bose polarons in ultracold atoms in one dimension: beyond ...€¦ · bose polarons in ultracold...
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Bose polarons in ultracold atoms in one dimension: beyond the Frohlich paradigm
Fabian Grusdt,1 Gregory E. Astrakharchik,2 and Eugene Demler1
1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA2Departament de Fisica, Campus Nord B4-B5, Universitat Politecnica de Catalunya, E-08034 Barcelona, Spain
(Dated: November 13, 2017)
Mobile impurity atoms immersed in Bose-Einstein condensates provide a new platform for explor-ing Bose polarons. Recent experimental advances in the field of ultracold atoms make it possibleto realize such systems with highly tunable microscopic parameters and to explore equilibrium anddynamical properties of polarons using a rich toolbox of atomic physics. In this paper we presenta detailed theoretical analysis of Bose polarons in one dimensional systems of ultracold atoms.By combining a non-perturbative renormalization group approach with numerically exact diffusionMonte Carlo calculations we obtain not only detailed numerical results over a broad range of pa-rameters but also qualitative understanding of different regimes of the system. We find that anaccurate description of Bose polarons requires the inclusion of two-phonon scattering terms whichgo beyond the commonly used Frohlich model. Furthermore we show that when the Bose gas isin the strongly interacting regime, one needs to include interactions between the phonon modes.We use several theoretical approaches to calculate the polaron energy and its effective mass. Theformer can be measured using radio-frequency spectroscopy and the latter can be studied experi-mentally using impurity oscillations in a harmonic trapping potential. We compare our theoreticalresults for the effective mass to the experiments by Catani et al. [PRA 85, 023623 (2012)]. In theweak-to-intermediate coupling regimes we obtain excellent quantitative agreement between theoryand experiment, without any free fitting parameter. We supplement our analysis by full dynamicalsimulations of polaron oscillations in a shallow trapping potential. We also use our renormalizationgroup approach to analyze the full phase diagram and identify regions that support repulsive andattractive polarons, as well as multi-particle bound states.
I. INTRODUCTION
When a mobile particle interacts with a surroundingbath of bosons, it becomes dressed by a cloud of exci-tations and forms a polaron [1, 2]. As a result many ofits properties, like the effective mass, are strongly mod-ified compared to those of the bare particle. Impurityatoms immersed in a Bose gas provide a promising newplatform for studying the long standing polaron prob-lem. Advantages of such systems include the tunabilityof both interactions [3–5] and the single particle disper-sion [6]. For example, both impurity and host atomscan be realized in a quasi one-dimensional (1D) geom-etry. This situation was realized experimentally by theFlorence group [7] and will be considered throughout thispaper. Recent experiments also demonstrated the exis-tence of strongly coupled Bose polarons in one [7] andthree dimensional systems [8, 9].
Numerous theoretical works have addressed the prob-lem of a mobile impurity in an ultracold quantum gas,see Refs. [10, 11] for reviews. However, they were eitherbased on an effective Frohlich Hamiltonian to describethe polaron [5, 12–19] or used truncated wave functionswith only a few excitations [20, 21]. Notable exceptionsinclude a third-order perturbative treatment of the prob-lem [22], a self-consistent T-matrix calculation [23], amean-field (MF) analysis beyond the Frohlich Hamilto-nian [24], diffusion Monte Carlo calculations based onthe full microscopic Hamiltonian [25, 26] and approxi-mate analytical descriptions [27, 28]. Recently Virial ex-pansion techniques have also been used to study spectra
of Bose polarons [29] and a flow-equation approach hasbeen applied to the problem [28].
A number of important questions remain open and acomplete theoretical understanding of Bose polarons atarbitrary couplings is lacking. Most strikingly, the phasediagram in the strongly interacting regime is still a sub-ject of debate. In this paper we focus on a system whereboth the impurity and the Bose gas are constrained to onedimension. When the impurity is interacting with onlya single boson, a two-particle bound state exists alreadyfor infinitesimal attractive interactions. If the mass of theimpurity is infinite and multiple bosons without mutualinteractions are considered, this gives rise to an infiniteseries of multi-particle bound states. The fate of thesemany-body eigenstates in a regime where the impurityis mobile and the Bose gas is interacting, is unclear. Arelated question concerns the regimes of validity of dif-ferent effective polaron models, including the celebratedFrohlich Hamiltonian.
In two and three dimensional systems, the MF ap-proach [24] is a convenient theoretical tool that canbe used to study models beyond the simplified FrohlichHamiltonian. It is a non-perturbative method which in-cludes strong correlations between the phonons and theimpurity, whereas phonon-phonon correlations are ne-glected. For example, one can include two phonon scat-tering terms that are crucial for the accurate descrip-tion of few-body aspects of the system including the ex-istence of bound states between the impurity and hostbosons [24, 25]. The spectral function of the impurityin three dimensional systems obtained using the MF ap-proach [24] was in good agreement with experimental re-
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Diffusion Monte Carlo (DMC)Renormalization group (RG)Mean-field theory (MF)
Diffusion Monte Carlo (DMC)Renormalization group (RG)
Experiment Catani et al.
FIG. 1. Strongly coupled polarons in one dimension: (a) System sketch: we consider the problem of a mobile impurityinteracting with a quasi-one dimensional Bose gas. We perform calculations based on various theoretical methods, includinga RG approach and diffusion Monte Carlo (DMC) calculations. (b) Deep in the Bogoliubov regime the RG method yieldsaccurate results for the density profile of bosons around the impurity (i.e. the impurity-boson correlation function) even forstrong impurity-boson interactions. Importantly the RG improves predictions by simpler mean-field (MF) calculations basedon an ansatz of uncorrelated phonons. Here we used a ratio of impurity-boson to boson-boson interaction strengths of η = 1000(very strong repulsion) and considered the case of high density with n0|aBB| = 144, where n0 is the density of the Bose gas andaBB the one-dimensional boson-boson scattering length. (c) We also calculate the effective polaron mass Mp and compare ourresults to experimental measurements by Catani et al. [7]. In the weak-to-intermediate coupling regimes we obtain excellentquantitative agreement between theory and experiment, without any free fitting parameters.
sults [8, 9]. The applicability of the MF approach to 1Dsystems has not been clarified yet. One of the indicationsthat 1D systems are special is the unphysical logarithmicinfrared divergence of the polaron energy, which is ab-sent in higher dimensions. The physical origin of thisdivergence is the enhanced role of quantum fluctuationsin 1D systems. These are essentially the same fluctua-tions that are responsible for the absence of true Bose-Einstein condensation in homogeneous 1D systems evenat zero temperature [30, 31].
Theoretical issues raised above provide a considerablechallenge for quantitative analysis of the experiment [7],where 1D Bose polarons have been realized at strong cou-plings for the first time. In fact, even in the weak couplingregime, the agreement of the measured effective polaronmass with earlier theoretical calculations based on theeffective Frohlich Hamiltonian has not been satisfactory.In order to obtain quantitative agreement, the impurity-boson coupling had to be multiplied by a factor of 3.15in Ref. [7]. Moreover, at stronger couplings a saturationof the effective mass has been observed [7], which lackedtheoretical explanation so far.
Most of the earlier theoretical work focused on equi-librium properties of polarons. The experiments withpolarons that have been carried out in 1D quantumgases so far, including measurements by Catani et al. [7],all probed non-equilibrium impurity dynamics [32, 33].Thus theoretical analysis has to study not only stronglyinteracting systems, but also understand its dynamicalproperties and their connection to equilibrium quanti-ties. This provides an additional challenge, since mostof the standard tools, such as Monte-Carlo methods, are
not applicable. First steps in this direction have beentaken in Refs. [34–37].
In this paper we address the questions raised aboveand provide a detailed theoretical analysis of the Bosepolaron problem in one dimension. We consider a mo-bile impurity of mass M interacting with a 1D Bosegas, see Fig. 1(a). We then compare our theoreticalmethods, which leads us to an understanding of whichterms in the microscopic Hamiltonian contribute most tothe polaron properties. As an example, in Fig. 1(b) weshow that the depletion of the Bose gas around the im-purity can be described accurately by a semi-analyticalrenormalization-group (RG) approach [16, 38, 39] whenthe Bose gas is deep in the Bogoliubov regime. Moreover,we use our theoretical methods to analyze the experimentby Catani et al. [7] in detail. In particular we calculatethe effective polaron mass. In the weak-to-intermediatecoupling regimes we obtain excellent agreement with theexperimental data, see Fig. 1 (c). Our results moreoverprovide an important test case for theories of Bose po-larons at strong couplings, applicable also in higher di-mensions [39].
A special feature of our work is the comparison of ana-lytical analysis with numerical calculations based on thediffusion Monte Carlo (DMC) method [40–42], supple-mented by variational Monte Carlo (VMC) calculations.In addition we present results from time-dependent MFsimulations, following Refs. [24, 43], to study impuritydynamics.
Our paper is organized as follows. After briefly sum-marizing our main results in the following section, weintroduce the model in Sec. III. In Sec. IV we consider
3
Bogoliubov-Fröhlich
expe
rim
ent
Cat
ani e
t al
.
beyond Fröhlich
beyondBogoliubov:
phononinteractions
FIG. 2. A diagram with different physical regimes of Bose po-larons: For weak impurity-boson interactions η an impurity ina Bose gas can be described by the Frohlich Hamiltonian. Forstrong interactions, two-phonon terms beyond the Frohlichmodel have to be included. For strongly interacting bosons,γ & 1 where γ = 2/n0|aBB| is the dimensionless interactionstrength of the Bose gas [6], the Bogoliubov approximationbreaks down and phonon-phonon interactions also have to beincluded. For weakly interacting bosons, γ 1, we identify aregime where the Bogoliubov approximation for the Bose gasis justified (beyond Frohlich regime).
the weak coupling limit and compare our calculations tothe experimental data from Ref. [7]. We discuss the effectof two-phonon terms in Sec. V. A detailed RG analysisis presented in Appendix B, from which we derive thepolaron phase diagram. In Sec. VI a time-dependent MFtheory is applied to analyze polaron dynamics as in theexperiment by Catani et al. Section VII is devoted to adiscussion of phonon-phonon interactions in the polaroncloud. The variational and diffusion Monte Carlo meth-ods are presented in Sec. VII A. We close with a summaryand an outlook in Sec. VIII.
II. SUMMARY OF RESULTS
Bose polarons have been commonly investigated the-oretically using an effective Frohlich Hamiltonian HF
[5, 7, 14]. This is justified for weak interactions betweenthe impurity and bath particles, see Fig. 2. In the presentwork we show that in this regime, the effective Frohlichmodel describes accurately the experimental results forthe effective mass [7], without any free fitting parameter.We point out the importance of high-energy phonons (atmomenta of the order of the inverse healing length ξ),which have not been treated accurately in the previousanalysis of the experimental data [7, 35, 36]. When theBose gas is weakly interacting, we find a regime whereperturbative treatments of impurity-phonon interactions[25, 26] fail. The Frohlich model is still valid for theseparameters however.
For stronger impurity-boson interactions, the FrohlichHamiltonian is no longer sufficient and two-phonon pro-cesses H2ph (see Eq. (9) for details) have to be includedto describe the depletion of the condensate correctly [23],allowing also for molecular states [23, 24, 39]. To solve
the extended polaron Hamiltonian, we first use MF the-ory in the spirit of Ref. [24] and show that it predictsa logarithmic divergence of the polaron energy with theinfrared momentum cut-off. The divergence can be reg-ularized by a more accurate RG calculation, which webenchmark in 1D by comparing to our DMC results. Forlarge boson densities, corresponding to a regime wherethe Bogoliubov approximation can be used to describethe Bose gas, we find good agreement of the RG withexact DMC predictions, see Fig. 1 (b).
Due to the presence of the two-phonon terms, we findfrom all our theoretical approaches that the effective po-laron mass saturates to a large but constant value forstrong interactions with the impurity. Qualitatively, thiseffect has been observed experimentally [7] using breath-ing oscillations of the impurity in a harmonic trap. Toanalyze these measurements further, we perform full dy-namical simulations of the polaron trajectory using time-dependent MF theory [24, 43]. Our calculations showthat while the impurity oscillations provide a powerfultool to determine the effective polaron mass, the accu-racy can be limited by the inhomogeneity of the Bosegas.
When the boson density n0 becomes small, the Bo-goliubov theory of the interacting Bose gas breaks down.In this regime phonon-phonon interactions described byHph−ph (see Eq. (12) for details) have to be included fora valid description of the polaron, see Fig. 2. From thecomparison of our DMC calculations (including Hph−ph)
with RG predictions (without Hph−ph) we find thatphonon-phonon interactions always need to be includedto obtain quantitative agreement for the polaron energyand mass.
From the comparison of our DMC and RG calcula-tions, we also conclude that the experiment by Catani etal. [7] has been performed in a regime where all terms in
the Hamiltonian (HF , H2ph and Hph−ph) are relevant. Inparticular the interactions between Bogoliubov phononsin the bath already play a role. Moreover our simula-tions of polaron dynamics suggest that the inhomogene-ity of the Bose gas should be included when analyzingresults of the experiment. We point out how additionalexperiments can shed new light on the physics of stronglycoupled Bose polarons in one dimension.
III. MODEL
Our starting point is a single impurity interactingwith a Bose gas in one dimension, see Fig. 1 (a). Thebosons also have mutual interactions. This situation canbe described by the following microscopic Hamiltonian(~ = 1),
H =
∫dx φ†(x)
[− ∂2
x
2mB+gBB
2φ†(x)φ(x)
]φ(x)+ (1)
4
+
∫dx ψ†(x)
[− ∂2
x
2M+ gIBφ
†(x)φ(x)
]ψ(x), (2)
where φ(x) stands for the Bose field operator and ψ(x)is the impurity field. The boson (impurity) mass is mB
(M) and gBB (gIB) denote the boson-boson and impurity-boson coupling constants respectively. We assume thatonly a single impurity is present in the homogeneous Bosegas with density n0. Experimentally this corresponds toa situation with sufficiently low impurity concentration,ideally with less than one impurity per healing length ξ.
The ground state of this Hamiltonian can be calculatedefficiently for up to N & 200 bosons by means of theDMC method. We provide the technical details of thisapproach in Sec. VII A, although we compare to DMCresults in earlier sections.
A. Polaron description
To arrive at a polaron description of the impurity prob-lem described above, we express the boson field operator
φ(x) in terms of Bogoliubov phonons ak [5, 10, 11, 44].
To this end we write the Fourier components φk =
(2π)−1/2∫dx eikxφ(x) as
φk = cosh θkak − sinh θka†−k, (3)
where θk is chosen as in the usual Bogoliubov theory fora weakly interacting Bose gas [45],
cosh θksinh θk
=1√2
(k2/2mB + gBBn0
ωk± 1
)1/2
. (4)
Here the Bogoliubov dispersion is given by
ωk = ck
(1 +
1
2k2ξ2
)−1/2
, (5)
where c =√gBBn0/mB and ξ = 1/
√2mBgBBn0 are the
speed of sound and the healing length in the limit of weakinteractions.
So far we have only applied a basis transformation,
allowing us to express the Bose field φ(x) in terms of Bo-goliubov phonons ak. Thereby the Hamiltonian in Eq. (2)can be written as
H = HF + H2ph + Hph−ph (6)
without any approximation. The first term correspondsto the effective Frohlich Hamiltonian,
HF =
∫dk ωka
†kak +
p2
2M+
+ gIBn0 +
∫dk Vke
ikx(a†k + a−k
). (7)
Here p and x denote the momentum and position opera-tors of the impurity in first quantization. The scatteringamplitude is given by [5, 11]
Vk =√n0(2π)−1/2gIBWk, Wk =
((ξk)2
2 + (ξk)2
)1/4
.
(8)
The second term, H2ph, describes two-phonon scatteringprocesses [23, 24] and reads
H2ph =gIB
2π
∫dkdk′
(cosh θka
†k − sinh θka−k
)×(
cosh θk′ ak′ − sinh θk′ a†−k′
)ei(k−k
′)x. (9)
The two terms above, HF and H2ph, provide an accu-rate model for the polaron problem when the Bose gascan be treated within Bogoliubov theory. This mean-fielddescription of the interacting bosons assumes a macro-scopic occupation of the condensate which is absent inone dimension [30, 31]. As pointed out by Lieb and Lin-iger [46, 47], some quantities including the total energycan nevertheless be calculated accurately using Bogoli-ubov theory in the regime of weak interactions. This isthe case when the dimensionless coupling strength γ . 2is sufficiently small [46], where
γ =2
n0|aBB|=mBgBB
n0(10)
and the 1D boson-boson s-wave scattering length is givenby the relation
aBB = −2/(mBgBB). (11)
In contrast, for strong interactions gBB or small densi-ties n0, when γ & 2 is large, the Bogoliubov descriptionof the Bose gas breaks down. In this case additional in-teractions between the Bogoliubov phonons ak have to beincluded, which we summarize in Hph−ph. These termsare of order
Hph−ph = O(a3k) +O(a4
k), (12)
but do not involve impurity operators. Since we do notneed the expression for Hph−ph in terms of Bogoliubovphonons in the following, we will not write them out ex-plicitly.
The terms in Hamiltonian (6) give rise to differentphysical regimes of the Bose polaron. For small gIB, theground state corresponds to a free impurity and phononcontributions are negligible. In this regime the polaronenergy is determined by gIBn0, sometimes referred to asthe mean-field shift. For stronger couplings states withone phonon need to be included, which has been done sys-tematically by using perturbation theory in Refs. [25, 26].This approach is valid when γ 1 and
η 2γ3/4 ≈ 3(n0|aBB|)−3/4. (13)
5
Beyond this coupling strength multi-phonon states needto be added in the ground state wavefunction to describecorrectly the polaron energy its effective mass [16, 43]. Togo beyond the Frohlich model, two-phonon terms need tobe included in the Hamiltonian [23, 24], see Fig. 2. Whenthe polaron cloud contains many phonons and boson-boson interactions become large, their interactions needto be taken into account as well. On a mean-field levelthis can be done by using the Gross-Pitaevskii equation,see Sec. VII B for a discussion.
B. Experimental considerations
Quasi-1D Bose gases have been realized in several ex-periments (see Ref. [48] for a review). Mobile impuritiescan be realized by using a second atomic level, as in therecent experimental observation of Bose polarons in a3D Bose-Einstein condensate [9]. Alternatively a secondatomic species can be added, as has been demonstratedin Refs. [7, 8, 49, 50].
In this paper we analyze the experimental results fromRef. [7]. We use the dimensionless parameter
η = gIB/gBB (14)
introduced therein to quantify the interaction strengthbetween impurity and boson with respect to the fixedstrength of interactions in the bath. The mass ratio inthe experiment was M/mB = 41/87 = 0.47 and for n0 weuse the peak density of the Bose gas, which was estimatedto be n0 = 7/µm in Ref. [7]. The coupling constant inthe bath is taken equal to gBB = 2.36 × 10−37Jm as inRef. [7].
Using Bogoliubov theory this yields the following esti-mates for the healing length, ξ = 0.15µm and the speedof sound, c = 3.38mm/s. Using ξ as a unit of lengthyields n0ξ = 1.05. A characteristic energy can be definedby c/ξ = 3.6 × 2πkHz. Another important length scaleis the 1D boson-boson scattering length aBB = −3.6µm,see Eq. (11). The speed of sound calculated from theBethe ansatz [46, 47] at γ = 2/(n0aBB) = 0.438, seeEq. (10), equals to c = 3.20mm/s so that the contribu-tion of quantum fluctuations is about 5%. In order toavoid a possible issue which value (Bogoliubov theory orBethe ansatz) is used, we take the healing length ξ as aunit of length for comparison with the MF theory andaBB for the comparison with Monte Carlo results.
The temperature of the Bose gas T = 350(50)nK mea-sured in the experiment corresponds to kBT ≈ 2c/ξ.The transverse confinement frequency of the bosons(the impurity) stated in Ref. [7] corresponds to ω⊥ ≈9.4 (12.5)c/ξ and justifies a description as a 1D system.
IV. WEAK COUPLING LIMIT: THEFROHLICH MODEL
We start by discussing the weak coupling limit whereexact analytical results can be obtained within theFrohlich model. This enables a direct comparison be-tween theory and experiment and moreover providesan important benchmark for subsequent analysis of thestrong coupling regime.
Experimental studies by Catani et al. [7] included anindirect measurement of the effective polaron mass Mp
for various values of the interaction strength η. The lat-ter was tuned using a Feshbach resonance. The value ofMp was extracted from observations of breathing oscilla-tions of the impurity interacting with the trapped Bosegas. The amplitude σ of such oscillations is renormalizedby a factor of
√M/Mp. This can be understood by con-
sidering an initially localized cloud of impurity atoms asin the experiment of Ref. [7], with average kinetic energy√〈p2〉/2M . When the impurities are released their mo-
mentum distribution is adiabatically mapped to an iden-tical distribution of polaron momenta. The resulting ki-netic energy
√〈p2〉/2Mp of polarons is subsequently con-
verted into potential energy MΩ2Iσ
2/2 of the expandedatoms in a harmonic potential with trapping frequencyΩI. The amplitude σ thus provides a measure of thepolaron mass, σ = σ0
√M/Mp where σ0 corresponds to
non-interacting impurities. For a more detailed discus-sion see Ref. [7] and Sec. VI below.
In Fig. 3 we compare experimental results for the ef-fective polaron mass Mp with predictions of our own nu-merical calculations. Before presenting a detailed tech-nical analysis, we note that our data from four differenttheoretical methods show good agreement with the ex-perimental results in the weak-to-intermediate couplingregimes. This is true for both repulsive and attractiveinteractions, without any free fitting parameter.
In their original analysis of the experiment, Catani andco-workers [7] performed numerical calculations starting
from an effective Frohlich Hamiltonian H′F (defined inSec. IV A) which they derived using bosonization tech-niques [51]. Then they applied Feynman’s variationalpath integral method [52] to obtain the effective polaronmass within this model. Surprisingly, their prediction(thick purple line in Fig. 3) showed appreciable disagree-ment with experimental results. Thus, either the evalu-ation of the polaron mass within the effective model wasinaccurate, or the effective model itself is insufficient. Toclarify, we provide the answers to the following questionsbefore explaining them in detail below.
(i) It has been suggested earlier that Feynman’s varia-tional path integral method becomes inadequate forintermediate coupling strengths η if the impurity islight [16–18]. Is this method accurate for solving
the effective Frohlich model H′F for the experimen-tal parameters? – We will show that Feynman’sapproach is accurate in the regime considered here.
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Feynman,Bogoliubov
RG, Fröhlich
MF, beyond Fröhlich
Feynman, exponential cut-off
DMC
RG, beyond Fröhlich
Experiment Catani et al.
attractive interactions repulsive interactions
FIG. 3. Effective polaron mass Mp: We compare theoretical predictions for Mp with the experimental data by Catani et al. [7],
obtained from an indirect measurement of√M/Mp (for details see Ref. [7] and discussion below). The solid purple theoretical
curve (Feynman, exponential cut-off) was also taken from Ref. [7] and corresponds to calculations based on Feynman’s varia-
tional path integral approach, using the effective Frohlich Hamiltonian H′F defined by Eqs. (15), (16). Other theoretical curvesare obtained from our own calculations, based on the Hamiltonians indicated in the legend. The shaded area on the attractiveside corresponds to the regime where the RG (assuming the Hamiltonian HBF = HF + H2ph) breaks down. DMC calculationswere performed for N = 50 particles and we checked that finite-size corrections are small.
(ii) To calculate the polaronic mass shift from Feyn-man’s approach applied to the effective Frohlichmodel H′F , Catani et al. [7] used an estimate basedon variational parameters instead of a full evalua-tion of the Green’s function. This approximationcan lead to sizable errors for polarons in ultracoldquantum gases [14, 16]. Does the estimate work for
the Frohlich model H′F for the experimentally rel-evant parameters? – The estimate leads to sizabledeviations from the value of the effective polaronmass expected from the Frohlich Hamiltonian H′F .Nevertheless this effect is not sufficient to fully ex-plain the disagreement with the experiment.
(iii) The Frohlich model is only applicable for suffi-ciently weak impurity-boson interactions. Wheredoes this approximation break down? – For |η| . 3,given the parameters form the experiment [7], a de-scription in terms of a Frohlich Hamiltonian leadsto quantitatively correct description.
(iv) The effective Frohlich Hamiltonian H′F derived
from bosonization differs from HF obtained fromBogoliubov theory in the way how the UV regu-larization is performed. Does this explain the dis-agreement between theory and experiment? – Inthe 1D polaron cloud, phonons from all energiescontribute to polaronic dressing and have strong ef-fect on the polaron energy, effective mass, and dy-namics. Therefore the polaron properties are sen-sitive to the details of the UV regularization. Thiscauses the large deviations between earlier theoret-ical calculations based on H′F and the experimentshown in Fig. 3.
A. UV regularization of the Frohlich model
The effective Frohlich Hamiltonian derived from Bo-goliubov theory, see Eqs. (5), (7) and (8), does not re-quire any UV regularization in one dimension and wecan simply set the UV cut-off Λ0 = ∞. At momentak & 1/ξ the Bogoliubov dispersion becomes quadratic,ωk ' k2/2mB, making all quantities (the effective mass,the polaron energy, etc.) well behaved.
On the other hand, bosonization can be used to derivean effective Frohlich Hamiltonian [7] for the impurity. Itdescribes the interaction of the impurity with the Lut-tinger liquid formed by the 1D Bose gas [48, 51]. Atlong wavelengths, or low energies, the resulting FrohlichHamiltonian H′F has the same asymptotic form as HF inEq. (7) which was derived from Bogoliubov theory. Thisprovides additional evidence that the use of the Bogoli-ubov theory for treating the 1D Bose gas is justified.
The Frohlich Hamiltonian H′F is obtained from Eq. (7)by replacing ωk and Vk with different expressions ω′k andV ′k, respectively. The bosonization approach relies on alinear phonon dispersion,
ω′k = ck, (15)
and determines the scattering amplitude V ′k at small k(long wavelengths). Its ultraviolet (UV) cut-off, requiredfor regularization of the model, is commonly representedby an exponential decay at a characteristic scale kc [7]which is usually taken to be kc ∼ 1/ξ,
V ′k =gIB
2π
√K|k|e−k/2kc . (16)
The dimensionless Luttinger parameter K can be deter-mined from Bethe-ansatz calculations. It is given by the
7
ratio of the Fermi velocity vF = πn0/mB and the speedof sound c,
K =vFc
=πn0
cmB, (17)
By using Eq. (17) one confirms that the asymptotic be-haviors of Vk (ωk) and V ′k (ω′k) in the infrared (IR) limitk 1/ξ are identical. For weakly interacting bosons,where the gas parameter γ 1, see Eq. (10), is small, Bo-goliubov theory provides an accurate value for the speedof sound as c =
√gBBn0/mB. For a strongly interact-
ing bath (Tonks-Girardeau regime), the speed of soundequals to the Fermi velocity, c = vF . For intermediate in-teractions, Bethe ansatz can be used to obtain the speedof sound c and Luttinger parameter K.
The main difference between the bosonization ap-proach and Bogoliubov theory is in the treatment ofphonons at high energies. In many transport and dy-namic phenomena in 1D systems the main contributionscome from the longest wavelength excitations [51] and theresults are insensitive to the UV cut-off. In this spirit,Ref. [7] focused on low-energy phonons and introducedthe exponential cut-off at kc in Eq. (16) by hand. In theBogoliubov theory, on the other hand, the phonon dis-persion ωk becomes non-linear for k & 1/ξ, providing anatural UV cut-off scale in the model.
To obtain reasonable results from the effective FrohlichHamiltonian H′F , the momentum cut-off kc has to be ofthe order 1/ξ. Even for this choice the properties of UVphonons at momenta around k ∼ 1/ξ differ in the two
models HF and H′F . As has been shown by a dimensionalanalysis in Refs. [16, 39], the high-energy modes withk ∼ 1/ξ play a crucial role in determining propertiesof the Bose polaron. Therefore we expect that there canbe sizable quantitative differences between predictions bythe two polaron Hamiltonians.
In Fig. 3 we calculate the polaron mass Mp starting
from the Frohlich model HF and using Feynman’s varia-tional path integral formalism [5, 14, 52] (purple dashed-dotted line in the figure). The same calculation was per-
formed in Ref. [7], but using the model H′F with an expo-nential UV cut-off (purple solid line in the figure). Sur-prisingly, the Bogoliubov theory predicts a polaron masswhich is about a factor of two larger than expected fromassuming an exponential UV cut-off. This also explainsthe large deviations between the experimental data andtheory based on H′F .
B. Validity of the Frohlich model
The Frohlich model is only valid when two-phonon pro-cesses can be ignored. This is the case when the deple-tion of the quasi-1D condensate is small, justifying alsothe assumption that phonon-phonon interactions cannotmodify the polaron cloud substantially in this regime.
To derive an estimate when the Frohlich model is ac-curate, we apply standard MF theory [10, 13, 53] (see
also Sec. V A) to the combined Hamiltonian HF + H2ph.In Appendix A we derive for the phonon number in thepolaron cloud,
NMFph = β2
MF
∫dk
(Vk
ωk + k2/2M
)2
. (18)
Here we assumed that the total momentum carried bythe polaron vanishes, p = 0. The result in Eq. (18) issimilar to the expression from the Frohlich model, whichis obtained by setting βMF = 1. When two-phonon termsare included, we obtain
βMF =
[1 +
gIB
2π
∫dk
W 2k
ωk + k2/2M
]−1
. (19)
Therefore the depletion of the quasi-1D condensate canbe described accurately by the Frohlich Hamiltonian pro-vided that βMF 1.
This yields the condition
|gIB| gcIB = 2π
[∫dk
W 2k
ωk + k2/2M
]−1
(20)
for the validity of HF . The integral can be estimated as
ηc = gcIB/gBB ≈√
2πc/gBB = π√n0|aBB|. (21)
As indicated in Fig. 2, the border of the weak couplingregion scales as ηc ∼
√n0|aBB|.
For the experimental parameters of Ref. [7], the crit-ical coupling strength where the Frohlich model breaksdown is given by ηc ≈ 6. Indeed, from Fig. 3 we see thattheoretical calculations based on the Frohlich Hamilto-nian only describe the experimental data for η . 3, andthe deviations become large around ηc.
Condition (21) provides an estimate when the FrohlichHamiltonian is valid. Note that this is different from con-dition (13) which describes in which regime the lowest-order perturbation theory in boson-boson and impurity-boson interactions is valid. Comparison of the two ex-pressions shows that there exists a parameter regime atlarge n0|aBB| and η < ηc where the Frohlich model isvalid but has to be solved non-perturbatively.
C. Feynman variational approach to the Frohlichmodel
In Refs. [16–18] the validity of Feynman’s variationalpath integral description of Frohlich polarons has beenquestioned. When the impurity mass is small and inter-actions are moderate, a new regime has been identifiedwhere the correlations between phonons in the polaroncloud become important. Because Feynman’s methodmerely interpolates between the two extremes of weakand strong coupling [54], it cannot capture the physicsaccurately in this situation.
8
10-1 100 1010
0.2
0.4
0.6
0.8
1
Feynman, estimateMF theory
Feynman, accurateRG method
weak coupling strong coupling
FIG. 4. Theoretical results for the effective polaron mass: Wecompare predictions by different theoretical approaches, allobtained assuming the Frohlich Hamiltonian HF . This modelis only valid for |η| ηc indicated in the plot. Parameterswere chosen as in the experiment by Catani et al. [7].
In the Florence experiment [7] the impurity mass isa factor of two smaller than the boson mass; It is thennatural to ask how accurate Feynman’s approach worksin this case. To answer this question we compare pre-dictions by the RG introduced in Ref. [16] to Feynman’svariational path integral method, see Fig. 4. Note thatwe only consider the Frohlich Hamiltonians HF and H′Fin this figure. For the sake of comparison we also presentresults at strong couplings, beyond the critical value ηcwhere the Frohlich model is not sufficient anymore.
To extract the effective polaron mass Mp, Feynmansuggested a simple estimate where one of his variationalparameters serves as a direct approximation for the massrenormalization δM = Mp −M . In addition he deriveda more accurate expression for the polaronic mass en-hancement from the imaginary-time Green’s function, seeRefs. [5, 11, 14, 52] for details. As can be seen from Fig. 4,the estimated polaron mass has a large relative error, inparticular in the regime of weak interactions.
The accurate expression for the polaron mass deter-mined from the imaginary-time Green’s function agreesremarkably well with our RG calculation. Only for in-termediate couplings, small differences between the twomethods can be observed. For weak and strong cou-plings both predictions coincide exactly. We concludethat Feynman’s ansatz provides an accurate descriptionof Frohlich polarons in the experiment by Catani et al. [7]
when the Frohlich Hamiltonian HF is used.
V. STRONG COUPLING: EFFECTS OFTWO-PHONON TERMS
When the coupling η ≈ ηc becomes too large, theFrohlich model is no longer valid. In the following we
will analyze the effect of two-phonon scattering termsH2ph which modify the properties of the polaron in thisregime [23]. We consider the beyond-Frohlich Hamilto-nian defined by
HBF = HF + H2ph. (22)
We compare our theoretical predictions based on HBF
with results for the full microscopic model H obtainedby Monte Carlo simulation.
A. MF theory
We start by highlighting some of the polaron prop-erties specific to 1D systems, using MF formalism dis-cussed in Ref. [24]. The formalism for 1D Bose polaronsis presented in Appendix A. In short, firstly one utilizesconservation of the total momentum p by applying theunitary transformation ULLP introduced by Lee, Low andPines [53]. This transforms the original Hamiltonian HBF
into the new one, HBF = ULLPHBFU†LLP. Next, a prod-
uct wave function of coherent phonon states is assumedto describe the ground state of HBF. This means thatthe polaron state can be written as
|ψMF〉 = U†LLP
∏k
|αMFk 〉. (23)
To find the minimum variational energy EMF[αk] =
〈ψMF|HBF|ψMF〉 one solves the saddle point equationsδEMF[αk]/δαk = 0, which yields
αMFk = − βMFVk
ωk + k2/2M − k(p− PMFph )/M
. (24)
The total phonon momentum PMFph =
∫dk k|αMF
k |2 andβMF have to be determined self-consistently. The MFvariational polaron energy is given by
EMF0 = n0gIBβMF +
gIB
2π
∫dk sinh2 θk +
p2 − (PMFph )2
2M.
(25)Note that MF calculations go beyond a straightforwardperturbative treatment of impurity-boson interactions aspresented e.g. in Ref. [26].
Effective polaron mass.– The effective mass of the po-laron, Mp, can be obtained from the momentum depen-dence of the MF polaron energy. As discussed e.g. inRef. [11], one finds
EMF0 (p) =
p2
2Mp+O(p4). (26)
In Fig. 3 we compare the MF effective polaron mass(dashed line) to the experimental results. At large re-pulsive couplings around η ≈ ηc we expect a saturationof Mp at some large but finite value. Qualitatively wefind a similar behavior from our DMC calculations and
9
the measurements by Catani et al. [7]. The values of theeffective mass predicted by different approaches at largecouplings are different.
In Fig. 5 we also show results from a quasi-1D DMCcalculation. It takes into account the finite extent ofthe trap in radial direction. The prediction for the ef-fective polaron mass is in excellent agreement with ourstrictly-1D DMC calculations. This suggests that thelarge discrepancy between theory and experiment in thestrong coupling regime cannot be explained by the in-fluence of radially excited states in the trap. Hence weconclude that the relevant physics remains strictly one-dimensional. Details of this calculation can be found inSec. VII A.
Multi-particle bound states and in-medium Feshbachresonance.– For a situation where the polaron momen-tum p = 0 vanishes, βMF is given by Eq. (19). Thismean-field expression suggests the existence of a tran-sition where βMF → ∞ diverges. It takes places at acritical attractive coupling strength
gMFIB,c = −2π
[∫dk
W 2k
ωk + k2/2M
]−1
< 0. (27)
At this interaction strength the number of phonons in thepolaron diverges, see Eq. (18). This suggests an instabil-ity of the system towards a state with a large number ofbosons accumulating around the impurity. Such behav-ior has been associated with multi-particle bound statesin higher dimensions [24, 25, 39].
At gMFIB,c the MF polaron energy also diverges and
changes sign, see Eq. (25). This indicates a transitionfrom an attractive polaron with negative energy, to arepulsive polaron with positive energy. In Fig. 6 the
10-1
100
101
0
0.2
0.4
0.6
0.8
1
RG
MF
RG
DMC
DMC, quasi 1D
FIG. 5. The effective polaron mass Mp is compared for differ-ent model Hamiltonians and using different theoretical meth-ods, as indicated in the legend. The Frohlich model is onlyvalid for |η| |ηc|. Parameters are chosen as in the experi-ment by Catani et al. [7], and all DMC calculations were per-formed for N = 50 particles. The quasi-1D calculations arecomputationally more demanding, explaining the increasedamount of noise in the data.
attractive polaron repulsive
polaron
multi-particle bound states
repulsive polaron
-15 -10 -5 0 5
MF
RG
DMC
200
100
0
-100
-200
FIG. 6. The polaron energy calculated from MF theory andusing the RG method, for parameters as in the Florence ex-periment [7]. For comparison, results from our DMC calcu-lations are shown for which we performed extrapolations tothe thermodynamic limit N →∞, see Appendix C for details.Different regimes discussed in the text are indicated in the toprow. In the shaded area the RG approach breaks down. Notethat DMC calculations fully include all microscopic terms inthe Hamiltonian, in particular boson-boson interactions whichlead to a stable solution in the regime where the RG breaksdown. For weak attractive interactions, ηc,RG < η < 0, anattractive polaron exists. On the other hand, for strong at-tractive interactions, η < ηc,MF, two branches can be realized.The energetically lower branch (energies not calculated) con-tains multi-particle bound states [24, 25, 55], while the en-ergetically higher branch corresponds to a repulsive polaronwith attractive microscopic interactions. For more details onthe phase diagram, see Appendix B and Ref. [39].
MF polaron energy is shown and the MF critical valueηc,MF = gMF
IB,c/gBB is indicated in the plot. In addition,we calculate the density profile of the bosons in Fig. 7.When ηc,MF is approached from η = 0, as expected, alarge number of bosons accumulates around the impu-rity. This effect is also observed by our full numericalMonte Carlo calculations.
To understand the physics of this transition, we firstanalyze the limits gIB → ±∞. Because the MF wavefunc-tion coincides for gIB = +∞ and gIB = −∞, we concludethat the two repulsive polaron branches at strong at-traction and repulsion are adiabatically connected. Thisbehavior is reminiscent of the meta-stable super-Tonks-Girardeau state, which can be realized by a strongly in-teracting Bose gas (without impurities) when the inter-actions are quickly changed from strongly repulsive tostrongly attractive [56, 57].
For an impurity inside a non-interacting Bose gas,gBB = 0, the critical value becomes zero, gMF
IB,c = 0. In
this case, gMFIB,c corresponds to the point where we ex-
pect the appearance of an infinite series of multi-particleimpurity-boson bound states in 1D, if the impurity massis infinite. This last condition guarantees that no cor-relations can be induced between the bosons by scat-tering off the impurity. Note that such processes arenot included in the MF wavefunction, even for finite
10
attractive polaron
repulsive polaron
0 2 4 6 8 100
1
2
3
4
5
6
RG
MF
DMC
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
RGMFDMC
MF
DMC
RG
FIG. 7. Density profile around the impurity: The boson density around the impurity (at x = 0) is calculated using the MF(dashed) and RG (solid) methods and compared to DMC simulations (dots) for N = 50 bosons. The parameters correspond
to the experiment by Catani et al. [7]. RG and MF calculations are based on the beyond-Frohlich Hamiltonian HBF, whichneglects phonon-phonon interactions. Their effect leads to the observable discrepancies in comparison to DMC calculationswhich fully include the boson-boson interactions.
impurity mass M < ∞. This explains why gMFIB,c = 0
vanishes in the non-interacting Bose gas, independentof M . In contrast, the RG approach presented in Ap-pendix B includes impurity-induced interactions betweenthe phonons. There we show that gRG
IB,c 6= 0 becomes non-zero in the non-interacting Bose gas when the impuritymass is finite.
In the limit of a non-interacting Bose gas, gBB = 0,and for a localized impurity, M = ∞, we note that the1D scattering length aIB = −1/mBgIB [6] diverges whengIB approaches the critical value gMF
IB,c = 0. This effectcan be associated with a 1D Feshbach resonance. Onthe other hand, when the Bose gas is interacting or theimpurity becomes mobile, the position of the Feshbachresonance is shifted. It is now located at the bare two-particle interaction gIB,c. This behavior is reminiscentof the in-medium shift of the Feshbach resonance pre-dicted for Bose polarons in 3D. In this case, too, theFeshbach resonance is associated with the appearance ofmulti-particle bound states in the spectrum [24, 39].
In general one should be cautious that the inclusionof quantum fluctuations might change the position ofthe transition compared to the MF prediction. In somecases, transitions predicted by MF theory even disappearcompletely. We address this problem in Appendix B,where quantum fluctuations are included using an RGapproach. See also Ref. [39] for a detailed discussion.
Logarithmic IR divergence of the polaron energy.– Incontrast to the 3D case [24], expression (25) for theMF polaron energy is fully convergent when the large-momentum cut-off Λ0 is sent to infinity. It has a loga-rithmic divergence when the small-momentum cut-off λis sent to zero however,
gIB
2π
∫ Λ0
λ
dk sinh2 θk ∼ −gIB log λλ→0−→ sgn(gIB)×∞.
(28)This IR divergence can be regularized by including quan-tum fluctuations using the RG approach presented in
Sec. V B. In our MF calculations of the polaron energy,e.g. in Fig. 6, we ignore this log-divergent term.
Orthogonality catastrophe.– In one dimension, the MFtheory of Bose polarons has even more noteworthy pecu-liarities associated with the IR cut-off. First of all, thephonon number in the polaron cloud diverges logarith-mically,
NMFph =
∫λ
dk
(βMFVkΩMFk
)2
∼ − log λλ→0−→ ∞. (29)
This is directly related to the log-divergence of the MFenergy. Depleting the quasi-1D condensate and creat-ing infinitely many phonons costs an interaction energywhich scales like ∼ gIBNph.
The diverging phonon number can also be understoodas a manifestation of Anderson’s orthogonality catastro-phe [58] for a mobile impurity in a 1D Bose gas. WithinMF theory, the quasiparticle residue is determined by thesimple relation (see e.g. Ref. [11])
Z = e−NMFph ' λ→ 0. (30)
Therefore already an infinitesimally small interactionleads to a vanishing quasiparticle weight Z = 0 in aninfinite system.
Supersonic polarons.– The large number of phononsin the 1D polaron cloud also affects the dependence ofpolaron properties on the total momentum [59]. Whilethere exists a phase transition from the subsonic to thesupersonic regime at large momenta in higher dimen-sions [43], this transition is absent in 1D. To calculatethe critical momentum pMF
c where the subsonic to super-sonic transition for the impurity takes place, we employLandau’s criterion for superfluidity. It states that thetransition takes place when the polaron velocity vp ex-ceeds the speed of sound c in the Bose gas. From the
11
expression vp = (p− PMFph )/M [43] we derive
pMFc = Mc+
∫dk
kβ2MFV
2k
(ωk + k2/2M − kc)2 '1
λ→∞. (31)
Here λ is the IR momentum cut-off. We find that po-larons are subsonic in 1D, due to the dressing of the im-purity with an infinite number of phonons.
B. RG approach
In Appendix B we extend our analysis and includequantum fluctuations on top of the MF solution by ap-plying the non-perturbative RG approach introduced inRefs. [16, 38, 39]. The RG procedure allows us to regu-larize the IR divergence of the polaron energy identifiedin Eq. (28). We use this method to make predictionsfor the energy shift between an interacting and a freeimpurity, which can be measured directly using radio-frequency spectroscopy [8, 9]. On the other hand, theRG method provides new insights into the polaron phasediagram. We present a detailed discussion of our RGanalysis of 1D Bose polarons in Appendix B.
From the solution of the RG flow equations (see Ap-pendix B) we obtain the fully regularized polaron energyERG
0 . Explicit calculation in the appendix demonstratesIR divergence of the polaron energy is regularized withthis approach. We further benchmark the RG approachby comparing to the DMC calculations and working inthe regime of large boson density n0. More concretely,we assume that the dimensionless interaction strengthγ 1, see Eq. (10), is small. In this regime the Bogoli-ubov approximation is justified and the effect of phonon-phonon interactions, included only by DMC, is expectedto be weak.
We estimate the importance of non-linear interactionterms between Bogoliubov phonons by the relative sizeof corrections ∆εLHY to the ground state energy ε0 ofthe homogeneous Bose gas caused by quantum fluctua-tions [46, 47]. Similar to Lee-Huang-Yang corrections in3D [60, 61], one obtains
∆εLHY/ε0 = −25/2
3π
1√n0|aBB|
= − 4
3π
√γ, (32)
see e.g. Ref. [6]. In the second expression, γ is the dimen-sionless interaction strength in the Bose gas, see Eq. (10).
In Fig. 8 we compare our result for the polaron en-ergy to DMC calculations. We used the same mass ratioM/mB = 41/87 as in the experiment by Catani et al. [7],but the density was chosen to be n0|aBB| = 144 (cor-responding to n0 = 6/ξ = 144/|aBB| or γ = 0.014) sothat the corrections of quantum fluctuation to the en-ergy is ∆εLHY/ε0 = 0.05 1. For weak-to-intermediatecouplings we obtain excellent agreement between RGand DMC predictions, validating the use of the effectivebeyond-Frohlich Hamiltonian HBF in this regime. For
MFRGDMC
VMC
-10 -8 -6 -4 -2 0-4000
-3500
-3000
-2500
-2000
-1500
-1000
-500
0
10-1
100
101
102
103
0
2000
4000
6000
8000
10000
MFRGDMC
FIG. 8. We compare the polaron energy, calculated fromdifferent methods, for attractive (a) and repulsive (b) mi-croscopic interactions. We used a value of γ = 0.014 (orn0|aBB| = 144) corresponding to the weakly interactingregime, where Bogoliubov theory is expected to be applicable.Parameters are chosen similarly to those of Florence experi-ment [7], except that we are assuming a larger boson density.The DMC results are obtained by 1/N extrapolation to thethermodynamic limit N →∞, see Appendix C for details.
the case of repulsive microscopic interactions, gIB > 0,we obtain sizable deviations in the regime of very strongcouplings. Yet the correction by the RG accounts forabout half of the deviation between MF and DMC re-sults in this case.
In the case of attractive microscopic interactions gIB <0, see Fig. 8 (a), an additional regularization of the po-laron energy E0(Λ) is required in the RG to deal withdivergencies of one of the coupling constants (G−) duringthe RG. This procedure is described in detail in Ref. [39].The agreement for the energy of attractive polarons isreasonable. It should also be noted that the attractivepolaron is not expected to be the ground state of theHamiltonian HBF based on the Bogoliubov approxima-tion for the Bose gas in this case, as discussed in detail inAppendix B. Instead it has been argued in Ref. [39] thatthe true ground state contains a large number of bosonsaccumulating around the impurity. This prediction isconsistent with our DMC calculations in this regime.
In Fig. 1 (b) we calculate how the density profile ofthe bosons is modified by impurity-boson interactions.
12
Again we consider the high-density regime where the Bo-goliubov approximation for the Bose gas is justified. Incontrast to the energies shown in Fig. 8, the depletion ofthe Bose gas is accurately described by the RG approachin this case, even for very large interactions (η = 103 inthe figure). This serves as another important benchmarkfor the RG and the use of the effective beyond-FrohlichHamiltonian HBF in the regime where quantum fluctua-tions in the bath are weak.
In contrast, for the experiment by Catani et al. [7] thecorrection to the energy due to quantum fluctuations issizable, ∆εLHY/ε0 ≈ −0.30, indicating that Bogoliubovtheory is no longer quantitatively accurate. Indeed thepredictions for the effective polaron mass in the strongcoupling regime differ by a factor of 4, see Fig. 5. Below,in Fig. 15, we also compare predictions for the polaronenergy in this case and find large quantitative deviationsbetween the two approaches on the repulsive side.
VI. DYNAMICS OF STRONG COUPLINGPOLARONS
Comparison of the effective polaron mass in Figs. 3and 5 suggests a good agreement between theory and ex-periment for weak and intermediate interaction strength.However, in the strong coupling regime the value of theplateau is not reproduced. Here we investigate the mea-surement method itself, which relies on the analysis ofoscillations of impurities subject to a shallow trappingpotential. We show that inside a homogeneous gas, boththeir amplitude and frequency renormalization can beused to measure the effective mass. Surprisingly, no fre-quency renormalization was observed by Catani et al. [7],who investigated the reduction of the amplitude instead.We speculate here that this can be caused by the inho-mogeneity of the Bose gas. Indeed, its size is below theextent of the largest oscillation amplitudes contributingto the measurement of the effective mass.
A. Time-dependent MF theory
To describe the dynamics of strongly coupled Bose po-larons inside a shallow trapping potential, we employ atime-dependent variant of MF theory as in Refs. [24, 43].We supplement this approach with the local-density ap-proximation (LDA). This allows us to treat the problemin the Lee-Low-Pines frame despite the external poten-tial.
Our variational ansatz corresponds to a product wavefunction of coherent states,
|ψMF(t)〉 = e−iχ(t)ULLP(P (t))∏k
|αk(t)〉, (33)
in the frame which is co-moving with the impurity, seeAppendix A. This is achieved by the unitary transfor-mation ULLP(P (t)) [53], where P (t) denotes the time-
dependent total system momentum. χ(t) denotes anoverall phase which guarantees conservation of the totalenergy.
The position of the impurity X(t) = 〈x〉 can be calcu-lated in the LLP frame by using the Ehrenfest theorem,
d
dtX(t) = (P (t)− Pph(t)) /M, (34)
where Pph(t) =∫dk k|αk(t)|2 is the expectation value
of the phonon momentum at time t. The equations ofmotion for αk(t) can be derived from the Hamiltonian
HBF, see Eq. (22), using Dirac’s time-dependent varia-tional principle [62]. As discussed in Ref. [24] this yields
i∂tαk = Vk +
(ωk +
k2
2M− k
M(P (t)− Pph(t))
)αk
+gIB
2π
∫dk′[Re(αk′)WkWk′ + i
Im(αk′)
WkWk′
]. (35)
We note that the log-divergent term in the MF polaronenergy, see Eq. (28), does not enter in the equations ofmotion (35). Therefore the time-dependent MF theory iswell-behaved in the long-wavelength limit. We note thatstarting from a free impurity we observe numerically thatthe phonon number grows logarithmically in time,
Nph(t) ' log t. (36)
This can be understood as a dynamical manifestation ofthe orthogonality catastrophe in a 1D Bose gas.
B. Local-density approximation
In the absence of an external force, the total systemmomentum is conserved, dP (t)/dt = 0. Now we con-sider the effect of an additional (confinement) potentialV (x) on the impurity. In the limit when its changes aresmall over a length scale set by the size of the polaronwavepacket, we can still treat the system as a homoge-neous one. In this LDA, the equation of motion for thetotal momentum is given by
d
dtP (t) = −∂xV (X(t)). (37)
Potentials from the Florence experiment.– In the fol-lowing we will consider different potentials, which haveall played a role in the experiment by Catani et al. [7].Firstly, both the impurity and the bosons are trappedinside a shallow parabolic potential,
VI,B(x) =1
2mI,BΩ2
I,Bx2, (38)
where mI = M is the impurity mass and ΩI,B are thetrapping frequencies. The oscillator lengths in Ref. [7]are `I ≈ 11ξ and `B ≈ 9ξ respectively, justifying the
13
0 20 40 60 80 100 120 1400
0.5
1
1.5
FIG. 9. The effective potential, including VI(x) (dashed line)and the polaronic part Vpol(x), is calculated for parametersas in the experiment by Catani et al. [7]. The size of the Bosegas was on the order of R ≈ 120ξ in Ref. [7].
LDA in this case because the size of the polaron is of theorder ξ `I,B.
In addition, Catani et al. [7] used a tight species-selective dipole trap (SSDP) to localize the impuritiesin the beginning. This gives rise to a steeper harmonictrap, VSSDP(x) = Mω2
I x2/2, where ωI = 0.28c/ξ in our
units. In this case the oscillator length is ` ≈ 3.3ξ andthe LDA is less justified.
Finally, the impurity energy depends on the densityn(x) of the Bose gas. This gives rise to an effective po-tential which is determined by the interaction strengthgIB. Assuming that the impurity can adiabatically fol-low its ground state at P = 0, we obtain the followingpolaronic potential,
Vpol(x) ≡ EMF0 (x) = gIBβMF(x)n(x). (39)
To calculate βMF(x) we note that the healing length ξand the speed of sound c both depend on the densityn(x), and thus on x. For a weakly interacting quasi-1Dcondensate in a shallow trap we can apply the Thomas-Fermi approximation to estimate the boson density,
n(x) ≈ n0 − VB(x)/gBB. (40)
In Fig. 9 the potentials are shown for parameters asin the experiment by Catani et al. [7] and assuming re-pulsive impurity-boson interactions. We note that in thestrong coupling regime, for η & 1, the effective pola-ronic potential Vpol(x) has its minimum at the edge ofthe Bose gas. In this case we expect that impurities be-come trapped in this region after they have had enoughtime to equilibrate with the quasi-1D condensate. Suchbehavior has indeed been observed in Ref. [7] for η & 1.
C. Polaron oscillations: homogeneous Bose gas
We begin by studying the dynamics of an impurityinteracting with a homogeneous Bose gas, where the im-
purity is subject to a shallow trapping potential. Exper-imentally this situation corresponds to the assumptionthat ΩB ΩI is small. In this case we can ignore thepolaronic potential in Eq. (39), and we will consider onlyVI(x) now.
Adiabatic limit.– In the limit where the polaron followsits local ground state adiabatically, it can be describedby an effective Hamiltonian
Heff =p2
2Mp+
1
2MΩ2
I x2. (41)
If we start from a wavepacket in the origin, with mo-mentum p0, it will undergo harmonic oscillations. Theirfrequency is renormalized,
Ωp = ΩI
√M/Mp, (42)
due to the enhanced polaron mass.The amplitude of the harmonic oscillations, σ, is
easily obtained from energy conservation, p20/2Mp =
MΩ2Iσ
2/2. By the same polaronic mass enhancement,this amplitude is also renormalized compared to the caseof a free impurity,
σ =
√M
Mp
p0
MΩI. (43)
Now we will use full dynamical simulations to showthat both the frequency and the amplitude renormaliza-tion of polaron oscillations can serve as indicators of theeffective polaron mass. In the case of a homogeneousBose gas, Eqs. (42), (43) provide an accurate descriptionof polaron trajectories at weak couplings.
Initial conditions.– We use initial conditions as closeas possible to the experimental situation described inRef. [7]. There the impurities were tightly confined in thespecies-selective dipole trap, before they were releasedand their breathing oscillations were recorded. To modelthis situation, we consider polaron wavepackets localizedin the origin x = 0. We assume that the distributionf(p0) of their momenta p0 = P (0) can be derived fromthe thermal state of the impurity in the initial, tight trap-ping potential. Then we perform numerical calculationsfor different initial system momenta P (0) = p0 and addup the resulting trajectories with their respective thermalweights.
We also need the initial phonon configuration in thepolaron frame. Because the species-selective dipole trapprovides a rather deep trapping potential, ωI = 0.28c/ξ,we think it is reasonable to assume that the impurity isinitially localized, corresponding to M = ∞. Thereforethe phonon cloud at time t = 0 is chosen as the MFsolution at M =∞,
αk(0) = −β∞MF
Vkωk, β∞MF =
(1 +
gIB
2π
∫dkW 2k
ωk
)−1
.
(44)
14
We study the influence of the initial phonon distribu-tion on the long-time behavior in Fig. 10. We comparedcases where the MF solution at finite mass M , infinitemass M =∞, or the phonon vacuum αk = 0, was choseninitially. In Fig. 10 we show typical polaron trajectoriescalculated within LDA for moderate interactions. Un-less the MF solution at finite M > 0 is chosen as initialstate, the polaron loses a substantial part of its momen-tum at short times (see inset), until its velocity dropsbelow the speed of sound c. This dissipation leads to areduced oscillation amplitude, which manifests itself inthe trajectories even at long times.
Numerical results.– To study polaron oscillations sys-tematically, we simulated the experiment by Catani etal. [7] but assumed a homogeneous Bose gas first. InFig. 10 a typical polaron trajectory is shown, startingfrom an initial momentum P (0) which is characteris-tic for the temperatures in the experiment. Note thatit corresponds to a supersonic impurity with velocityX(0) > c. At long times we observe long-lived but decay-ing oscillations with polaron velocities which are alwaysbelow the speed of sound c.
We repeated these calculations for different interac-tions η and averaged over the thermal distribution of ini-tial momenta P (0) corresponding to Ref. [7]. Then theresulting trajectories
σ(t) =√〈X2(t)〉T , (45)
where 〈·〉T corresponds to thermal averaging, were fittedto a function,
σ(t) = σ| sin(ωt)|e−γt. (46)
0 100 200 300 400 500 600 700 800-150
-100
-50
0
50
100
150
0 5 100
5
10
free impurityadiabatic approximation
MF,
MF,
MF,
FIG. 10. Typical polaron trajectories are shown for differentinitial conditions of the phonon cloud around the impurity.We used a trapping potential ΩI = 0.025c/ξ as in Ref. [7],which we treated in LDA. The Bose gas was assumed to behomogeneous in this case and the initial momentum p0 =3.5Mc of all trajectories corresponds to the typical thermalenergy kBT = p2
0/2M in the experiment by Catani et al. [7].The impurity-boson interaction was η = 4, and the remainingparameters were chosen as in Ref. [7]. We used a UV cut-offΛ0 = 3/ξ and checked that the results have converged.
In Fig. 11 the fit parameters σ, ω and γ are shown as afunction of the interaction strength η. Some of the tra-jectories from which these plots were derived are plottedin Fig. 12.
In Fig. 11(a) we compare results for the frequencyrenormalization with predictions by the adiabatic ap-proximation, see Eq. (42), and obtain excellent agree-ment. Our findings demonstrate that a frequency mea-surement of polaron oscillations provides an accuratemethod to detect the effective polaron mass. In particu-lar, it also works in the strongly interacting regime andshows the saturation of the polaron mass in this case.
In Fig. 11(b) we compare results for the amplituderenormalization to predictions by the adiabatic approxi-mation, see Eq. (43). The amplitude σ0 is determined bythe temperature of the impurity in this case; It is on theorder of σ0 ≈ Ω−1
I
√2kBT/M . Both for weak and strong
impurity-boson interactions we find that the amplitudeof impurity oscillations is directly related to the effec-tive polaron mass according to the adiabatic expressionin Eq. (43). In the intermediate regime large deviationsfrom this prediction can be observed.
To understand this behavior we take a closer look atthe impurity trajectories. In Fig. 12 (b) we observe that,for small interactions, the impurity oscillates back andforth several times at supersonic speeds. It is sloweddown continuously until eventually it becomes subsonic.In this regime the amplitude of oscillations is well approx-imated by the adiabatic expression (43). At intermedi-ate interactions the impurity quickly loses kinetic energyand becomes subsonic, see Fig. 10 and 12(d). This dis-sipative effect goes beyond the adiabatic approximationand explains the strongly enhanced amplitude renormal-ization found in Fig. 11(b) for this regime. For stronginteractions, the impurity velocity drops below the speedof sound almost instantly for all initial velocities, seeFig. 12 (f). This suggests that the initial impurity mo-mentum is almost reversibly transformed into polaronmomentum in this regime. Afterwards the propagationof the heavy polaron can be described by the adiabaticapproximation again.
In Fig. 11(c) we present results for the damping rateof polaron oscillations in a harmonic trap. As before,three regimes of weak, intermediate and strong interac-tions can be identified. In agreement with our previousanalysis, the decay is largest for moderate interactionswhere deviations from the adiabatic approximation areexpected to be maximized.
Relation to the Florence experiment.– In this sectionwe neglect the additional complication arising from aninhomogeneous Bose gas and compare experimental re-sults to the theoretical analysis that assumes uniformboson density. We discuss the role of inhomogeneity inthe following section VI D.
Our analysis in Fig. 11 suggests that the oscillationfrequency is the most useful observable for measurementof the effective polaron mass. Surprisingly no frequencyrenormalization has been found by Catani et al. [7]. On
15
10-2
10-1
100
101
102
0
0.005
0.01
0.015
0.02
0.025
10-2
10-1
100
101
102
0
0.2
0.4
0.6
0.8
1
10-2
10-1
100
101
102
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8×10-3(a) (b) (c)
MF dynamics MF dynamics
MF dynamics
FIG. 11. Polaron oscillations in a homogeneous Bose gas: We simulate impurity trajectories as expected in a situation similarto the experiment by Catani et al. [7]. To this end we average over wavepackets, which we treat in LDA, with a momentumdistribution determined by the temperature kBT ≈ 2c/ξ. The resulting trajectories σ(t) are fitted to Eq. (46). Here we plot thefitting parameters for the frequency ω (a), the amplitude σ (b) and the damping rate γ of the resulting polaron oscillations asa function of the coupling strength. Except for the assumption of a homogeneous Bose gas we used parameters to describe the
situation in Ref. [7]. To facilitate the dynamical simulations, we chose a soft UV cut-off Wk →Wke−4k2/Λ2
0 and used Λ0 = 3/ξ.
(a) (c) (e)
(b) (d) (f)
0 100 200 300 400 500 600 700 8000
20
40
60
80
100
0 100 200 300 400 500 600 700 8000
20
40
60
80
100
0 100 200 300 400 500 600 700 8000
20
40
60
80
100
0 100 200 300 400 500 600 700 800-400
-200
0
200
400
0 100 200 300 400 500 600 700 800-400
-200
0
200
400
0 100 200 300 400 500 600 700 800-400
-200
0
200
400
MF dynamics
free impurity
fit
MF dynamics
free impurity
fit
MF dynamics
free impurity
fit
FIG. 12. Polaron oscillations inside a homogeneous Bose gas are simulated using the LDA. The top row shows the expansionof the impurity cloud σ(t), see Eq. (45), overaged over initial polaron momenta, along with the fit according to Eq. (46) andthe case of a free impurity. In the bottom row trajectories in the ensemble are shown, with a width corresponding to theirthermal weight and starting only with positive momenta. The trajectories with the largest initial velocity are highlighted bythe dash-dotted lines. In the weak coupling regime (a), (b) for η = 0.139 we observe long-lived but decaying oscillations. Forintermediate interactions (c), (d) at η = 1.9 the velocity of almost all trajectories quickly drops below the speed of sound c.This happens almost instantly in the strong coupling regime (e), (f) where η = 27. Parameters are the same as in Fig. 11.
the other hand, the amplitude renormalization has beenobserved and as shown in Fig. 3 its comparison withour different theoretical calculations is excellent at weakcouplings. At larger interactions the agreement is notas good, and the deviations are opposite from what wewould have expected according to Fig. 11(b).
From Fig. 11(c) we read off a typical damping rateγ ≈ 10−3c/ξ around η = 1. The breathing frequency ωb,which is twice the oscillation frequency ω, is on the orderof ωb = 0.04c/ξ in this regime. Therefore we expect afriction coefficient γ := γ/2ωb, as introduced in Ref. [7],of γ ≈ 0.025. This value agrees well with the measuredvalue of γ = 0.03(2) for this interaction strength.
D. Polaron oscillations: inhomogeneous Bose gas
As we discussed above, the experimental observationsin Ref. [7] are not consistent with our predictions for po-laron oscillations inside a homogeneous Bose gas. Now wediscuss possibilities how the inhomogeneity of the Bosegas can affect the oscillations of the impurity.
First we start by including the effective polaronic po-tential from Eq. (39) in the simulations. Some commentsare in order. The use of this potential can only be jus-tified in the limit where the polaron cloud follows theimpurity adiabatically. Moreover, the treatment becomesmeaningless close to the boundary of the Bose gas. In thisregime the boson density is small and the Bogoliubov ap-proximation breaks down. Moreover the Thomas-Fermiapproximation is not accurate in this regime. Thereforethe following results should be considered as a qualitative
16
0 100 200 300 4000
20
40
60
80
100
FIG. 13. Polaron trajectories σ(t) for an impurity describedby the effective polaronic potential in Eq. (39). Calculationsinclude inhomogeneous density of the Bose gas in a trap andaveraging over thermal distribution of the initial momentumof the impurity. We used the same parameters as in Fig. 11,except that the potential ΩB = 0.0175c/ξ was included. TheThomas-Fermi radius of the quasi-1D condensate in this caseis R ≈ 80ξ.
guide to understanding experiments rather than accuratequantitative analysis.
The polaron trajectories in the presence of the pola-ronic trapping potential are shown in Fig. 13. For weakimpurity-boson interactions, the result is similar to thecase of a homogeneous Bose gas and we obtain long-livedoscillations, slightly renormalized compared to the freeimpurity. For larger interactions all trajectories quicklybecome localized at the boundary of the Bose gas. Simi-lar behavior has been reported in Ref. [7] for long times,although pronounced oscillations have also been seen inthat case.
We now discuss that absence of frequency renormaliza-tion observed in Ref. [7] in the strongly interacting regimemay be a result of inhomogeneous density of bosons.Let us consider an impurity trajectory with an ampli-tude which exceeds the size of the Bose gas by a siz-able amount. Moreover, when the particle is interactingwith the bosons, let us apply the adiabatic approxima-tion. We may thus assume that all the initial momentumof the impurity is adiabatically transferred into polaronmomentum. Indeed, we have shown for the case of a ho-mogeneous gas that this simple picture explains why theamplitude of polaron oscillations is renormalized by a fac-tor of
√M/Mp for very strong couplings. This argument
only relies on the conservation of energy, and thereforeit also applies when the impurity exists the region wherethe Bose gas is confined.
For sufficiently large interactions, Fig. 9 shows that theeffective polaronic potential can be steeper than the har-monic confining potential for the impurity. Therefore itis conceivable that the impurity spends more time out-side the Bose gas, than inside. In an extreme case it mayeven be reflected off the boundary. In such a situationthe breathing frequency ωb is given by twice the baretrap frequency, ωb = 2ΩI, as observed in Ref. [7]. The
oscillation amplitude, on the other hand, is determinedby the initial energy of the impurity inside the Bose gas,which can be related to the effective polaron mass.
VII. BEYOND THE BOGOLIUBOVAPPROXIMATION
So far we mostly ignored the effects of interactions be-tween phonons. There are at least two good reasons toinclude them in our discussion and investigate the fullHamiltonian,
H = HF + H2ph + Hph−ph.
First of all, the experiment by Catani et al. [7] is in arather strongly interacting regime. The LHY correctionsto the ground state energy, see Eq. (32), are on the orderof 30% in this case. This indicates that the Bogoliubovapproximation may not be justified for describing theexperimental observations.
Secondly, we are interested in regimes where theimpurity-boson coupling becomes dominant. As we haveshown, this leads to an accumulation of a large numberof phonons (bosons) around the impurity, see e.g. Fig. 7.Even if the boson interaction strength gBB gIB is neg-ligible compared to the impurity-boson coupling, the in-teraction energy ∼ gBBn(x)2 can become comparable tothe polaron energy ∼ gIBn(x). This is possible becauseof the quadratic scaling of the boson’s interaction energywith their density n(x) around the impurity. The break-down of the RG (see Appendix B) also indicates thatinteractions between the phonons have to be included toprevent the system from becoming unstable to quantumfluctuations.
In this section we explain the DMC method (VII A)which we used for calculating the ground state energyof the full Hamiltonian H, as well as the effective massof the impurity. We benchmark this method (in VII B)by discussing the exactly solvable case of an impenetrableimpurity inside a weakly interacting Bose gas. This leadsus to a final discussion of the experiment by Catani etal. [7] (in VII C), where we focus on the effect of boson-boson interactions.
A. The DMC method for Bose polarons
Monte Carlo methods provide an efficient tool for theevaluation of multidimensional integrals. Expectationvalues of quantum mechanical operators can be writtenas integrals over the ground state wave function and canbe efficiently evaluated using Monte Carlo techniques.Here we resort to the diffusion Monte Carlo method inorder to obtain the ground state properties of the Bosepolaron problem. For a general reference on the DMCmethod, see for example [41]. An important advantageof the non-perturbative DMC method is that it can beapplied to arbitrary interaction strength η between the
17
impurity and the surrounding Bose gas. This is true forarbitrary values of the gas parameter γ of the bath, seeEq. (10).
Another question which can be addressed by perform-ing DMC simulations concerns the dimensionality of thesystem. Specifically, we study the effect of the transverseconfinement potential in the Florence experiment [7] onthe energetic and dynamical properties of the system.To this end we consider both the effective 1D Hamilto-nian with contact interactions, see Eq. (2), and a full 3DHamiltonian. The latter takes into account the geome-try of the optical lattice potential as well as the three-dimensional scattering length. Even though the exper-imental geometry is rather complicated, with differentoptical lattice potentials felt by the Rb and K atoms, theDMC method can still be applied.
We focus on two densities, first corresponding to stronginteractions in the bath, n0|aBB| = 4.56 or γ = 0.44, asin the Florence experiment [7]. The second consideredvalue, n0|aBB| = 144, corresponds to γ = 0.014 deep inthe Bogoliubov regime. For the first case we consider im-purities (Rb) and bath particles (K) of different mass. Inthe second case we will also consider the limit of an in-finitely repulsive pinned impurity, in which Bethe ansatzcan be used in order to find the ground state energy of theBose polaron. This permits us to verify the consistencyof the DMC energy with the Bethe ansatz result in thisexactly integrable limit. In addition we obtain the corre-lation functions and the density profile of bosons aroundthe impurity from DMC calculations.
Three-dimensional DMC calculations.– In first quanti-zation the 3D model Hamiltonian is
H = − ~2
2M∆I −
~2
2mB
N∑i=1
∆i +
N∑i<j
VBB(ri − rj) +
N∑i=1
VIB(ri − rI) + V extI (rI) +
N∑i=1
V extB (ri). (47)
Here VBB(r) and VIB(r) are the boson-boson andimpurity-boson interaction potentials, and ∆I , ∆i de-note the Laplace operators with respect to the impurityand the boson labeled by i, respectively. The externalpotentials, V ext
B and V extI , are felt by bosons and the im-
purity respectively. In the experiment [7] they have beencreated by two-dimensional lattices forming an array of1D tubes. We consider the case when a single tube inthe array is populated.
The depth of the Rb lattice potential Vlatt(r) is s =60 recoils with the lattice wavelength λ = 1064 nm,VB(r) = s~2/(2mRbλ
2). We have checked that similarresult can be obtained by considering a simple harmonicexternal potential, Vosc(r), with the transverse oscillatorfrequency ω⊥/2π = 34(45) kHz for Rb(K) atoms. We ig-nore the residual shallow trapping along the longitudinaldirection in DMC calculations.
The relation of the three-dimensional s-wave scatter-ing length, a3D, to the one-dimensional one, a1D, forthe tight transverse confinement with oscillator lengthaosc, is given by Olshanii’s formula a1D = −a2
osc/a3D(1−1.0326a3D/aosc) [4]. In an optical lattice geometry nosimple analytical result is known and the correspondingrelation is obtained following [63]. In the Florence exper-iment [7] the boson-boson s-wave scattering length wasfixed to a1D,Rb = −652.2nm and was not changed. Theboson-impurity s-wave scattering length, in contrast, istunable over a wide range by changing the strength ofthe applied magnetic field.
In our quasi-1D simulations we model the three-dimensional interaction potential by hard-spheres,VHS(r) = +∞ when |r| < aHS and VHS(r) = 0 otherwise.The diameter of the hard-sphere potential coincides withits s-wave scattering length and is set to reproduce Rb-
Rb (Rb-K) value for the boson-boson (boson-impurity)scattering amplitude. In our simulations we consider asingle impurity and impose periodic boundary conditionsalong the longitudinal direction of the tube.
The statistical fluctuations in Monte Carlo simulationcan be greatly reduced by using importance sampling.This is done on the basis of a distribution function whichwe derive from a trial guiding wave function. Motivatedby our physical insights, it is chosen as a product of one-and two-body terms,
ψT (r1, · · · , rN ; rI) = hI(rI)
N∏i=1
hB(ri)
N∏i<j
fBB(|ri − rj |)N∏i=1
fIB(|ri − rI |). (48)
The Gaussian one-body terms hI(r) = exp[−(x2 +y2)/2a2
osc,K] and hB(r) = exp[−(x2 + y2)/2a2osc,Rb] local-
ize the particles inside the central tube. The two-bodyJastrow terms are taken in the following form
fα(r) =
0, r < aαAr sin(B(r − aα))
1, r > L/2
(49)
where parameters A and B are chosen such that bothfα(r) and its derivative are continuous at the half size ofthe box, r = L/2. Here α = BB corresponds to boson-boson and α = IB to the case of impurity-boson scatter-ing. The Jastrow terms (49) are obtained as the solutionsof the two-body scattering problem on the hard spherepotential. Although the guiding wave function (48) - (49)does not contain any variational parameters, it providesa sufficient quality.
18
One-dimensional DMC calculations.– We also performcalculations for the 1D Hamiltonian from Eq. (2). Inthis case the guiding wave function can be convenientlywritten in a pair-product form
ψT (x1, · · · , xN ;xI) =
N∏i<j
fBB(|xi − xj |)N∏i=1
fIB(|xi − xI |).
We chose the two-body terms as
fα(x) =
Aα cos(kα(x−Bα)), |x| ≤ Rpar
α
| sin(πx/L)|1/Kparα , Rpar
α < |x|(50)
where α = BB, BI denotes boson-boson (boson-impurity)terms. Parameters Aα and Bα are chosen in such away that fα(x) and its first derivative are continuous for|x| > 0 and satisfy Bethe-Peierls boundary condition atthe contact, f ′α(0) = −fα(0)/aα. Here aα denotes thecorresponding 1D s-wave scattering length. The short-range part in Eq. (50), for |x| ≤ Rpar
α , is the two-bodyscattering solution for a contact δ-function potential. In-stead the “phononic” long-range part in Eq. (50), for|x| > Rpar
α , is obtained from the hydrodynamic approach[64].
The variational parameter Rparα corresponds to the
crossover distance between the two-body and phononicregimes. It is optimized by minimizing the variational en-ergy, which leads to the variational Monte Carlo (VMC)results presented earlier in this paper. The variation pa-rameter Kpar
BB for large system size coincides with theLuttinger parameter of the bath K. Its dependence onthe gas parameter, K(γ), is known from the Bethe ansatzsolution to the Lieb-Liniger model [65]. We use the ther-modynamic value of parameter Kpar
BB for the bath andoptimize parameter Kpar
IB by minimizing the variationalenergy.
Variational Monte Carlo method.– The variationalMonte Carlo method evaluates averages over the trialwave function ψT . The Metropolis algorithm [66] is usedto sample its square, |ψT |2 by generating a Markov chainwith corresponding probability distribution. The averageof the Hamiltonian, Evar = 〈ψT |H|ψT 〉/〈ψT |ψT 〉, pro-vides an upper bound to the ground state energy E0. Itis interesting to compare the value of Evar with predictionof MF theory.
Diffusion Monte Carlo method.– The diffusion MonteCarlo method [41] is based on solving the Schrodingerequation in imaginary time. For large times, the contri-bution to the energy from the excited states is exponen-tially suppressed, permitting to obtain the exact groundstate energy E0. The density profile of the polaron can becalculated using the technique of pure estimators [67, 68].The effective mass of the polaron, Mp, is obtained by cal-culating the diffusion coefficient D of the impurity in theimaginary time τ , D = limτ→∞〈[rI(τ) − rI(0)]2〉/τ , ac-cording to the relation D = ~2/2Mp. The variationalvalue for Mp is obtained from DMC algorithm with-out branching, which is an alternative method to the
Metropolis algorithm to generate the probability distri-bution according to |ψT |2.
B. The GPE limit: Dark solitons
A physically important limit is that of an impurity withan infinite mass, M =∞. It corresponds to two realisticsituations: (i) a pinned impurity, (ii) a static potentialcreated by a focused laser beam. Furthermore this limitis interesting as it is allows to obtain physical insights tothe effects of phonon-phonon interactions on the polaroncloud.
When the Bogoliubov approximation is justified –which is the case for γ 1 or n0|aBB| 1 – we canstudy this situation using the Gross-Pitaevskii mean-fieldequation (GPE) [45]:
EBφ(x) =
[− ∂2
x
2mB+ gBB|φ(x)|2 + gIBδ(x)
]φ(x). (51)
Here φ(x) = 〈φ(x)〉 describes the boson field in the 1Dsystem, and EB denotes the total energy EB of the com-bined impurity-boson system.
The GPE is valid even for strong impurity-boson in-teractions and goes beyond perturbative expansions inorders of gIB. This is different from perturbative theo-ries which linearize boson-impurity interactions, which isjustified when gIB is small. Because the boson-boson in-teractions are treated on a mean-field level, the validityof GPE is limited to the Bogoliubov regime n0|aBB| 1(γ 1) as we discuss below.
Dark soliton solution.– The repulsive non-linear GPEpossesses a famous class of solutions known as gray soli-tons. These correspond to a depletion in the boson den-sity which maintains its shape while propagating witha constant speed. In the case of zero velocity the den-sity completely vanishes in a single point, and the so-lution is referred to a a dark soliton. Thus we expectthat the effect of a massive impenetrable impurity, withM, gIB →∞, is to localize a dark soliton in the Bose gas.
The wave function of a dark soliton is given byφ0(x) =
√n0 tanh(x/2ξ) with the corresponding energy
EB = gBBn0N/2 − 2n0c/3. If we take into account that
∆Nsol = 2√
2n0ξ bosons are repelled from the homoge-neous Bose gas, we find that the energy E0 of the impu-rity in a system with fixed total particle number N anddensity n0 is
Esol0 =
4
3n0c. (52)
This energy has the same physical meaning as other po-laron energies E0 which we calculated before.
On the other hand, we can calculate the polaron en-ergy starting from an impurity and using MF theory asdescribed in Sec. V A. In that case we obtain
EMF0 = 2n0c =
3
2Esol
0 (53)
19
0.001 0.01 0.1 1 10 1000
1
2
3
4
GPE dark soliton
Bethe ansatzDMC
MFRG
Tonks-Girardeau
FIG. 14. The energy shift E0 caused by an impenetrable staticimpurity (M = ∞ and gIB = ∞) is calculated as a functionof the interaction parameter n0|aBB| of the surrounding Bose
gas. While MF and RG rely on the Hamiltonian HBF, theother approaches start from the full model H. The dash-dotted line corresponds to the dark-soliton solution of theGross-Pitaevskii equation (GPE). To ensure proper finite-sizescaling, DMC results are compared to exact analytical calcu-lations using Bethe ansatz and in the Tonks-Girardeau limit.
by sending gIB → ∞ and M → ∞. The obtained ex-pression overestimates the correct energy by the factorof 3/2. The reason is that in our MF theory we ig-nore the non-linearity in Eq. (51). By making the Bo-goliubov approximation we effectively linearize |φ|2φ ≈n
3/20 + 2n0δφ(x) + n0δφ
∗(x).
Comparison to DMC.– To benchmark our theoreticalmethods we calculated the energy of an impenetrable,localized impurity for different parameters of the Bosegas. Our comparison in Fig. 14 shows that DMC is inperfect agreement with the exact result in the limit whereBogoliubov theory is valid, n0|aBB| 1 or γ 1. Asexpected, MF theory and RG differ by the factor 2/3 inthis regime.
In Sec. V B we suggested to use the relative size of LHYcorrections to the ground state energy of the Bose gas asan indicator where the Bogoliubov theory can be appliedto describe polarons. This leads to the condition thatn0|aBB| 1 (or γ 1). Although in Fig. 14 we neverobtain quantitative agreement of the Bogoliubov approx-imation with the numerically exact DMC results, we findthat the qualitative behavior of the impurity energy E0
is correctly described in this framework for n0|aBB| 1(or small γ 1). In the opposite limit n0|aBB| . 1 (orγ & 1) in contrast, the Bogoliubov description completelyfails. This shows that the LHY corrections provide a re-liable measure for the accuracy of the truncated descrip-tion of the polaron cloud using only the beyond-FrohlichHamiltonian HBF.
10-1 100 101 1020
10
20
30
40
50
60
MF, beyond Fröhlich
RG, beyond Fröhlich
DMC, DMC,
RG, Fröhlich
MF, Fröhlich
VMC,
FIG. 15. Comparison of the polaron energy E0 computed bydifferent methods and using Hamiltonians as indicated in thelegend. We have chosen parameters as in the experiment byCatani et al. [7]. Note that the RG predicts a larger energythan MF theory for weak interactions because for the latterwe ignored the logarithmically divergent term Eq. (28). It isincluded and properly regularized in the RG. We show DMCand VMC results for N = 50 particles. For DMC also theextrapolated value expected in the thermodynamic limit N →∞ is shown, see Appendix C for details of our analysis.
C. Comparison to Florence experiment
In Fig. 15 we compare the polaron energy for repulsiveinteractions gIB > 0 and parameters as in the experimentof Ref. [7]. For weak-to-intermediate impurity-boson in-teractions, |η| . 1, where the Frohlich model is validand the density modulation in the Bose gas is small,all theoretical predictions agree with each other. Forstronger couplings we observe sizable quantitative differ-ences. Nevertheless, the qualitative behavior of all resultsgoing beyond the Frohlich Hamiltonian is the same. Thecorrections of the RG to the MF results is pronounced atlarge couplings, but DMC predicts even smaller polaronenergies in this regime.
In Fig. 3 we compared predictions for the effective po-laron mass to the experimental results from analyzingpolaron oscillations. There we have found large devia-tions for η & 3. Interestingly this is exactly where thestrong coupling regime starts and DMC predicts differ-ent energies than our RG approach. In this regime thedensity modulations of the Bose gas around the impurityare expected to become large. As a result, our analysisof polaron oscillations showed that large deviations fromthe adiabatic result can be expected. We speculate thatthis may be related to the large difference between theoryand experiment at strong couplings in Fig. 3.
In Ref. [7] it was moreover suggested that η ≈ 15could also be the point where higher transverse modesare important. Population of such modes would im-ply that the system can no longer be treated as strictlyone-dimensional. However, the argument of Ref. [7] wasbased on a comparison of the bare energy gIBn0 withthe transverse trapping frequency ω⊥ ≈ 126/mBa
2BB. As
can be seen from Fig. 15, the relevant polaron energies
20
are well below ω⊥ for all interaction strengths. Thereforewe conclude that a cross-over into a higher-dimensionalregime cannot explain the experimental observations.This supplements our analysis of higher transverse modesin Fig. 6, where we arrived at the same conclusion.
We conclude by noting that phonon-phonon inter-actions play an important role for understanding po-larons at strong couplings in the experiment by Cataniet al. [7]. Not only do two-phonon terms beyond theFrohlich Hamiltonian become important, but also thenon-linear interactions between phonons are required todescribe correctly the polaron energy for repulsive inter-actions. On the attractive side, where gIB < 0, we expecttheir influence to be even more dramatic, because largerdeformations of the Bose gas around the impurity arepossible.
VIII. SUMMARY AND OUTLOOK
In this paper we studied theoretically mobile impuri-ties interacting with a 1D Bose gas. We provided generaltheoretical analysis of such problems and considered aspecific experimental system realized in experiments byCatani et al. [7]. We showed that in the weak couplingregime the Frohlich model provides an accurate descrip-tion of the system. We extended our analysis to includetwo-phonon scattering terms, which become importantfor stronger impurity-boson interactions. Finally we alsodiscussed the effects of boson-boson interactions on thepolaron cloud.
Main new results.– The main new theoretical insightsof our work are related to how two-phonon terms affectBose polarons at strong coupling. Simple mean-field doesnot work in one dimension and needs to be corrected byRG calculations. For sufficiently weak boson-boson inter-actions we find that qualitative features of the polaronphase diagram remain the same as obtained from mean-field description of polarons in three dimensions [24, 39].
In particular we find a repulsive polaron branch forrepulsive couplings, η > 0, and an attractive polaronbranch for sufficiently weak attractive interactions, η .0. In the weakly interacting limit, both branches canbe described by the Frohlich polaron, which has been ob-served in the Florence experiment [7]. On the other hand,for sufficiently strong attractive interactions, η 0, weexpect multi-particle bound states at low energies anda meta-stable repulsive polaron branch at high energies.The latter is adiabatically connected to the polaron atinfinitely repulsive microscopic interactions η → ∞. Asa result, we find that the effective polaron mass saturatesat a finite value Mp <∞ when the repulsive interactionsdiverge, η → ∞. This is confirmed by our Monte Carlocalculations using the full microscopic Hamiltonian andshould be contrasted with a self-trapping scenario as pre-dicted by Landau and Pekar for Frohlich polarons [1].
The enhanced role of quantum fluctuations in 1D man-ifests itself in the logartihmic divergence (with system
size) of the MF polaron energy. In Appendix B we gen-eralized the RG approach from Ref. [39] to one dimen-sion. We showed that the resulting polaron energy isregularized in the RG approach and converges to finitevalue when the system size is increased. Furthermore weshowed that RG analysis can be used to study other prop-erties of 1D Bose polarons, incudling the effective massand impurity boson correlations. We compared thesepredictions to our numerically exact DMC calculationsand found good agreement for a weakly interacting Bosegas in the Bogoliubov regime. We concluded that fora full quantitative description of Bose polarons at strongcouplings, phonon-phonon interactions always need to beincluded. In addition we identified regimes in the phasediagram where the full microscopic Hamiltonian is re-quired for reaching even a qualitative understanding ofthe polaron properties (Fig. 2).
Analysis of the experiment by Catani et al. [7].– Orig-inal analysis of the experiments showed a disagreementbetween theoretical results and experimentally measuredeffective mass already for weak impurity-boson interac-tion. We explained that this disagreement results fromthe high energy modes with k ∼ 1/ξ that were not in-cluded properly in earlier analysis. We showed that bothanalytical RG and numerical DMC methods give resultsin good agreement with experiments when the high en-ergy modes are included more accurately.
Our theoretical methods predict a saturation of thepolaron mass at a finite value of the impurity-boson in-teraction strength. Although qualitatively this behaviorhas been observed in the experiment, large quantitativedeviations from our theoretical calculations are found inthis regime. We performed full numerical simulations ofthe polaron trajectories in a harmonic trapping potentialand argued that the disagreement between theory andexperiment could be related to the inhomogeneity of theBose gas.
By performing DMC simulations for two differentHamiltonians: (i) strictly one-dimensional, see Eq. (2),(ii) three-dimensional, see Eq. (47), with strong trans-verse confinement, we found no significant differences forthe dynamic and static properties, even in the regime ofstrong interactions. This means that the use of a strictlyone-dimensional Hamiltonian is justified.
Closer inspection of different theoretical models re-vealed that in the strong coupling regime phonon-phononinteractions need to be included if one wants to do ac-curate comparison to experiments. In contrast to Bosepolarons in three dimensions [8, 9], the relative size ofquantum fluctuation corrections to the ground state en-ergy of the Bose gas was sizable in Ref. [7]. We showedhere for 1D systems that this is a suitable indicator forthe applicability of the Bogoliubov approximation for de-scribing Bose polarons.
We conclude that a detailed quantitative analysis ofthe experiment by Catani et al. [7] at strong couplingsis extremely challenging. We showed that it requires fullinclusion of two-phonon terms H2ph as well as phonon-
21
phonon interactions Hph−ph, both going beyond theFrohlich Hamiltonian that has been used previously toanalyze the experiment [7, 36]. Moreover, from full dy-namical simulations of this problem we found indicatorsthat the inhomogeneity of the Bose gas needs to be takeninto account as well. We expect that numerical DMRGor TEBD [69, 70] calculations could shed new light onthis problem in the future.
Possible future experiments.– Here we performed fulldynamical simulations of polaron trajectories inside ashallow trapping potential. We showed by using a time-dependent MF ansatz combined with the local-densityapproximation that polaron oscillations inside a homo-geneous trap provide a powerful means for measuringthe effective polaron mass. When the Bose gas can beassumed to be homogeneous, the frequency renormaliza-tion provides accurate results. We suggest to use species-selective optical traps in the future to perform such mea-surements, in a regime where the Bose gas is as largeas possible to avoid effects of the inhomogeneous densityprofile.
The energy provides another important quantity tocharacterize Bose polarons. It can be obtained directlyfrom the impurity’s radio-frequency spectrum, which hasbeen measured in 3D [8, 9]. We suggest to repeat theseexperiments in 1D systems, possibly even in the time-domain [71]. It would be particularly interesting to studyquenches from strong repulsive to strong attractive in-teractions and show the existence of a repulsive polaronbranch for attractive microscopic interactions gIB < 0 inone dimension.
ACKNOWLEDGEMENTS
We acknowledge useful comments by Adrian Kantianand Thierry Giamarchi, as well as fruitful discussionswith Yulia Shchadilova, Richard Schmidt, Thierry Gi-amarchi and Artur Widera. F.G. acknowledges supportfrom the Gordon and Betty Moore foundation. G.E.A.acknowledges partial financial support from the MICINN(Spain) Grant No. FIS2014-56257-C2-1-P. The authorsacknowledge support from Harvard-MIT CUA, NSFGrant No. DMR-1308435, AFOSR Quantum SimulationMURI, AFOSR grant number FA9550-16-1-0323 and theMoore foundation. Authors thankfully acknowledge thecomputer resources at MareNostrum and the technicalsupport provided by Barcelona Supercomputing Center(FI-2017-1-0009). The authors gratefully acknowledgethe Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing comput-ing time on the GCS Supercomputer SuperMUC at Leib-niz Supercomputing Centre (LRZ, www.lrz.de).
Appendix A: Mean-field theory for Bose polarons
in one dimension
In this appendix we briefly derive the mean-field (MF)theory for Bose polarons in 1D. Our starting point is thebeyond-Frohlich Hamiltonian HBF from Eq. (22), i.e. weapply the Bogoliubov approximation for describing theBose gas and neglect interactions between the phonons.The standard MF theory developed for Frohlich polarons[10, 13, 53] has been generalized to this case in higherdimensions [24].
To solve the full Hamiltonian (7), (9) we start by ap-plying the unitary Lee-Low-Pines (LLP) transformation[53]. Following Refs. [10, 11, 24] this gives rise to theHamiltonian
HLLP(p) = gIBn0 +1
2M
(p−
∫dk ka†kak
)2
+
∫dk ωka
†kak +
∫dk Vk
(a†k + a−k
)+
+gIB
2π
∫dk sinh2 θk +
gIB
2π
∫dk dk′ :
(cosh θka
†k − sinh θka−k
)(cosh θk′ ak′ − sinh θk′ a
†−k′
):, (A1)
where p is the total conserved system momentum. Notethat we normal-ordered the two-phonon scattering terms(denoted by : ... :), which gives rise to the constant energyshift in the first term of the second line.
In the MF theory of Bose polarons one makes an ansatzof coherent states
∏k |αMF
k 〉 in the LLP frame. As shownin Ref. [24], minimization of the variational energy with
respect to αMFk leads to the MF solution
αMFk = −V
MFk
ΩMFk
, (A2)
where we defined
ΩMFk = ωk +
k2
2M− k
M
(p− PMF
ph
), V MF
k = βMFVk.
(A3)
22
In contrast to the Frohlich case [13, 43], the scatteringamplitudes V MF
k are renormalized as well. The two re-maining parameters PMF
ph and βMF are obtained by solv-ing the following set of coupled self-consistency equa-tions,
PMFph = (βMF)2
∫dk k
V 2k(
ΩMFk
)2 , (A4)
and
βMF =
[1 +
gIB
2π
∫dk
W 2k
ΩMFk
]−1
. (A5)
Appendix B: RG approach to Bose polaronsin one dimension
In this appendix we extend the MF analysis fromSec. V A using a RG approach. We also study the po-laron phase diagram of the Bogoliubov polaron model,where phonon-phonon interactions are neglected.
The non-perturbative RG method was developed forthe Frohlich model in Ref. [16]. It was checked by show-ing excellent agreement with numerically exact diagram-matic Monte Carlo calculations [18] for the polaron en-ergy. In Ref. [39] we extended this approach to Bosepolarons, including two-phonon terms. To benchmarkthe method also in this case, we compare to our DMCcalculations in Fig. 1 (b). In the considered regime oflarge Bose gas densities n0, corresponding to a small gasparameter γ . 1, we find good quantitative agreement.
a. RG equations
Starting from the MF polaron solution αMFk one can
rewrite the Hamiltonian HBF = HF + H2ph by an ef-
fective Hamiltonian H describing quantum fluctuationsaround the MF solution. Then the high-energy phonons,with momenta larger than a running UV cut-off Λ, areeliminated step by step in the RG. This modifies the MFsolution αk(Λ) at small energies [38], and gives rise toan RG flow of the coupling constants when Λ changes.Using the same notation and following the derivation ofRef. [39] we obtain an effective Hamiltonian
H(Λ) = E0(Λ)+
∫ Λ
dk
Ωka
†kak+
∫ Λ
dk′kk′
2M: ΓkΓk′ :
+G+
2n0: δnkδnk′ : +2n0G− : ϑkϑk′ :
. (B1)
A few explanations are in order. First of all, note thatthe impurity operators x and p have been eliminated byapplying the Lee-Low-Pines transformation [53] and con-sidering a polaron with vanishing total momentum. Thepolaron energy E0(Λ), starting at EMF
0 for Λ = Λ0, de-creases until it reaches the ground state energy of the
polaron for Λ → 0. The effective phonon frequency inthe frame co-moving with the impurity is given by
Ωk(Λ) = ωk + k2/2M(Λ), (B2)
where M(Λ) denotes the renormalized mass of the im-purity. Note that M(Λ → 0) ≈ Mp can be used as anapproximation for the effective polaron mass [38].
The last term in the first line of Eq. (B1) describesphonon-phonon interactions induced by the mobile im-purity, where the operators Γk are defined as Γk =
a†kak + αk(a†k + ak). Note that the MF amplitude isflowing in the RG,
αk(Λ) = −β(Λ)VkΩk(Λ)
. (B3)
The second line in Eq. (B1) is an alternative formula-
tion of the two-phonon scattering terms H2ph in Eq. (9),with coupling constants G±(Λ) running in the RG. Weintroduced the following pairs of conjugate operators,
δnk =√n0Wk
(ak + a†−k
), (B4)
ϑk =1
2i√n0W−1k
(ak − a†−k
), (B5)
describing particle number and phase fluctuations of theBose gas.
The initial conditions for the coupling constants flow-ing in the RG are given by
G±(Λ0) =gIB
4π, M(Λ0) = M (B6)
E0(Λ0) = EMF0 , β(Λ0) = βMF. (B7)
In one dimension the RG flow equations read [39],
∂ΛM = −4Λ2α2Λ/ΩΛ, (B8)
∂ΛG−1± = −4W±2
Λ /ΩΛ, (B9)
and the RG flow of β(Λ) is described by
β(Λ) = G−1+ (Λ0)
[G−1
+ (Λ) + 2
∫ Λ
−Λ
dp W 2p /Ωp
]−1
.
(B10)The ground state energy can be determined from
∂ΛE0 =1
2∂ΛM−1
∫ Λ
−Λ
dp α2pp
2+
+
∫ Λ
−Λ
dp2
ΩΛ
(WΛWpG+ −W−1
p W−1Λ G−
)2. (B11)
In three dimensions [39] all coupling constants con-verge when the cut-off Λ→ 0. In that case the RG flowsstop when the dispersion relation ωk becomes linear forΛ ≈ 1/ξ. Except for G−, this is also true in one dimen-sion. Here the coupling constant G− always flows to theweak coupling fixed-point in the IR limit,
G−(Λ→ 0) = 0+. (B12)
To see this, note that we obtain a divergent RG flow∂ΛG
−1− ' −1/Λ2 when Λ→ 0.
23
b. Regularization of the IR log-divergence
The MF polaron energy EMF0 from Eq. (25) diverges
when the IR cut-off λ is sent to zero. The reason for thisdivergence is the unphysical assumption of MF theorythat the coupling constant gIB is unmodified by quan-tum fluctuations. Now we show that the RG flow of G−to the universal weak coupling fixed point G− = 0, seeEq. (B12), leads to a regularized polaron energy.
We assume an IR cut-off λ, where the RG flow isstopped. As discussed around Eq. (28) the MF energyEMF
0 (λ) has a contribution
EMF0 (λ)
IR' gIB
π
∫ Λ0
λ
dk sinh2 θkIR' −gIB log(λ)
2√
2πξ, (B13)
which diverges logarithmically when λ → 0. Fromthe RG we can calculate the polaron energy ERG
0 (λ)by solving Eq. (B11) for Λ flowing from Λ0 to λ, i.e.ERG
0 (λ) = E0(Λ = λ). From the terms in the second lineof Eq. (B11) we obtain a contribution
∂ΛE0IR'
4G2−(Λ)
W 2ΛΩΛ
∫ Λ
λ
dp W−2p = (∂ΛG−)
∫ Λ
λ
dp W−2p
IR' −(∂ΛG−)√
2ξ−1 log(λ). (B14)
Integrating this equation yields
ERG0 (λ)
IR' EMF0 (λ) +
√2 log(λ)
ξ[G−(Λ0)−G−(λ)] .
(B15)From the explicit solution of the RG flow of G−1
− (Λ)we obtain the exact expression
G−1− (λ) = G−1
− (Λ0) + 4
∫ Λ0
λ
dΛW−2
Λ
ΩΛ
IR' λ−1, (B16)
from which it follows that G−(λ) = O(1/λ). I.e. G−approaches the weak coupling fixed point G− = 0 witha power-law in λ. Because log(λ)/λ → 0 for λ → 0, thelast term in Eq. (B15) is irrelevant in the IR limit.
Finally, combining Eqs. (B13), (B15) and usingG−(Λ0) = gIB/4π we obtain
ERG0 (λ)
IR'√
2 log(λ)
ξ
[G−(Λ0)− gIB
4π
]= 0× log(λ).
(B17)I.e. the two log-divergent terms cancel exactly and thepolaron energy obtained from the RG is fully convergentwhen the IR cut-off λ→ 0 becomes small.
Before moving on, a comment is in order about thenumber of phonons in the polaron cloud, which accordingto MF theory diverges logarithmically with the IR cut-off. In Sec.V A we argued that this is directly connectedto the log-divergence of the MF polaron energy. Now wehave proven that the coupling constant gIB gives rise totwo different couplings G± flowing in the RG, where the
renormalization of G− to zero at low energies regularizesthe polaron energy.
We emphasize that the number of phonons in the po-laron cloud is still diverging as λ → 0, as can be readilychecked from analyzing the IR behavior of the renormal-ized MF amplitudes in Eq. (B3). As a consequence weexpect that the quasiparticle weight Z = 0 for λ→ 0 alsowithin the RG formalism. Therefore we conclude that theorthogonality catastrophe [58] also exists for mobile im-purities interacting with 1D quantum gases. As a directway of detecting this effect for ultracold atoms, Ramseyinterferometry can be used as suggested in Ref. [71].
c. Polaron phase diagram
Now we analyze the RG flows of the coupling con-stants more closely and derive the polaron phase dia-gram. We work in a regime where phonon-phonon inter-actions Hph−ph can be neglected. We will show that thephase diagram shares all qualitative features with the 3Dcase discussed in Ref. [39].
Static impurity in a non-interacting Bose gas.– Let usstart by considering the exactly solvable case of an in-finitely heavy impurity, M = ∞, localized in the ori-gin. Furthermore we assume that the bosons are non-interacting. For repulsive impurity-boson interactions,gIB > 0, the ground state corresponds to a wave func-tion where all bosons populate the same single-particlestate, forming a repulsive polaron. For arbitrarily weakattraction, gIB < 0, a bound state ψb(x) of bosons to theimpurity always exists in one dimension. In this regimethe spectrum is unbounded, because ψb(x) can be occu-pied by any integer number of bosons. Note that theMF and RG theories provide a description of the polaronstate at finite energy, where no bosons are bound to theimpurity [39]. The polaron is meta-stable on the attrac-tive side because it can decay and form a molecule.
In the RG theory the existence of a bound state is indi-cated by a divergence of the effective interaction strengthduring the RG flow, G±(Λ) → ∞. For the case with-out boson-boson interactions described above it holdsG+(Λ) = G−(Λ). As shown in Eq. (B12), G− → 0+
always flows to the repulsive weak coupling fixed point.Because the RG flow starts at G±(Λ0) = gIB/4π, therealways exists a divergence G−(Λc) → ∞ at some inter-mediate Λc on the attractive side gIB < 0. As explainedin detail in Ref. [39] this is a direct manifestation for thebound state existing at low energies in this regime.
When the mass of the impurity is slowly decreased, weexpect the bound states to remain stable because theirenergy spacings are sizable. Note however that the finitemass of the impurity introduces correlations between thebosons and requires us to solve a full many-body problem.
Stable repulsive polarons.– Now we extend our discus-sion to finite mass M < ∞ and non-vanishing boson-boson interactions gBB > 0. The RG flows of G±(Λ),which differ in this case, are shown in Fig. 16. On the re-
24
-20 -15 -10 -5 0 5 10 15 20
-0.6
-0.4
-0.20
0.2
0.4
0.6
repulsive polaronphonon
instabilityRG breaks
downmolecule
FIG. 16. The RG flows of G±(Λ) (blue, yellow) are shown,which start form gIB/4π (indicated by the dashed line).The IR values G±(0) are shown by thick lines; Note thatG−(0) ≡ 0+ and the RG flows have divergencies on the at-tractive side. The ground states in the different parameterregimes are indicated in the bottom row. In the regime be-tween ηc,RG < η < ηc,MF the RG breaks down. We usedparameters as in the experiment by Catani et al. [7].
pulsive side, gIB > 0, the only qualitative change is thatG+(0) > 0 saturates at a finite value in the IR limit. Inthis regime the ground state is a repulsive polaron. InFig. 7 we show the density profile of the Bose gas aroundthe impurity, and indeed the impurity repels bosons inthis regime. For very strong repulsive interactions, thequasi-1D Bose gas is completely depleted around the im-purity, reminiscent of the bubble polarons predicted inthis regime in Ref. [72] or, equivalently, a dark soliton asdescribed in Sec. VII B.
Attractive polarons.– As discussed above, G− divergesduring the RG flow for arbitrary attractive interactions.On the other hand, the flow of the coupling constantG+(Λ) stops in the IR limit due to finite gBB > 0. Forsufficiently weak attraction,
0 > gIB > gRGIB,c = −2π
(∫ Λ0
−Λ0
dΛ W 2Λ/ΩΛ
)−1
, (B18)
this effect is sufficient to prevent G+(Λ) from divergingduring the RG and we obtain G+(0) < 0. This cor-responds to attractive interactions of the impurity withdensity fluctuations in the Bose gas and gives rise to anattractive polaron. Compared to the MF result for thecritical interaction strength gMF
IB,c, see Eq. (27), the RGpredicts a transition already somewhat earlier,
0 ≥ gRGIB,c ≥ gMF
IB,c, (B19)
as a consequence of the renormalized mass M≥M .In Fig. 16 the attractive polaron regime, where 0 >
η > ηc,RG = gRGIB,c/gBB, can be identified. Comparison
with Fig. 6 shows that, indeed, this is the regime where
the polaron energy is negative. The density profile of theBose gas around the impurity also shows a pronouncedpeak for this set of parameters, see Fig. 7.
It is worth emphasizing that the existence of attractivepolarons in one dimension is due solely to non-vanishingboson-boson interactions. As pointed out before, whengBB = 0 the interactions are always repulsive, G±(0) > 0,in the long-wavelength limit Λ → 0. The effect man-ifests in the relation gRG
IB,c = gMFIB,c = 0 for gBB = 0. In
view of this conclusion, the agreement between attractivepolaron energies in Fig. 8 (a) is remarkable, because itsuggests that indeed a meta-stable polaronic eigenstateexists which has negative energy E0 < 0.
The divergence of G− during the RG in the attractivepolaron regime suggests that there exists a mode boundto the impurity at energies below the polaron. In Ref. [39]this effect is discussed in detail and it is shown that thepolaron becomes dynamically unstable in this case. Be-cause only G− is negative, while G+ remains positive, the
spectrum of HBF is continuous and unbounded in the at-tractive polaron regime [39]. In our DMC calculationswe fully included phonon-phonon interactions, which areexpected to stabilize the attractive polaron [39]. Indeedwe find a nodeless state at energies corresponding to theattractive polaron, see Fig. 8 (a).
Break-down of the RG.– We find that the numberof phonons increases dramatically when gIB approachesgRG
IB,c. In the regime
gRGIB,c > gIB > gMF
IB,c, (B20)
the RG breaks down because the MF amplitude divergesat a finite value of Λ, β(Λ) → ∞. This is a result ofquantum fluctuations of the mobile impurity, because forM = ∞ it holds gRG
IB,c = gMFIB,c. Unlike for G±(Λ), the
divergence of β cannot be regularized. As discussion fur-ther in Ref. [39] phonon-phonon interactions are requiredto stop the divergence of the MF amplitude.
Metastable repulsive polarons.– When gIB < gMFIB,c both
coupling constants G±(Λ) diverge during the RG flow,see Fig. 16. While they both start out attractive at highenergies, they become repulsive in the long-wavelengthlimit. Therefore the polaron energy E0 > 0 is positivein this regime, corresponding to a repulsive polaron, seeFig. 6. The repulsive polaron branch is adiabatically con-nected to the polaronic states realized for repulsive mi-croscopic interactions, similar to the physics of the super-Tonks-Girardeau metastable state [56, 57].
The accumulation of bosons around the impurity is anindicator for molecule formation at gIB < gMF
IB,c. Already
for gIB & gRGIB,c we find pronounced oscillations in the
impurity-boson correlation function, which decay withthe distance from the impurity. Indeed, when both cou-pling constants G±(Λ) diverge during the RG, the ap-pearance of a bound state with a discrete energy is ex-pected [24, 39]. This state is adiabatically connected tothe molecular bound state discussed above at M = ∞and gBB = 0. In the spectral function it is expected to
25
0 0.005 0.01 0.015 0.02 0.0250
5
10
15
20
25
30
35
FIG. 17. Finite-size dependence of the ground-state energyE(N)−E(N =∞) of the bath deep in the Bogoliubov regime,n0|aBB| = 144, and in the absence of the impurity (η = 0).The dependence on the number of atoms in the bath N isobtained from Bethe ansatz theory [46]. The asymptotic 1/Ndependence is shown with a solid line obtained as a fit.
give rise to a series of peaks separated by the bound stateenergy [24].
The possibility to decay into molecular states leads toa finite life-time of repulsive polarons when gIB < 0, awell-known phenomenon close to a Feshbach resonancein three dimensions [8, 9, 23, 24]. This makes a directcalculation of the polaron energy using DMC difficult,because the polaron is no longer the ground state.
To observe repulsive polarons at gIB < 0 experimen-tally, we suggest to study quenches from the strongly re-pulsive side gIB = +∞ to −∞ where the microscopic in-teractions are attractive. This approach has successfullybeen used to realize the super-Tonks-Girardeau regime ofan interacting 1D gas, see Refs. [56, 57]. We expect thatthe finite life-time of the repulsive polaron at gIB < 0should be observable, for example by using Ramsey in-terferometry between two spin states which interact dif-ferently with the bosons.
Comparison to the experiment.– In Fig. 3 we plottedthe critical values ηc,RG and ηc,MF corresponding to theparameters in Ref. [7]. For weakly attractive interac-tions, 0 > η & ηc,RG, the measured values for the ef-fective mass are in good agreement with predictions foran attractive polaron. Around ηc,RG the qualitative be-havior of the data changes. The range of parametersηc,MF < η < ηc,RG where the RG breaks down and weexpect a polaron cloud with many phonons is too narrowto draw any conclusions from the comparison.
Appendix C: Finite-size scaling in Monte Carlocalculations
The MF and RG theories discussed in the main textpredict system properties in the thermodynamic limit ofthe bath. Instead, quantum Monte Carlo (QMC) simu-
lations are carried out for a finite-size system in a box
0 0.005 0.01 0.015 0.02 0.0250
2000
4000
6000
8000
10000
12000
FIG. 18. Finite-size dependence of the polaron energyE(N, η) − E(N = ∞, η = 0) deep in the Bogoliubov regime,n0|aBB| = 144, and for the strongest interaction with the im-purity, η =∞. The thermodynamic N →∞ value is obtainedfrom 1/N fits, shown with lines.
with periodic boundary conditions. Thus, for makinga comparison between different theories it is preferablefirst to do the extrapolation of the QMC results to thethermodynamic limit. Between two considered densities,corresponding to the Florence experiment [7] and to deepBogoliubov regime, the latter is expected to have thestrongest finite-size effects and we analyze it here.
It is instructive first to study how the energy of thebath depends on the system size in the absence of theimpurity. The energy of a single-component Bose gaswith δ-pseudopotential interaction can be exactly foundusing Bethe ansatz approach [46]. Figure 17 shows howthe difference of the total energy of the bath and its ther-modynamic value E(N) − E(N = ∞) depends on thenumber of particles N . Even if the convergence in theenergy per particle E(N)/N has a fast 1/N2 dependencefor large system sizes, in the total energy the asymptoticdependence is weaker, as can be seen from the 1/N fitin Fig. 17. It should be noted, that the polaron energyis obtained from the total energy E(N), which divergeslinearly with number of particles N . This imposes severerequirements for the numerical accuracy goal, especiallywhen large system sizes are used.
For the same high density, n0|aBB| = 144, we now adda finite interaction with the impurity. We consider theextreme case of η = ∞, corresponding to the strongestinteraction. The resulting energy scaling obtained fromQMC calculations is reported in Fig. 18. By comparisonwith the non-interacting case of η = 0 shown in Fig. 17one can see that the effect is greatly enhanced by a finiteinteraction with the impurity, as can be perceived bycontrasting the scales of the vertical axis. At the sametime, the asymptotic 1/N convergence law is clearly seen.
The polaron energy which we report in the thermo-dynamic limit in the main part of the paper is ob-tained from DMC by adjusting a 1/N fit to system sizesN = 50, 100, 150, 200, 250.
26
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