Theory of Excitation of Rydberg Polarons in an Atomic Quantum Gas

Theory of excitation of Rydberg polarons in an atomic quantum gas

R. Schmidt,1, 2, 3, ∗ J. D. Whalen,4 R. Ding,4 F. Camargo,4 G. Woehl Jr.,4, 5 S. Yoshida,6 J. Burgdörfer,6 F. B. Dunning,4 E. Demler,2 H. R. Sadeghpour,1 and T. C. Killian4 1ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA 2Department of Physics, Harvard University, Cambridge, MA 02138, USA 3Institute of Quantum Electronics, ETH Zürich,CH-8093 Zürich,Switzerland 4Department of Physics & Astronomy, Rice University, Houston, TX 77251, USA 5Instituto de Estudos Avan¸cados,12.228-001 SãoJosédos Campos, SãoPaulo, Brazil 6Institute for Theoretical Physics, Vienna University of Technology, Vienna, Austria, EU (Dated: January 19, 2018) We present a quantum many-body description of the excitation spectrum of Rydberg polarons in a Bose gas. The many-body Hamiltonian is solved with a functional determinant approach, and we extend this technique to describe Rydberg polarons of finite mass. Mean-field and classical descriptions of the spectrum are derived as approximations of the many-body theory. The various approaches are applied to experimental observations of polarons created by excitation of Rydberg atoms in a strontium Bose-Einstein condensate.

I. INTRODUCTION FDA theory for both the observed few-body molecular spectra and the many-body polaronic states is excellent. When an impurity is immersed in a polarizable In the companion paper [9], we provide experimental medium, the collective response of the medium can form evidence for the observation of polarons created by ex- quasi-particles, labelled as polarons, which describe the citation of Rydberg atoms in a strontium Bose-Einstein dressing of the impurity by excitations of the background condensate, with an emphasis on the determination of medium. Polarons play important roles in the con- the excitation spectrum in the absence of density inho- duction in ionic crystals and polar semiconductors [1], mogeneity. Here we provide a detailed discussion of the spin-current transport in organic semiconductors [2], dy- theoretical methods and experimental analysis. namics of molecules in superfluid helium nanodroplets [3–5], and collective excitations in strongly interacting fermionic and bosonic ultracold gases [6–8]. II. BOSE POLARON HAMILTONIANS In this paper, which accompanies the publication heralding the observation of Rydberg Bose polarons [9], With ultracold atomic systems, various quantum im- we present details of calculations and the interpreta- purity models can be studied in which an impurity in- tion of this observation as a new class of Bose po- teracts with a bosonic bath. Here we focus on models 3 larons, formed through excitation of Sr(5sns S1) Ry- that follow from the general Hamiltonian describing an dberg atoms in a strontium Bose-Einstein condensate impurity of mass M interacting with a gas of weakly in- (BEC). We begin with a general outline of different po- teracting bosons of mass m: laron Hamiltonians, and construct the Rydberg polaron X I † X † gbb X † † Hˆ = dˆ dˆ + aˆ aˆ + aˆ aˆ aˆ 0 aˆ Hamiltonian used in this work. The spectral response p p p k k k 2V k0+q k−q k k function in the linear response limit is derived and the p k kk0q many-body mean field shift of the spectral response is 1 X ˆ† ˆ † + V (q)d 0 dk0 aˆ aˆk . (1) obtained. The mean field theory particularly fails to V k −q k+q kk0q describe the limits of small and large detuning, where ˆ the detailed description of quantum few- and many-body HIB processes is particularly relevant. These processes are Here, the first two terms describe the kinetic energy of fully accounted for by the bosonic functional determinant ˆ the impurities (dp) and bosons (âk) with dispersion rela- approach (FDA) [10] that solves an extended Fröhlich I p2 k2 tions p = 2M and k = 2m , respectively (unless explic-

arXiv:1709.01838v3 [physics.atom-ph] 17 Jan 2018 Hamiltonian for an impurity in a Bose gas. In the fre- itly stated we set = 1). The third term accounts for quency domain, the FDA predicts a gaussian shape for ~ the interaction between the bosons. Assuming weak cou- the intrinsic spectrum, which is a hallmark of Rydberg pling between bosons, the microscopic coupling constant polarons. A classical Monte Carlo simulation [11] which is given by the relation g = πa /m, with a the s-wave reproduces the background spectral shape is shown to bb bb bb scattering length describing the low-energy boson-boson miss spectral features arising from quantization of bound interactions. The last term, Hˆ describes the impurity- states. Agreement between experimental results and IB boson interaction in momentum space, which, in the real space, reads Z ˆ 3 3 0 0 0 ∗ corresponding author: [email protected] HIB = d rd r nˆI (r ) V (r − r)n ˆB(r). (2) 2

ˆ† ˆ 1 P ˆ† ˆ −iqr Heren Î (r) = ψI (r)ψI (r) = V kq dk+qdke and [12]: ˆ† ˆ 1 P † −iqr nˆB(r) = ψB(r)ψB(r) = V kq aˆk+qaˆke are the im- 2 pˆ X 1 X ˆ purity and boson density, respectively. Hˆ = + ω ˆb† ˆb + √ g(q)e−iqR ˆb† + ˆb 2M k k k q −q As we consider the limit of a single impurity it is con- k V q venient to switch to the first quantized description of (6) the impurity, which is characterized by its position and momentum operator Rˆ and pˆ. The density becomes In initial attempts to describe impurities in weakly in- (3) nÎ (r) → δ (r − Rˆ ) and Eq. (1) takes the form teracting Bose gases, this Hamiltonian was used for systems close to Feshbach resonances [13, 14]. However, as 2 pˆ X † gbb X † † shown in [15], the Fröhlich Hamiltonian alone is insuffi- Hˆ = + aˆ aˆ + aˆ aˆ aˆ 0 aˆ (3) 2M k k k 2V k0+q k−q k k cient to describe impurities that interact strongly with k kk0q atomic quantum gases (for an explicit third-order per- 1 X ˆ † turbation theory analysis see Ref. [16]). Indeed the ma- + V (q)e−iqRaˆ aˆ . V k+q k jor theoretical shortcoming of Eq. (6) is that from this kq model one cannot recover the Lippmann-Schwinger equation for two-body impurity-bose scattering. Hence it fails From this Hamiltonian various polaron models can to describe the underlying two-body scattering physics be derived, and we briefly explain their relevance and between the impurity and bosons including molecule for- regimes of validity. At T = 0 a large fraction of bosons mation. is condensed in a BEC. This condensate can be regarded Similarly the Fröhlich Hamiltonian cannot account for as a coherent√ state such that the zero-momentum mode the intricate dynamics leading to the formation of Ry- haˆki = N0δk,0 takes on a macroscopic expectation dberg polarons. For such strongly interacting systems value. the inclusion of the last term in Eq. (5) becomes cru- Within the Bogoliubov approximation the bosonic cre- cial. This term accounts for pairing of the impurity with ation and annihilation operators in Eq. (3) are expanded √ the atoms in the environment and accounts for the de- in fluctuations Bˆp around this expectation value N0 tailed ‘short-distance’ physics of the problem, which is and terms of higher than quadratic order are neglected completely neglected in the Fröhlich model that is tai- in the resulting Hamiltonian. The purely bosonic part of lored for the description of long-wave length (low-energy) the Hamiltonian can then be diagonalized by the Bogoli- physics. ubov rotation So far it has been experimentally verified that the inclusion of this term is relevant for the observation of Bose ˆ ˆ ∗ ˆ† ˆ† ∗ ˆ† ˆ Bp = upbp + v−pb−p , Bp = upbp + v−pb−p, (4) polarons [7, 8] where the impurity-Bose interaction can be modeled by a potential that supports only a single, where ω = p ( + 2g ρ) is the Bogoliubov disper- weakly bound two-body molecular state. In the present k p p bb work we encounter a new type of impurity problem where q p+gbbρ 1 sion relation and up, v−p = ± ± . The den- 2ωp 2 the impurity is dressed by large sets of molecular states sity of the homogeneous condensate is given by ρ. The that have ultra-long-range character. This yields the transformation (4) yields the Bose-impurity Hamiltonian novel physics of Rydberg polarons that is beyond the physics of Bose polarons so far observed in experiments pˆ2 X and discussed in the literature. To account for the rele- Hˆ = + ω ˆb† ˆb 2M k k k vant Rydberg molecular physics theoretically, we analyze k the full Hamiltonian Eq. (5). 1 X −iqRˆ ˆ† ˆ We note that the physics of the Rydberg oligomer + ρV (0) + √ g(q)e bq + b−q V q states is different from that of Efimov states in the Bose polaron problem [15, 17–19]. While Efimov states are Fröhlich interaction also multi-body bound states, they arise due a quantum 1 X ˆ + V (q)e−iqRBˆ† Bˆ (5) anomaly of the underlying quantum field theory [20–22]. V k+q k kq Indeed, the Efimov effect [23–25] gives rise to an infinite series of three-body states (and also states of larger atom extended Fröhlich interaction number [26, 27]) that respect a discrete scaling symme- where we introduced the ‘Fröhlich coupling’ g(q) = try. The Efimov effect arises for resonant short-range q two-body interactions in three dimensions. In previous ρq V (q) and in the last term we kept the untrans- ωq experiments studying Bose polarons close to a Feshbach formed expression for notational brevity. In Eq. (5) the resonance it was found that these Efimov states do not term ρV (0) = ρ R d3rV (r) describes the mean-field en- strongly influence the polaron physics [7, 8]. In our case ergy shift of the polaron in the Born approximation. Ne- the multi-molecular states are not related to the Efimov glecting the constant mean-field shift and the last term effect and dimer, trimer, etc. states are rather comprised in Eq. (5) one arrives at the celebrated Fröhlich model of nearly independently bound two-body molecular states 3

impurity Green‘s function impurity A Fourier transformation of the parameter αk to real = space reveals the creation of density modulations in the BEC: boson medium, which represent the formation of a dressing + + cloud that is build around the impurity particle. In Fröhlich model fact, wave functions of the type in Eq. (7) were found + + . . . to yield a rather accurate description of impurities cou- pled to a fermionic bath of cold atoms via Feshbach res- + onances [29–33] and also describe exciton-impurities in

beyond Fröhlich / two-dimensional semiconductors [34]. When truncated at + molecule formation the one-excitation level, such an ansatz leads to limited agreement with experimental observations for impurities + + . . . in a Bose gas [35, 36]. Dressing can also be understood from a perturbative FIG. 1. Illustration of polaron formation in terms of expansion in terms of Feynman diagrams as shown in Feynman diagrams. The impurity Green’s function fully Fig. 1. Here solid lines represent the propagating impu- determines the impurity spectral function and hence its ab- rity, and the dashed lines denote Bosons that are excited sorption spectrum, mobility, and other single particle proper- from the BEC. The Fröhlich model considers only pro- ties. The impurity properties are changed due to the dressing cesses where bosons are excited out of the condensate by bosonic excitations signified by the self-energy corrections and then re-enter the BEC in their next scattering event depicted here as diagrams. The extended Fröhlich model fea- with the impurity. tures two types of excitations. In “Fröhlich processes” bosons However, these ‘Fröhlich scattering processes’ can nei- are excited out of the BEC by scattering with the impurity √ (square box). Such scattering contributes a factor of ρ. ther account for molecular bound-state formation, nor for The excited bosons then reenter the BEC after an additional intricate strong-coupling physics found close to Feshbach scattering off the impurity. A second type of process (gray resonances. To describe these phenomena, the last term disks) is ignored in the Fröhlich model. Here an excited bo- in Eq. (5) has to be considered. This term allows for scat- son can scatter off the impurity repeated times and in each tering of bosons where an excited boson does not directly such scattering process it changes its momentum. These pro- reenter the BEC but can scatter off the impurity arbi- cesses account for strong coupling physics in cold atomic gases trarily many times. In this process, illustrated as gray including bound state formation. disk in Fig. 1, the boson changes its momentum. The infinite repetition of such scattering processes represents the Lippmann-Schwinger equation in terms of Feynman that feature only very weak three- and higher-body cor- diagrams and accounts for molecular bound state forma- relations. In contrast, in Efimov physics, a single impu- tion. rity potentially mediates strong correlations between two particles in the Bose gas. Due to large binding energies of Rydberg molecular states, this induced interaction is III. RYDBERG POLARON HAMILTONIAN expected to be small for Rydberg polarons [28]. We now specialize to the case of Rydberg impurities. A. Polaron formation When a bath atom is excited into a Rydberg state of principal quantum number n, it interacts with the sur- rounding ground-state atoms through a pseudopotential, The formation of a polaron can be understood as dress- first proposed by Fermi [37], in which molecular binding of the impurity by excitations of the bosonic bath, ing occurs through frequent scattering of the nearly free which entangles the momentum of the impurity with that and zero-energy Rydberg electron from the ground-state of the bath excitations. This can be illustrated with a atom. In this picture, the Born-Oppenheimer potential simple wave function expansion for a polaron of zero mo- for a ground-state atom at distance r from the Rydberg mentum impurity, retaining s-wave and p-wave scattering partial √ X ˆ† ˆ waves, is given as [37–41] |Ψi = Z |BECi |p = 0iI + αkb−kb0 |BECi |kiI k 2 2 2π~ 2 6π~ 3 −→ 2 X ˆ† ˆ† ˆ ˆ VRyd(r) = as|Ψ(r)| + ap|∇Ψ(r)| , (8) + αkqb−kbk−qb0b0 |BECi |qiI + ... (7) me me kq where we kept explicit factors of ~ and Ψ(r) is the where |piI denotes impurity momentum states. The un- Rydberg electron wave function, as and ap are the perturbed impurity-bath state in the first terms becomes momentum-dependent s-wave and p-wave scattering supplemented by ‘particle-hole’ fluctuations of the BEC lengths, and me is the electron mass. When as < 0, V (r) state given by the second, etc. terms. These terms de- can support molecular states with one or more ground- scribe the polaronic dressing of the impurity by bath state atoms bound to the impurity [38, 42, 43]. A Ry- excitations that leads to the formation of the polaron. dberg polaron is formed when the Rydberg impurity is 4 dressed by the occupation of a large number of these state |nsi the potential V (q) is switched on. In experi- bound states in addition to finite momentum states. This ments [9], transitions between both states are driven by process gives rise to an absorption spectrum of a distribu- a two-photon excitation. tion of molecular peaks with a Gaussian envelope, which Within linear response, the corresponding absorption is the key spectral signature of Rydberg polaron forma- of laser light at frequency ν is given by Fermi’s Golden tion. rule The large energy scale of the Rydberg molecular states X 2 involved in the formation of Rydberg polarons allow us to A(ν) = 2π wi| hf| VˆL |ii | × δ(ν − (Ei − Ef )). (10) simplify the extended Fröhlich Hamiltonian Eq. (5): the if typical energy range of Rydberg molecules is 0.1−10 MHz for high quantum number n, while the typical energy Here the sum extends over all initial and final states of scale for Bose-Bose interactions is 1-10 KHz. Therefore the impurity plus bosonic bath that fulfill Hˆ0 |ii = Ei |ii bosons that are bound to a Rydberg impurity probe mo- and Hˆ |fi = Ef |fi, where Hˆ is given by Eq. (9) so that mentum scales deep in the particle branch of the Bogoli- the final states |fi are eigenstates in the presence of the ubov dispersion relation, in a regime where the Bogoli- strong perturbation from Rydberg-boson interactions. In ubov factors up = 1 and vp = 0. Hence we can neglect the states |ii, all atoms (including the impurity atom) are the Bose-Bose interactions and Eq. (5) reduces to the in the |5si state, while in final states |fi the impurity is in simplified, extended Fröhlich model [11] the atomic |nsi state. The laser operator that drives the 2 two-photon transition between these two atomic states pˆ X † 1 X −iqRˆ † ˆ Hˆ = + aˆ aˆ + V (q)e aˆ aˆ |5si and |nsi, is given by VL ∼ |nsi h5s| + h.c.. 2M k k k V k+q k k kq Using the Fourier-representation of the delta function 2 in Eq. (10), one can show that the absorption spectrum pˆ X = + ˆb† ˆb + ρV (0) follows from [10] (for details see [45]) 2M k k k k Z ∞ 1 X √ −iqRˆ ˆ† ˆ iνt + √ ρV (q)e bq + b−q A(ν) = 2 Re dt e S(t) (11) 0 V q Fröhlich interaction with the many-body overlap (also called the Loschmidt 1 X −iqRˆ ˆ† ˆ echo) S(t). + V (q)e bk+qbk . (9) V In the general case of finite temperature, the wi in kq Eq. (10) are the thermodynamic weights of each initial extended Fröhlich interaction state given by the diagonal elements of the density matrix −βH0 where in the second equivalent expression we expanded ρîni = e /Zp, for sample temperature kBT = 1/β † and partition function Zp. For finite temperature the the boson operatorsa ˆk around their expectation value √ Loschmidt echo is then given by N0 which highlights the connection to the Fröhlich model. In order to describe Rydberg impurities the inter- iHˆ0t −iHtˆ action V (q) is given as the Fourier transform of the Ryd- S(t) = Tr[ˆρini e e ] (12) R 3 iq·r berg molecular potential V (q) = d rVRyd(r)e . The molecular potentials and interacting single-particle wave where Tr denotes the trace over the complete many-body Fock space. This expression shows that the overlap func- functions (|βii) are calculated as described in [10, 43]. We note that the depletion of a BEC by a single electron in tion S(t) encodes the non-equilibrium time evolution of a BEC has been studied in [44] including only the linear the system following a quantum quench represented by Fröhlich term in Eq. (9). While such an approach can the introduction of the Rydberg impurity at time t = 0. approximately account for the rate of depletion of the From this time evolution then follows the absorption condensate it fails to describe the formation of molecules spectrum by Fourier transformation. that is essential for the formation of Rydberg polarons. The relevant time scales for the dynamics of Rydberg impurities in a BEC is given by the Rydberg molecular energies which exceed both the temperature scale kBT IV. LINEAR-RESPONSE ABSORPTION FROM and the energy scale associated with boson-boson inter- QUENCH DYNAMICS actions. Thus temperature and boson interaction effects can be neglected in the calculation of S(t) for Rydberg One way to probe polaron structure and dynamics is polarons. In this limit the initial state of the system is absorption spectroscopy. Here one utilizes the fact that given by |ii = |BECi ⊗ |p = 0iI ⊗ |5siI and Rydberg before being excited to a Rydberg state, an atom is in polarons are well described by the T=0 limit of Eq. (12), its ground state |5si, and its interaction with the sur- iHˆ0t −iHtˆ rounding Bosons is negligible. The system is described S(t) = I hp = 0| hΨBEC| e e |ΨBECi |p = 0iI , by the Hamiltonian Hˆ0 given by the first two terms in (13) Eq. (9). In contrast, when the atom is in its Rydberg where |ΨBECi represents an ideal BEC of atoms. 5

The accurate calculation of the absorption response An intuitive way to express this shift is in terms of presents a formidable theoretical challenge. In the fol- an effective s-wave electron-atom scattering length that lowing we will present three different approaches of dif- reflects the average of the interactions over the Rydberg ferent degrees of sophistication. In a simple approxima- wave function [47], tion one may employ mean-field theory (Section V) where Z 2πm the impurity-boson interactions are solely described by 3 0 e 0 as,eff = d r 2 V (r ), (18) the mean-field result ρV (0) in Eq. (9). In this approach h all quantum effects and fluctuations are neglected. Fur- yielding thermore one may employ a classical stochastic model h2a (Section VII), which at least can treat fluctuations to a h∆(r) = s,eff ρ(r). (19) certain degree. Finally, we employ a functional deter- 2πme minant approach, which was developed in [10] and which as,eff varies with principal quantum number through the we review in Section VI. This approach treats all interac- variation in the Rydberg electron wavefunction and the tion terms in Eq. (9) fully on a quantum level and allows classical dependence of the electron momentum on posi- us to explain the intricate Rydberg polaron formation tion. Figure 2 shows calculated values of as,eff for the 3 dynamics observed in our experiments. interaction of Sr(5sns S1) Rydberg atoms with background strontium atoms [43]. To obtain this result, 0 2 Eq. (18) is only integrated over |r | > 0.06n a0 because V. MEAN FIELD APPROACH the approximation breaks down near the Rydberg core, where, among other modifications, the ion-atom polar- The simplest treatment of the polaron excitation spec- ization potential becomes important. trum is to neglect all interaction terms other than ρV (0) Figure 3 shows the mean-field description of exper- 3 in Eq. (9), which yields the mean-field approximation. imental results for excitation of Sr(5sns S1) Rydberg Hence at a given constant density ρ(r) in the trap the atoms in a strontium Bose-Einstein condensate (BEC). absorption spectrum is obtained by Fourier transforma- For the prediction of the absorption response the effect tion of (cf. Eq. (13)) of temperature enters by determining the local density of the cloud. In fact, in the local-density-approximation S(t, ~r) = e−iρ(r)V (0)t = e−i∆(r)t, (14) model of the impurity excitation spectrum (Eq. (17)), the density of atoms in the trap, ρ(r) = ρBEC (r)+ρth(r), has which yields a delta function Rydberg-absorption re- contributions from thermal, non-condensed atoms and sponse A(ν, r) = δ(ν − ∆(r)) for a given local, homo- condensed atoms. The contribution from the thermal gas geneous density ρ(r) at a detuning from the atomic tran- is restricted to the sharp peak near zero detuning and a sition very small contribution towards the red in each data set Z in Fig. 3. Thus the BEC and thermal contributions can ∆(r) = ρ(r)V (q = 0) = ρ(r) d3r0V (r − r0). (15) be calculated separately in the mean-field approximation. The profile of the condensate density is well- In the experiment, the density ρ(r) varies with position approximated for our conditions with a Thomas-Fermi r. Since the excitation laser illuminates the entire atomic (TF) distribution for a harmonic trap [48]. If we ne- cloud it excites Rydberg atoms in regions of varying den- glect the contribution of thermal atoms to the density, sity. To theoretically model the resulting average over various contributions from the atomic cloud, we perform 12 a local density approximation (LDA), which assumes the density variation is negligible over the range of the Ryd- 11 berg interaction V (r) given by Eq. 8. ) 0

a 10

The spectrum for creation of a Rydberg impurity is f o

then given by s

t 9 i n

Z u (

3 8 f A(ν) ∝ dr ρ(r)A(ν, r). (16) f e , s

a 7 In the mean-field approximation this yields the response 6 Z 3 A(ν) ∝ d r ρ(r)δ(ν − ∆(r)) (17) 5 20 30 40 50 60 70 80 n* for laser detuning ν from unperturbed atomic resonance. Note that the mean-field treatment is similar to the de- ∗ scription of 1S − 2S spectroscopy of a quantum degener- FIG. 2. Calculated values of as,eff from Eq. (18) versus n = ate hydrogen gas given in [46]. n − δ, where δ = 3.371 is the quantum defect. 6

0.08 0.08

) ) ) 0.3 s s s t t t i i i

n 0.06 n n

u u 0.06 u

. . .

b b b 0.2 r r r

a 0.04 a a ( ( (

0.04 ) ) ) ( ( (

A A A 0.1 0.02 0.02

0.00 0.00 0.0 30 20 10 0 30 20 10 0 30 20 10 0 Laser Detuning (MHz) Laser Detuning (MHz) Laser Detuning (MHz)

3 FIG. 3. Mean-field description of the excitation spectrum of Sr(5sns S1) Rydberg atoms in a strontium Bose-Einstein condensate (BEC) for (Left) n=49, (Middle) n=60, (Right) n=72. Symbols are the experimental data. Blue bands represent the confidence interval for the mean-field fit of the BEC contribution to the spectrum corresponding to the uncertainties in parameters given in Tab. I. The central black line indicates the best fit. The dashed red line is the mean-field prediction for the contribution from the low-density region of the atomic cloud formed by thermal atoms. This contribution is calculated using the parameters given in Tab. I including a sample temperature adjusted to reproduce the observed BEC fraction.

−3 n ∆max [MHz] η NBEC ω/¯ 2π [Hz] ρmax [cm ] µ/kB [nK] ∆max/∆˜ max T [nK] 49 −19 ± 2 0.72 ± 0.06 (2.8 ± 0.3) × 105 107 ± 10 (3.2 ± 0.3) × 1014 (150 ± 20) 0.95 ± 0.1 160 ± 10 60 −24 ± 1 0.77 ± 0.03 (3.7 ± 0.4) × 105 112 ± 10 (3.8 ± 0.4) × 1014 (180 ± 20) 0.93 ± 0.05 170 ± 10 72 −24 ± 1 0.80 ± 0.03 (3.6 ± 0.4) × 105 117 ± 10 (3.9 ± 0.4) × 1014 (190 ± 20) 0.88 ± 0.05 170 ± 10

TABLE I. Parameters for data and fits shown in Fig. 3: Peak detuning of the mean-field fit (∆max) and BEC fraction (η) are determined from the spectra. Number of atoms in the condensate (NBEC ) and mean trap frequencyω ¯ are determined from time-of-flight-absorption images and measurements of collective mode frequencies for trapped atoms respectively, and they determine the peak condensate density (ρmax) and chemical potential (µ). The second-to-last column provides the ratio 2/5 ω 15Nabb m as,eff of ∆max to ∆˜ max = , the peak shift predicted using information independent of the mean-field fits. 4π aho me abb The sample temperature (T ) is the result of a fit of the observed BEC fraction using a numerical calculation of the number of non-condensed atoms in the trap fixing all the BEC parameters at values given in the table. the condensate contribution to the spectrum is thus temperature of very cold Bose gases. Once the BEC parameters are set, for a consistency check, we then cal- −ν p AMF(ν) ∝ NBEC 2 1 − ν/∆max, (20) culate the contribution to the spectrum from thermal ∆max atoms using the mean-field approximation and adjust- for ∆max ≤ ν ≤ 0 and zero otherwise [46]. Here, ing the sample temperature to match the BEC fraction. has,eff Additional details of the fitting process are described in ∆max = ρmax, and the peak BEC density is 2πme App. A. ρmax = µTF /g, where the chemical potential is µTF = 2/5 2 (~ω/2) (15NBEC abb/aho) and g = 4π~ abb/m for har- 1/2 monic oscillator length aho = (~/mω) and mean trap 1/3 radial frequency ω = (ω1ω2ω3) . This functional form is used to fit several data sets in Fig. 3 with ∆max and the overall signal amplitude as the only fit parameters. When For n = 72, the mean-field approach is quite accurate frequency is scaled by ∆max, the mean-field prediction for in describing the overall shape of the response across the a BEC in a harmonic trap is universal, making ∆max an entire spectrum. However, at lower principal quantum important parameter for describing the spectrum. The numbers, the data and fit deviate significantly, especially resulting fit parameters are given in table I. We ascribe at larger detunings. Generally, mean-field fails both at the difference between the mean-field fit and the observed small detuning (i.e. for response from low-density re- spectra at small detuning to the contribution from the gions of the cloud), where the deviation arises from the low-density region of the atomic cloud formed by thermal formation of few-body molecular states, and at large de- atoms. From the ratio of the areas of these two signal tuning, where fluctuations in the macroscopic occupation components, we extract the condensate fraction. Given of molecular states become important. Fluctuations cor- the clean spectral separation between the contributions respond to a spread in binding energies of the polaron from thermal and condensed atoms, this is a promising states excited for a given average density. All of these technique for measuring very small thermal fractions and effects are neglected in the mean-field description. 7

VI. FUNCTIONAL DETERMINANT which is inserted in the time evolution APPROACH −1 −iHtˆ −1 S(t) = hp = 0|I hΨBEC| UU e UU |ΨBECi |p = 0iI −iHˆt The mean-field approach is insufficient to describe the = hp = 0|I hΨBEC| e |ΨBECi |p = 0iI . (22) main features found in the absorption spectrum both ˆ at small and large detuning from the atomic transition. In the first line we used that the term eiH0t can be For instance its failure at low detuning (low densities) dropped as the initial state is a zero energy state. Fur- has its origin in the formation of Rydberg molecular thermore, in the second line we made use of the fact that states, which is not described in mean-field theory. The the total boson momentum of the BEC state is zero and low density regime can be most conveniently studied in we defined the transformed Hamiltonian detail by working at a low principal quantum number 2 (here n = 38) where only few atoms are in the Ry- P † pˆ − kaˆ aˆk dberg orbit, and the signal is thus dominated by the −1 k k X † Hˆ = U HUˆ = + kaˆ aˆk low-density response. To calculate S(t) efficiently we 2M k k evaluate it within the FDA. In [10] the FDA was de- 1 X veloped to include (for impurities of infinite mass) also + V (q)â† aˆ (23) V k+q k the finite-temperature corrections that are however ir- kq relevant for our experiment. Thus we can restrict our- selves to the description of the zero-temperature response By virtue of the transformation U the impurity coordi- given by Eq. (13), where the bosonic ground state can be nate is eliminated in the interaction and only the impu- expressed as a state of fixed particle number |BECi = rity momentum operator pˆ remains. It thus commutes √ † N with the many-body Hamiltonian and is replaced by a (â0)0 / N0! |0i. For sufficiently large particle number we may equally use its representation as a coherent state c-number, pˆ → p. Since in our case we are interested √ † in the polaron at zero momentum, p → 0. After normal |BECi = exp[ N0(â0 − aˆ0)] |0i where |0i is the boson vacuum. The FDA, as developed in Ref. [10], did not in- ordering, we arrive at clude the description of impurity recoil which is, however, 0 2 ˆ X kk † † X k † essential for an accurate prediction of Rydberg molecular H = aˆk0 aˆkaˆkaˆk0 + aˆkaˆk 2M 2µred spectra. In the following subsection we develop a exten- kk0 k sion of the FDA approach that overcomes this limitation. 1 X + V (q)â† aˆ (24) V k+q k kq

The above transformation has the effect that the boson dispersion relation → k2/2µ now has the re- A. Canonical transformation: mobile Rydberg k red impurity duced mass µred = mM/(M + m) of boson-impurity partners, and the Hamiltonian includes an induced interaction of bosons described by the first term in Eq. (24). The FDA can be applied to time evolutions described Due to spherical symmetry and the large energy scale by a Hamiltonian bilinear in creation and destruction op- of the Rydberg impurity-Bose gas interaction, we may erators. This is fulfilled for the Hamiltonian Eq. (9) in neglect this term. The reliability of this approximation the case of an infinitely heavy impurity M = ∞. How- has been demonstrated in recent work of some of the au- ever, for a mobile Rydberg impurity, the presence of the thors [36] where a time-dependent variational principle non-commuting operators pˆ and ˆr effectively gives rise has been applied to the evaluation of the time-evolution to non-bilinear terms. of Eq. (24). In the approximation where the time- To remedy this challenge we combine the FDA with dependent wave function |Ψ(t)i evolved by exp(−iHˆt) a canonical transformation. Here we focus on the zero- is taken to be a product of coherent states of the form P (γ (t)â† +h.c.) temperature case. In this case the response is obtained |Ψ(t)i = e k k k |0i, one finds from the equation from the time evolution using Eq. (9), where the Hamil- of motions of the variational parameters γq that the ex- tonian includes bosonic and impurity operators. To deal P kk0 † † pectation value hΨ(t)| kk0 2M aˆk0 aˆkaˆkaˆk0 |Ψ(t)i always with the impurity motion we perform a canonical trans- remains zero and this term hence does not contribute to formation first proposed by Lee, Low, and Pines [49] the dynamics. For our case of Rydberg impurities the which effectively transforms into the system comoving FDA solution presented here reproduces the variational with the impurity. To this end we define the translation result of Ref. [36] and again shows excellent agreement operator [49] with experimental data attesting a postiori to the accu- racy of the method and the neglect of the first term in ( ) Eq. (24). To which extent this remains valid for other ˆ X † Bose polaron scenarios is an open question and subject U = exp iR kaˆkaˆk (21) k to ongoing theoretical studies [50]. In summary, we fi- 8

(a) (b) 1.0

10 - 3

|S(t)| 0.5

- 7 trast 10 0.0 co n

10 - 11 phase - 0.5 Ramse y 10 - 15 - 1.0 0.0 0.5 1.0 1.5 2.0 0.0 0.1 0.2 0.3 0.4 time (µs) time (µs)

FIG. 4. Exemplary Ramsey signal S(t) = |S(t)|eiϕ(t) including contrast |S| and phase ϕ underlying the calculation of the Rydberg polaron absorption spectrum for n = 49 at peak density in absence of a ﬁnite Rydberg lifetime. A fast decay visible in the contrast accompanies fast oscillations of the full complex system. The combination of both gives rise to the distinct Rydberg polaron features of molecular peaks that are distributed according to a gaussian envelope. The Ramsey signal thus provides an alternative pathway for observing Rydberg polaron formation dynamics in real-time. The time scales of this coherent dressing dynamics are ultrafast compared to the typical time scales of collective low-energy excitations of ultracold quantum gases allowing study of ultrafast dynamics in a new setting. nally arrive at the Hamiltonian S(t) becomes [10]

N 2   X k † 1 X † ˆ X 2 i(s−ωβ )t H = aˆkaˆk + V (q)âk+qaˆk |nsi hns| (25) S(t) =  |hβ|si| e  , (26) 2µred V k kq β whose dynamics we simulate to obtain Rydberg polaron where |si denotes the lowest single particle eigenstate [55] ˆ excitation spectra. Note, in Eq. (25), we make explicit of h0, and N is the number of atoms in the spherical box that the impurity-boson interaction is only present when of radius R chosen to reproduce a given local experimen- 6 the impurity is in its excited atomic Rydberg state |nsi. tal density of a Bose gas. We choose R = 10 a0 with In this state the Rydberg electron scatters off ground a0 the Bohr radius so that we find negligible finite size state atoms in the environment, leading to the strong corrections. We emphasize again that the temperature Rydberg Born-Oppenheimer potential V (q). For sim- enters the calculation only in determining the local den- plified notation we will now switch again to the symbol sity of the atomic cloud. For the dynamics and thus the Hˆ → Hˆ . prediction of the absorption spectra at a given local density the temperature T is irrelevant. This is due to the fact that the energy scales of the dynamics of the system is determined by the binding energy of Rydberg molecu- B. Time-domain S(t) lar states, which exceed kBT . We obtain absorption spectra by predicting the many- body overlap, or Loschmidt echo, S(t). This calculation The strength of the FDA is that it allows one to express corresponds to solving the full time evolution of the sys- overlaps of many-body states in terms of single-particle tem following a quantum quench, where at time t = 0 the eigenstates [10, 51–54]. In our case this implies that we Rydberg impurity is suddenly introduced into the BEC. must calculate single particle eigenstates and energies of The overlap S(t) describes the dephasing dynamics of the ˆ the free and ‘interacting’ Hamiltonian, i.e. h0 |ni = n |ni many-body system and thus the evolution of the dress- ˆ ˆ ˆ and h |βi = ωβ |βi, respectively (h0 and h are the single- ing of the Rydberg impurity by bath excitations. It is particle representatives of the many-body operators Hˆ0 one of the virtues of cold atomic systems that the signal and Hˆ , respectively). The single-particle eigenstates and S(t) can be directly measured experimentally by Ram- energies |βi and ωβ are calculated using exact diagonal- sey spectroscopy where via a π/2 rotation at time t = 0, ization. Here we solve the radial Schrödingerequation the impurity is prepared in a superposition state of |5si for a spherical box of radius R discretized in real-space. and |nsi. Following a time evolution of duration t, a fur- Since the BEC is initially in a zero angular state and ther π rotation is performed and σz is measured. This the Rydberg molecular potential is spherically symmet- gives the Ramsey contrast |S(t)|. Changing the phase ric we can restrict the analysis to single-particle states of the final π rotation, the full signal S(t) = |S(t)|eiϕ(t) with zero-angular momentum. We include the 300 en- can be measured [45, 56, 57]. In Fig. 4 we show the ergetically lowest eigenstates, which leads to convergent predicted Ramsey signal that underlies the calculation of results. In terms of these states and energies the overlap the Rydberg polaron absorption spectrum for n = 49 at 9

takes place on a µs timescale, which is short compared to typical ultracold-atom time scales.

C. FDA absorption spectra

The accurate description of molecular formation underlies the precise analysis of Rydberg-polaron absorption response. As the principal quantum number is in- creased to n = 49, more atoms are situated on average within the Rydberg orbit and many-body effects become relevant. The density-averaged prediction using FDA is shown in Fig. 6. For all density-averaged spectra, we rely on density profiles such as shown in Fig. 5 for pa- FIG. 5. Density profile of bosonic atoms in a harmonic trap rameters used for the FDA prediction of the n = 60 5 5 for parameters T = 180 nK, NBEC = 3.7·10 , Ntot = 5.2·10 , Rydberg-absorption spectrum. We include Hartree-Fock ωr = 104 Hz, and ωa = 111 Hz, used for the prediction of the corrections [48, 58] as discussed in Appendix A. Further- spectrum of the n = 60 Rydberg excitation shown in Fig. 8. more, the absorption spectrum is calculated from the In blue is shown the contribution from the condensed atoms, Fourier transformation according to Eq. (11), where in while the red area represents the contribution from thermal the time-evolution of S(t) we choose a temporal cutoff atoms. determined by the experimental laser-pulse duration, and in the Fourier transform we account for the finite laser peak density, here shown in absence of a finite Rydberg line width (400 kHz). lifetime. We observe a fast decay in the contrast that In Fig. 7, we show again the n = 49 absorption spec- indicates strong dephasing and thus efficient creation of trum obtained theoretically, but with a simulated laser polaron dressing by particle-hole excitations. This de- linewidth of 100 kHz (solid blue line). The shaded region cay is accompanied by fast oscillations of the complex shows the result of an FDA calculation of the spectrum signal S(t) as visible in the shown evolution of the Ram- for a central region of the atomic cloud with roughly con- sey phase ϕ(t) restricted to the branch (−π, π). The stant density. We find that the signal from this region combination of the oscillations at the Rydberg molecu- shows a series of molecular peaks whose weights follow a lar binding energies and the decay of the Ramsey signal Gaussian envelope. This distribution of molecular peaks S(t) gives rise to the distinct Rydberg polaron features is one of the key signatures of Rydberg polarons as pre- of molecular peaks that are distributed according to a dicted theoretically in the accompanying work [10]. The gaussian envelope, to be discussed below. The Ramsey comparison of experiment and theory in Fig. 6 in turn signal thus provides an alternative pathway to observ- shows that the response is indeed composed of many in- ing Rydberg polaron formation dynamics in real-time. dividual molecular lines. While due to the finite lifetimes The time scales of this coherent dressing dynamics are of Rydberg molecules and the fact that their binding en- ultrafast compared to the typical time scales of collective ergies decrease for increasing principal number, these in- low-energy excitations of ultracold quantum gases. This dividual molecular peaks cannot be resolved experimen- again highlights that effects arising from Bose-Bose interaction as well as finite temperature will play only a minor role in the prediction of the absorption response FDA as those start to influence dynamics only on much longer experiment time scales. 0.06

Using the data of the non-equilibrium quench dynam- (arb. units) ics, the FDA can accurately capture the formation of Rydberg molecular dimers, trimers, tetramers, etc. This 0.04 is demonstrated by comparing the FDA spectrum and experimental measurement for low principal quantum 0.02 number, as shown in [9]. Indeed, from a many-body wave-function perspective capturing the trimer, tetrame- ter and higher-order oligamer state requires the inclusion Absorption 0.00 of the corresponding higher order terms in Eq. (7). This - 25 - 20 - 15 - 10 - 5 0 attests to the challenge of describing Rydberg polarons Detuning (MHz) which are formed by the dressing with many deeply bound atoms, rather than just a small number of bath ex- FIG. 6. (a) LDA absorption spectrum for n = 49 as calculated citations. In fact due to the magnitude of the binding en- by FDA (solid blue) in comparison with experimental data ergies involved, coherent formation of Rydberg polarons (symbols). 10

serve this Gaussian response experimentally for Rydberg quantum (FDA) polarons of large principal number n = 60 and n = 72. 0.100 classical (CMC) While due to the ﬁnite lifetimes of Rydberg molecules 0.050 and the fact that their binding energies decrease for in-

(arb. units) 14 atom-bound creasing principal number, these individual molecular state peaks cannot be resolved experimentally for high prin- 0.010 cipal numbers, their Gaussian envelope is a robust sig- molecular peaks 0.005 nature of the formation of Rydberg polarons. The comparison of experiment and theory in Fig. 6 in turn shows Rydberg polaron response that the response is indeed composed of many individual from center of cloud Absorption 0.001 molecular lines. - 25 - 20 - 15 - 10 - 5 0 The FDA calculation of the absorption spectra takes Detuning (MHz) as input the experimentally determined number of atoms in the condensate NBEC and the trap frequencies ωi. The temperature T and the total number of atoms N are FIG. 7. LDA absorption spectrum for n = 49 as calculated tot taken as fit parameters, and results are consistent with from the classical Monte Carlo sampling model (black) compared to the FDA prediction. The shaded area, which cor- their determination using the mean-field model and ab- responds to the prediction of absorption response from the sorption imaging described in Section V. For the n = 60 cloud center at peak density, reveals that the Gaussian line spectrum, we also show a range of FDA predictions re- shape of the response is due to discrete, Gaussian-distributed sulting from varying the trap frequencies within experi- molecular peaks and not a continuous response as described mental uncertainty, which illustrates the impact of these by a simple classical model. uncertainties. Table II lists the parameters used in the theoretical simulation. Note that the requirement to 0.08 explain the spectrum over the whole frequency regime x10-2 (Hz) places tight constraints on the fit parameters T , and Ntot. 5.0 111.6 (For Rydberg polaron response at n = 72, refer to Ref. 0.06 1.0 106.0 0.5 100.4 [9]). As discussed in Ref. [10], the emergence of the Gaus-

(arb. units) 0.1 sian response can be understood from a direct, analytical 0.04 calculation of S(t) in terms of single particle eigenstates - 30 - 20 - 10 0 and energies. Indeed expanding the sum in Eq. (26) ex- 0.02 plicitly in a multinominal form leads, after Fourier transform, to the expression

Absorption 2n1 2nM 0.00 X |hα1|si| · ... · |hαM |si| · ... - 30 - 20 - 10 0 A(ν) = N! n1! . . . nM ! ... Detuning (MHz) Σni=N ×δ(−ν + n1ω1 + ... + nM ωM + ...). (27) FIG. 8. Experimentally observed absorption spectrum (symbols) for n = 60 in comparison with the theoretical prediction This expression captures not only the bound states but from FDA (lines). The experimental input parameters such also continuum states. Expressing the spectrum in this as the trap frequencies are varied within the experimental explicit form again highlights the fact that the actual uncertainty, giving rise to the gray band around the solid red spectrum is indeed built by δ-peaks corresponding to the line. The specific set of values is given in Table II. The inset many configurations in which bound molecules can be shows the signal on a logarithmic scale. formed and thus dress the impurity. It also explains the emergence of the gaussian lineshape as a limit of the multinominal distribution of delta-function peaks for tally for high principal numbers, their Gaussian envelope large particle number N. We note that the fact that the is a robust signature of the formation of Rydberg po- spectrum is built by molecular excitation peaks (of up to larons. hundreds of atoms bound to a single impurity) is missed As our simulation shows, the broad tail to the red by the classical description of the Rydberg polarons dis- and the characteristic gaussian profile of the signal are cussed in Section VII. both clear signatures of Rydberg polarons. The excel- We note that while Eq. (27) is appealing as it quali- lent agreement between many-body theory and experi- tatively explains the observed spectral features, it is less ment confirms the presence of Rydberg polarons in the useful for quantitative calculations due to the exponen- Sr experiment. tial growth of the number of terms in the sum (reflect- The Gaussian signature of the Rydberg-polaron spec- ing the exponential growth of Hilbert space for increas- trum becomes more pronounced when exciting n = 60 ing particle number). In contrast, the calculation of the states (Fig. 8). In the accompanying work [10] we ob- time-evolution in Eq. (26) is numerically efficient (when 11

Data ω/¯ 2π [Hz] ωi/2π [Hz] η NBEC Ntot ρpeak,BEC ρpeak,tot µ/kB [nK] T [nK] Experiment 112 (109, 109, 117) 0.77 3.7 × 105 4.8 × 105 3.8 × 1014 - 170 - Fig. 5(gray,solid) 111.6 (109, 109, 117) 0.77 3.7 × 105 4.8 × 105 3.72 × 1014 3.78 × 1014 175.5 166 Fig. 5(red) 106.0 (103.6, 103.6, 111.2) 0.71 3.7 × 105 5.2 × 105 3.5 × 1014 3.57 × 1014 165 180 Fig. 5(gray) 100.4 (98.1, 98.1, 105.3) 0.77 3.7 × 105 4.8 × 105 3.3 × 1014 3.58 × 1014 155 150

TABLE II. Parameters used for the prediction of the theoretical spectrum shown in Fig. 5(b) for the Rydberg n = 60 excitation. Inferred quantities are the condensate fraction η = NBEC/Ntot, peak BEC density ρpeak,BEC, and chemical potential µ. Densities are given in cm−3.

limited to finite evolution times) and only requires a final and bath atoms (non-commuting pˆj operators are now Fourier transformation. absent), and hence dynamics becomes completely classical. In this approximation the Hamiltonian becomes (without loss of generality, we assume that the impurity VII. FROM QUANTUM TO CLASSICAL is at the center of the coordinate system) DESCRIPTION OF RYDBERG ABSORPTION RESPONSE Z Hˆ = d3rV (r)â†(r)â(r) (28) For a sufficiently large average number of atoms within the Rydberg electron orbit, the overall line shape of Ry- Furthermore, Hˆ0 = 0, and henceρ îni = 1/Z in Eq. (12). dberg spectra can be described with classical statistical Since all boson coordinates ˆrj now commute with the arguments. In Fig. 9, we show the experimental spec- Hamiltonian, the many-body eigenstates are given by trum obtained for n = 60 in comparison with the pre- (j) (j) (j) |φ(j)i = |r , r ,..., r i. The trace in Eq. (12) reduces diction from the FDA (dashed red line) and a classi- 1 2 N to the sum over the set of all basis states {|φ(j)i}. The cal statistical approach (solid black). In the latter ap- (j) ˆ ˆ (j) proach, using a classical Monte Carlo (CMC) algorithm |φ i are eigenstates of H with eigenvalues H |φ i = P (j) (j) [11], we randomly distribute atoms in three-dimensional {r(j)} V (ri ) |φ i and S(t) becomes space around the Rydberg ion such that the correct den- i sity profile is obtained. Due to statistical fluctuations in P (j) X i ν− i V (ri ) t the random sampling of coordinates (sampled from a uni- Scl(t) = N e , (29) form distribution), the local density within the Rydberg- (j) {ri } electron radius fluctuates. For each of the random configurations of atoms C = (r1, r2,...) we calculate the classi- where the sum extends over all possible atomic config- cal energy of the configuration E = P V (r ), where r C i i i urations in real-space, and N is a normalization factor are the atom coordinates. The resulting energies EC are (representing the partition function Z in the density op- collected in an energy histogram and shown as the black erator ρ ). Finally, performing the Fourier transform of solid line in Fig. 9. ini The validity of the classical description reflects the fact that the macroscopic occupation of the molecular bound and scattering states probes the Rydberg molecular po- quantum (FDA) n=60 tential uniformly over its entire range. This process is 0.08 classical (CMC) also well described by a repeated sampling of different -2 classical atom configurations drawn from a homogeneous x10 (arb. units) 0.06 1.0 density distribution. The quantum-classical correspondence also becomes 0.1 evident when considering the specific approximation that 0.04 underlies the classical statistical approach. The Ryd- - 30 - 25 - 20 - 15 - 10 - 5 0 5 berg absorption response is given as a Fourier trans- 0.02 form of the full quantum quench dynamics as given by Eq. (13). The classical statistical model arises as an ap- Absorption 0.00 proximation of the time evolution of S(t) where the ki- - 30 - 25 - 20 - 15 - 10 - 5 0 5 netic energy of both the impurity and the Bose gas is Detuning (MHz) completely neglected. Hence both bosons and the impurity are treated as effectively infinitely heavy objects; FIG. 9. LDA absorption spectrum for n = 60 as calculated their motion is ‘frozen’ in space. The disregard of the by the FDA (dashed red) and classical statistical Monte Carlo kinetic terms has the consequence that the Hamiltonian sampling method (black). The inset shows the signal on a logarithmic scale. now commutes with the position operators ˆrj of impurity 12

S(t) one arrives at whose Fourier transform leads to the spectral function ! for Rydberg impurity excitation in a Bose gas, Eq. (11). X X (j) We show how different approximations to the many-body A(ν) = N δ ν − V (r ) . (30) i quantum description, such as mean field and classical (j) i {ri } Monte Carlo treatments can be derived. Here we extend the bosonic functional determinant approach to the Ry- A finite lifetime of the Rydberg or laser excitation, as well dberg polaron problem to account for recoil of the impu- as a finite linewidth of the laser leads to a broadening of rity. We show that the FDA approach can correctly and the delta function in Eq. (30). The sum in Eq. (30) is accurately account for the few-body molecular bound- exactly the object sampled in the classical Monte Carlo state formation, as well as the macroscopic occupation (CMC) approach derived from first principles. of many-body bound and continuum states. The vari- Moreover we note that for impurities interacting with ous treatments are compared with experimental data for an ideal Bose gas with contact interactions, the gaus- Rydberg excitation in a 84Sr BEC [9] with the FDA re- sian spectral signature of polarons [10] would remain, sults reproducing the observed data over a wide range of while in the classical model, a sharp excitation at the Rydberg line spectral intensities. atomic transition frequency is expected. Indications of a Gaussian response have been seen in a recent experiment performed independently at Aarhus [8] and JILA Rydberg polarons are exemplified on the one hand by [7], in agreement with theory [36]. The classical statisti- their large energy scales, 1-10 MHz, that allow for co- cal approach fails to describe the Bose polarons close to herent polaronic dressing on correspondingly short time a Feshbach resonance [7, 8]. Here the impurity indeed in- scales, and on the other hand large spatial extent, 1 µm, teracts with the bath via a contact interaction, and, due leading to large-scale variations of the density in the to a diverging scattering length, the single bound state many-body medium. Another key feature of Rydberg present in the problem is highly delocalized and extends polarons is the coherent dressing of the quantum impu- in size far beyond the range of the potential. This effect rity not by collective low-energy excitations, but by a cannot be captured by a classical approach. large number of molecular bound states. This is a new While Rydberg absorption spectra find an effective, yet dressing mechanism, distinguishing them from polarons approximate description in terms of classical statistics, encountered so far in the solid state physics context. such an approach does not reveal the quantum mechanical origin of Rydberg polarons. The classical sampling model treats both the Rydberg and the ground state Our FDA time domain analysis suggests a way to inves- atoms as infinitely heavy objects. Zero-point motion and tigate the dynamical formation of the Rydberg polarons. the discrete nature of the bound energy levels are absent A natural extension will be to explore the feasibility of in this treatment, so that it cannot describe the forma- observing Pauli blocking in a degenerate Fermi atomic tion of Rydberg molecular states. This short-coming of gas with non-trivial spatial correlations. Note that one the classical approach is evident in Fig. 7, where we show can study polarons with nonzero momentum by keeping the absorption spectrum for n = 49. The FDA (solid all terms in Eq. (23) and allowing p to be finite. The ef- blue) fully describes the quantum mechanical formation fective polaron mass can then be analyzed as previously of molecules (dimers, trimers, tetrameters, excited molec- demonstrated [59]. ular states, etc.), which gives rise to discrete molecular peaks visible in the spectrum. In contrast, and as evident Acknowledgements: Research supported by from Fig. 7, the existence of molecular states, which are the AFOSR (FA9550-14-1-0007), the NSF (1301773, the quantum mechanical building block of Rydberg po- 1600059, and 1205946), the Robert A, Welch Foundation larons, is not described by the classical approach (solid (C-0734 and C-1844), the FWF(Austria) (P23359-N16, black). The fact that the classical model can only de- and FWF-SFB049 NextLite). The Vienna scientific scribe the envelope of the Rydberg polaron response but cluster was used for the calculations. H. R. S. was not the underlying distribution of molecular peaks, is fur- supported by a grant to ITAMP from the NSF. R. S. ther emphasized by the comparison of FDA and CMC and H. R. S. were supported by the NSF through a grant simulation from the center of the atomic cloud shown as for the Institute for Theoretical Atomic, Molecular, and shaded region in Fig. 7. Optical Physics at Harvard University and the Smith- sonian Astrophysical Observatory. R. S. acknowledges support from the ETH Pauli Center for Theoretical VIII. CONCLUSION Studies. E. D. acknowledges support from Harvard-MIT CUA, NSF Grant (DMR-1308435), AFOSR Quantum In this work, we have detailed the descriptions of po- Simulation MURI, the ARO-MURI on Atomtronics, larons as encountered in condensed matter and in ul- and support from Dr. Max Rössler,the Walter Haefner tracold atomic systems. We time-evolve an extended Foundation and the ETH Foundation. T. C. K acknowl- Fröhlich Hamiltonian as relevant for Rydberg excitations, edges support from Yale University during writing of Eq. (9), unitarily to obtain the overlap function, Eq. (12), this manuscript. 13

Appendix A: Determination of BEC parameters and uation is much more complicated than this for an inter- Mean-Field description of the Impurity Excitation acting gas. There are corrections to the condensate frac- Spectrum in a Strontium BEC tion that are independent of the trapping potential, and these are often discussed in terms of the shift of the crit- The mean-field approximation neglects fluctuations in ical temperature for condensation [48, 61, 62], but these the density around the Rydberg impurity, which cor- are all small in our case, reflecting the smallness of rel- 3 −4 respond to a spread in binding energies of the polaron evant expansion parameters: ρmaxabb = 10 , abb/λth = −2 −1 states excited for a given average density. This explains 10 at 100 nK, and aho/RTF = 10 , where λth is the the discrepancy between data and fit at large detuning. thermal de Broglie wavelength and aho is the trap har- But as shown in [9], the broadening resulting from these monic oscillator length. For a gas trapped in an inho- √ fluctuations is proportional to ρ and vanishes as detun- mogeneous external potential Vext(r), the Hartree-Fock ing approaches zero. Thus, we assume that the difference approximation (as described in detail in Refs. [60, 63]) between data and the mean-field fit in Fig. 3 for small de- yields a mean-field interaction between thermal and BEC tuning (ν/∆max < 0.5) arises from non-condensed atoms. atoms that creates an effective potential for thermal We adjust the area of the mean-field BEC contribution atoms, Veff(r) = Vext(r) + 2g[ρBEC (r) + ρth(r)] that is of so the sum of the non-condensed and mean-field-BEC a “Mexican hat” shape rather than parabolic [60, 63, 64]. signal matches the total experimental spectral area. Be- This effect is particularly important when the sample cause the deviations between the mean-field fit and the temperature is close to or lower than the chemical po- BEC spectrum are significant, the fit of the peak shift tential, which is the case here. It increases the volume available to non-condensed atoms, and implies a lower ∆max is less rigorous. We adjust it to qualitatively match the data and so that the mean-field fit and the data have sample temperature for a given value of η compared to the non-interacting case. In other words, the number of approximately equal area for (ν/∆max > 0.5). This uncertainty in the fitting procedure decreases with increas- thermal atoms can significantly exceed the critical num- ing principal quantum number as the experimental data ber for condensation predicted for an ideal gas [58, 64]. converges towards the mean-field form. In the mean-field 240 nK is thus an upper limit of the sample temperature. description we neglect the laser linewidth of 400 kHz in Sample temperatures extracted from bimodal fits to the this analysis. time-of-flight absorption images are 100-190 nK. The fit parameters are given in Table I. The conden- Both the mean-field and the FDA fits use a local densate fraction η is directly determined from the spectrum. sity approximation for the spectrum. Here the tempera- The uncertainties correspond to values calculated for the ture and interaction between bosons enter by determin- extremes of the confidence intervals shown as bands in ing the precise form of the overall density distribution of the atomic gas. As discussed above, the density profile Fig. 3, except for uncertainties in ωi and total atom number, which reflect uncertainties from the independent has contributions from both the condensed and thermal, procedures for measuring these quantities. The fit value non-condensed atoms of the peak shift, ∆max, can be compared to the pre- ρ(r) = ρBEC(r) + ρth(r). (A1) dicted peak shift, ∆˜ max, calculated from independent, a priori information: the number of atoms in the conden- The BEC density ρBEC and resulting chemical potential sate NBEC determined from time-of-flight-absorption im- are assumed to be given by the Thomas-Fermi expres- ages, the trap oscillation frequencies ωi determined from sions [63] measurements of collective mode frequencies for trapped atoms, and the value of as,eff found theoretically from m ρBEC (r) = 2 [µTF − Vext(r)], (A2) VRyd(r). The values of ∆max are all about 10% below 4π~ abb ∆˜ . This may point to something systematic in our max 2/5 analysis procedure, but it is also a reasonable agreement with µTF = ~ω/2(15NBECabb/aho) . NBEC is the total 84 given our uncertainty in determination of ω and total number of condensed atoms, abb = 123a0 for Sr, and i 1/2 aho = ( /mω) is the harmonic oscillator length. (In atom number. The calculation of as,eff from VRyd(r) ~ (Eq. (18)) could also account for some of this discrepancy this section we make explicit factors of ~ and kB.) through approximations made to describe the Rydberg- The density distribution for atoms in the thermal gas atom potential at short range. Residual deviations in can be calculated from known trap and BEC parameters the description of the short-range details of the Ryd- using a Bose-distribution berg molecular potential lead, for instance, also to the 1 n 1 o − k T [Veff(r)−µTF ] minor discrepancies between the FDA and CMC simu- ρth(r) = g e B , (A3) λ3 3/2 lation of the Rydberg response from the center of the T atomic cloud, shown in Fig. 7. q 2π 2 where λT = ~ is the thermal wavelength and For a non-interacting gas in a harmonic trap, the tem- mkB T 3 perature is easily found from η, andω ¯ [60], and the stan- g3/2(z) the polylogarithm PolyLog[ 2 ; z] [63], and the dard expressions imply a temperature of approximately chemical potential is set to µTF . The parameters T 240 nK for the data presented in Fig. 3. However, the sit- and µTF are varied self-consistently to fit that spectrum, 14 which can be seen as matching the experimentally ob- trap center and we neglect it in discussions of the peak served total particle number Ntot = NBEC + Nth and mean-field shift in the spectrum. BEC fraction η = NBEC/Ntot. The mean-field calculations use the full geometry of the For the FDA simulation, the dipole trap potential 2 2 2 2 external trapping potential as determined by the optical- is approximated by Vext(r) = m(ωr r + ωz z )/2. In dipole-trap lasers. Integrals involving Vext(r) are eval- Fig. 5(a) we show the density profile along the radial uated numerically. This procedure yields temperatures direction for parameters as used for the FDA prediction of 155 − 170 nK, in good agreement with other deter- of the n = 60 Rydberg absorption spectrum including minations. The modification of the potential seen by Hartree-Fock corrections. We emphasize again that the thermal atoms due to Hartree-Fock corrections has an temperature and interaction between bosons are relevant important consequence for our analysis. The peak shift only for the determination of the density profile of atoms. in the Rydberg excitation spectrum should in principle In the simulation of the absorption spectrum, which is ob- reflect the peak condensate density plus the density of tained within a local density approximation, this density thermal atoms at the center of the trapping potential. distribution of atoms enters as input. In this simula- The contribution from thermal atoms would be a sig- tion, then locally performed for constant density, both nificant correction if the mean-field repulsion of thermal temperature effects and boson-interaction effects can be atoms from the center of the trap were ignored. But be- neglected due to the short time scales associated with cause the sample temperature is close to the chemical Rydberg polaron dressing dynamics (that are given by potential, the density of thermal atoms is suppressed at the Rydberg molecular binding energies).

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