Calculation of Rydberg Interaction Potentials

Tutorial Calculation of Rydberg interaction potentials Sebastian Weber,1, ∗ Christoph Tresp,2, 3 Henri Menke,4 Alban Urvoy,2, 5 Ofer Firstenberg,6 Hans Peter Büchler,1 and Sebastian Hofferberth2, 3, † 1Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, UniversitätStuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany 25. Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany 3Department of Physics, Chemistry and Pharmacy, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark 4Max Planck Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany 5Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 6Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel (Dated: June 7, 2017) The strong interaction between individual Rydberg atoms provides a powerful tool exploited in an ever-growing range of applications in quantum information science, quantum simulation, and ultracold chemistry. One hallmark of the Rydberg interaction is that both its strength and angular dependence can be fine-tuned with great flexibility by choosing appropriate Rydberg states and applying external electric and magnetic fields. More and more experiments are probing this interaction at short atomic distances or with such high precision that perturbative calculations as well as restrictions to the leading dipole-dipole interaction term are no longer sufficient. In this tutorial, we review all relevant aspects of the full calculation of Rydberg interaction potentials. We discuss the derivation of the interaction Hamiltonian from the electrostatic multipole expansion, numerical and analytical methods for calculating the required electric multipole moments, and the inclusion of electromagnetic fields with arbitrary direction. We focus specifically on symmetry arguments and selection rules, which greatly reduce the size of the Hamiltonian matrix, enabling the direct diagonalization of the Hamiltonian up to higher multipole orders on a desktop computer. Finally, we present example calculations showing the relevance of the full interaction calculation to current experiments. Our software for calculating Rydberg potentials including all features discussed in this tutorial is available as open source. I. INTRODUCTION interaction over long range and facilitates the extension to multi-qubit Rydberg-mediated gates in such registers Among the many fascinating properties of highly ex- [34–36]. Such tailored atomic ensembles are also ideal for cited Rydberg atoms [1], the strong interaction between investigating processes such as excitation transfer [5, 37– pairs of Rydberg atoms has proven to be the key feature 40] or simulation of spin systems [41, 42]. for diverse applications in quantum information process- The mapping of Rydberg interactions onto photons ing and quantum simulation [2]. A particularly important by means of electromagnetically induced transparency concept is the Rydberg blockade [3], where the excitation (EIT) [43] has emerged as a powerful approach to realiz- of two or more atoms to a Rydberg state is prevented ing few-photon optical nonlinearities [44–50], enabling a due to the interaction. The Rydberg blockade of atomic variety of optical quantum information applications such arXiv:1612.08053v2 [quant-ph] 6 Jun 2017 ensembles [4] has been observed in ultracold atomic sys- as highly efficient single-photon generation [51], entan- tems in the frozen Rydberg-gas regime [5, 6], both in bulk glement generation between light and atomic excitations ensembles [7–18] and in small systems supporting only [52], single-photon all-optical switches [53] and transistors a single excitation [19–22]. More recently, experiments [54–56], single-photon subtraction [57], and interaction- have also begun to probe Rydberg interaction effects induced π-phase shifts [58]. Additionally, Rydberg EIT in room-temperature thermal vapor [23–25]. Based on provides access to novel phenomena such as attractive the Rydberg blockade, atomic two-qubit gates have been interaction between single photons [59], crystallization demonstrated [2, 26–28] as basic building blocks for large- of photons [60], or photonic scattering resonances [61], scale neutral atom quantum registers [29–33]. In turn, as well as spatially resolved detection of single Rydberg the power-law decay of the Rydberg interaction provides atoms in a bulk medium [37, 62, 63]. Recent Rydberg EIT experiments, which simultaneously use Rydberg S- and P -states [64] or two different S-states [37, 54, 55] ∗ [email protected] further increase the flexibility of manipulating weak light † hoff[email protected] fields [65, 66]. 2 Detailed understanding of the Rydberg interaction is calculations including the external fields reveal the num- also required for the concept of Rydberg dressing, where ber of states which must be included in specific cases for a small Rydberg admixture modifies the interaction be- accurate results. tween ground-state atoms in ultracold gases [67–72]. In As consequence of the rapid evolution of the field, it particular, by the choice of the Rydberg state, one can becomes more and more common to rely on numerical map the anisotropy of the Rydberg interaction onto the diagonalization of the Rydberg interaction Hamiltonian ground-state atoms [73–75]. Experimental demonstra- including higher orders of the multipole expansion [81, tions of Rydberg dressing have recently been performed 82, 94, 97]. One example is the recently released ARC using individual atoms [76] or atomic ensembles in an library [111], which provides powerful tools for calculating optical lattice [77]. Finally, the rich structure of the Ry- alkali Rydberg properties. In this tutorial, we discuss all dberg interaction potentials supports bound molecular relevant steps required for the numerical calculation of states formed by two Rydberg atoms [78], which have pairwise Rydberg interaction potentials: been observed in experiments [79–81]. Prediction of the 1. Construction of the single-atom Hamiltonian from equilibrium distance and vibrational spectra for these the orbital wave functions (A) and their transition macro-dimers requires precise knowledge of the interac- matrix elements (B andC) in the absence of external tion potential [81–84]. fields. The physics of Rydberg interaction has been well- established for decades [85]. As long as the two atoms are 2. Derivation of the interaction Hamiltonian for a well separated and their wave functions do not overlap, pair of Rydberg atoms by multipole expansion (Sec. one needs to consider only the electrostatic interaction II A). between two localized charge distributions, most conve- 3. Inclusion of external electric and magnetic fields in niently utilizing the well-known electric multipole expan- arbitrary directions relative to the inter-atomic axis sion in spherical coordinates [86–88]. The leading relevant (Sec. IIB). term in this expansion is the dipole-dipole interaction [89], which for unperturbed Rydberg atoms at large separation 4. Application of selection rules and symmetry argu- results in the extensively studied van-der-Waals interac- ments to reduce the size of the Hilbert space to the tion [90–93]. More generally, as long as the interaction relevant states (Sec. IIC). energies are small compared to the level spacing of the 5. Rotation of the interaction Hamiltonian to a par- unperturbed Rydberg pair states, perturbative calcula- ticular coordinate system, given for example by the tions offer a very convenient method for determining the direction of an incident excitation laser (Sec. IID). radial [90] and angular [91, 92] behavior of the Rydberg potentials. 6. Diagonalization and extraction of the interaction Nevertheless, the rapid experimental progress in re- potentials (Sec. IIE). In the context of two practi- cent years has led to a growing number of experiments cal applications (Sec. III), we discuss best practices for which the perturbative calculation is no longer suf- and specific considerations in the relevant scenar- ficient. For example, this approach fails when shorter ios. These examples illustrate the capabilities of atomic distances are probed and state-mixing due to the the presented approach and the agreement with interaction becomes significant. The dipole-quadrupole experiments. contribution to the interaction has recently been observed Our software for numerical Rydberg potential both in ultracold [94] and room-temperature [25] systems calculations, which includes all features we dis- in experiments with large excitation bandwidth, while the cuss in this review, is available open source from correct prediction of macro-dimer photo-association spec- https://pairinteraction.github.io/. tra required the inclusion of terms up to octupolar order in the potential calculation [81, 84]. In experiments utilizing very high principal quantum numbers n > 100 [95–97], II. RYDBERG INTERACTION state-mixing and additional molecular resonances [98] be- come relevant already at large interatomic distances and A. Hamiltonian make non-perturbative potential calculations necessary. The nature of the interaction also changes when cou- We study two neutral atoms, each having one electron pled pair states are (nearly) resonant [99, 100]. Such excited into a Rydberg state, as depicted in Figure1 (a). degeneracies,

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