week ending PRL 116, 105302 (2016) PHYSICAL REVIEW LETTERS 11 MARCH 2016 Mesoscopic Rydberg Impurity in an Atomic Quantum Gas Richard Schmidt,1,2,* H. R. Sadeghpour,1 and E. Demler2 1ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA 2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 2 December 2015; revised manuscript received 29 February 2016; published 10 March 2016) Giant impurity excitations are powerful probes for exploring new regimes of far out of equilibrium dynamics in few- and many-body quantum systems, and in situ observations of correlations. Motivated by recent experimental progress in spectroscopic studies of Rydberg excitations in ultracold atoms, we develop a new theoretical approach for describing multiscale dynamics of Rydberg excitations in quantum Bose gases. We find that the crossover from few- to many-body dynamics manifests in a dramatic change in spectral profile from resolved molecular lines to broad Gaussian distributions representing a superpolaronic state in which many atoms bind to the Rydberg impurity. We discuss signatures of this crossover in the temperature and density dependence of the spectra. DOI: 10.1103/PhysRevLett.116.105302 Introduction.—Until recently the main motivation for Gaussian profile at T ¼ 0 in a Bose-Einstein condensate studying impurity dynamics came from the study of (BEC) signifying the formation of a giant superpolaronic electron transport in mesoscopic and nanoelectronic sys- state at higher densities. This dressed impurity state is tems [1–4]. Theoretical work focused almost exclusively on comprised of a mesoscopic number of bound atoms and fermionic systems [5,6] discussing such spectacular phe- exhibits a shell structure in real space, see Fig. 1. nomena as the Kondo effect [7,8] and the orthogonality The present method applies generally to thermal and catastrophe [9–11]. In contrast, studies of impurities in degenerate gases, is readily extendable to fermionic mixed- bosonic systems were limited to equilibrium properties species systems, and allows us to predict a wide range of such as the polaron energy and effective mass [12–14]. time-dependent observables such as spin echo or Ramsey However, recent efforts to probe nonequilibrium polaronic signals. Model for Rydberg impurities.—We investigate a single dynamics in bosonic gases have renewed theoretical Rydberg impurity, localized in space, immersed in a Bose interest [15–21]. The realization of Rydberg impurities in ultracold quantum gases [22–24] presents a new frontier where microscopic atomic physics meets condensed matter and mesoscopic physics, going beyond quantum simulations of strongly correlated states [25]. At high principal quantum numbers, Rydberg blockade ensures that only one excita- tion is possible [26–28], promoting the Rydberg atom to a novel realization of an impurity problem. At the same time Rydberg excitations in ultracold gases are predicted to form exotic molecular bound states [29,30]. Their mesoscopic size with yet large binding energies, represent a multiscale challenge in impurity physics, requiring the confluence of powerful theoretical tools, from atomic and condensed matter physics. In this Letter, we address the challenge of nonequili- brium dynamics of Bose polarons arising from Rydberg impurities and demonstrate how their dynamics manifests FIG. 1. When a Rydberg impurity is excited in a Bose gas, a itself in the absorption spectra. Our approach employs molecular potential supporting ultra-long-range vibrational states is formed. In a thermal gas these states are observed as few-body microscopic molecular potentials calculated from Rydberg absorption lines (dimer, trimer, tetramer, etc.). The occupation of wave functions [31]. Using exact bound and scattering vibrational modes results in a shell structure as illustrated in (b). solutions, we solve the full time evolution of the many- Deep in the condensed gas, the spectrum becomes Gaussian body density matrix, based on a novel functional determi- [dashed curve in (a)] representing a superpolaronic state. In nant approach (FDA) to bosonic systems. We show in detail typical experiments density averaged spectra [blue curve in (a)] how few-body molecular lines at low densities evolve into a are observed. 0031-9007=16=116(10)=105302(6) 105302-1 © 2016 American Physical Society week ending PRL 116, 105302 (2016) PHYSICAL REVIEW LETTERS 11 MARCH 2016 gas of ultracold atoms at arbitrary temperature T and excitations, the interboson interaction can be neglected particle density ρ. The Rydberg atom is initially in its as well. electronic ground state, j↓i. In a two-photon laser excita- Rydberg impurity spectra from a quantum quench tion, the atom is transferred to an excited state, j↑i¼jnsi, problem.—We compute the time-dependent overlap of principal quantum number n, and whose orbital ψ eðrÞ −1=3 ˆ ˆ has a size comparable to the interparticle distance ρ . SðtÞ¼tr½eiH0te−iHtρˆð3Þ The scattering of the Rydberg electron from the surround- ing ground state atoms is described by the Fermi pseudo- where ρˆ is the density matrix representing the initial state of ˆ potential [32]. A host atom situated at a distance r from the system, and H0 is the Hamiltonian in the absence of the the impurity ion core at R ¼ 0 is then subject to the impurity. The expression Eq. (3) describes the many-body potential [23] dephasing dynamics following the sudden quench of the Rydberg potential and can be directly measured using 2πℏa – V ðrÞ¼ e jψ ðrÞj2: ð Þ Ramsey spectroscopy [38 41]. The Fourier transform of Ryd e 1 me Eq. (3)R yields the two-photon absorption spectrum AðωÞ¼ ∞ iωt 2Re 0 dte SðtÞ [39,42]. This Born-Oppenheimer potential supports bound vibra- Zero temperature description.—We focus first on a Bose tional states [29], when the Rydberg-electron perturber- gas at T ¼ 0. The Fock statepffiffiffiffiffiffi representing the atoms in a ˆ † N atom scattering length ae is negative; cf. Fig. 1. BEC is given by jΨ0i¼1= N!ðb0Þ jvaci where N is the The main knob for the control of interactions in quantum particle number. The time-dependent overlap [Eq. (3)]is ρˆ ¼ gases has universally been the zero-momentum scattering then evaluated with respect to the density matrix BEC length which can be manipulated by magnetic or optical jΨ0ihΨ0j and gives Fano-Feshbach spectroscopy [33]. In Rydberg impurity 2 X N excitations, it is the variation of jψ eðrÞj which brings in the S ðtÞ¼ jhαjsij2eiðϵs−ωαÞt ; ð4Þ possibility to probe the quantum gas in a completely novel BEC α way, i.e., by changing n. Theoretical analyses of Rydberg molecule excitations where the collective index α ¼ðk; l; mÞ labels the [34–37] have thus far relied on few-body atomic methods, interacting single particle states jαi with nodal number which by construction cannot account for many-body k, angular momentum l of projection m, and energy ωα. The ˆ † quantum dynamics. In contrast, to describe the physics lowest noninteracting scattering state is jsi¼b0jvaci with on all length scales, arbitrary temperature, and density, one energy ϵs. must rigorously include the full quantum statistics, encoded We demonstrate the utility of our method for excitations in the many-body density matrix. This applies in particular in a 87Rb gas, as recently realized [23,24,30], by focusing to high Rydberg excitations where the ultracold medium is on excitations into Rb(71s). Vibrational bound and scatter- probed on mesoscopic length scales and standard many- ing states up to large angular momenta are calculated in a body techniques such as mean-field or variational spherical box of radius 10 μm [43]. We incorporate the approaches fail due to the multiscale and nonperturbative finite length τp of excitation pulses in Eq. (3), and the finite character of the underlying microscopic physics. lifetime of Rydberg excitations [44] can be included To this end, we study the dynamics of the Rydberg as well. impurity governed by the many-body Hamiltonian In Fig. 2 we show the absorption spectrum for a Rydberg X X excitation in the BEC phase at T ¼ 0, for different densities ˆ ˆ † ˆ ˆ † ˆ ρ ¼ ρðr ¼ 0Þ H ¼ ϵkbkbk þ VðqÞbkþqbkj↑ih↑jð2Þ 0 . The low density response is shown in k kq Fig. 2(a). Here the interparticle spacing exceeds the range of the Rydberg potential, r0 ≈ 8850a0. The physics is ˆ † ˆ where bk, bk represent the creation and annihilation dominated by few-body interactions resulting in a series of operators of the bath bosons of mass m, momentum k, molecular lines corresponding to one, two, or more 2 ϵk ¼ k =2m, and VðqÞ is the Fourier transform of Eq. (1). medium atoms bound and localized inside the Rydberg The Fock space is constructed from the bosonic single- orbit. These molecular lines were recently observed in a particle orbitals, obtained from the bound and continuum thermal Rb gas [23,30], and are identified in Fig. 2(a) with a eigensolutions of the Schroedinger equation for a localized red sphere for the fundamental dimer line, with two red Rydberg impurity with infinite mass. The scattering length spheres for the associated trimer line, and similarly for the in the pseudopotential in Eq. (1) is rescaled to reproduce the tetramer and pentamer lines. bound molecular dimer energies. Since the Rydberg mol- The Rydberg potential Eq. (1) supports several vibra- ecule formation takes place on time scales of the vibrational tional bound states. This leads to additional spectral energies, the Rydberg ion recoil can be ignored. Since this features represented by a combination of green and red time is short compared to the time for collective bath spheres, which are associated with exotic bound dimers, 105302-2 week ending PRL 116, 105302 (2016) PHYSICAL REVIEW LETTERS 11 MARCH 2016 interacting states contribute, leading to a multinominal spectral decomposition of AðωÞ which also evolves into a Gaussian.