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![Lu-Fano plots for (a) J = 0, (b) J = 1, (c) J = 2, and (d) J = 3 symmetries. Blue points are l ≈ 0 odd parity; red are l ≈ 1 even parity, and green are l ≈ 2 odd parity. Intersections of the solid curves (equation (30)) with the diagonal lines (equation (34); only a few representative ones are shown) give the positions of bound states (points). This figure is modified from [78].](publication/332451196/figure/fig1/AS:1132493621215306@1647018928635/Lu-Fano-plots-for-a-J0-b-J1-c-J2-and-d-J3-symmetries-Blue-points-are-l0.jpg)
Lu-Fano plots for (a) J = 0, (b) J = 1, (c) J = 2, and (d) J = 3 symmetries. Blue points are l ≈ 0 odd parity; red are l ≈ 1 even parity, and green are l ≈ 2 odd parity. Intersections of the solid curves (equation (30)) with the diagonal lines (equation (34); only a few representative ones are shown) give the positions of bound states (points). This figure is modified from [78].
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![Scattering phase shifts for Cs (a) and Rb (b), extracted from [114]. In...](publication/332451196/figure/fig3/AS:1132493625401367@1647018929127/Scattering-phase-shifts-for-Cs-a-and-Rb-b-extracted-from-114-In-panel-a-the_Q320.jpg)
This PhD tutorial discusses ultra-long-range Rydberg molecules, the exotic bound states of a Rydberg atom and one or more ground state atoms immersed in the Rydberg electron’s wave function. This novel chemical bond is distinct from an ionic or covalent bond, and is accomplished by a very different mechanism: the Rydberg electron, elastically scatt...
Citations
... Interest in ultralong-range Rydberg molecules (ULR-RMs), which comprise a Rydberg atom in whose electron cloud are embedded one (or more) weakly-bound groundstate atoms, has increased steadily over the years [1][2][3][4][5]. Such molecules, which represent a new molecular class, have provided a valuable microscale laboratory in which to study low-energy electron-atom scattering at energies not readily accessible using alternate techniques [6,7], have furnished a powerful probe of non-local spatial correlations in cold quantum gases [8,9], illuminating the important role played by quantum statistics, and have, through measurements of ULRRM formation in dense Bose-Einstein condensates, provided the opportunity to study many-body phenomena, such as the creation of Rydberg polarons [10][11][12][13]. ...
Spectroscopic measurements of the rotational distribution of $^{84}$Sr and $^{86}$Sr 5sns $^1S_0$ ultralong-range Rydberg molecular dimers created via photoassociation in a cold gas are reported. The dimers are produced by two-photon excitation via the 5s5p $^1P_1$ intermediate state. The use of singlet states permits detailed study of the roles that the initial atom-atom interaction, photon momentum transfer during Rydberg excitation, and sample temperature play in determining the spectral lineshape and final dimer rotational distribution. The results are in good agreement with the predictions of a model that includes these effects. The present work further highlights the sensitivity of ultralong-range Rydberg molecule formation to the state of the initial cold gas.
... An ultralong-range Rydberg molecule consists of a Rydberg atom with principal quantum number n and a distant (located R ∼ n 2 a 0 away) "perturber" atom in its electronic ground state. Nonadiabatic physics are a particularly interesting aspect of this system due to the close connection between the potential energy curves and the Rydberg wave functions [17][18][19][20]. Additionally, the large size of the molecules makes them an ideal laboratory to explore beyond Born-Oppenheimer physics on exaggerated scales, and the flexibility provided by Rydberg state parameters allows for controllable enhancement or suppression of nonadiabatic effects and the possibility to steer ultracold chemical reactions. ...
... We additionally neglect coupling to additional n levels. Both of these assumptions 2 are well justified here [20,36] and permit the replacement ofĤ e (r) by the number E n = −1/(2n 2 ) and the development of analytic expressions. In the following we define all such energies relative to E n . ...
... (A4)] as a sum over the degenerate and m states. This summation can be performed analytically, as described in [20,28,29]. In doing so, allQ terms can be defined in terms of only the s-wave radial wave function and its derivative, as summarized below. ...
We consider nonadiabatic coupling in the “trilobite”-like long-range Rydberg molecules created by perturbing degenerate high-ℓ Rydberg states with a ground-state atom. Due to the flexibility granted by the high Rydberg level density, the avoided crossings between relevant potential energy curves can become extremely narrow, leading to highly singular nonadiabatic coupling. We find that the gap between the trilobite potential curve and neighboring “butterfly” or “dragonfly” potential curves can even vanish, as in a conical intersection, if the gap closes at an internuclear distance which matches a node of the s-wave radial wave function. This is an unanticipated outcome of Kato's theorem.
... the last term describes its interaction with the scatterers using the Fermi pseudopotential, valid in the low-energy scattering limit. The interaction strength is determined by the s-wave scattering length a s [37,38], and is is too weak to mix Rydberg states with different principal quantum numbers ν [39,40]. Instead, it splits the degenerate Rydberg levels with different angular momentum l but the same ν into two subspaces [41]. ...
... Instead, it splits the degenerate Rydberg levels with different angular momentum l but the same ν into two subspaces [41]. The first, of size ν 2 − M, remains degenerate and unshifted, while the second, of size M, splits away [34,38]. It is in this shifted manifold that topologically protected edge states can be realized. ...
... Since the spectrum of this Hamiltonian coincides exactly, within the stated approximations, with that of the Hamiltonian H e it is possible to realize a given H by tuning the parameters of H e to achieve the desired matrix elements E q and V qq . In the high ν limit where the effect of quantum-defect-shifted states is negligible, these matrix elements are given in closed form [38]: ...
We examine topological phases and symmetry-protected electronic edge states in the context of a Rydberg composite: a Rydberg atom interfaced with a structured arrangement of ground-state atoms. We show that the spectrum of the electronic Hamiltonian of such a composite possesses a mapping to that of a tight-binding Hamiltonian, which can exhibit nontrivial topology depending on the arrangement of the ground-state atoms and the principal quantum number of the Rydberg state. The Rydberg electron moves in a combined potential including the long-ranged Coulomb interaction with the Rydberg core and short-ranged interactions with each neutral atom; the effective hopping amplitudes between sites are determined by this combination. We first confirm the existence of topologically-protected edge states in a Rydberg composite by mapping it to the paradigmatic Su-Schrieffer-Heeger dimer model. Following that, we show that more complicated systems with trimer unit cells can be studied in a Rydberg composite.
... In the spin-independent description of LRRMs, perturbation theory provides a way of finding very good approximated analytic Rydberg electron wave functions and PECs [1,28,47]. Since states with low-have nonzero quantum defects with large noninteger parts they are energetically well differentiated. As increases the quantum defects rapidly get smaller and are almost j-independent. ...
... Here we extend the perturbative analysis presented in Refs. [1,47] to include spin effects. The first step is to define which part of the electronic Hamiltonian in Eq. (1) is identified as the unperturbed Hamiltonian. ...
... First, we express the electronic state with the spin of the Rydberg electron written explicitly. Starting from Eq. (18), whenever the j dependence of the high-quantum defects is neglected, the electronic orbitals can be approximately written as where | (n) LM L are the spin-independent trilobite and butterfly orbitals [47]. Substituting Eq. (22) in the expressions for the perturbative states, we find molecular states in which the spatial and spin degrees of freedom are completely separated. ...
An operator that generates an approximate symmetry of long-range Rydberg molecules (LRRMs) formed by two alkali atoms, one in a Rydberg state and the other in the ground state, is identified. This is first done by evaluating the natural orbitals associated with a variational calculation of the binding wave function within the Born-Oppenheimer description of the molecule including s and p Fermi pseudopotential and the hyperfine structure energy terms. The resulting orbitals with the highest occupation number are shown to be identical to those obtained by a perturbative model for high angular momentum—trilobite and butterfly—LRRMs. Whenever the slight dependence of the quantum defects of the Rydberg electron on its total momentum j⃗=ℓ⃗+s⃗1 can be neglected, the symmetry operator of the high angular momentum LRRMs orbitals is identified as the sum of the spin of the Rydberg electron s⃗1, spin of the valence electron s⃗2, and the spin of nucleus i⃗ of the ground-state atom, N⃗=s1⃗+s2⃗+i⃗. The spin orbitals that diagonalize N⃗ define compact basis sets for the description of LRRMs beyond the aforementioned approximations. The matrix elements of the Hamiltonian in these basis sets have simple expressions, so that the relevance of triplet and singlet contributions can be directly estimated. The expected consequences of this approximate spin-symmetry on the spectra of LRRMs are briefly described.
... Ultralong-range Rydberg molecules (ULRMs) [10][11][12] are a platform for creating such dipolar molecules in ultracold environments. In these molecules, a neutral ground state atom is trapped inside the giant electronic wavefunction of a Rydberg state by a binding mechanism stemming from the electron-ground state scattering interaction. ...
In trilobite Rydberg molecules, an atom in the ground state is bound by electron-atom scattering to a Rydberg electron that is in a superposition of high angular momentum states. This results in a homonuclear molecule with a permanent electric dipole moment in the kilo-debye range. Trilobite molecules have previously been observed only with admixtures of low-l states. Here we report on the observation of two vibrational series of pure trilobite Rubidium-Rydberg molecules that are nearly equidistant. They are produced by three-photon photoassociation and lie energetically more than 15 GHz below the atomic 22F state of rubidium. We show that these states can be used to measure the electron-atom scattering length at low energies in order to benchmark current theoretical calculations. In addition to measuring their kilo-Debye dipole moments, we also show that the molecular lifetime is increased compared to the 22F state due to the high-l character. The observation of an equidistant series of vibrational states opens the way to observe coherent molecular wave packet dynamics.
... The first two terms govern the electron's motion in the Coulomb field of the Rydberg core, while the last term describes its interaction with the scatterers using the Fermi pseudopotential, valid in the lowenergy scattering limit. The interaction strength is determined by the S-wave scattering length a s [37,38]. We neglect the role of quantum defects, as we focus on the perturbation of the degenerate hydrogen-like states with angular momentum greater than 3 [34]. ...
... This mapping relies on the fact that the electron-scatterer interaction [39,40] is too weak to mix Rydberg states with different principal quantum numbers ν, but nevertheless it splits the degenerate energy levels with different angular momentum l but the same ν into two subspaces. One, of size ν 2 − M , remains degenerate and unshifted, while the second, of size M , splits away [34,38]. The spectrum of the Hamiltonian H e restricted to this non-trivial subspace coincides exactly with that The black (pink) curve shows the interaction V as a function of the distance D6 (D5) away from site 6 (5). ...
We examine topological phases and symmetry-protected electronic edge states in the context of a Rydberg composite: a Rydberg atom interfaced with a structured arrangement of ground-state atoms. The electronic Hamiltonian of such a composite possesses a direct mapping to a tight-binding Hamiltonian, which enables the realization and study of a variety of systems with non-trivial topology by tuning the arrangement of ground-state atoms and the excitation of the Rydberg atom. The Rydberg electron moves in a combined potential including the long-ranged Coulomb interaction with the Rydberg core and short-ranged interactions with each neutral atom; the effective interactions between sites are determined by this combination. We first confirm the existence of topologically-protected edge states in a Rydberg composite by mapping it to the paradigmatic Su-Schrieffer-Heeger dimer model. Following that, we study more complicated systems with trimer unit cells which can be easily simulated with a Rydberg composite.
... Here we report the experimental realisation of this platform, using a BEC to herald the internal conversion of an electronic bound state, initialised in a Rydberg atomic orbital, into a Rydberg molecular trilobite orbital [10][11][12]. These exotic molecules are bound through elastic collisions of a highly excited Rydberg electron [13] with condensate atoms inside its orbital volume, and exist in a diverse zoo of molecular states [10][11][12]. ...
... Here we report the experimental realisation of this platform, using a BEC to herald the internal conversion of an electronic bound state, initialised in a Rydberg atomic orbital, into a Rydberg molecular trilobite orbital [10][11][12]. These exotic molecules are bound through elastic collisions of a highly excited Rydberg electron [13] with condensate atoms inside its orbital volume, and exist in a diverse zoo of molecular states [10][11][12]. The conversion is driven by non-adiabatic motion of the atoms in the condensate within the orbital sphere of the Ryd- (d) Data from (c) but with the BEC background in the absence of Rydberg excitations (ρ bg ) subtracted, yielding the change of the column density ∆ρc = ρc − ρ bg due to the impact of the Rydberg electron inside the marked region. ...
We often infer the state of systems in nature indirectly, for example in high energy physics by recording the tracks particles leave behind in an ambient medium. We adapt this principle to energies $9$ orders of magnitude smaller, to classify the final state of exotic molecules after internal conversion of their electronic state, through their interaction with an ambient quantum fluid, a~Bose-Einstein condensate. The BEC is the ground-state of a million bosonic atoms near zero temperature, and a single embedded ultra-long range Rydberg molecule can coherently excite waves in this fluid, which carry tell-tale signatures of its dynamics. Bond lengths exceeding a micrometer allow us to observe the molecular fingerprint on the BEC in situ, via optical microscopy. Interpreting images in comparison with simulations shows that the molecular electronic state rapidly converts from the initially excited S- and D-orbitals to a much more complex molecular state (called ``trilobite''), marked by a maximally localized electron. This internal conversion liberates energy, such that one expects final state particles to move rapidly through the medium, which is however ruled out by comparing experiment and simulations. The molecule thus must strongly decelerate in the medium, for which we propose a plausible mechanism. Our experiment demonstrates a coherent medium that facilitates and records an electronic state change of embedded exotic molecules in ultra-cold chemistry, with sufficient sensitivity to constrain velocities of final state particles.
... The presence of occasionally divergent terms in the Hamiltonian produces instabilities that require renormalization when standard methods for computing energy eigenvalues are utilized, such as diagonalization of H in a truncated expansion into an orthonormal basis set. That diagonalization approach has until now been the method of choice for calculations of the Rydberg molecule Born-Oppenheimer potential energy curves; this preference is in part because of the relative ease with which one can add additional spin-spin and spin-orbit interaction terms to the Hamiltonian, when higher precision is desired [30][31][32][33][34]. (While Ref. [34] and Ref. [30] have both developed Hamiltonians for the diagonalization method which include all of the spin degrees of freedom treated in the present study, we recommend that calculations using this method should preferably implement the final result of Ref. [30], for the reasons discussed in that article.) ...
... The hyperfine splitting, atomic quantum defects, and polarizabilities utilized in our calculations have been determined via precision spectroscopy to very high accuracy and are collected in Ref. [31]. The electron-atom scattering phase shifts, on the other hand, are only available from theoretical calculations and the values of key properties, such as the zero-energy scattering lengths and shape resonance widths and positions, vary from source to source [31,33]. ...
... The hyperfine splitting, atomic quantum defects, and polarizabilities utilized in our calculations have been determined via precision spectroscopy to very high accuracy and are collected in Ref. [31]. The electron-atom scattering phase shifts, on the other hand, are only available from theoretical calculations and the values of key properties, such as the zero-energy scattering lengths and shape resonance widths and positions, vary from source to source [31,33]. For example, relativistic e-Rb phase shifts for L ≤ 1 were published by Fabrikant and coworkers in Ref. [36], and are shown as black(dark) curves in Fig. 1. ...
The determination of ultra-long-range molecular potential curves has been reformulated using the Coulomb Greens function to give a solution in terms of the roots of an analytical determinantal equation. For a system consisting of one Rydberg atom with fine structure and a neutral perturbing ground state atom with hyperfine structure, the solution yields potential energy curves and wavefunctions in terms of the quantum defects of the Rydberg atom and the electron-perturber scattering phaseshifts and hyperfine splittings. This method provides a promising alternative to the standard currently utilized method of diagonalization, which suffers from problematic convergence issues and nonuniqueness, and can potentially yield a more quantitative relationship between Rydberg molecule spectroscopy and electron-atom scattering phaseshifts.
... Unlike the spherically symmetric eigenstates of the bare Rydberg atom, which extend over the entire scatterer array [ Fig. 1(d)], the trilobite state is peaked at the scatterer's position [ Fig. 1(e)]. The matrix elements H qq are proportional to the trilobite overlaps T q |T q or, equivalently, the amplitudes R q |T q [27][28][29]. This provides a convenient means to pictorially estimate the properties of the tight-binding Hamiltonian, as in Fig. 1(e). ...
... This provides a convenient means to pictorially estimate the properties of the tight-binding Hamiltonian, as in Fig. 1(e). Furthermore, closed-form expressions for T q |T q simplify calculations and facilitate asymptotic expansions, as discussed in Appendix E [28,29]. ...
... where V sr (r) is an empirically derived potential parametrizing the effect of the multielectron core [29]. Because of this non-Coulombic potential, the eigenenergies of H Ryd are in general different from those of hydrogen. ...
Highly excited Rydberg atoms inherit their level structure, symmetries, and scaling behavior from the hydrogen atom. We demonstrate that these fundamental properties enable a thermodynamic limit of a single Rydberg atom subjected to interactions with nearby ground-state atoms. The limit is reached by simultaneously increasing the number of ground-state atoms and the level of excitation of the Rydberg atom, for which the Coulomb potential supplies infinitely many and highly degenerate excited states. Our study reveals a surprising connection to an archetypal concept of condensed matter physics, Anderson localization, facilitated by a direct mapping between the Rydberg atom's electronic spectrum and the spectrum of a tight-binding Hamiltonian. The hopping amplitudes of this tight-binding system are determined by the arrangement of ground-state atoms and can range from oscillatory and long-ranged to nearest-neighbor. In the latter we identify clear signatures of the Anderson localization of the Rydberg electron.
... Ultralong-range Rydberg molecules (ULRMs) [10][11][12] are a platform for creating such dipolar molecules in ultracold environments. In these molecules a neutral ground state atom is trapped inside the giant electronic wavefunction of a Rydberg state by a binding mechanism stemming from the electron-ground state scattering interaction. ...
We report on the observation of two vibrational series of pure trilobite rubidium Rydberg molecules. They are created via three-photon photoassociation and lie energetically more than 15 GHz below the atomic 22$F$ state of rubidium. In agreement with theoretical calculations, we find an almost perfect harmonic oscillator behavior of six vibrational states. We show that these states can be used to measure electron-atom scattering lengths for low energies in order to benchmark current theoretical calculations. The molecules have extreme properties: their dipole moments are in the range of kilo-Debye and the electronic wave function is made up of high angular momentum states with only little admixture from the nearby 22$F$ state. This high-$l$ character of the trilobite molecules leads to an enlarged lifetime as compared to the 22$F$ atomic state. The observation of an equidistant series of vibrational states opens an avenue to observe coherent molecular wave-packet dynamics.
