Universality of an impurity in a Bose-Einstein condensate

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, Japan2School of Physics and Astronomy, Monash University, Victoria 3800, Australia

(Dated: October 10, 2017)

Universality is a powerful concept in physics, allowing one to construct physical descriptions ofsystems that are independent of the precise microscopic details or energy scales. A prime exampleis the Fermi gas with unitarity limited interactions, whose universal properties are relevant tosystems ranging from atomic gases at microkelvin temperatures to the inner crust of neutron stars.Here we address the question of whether unitary Bose systems can possess a similar universality.We consider the simplest strongly interacting Bose system, where we have an impurity particle(“polaron”) resonantly interacting with a Bose-Einstein condensate (BEC). Focusing on the groundstate of the equal-mass system, we use a variational wave function for the polaron that includes upto three Bogoliubov excitations of the BEC, thus allowing us to capture both Efimov trimers andassociated tetramers. Unlike the Fermi case, we find that the length scale associated with Efimovtrimers (i.e., the three-body parameter) can strongly affect the polaron’s behaviour, even at bosondensities where there are no well-defined Efimov states. However, by comparing our results withrecent quantum Monte Carlo calculations, we argue that the polaron energy is a universal functionof the Efimov three-body parameter for sufficiently low boson densities. We further support thisconclusion by showing that the energies of the deepest bound Efimov trimers and tetramers atunitarity are universally related to one another, regardless of the microscopic model. On the otherhand, we find that the quasiparticle residue and effective mass sensitively depend on the coherencelength ξ of the BEC, with the residue tending to zero as ξ diverges, in a manner akin to theorthogonality catastrophe.

I. INTRODUCTION

Quantum gases are expected to exhibit universal ther-modynamic and dynamical properties when the scatter-ing length a→ ±∞ and the interparticle spacing greatlyexceeds the range of the interactions. Here, the largeseparation of length scales implies that the unitary gas isindependent of an interaction length scale, thus makingit insensitive to its microscopic properties. This scenarioexists for the two-component Fermi gas at unitarity [1],where highly controlled cold-atom experiments have re-cently determined its universal equation of state [2–4],pairing correlations [5–12] and transport properties [13].Moreover, Bose gases in the unitary regime are now alsobeing experimentally probed using ultracold atomic gases[14–20]. However, it remains an open question whetherthe unitary Bose system can also display universal be-havior that has relevance to a range of different physicalsystems.

In contrast to the Fermi case, bosonic systems at uni-tarity always support few-body Efimov bound states [21–26], whose energy and size depend on short-distance pa-rameters such as the interaction range r. Thus, there isan additional interaction length scale that can impact thebehavior of the unitary Bose system and potentially makeit sensitive to microscopic details. Indeed, the deepestbound trimers and larger few-body clusters typically can-not be universally related to each other without invokinga specific model for the short-range interactions [24, 25].

To investigate the universality of a Bose system atunitarity, we consider an impurity immersed in a Bose-Einstein condensate (BEC), where the boson-impurity

interactions can be tuned to unitarity, while the restof the system remains weakly interacting. This so-called “Bose polaron” has received much theoretical at-tention [27–41] and has very recently been observedin cold-atom experiments [15, 16]. Moreover, it hasbeen extended to impurities with more complex inter-nal structure [42–44], and it has potential relevance topolaron problems in solid-state systems — for instance,the limit of weak impurity-boson interactions can be di-rectly mapped to the Frohlich model [27, 45]. The Bosepolaron is a promising system in which to search for uni-versal behavior, since the Efimov trimers consisting ofthe impurity and two bosons are orders of magnitudelarger than the short distance scale r (provided the im-purity mass is not too small) [46, 47]. If we parameterizethe size of the deepest Efimov trimer by |a−|, where a−is the scattering length at which the trimer crosses thethree-atom continuum (see Fig. 1), then the hierarchy oflength scales at unitarity is r � n−1/3 � |a−| for thetypical densities n in experiment [15, 16]. This suggeststhat there is a regime where Efimov physics is irrelevant,such that the ground-state properties of the polaron onlydepend on the density, like in the case of an impurity res-onantly interacting with a Fermi gas [48–52].

In this paper, we address this question using impuritiesthat have a mass equal to that of the bosons — a situ-ation which has already been realized in the 39K atomicgas experiments in Aarhus [16]. We first investigate thefew-body limit, and determine the Efimov trimer and as-sociated tetramer states that involve the impurity. Forvanishing boson-boson interactions, we show that the ra-tios of the binding energies for the deepest bound states

arX

iv:1

710.

0296

8v1

[co

nd-m

at.q

uant

-gas

] 9

Oct

201

7

2

are universal, unlike the identical-boson case. We theninclude this Efimov physics in the many-body system byconstructing a variational wave 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 unitarity sensi-tively depend on the Efimov scale |a−|, even in the regimewhere |a−| far exceeds the interparticle spacing. How-ever, we show that the ground-state polaron energy is auniversal function of n1/3|a−| that is model independentin the regime n1/3r � 1. We corroborate this finding bycomparing our results with recent quantum Monte Carlo(QMC) calculations [33, 36].

In the non-universal, high-density limit n1/3r � 1, weconsider the case of a narrow Feshbach resonance and wederive perturbative expressions for the polaron energyand contact at unitarity. This allows us to demonstrate,for the first time, that the Bose polaron at unitarity canbe thermodynamically stable even in the limit of van-ishing boson-boson interactions. On the other hand, wefind that the quasiparticle residue and effective mass sen-sitively depend on the BEC coherence length. For thecase of an ideal BEC, the effective mass converges to afinite value as we increase the number of excitations ofthe condensate, while the residue vanishes in a mannerreminiscent of the orthogonality catastrophe for a staticimpurity in a Fermi gas [53].

II. MODEL AND VARIATIONAL APPROACH

We consider an impurity atom immersed in a weaklyinteracting homogeneous Bose-Einstein condensate atzero temperature. The interactions in the medium arecharacterised by the boson-boson scattering length aB,and we thus require n0a

3B � 1, with n0 the conden-

sate density (which essentially equals the total densityn in this regime). This allows us to treat both theground state and the excitations of the BEC within Bo-goliubov theory [54]. Note that we always implicitly as-sume aB > 0 to ensure the stability of the condensate.

The presence of the impurity in the medium adds twolength scales associated with the impurity-boson inter-action. The first of these is the scattering length a be-tween the impurity and a boson from the medium. Wedisregard this length scale since we will focus on thestrong-coupling unitary regime close to a Feshbach reso-nance, where |a| greatly exceeds all other length scales,i.e., we generally consider 1/a = 0. Here, we assumeshort-range contact interactions, which is well realized incurrent cold-atom experiments.

The second additional length scale is the so-calledthree-body parameter, which characterizes the size of thesmallest (ground-state) Efimov trimer consisting of twobosons and the impurity. To demonstrate the model in-dependence of our results, we fix this Efimov length scalein two different ways:

• r0-model — we introduce an effective range r0,

which directly relates the three-body parameter tothe two-body interaction [55]

• Λ-model — we take r0 → 0 and instead apply an ex-plicit ultraviolet cutoff Λ to the momenta involvedin Efimov physics, which is equivalent to includinga three-body repulsion [56]

As we explicitly show in Section III, for an impurity ofthe same mass as the medium atoms — the scenario con-sidered here — the precise manner of regularizing Efi-mov physics is unimportant owing to the large separa-tion of scales between the short-range physics and Efimovphysics.

Given the above considerations, we model the systemusing a two-channel description of the Feshbach reso-nance [57], which corresponds to the Hamiltonian (set-ting ~ and the volume to 1):

H =∑

k

[Ekβ

†kβk + εkc

†kck +

(εdk + ν0

)d†kdk

]

+ g√n0∑

k

(d†kck + h.c.

)+ g

∑

k,q

(d†qcq−kbk + h.c.

). (1)

Here, b†k and c†k are the creation operators of a boson andthe impurity, respectively, with momentum k and singleparticle energy εk = k2/2m, where the mass m of a bo-son and of the impurity are taken to be equal. A bosonand the impurity interact by forming a closed-channel

dimer created by d†k, with εdk = εk/2. The inter-channelcoupling g and the bare detuning ν0 are chosen such thatthey reproduce the two-body scattering amplitude in vac-uum at relative momentum k:

f(k) = − 1

a−1 − 12r0k

2 + ik. (2)

Carrying out the renormalization procedure with highmomentum cutoff k0, one obtains [58, 59]

a =mg2

4π

1g2mk02π2 − ν0

, r0 = − 8π

m2g2, (3)

which relates the physical low-energy observables — thescattering length a and effective range r0 ≤ 0 — to thebare parameters of the model. In all calculations, wetake the limit k0, ν0 →∞ for a given set of values for theobservables. We recover the single-channel model (wherer0 = 0) by taking g →∞ while holding a fixed.

In writing the Hamiltonian (1) we already appliedthe Bogoliubov approximation for the weakly-interacting

condensate: The operator β†k creates a Bogoliubov ex-

citation, and is related to the bare boson operator b†kby the transformation b†k = ukβ

†k − vkβ−k. Here, uk =√

[(εk + µ)/Ek + 1]/2, vk =√

[(εk + µ)/Ek − 1]/2, the

Bogoliubov dispersion is Ek =√εk(εk + 2µ), and the

boson chemical potential is µ = 4πn0aB/m ≡ 1/(2mξ2),where ξ is the coherence length of the condensate. Note

3

that Eq. (1) is defined with respect to the energy of theBEC in the absence of the impurity.

To explore the ground-state properties of the unitaryBose polaron, we apply the variational principle, using awave function consisting of terms |ψN 〉 with an increasingnumber N of Bogoliubov modes excited from the conden-sate by the impurity:

|Ψ〉 = |ψ0〉+ |ψ1〉+ |ψ2〉+ |ψ3〉 . (4)

Explicitly, we have

|ψ0〉 =α0c†0 |Φ〉

|ψ1〉 =

(∑

k

αkc†−kβ

†k + γ0d

†0

)|Φ〉

|ψ2〉 =

(1

2

∑

k1k2

αk1k2c†−k1−k2

β†k1β†k2

+∑

k

γkd†−kβ

†k

)|Φ〉

|ψ3〉 =

(1

6

∑

k1k2k3

αk1k2k3c†−k1−k2−k3

β†k1β†k2

β†k3

+1

2

∑

k1k2

γk1k2d†−k1−k2

β†k1β†k2

)|Φ〉 ,

(5)

with |Φ〉 the ground state wave function of the weaklyinteracting BEC. Including up to three excitations al-lows us to investigate the effect of both Efimov trimersand tetramers on the unitary Bose polaron. This goesbeyond previous works on the Bose polaron: The effectof dressing the impurity by a single excitation was in-vestigated in Ref. [28] using a T -matrix approach andin Ref. [29] with a variational approach. Furthermore,two of us previously used two-excitation dressing to in-vestigate the relationship between polaronic and Efimovphysics [32], as well as to successfully compare with theexperimentally obtained polaron spectral function [16](see also Ref. [60]).

To investigate the properties of the ground-state Bosepolaron, we determine the variational parameters by thestationary condition ∂α∗,γ∗ 〈Ψ| (H − E) |Ψ〉 = 0, wherethe energy E can be viewed as the Lagrange multiplierensuring normalization. This yields a set of coupled equa-tions, from which we can eliminate the α coefficients andobtain coupled integral equations for the γ’s only. Theresulting equations are given in Appendix A. In the r0-model, we keep the effective range r0 in the two-bodyinteraction, while in the Λ-model, we set r0 = 0 andapply the cutoff Λ to the momenta in the γ coefficients.

III. UNIVERSAL FEW-BODY EFIMOV STATES

Before proceeding to the many-body polaron physics,it is important to first discuss the few-body limit since —as we shall explicitly demonstrate in Section IV — this

plays a crucial role in understanding the polaron proper-ties. To determine the few-body spectrum, we use wavefunctions of the form (5) in the limit n → 0, such that|Φ〉 now corresponds to the Fock vacuum. Here, the state|ψN 〉 with N excitations is identified as a few-body statecontaining N bosons and the impurity. Consequently,the varational approach is exact and the sectors of differ-ent particle numbers completely decouple. Ignoring thetrivial equation arising from the single-particle problem,the resulting few-body equations take the form

T −10 (E,0) = 0, (6)

T −10 (E − εk,k) γk =∑

q

γqE − εkq

, (7)

T −10 (E − εk1 − εk2 ,k1 + k2)γk1k2 =∑

q

γk1q + γqk2

E − εk1k2q,

(8)

which may be obtained as the zero-density limit of thepolaron equations in Appendix A. Here we have definedεk1k2 ≡ εk1 + εk2 + εk1+k2 and εk1k2q ≡ εk1 + εk2 + εq +εk1+k2+q, while T0 is the vacuum two-body T -matrix,

T −10 (E,k) =m

4πa− m2r0

8π(E − εdk)− m

32

4π

√εdk − E − i0.

(9)

Equation (6) gives the condition for a two-body boundstate, while the solutions of Eqs. (7) and (8) yield, respec-tively, the trimer spectrum and associated tetramers [61].Note that boson-boson interactions are neglected inthis limit, since they are only included via a density-dependent mean-field shift in the boson chemical poten-tial. However, this approximation is reasonable when theEfimov states are insensitive to aB, i.e., when the micro-scopic length scale determining the three-body parametergreatly exceeds aB.

In Figure 1, we show the spectrum of bound statesobtained by solving Eqs. (6–8) within the r0-model. Forpositive scattering length, a single two-body bound dimerexists, with energy

εB = −

(√1− 2r0/a− 1

)2

mr20, (10)

as found by solving Eq. (6). The three-body problem de-scribed by Eq. (7) is distinctly different: For strong inter-actions, the impurity and two bosons can form an infiniteseries of three-body bound states, even in the absence ofthe two-body bound state. Remarkably, these so-calledEfimov trimers [21, 62] satisfy a discrete scaling symme-try whereby the energy E of one trimer can be mappedonto the next via the discrete transformation E → λ−2l0 Eand a → λl0a, with l integer and λ0 a scaling parameterthat depends on the particular three-body problem (forreviews, see Refs. [22, 24–26]). In our case of two non-interacting identical bosons with the same mass as the

4

-0.2 -0.1 0 0.1 0.2-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0

Four atoms continuum

Two atoms + dimer continuum

Atom + trimer continuum

a−a(1)4B,− a

(2)4B,− a

(1)−

E−

E(2)4B,−

E(1)4B,−

E(1)−

sgn(a) |r0/a|1/4

−! ! m

r2 0E

! !1/8

a(1)∗

FIG. 1. Few-body energy spectrum calculated from equations (6-8). The ground (first excited) trimer is shown as a red

solid curve, which first appears at a negative scattering length a− (a(1)− ) and then dissociates into the atom-dimer continuum

(delimited by the black solid line) at a positive scattering length a∗ (a(1)∗ ), where a∗ is outside the plotted region. We only show

the ground and first excited trimers, but there exists an infinite series of higher excited trimers with an accumulation point atunitarity, 1/a = 0. We also find two tetramer states (blue dashed lines) tied to the ground Efimov trimer. The excited tetrameris very weakly bound, but it persists at unitarity and disappears into the atom-trimer continuum at the positive scatteringlength a ' 9× 103|r0|.

third particle, the scaling factor is found from a tran-scendental equation to be λ0 = eπ/s0 ' 1986.1 [62]. Dueto this large scaling factor, we only show the two lowesttrimer states in the figure (indeed, for realistic experi-mental parameters, the size of the first excited Efimovtrimer is of order 1cm!).

Below the ground-state Efimov trimer, we predict theexistence of two tetramer states, even at unitarity (seeFig. 1). This mirrors previous results for four identicalbosons, where it was found, both theoretically [63–66]and experimentally [67], that there are two lower-lyingtetramers associated with the ground-state trimer. Thepresent scenario more closely resembles studies of a lightatom resonantly interacting with three identical heavyatoms [47, 68, 69]. Our findings are generally consis-tent with the results of these previous works, althoughno excited tetramer was reported for small mass imbal-ance [70]. Note that, within our theory, we do not re-quire an additional length scale to fix the positions ofthe tetramers relative to the Efimov trimers [65].

A crucial point is that the few-body physics describedhere is essentially universal, in the sense that the entirespectrum is independent of the details of the short-rangephysics. This universality originates from the large sepa-ration of scales between the short-distance physics, char-acterized here by the effective range r0, and the typicalsize of the ground-state Efimov trimer. The latter maybe conveniently characterized by the “critical” scattering

length a− at which the trimer crosses into the three-atomcontinuum. In our model, we have a−/r0 ' 2467 [71] (seeTable I for a summary of the length scales and energyscales associated with Efimov physics). This large ratioensures that the energy of the ground-state Efimov trimerat unitarity is far smaller than that required to probe theshort-range physics. Indeed, we see that the ratio of crit-ical scattering lengths for the excited and ground-state

trimers, i.e., a(1)− /a− ' 1991, is very close to the pre-

dicted scaling factor of 1986.1. Likewise, at unitarity,the ratio of the ground-state trimer energy to that of

the excited state is E−/E(1)− ' (1986.1)2, i.e., it matches

the universal prediction to 5 significant digits. This isin remarkable contrast to the situation for three identi-cal bosons, where deviations from the universal scalingfactor of 22.7 are usually on the order of 10-20% due tofinite-range corrections [22, 24, 72].

To test the model independence of the few-bodyphysics, we also consider the Λ-model, where r0 = 0 andwe apply an explicit cutoff Λ to the momentum sums inthe three- and four-body equations (7–8). Here we findthat after fixing the three-body parameter, the spectrumis essentially indistinguishable from that in Fig. 1, in thesense that the differences are within the thickness of thelines [73]. The agreement between the two models is alsoevident in the summary of length scales and energy scalesin Table I. Thus, we conclude that our description of the

5

|a−| ma2−|E−| a(1)− /a−

√E−/E

(1)− a−/a

(1)4B,− E

(1)4B,−/E− a−/a

(2)4B,− E

(2)4B,−/E−

r0-model 2467|r0| 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)

TABLE I. Comparison of few-body data for the r0- and the Λ-model. While the overall length scale set by a− depends on thedetails of the models, the dimensionless ratios characterizing the few-body spectra agree to an accuracy of 1% or less. Thesequantities include the scattering lengths at which the few-body bound states cross into the continuum, as well as their energiesat unitarity, and are illustrated in Fig. 1.

few-body spectrum can be accurately characterized byuniversal Efimov theory.

Further evidence of the universality of our few-bodyresults is found when comparing the ratio of our ground-state tetramer and trimer energies with recent QMCresults [33]. There, this ratio was reported to be

E(1)4B,−/E− ' 9.7, while we find E

(1)4B,−/E− ' 9.35. Im-

portantly, the QMC calculation used a completely dif-ferent model involving a hard-sphere boson-boson inter-action, characterized by the scattering length aB, andattractive square-well interactions between the impurityand the bosons. The range of the square well was takento be much smaller than aB, and consequently the en-ergies of the ground-state Efimov trimer and associatedtetramers are set by aB. The universality of the spectrumthus means that we can relate the three-body parameterin the QMC calculation to the boson-boson scatteringlength [74]:

a(QMC)

− ' −2.09(36)× 104a(QMC)

B . (11)

The agreement between the results from the differentmodels also reinforces the point that we do not need afour-body parameter to fix the tetramer relative to thetrimer. This is consistent with theoretical results foridentical bosons [63, 64, 66] and for heteronuclear sys-tems [47, 68, 69].

We end this section by contrasting the few-body uni-versality found here with the so-called van der Waalsuniversality of Efimov physics [75–79]. The latter refersto the phenomenon where the Efimov states of variousatomic species are universally related to their van derWaals length, e.g., a− ' −(9.1 ± 1.5)rvdW. This behav-ior, however, is rather different from that predicted byuniversal zero-range Efimov theory, in particular for the

ground-state trimer. Indeed, the ratio a(1)− /a− ' 20.7

found in van der Waals universal theory [79] significantlydeviates from the universal ratio 22.7, in contrast to thescenario considered here.

IV. POLARON ENERGY AT UNITARITY

We now turn to the many-body system and investi-gate the zero-temperature equation of state for the po-laron at unitarity. Since we wish to properly describeEfimov physics within our models, we will focus on the

regime where the boson-boson scattering length aB isalways much smaller than all other length scales. Totest the accuracy of our variational approach outlined inEqs. (4) and (5), we will consider wave functions with upto one, two or three Bogoliubov excitations, correspond-ing to |ψ0〉+|ψ1〉, |ψ0〉+|ψ1〉+|ψ2〉, and |Ψ〉, respectively.For each model, we will convert the short-distance scale(e.g., r0 or Λ−1) into the dimensionless three-body pa-rameter n1/3|a−| in order to expose the role of univer-sal Efimov physics. The numerical results are displayedin Fig. 2 and, in the following, we analyze the differentregimes defined by particle density.

A. Low-density limit

In the limit of vanishing density n1/3|a−| → 0, theground-state polaron energy should reduce to that of thedeepest bound few-body cluster. If the polaron varia-tional wave function only includes one Bogoliubov ex-citation, the largest few-body state it can describe isthe dimer, which has zero binding energy at unitar-ity. Thus, as shown in Fig. 2, we obtain an energythat simply tends to zero with decreasing density, i.e.,E = −(4πn)2/3/m ' −5.4n2/3/m [28, 29]. Note thatthis is independent of the microscopic model we use sincethe interparticle spacing greatly exceeds any length scaleof the impurity-boson interactions.

For polaron wave functions with up to two (three)Bogoliubov excitations, the low-density limit recoversthe deepest bound universal trimer (tetramer) state dis-cussed in Sec. III. Here, the density drops out of the prob-lem and we have polaron energy E = − η

m|a−|2 , where η

is a universal, model-independent constant that dependson the Efimov cluster size (see Fig. 2).

We also expect there to be larger bound clusters be-yond the tetramer, similarly to the case of identicalbosons where larger clusters associated with the Efimoveffect have been predicted [80–82] and, indeed, exper-imentally observed [83]. For the impurity case, recentQMC calculations already demonstrate the existence ofa pentamer with η ' 300 [33]. However, since there isa high-energy cutoff in the problem (e.g., set by Λ orr0), the system cannot support arbitrarily deeply boundEfimov clusters, and the deepest bound state should beuniversal as long as η � 106. Indeed, as suggestedby the QMC calculations [33], it is possible that there

6

FIG. 2. Energy equation of state for the Bose polaron at unitarity 1/a = 0, obtained using the variational wave functionwith one (gray), two (red) and three (blue) Bogoliubov excitations for vanishing boson-boson interactions aB → 0. We showthe results of the r0-model (dashed) and the Λ-model (solid). In the low-density limit, the dotted straight lines correspond

to the Efimov trimer and associated tetramer energies, with E− = −9.91/m|a−|2 and E(1)4B,− = −92.7/m|a−|2, respectively.

The black dotted line in the high-density regime shows the result of a perturbative expansion, Eq. (13). The gray shadedregion depicts the low-density few-body dominated regime, while the blue shaded region corresponds to the high-density regime|r0|,Λ−1 > n−1/3, where Efimov physics is suppressed and the system is sensitive to the microscopic model.

is no bound cluster larger than the pentamer for theequal-mass impurity case. Firstly, the pairwise attrac-tion between the impurity and the bosons scales withN rather than N(N − 1), unlike the case for N iden-tical bosons [25], so larger bound states will be less fa-vored. Secondly, an effective short-range repulsion be-tween bosons could mean that the pentamer has a closed-shell structure, where any additional bosons must occupyhigher energy orbitals, similar to what has been arguedfor mass-imbalanced fermions [84, 85]. Such an effectiverepulsion naturally emerges in both the r0- and the Λ-models: In the former, this arises from the fact that onlyone boson at a time can occupy the closed-channel dimer,while in the latter it is due to the three-body cut-off beingequivalent to a three-body repulsion [56].

From the above considerations, it is reasonable to con-clude that the low-density limit is universal. Moreover,since there is a maximum size to the deepest bound Efi-mov cluster, we expect the polaron energy to have a well-defined thermodynamic limit, wheremE/n2/3 is finite fornon-zero density.

B. High-density limit

To further support the claim that the polaron energyis finite at unitarity, we consider the opposite limit ofhigh density, n→∞. Here, the interparticle spacing be-comes smaller than the range of the interactions and thus

the system will depend on the microscopic details of themodel. However, in the case of the r0-model, it allowsus to perform a controlled perturbative expansion in theinter-channel coupling g (or, equivalently, 1/n1/3|r0|) atunitarity [86]. Physically, this corresponds to the limitof a narrow Feshbach resonance, a scenario which has al-ready been successfully investigated for an impurity ina Fermi sea, both theoretically [87–89] and experimen-tally [90, 91].

Our starting point is the self-energy of the unitary Bosepolaron, which, in the limit g → 0 and aB → 0, consists ofthe lowest order diagrams shown in Fig. 3. Such diagramsare included in the self-consistent T -matrix approach tothe Bose polaron [28]. For an impurity with momentum pand frequency ω, the self-energy in this limit is explicitly

Σ(p, ω) ' n0g2

ω − εdp+

n0g4

(ω − εdp)2×

∑

k

1

ω − εk+p − εk − n0g2

ω−εdk+p−εk+

1

2εk

, (12)

where the first and second terms correspond to the di-agrams in Fig. 3(a) and (b), respectively. The ground-state polaron energy is then determined by taking thezero-momentum pole of the impurity propagator, whichgives E = Σ(0, E). Note that we must include self-energyinsertions in the impurity propagator even at lowest or-der (see Fig. 3), since the we cannot simply take ω = 0at the pole when aB → 0. This is in contrast to previous

7

perturbative treatments of the weakly interacting Bosepolaron which focussed on aB > 0 [31].

In the regime of weak boson-boson interactions, wherena3B � aB/|r0| � 1, we obtain the ground-state polaronenergy

E ' − 1

m

√8πn

|r0|+

1

m

√3

7

(8πn

|r0|5)1/4

. (13)

This demonstrates that the energy is well-defined andbounded from below in this limit, even when the boson-boson interactions are vanishingly small. Such behaviorarises from the fact that only one boson at a time canscatter into the closed channel, thus producing an effec-tive boson-boson repulsion that restricts the density ofbosons that can cluster around the impurity. As shown inFig. 2, the perturbative expression correctly reproducesthe energy for the r0-model in the high-density limit.Note that the one-excitation wave function only capturesthe leading order term in Fig. 3(a), while the wave func-tions with two or three excitations correctly describe thenext order correction in 1/n1/3|r0|.

+=

FIG. 3. Lowest order diagrams for the unitary ground-statepolaron in the high-density limit n1/3|r0| � 1, where (a) and(b) give, respectively, the first and second order terms for thepolaron energy in a weakly interacting BEC. Here, the dashedlines correspond to particles emitted or absorbed from thecondensate, while the blue solid line is the boson propagator,the wavy lines represent closed-channel molecule propagators,and the filled circles correspond to the inter-channel couplingg. The black double line, defined in (c), represents the dressedimpurity to lowest order, where the black single line is the bareimpurity propagator.

In the case of the Λ-model, the high-density limit cor-responds to n1/3/Λ → ∞, which is equivalent to tak-ing the cutoff Λ to zero. Therefore, within our varia-tional approach, only two-body correlations survive andwe obtain the equations for the wave function with oneexcitation (see Appendix A). For vanishing boson-bosoninteractions, this yields E = −(4πn)2/3/m, which is thepolaron energy of the one-excitation wave function acrossall densities, as shown in Fig. 2.

C. Many-body universal regime

In the intermediate regime r � n−1/3 � |a−| (wherer can represent |r0| or Λ−1), the interparticle distance iswell separated from all length scales associated with theinteractions. If the medium were fermionic rather thanbosonic, then the polaron energy at unitarity would be a

universal value that only depends on the medium density,i.e., Epol ' −4.6n2/3/m [48]. Remarkably, for the Bosepolaron, we see in Fig. 2 that the energy strongly dependson n1/3|a−| at intermediate densities. This suggests thatthere exist resonant three-body interactions, such thatthere are strong three-body correlations in the systemeven when the trimer binding energy is comparativelysmall.

Even though the energy of the Bose polaron cannotbe assigned a universal value at unitarity, we argue thatit is, in fact, a universal function of n1/3|a−| away fromthe high-density regime. We have previously argued thatthe low-density limit of the polaron energy universallydepends on n1/3|a−| due to the universal behavior ofthe few-body states demonstrated in Sec. III. Here, forthe intermediate density regime shown in Fig. 4, we seethat different microscopic models of the Bose polaron canessentially be collapsed onto the same curve when theground-state energy is plotted versus n1/3|a−|. In par-ticular, both the results for the Λ- and the r0-models areconsistent with the QMC calculation from Ref. [36] whenEq. (11) is used to plot the QMC data in terms of theEfimov three-body parameter. This demonstrates thatthe polaron energy can be universally described in terms

of n1/30 |a−|, regardless of the microscopic details. Fur-

thermore, it suggests that our variational approach withthree Bogoliubov excitations captures the dominant cor-relations in the full many-body problem, and that the Efi-mov three-body parameter plays a more important rolein the polaron energy than the coherence length of thecondensate (see Appendix B). We emphasize that ourcomparison in Fig. 4 constitutes a complete reinterpre-tation of the QMC results of Ref. [36], since that workdid not consider the possibility that the variation of thepolaron energy with boson-boson interaction was due toEfimov physics.

V. CHARACTERIZATION OF THE POLARON

Having shown that the polaron energy is a universalfunction of n1/3|a−| for sufficiently low densities, we nowturn to the other quasiparticle properties which charac-terize the polaron: the residue, effective mass, and con-tact. We will generally focus on the r0-model, which isthe more physical model and which allows a perturbativeanalysis in the high-density limit.

A. Residue

The residue Z is the squared overlap of the polaron

wave function with the non-interacting state c†0 |Φ〉. Mak-ing reference to the wave function in Eq. (5), this is

Z = |α0|2, (14)

where we assume 〈Ψ|Ψ〉 = 1 for normalization (see Ap-pendix C). The residue can be accessed directly in experi-

8

FIG. 4. Comparison of different models for the unitary Bosepolaron in the regime r � n−1/3 � |a−|, where r is the rangeof the interactions. We show the results for the polaron energyfrom the Λ-model (solid) and the r0-model (dashed), whichinclude up to three excitations of the condensate and whichtake aB → 0. The QMC results of Ref. [36] are shown as thegray dots. The recent Aarhus experiment [16] is estimated to

have n1/3|a−| ' 100.

ment; indeed, in the case of the Fermi polaron it has beenmeasured using both radio-frequency spectroscopy [92]and Rabi oscillations [90].

In Fig. 5 we show our results for the polaron residueat unitarity as a function of the three-body parameter,where the boson-boson interactions are taken to be van-ishingly small. We find that there are clear differencesbetween the results depending on the number of Bogoli-ubov excitations used in the variational approach, andalso between the different models. Within the r0-modelthe residue takes the value 1/2 in the high-density regimewhere the polaron wave function is an equal superposi-tion of the impurity and the closed-channel dimer. How-ever, while the residue within the one-excitation approxi-mation is seen to increase from 1/2 to 2/3 with increasingdensity, we observe a rapid decrease of Z towards the low-density regime when including several excitations. Onthe other hand, for the Λ-model, the residue is 2/3 if weconsider only one Bogoliubov excitation. By increasingthe number of Bogoliubov excitations, it is suppressednot only in the few-body dominated regime, but also inthe high-density limit, where the residue takes the values1/3 and 1/9, respectively, when two and three excitationsare taken into account. Thus, unlike the energy, the po-laron residue does not appear to converge to a universalfunction of the three-body parameter.

The origin of this behavior can be understood withinperturbation theory for the r0-model in the high-densitylimit. Considering again the diagrams in Fig. 3 and theexplicit expression for the self-energy in Eq. (12), we find

FIG. 5. Polaron residue Z at unitarity 1/a = 0 as a function ofthe dimensionless three-body parameter. We show the resultsfor the Λ-model (solid) and the r0-model (dashed) for aB → 0.In order of decreasing residue, the results are obtained forwave functions including up to one, two or three excitationsof the condensate. The grey and blue shaded regions are thesame as in Fig. 2, i.e., the low- and high-density regimes,respectively.

at small g:

Z−1 = 1− ∂Σ(0, ω)

∂ω

∣∣∣∣ω=E

' 2−√

3

7

2

(8πn|r0|3)1/4+

L

|r0|, (15)

where L is a low-energy length scale that provides aninfrared cutoff of low momenta. In the thermodynamiclimit, this is set by the coherence length of the BEC,i.e., we have L ∝ ξ. Thus, we find that the residuevanishes when the coherence length ξ → ∞, and conse-quently the polaron wave function is orthogonal to thenon-interacting impurity state in the case of an idealBEC. This feature cannot properly be captured withinour variational approach, since this requires an infinitenumber of Bogoliubov excitations; hence the observedlack of convergence in Fig. 5. On the other hand, whenaB 6= 0 we find that the wave functions with two andthree excitations both give the same leading order cor-rection to Z:

Z − 1

2∝√aB|r0|

, (16)

in the limit n1/3aB � 1 and n1/3|r0| → ∞ [93]. Thus, inthis case, the residue is finite and initially increases withdecreasing |r0|. Note, however, that the residue alwaysvanishes in the low-density regime, as shown in Fig. 5,since the polaron evolves into a few-body bound statewhich has zero overlap with the non-interacting impuritystate.

The simultaneous convergence of the energy but sup-pression of the residue in the ideal BEC arises from the

9

fact that the impurity affects the condensate at arbitrar-ily large distances and thus excites an infinite number oflow-energy modes in the condensate. This result is akinto the orthogonality catastrophe for a static impurity in aFermi gas. Indeed, in that case, the orthogonality of theinteracting and non-interacting impurity wave functionsis also apparent in perturbation theory, already at lowestorder beyond mean-field theory [94]. We also note thatthe vanishing residue of the Bose polaron is not confinedto the strongly interacting impurity at unitarity: Evenfor weak impurity-boson interactions, the residue of theattractive polaron was found within perturbation theoryto approach zero as 1/ξ when aB → 0 [31].

B. Effective mass

The interactions between the impurity and the bosonicmedium also modify the mobility of the impurity and giverise to an effective mass m∗. This can be derived fromthe impurity’s self-energy as follows:

m

m∗= Z

[1 +m

∂2Σ(q, E)

∂q2

∣∣∣∣q=0

]. (17)

Note that this gives m∗ = m for the non-interacting im-purity state, as required. Our calculation of m∗ withinthe variational approach is described in Appendix C, andthe results for aB → 0 at unitarity are plotted in Fig. 6.Due to the complexity of the calculation, we restrict our-selves to wave functions with up to two Bogoliubov exci-tations.

For the case of an ideal BEC, we have residue Z → 0,and thus one might conclude from Eq. (17) that the ef-fective mass diverges. However, if we once again considerthe high-density limit of the r0-model and use the self-energy in Eq. (12), we obtain

m

m∗' Z

(3

2−√

3

7

1

(8πn|r0|3)1/4+ α

L

|r0|

), (18)

where α is a constant that must be determined from aproper finite-aB perturbative analysis. Thus, we see thatthere is a term proportional to L that cancels the corre-sponding term in the residue in Eq. (15), implying thatthe effective mass is finite.

Indeed, in the low-density few-body regime, we alreadyhave the situation where the residue vanishes, while theeffective mass remains finite with m∗ = (N +1)m, whereN is the number of bosons bound to to the impurity.This behavior is captured by both r0- and Λ-models, asshown in Fig. 6, where the wave function with two Bo-goliubov excitations has m∗/m→ 3 as n→ 0. Likewise,we expect the wave function with three excitations to re-cover the mass of the tetramer in this limit. For higherdensities, we find that the results of our variational ap-proach become sensitive to the specific model used, aswell as to the number of excitations of the ideal BEC

considered. Here, in the many-body limit, one requiresan infinite number of low-energy excitations to correctlydescribe the effective mass for aB → 0, similarly to thecase of the residue.

For a BEC with 0 < n1/3aB � 1, the behavior of theeffective mass in the high-density limit of the r0-modelcan once again be captured using just two Bogoliubovexcitations of the BEC. In the limit |r0| → ∞, the po-laron effective mass m∗/m → 4/3, which correspondsto twice the reduced mass of the impurity and closed-channel dimer. Moreover, we find that the leading ordercorrection behaves as

m

m∗− 3

4∝√aB|r0|

. (19)

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

FIG. 6. Inverse effective mass of the Bose polaron at unitarity,1/a = 0, for aB = 0. We calculate it with the Λ-model (solid)and the r0-model (dashed), taking into account one (gray) andtwo (red) Bogoliubov excitations. The grey and blue shadedregions are the same as in Fig. 2

C. Contact

Another quantity of interest is the unitary Tan contact,which is defined from the polaron energy as [95, 96]

C = m2g2∂E

∂ν

∣∣∣∣ν=0

. (20)

Here we have used the physical detuning ν ≡ −g2m/4πa,which can be related to the bare detuning ν0 usingEq. (3). Physically, the contact gives a measure of thedensity of pairs at short distances. For the two-channelHamiltonian in Eq. (1), one can show that the con-tact is proportional to the population of closed-channeldimers [96]:

C = m2g2∑

k

〈Ψ| d†kdk |Ψ〉 . (21)

10

Within our variational ansatz, this corresponds to

C = m2g2

(|γ0|2 +

∑

k

|γk|2 +1

2

∑

k1k2

|γk1k2 |2). (22)

Note that the limit g →∞ is well defined, and thus thisexpression also captures the single-channel Λ-model.

FIG. 7. Contact of the Bose polaron at unitarity, 1/a = 0,for aB = 0. The solid and dashed curves correspond to theΛ- and r0-model, respectively. We show the results obtainedusing wave functions with up to one (gray), two (red) andthree (blue) excitations. The grey and blue shading indicatethe same regions as in Fig. 2

Figure 7 displays the contact at unitarity for both Λ-and r0-models in the case of vanishing boson-boson inter-actions. Similarly to the polaron energy, we find that thecontact in the high-density limit is well converged withrespect to the number of Bogoliubov excitations. Forthe r0-model, we can evaluate the contact perturbativelyfor large n1/3|r0| using the self-energy in Eq. (12). Herewe simply shift the closed-channel dispersion from εdp to

εdp + ν and then determine the ground-state energy fromthe equation E = Σ(0, E) as a function of detuning ν.Applying Eq. (20) then yields the high-density expression

C ' 8π

|r0|

(1

2− 29

28

√3

7

1

(8πn|r0|3)1/4

). (23)

We find that this perturbative result is correctly repro-duced by both two- and three-excitation approximationsin the high-density limit. The leading order term is in-dependent of density since the closed-channel fractiontends to 1/2 at unitarity as |r0| → ∞ (or, equivalently,as g → 0), see Eq. (15).

For lower densities n1/3|a−| . 103, the contact be-comes insensitive to the underlying model and thus ap-pears to be a universal function of the dimensionlessthree-body parameter (see Fig. 7). In the limit of vanish-ing density n→ 0, we recover the contact of the deepestbound few-body cluster, where C|a−| is a universal num-ber.

VI. CONCLUSIONS AND OUTLOOK

Using a variational approach, we have investigated theground state of the Bose polaron with unitarity limitedimpurity-boson interactions. For sufficiently low bosondensities, we have demonstrated that the polaron equa-tion of state is a universal function of the three-bodyparameter associated with Efimov physics. This is ahighly non-trivial result, since the regime of universalityextends to densities at which the binding energies of thefew-body states are several orders of magnitude smallerthan the polaron energy. Our findings are corroboratedby the fact that we observe the same behavior for twodifferent microscopic models, and by the fact that ourcalculations agree well with recent QMC results [33, 36]when they are appropriately reinterpreted in terms of thethree-body parameter. We have also demonstrated thatthe few-body spectrum of Efimov trimers and associatedtetramers is model independent, further highlighting theuniversality of our results.

In the non-universal, high-density limit, we have per-formed a controlled perturbative analysis of the r0-modeland we have derived new analytic expressions for the uni-tary Bose polaron at a narrow Feshbach resonance. Ourresults demonstrate that even when we take the idealBose gas limit, the polaron energy and contact can re-main well defined thermodynamically. This is a conse-quence of the fact that only one boson at a time canoccupy the closed-channel dimer state, which thus gen-erates an effective repulsion between bosons that acts tostabilize the system. We expect a similar scenario for theΛ-model since the high-momentum cut-off in the three-and four-body terms of the variational equations is equiv-alent to an explicit three-body repulsion [56]. However,in general, it is not clear that a well-defined thermody-namic limit exists for treatments of the Bose polaron thatfocus on single-channel Hamiltonians and neglect Efimovphysics [39, 97].

While the polaron equation of state at unitarity is rel-atively insensitive to boson-boson interactions, we findthat the quasiparticle residue and effective mass stronglydepend on the coherence length of the BEC. In partic-ular, the residue vanishes as ξ → ∞, a feature whichresembles the orthogonality catastrophe [53]. Thus, aninteresting remaining question is how the residue tends tozero with the number of excitations of the condensate in-cluded in the wave function. A related question is: whatfinite result does the effective mass converge towards asξ → ∞ and is it a universal function of the three-bodyparameter?

We emphasize that the (two-channel) r0-model faith-fully describes the physics of an actual Feshbach reso-nance in an ultracold atomic gas [57], and thus we ex-pect our results to accurately describe current and fu-ture cold atom experiments. Furthermore, we expect theuniversal behavior presented here to also exist for heavyimpurities, a scenario which can be realized by choosingatomic species with a large mass ratio or by pinning the

11

impurity in an optical lattice. In the particular case ofa static impurity, there are no Efimov few-body states,and therefore it would be of great interest to understandhow the system behaves in the limit n1/3|r0| � 1 of thetwo-channel model.

Our results may be probed in experiments similar tothose carried out recently at JILA [15] and in Aarhus [16].In particular, our results directly apply to the latter ex-periment which used 39K as both the impurity and hostatoms. Furthermore, we estimate that n1/3|a−| ' 100 inthat experiment, which indicates that it is already in aregime where the non-trivial dependendence on the in-terparticle spacing is strongly pronounced. Indeed, in arecent JILA experiement [19], the density of the Bosegas was changed by two orders of magnitude, and thusan experimental investigation of the density dependenceof the unitary Bose polaron appears well within reach.

ACKNOWLEDGMENTS

We gratefully acknowledge fruitful discussion with LuisA. Pena Ardila, Jan J. Arlt, Georg M. Bruun, Ras-mus S. Christensen, Victor Gurarie, Nils B. Jørgensen,Richard Schmidt, Zhe-Yu Shi, and Bing Zhu. We alsothank Luis A. Pena Ardila for the QMC data of Ref. [36]and for helpful comments on the manuscript. SMY ac-knowledges support from the Japan Society for the Pro-motion of Science through Program for Leading Grad-uate Schools (ALPS) and Grant-in-Aid for JSPS Fel-lows (KAKENHI Grant No. JP16J06706). JL and MMPacknowledge financial support from the Australian Re-search Council via Discovery Project No. DP160102739.JL is supported through the Australian Research CouncilFuture Fellowship FT160100244. JL and MMP acknowl-edge funding from the Universities Australia – GermanyJoint Research Co-operation Scheme. This work was per-formed in part at the Aspen Center for Physics, whichis supported by the National Science Foundation GrantNo. PHY-1607611.

12

Appendix A: Coupled integral equations

Taking the derivative 〈∂Ψ| (H − E) |Ψ〉 = 0 with respect to α0, αk, αk1k2, αk1k2k3

, γ0, γk, and γk1k2for our

variational wave function in Eq. (4) yields the following linear equations

Eα0 =g√n0γ0 − g

∑

k

vkγk, (A1)

(E−εk−Ek)αk =gukγ0 + g√n0γk − g

∑

q

γkqvq, (A2)

(E−Ek1k2)αk1k2

=g (γk1uk2

+γk2uk1

) + g√n0γk1k2

, (A3)

(E−Ek1k2k3)αk1k2k3

=g (γk1k2uk3

+γk2k3uk1

+γk1k3uk2

) , (A4)

(E − ν0)γ0 =g√n0α0 + g

∑

k

ukαk, (A5)

(E −εdk−ν0−Ek

)γk =g

√n0αk − gvkα0 + g

∑

k′

uk′αkk′ , (A6)

(E −εdk1+k2

−ν0−Ek1−Ek1

)γk1k2

=g√n0αk1k2

− g (αk1vk2

+αk2uk1

) + g∑

k3

αk1k2k3uk3

, (A7)

where Ek1k2 = Ek1+Ek2 + εk1+k2 and Ek1k2k3 = Ek1+Ek2+Ek3+εk1+k2+k3 . Using the first four equations to removethe α coefficients, we obtain

T −1(E,0)γ0 =n0Eγ0 +

√n0∑

k

(ukγk

E − εk − Ek− vkγk

E

)−∑

kq

ukvqγkqE − εk − Ek

, (A8)

T −1(E − Ek,k)γk =√n0

(uk

E − εk − Ek− vkE

)γ0 +

n0E − εk − Ek

γk +∑

q

(ukuqγqE − Ekq

+vkvqγqE

)

+√n0∑

q

(uqγkqE − Ekq

− vqγkqE − εk − Ek

), (A9)

T −1(E − Ek1 − Ek2 ,k1 + k2)γk1k2 =n0

E − Ek1k2

γk1k2

+

[{√n0

(uk1

γk2

E − Ek1k2

− vk1γk2

E − εk2− Ek2

)− uk1

vk2γ0

E − εk1− Ek1

+∑

q

(uk1

uqγk2q

E − Ek1k2q+

vk1vqγk2q

E − εk2− Ek2

)}+ (k1 ↔ k2)

].

(A10)

Here, the two-body T matrix T (E,k) in the BEC medium is

T −1(E,k) =m

4πa− mr0

8π(E − εdk)−

∑

q

[u2q

E − Eq − εk+q+

1

2εq

]

= T −10 (E,k) +∑

q

[1

E − εq − εk+q− u2qE − Eq − εk+q

], (A11)

where the vacuum T matrix T0(E,k) was defined in Eq. (9).In this paper, we show results obtained from variational wave functions which include 1, 2, or 3 excitations of the

condensate, see Eq. (5). These approximations correspond to solving Eq. (A8) with γk = γkq = 0, Eqs. (A8-A9) withγkq = 0, and all of Eqs. (A8-A10), respectively.

Appendix B: Effect of finite coherence length and number of Bogoliubov excitations

In Section IV C we compared the results of our variational approach with those obtained in quantum Monte Carlocalculations by Pena Ardila and Giorgini [36], and found very good agreement across more than an order of magnitudein interparticle spacing, see Fig. 2. In this comparison we highlighted the fact that the dependence of the polaron

13

FIG. 8. Polaron energy calculated within the Λ-model as a function of n1/3|a−| for aB = 0 (solid line) and for aB = 2.1×104|a−|.The latter choice of aB ensures that we take the same ratio of a− and aB as in the QMC calculation (gray dots), see Eq. (11),and the corresponding scale for aB is shown on the top x-axis. Inset: Polaron energy at unitarity as a function of inversenumber N of Bogoliubov excitations. We present the results of the Λ-model (solid) and r0-model (open) for n1/3|a−| = 100

(blue circles) and n1/3|a−| = 500 (red triangles). resonance, 1/a = 0, for aB = 0.

energy on the three-body parameter a− is much stronger than the dependence on aB, and to emphasize this we tookaB = 0. However, as we have argued in Section III, in the QMC calculation the ratio between a− and aB was kept fixedat a value of a(QMC)

− ' −2.09(36)× 104a(QMC)

B as identified in Eq. (11). Therefore, we now consider the residual effectof keeping a−/aB fixed in the same manner as in the QMC. Our results for the polaron energy within the Λ-model areshown in Fig. 8. We see that the polaron energy is almost unaffected in the low-density, few-body dominated regime,while aB becomes more important with increasing density. However, throughout the range plotted, i.e., the universalmany-body regime, the relative change in energy arising from the change in n1/3|a−| clearly greatly exceeds that fromthe increase in n1/3aB. We furthermore see that both sets of results are consistent with our reinterpretation of theQMC results in terms of a three-body parameter.

In the inset of Fig. 8 we furthermore illustrate the convergence of the polaron energy as a function of (inverse)number of Bogoliubov excitations included in our variational approach. In the many-body dominated regime, atn1/3|a−| = 500, we see that the convergence appears to be nearly linear, with a slope (and extrapolation to an infinitenumber of excitations) that is consistent with the difference between the finite aB dotted line and the QMC resultsin the main figure. Closer to the few-body dominated regime at n1/3|a−| = 100 our approach converges more slowlydue to the strong few-body correlations induced by the trimer and tetramer bound states. However, it is likely thatincluding four Bogoliubov excitations in our approach to account for the hypothesized pentamer (see Sec. IV A) wouldyield a nearly converged result.

Appendix C: Calculation of observables

Once the wave function |Ψ〉 is obtained, we can calculate physical quantities of interest. In the following, we assumethat the variational wave function is normalized, 〈Ψ|Ψ〉 = 1. When we solve the renormalized equations, (A8-A10),this implies that

1 = |α0|2 +∑

k

|αk|2 +1

2

∑

k1k2

|αk1k2 |2 +1

6

∑

k1k2k3

|αk1k2k3 |2 + |γ0|2 +∑

k

|γk|2 +1

2

∑

k1k2

|γk1k2 |2. (C1)

Since we solve for γ0, γk, and γk1k2, we relate the α-coefficients to these through Eqs. (A1-A4).

14

1. Effective mass

For a system with rotational symmetry, the inverse effective mass is defined as

1

m∗=∂2E(P )

∂P 2, (C2)

where E(P ) is the energy dispersion of the Bose polaron. The variational ansatz can be easily extended to a Bosepolaron with a finite momentum, but solving the resulting integral equations becomes numerically expensive becausethe equations do not have rotational symmetry. Here, following a similar line of argument to Ref. [88], we present amethod to calculate the effective mass without solving the integral equations for a finite-momentum Bose polaron.

In general, the coupled integral equations for γ’s for a finite momentum polaron have the following structure (similarto Eqs. (A1-A4)):

X[E(P ),P] |γ〉 = 0. (C3)

Here, |γ〉 is an ordered set of γ0, γk and γk1k2, and the center-of-mass momentum can be taken to be Pez without

loss of generality. While the matrix X is not the Hamiltonian, it is still Hermitian. The energy dispersion E(P )

is determined by the zero-crossing of the lowest (or highest, depending on the sign convention) eigenvalue of X.

Therefore, for a small P , we can expand X up to O(P 2) and evaluate the lowest eigenvalue in the same way as theRayleigh-Schrodinger perturbation theory. It is straightforward to show that

X[E(P ), P ] = X0 + PXP +P 2

2

(XPP + (1/m∗)XE

)+O(P 3), (C4)

where

X0 = X[E(0), 0], XP =∂X

∂P[E,P ]

∣∣∣∣∣P=0E=E(0)

, XPP =∂2X

∂P 2[E,P ]

∣∣∣∣∣P=0E=E(0)

, XE =∂X

∂E[E,P ]

∣∣∣∣∣P=0E=E(0)

. (C5)

Then the lowest eigenvalue λ(P ) of X[E(P ), P ] can be expressed as a series expansion in terms of P ,

λ(P ) = λ0 + λ1P + λ2P2 +O(P 3), (C6)

and its coefficients λi are determined perturbatively:

λ1 =⟨γ(0)

∣∣∣ XP

∣∣∣γ(0)⟩

= 0, (C7)

λ2 =1

2

⟨γ(0)

∣∣∣(XPP + (1/m∗)XE

) ∣∣∣γ(0)⟩

+∑

i>0

|⟨γ(i)∣∣ XP

∣∣γ(0)⟩|2

λ(0)0 − λ

(i)0

. (C8)

Here, λ(i)0 and

∣∣γ(i)⟩

are the i-th eigenvalue and its corresponding eigenvector of X0, and i = 0 corresponds to the

state that satisfies X0[E(0),0]∣∣γ(0)

⟩= λ

(0)0

∣∣γ(0)⟩

= 0. λ1 = 0 follows from the rotational invariance of∣∣γ(0)

⟩. Now,

recalling that the energy dispersion is found by zero-crossing of the lowest eigenvalue of X[E(P ), P ], we can set

λ(P ) = λ(0)0 = 0, (C9)

which in turn, implies that λ2 = 0. From this, the inverse effective mass is obtained as follows:

1/m∗ =⟨γ(0)

∣∣∣ XE

∣∣∣γ(0)⟩−1

[∑

i>0

2

λ(i)0

∣∣∣⟨γ(i)∣∣∣ XP

∣∣∣γ(0)⟩∣∣∣

2

−⟨γ(0)

∣∣∣ XPP

∣∣∣γ(0)⟩]

(C10)

=⟨γ(0)

∣∣∣ XE

∣∣∣γ(0)⟩−1 [

2⟨γ(0)

∣∣∣ XP QX−10 QXP

∣∣∣γ(0)⟩−⟨γ(0)

∣∣∣ XPP

∣∣∣γ(0)⟩], (C11)

where Q = 1−∑i>0

∣∣γ(i)⟩ ⟨γ(i)∣∣.

[1] M. Randeria, W. Zwerger, and M. Zwierlein, The BCS–BEC Crossover and the Unitary Fermi Gas, in The BCS-

BEC Crossover and the Unitary Fermi Gas, edited by

15

W. Zwerger (Springer Berlin Heidelberg, Berlin, Heidel-berg, 2012) pp. 1–32.

[2] M. J. Ku, A. T. Sommer, L. W. Cheuk, and M. W.Zwierlein, Revealing the superfluid lambda transition inthe universal thermodynamics of a unitary Fermi gas,Science 335, 563 (2012).

[3] S. Nascimbene, N. Navon, K. Jiang, F. Chevy, andC. Salomon, Exploring the thermodynamics of a universalFermi gas, Nature 463, 1057 (2010).

[4] M. Horikoshi, S. Nakajima, M. Ueda, andT. Mukaiyama, Measurement of Universal Thermo-dynamic Functions for a Unitary Fermi Gas, Science327, 442 (2010).

[5] G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W.Jack, and R. G. Hulet, Molecular Probe of Pairing inthe BEC-BCS Crossover, Phys. Rev. Lett. 95, 020404(2005).

[6] J. T. Stewart, J. P. Gaebler, and D. S. Jin, Using Pho-toemission Spectroscopy to Probe a Strongly InteractingFermi Gas, Nature 454, 744 (2008).

[7] A. Schirotzek, Y.-i. Shin, C. H. Schunck, and W. Ket-terle, Determination of the Superfluid Gap in AtomicFermi Gases by Quasiparticle Spectroscopy, Phys. Rev.Lett 101, 140403 (2008).

[8] Y. Inada, M. Horikoshi, S. Nakajima, M. Kuwata-Gonokami, M. Ueda, and T. Mukaiyama, Critical Tem-perature and Condensate Fraction of a Fermion PairCondensate, Phys. Rev. Lett. 101, 180406 (2008).

[9] G. Veeravalli, E. Kuhnle, P. Dyke, and C. J. Vale, BraggSpectroscopy of a Strongly Interacting Fermi Gas, Phys.Rev. Lett. 101, 250403 (2008).

[10] N. Navon, S. Nascimbene, F. Chevy, and C. Salomon,The Equation of State of a Low-Temperature Fermi Gaswith Tunable Interactions, Science 328, 729 (2010).

[11] J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin,Verification of Universal Relations in a Strongly Inter-acting Fermi Gas, Phys. Rev. Lett. 104, 235301 (2010).

[12] E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D.Drummond, P. Hannaford, and C. J. Vale, UniversalBehavior of Pair Correlations in a Strongly InteractingFermi Gas, Phys. Rev. Lett. 105, 070402 (2010).

[13] C. Cao, E. Elliott, J. Joseph, H. Wu, J. Petricka,T. Schafer, and J. E. Thomas, Universal Quantum Vis-cosity in a Unitary Fermi Gas, Science 331, 58 (2011).

[14] P. Makotyn, C. E. Klauss, D. L. Goldberger, E. A. Cor-nell, and D. S. Jin, Universal dynamics of a degenerateunitary Bose gas, Nature Physics 10, 116 (2014).

[15] M.-G. Hu, M. J. Van de Graaff, D. Kedar, J. P. Cor-son, E. A. Cornell, and D. S. Jin, Bose Polarons inthe Strongly 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, andJ. J. Arlt, Observation of Attractive and Repulsive Po-larons in a Bose-Einstein Condensate, Phys. Rev. Lett.117, 055302 (2016).

[17] R. J. Fletcher, R. Lopes, J. Man, N. Navon, R. P. Smith,M. W. Zwierlein, and Z. Hadzibabic, Two- and three-body 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 In-teracting Homogeneous Bose-Einstein Condensate, Phys.

Rev. Lett. 118, 210401 (2017).[19] C. E. Klauss, X. Xie, C. Lopez-Abadia, J. P. D’Incao,

Z. Hadzibabic, D. S. Jin, and E. A. Cornell, Observationof Efimov molecules created from a resonantly interactingBose gas, ArXiv e-prints (2017), arXiv:1704.01206 [cond-mat.quant-gas].

[20] C. Eigen, J. A. P. Glidden, R. Lopes, N. Navon, Z. Hadz-ibabic, and R. P. Smith, Universal Scaling Laws in theDynamics of a Homogeneous Unitary Bose Gas, ArXive-prints (2017), arXiv:1708.09844 [cond-mat.quant-gas].

[21] V. Efimov, Energy levels arising from resonant two-bodyforces in a three-body system, Phys. Lett. B 33, 563(1970).

[22] E. Braaten and H.-W. Hammer, Universality in few-body systems with large scattering length, Physics Reports428, 259 (2006).

[23] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl,C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola,H.-C. Nagerl, and R. Grimm, Evidence for Efimov quan-tum states in an ultracold gas of caesium atoms, Nature440, 315 (2006).

[24] P. Naidon and S. Endo, Efimov Physics: a review, Re-ports on Progress in Physics 80, 056001 (2017).

[25] C. H. Greene, P. Giannakeas, and J. Perez-Rıos, Uni-versal few-body physics and cluster formation, Rev. Mod.Phys. 89, 035006 (2017).

[26] J. P. D’Incao, Few-body physics in resonantly in-teracting ultracold quantum gases, arXiv preprintarXiv:1705.10860 (2017).

[27] J. Tempere, W. Casteels, M. K. Oberthaler, S. Knoop,E. Timmermans, and J. T. Devreese, Feynman path-integral treatment of the BEC-impurity polaron, Phys.Rev. B 80, 184504 (2009).

[28] S. P. Rath and R. Schmidt, Field-theoretical study of theBose polaron, Phys. Rev. A 88, 053632 (2013).

[29] W. Li and S. Das Sarma, Variational study of polaronsin Bose-Einstein condensates, Phys. Rev. A 90, 013618(2014).

[30] A. Shashi, F. Grusdt, D. A. Abanin, and E. Dem-ler, Radio-frequency spectroscopy of polarons in ultracoldBose gases, Phys. Rev. A 89, 053617 (2014).

[31] R. S. Christensen, J. Levinsen, and G. M. Bruun,Quasiparticle Properties of a Mobile Impurity in a Bose-Einstein Condensate, Phys. Rev. Lett. 115, 160401(2015).

[32] J. Levinsen, M. M. Parish, and G. M. Bruun, Impurityin a Bose-Einstein Condensate and the Efimov Effect,Phys. Rev. Lett. 115, 125302 (2015).

[33] L. A. Pena Ardila and S. Giorgini, Impurity in a Bose-Einstein condensate: Study of the attractive and repulsivebranch using quantum Monte Carlo methods, Phys. Rev.A 92, 033612 (2015).

[34] F. Grusdt, Y. E. Shchadilova, A. N. Rubtsov, andE. Demler, Renormalization group approach to theFrohlich polaron model: application to impurity-BECproblem, Scientific Reports 5, 12124 (2015).

[35] F. Grusdt and E. A. Demler, New theoretical ap-proaches to Bose polarons, ArXiv e-prints (2015),arXiv:1510.04934 [cond-mat.quant-gas].

[36] L. A. Pena Ardila and S. Giorgini, Bose polaron problem:Effect of mass imbalance on binding energy, Phys. Rev.A 94, 063640 (2016).

[37] Y. E. Shchadilova, R. Schmidt, F. Grusdt, and E. Dem-ler, Quantum Dynamics of Ultracold Bose Polarons,

16

Phys. Rev. Lett. 117, 113002 (2016).[38] N. J. S. Loft, Z. Wu, and G. M. Bruun, Mixed-

dimensional Bose polaron, Phys. Rev. A 96, 033625(2017).

[39] F. Grusdt, R. Schmidt, Y. E. Shchadilova, and E. Dem-ler, Strong-coupling Bose polarons in a Bose-Einsteincondensate, Phys. Rev. A 96, 013607 (2017).

[40] M. Sun, H. Zhai, and X. Cui, Visualizing the Efimov Cor-relation in Bose Polarons, Phys. Rev. Lett. 119, 013401(2017).

[41] A. Lampo, S. H. Lim, M. A. Garcıa-March, andM. Lewenstein, Bose polaron as an instance of quantumBrownian motion, Quantum 1, 30 (2017).

[42] R. Schmidt and M. Lemeshko, Rotation of Quantum Im-purities in the Presence of a Many-Body Environment,Phys. Rev. Lett. 114, 203001 (2015).

[43] R. Schmidt, H. R. Sadeghpour, and E. Demler, Meso-scopic Rydberg Impurity in an Atomic Quantum Gas,Phys. Rev. Lett. 116, 105302 (2016).

[44] F. Camargo, R. Schmidt, J. D. Whalen, R. Ding,G. Woehl, Jr., S. Yoshida, J. Burgdorfer, F. B. Dunning,H. R. Sadeghpour, E. Demler, and T. C. Killian, Cre-ation of Rydberg Polarons in a Bose Gas, ArXiv e-prints(2017), arXiv:1706.03717 [quant-ph].

[45] T. Rentrop, A. Trautmann, F. A. Olivares, F. Jendrze-jewski, A. Komnik, and M. K. Oberthaler, Observationof the Phononic Lamb Shift with a Synthetic Vacuum,Phys. Rev. X 6, 041041 (2016).

[46] Y. Wang, J. Wang, J. P. D’Incao, and C. H. Greene, Uni-versal Three-Body Parameter in Heteronuclear AtomicSystems, Phys. Rev. Lett. 109, 243201 (2012).

[47] D. Blume and Y. Yan, Generalized Efimov Scenariofor Heavy-Light Mixtures, Phys. Rev. Lett. 113, 213201(2014).

[48] F. Chevy, Universal phase diagram of a strongly inter-acting Fermi gas with unbalanced spin populations, Phys.Rev. A 74, 063628 (2006).

[49] R. Combescot and S. Giraud, Normal State of Highly Po-larized Fermi Gases: Full Many-Body Treatment, Phys.Rev. Lett. 101, 050404 (2008).

[50] N. Prokof’ev and B. Svistunov, Fermi-polaron problem:Diagrammatic Monte Carlo method for divergent sign-alternating series, Phys. Rev. B 77, 020408 (2008).

[51] J. Vlietinck, J. Ryckebusch, and K. Van Houcke, Quasi-particle properties of an impurity in a Fermi gas, Phys.Rev. B 87, 115133 (2013).

[52] P. Massignan, M. Zaccanti, and G. M. Bruun, Polarons,dressed molecules and itinerant ferromagnetism in ultra-cold Fermi gases, Reports on Progress in Physics 77,034401 (2014).

[53] P. W. Anderson, Infrared Catastrophe in Fermi Gaseswith Local Scattering Potentials, Phys. Rev. Lett. 18,1049 (1967).

[54] A. L. Fetter and J. D. Walecka, Quantum Theory ofMany-Particle Systems (McGraw-Hill, 1971).

[55] D. S. Petrov, Three-boson problem near a narrow Fesh-bach resonance, Phys. Rev. Lett. 93, 143201 (2004).

[56] P. Bedaque, H.-W. Hammer, and U. van Kolck, Thethree-boson system with short-range interactions, Nucl.Phys. A 646, 444 (1999).

[57] E. Timmermans, P. Tommasini, M. Hussein, and A. Ker-man, Feshbach resonances in atomic Bose-Einstein con-densates, Phys. Rep. 315, 199 (1999).

[58] G. M. Bruun and C. J. Pethick, Effective Theory of Fes-hbach Resonances and Many-Body Properties of FermiGases, Phys. Rev. Lett. 92, 140404 (2004).

[59] J. Levinsen and D. Petrov, Atom-dimer and dimer-dimerscattering in fermionic mixtures near a narrow Feshbachresonance, European Physical Journal D 65, 67 (2011).

[60] M. M. Parish and J. Levinsen, Quantum dynamics of im-purities coupled to a Fermi sea, Phys. Rev. B 94, 184303(2016).

[61] L. Pricoupenko, Isotropic contact forces in arbitrary rep-resentation: Heterogeneous few-body problems and low di-mensions, Phys. Rev. A 83, 062711 (2011).

[62] V. Efimov, Energy levels of three resonantly interactingparticles, Nucl. Phys. A 210, 157 (1973).

[63] H.-W. Hammer and L. Platter, Universal properties ofthe four-body system with large scattering length, Euro-pean Physical Journal A 32, 113 (2007).

[64] J. von Stecher, J. P. D’Incao, and C. H. Greene, Signa-tures of universal four-body phenomena and their relationto the Efimov effect, Nature Physics 5, 417 (2009).

[65] M. R. Hadizadeh, M. T. Yamashita, L. Tomio,A. Delfino, and T. Frederico, Scaling Properties of Uni-versal Tetramers, Phys. Rev. Lett. 107, 135304 (2011).

[66] A. Deltuva, Shallow Efimov tetramer as inelastic virtualstate and resonant enhancement of the atom-trimer re-laxation, EPL (Europhysics Letters) 95, 43002 (2011).

[67] F. Ferlaino, S. Knoop, M. Berninger, W. Harm, J. P.D’Incao, H.-C. Nagerl, and R. Grimm, Evidence for Uni-versal Four-Body States Tied to an Efimov Trimer, Phys.Rev. Lett. 102, 140401 (2009).

[68] Y. Wang, W. B. Laing, J. von Stecher, and B. D.Esry, Efimov Physics in Heteronuclear Four-Body Sys-tems, Phys. Rev. Lett. 108, 073201 (2012).

[69] C. H. Schmickler, H.-W. Hammer, and E. Hiyama,Tetramer bound states in heteronuclear systems, Phys.Rev. A 95, 052710 (2017).

[70] We have not investigated tetramer states tied to the ex-cited Efimov trimer, but we expect that they will exist asresonant states embedded in the atom-trimer continuum,as is the case for identical bosons [66, 98].

[71] This number had a typo in Ref. [32] which did not affectany of the other results of that paper.

[72] B. Huang, L. A. Sidorenkov, R. Grimm, and J. M. Hut-son, Observation of the Second Triatomic Resonance inEfimov’s Scenario, Phys. Rev. Lett. 112, 190401 (2014).

[73] The exception to this expected universality is the crossingof the ground-state trimer with the atom-dimer contin-uum (not shown in Fig. 1). This crossing is in generalknown to be the least universal aspect of Efimov trimers,as it corresponds to the largest energy scale. We indeedfind that the ground state trimer crossing is at a∗/|a−| '4.86×10−4 in the r0-model and at a∗/|a−| ' 5.58×10−4

in the Λ-model, neither of which quite fits the universalprediction: a∗/|a−| ' 0.72418e−π/s0 = 3.6 × 10−4 [99].On the other hand, we find that the first excited trimer,

with a(1)∗ /|a−| ' 0.71, matches reasonably well.

[74] Under the assumption of universality where the quantity

ma2−E(1)4B,− is the same in all models, the ratio is given by

a(QMC)

− /a(QMC)

B '√r20E

(1,2ch)4B,− /a2BE

(1,QMC)4B,− a

(2ch)− /r

(2ch)0 .

All parameters on the right-hand-side are known fromthe results of the QMC and the r0-model.

[75] M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C.Nagerl, F. Ferlaino, R. Grimm, P. S. Julienne, and J. M.

17

Hutson, Universality of the Three-Body Parameter forEfimov States in Ultracold Cesium, Phys. Rev. Lett. 107,120401 (2011).

[76] J. Wang, J. P. D’Incao, B. D. Esry, and C. H. Greene,Origin of the Three-Body Parameter Universality in Efi-mov Physics, Phys. Rev. Lett. 108, 263001 (2012).

[77] P. Naidon, S. Endo, and M. Ueda, Physical origin of theuniversal three-body parameter in atomic Efimov physics,Phys. Rev. A 90, 022106 (2014).

[78] P. Naidon, S. Endo, and M. Ueda, Microscopic Originand Universality Classes of the Efimov Three-Body Pa-rameter, Phys. Rev. Lett. 112, 105301 (2014).

[79] Y. Wang and P. S. Julienne, Universal van der Waalsphysics for three cold atoms near Feshbach resonances,Nature Physics 10, 768 (2014).

[80] J. Von Stecher, Weakly bound cluster states of Efimovcharacter, J. Phys. B 43, 101002 (2010).

[81] M. Gattobigio and A. Kievsky, Universality and scalingin the N-body sector of Efimov physics, Phys. Rev. A 90,012502 (2014).

[82] Y. Yan and D. Blume, Energy and structural proper-ties of N-boson clusters attached to three-body Efimovstates: Two-body zero-range interactions and the role ofthe three-body regulator, Phys. Rev. A 92, 033626 (2015).

[83] A. Zenesini, B. Huang, M. Berninger, S. Besler, H.-C.Nagerl, F. Ferlaino, R. Grimm, C. H. Greene, and J. vonStecher, Resonant five-body recombination in an ultracoldgas of bosonic atoms, New J. Phys. 15, 043040 (2013).

[84] B. Bazak and D. S. Petrov, Five-Body Efimov Effect andUniversal Pentamer in Fermionic Mixtures, Phys. Rev.Lett. 118, 083002 (2017).

[85] B. Bazak, Mass-imbalanced fermionic mixture in a har-monic trap, Phys. Rev. A 96, 022708 (2017).

[86] V. Gurarie and L. Radzihovsky, Resonantly pairedfermionic superfluids, Annals of Physics 322, 2 (2007).

[87] P. Massignan, Polarons and dressed molecules near nar-row Feshbach resonances, EPL (Europhysics Letters) 98,10012 (2012).

[88] C. Trefzger and Y. Castin, Impurity in a Fermi sea ona narrow Feshbach resonance: A variational study ofthe polaronic and dimeronic branches, Phys. Rev. A 85,053612 (2012).

[89] R. Qi and H. Zhai, Highly polarized Fermi gases acrossa narrow Feshbach resonance, Phys. Rev. A 85, 041603

(2012).[90] C. Kohstall, M. Zaccanti, M. Jag, A. Trenkwalder,

P. Massignan, G. M. Bruun, F. Schreck, and R. Grimm,Metastability and coherence of repulsive polarons in astrongly interacting Fermi mixture, Nature (London)485, 615 (2012).

[91] M. Cetina, M. Jag, R. S. Lous, I. Fritsche, J. T. M. Wal-raven, R. Grimm, J. Levinsen, M. M. Parish, R. Schmidt,M. Knap, and E. Demler, Ultrafast many-body interfer-ometry of impurities coupled to a Fermi sea, Science 354,96 (2016).

[92] A. Schirotzek, C.-H. Wu, A. Sommer, and M. W. Zwier-lein, Observation of Fermi Polarons in a Tunable FermiLiquid of Ultracold Atoms, Phys. Rev. Lett. 102, 230402(2009).

[93] The high-density expansion for Z at finite aB can, in prin-ciple, be calculated using an approach similar to that inRef. [31]. However, such a lengthy perturbative analysisgoes beyond the scope of this work.

[94] P. W. Anderson, Infrared Catastrophe: When Does ItTrash Fermi Liquid Theory?, in The Hubbard Model.NATO ASI Series (Series B: Physics), vol 343, editedby D. Baeriswyl, D. K. Campbell, J. M. P. Carmelo,F. Guinea, and E. Louis (Springer, Boston, MA, 1995)p. 217.

[95] S. Tan, Energetics of a strongly correlated Fermi gas, An-nals of Physics 323, 2952 (2008); Large momentum partof fermions with large scattering length, Annals of Physics323, 2971 (2008); Generalized Virial Theorem and Pres-sure Relation for a strongly correlated Fermi gas, Annalsof Physics 323, 2987 (2008).

[96] E. Braaten, D. Kang, and L. Platter, Universal relationsfor a strongly interacting Fermi gas near a Feshbach res-onance, Phys. Rev. A 78, 053606 (2008).

[97] Y. E. Shchadilova, F. Grusdt, A. N. Rubtsov, andE. Demler, Polaronic mass renormalization of impuri-ties in Bose-Einstein condensates: Correlated Gaussian-wave-function approach, Phys. Rev. A 93, 043606 (2016).

[98] J. von Stecher, Five- and Six-Body Resonances Tied toan Efimov Trimer, Phys. Rev. Lett. 107, 200402 (2011).

[99] K. Helfrich, H.-W. Hammer, and D. S. Petrov, Three-body problem in heteronuclear mixtures with resonant in-terspecies interaction, Phys. Rev. A 81, 042715 (2010).