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Rydberg Electrons in a Bose-Einstein Condensate

We investigate a hybrid system composed of ultracold Rydberg atoms immersed in an atomic Bose-Einstein condensate (BEC). The coupling between the Rydberg electrons and BEC atoms leads to the excitation of phonons, the exchange of which induces Yukawa interaction between Rydberg atoms. Due to the small electron mass, the effective charge associated with this quasi-particle-mediated interaction can be large, while its range is equal to the healing length of the BEC, which can be tuned by adjusting the scattering length of the BEC atoms. We find that for small healing lengths, the distortion of the BEC can"image"the wave function density of the Rydberg electron, while large healing lengths induce an attractive Yukawa potential between the two Rydberg atoms that can form a new type of ultra-long-range molecule. We discuss both cases for a realistic system.

Impurities in a Bose-Einstein condensate (BEC) have attracted much attention and motivated the investigation of a wide range of phenomena. For example, the motion of a single impurity in a BEC can probe the superfluid dynamics [1][2][3], while an ionic impurity in a BEC can form a mesoscopic molecular ion [4]. Due to the selfenergy induced by phonons (excitations of the BEC), a neutral impurity can self-localize in both a homogeneous and a harmonically trapped BEC [5][6][7], which sheds light on polaron physics [8,9]. Exchanging phonons between multiple impurities induces an attractive Yukawa potential between each pair of impurities [10,11], which leads to the so called "co-self-localization" [12] and is related to forming bipolarons and multipolarons [13]. Recent experiments, where an atom of a BEC is excited into a Rydberg state [14] to study phonon excitations and collective oscillations, open the door to exploration of the electron-phonon coupling in ultracold degenerate gases, a phenomenon responsible for the formation of Cooper pairs of two repelling electrons in BCS superconductivity [15].
In this Letter, we study Rydberg atoms immersed in a homogeneous BEC, as sketched in Fig. 1(a). Rydberg atoms consist of an ion core and a highly excited electron with its oscillatory wave function Ψ e extending to large distances of the order of ∼n 2 a 0 (n: principle quantum number, a 0 : Bohr radius). As pointed out by Fermi [16], the interaction between the quasi-free electron at x and a ground state atom at r can be approximated at low scattering energies by a contact interaction parametrized by an energy-dependent s-wave scattering length A s (k) = −k −1 tan δ s (k), While the s-wave approximation is valid for qualitative analysis, we include higher-partial wave contributions for quantitative results [17]. A s (k) depends on the scattering energy via the local wave number k(r) given by where R y is the Rydberg constant, 0 the vacuum permittivity, e and m e the charge and mass, respectively, of the electron with angular momentum e and quantum defect δ e . For lowe state, Eq.(1) gives an effective interaction between Rydberg and ground state atoms as which leads to an attraction and formation of ultra-longrange Rydberg molecules for A s < 0 [18]. The electron density and corresponding oscillatory potential are sketched in Fig. 1 [19][20][21][22][23], has motivated the studies of the p-wave electron (leading to "butterfly states" [24]), and Rydberg electrons scattering off a perturber with a permanent dipole moment [25].
In our system, Rydberg electrons interact with the coherent ground state of BEC, i.e. with many atoms, resulting in collective excitations described as phonons (scalar bosons). One of the most profound conceptual advances in physics is that the exchange of particles can produce a force (e.g., the electromagnetic force is generated by charges exchanging virtual photons). Exchanging phonons in a BEC will lead to a Yukawa potential. As we describe below, under appropriate conditions, we find two regimes. For a BEC with a small healing length ξ, the Yukawa potential is short-ranged, and distorts the BEC locally, "mapping" the electron density onto the BEC density. For a large ξ, the Yukawa potential is longranged and can bind Rydberg atoms and form a new type of "ultra-long-range" molecule.
We first consider a homogeneous BEC in the absence of impurities, described by the Hamiltonian where u B = 4π 2 a B /m B is the coupling constant between the atoms of mass m B and scattering length a B , Ω V is the quantization volume, and c † k (c k ) is the creation (annihilation) operator of bosonic atoms with momentum k. If most atoms occupy the ground state (k=0), one can replace c † 0 and c 0 by the c-number √ N 0 and expand Eq.(4) in the decreasing order of N 0 . The number of atoms is given by N = N 0 + k =0 c † k c k . By keeping the terms of the order √ N 0 or higher, H BEC can be diagonalized via the Bogoliubov transformation creates (annihilates) a quasi-particle (or phonon) of momentum q when applied to the ground e iq·r c † p+q c p can be written as, and the interaction between a Rydberg electron and BEC atoms where V q = d 3 rV R (r)e iq·r is the Fourier transform of the potential. Applying the perturbation theory gives the first and second order corrections to the ground state energy: and (7) by taking the thermal limit of V −1 q → (2π) −3 d 3 q and integrating over q, where ρ B is assumed to be a constant. Note that N 0 can be replaced by the total atom number N at this level of approximation.
Under approximation in Eq. (1), e /m e is the mean-field energy shift given in terms of the average scattering lengthā e = d 3 rA s [k(r)]|Ψ e (r)| 2 , while where its range ξ = 1/ √ 16πρ B a B is exactly equal to the BEC healing length, andQ 2 ≈ 4π 2ā2 e ρ B m B /m 2 e characterizes its strength; the "effective charge"Q emphasizes the analogy with Coulomb interactions. The term E (2) can be understood as the self-interaction of electrons by a Yukawa potential induced via phonon exchange at two different positions. This term is crucial in studies of self-localization of impurities in a BEC. However, in our system, the Rydberg electrons are already localized by strong Coulomb forces with ion cores. Therefore, the distorted BEC density, under appropriate conditions, can reflect the oscillatory nature of Ψ e and "image" the Rydberg electron.
To the first order, the perturbed ground state given Eq. (9) shows that δρ(r) is affected by "averaging" the effective interaction V R within the range ξ. The oscillatory nature of Ψ e can be imaged onto δρ(r) [26]. However, if ξ is larger than the local wavelength of the Rydberg electron, the averaging will erase this signature. This effect is illustrated in Fig. 2 for a 87 Rb(160s) Rydberg atom in a 87 Rb BEC with ρ B = 2 × 10 13 cm −3 , by comparing the radial probability density P e (x) = 4πx 2 |Ψ e (x) | 2 with ∆P (r) = 4πr 2 δρ (r) for different scattering lengths a B . Larger values of a B produce a sharper signature of the oscillation, albeit an overall smaller distortion amplitude. This effect is better illustrated in a 2D density plot (Fig. 3), where different quadrants represent the "normalized" 2D distortion densities ∆P 2D / max (∆P 2D ) in the x-z plane for four different a B . Here, ∆P 2D = 2πrδρ, and x and z are rescaled byx = exp(x/l 0 ) andz = exp(z/l 0 ), where l 0 = 1k a.u., so that the effects at large distances are emphasized. It is evident that the oscillations for a B = 5k a.u. (fourth quadrant) are much blurrier than for a B = 20k a.u. (first quadrant). For a large healing length ξ, the averaging of V R masks the effect of the electron self-interaction due to the phonon exchange. However, the phonon exchange still mediates non-trivial interactions between the Rydberg atoms. Without a BEC, two Rydberg atoms experience strong long-range interactions, leading to formation of macrodimers [27] and the interaction blockade [28][29][30][31][32]. For two ns Rydberg atoms separated by R, this interaction is repulsive with its leading term being the van-der-Waals (vdW) +C 6 /R 6 term, where C 6 ∝ n 11 [33]. Immersed in a BEC, however, the exchange of phonons between two Rydberg atoms gives rise to a Yukawa potential. We derive this potential within the Born-Oppenheimer (BO) approximation, starting from the interaction of two Rydberg atoms, located at R 1 and R 2 , and BEC atoms (after applying the Bogoliubov transformation) Here, i V q ≡ d 3 rV i (r)e iq·r , where V i (r) describes the interaction of "impurity" i and the BEC atoms in coordinate space. Within perturbation theory, the first order correction E (1) gives a mean-field energy shift similar to the single Rydberg atom case in the thermal limit. For spherically symmetric interactions (where i V q = i V −q = i V q is real), the second order correction is where is the Rydberg atoms separation, and θ q is the angle between the vector R and q. The term containing A q can be understood as the self-localizing energy for both Rydberg atoms calculated previously, and will be neglected together with the mean-field energy shift E (1) for the study of relative dynamics of Rydberg atoms, since they simply contribute a constant energy shift. The term containing B q leads to the BO potential which can be easily generalized to interactions between any two impurities immersed in a BEC [11]. The BO approach allows the study of the adiabatic corrections induced by the motion of Rydberg atoms. The diagonal adiabatic correction ∆E (2) q|HINT|0 ωq |q is the perturbed ground state.
0|∂ R HINT|q q|∂ R HINT|0 2 ω 2 q , so that, in the thermal limit and neglecting the constant energy shift terms, the diabatic correction to U (R) is .
(13) The e iqR cos θq term in Eqs. (12) and (13) implies that, for a large R, only the terms where q is small are important, leading to the asymptotic behavior of the potential U (R) → −Q 2 e −R/ξ /R, whereQ 2 ≈ ρ B m B B q=0 /π. For the s-wave, the effective charge isQ 2 ≈ 4π 2ā2 e ρ B m B /m 2 e . Not surprisingly, we obtain the same Yukawa potential as in Eq. (8), since phonon-exchange mediates the interaction. Note that Q is inversely proportional to m e for Rydberg atoms, since the electrons are really the perturbers, as opposed to more massive neutral impurities for whichQ is inversely proportional to m I . Hence, the induced interaction is much stronger for Rydberg atoms. Under the same approximation, the adiabatic correction is given by is given in terms of the modified Bessel function of the first kind I n (x) and the modified Struve function L n (x). The asymptotic behaviors are F (R/ξ) → 4/(3π) for R ξ, and F (R/ξ) → 12ξ 4 /(πR 4 ) for R ξ. We note that for R ξ, these imply a vanishing adiabatic potential U (R) so that the adiabatic correction becomes dominated by a repulsive 1/R 4 term. As expected, the diabatic correction can be neglected if m B m I . A surprising limit is reached for a very large healing length ξ, which can be achieved by using a Feshbach resonance to tune a B → 0. Then, the diabatic correction also vanishes: lim ξ→∞ ∆U (R) = 0, even when m B is larger than m I . Notice that, in this limit, U (R) reduces to the Coulomb potential.
To illustrate these predictions, we consider a realistic system of two 87 Rb(50s) Rydberg atoms immersed in a BEC of 87 Rb atoms of density ρ B = 10 13 cm −3 . To ensure a healing length ξ much larger than the Rydberg atoms, the scattering length between the BEC atoms is tuned to a B = 10 a.u. (e.g., via a Feshbach resonance), so that ξ = 3.66 × 10 4 a.u. The numerical "trilobitelike" interaction shown in Fig. 4(a) is constructed us- ing the first-order perturbative model [16,34] including s-and p-contributions [17], with zero-energy scattering lengths, A s = −16.05 a 0 and A p = −21.15 a 0 , respectively [35]. The states in the range n = 47 − 51 were included and the resulting Hamiltonian diagonalized to obtain the 50s eigenstate [36]. Fig. 4(b) shows that B q computed using this potential converges to a constant B q=0 ≈ 2 × 10 4 a.u. for a small q, yielding an effective chargeQ 2 ≈ 1.537 × 10 −3 a.u. Fig. 4(c) depicts the effect of immersing Rydberg atoms in a BEC: without the BEC, two Rydberg atoms interact via the vdW potential −C 6 /R 6 − C 8 /R 8 − C 10 /R 10 (repulsive solid-red curve with C 6 = −1.074 × 10 20 , C 8 = 7.189 × 10 26 , and C 10 = −7.162 × 10 33 , in a.u. for 87 Rb(50s) [37].) In a BEC, the BO potential (solid black curve) is attractive at large separations, in agreement with the Yukawa potential −Q 2 e −R/ξ /R (dash-dotted curve) at large distances, before becoming repulsive at shorter range where the "bare" repulsive vdW interaction dominates the phonon-exchange contribution. The well produced by the phonon-exchange can support bound levels; in the example above, its depth is about -17.77 MHz, while the equilibrium separation is about 60k a.u. (much larger than the 5k a.u. extension of the "trilobite-like" potentials). The large mass of Rb atoms leads to many bound levels; the three lowest are at about -17.64 MHz, -17.56 MHz and -17.47 MHz. The ground state wave function with a spatial width about 2k a.u. is also shown in Fig. 4(c), These results show how phonon-exchange modifies an otherwise repulsive interaction into a potential well capable of binding two Rydberg atoms. Fig. 5 explores the sensitivity of the BO potentials to variations in density ρ B and scattering length a B , and compares them to the "bare" case (without BEC). The behavior of the BO curves can be understood qualitatively from the s-wave approximation with average scattering lengthā e :Q 2 is proportional to ρ B and ξ to ρ −1/2 B . The competition between the two effects as ρ B varies leads to a deeper BO curve for a moderate density (see Fig. 5(a)). However,Q 2 is independent of a B while ξ is proportional to a −1/2 B , giving deeper BO curves as a B gets smaller (see Fig. 5(a)). Hence, the BEC-induced interaction can be conveniently controlled by tuning a B via a Feshbach resonance; in the limit a B = 0, the long-range Yukawa potential becomes an attractive Coulomb potential.
In summary, we studied BEC-induced interactions involving Rydberg impurities due to phonon-exchange, and found two limiting cases depending on the healing length ξ of the BEC. For a small ξ, the BEC modulation can be used to "image" the wave function of the Rydberg electron, while large ξ leads to the formation of ultra-longrange diatomic molecules. By tuning a B , "synthetic" Coulomb potentials can be generated between neutral particles and their sign can be modified by using different Rydberg states for the two impurity atoms. This long range interaction is well-behaved and easily controlled and, hence, opens promising avenues of research. For example, for systems containing many Rydberg atoms impurities, this interaction might lead to crystallization [38], and be used to study the phase diagram of Yukawa bosons [39]. This