Polaronic mass renormalization of impurities in BEC: correlated Gaussian wavefunction approach

Yulia E. Shchadilova,1, 2 Fabian Grusdt,3, 4, 2 Alexey N. Rubtsov,1, 5 and Eugene Demler2

1Russian Quantum Center, Skolkovo 143025, Russia2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

3Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, Germany4Graduate School Materials Science in Mainz, Gottlieb-Daimler-Strasse 47, 67663 Kaiserslautern, Germany

5Department of Physics, Moscow State University, 119991 Moscow, Russia(Dated: October 22, 2014)

We propose a class of variational Gaussian wavefunctions to describe Frohlich polarons at finite momenta.Our wavefunctions give polaron energies that are in excellent agreement with the existing Monte Carlo resultsfor a broad range of interactions. We calculate the effective mass of polarons and find smooth crossover betweenweak and intermediate impurity-bosons coupling. Effective masses that we obtain are considerably larger thanthose predicted by the mean-field method. A novel prediction based on our variational wavefunctions is a specialpattern of correlations between host atoms that can be measured in time-of-flight experiments. We discussatomic mixtures in systems of ultracold atoms in which our results can be tested with current experimentaltechnology.

PACS numbers: 71.38.Fp,67.85.Pq

Renormalization of particle masses due to their interac-tion with the environment is a ubiquitous phenomenon inphysics. In the standard model of high energy physics ele-mentary particles acquire a mass through interaction with theHiggs field [1]. In solid state systems heavy fermion mate-rials exhibit renormalization of electron masses of up to twoorders of magnitude due to interaction of electrons with local-ized spins [2]. Complete localization of quantum degrees offreedom caused by interaction with the environment has beendiscussed in spin-bath models [3, 4] and quantum Joseph-son junctions [5–7]. Surprisingly one of the first systems inwhich strong mass renormalization due to particle-bath inter-action has been predicted, the so-called polaron model intro-duced by Landau in 1933 [8, 9], remains a subject of con-siderable debate. This model describes interaction of a quan-tum particle with a bosonic bath, such as an electron inter-acting with phonons in a crystal (see Refs. [10–12] for re-views). While the limiting cases of weak and strong couplingcan be analyzed using controlled perturbtative expansions (seeRefs. [1, 13] for weak coupling analysis and Refs. [9, 15] forstrong coupling expansion), the intermediate coupling regimeremains poorly understood with the effective mass of the po-laron being the most contentious issue [16]. For example, con-siderable disagreement between different approximations forthe effective mass of polarons in BEC has been reported inthe literature (see Fig.1). Perturbative expansion for small in-teraction strength suggest a divergence of the effective massbeyond a certain interaction strength, indicating localizationof the impurity particle [17]. Variational method based on theFeynman path integral approach does not have a phase tran-sition but exhibits a sharp crossover in the effective mass [3].By contrast mean-field approach to the problem shows only agradual evolution of the effective mass [4]. Recent experimen-tal progress in the field of ultracold atoms brought new interestin the study of impurity problems. Feshbach resonances madeit possible to realize both Fermi [20–23] and Bose polarons

Ê Ê Ê Ê Ê Ê ÊÊ

Ê

Ê

Ê

Ê2nd order PTMean field

Ê Feynman's variationalCGWRG

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.51

2

3

4

5

a

MpêM

Figure 1: The polaron effective mass computed using different ap-proaches as a function of the dimensionless interaction parameter αdefined by Eq. (12). Our result (CGWs) is compared with Feynman’svariational method [3], mean-field [4], and RG [6]. We used param-eters corresponding to 6Li impurities in 23Na BEC for comparisonwith Refs. [3, 32].

[24–30] with tunable interactions between impurity and hostatoms and the rich toolbox of atomic physics has been used tostudy their properties including the effective mass [20, 23].

In this letter we propose an analytical class of wavefunc-tions based on the correlated Gaussian ansatz that can describeFrohlich polarons at finite momentum for a wide range of pa-rameters (see Refs. [33, 34] for earlier work). Frohlich typeHamiltonians can be used to describe several different fami-lies of physical systems including electrons interacting withlattice phonons in polar [35], organic [36], and piezoelec-tic [37–39] semiconductors, magnetic polarons in stronglycorrelated electron systems [40, 41], 3He atoms in super-fluid 4He [42], and impurity atoms in Bose-Einstein conden-sates [3–6, 32, 43–49, 51–53]. In this paper we focus on theBEC polarons realized with ultracold atoms since these sys-

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tems have highly tunable paramaters and should allow mostdetailed comparison with our theoretical analysis. Howeverour method can be easily adapted to other systems and gener-alized to dynamics.

The essence of our approach is an extension of the earliermean-field variational wavefunction (after performing Lee-Low-Pines transformation on the Hamiltonian, see [54] anddiscussion below) to Gaussian wavefunctions that include en-tanglement between different phonon modes. The explicitform of these wavefunctions is given in equation (5) and fromnow on we will refer to them as correlated Gaussian wave-functions (CGW) [55]. We demonstrate that CGWs show ex-cellent agreement with the Monte Carlo (MC) results for thepolaron binding energy at zero momentum for the range of in-teractions relevant to ultracold atoms: from weak to interme-diate coupling (see Fig.2). Compared to the mean-field solu-tion the number of varational parameters in CGWs increasesonly by a factor of three, which keeps the number of self-consistent equations reasonably small. In the context of BECpolarons MC method has only been used to calculate polaronbinding energies at zero momentum, which makes it impossi-ble to obtain the effective mass (for other types of polaronshowever effective mass analysis based on MC calculationshas been done, see e.g. [56, 57].) CGW analysis at finite po-laron momentum does not introduce additional complicationswhich allows us to calculate the effective mass of polarons.The key ingredient of our new class of wavefunctions is theappearance of additional correlations for the host atoms intro-duced through their interaction with mobile impurities. Thesecorrelations can be observed in the time of flight experimentsand should provide a quantitative test of our variational wave-function. Intermediate coupling regime of polarons should beaccessible, e.g., with 41K or 133Cs atoms in 87Rb BEC, bothof which have Feshbach resonances that can be used to tunethe impurity-boson interaction. In Fig.3 we present our pre-dictions for polaron masses for the two cases and in Fig.4 wepresent predictions for the BEC atoms correlations.

Frohlich model for the impurity atom interacting with BEC.We use Bogoliubov model to describe BEC of the host atomsand limit ourselves to small deviations of the BEC densityfrom the homogeneous case. In this case interaction of theimpurity with phonons of the BEC can be described using theFrohlich Hamiltonian [3, 43, 58]

H =~P2

2M+

∑k

Vk

(bk + b†

−k

)ei~k~R +

∑k

ωkb†k bk (1)

Here ~P and ~R are momentum and position operators of theimpurity atom with mass M, b†k is the annihilation opera-tor of the Bogoliubov phonon excitation with momentum ~k,ωk = ck

√1 + (ξk)2/2 is the Bogoliubov mode dispersion,

with c being the sound velocity and ξ is the coherence lengthof the condensate. The impurity-phonon interaction strengthis given by Vk = gIB

√n0V

−1ξk(2 + (ξk)2

)−1/4, where n0 is the

BEC density, gIB denotes the interaction strength between the

impurity atom and host atoms with mass m, andV is the vol-ume of the system. From now on we will setV = 1 in the restof the paper. In the first-order Born approximation this inter-action strength can be related to the imurity/BEC atom scat-tering length via gIB = 2πaIB

(m−1 + M−1

). Conditions on the

applicability of the Frohlich model for describing impurity-boson systems are discussed below when we consider experi-mental systems.

To utilize translational symmetry of the Frohlich Hamil-tonian (1) we apply the Lee-Low-Pines (LLP) transforma-tion [1] ULLP = eiS , S = ~R

∑k~kb†k bk. The transformed Hamil-

tonian HLLP = ULLPHU†LLP is

HLLP =1

2M

~P −∑k

~kb†k bk

2

+∑k

ωkb†k bk +∑

k

Vk

(bk + b†

−k

)(2)

Here ~P is a conserved total momentum of the system whichcan be treated as a c-number. Equation (2) no longer has de-grees of freedom corresponding to the impurity: they were in-tegrated out using conservation of the total momentum. Thisgenerated an interaction term between phonon modes whichis proportional to 1/M. The appearance of the phonon-phononinteraction can be understood as exchange of momentum be-tween phonons via the impurity.

Mean-field solution. To motivate the mean-field solutionwe first discuss the limit of infinitely heavy impurity, M → ∞.In this case interactions between phonon modes in eq. (2)vanish and the Hamiltonian can be transformed to the canon-ical form using the displacement transformation D(β0

k) =

exp(∑

k β0k b†k − H.c.

)with β0

k = −Vk/ωk. Then the groundstate is given by a coherent state D(β0

k)|0〉, where |0〉 is thephonon vacuum. Note the key feature of this solution: it fac-torizes into a product of wavefunctions for different k-modes.Now we can generalize this result to the interacting case atfinite M. The mean-field approach to polarons assumes a sim-ilar structure of the polaron wavefunction even in the interact-ing case of finite impurity mass [59]. In this method a productof coherent states for different phonon modes is taken as avariational ansatz

|MF〉 = D(βk)|0〉 (3)

and coefficients βk are determined from minimizing the energy〈MF|HLLP|MF〉. Straightforward calculation [4] gives βk =

−Vk/Ωk, where the renormalized dispersion Ωk is given by

Ωk = ωk +k2

2M−~kM

(~P − ~Pph

)(4)

The parameter ~Pph describes the part of the total polaron mo-mentum which is carried by the phonon cloud, and in themean-field approximation reads ~Pph =

∑q ~q

∣∣∣βq

∣∣∣2. A majorlimitation of the mean-field state is that it does not include

3

correlations between different phonon modes since the wave-function factorizes into a product of wavefunctions for indi-vidual ks. Different modes affect each other only through theself-consistency condition on βk.

Correlated Gaussian wavefuction. To account for correla-tions between different phonon modes in the polaron problemwe introduce a Gaussian wavefunction

|CGW〉 = D(β)S (Q) |0〉 (5)

where S (Q) = exp( 12∑

k,k′ Qkk′ b†

k b†k′ − H.c.). A convenientway of understanding this ansatz is to interpret it as a general-ized Bogoliubov transformation with

S †(Q)D†(β)bkD(β)S (Q) =

βk +∑

k′[cosh Q]kk′ bk′ +

∑k′

[sinh Q]kk′ b†k′ (6)

A new feature of wavefunction (5) is that expectation valuesof boson creation and annihilation operators no longer factor-ize. We have

⟨bk

⟩= βk,

⟨b · b

⟩= βk · βk′ + (cosh Q sinh Q)kk′ ,

and⟨b† · b

⟩= βk · βk′ + (sinh2 Q)kk′ . All higher order expec-

tation values of bk and b†k operators can be computed usingWick’s theorem. Variational parameters Qkk′ and βk should bedetermined by minimizing the energy.

Explicit expression for the expectation value of HLLP instate (5) is given in Supplementary Materials (SM). In theregime of interest for cold atoms systems it is sufficient toexpand the hyperbolic functions in (6) up to second order inmatrices Qkk′ [67]. We find

〈CGW |HLLP|CGW〉 =P2

2M−

P2ph

2M+

∑k

(2Vkβk + Ωkβ

2k

)+

∑kk′

Ωk(Qkk′ )2 +~k~k′

M

(βkβk′Qkk′ + βkβk′QkqQqk′ + (Qkk′ )2

)(7)

In this approximation the momentum of phonon cloud is de-fined as ~Pph =

∑k~kβ2

k +∑

kk′~k(Qkk′ )2, where (Qkk′ )2 is the

square of the matrix element. Minimizing expression with re-spect to Qkk′ we obtain equationsΩk +

~k~k′

M+ Ωk′

Qkk′ +~k~k′

Mβkβk′+

∑q

~k′~qMβqβk′Qkq +

~k~qMβqβkQqk′

= 0 (8)

where Ωk is still given by equation (4). At first sight thisintegral equation on the matrix Qkk′ appears quite challeng-ing. Fortunately, it can be reduced to a much simpler vectorequation by introducing ~Fk = −β−1

k∑

q βqQkq~q. Then equation(S.6) is equivalent to

~Fk =1M

∑k′

β2k′

Ωk +~k~k′2M + Ωk′

(~k~k′ − ~Fk~k′ − ~k ~Fk′

)~k′. (9)

Feynman's variational diagMC

Mean fieldCGWRG

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

a

Ep

Figure 2: Polaron binding energy Ep for static 6Li impurity (P =

0) in 23Na BEC predicted by different theoretical approaches asa function of the dimensionless coupling constant α. Our result(CGWs) is compared with MC calculations [32], Feynman’s varia-tional method [3], mean-field [4], and RG [6]. Note that polaronbinding energy depends on the UV cut-off. We choose the cut-off

2 · 103ξ−1 as in Refs. [3, 32].

Minimization of (S.5) with respect to βk gives

Vk + βkΩk −~kM

∑q

βq

MQqk

(~Fq − ~q

)= 0 (10)

Eqs. (9) and (S.17) can now be solved numerically. Details ofthe derivation of these equations can be found in SM.

To benchmark the approach we compare the polaron bind-ing energy Ep =

⟨HLLP

⟩for the system of Li impurity in Na

BEC with other known theoretical results in Fig.2. Our ap-proach shows excellent agreement with the MC approach anddrastically improves the mean-field solution.

Effective mass of the polaron can be obtained by taking thesecond derivative of the polaron energy with respect to the to-tal momentum. In practice this is not the most convenient wayof computing it because the polaron energy for the FrohlichHamiltonian (1) has UV divergencies. To regularize these di-vergencies we should not only use a full cut-off dependentrelation between the microscopic interaction gIB and the scat-tering length aIB [3–5], but also consider renormalization ofthe impurity mass that enters the mean-field part of the im-purity interaction with the condensate. Detailed discussion ofthese divergences is presented in Ref. [6]. One can howevercircumvent dealing with UV divergences if we use the follow-ing argument: When analyzing the variational wavefunction(5) we can calculate separately momentum of the polaron car-ried by the impurity ~Pimp = ~P− ~Pph. The velocity of the impu-rity, ~Pimp/M, should coincide with the velocity of the polaron,P/Mp. Thus we find

MMp

= 1 −Pph

P. (11)

For the comparison with other theoretical results we show thepolaron mass for the 6Li impurities in 23Na BEC in Fig.1 .

4

KêRb: MFKêRb: CGWKêRb: RGCsêRb: MFCsêRb: CGWCsêRb: RG

0 1 2 3 4 5 61

2

3

4

5

a

MpêM

Figure 3: Polaron mass for 41K and 133Cs impurities in BEC of 87Rbatoms (in units of bare impurity mass M). Increase of the interac-tion strength α between impurity atom and BEC enhances quantumfluctuations and results in stronger renormalization of Mp.

In contrast with Feynman’s variational approach the polaronmass calculated with CGWs shows smooth crossover from theregime of weak to intermediate coupling.

Relevant experimental systems and results. When selectingatomic mixtures for testing our theoretical analysis one needsto consider two factors. Firstly, a Feshbach resonance betweenthe impurity and BEC atoms is required, that can be used totune the interaction strength. It is common to describe thestrength of this interaction using a dimensionless parameter

α = 8πn0a2IBξ. (12)

Intermediate coupling regime requires α to be of the orderof one. Secondly, applicability of the Frohlich Hamiltonianrelies on the condition that the condensate density depletioncaused by the impurity is smaller than the density of the con-densate itself. This allows us to restrict ourselves to linearterms in Bogoliubov operators in the Frohlich Hamiltonian (1)and gives rise to the condition |gIB| 4cξ2 [60]. We find thatboth conditions can be satisfied for 41K impurities in 87RbBEC [24, 61] and 133Cs impurities in 87Rb BEC [29, 62].They correspond to the cases of moderately light impuritieswith M/mB = 0.46 for 41K/87Rb and M/mB = 1.53 for133Cs/87Rb. Fig.3 shows our predictions for the effective massof polarons in these two systems as a function of the impurity-boson interaction strength. We expect that mass enhance-ments up to a factor of three should be observable in 41K/87Rbsystems.

Experimental signatures of correlations. The main new fea-tures of the CGW (5) compared to the mean-field wavefuction(3) are correlations between different phonon modes. Suchcorrelations will also be present for atoms of the host BEC it-self and can be measured using noise correlation analysis inthe time-of-flight experiments (TOF) [63, 64]. The quantitythat can be extracted from TOF images is the second order

(a) (b)0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

kx

gH2L Hk,kL

CsêRbKêRb

0 2 4 6 8 100

2

4

6

8

kx

gH2L Hk,-

kL

Figure 4: Inset in the panel (a) shows the typical experimental setupfor measuring noise correlation functions g(2)(k, k′) (see Eq.(13)).TOF measurement should be performed with two detectors placedat relative angle θ between two directions of measurement. Panels(a) and (b) show noise correlations for (a) θ = 0 and (b) for θ = π for41K/87Rb and 133Cs/87Rb systems at α = 4. In the case θ = 0 BECatoms show antibunching. In the case of θ = π we find atom bunch-ing. For the 41K/87Rb mixture this bunching has a peak at k/ξ = 3.Large momentum asymptotics can be computed analytacally and areshown with dashed lines.

correlation function [65]

g(2)(k, k′) ≈

⟨b†k b†k′ bkbk′

⟩⟨b†k bk

⟩ ⟨b†k′ bk′

⟩ . (13)

Note that we will focus on the additional correlations amonghost atoms caused by the impurity and will not include corre-lations present in the BEC itself. The latter are expected onlyfor ±k atoms, as described by the Bogoliubov wavefunction[66]. We also point out that while our analysis consideredonly a single impurity, experiments are performed at finiteimpurity concentration. Assuming that impurities are suffi-ciently dilute and their polarization clouds do not overlap, wecan neglect interaction between polarons. Then changes in theoccupation number of host bosons at finite k due to severalimpurities will be proportional to the number of impurities.In the case of 41K/87Rb mixtures, interaction strength α = 4,and impurity concentration 5 per cent, we estimate the num-ber of atoms excited from the condensate to finite momentumstates by scattering on impurity atoms to be 3 per cent. Thissets the magnitude of the signal whose correlations we dis-cuss below. Fig.4 presents results of correlations describedby the equation (13) for the experimental systems 41K/87Rband 133Cs/87Rb. It is possible to obtain asymptotic values ofthese correlations in several regimes. In the long wavelengthlimit phonon modes decouple and g(2) approaches unity. Forhigh momenta occupation numbers of atoms nk decrease butg(2)(k, k′) saturates at values that depend on the angle θ be-tween ~k and ~k′: g(2)(k = ∞, k′ = ∞)

(1 +√

2mB/M)−1

and

g(2)(k = ∞, k′ = −∞) =(1 +√

2mB/M). This indicates anti-

bunching of bosons for small θ and bunching for θ = π. Cor-

5

relations induced between host atoms are stronger for lightimpurities. These results are consistent with our intuition thatan impurity colliding with one of the BEC atoms and giving itmomentum~k is more likely to scatter the next BEC atom in theopposite direction. One of the intriguing features in Fig.4(b)is a peak in the correlation function at stronger coupling α = 4and kξ ≈ 2.

Summary and outlook. We proposed a class of variationalGaussian wavefunctions for Frohlich polarons that gives ex-cellent agreement with Monte-Carlo results for the polaronenergy all the way from weak to strong coupling. In thisregime we find a smooth crossover of the effective polaronmass. Our wavefunction predicts a specific pattern of correla-tions between host atoms that can be measured in TOF experi-ments. We suggest that our predictions can be checked in suchsystems as 41K or 133Cs impurities in 87Rb BEC, in which thedimensionless coupling constant can reach the value as largeas 4, while the Frohlich polaron description remains appropri-ate. We point out that Gaussian wavefunctions can be used todescribe not only equilibrium states but also dynamics. Thusour formalism can be extended to compute spectral functionsof polarons and study response of polarons to external fields.

We acknowledge useful discussions with D. Abanin, I.Bloch, M. Fleischhauer, T. Giamarchi, W. Hofstetter, L.Keldysh, M. Oberthaler, D. Pekker, A. Polkovnikov, L. Pol-let, N. Prokof’ev, S. Das Sarma, R. Schmidt, A. Shashi, V.Stojanovic, L. Tarruell, N. Trivedi, A. Widera and M. Zwier-lein. The authors acknowledge support from the NSF grantDMR-1308435, Harvard-MIT CUA, AFOSR New QuantumPhases of Matter MURI, the ARO-MURI on Atomtronics,ARO MURI Quism program. Y.E.S. and A.N.R. thank theDynasty foundation and the RRBR grant 14-02-01219 for fi-nancial support. F.G. is a recipient of a fellowship throughthe Excellence Initiative (DFG/GSC 266) and is grateful forfinancial support from the ”Marion Koser Stiftung”.

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(World Scientific Singapore, 1995).[60] M. Bruderer, A. Klein, S. R. Clark, and D. Jaksch, Phys. Rev.

A 76, 11605 (2007).[61] J. Catani, L. De Sarlo, G. Barontini, F. Minardi, and M. Ingus-

cio, Phys. Rev. A 77, 11603 (2008).[62] D. J. McCarron, H. W. Cho, D. L. Jenkin, M. P. Koppinger, and

S. L. Cornish, Phys. Rev. A 84, 11603 (2011).[63] S. Folling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and

I. Bloch, Nature 434, 481 (2005).[64] T. Rom, T. Best, D. van Oosten, U. Schneider, S. Folling,

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13603 (2004).[66] J. Esteve, J.-B. Trebbia, T. Schumm, A. Aspect, C. I. West-

brook, and I. Bouchoule, Phys. Rev. Lett. 96, 130403 (2006).[67] Note that this expansion can not be justified on the ground

that matrix elements Qkk′ are of the order of inverse volume1/V. Summations implied in matrix multiplication

[Q2

]kk′

=∑p QkpQpk′ = V

∫p

QkpQpk′ show that higher order terms havethe same scaling in powers of 1/V. However analysis shows thateven for intermediate interaction strength elements of the matrix[Q2

]kk′

are numerically small justifying the expansion.

Supplementary Materials for ”Polaronic mass renormalization of impurities in BEC: correlated Gaussian wavefunctionapproach”

In this supplementary materials we present details of thecalculation of the polaron energy using CGW given in eq. (5)of the main text. We then show how one can minimize thisenergy to obtain self-consistency equations for the variationalparameters.

Variational Gaussian Approach. Our starting point is theFrohlich Hamiltonian after the Lee Low Pines [1] transforma-tion

⟨HLLP

⟩=

12M

~P −∑k

~kb†k bk

2

+∑

k

Vk

(b†k + b−k

)+

∑k

ωkb†k bk. (S.1)

Gaussian wavefunctions take into account entanglement be-tween different phonon modes, which are absent in mean fieldtheories. As a consequence pairwise averages, e.g., 〈bkbk′〉,have a nonzero irreducible part. Because of the Gaussianstatistics all higher-order correlators as

⟨b†kb†k′bkbk′

⟩can be re-

duced to simple two-point expressions using Wick’s theorem.In particular the average of Frohlich Hamiltonian over arbi-trary Gaussian trial state

⟨HLLP

⟩becomes

⟨HLLP

⟩=

P2

2M+

1√

V

∑k

Vk

(⟨b†k

⟩+

⟨bk

⟩)+

∑k

ωk +k2

2M−~P~kM

+~k

2M

∑k′

~k′⟨b†k′ bk′

⟩ ⟨b†k bk

⟩+

12M

∑kk′

~k~k′(⟨

b†k⟩ ⟨

b†k′⟩ ⟨

bkbk′⟩

c+

⟨b†k b†k′

⟩c

⟨bk

⟩ ⟨bk′

⟩+

⟨b†k

⟩ ⟨bk′

⟩ ⟨b†k′ bk

⟩c

+⟨b†k bk′

⟩c

⟨b†k′

⟩ ⟨bk

⟩)+

12M

∑kk′

~k~k′(⟨

b†k b†k′⟩

c

⟨bkbk′

⟩c

+⟨b†k bk′

⟩c

⟨b†k′ bk

⟩c

). (S.2)

where we defined the irreducible connected correlations as⟨AB

⟩c

=⟨AB

⟩−

⟨A⟩ ⟨

B⟩.

Our variational CGW given by eq. (5) of the main text givethe most general Gaussian wavefunctions. For the ground

7

state (equilibrium) problem under consideration it is sufficientto consider real vector β and real symmetric matrix Q, up toan overall phase, which provide minimum to the energy ineq. (S.2).

The unitary transformations D(β)S (Q) can be under-stood either as a transformation of the bosonic vacuum wave-function into a correlated Gaussian state |0〉 → |CGW〉, or as aBogoliubov rotations of the creation (annihilation) operators.To evaluate

⟨HLLP

⟩in Eq. (2) with the CGWs, we find it most

convenient to perform a Bogoliubov basis transformation,

Bk ≡ S †(Q)D†(β)bkD(β)S (Q) =

βk +∑

k′[cosh Q]kk′ bk′ +

∑k′

[sinh Q]kk′ b†

k′ (S.3)

and calculate the vacuum expectation value in the new ba-sis, e.g.

⟨HLLP

(b†k , bk

)⟩= 〈0| HLLP

(B†

k , Bk

)|0〉. Here, and in

what follows, functions of the matrix Q (e.g. cosh Q) shouldbe understood as being defined through their Taylor expan-

sion. Using the relation (S.3), we can now calculate the ir-reducible two-point functions required to evaluate the varia-tional energy,

⟨bk

⟩= βk, (S.4)⟨

bkb†k′⟩

c=

12

[cosh 2Q]kk′ ,⟨bkbk′

⟩c

=12

[sinh 2Q]kk′

In order to derive self-consistency equations for β and Q,we minimize the variational energy (S.2) with the expecta-tion values given by eq. (S.4). In addition, to obtain tractableequations, we consider only terms up to second order in Qin the energy

⟨HLLP

⟩. Physically, this corresponds to the as-

sumption that phonon-phonon correlations are small, albeitnon-vanishing. Thus we obtain the truncated variational en-ergy

〈H〉 =P2

2M+ 2

∑k

Vkβk +∑

k

ωk +k2

2M−~P~kM

β2k +

∑k′

Q2kk′

+∑kk′

~k~k′

2Mβ2

kβ2k′+

∑kk′q

~k~k′

Mβ2

k Q2k′q +

∑kk′

~k~k′

Mβkβk′Qkk′ +

∑kk′q

~k~k′

Mβkβk′QkqQqk′ +

∑kk′

~k~k′

2MQ2

kk′ (S.5)

To find the minimum of (S.5) we vary the last expression withrespect to β and Q, and derive the self-consistency equations.

Equations for Qkk′ . Minimization with respect to Q givesΩk +~k~k′

M+ Ωk′

Qkk′ +~k~k′

Mβkβk′+

∑q

~q~k′

Mβqβk′Qkq +

∑q

~k~qMβqβkQqk′ = 0 (S.6)

where the dispersion relation reads

Ωk = ωk +k2

2M−~P~kM

+~kM

∑k′

~k′β2k′ . (S.7)

It is similar to the mean field expression (see eq. (3) in themain text), except that the polarization βk is now determinedby a different self-consistent procedure.

To cast Eq. (S.6) into a more tractable form, we now definethe following auxiliary quantities, ηk,k′ and D(k, k′) [7] by thefollowing formulas

ηk,k′ = −MQkk′D(k, k′)βkβk′

, (S.8)

D(k, k′) = Ωk +~k~k′

M+ Ωk′ .

We express Qkk′ via ηkk′ and substitute it into the equa-tion (S.6):

ηk,k′ = ~k~k′ −∑

q

β2q

M

ηk,q~q~k′

D(k, q)+~k~qηq,k′

D(q, k′)

. (S.9)

Let us now introduce the vector

~F(k) =∑

q

β2q

MD(k, q)ηk,q~q, (S.10)

so that the equation (S.6) takes a particularly simple form

ηk,k′ = ~k~k′ − ~F(k)~k′ − ~k ~F(k′). (S.11)

We introduce the tensorial quantities A

A(0)k =

∑k′

β2k′

MD(k, k′)~k′ ⊗ ~k′, (S.12)

A(1)k =

∑k′

β2k′

MD(k, k′)~Fk′ ⊗ ~k′, (S.13)

A(2)k =

∑k′

β2k′

MD(k, k′)~Fk′ ⊗ ~Fk′ . (S.14)

where the outer product of two vectors is ~k ⊗ ~k′.

8

~P

~k~F (k)

~k0

~F (k0)

Figure S1: Schematic of the respect direction of vectors: total mo-mentum ~P, momentum of given mode ~k, and vector ~F(k). Vectors ~Pand ~k define a plane

(~P,~k

)and vector ~F(k) is in this plane.

Then a multiplication of the equation (S.11) byβ2

k′

MD(k,k′)~k′

and subsequent summation over k′ gives the self-consistencyequation for the vector ~Fk

~Fk =(~k − ~Fk

)A(0)

k −~k A(1)

k . (S.15)

This equation is solved numerically together with the equationfor β, which will be derived in the next subsection.

Let us first discuss the geometrical properties of the vector~Fk. In case of P = 0 vector ~Fk is collinear to ~k as there are noother vector quantities in the formalism. Formally this corre-sponds to ~Fk = Rk~k, with the proportionality coefficient Rk. Ingeneral case P , 0 ~Fk belongs to the plane of the vectors ~Pand ~k, and Rk is a tensor that describes a combination of rota-tion in this plane and rescaling ( see Fig.S1 for illustration).

Equations for βk. Variation of the expression (S.5) withrespect to β gives

Vk + βk

ωk +k2

2M−~kM

(~P − ~Pph

) +

βk~kM

∑q

β2q

MD(k, q)η(q, k)

(~F(q) − ~q

)= 0 (S.16)

where we substituted Q with the corresponding expres-sions (S.8) in terms of ηk,k′ after the variation. The total mo-mentum carried by the phonons is ~Pph =

∑kk′~k(β2

kδkk′ + Qkk′).

The last term on the left hand side of Eq. (S.16) can be inter-preted as a renormalizition of the phonon dispersion relationΩk. Let us rewrite the expressions so that this statement ismore clear. We use equations (S.11) and (S.15), and also re-call the geometrical properties of vector ~Fk discussed above:~Fk = Rk~k. We rewrite the expression (S.16) as follows

Vk + βk

ωk +k2

2M−~kM

(~P − ~Pph

)+

~k1M

((A(1)

k − A(0)k

)(I − Rk) − A(2)

k

)~k)

= 0. (S.17)

Thus equation for βk can be written in a compact form

βk = −Vk

ω(k) +~kM−1

k~k

2 +~kM

(~P − ~PB

) . (S.18)

Here the effective impurity mass

MM−1k = I − 2

((A(1)

k − A(0)k

)(I − Rk) − A(2)

k

)(S.19)

is a tensor quantity which is non-diagonal for P , 0.

Observables. Equations (S.15) and (S.18) for βk and ~Fk

form a self-consistent set for βk and ~Fk which we solve it-eratively. After obtaining βk and ~Fk all observables can becalculated using Wick’s theorem.

In particular the polaron binding energy

Ep =⟨HLLP

⟩=

P2

2M−~P2

B

2M+

∑k

Vkβk −∑

k

β2k

F2(k)2M

+∑

k

β2k

kµ(M−1δµν −M

−1µν (k)

)kν

2(S.20)

It is important to point out that only the total impurity energyE = gIBn0 + Ep is physically meaningful, where the first termis the bare impurity-condensate interaction energy. The bind-ing energy Ep by itself is UV-divergent. This divergence ap-pears at the mean field level and comes from the term

∑k Vkβk.

Indeed one can check that in UV-limit βk ∝ k−2 and Vk tendsto a constant value. Therefore in d > 2 this gives rise to apower-law divergency of the binding energy

∑k Vkβk ∝ Λd−2,

where Λ is a sharp UV momentum cut-off. This divergenceis resolved by the standard regularization procedure [3–5],expressing gIB in terms of the scattering length aIB and thecut-off Λ. When quantum fluctuations are taken into accountand an additional logarithmic divergence with Λ appears as wediscuss in detail in [6]. The presence of this logarithmic be-havior makes a direct comparison with the experimental datainvolved. Thus our results for polaron energy are only used tobenchmark the approach by comparing to other known theo-retical results.

[1] T. Lee, F. Low, and D. Pines, Phys. Rev. 341, 297 (1953).[2] The quantity D(k, k′) is in fact the two particle excitation energy

over the vacuum state |0 >. One can check this using the standard4-th order perturbation theory with respect to interaction Vk.

[3] J. Tempere, W. Casteels, M. Oberthaler, S. Knoop, E. Timmer-mans, and J. Devreese, Phys. Rev. B 80, 184504 (2009).

[4] A. Shashi, F. Grusdt, D. A. Abanin, and E. Demler, Phys. Rev.A 89, 053617 (2014).

[5] S. P. Rath and R. Schmidt, Phys. Rev. A 88, 053632 (2013).[6] F. Grusdt, Y. E. Shchadilova, A. N. Rubtsov, and E. Demler,

Arxiv Prepr. (2014), arXiv:1410.2203 .[7] The quantity D(k, k′) is in fact the two particle excitation energy

over the vacuum state |0 >. One can check this using the standard4-th order perturbation theory with respect to interaction Vk.