ γ γ Here, is an ordered set of 0, k, and k1k2 , and the be nearly linear, with a slope (and extrapolation to an center-of-mass momentum can be taken to be Pe without infinite number of excitations) that is consistent with the z loss of generality. While the matrix Xˆ is not the difference between the finite a dotted line and the QMC B Hamiltonian, it is still Hermitian. The energy dispersion results in the main figure. Closer to the few-body domi- ð Þ 1=3 E P is determined by the zero crossing of the lowest (or nated regime at n ja−j¼100, our approach converges highest, depending on the sign convention) eigenvalue of more slowly because of the strong few-body correlations Xˆ . Therefore, for a small P, we can expand Xˆ up to OðP2Þ induced by the trimer and tetramer bound states. However, it is likely that including four Bogoliubov excitations in our and evaluate the lowest eigenvalue in the same way as the approach to account for the hypothesized pentamer (see Rayleigh-Schrödinger perturbation theory. It is straightfor- Sec. IVA) would yield a nearly converged result. We ward to show that expect a small residual difference between the results of the 2 two interaction models, even in the N → ∞ limit, because ˆ ˆ ˆ P ˆ ˆ X½EðPÞ;P¼X0 þ PX þ ðX þð1=m ÞX Þ of the different manners in which the two-body physics is P 2 PP E implemented. A similar effect would be present for an þ OðP3Þ; ðC4Þ impurity in a Fermi gas. where APPENDIX C: CALCULATION OF OBSERVABLES ˆ ˆ ˆ ˆ ∂X X0 ¼ X½Eð0Þ; 0; XP ¼ ½E; P ; Once the wave function jΨi is obtained, we can calculate ∂ P¼0 P E¼Eð0Þ physical quantities of interest. In the following, we assume ∂2 ˆ ∂ ˆ that the variational wave function is normalized, hΨjΨi¼1. ˆ X ˆ X XPP ¼ ½E; P ; XE ¼ ½E; P : ðC5Þ ∂ 2 P¼0 ∂ P¼0 When we solve the renormalized equations (A8)–(A10), P E¼Eð0Þ E E¼Eð0Þ this implies that ˆ X 1 X Then, the lowest eigenvalue λðPÞ of X½EðPÞ;P can be 1 ¼jα j2 þ jα j2 þ jα j2 0 k 2 k1k2 expressed as a series expansion in terms of P, k k1k2 1 X X 1 X 2 3 2 2 2 2 λðPÞ¼λ0 þ λ1P þ λ2P þ OðP Þ; ðC6Þ þ jαk k k j þjγ0j þ jγkj þ jγk k j : 6 1 2 3 2 1 2 k1k2k3 k k1k2 λ ðC1Þ and its coefficients i are determined perturbatively:

Since we solve for γ0, γk, and γk k , we relate the α ð0Þ ˆ ð0Þ 1 2 λ1 ¼hγ jXPjγ i¼0; ðC7Þ coefficients to these through Eqs. (A1)–(A4).

1 XjhγðiÞj ˆ jγð0Þij2 1. Effective mass ð0Þ ˆ ˆ ð0Þ XP λ2 ¼ hγ jðX þð1=m ÞX Þjγ iþ : 2 PP E λð0Þ − λðiÞ For a system with rotational symmetry, the inverse i>0 0 0 effective mass is defined as ðC8Þ 1 ∂2 ð Þ ¼ E P ð Þ 2 ; C2 ðiÞ ðiÞ m ∂P Here, λ0 and jγ i are the ith eigenvalue and its corre- ˆ ¼ 0 where EðPÞ is the energy dispersion of the Bose polaron. sponding eigenvector of X0, and i corresponds to the ˆ ð0Þ ð0Þ ð0Þ The variational ansatz can be easily extended to a Bose state that satisfies X0½Eð0Þ; 0jγ i¼λ0 jγ i¼0. Note ð0Þ polaron with a finite momentum, but solving the resulting that λ1 ¼ 0 follows from the rotational invariance of jγ i. integral equations becomes numerically expensive because Now, recalling that the energy dispersion is found by zero the equations do not have rotational symmetry. Here, crossing of the lowest eigenvalue of Xˆ ½EðPÞ;P, we can set following a line of argument similar to Ref. [90], we present a method to calculate the effective mass without solving the λð Þ¼λð0Þ ¼ 0 ð Þ integral equations for a finite-momentum Bose polaron. P 0 ; C9 In general, the coupled integral equations for γ’s for a finite-momentum polaron have the following structure which, in turn, implies that λ2 ¼ 0. From this, the inverse [similar to Eqs. (A1)–(A4)]: effective mass is obtained as follows:

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ð0Þ ˆ ð0Þ −1 [14] P. Makotyn, C. E. Klauss, D. L. Goldberger, E. A. Cornell, 1=m ¼hγ jXEjγ i and D. S. Jin, Universal Dynamics of a Degenerate Unitary X 2 jhγðiÞjXˆ jγð0Þij2 − hγð0ÞjXˆ jγð0Þi Bose Gas, Nat. Phys. 10, 116 (2014). × ðiÞ P PP i>0 λ0 [15] M.-G. Hu, M. J. Van de Graaff, D. Kedar, J. P. Corson, E. A. Cornell, and D. S. Jin, Bose Polarons in the Strongly ð Þ C10 Interacting Regime, Phys. Rev. Lett. 117, 055301 (2016). [16] N. B. Jørgensen, L. Wacker, K. T. Skalmstang, M. M. Parish, J. Levinsen, R. S. Christensen, G. M. Bruun, and ð0Þ ˆ ð0Þ −1 ð0Þ ˆ ˆ ˆ −1 ˆ ˆ ð0Þ ¼hγ jXEjγ i ½2hγ jXPQX0 QXPjγ i J. J. Arlt, Observation of Attractive and Repulsive Polarons − hγð0Þj ˆ jγð0Þi ð Þ in a Bose-Einstein Condensate, Phys. Rev. Lett. 117, XPP ; C11 055302 (2016). [17] R. J. Fletcher, R. Lopes, J. Man, N. Navon, R. P. Smith, P where Qˆ ¼ 1 − jγðiÞihγðiÞj. M. W. Zwierlein, and Z. Hadzibabic, Two- and Three-Body i>0 Contacts in the Unitary Bose Gas, Science 355, 377 (2017). [18] R. Lopes, C. Eigen, A. Barker, K. G. H. Viebahn, M. Robert-de Saint-Vincent, N. Navon, Z. Hadzibabic, and R. P. Smith, Quasiparticle Energy in a Strongly Interacting Homogeneous Bose-Einstein Condensate, Phys. Rev. Lett. [1] M. Randeria, W. Zwerger, and M. Zwierlein, in The BCS- 118, 210401 (2017). BEC Crossover and the Unitary Fermi Gas, edited by W. [19] C. E. Klauss, X. Xie, C. Lopez-Abadia, J. P. D’Incao, Z. Zwerger (Springer, Berlin, Heidelberg, 2012), pp. 1–32. Hadzibabic, D. S. Jin, and E. A. Cornell, Observation of [2] M. J. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Efimov Molecules Created from a Resonantly Interacting Revealing the Superfluid Lambda Transition in the Bose Gas, Phys. Rev. Lett. 119, 143401 (2017). Universal Thermodynamics of a Unitary Fermi Gas, [20] C. Eigen, J. A. P. Glidden, R. Lopes, N. Navon, Z. Science 335, 563 (2012). Hadzibabic, and R. P. Smith, Universal Scaling Laws in [3] S. Nascimbène, N. Navon, K. Jiang, F. Chevy, and C. the Dynamics of a Homogeneous Unitary Bose Gas, Phys. Salomon, Exploring the Thermodynamics of a Universal 119 Fermi Gas, Nature (London) 463, 1057 (2010). Rev. Lett. , 250404 (2017). Energy Levels Arising from Resonant Two-Body [4] M. Horikoshi, S. Nakajima, M. Ueda, and T. Mukaiyama, [21] V. Efimov, Measurement of Universal Thermodynamic Functions for a Forces in a Three-Body System, Phys. Lett. B 33, 563 Unitary Fermi Gas, Science 327, 442 (2010). (1970). [5] G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack, and [22] E. Braaten and H.-W. Hammer, Universality in Few-Body R. G. Hulet, Molecular Probe of Pairing in the BEC-BCS Systems with Large Scattering Length, Phys. Rep. 428, 259 Crossover, Phys. Rev. Lett. 95, 020404 (2005). (2006). [6] J. T. Stewart, J. P. Gaebler, and D. S. Jin, Using Photoemis- [23] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, sion Spectroscopy to Probe a Strongly Interacting Fermi B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-C. Gas, Nature (London) 454, 744 (2008). Nägerl, and R. Grimm, Evidence for Efimov Quantum States [7] A. Schirotzek, Y.-i. Shin, C. H. Schunck, and W. Ketterle, in an Ultracold Gas of Caesium Atoms, Nature (London) Determination of the Superfluid Gap in Atomic Fermi Gases 440, 315 (2006). by Quasiparticle Spectroscopy, Phys. Rev. Lett. 101, [24] P. Naidon and S. Endo, Efimov Physics: A Review, Rep. 140403 (2008). Prog. Phys. 80, 056001 (2017). [8] Y. Inada, M. Horikoshi, S. Nakajima, M. Kuwata- [25] C. H. Greene, P. Giannakeas, and J. Pérez-Ríos, Universal Gonokami, M. Ueda, and T. Mukaiyama, Critical Few-Body Physics and Cluster Formation, Rev. Mod. Phys. Temperature and Condensate Fraction of a Fermion Pair 89, 035006 (2017). Condensate, Phys. Rev. Lett. 101, 180406 (2008). [26] J. P. D’Incao, Few-Body Physics in Resonantly Interacting [9] G. Veeravalli, E. Kuhnle, P. Dyke, and C. J. Vale, Bragg Ultracold Quantum Gases, arXiv:1705.10860. Spectroscopy of a Strongly Interacting Fermi Gas, Phys. [27] J. Tempere, W. Casteels, M. K. Oberthaler, S. Knoop, E. Rev. Lett. 101, 250403 (2008). Timmermans, and J. T. Devreese, Feynman Path-Integral [10] N. Navon, S. Nascimbène, F. Chevy, and C. Salomon, The Treatment of the BEC-Impurity Polaron, Phys. Rev. B 80, Equation of State of a Low-Temperature Fermi Gas with 184504 (2009). Tunable Interactions, Science 328, 729 (2010). [28] S. P. Rath and R. Schmidt, Field-Theoretical Study of the [11] J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin, Bose Polaron, Phys. Rev. A 88, 053632 (2013). Verification of Universal Relations in a Strongly Interacting [29] W. Li and S. Das Sarma, Variational Study of Polarons in Fermi Gas, Phys. Rev. Lett. 104, 235301 (2010). Bose-Einstein Condensates, Phys. Rev. A 90, 013618 [12] E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D. (2014). Drummond, P. Hannaford, and C. J. Vale, Universal Behav- [30] A. Shashi, F. Grusdt, D. A. Abanin, and E. Demler, ior of Pair Correlations in a Strongly Interacting Fermi Radio-Frequency Spectroscopy of Polarons in Ultracold Gas, Phys. Rev. Lett. 105, 070402 (2010). Bose Gases, Phys. Rev. A 89, 053617 (2014). [13] C. Cao, E. Elliott, J. Joseph, H. Wu, J. Petricka, T. Schäfer, [31] R. S. Christensen, J. Levinsen, and G. M. Bruun, and J. E. Thomas, Universal Quantum Viscosity in a Quasiparticle Properties of a Mobile Impurity in a Bose- Unitary Fermi Gas, Science 331, 58 (2011). Einstein Condensate, Phys. Rev. Lett. 115, 160401 (2015).

011024-14 UNIVERSALITY OF AN IMPURITY IN A BOSE- … PHYS. REV. X 8, 011024 (2018)

[32] J. Levinsen, M. M. Parish, and G. M. Bruun, Impurity in a [51] J. Vlietinck, J. Ryckebusch, and K. Van Houcke, Quasi- Bose-Einstein Condensate and the Efimov Effect, Phys. Rev. particle Properties of an Impurity in a Fermi Gas, Phys. Lett. 115, 125302 (2015). Rev. B 87, 115133 (2013). [33] L. A. Peña Ardila and S. Giorgini, Impurity in a Bose- [52] P. Massignan, M. Zaccanti, and G. M. Bruun, Polarons, Einstein Condensate: Study of the Attractive and Repulsive Dressed Molecules and Itinerant Ferromagnetism in Ultra- Branch Using Quantum Monte Carlo Methods, Phys. Rev. cold Fermi Gases, Rep. Prog. Phys. 77, 034401 (2014). A 92, 033612 (2015). [53] P. W. Anderson, Infrared Catastrophe in Fermi Gases with [34] F. Grusdt, Y. E. Shchadilova, A. N. Rubtsov, and E. Demler, Local Scattering Potentials, Phys. Rev. Lett. 18, 1049 Renormalization Group Approach to the Fröhlich Polaron (1967). Model: Application to Impurity-BEC Problem, Sci. Rep. 5, [54] A. L. Fetter and J. D. Walecka, Quantum Theory of Many- 12124 (2015). Particle Systems (McGraw-Hill, New York, 1971). [35] F. Grusdt and E. A. Demler, New Theoretical Approaches to [55] D. S. Petrov, Three-Boson Problem Near a Narrow Fes- Bose Polarons, arXiv:1510.04934. hbach Resonance, Phys. Rev. Lett. 93, 143201 (2004). [36] L. A. Peña Ardila and S. Giorgini, Bose Polaron Problem: [56] P. Bedaque, H.-W. Hammer, and U. van Kolck, The Three- Effect of Mass Imbalance on Binding Energy, Phys. Rev. A Boson System with Short-Range Interactions, Nucl. Phys. 94, 063640 (2016). A646, 444 (1999). [37] Y. E. Shchadilova, R. Schmidt, F. Grusdt, and E. Demler, [57] E. Timmermans, P. Tommasini, M. Hussein, and A. Quantum Dynamics of Ultracold Bose Polarons, Phys. Rev. Kerman, Feshbach Resonances in Atomic Bose-Einstein Lett. 117, 113002 (2016). Condensates, Phys. Rep. 315, 199 (1999). [38] N. J. S. Loft, Z. Wu, and G. M. Bruun, Mixed-Dimensional [58] G. M. Bruun and C. J. Pethick, Effective Theory of Feshbach Bose Polaron, Phys. Rev. A 96, 033625 (2017). Resonances and Many-Body Properties of Fermi Gases, [39] F. Grusdt, R. Schmidt, Y. E. Shchadilova, and E. Demler, Phys. Rev. Lett. 92, 140404 (2004). Strong-Coupling Bose Polarons in a Bose-Einstein [59] J. Levinsen and D. Petrov, Atom-Dimer and Dimer-Dimer Condensate, Phys. Rev. A 96, 013607 (2017). Scattering in Fermionic Mixtures Near a Narrow Feshbach [40] M. Sun, H. Zhai, and X. Cui, Visualizing the Efimov Resonance, Eur. Phys. J. D 65, 67 (2011). Correlation in Bose Polarons, Phys. Rev. Lett. 119, [60] M. M. Parish and J. Levinsen, Quantum Dynamics of 013401 (2017). Impurities Coupled to a Fermi Sea, Phys. Rev. B 94, [41] A. Lampo, S. H. Lim, M. Á. García-March, and M. 184303 (2016). Lewenstein, Bose Polaron as an Instance of Quantum [61] L. Pricoupenko, Isotropic Contact Forces in Arbitrary Brownian Motion, Quantum 1, 30 (2017). Representation: Heterogeneous Few-Body Problems and [42] R. Schmidt and M. Lemeshko, Rotation of Quantum Low Dimensions, Phys. Rev. A 83, 062711 (2011). Impurities in the Presence of a Many-Body Environment, [62] V. Efimov, Energy Levels of Three Resonantly Interacting Phys. Rev. Lett. 114, 203001 (2015). Particles, Nucl. Phys. A210, 157 (1973). [43] R. Schmidt, H. R. Sadeghpour, and E. Demler, Mesoscopic [63] H.-W. Hammer and L. Platter, Universal Properties of the Rydberg Impurity in an Atomic Quantum Gas, Phys. Rev. Four-Body System with Large Scattering Length, Eur. Phys. Lett. 116, 105302 (2016). J. A 32, 113 (2007). [44] F. Camargo, R. Schmidt, J. D. Whalen, R. Ding, G. [64] J. von Stecher, J. P. D’Incao, and C. H. Greene, Signatures Woehl, Jr., S. Yoshida, J. Burgdörfer, F. B. Dunning, of Universal Four-Body Phenomena and Their Relation to H. R. Sadeghpour, E. Demler, and T. C. Killian, Creation the Efimov Effect, Nat. Phys. 5, 417 (2009). of Rydberg Polarons in a Bose Gas, arXiv:1706.03717. [65] M. R. Hadizadeh, M. T. Yamashita, L. Tomio, A. Delfino, [45] T. Rentrop, A. Trautmann, F. A. Olivares, F. Jendrzejewski, and T. Frederico, Scaling Properties of Universal Tet- A. Komnik, and M. K. Oberthaler, Observation of the ramers, Phys. Rev. Lett. 107, 135304 (2011). Phononic Lamb Shift with a Synthetic Vacuum, Phys. [66] A. Deltuva, Shallow Efimov Tetramer as Inelastic Virtual Rev. X 6, 041041 (2016). State and Resonant Enhancement of the Atom-Trimer [46] Y. Wang, J. Wang, J. P. D’Incao, and C. H. Greene, Relaxation, Europhys. Lett. 95, 43002 (2011). Universal Three-Body Parameter in Heteronuclear Atomic [67] F. Ferlaino, S. Knoop, M. Berninger, W. Harm, J. P. Systems, Phys. Rev. Lett. 109, 243201 (2012). D’Incao, H.-C. Nägerl, and R. Grimm, Evidence for [47] D. Blume and Y. Yan, Generalized Efimov Scenario Universal Four-Body States Tied to an Efimov Trimer, for Heavy-Light Mixtures, Phys. Rev. Lett. 113, 213201 Phys. Rev. Lett. 102, 140401 (2009). (2014). [68] Y. Wang, W. B. Laing, J. von Stecher, and B. D. Esry, [48] F. Chevy, Universal Phase Diagram of a Strongly Interact- Efimov Physics in Heteronuclear Four-Body Systems, Phys. ing Fermi Gas with Unbalanced Spin Populations, Phys. Rev. Lett. 108, 073201 (2012). Rev. A 74, 063628 (2006). [69] C. H. Schmickler, H.-W. Hammer, and E. Hiyama, Tetramer [49] R. Combescot and S. Giraud, Normal State of Highly Bound States in Heteronuclear Systems, Phys. Rev. A 95, Polarized Fermi Gases: Full Many-Body Treatment, Phys. 052710 (2017). Rev. Lett. 101, 050404 (2008). [70] We have not investigated tetramer states tied to the excited [50] N. Prokof’ev and B. Svistunov, Fermi-Polaron Problem: Efimov trimer, but we expect that they will exist as resonant Diagrammatic Monte Carlo Method for Divergent Sign- states embedded in the atom-trimer continuum, as is the case Alternating Series, Phys. Rev. B 77, 020408 (2008). for identical bosons [66,71].

011024-15 YOSHIDA, ENDO, LEVINSEN, and PARISH PHYS. REV. X 8, 011024 (2018)

[71] J. von Stecher, Five- and Six-Body Resonances Tied to an J. von Stecher, Resonant Five-Body Recombination in an Efimov Trimer, Phys. Rev. Lett. 107, 200402 (2011). Ultracold Gas of Bosonic Atoms, New J. Phys. 15, 043040 [72] This number had a typo in Ref. [32], which did not affect (2013). any of the other results of that paper. [86] B. Bazak and D. S. Petrov, Five-Body Efimov Effect and [73] B. Huang, L. A. Sidorenkov, R. Grimm, and J. M. Hutson, Universal Pentamer in Fermionic Mixtures, Phys. Rev. Lett. Observation of the Second Triatomic Resonance in Efimov’s 118, 083002 (2017). Scenario, Phys. Rev. Lett. 112, 190401 (2014). [87] B. Bazak, Mass-Imbalanced Fermionic Mixture in [74] The exception to this expected universality is the crossing of a Harmonic Trap, Phys. Rev. A 96, 022708 (2017). the ground-state trimer with the atom-dimer continuum (not [88] V. Gurarie and L. Radzihovsky, Resonantly Paired shown in Fig. 1). This crossing is, in general, known to be Fermionic Superfluids, Ann. Phys. (Amsterdam) 322,2 the least universal aspect of Efimov trimers, as it corre- (2007). sponds to the largest energy scale. We indeed find that the [89] P. Massignan, Polarons and Dressed Molecules Near −4 ground-state trimer crossing is at a=ja−j ≃ 4.86 × 10 in Narrow Feshbach Resonances, Europhys. Lett. 98, 10012 −4 the r0 model and at a=ja−j ≃ 5.58 × 10 in the Λ model, (2012). neither of which quite fits the universal prediction: [90] C. Trefzger and Y. Castin, Impurity in a Fermi Sea on a −π=s0 −4 a=ja−j ≃ 0.72418e ¼ 3.6 × 10 [75]. On the other Narrow Feshbach Resonance: A Variational Study of the ð1Þ Polaronic and Dimeronic Branches, Phys. Rev. A 85, hand, we find that the first excited trimer, with a =ja−j ≃ 053612 (2012). 0.71, matches reasonably well. [91] R. Qi and H. Zhai, Highly Polarized Fermi Gases across a [75] K. Helfrich, H.-W. Hammer, and D. S. Petrov, Three-Body Narrow Feshbach Resonance, Phys. Rev. A 85, 041603 Problem in Heteronuclear Mixtures with Resonant Inter- (2012). species Interaction, Phys. Rev. A 81, 042715 (2010). [92] C. Kohstall, M. Zaccanti, M. Jag, A. Trenkwalder, P. [76] Under the assumption of universality where the quantity 2 ð1Þ Massignan, G. M. Bruun, F. Schreck, and R. Grimm, ma−E4B;− is the sameqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in all models, the ratio is given by Metastability and Coherence of Repulsive Polarons in a ðQMCÞ ðQMCÞ 2 ð1;2chÞ 2 ð1;QMCÞ ð2chÞ ð2chÞ Strongly Interacting Fermi Mixture, Nature (London) 485, a− =a ≃ r0E4 − =a E4 − a− =r0 . All B B; B B; 615 (2012). parameters on the right-hand side are known from the [93] M. Cetina, M. Jag, R. S. Lous, I. Fritsche, J. T. M. Walraven, results of the QMC and the r0 model. R. Grimm, J. Levinsen, M. M. Parish, R. Schmidt, M. Knap, [77] M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C. and E. Demler, Ultrafast Many-Body Interferometry of Nägerl, F. Ferlaino, R. Grimm, P. S. Julienne, and J. M. Impurities Coupled to a Fermi Sea, Science 354, 96 (2016). Hutson, Universality of the Three-Body Parameter for [94] A. Schirotzek, C.-H. Wu, A. Sommer, and M. W. Zwierlein, Efimov States in Ultracold Cesium, Phys. Rev. Lett. 107, Observation of Fermi Polarons in a Tunable Fermi Liquid 120401 (2011). of Ultracold Atoms, Phys. Rev. Lett. 102, 230402 (2009). [78] J. Wang, J. P. D’Incao, B. D. Esry, and C. H. Greene, Origin [95] The high-density expansion for Z at finite a can, in of the Three-Body Parameter Universality in Efimov Phys- B principle, be calculated using an approach similar to that ics, Phys. Rev. Lett. 108, 263001 (2012). in Ref. [31]. However, such a lengthy perturbative analysis [79] P. Naidon, S. Endo, and M. Ueda, Physical Origin of the goes beyond the scope of this work. Universal Three-Body Parameter in Atomic Efimov Physics, [96] P. W. Anderson, Infrared Catastrophe: When Does It Trash Phys. Rev. A 90, 022106 (2014). Fermi Liquid Theory?,inThe Hubbard Model. NATO ASI [80] P. Naidon, S. Endo, and M. Ueda, Microscopic Origin and Series (Series B: Physics), Vol. 343, edited by D. Baeriswyl, Universality Classes of the Efimov Three-Body Parameter, D. K. Campbell, J. M. P. Carmelo, F. Guinea, and E. Louis Phys. Rev. Lett. 112, 105301 (2014). (Springer, Boston, MA, 1995), p. 217. [81] Y. Wang and P. S. Julienne, Universal van der Waals [97] S. Tan, Energetics of a Strongly Correlated Fermi Gas, Physics for Three Cold Atoms Near Feshbach Resonances, Ann. Phys. (Amsterdam) 323, 2952 (2008); Large Momen- Nat. Phys. 10, 768 (2014). tum Part of Fermions with Large Scattering Length, Ann. [82] J. Von Stecher, Weakly Bound Cluster States of Efimov Phys. (Amsterdam) 323, 2971 (2008); Generalized Virial Character, J. Phys. B 43, 101002 (2010). Theorem and Pressure Relation for a Strongly Correlated [83] M. Gattobigio and A. Kievsky, Universality and Scaling in Fermi Gas, Ann. Phys. (Amsterdam) 323, 2987 (2008). the N-Body Sector of Efimov Physics, Phys. Rev. A 90, [98] E. Braaten, D. Kang, and L. Platter, Universal Relations for 012502 (2014). a Strongly Interacting Fermi Gas Near a Feshbach Reso- [84] Y. Yan and D. Blume, Energy and Structural Properties of nance, Phys. Rev. A 78, 053606 (2008). N-Boson Clusters Attached to Three-Body Efimov States: [99] Y. E. Shchadilova, F. Grusdt, A. N. Rubtsov, and E. Two-Body Zero-Range Interactions and the Role of the Demler, Polaronic Mass Renormalization of Impurities in Three-Body Regulator, Phys. Rev. A 92, 033626 (2015). Bose-Einstein Condensates: Correlated Gaussian-Wave- [85] A. Zenesini, B. Huang, M. Berninger, S. Besler, Function Approach, Phys. Rev. A 93, 043606 (2016). H.-C. Nägerl, F. Ferlaino, R. Grimm, C. H. Greene, and

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