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.
This Ph.D. 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 scattering off of the ground state atoms, exerts a weak attractive force sufficient to form the molecule in long-range oscillatory potential wells. In the last decade this topic has burgeoned into a vibrant and mature subfield of atomic and molecular ph…
We present quantum mechanical calculations of Auger decay rates for two Rubidium Rydberg atoms with weakly overlapping electron clouds. We neglect exchange effects and consider tensor products of independent atom states forming an approximate basis of the two-electron state space. We consider large sets of two-atom states with randomly chosen quantum numbers and find that the charge overlap between the two Rydberg electrons allows one to characterise the magnitude of the Auger decay rates. If the electron clouds overlap by more than one percent, the Auger decay rates increase approximately exp…
The Jaynes-Cummings (JC) model is a milestone in the theory of coherent interaction between a two-level system and a single bosonic field mode. This tutorial aims to give a complete description of the model, analyzing the Hamiltonian of the system, its eigenvalues and eigestates, in order to characterize the dynamics of system and subsystems. The Rabi oscillations, together with the collapse and revival effects, are distinguishing features of the JC model and are important for applications in Quantum Information theory. The framework of cavity quantum electrodynamics (cQED) is chosen and two f…
We experimentally study the effects of the anisotropic Rydberg-interaction on $D$-state Rydberg polaritons slowly propagating through a cold atomic sample. In addition to the few-photon nonlinearity known from similar experiments with Rydberg $S$-states, we observe the interaction-induced dephasing of Rydberg polaritons at very low photon input rates into the medium. We develop a model combining the propagation of the two-photon wavefunction through our system with nonperturbative calculations of the anisotropic Rydberg-interaction to show that the observed effect can be attributed to pairwise…
We investigate the electronic structure of a triatomic Rydberg molecule formed by a Rydberg atom and two neutral ground-state atoms. Taking into account the $s$-wave and $p$-wave interactions we perform electronic structure calculations and analyze the adiabatic electronic potentials evolving from the Rb$(n=35, l\ge 3)$ Rydberg degenerate manifold. We hereby focus on three different classes of geometries of the Rydberg molecules, including symmetric, asymmetric and planar configurations. The metamorphosis of these potential energy surfaces in the presence of an external electric field is explo…
Over the past years, machine learning has emerged as a powerful computational tool to tackle complex problems over a broad range of scientific disciplines. In particular, artificial neural networks have been successfully deployed to mitigate the exponential complexity often encountered in quantum many-body physics, the study of properties of quantum systems built out of a large number of interacting particles. In this Article, we overview some applications of machine learning in condensed matter physics and quantum information, with particular emphasis on hands-on tutorials serving as a quick-…
We calculate pair potential curves for interacting Rydberg atoms in a constant electric field and use them to determine the effective $C_3$ dipole-dipole and $C_6$ van der Waals coefficients. We compare the $C_3$ and $C_6$ with experiments where the angle of a polarizing electric field is varied with respect to the axis of a quasi-1-dimensional trap at ultracold temperatures. The dipoles produced via polarization of the atoms have an angular dependent dipole-dipole interaction. We focus on the interaction potential of two rubidium Rydberg atoms in $50S_{1/2}$ states in the blockade regime. For…
We show that ultracold chemical reactions can be manipulated and controlled by using Rydberg-dressed interactions. Scattering in the ultracold regime is sensitive to long-range interactions, especially when weakly bound (or quasi-bound) states exist near the collision threshold. We investigate how, by Rydberg-dressing a reactant, one enhances its polarizability and modifies the long-range van der Waals collision complex, which can alter chemical reaction rates by shifting the position of near threshold bound states. We carry out a full quantum mechanical scattering calculation for the benchmar…
We describe the calculation of hydrogenic (one-loop) Bethe logarithms for all states with principal quantum numbers n <= 200. While, in principle, the calculation of the Bethe logarithm is a rather easy computational problem involving only the nonrelativistic (Schroedinger) theory of the hydrogen atom, certain calculational difficulties affect highly excited states, and in particular states for which the principal quantum number is much larger than the orbital angular momentum quantum number. Two evaluation methods are contrasted. One of these is based on the calculation of the principal value…
In this paper a coherent superposition of angular momentum states created by absorption of polarized light by molecules is analyzed. Attention is paid to the time evolution of wave packets representing spatial orientation of internuclear axis of diatomic molecule. Two examples are considered in detail. Molecules absorbing light in a permanent magnetic field experiencing Zeeman effect and molecules absorbing light in a permanent electric field experiencing quadratic Stark effect. In a magnetic field we have a wave packet that evolves in time exactly as classical dipole oscillator in a permanent…
We propose a method for reducing the sensitivity of atomic ground to Rydberg transitions to stray dc electric fields, using microwave-induced dressing of Rydberg states. Calculations are presented for the Cs $\state{90S}{1/2}$ and $\state{90P}{3/2}$ states. With zero dc bias electric field, a two-frequency ac field is used to simultaneously reduce the sensitivity of both states to dc field variations. The sensitivity reduction is a factor of 95 for the $\state{90S}{1/2}$ state and a factor of 1600 for the $\state{90P}{3/2}, m_J=3/2$ state. We also show how the two-frequency ac field can be use…
We present our results on the experiments with cold Rb Rydberg atoms in a magneto-optical trap (MOT). Characteristic features of our experiment were the excitation of Rydberg atoms in a small volume within the cold atom cloud and sorting of the measured signals and spectra over the number of registered Rydberg atoms. We have measured the effective lifetime of the Rydberg state 37P, as well as its polarizability in a weak electric field. The results are in good agreement with the theoretical calculations. We have shown that localization of the small excitation volume around the zero-magnetic-fi…
We show that sizeable Abelian and non-Abelian gauge fields arise in the relative motion of two dipole-dipole interacting Rydberg atoms. Our system exhibits two magnetic monopoles for adiabatic motion in one internal two-atom state. These monopoles occur at a characteristic distance between the atoms that is of the order of one micron. The deflection of the relative motion due to the Lorentz force gives rise to a clear signature of the effective magnetic field. In addition, we consider non-adiabatic transitions between two near-degenerate internal states and show that the associated gauge field…
A tutorial on quantum scattering theory and coupled-channel calculations of bimolecular collisions
Freie Universität Berlin, Dahlem Center for Complex Quantum Systems and Fachbereich Physik（柏林自由大学，达勒姆复杂量子系统中心及物理系）；Technische Universität Dortmund（多特蒙德工业大学）
Results of theoretical calculations of ionization rates of Rb and Na Rydberg atoms by blackbody radiation (BBR) are presented. Calculations have been performed for nS, nP and nD states of Na and Rb, which are commonly used in a variety of experiments, at principal quantum numbers n=8-65 and at three ambient temperatures of 77, 300 and 600 K. A peculiarity of our calculations is that we take into account the contributions of BBR-induced redistribution of population between Rydberg states prior to photoionization and field ionization by extraction electric field pulses. The obtained results show…
We present calculations of the photoionization (PI) cross sections of rubidium and cesium Rydberg atoms for light with wavelengths ranging from the infrared to the ultraviolet, using model potentials from [M. Marinescu, H. R. Sadeghpour, and A. Dalgarno, Phys. Rev. A 49, 982 (1994)]. The origins of pronounced PI minima are identified by investigating the free-electron wavefunctions. These include broad PI minima in the $nS$ to $εP$ PI channels of both Rb and Cs, with free-electron energy $ε$, which are identified as Cooper minima. Much narrower PI minima in the $nD$ to $εF$ channels are due to…
We present an efficient method for producing $N$ particle entangled states using Rydberg blockade interactions. Optical excitation of Rydberg states that interact weakly, yet have a strong coupling to a second control state is used to achieve state dependent qubit rotations in small ensembles. On the basis of quantitative calculations we predict that an N=8 Schrödinger cat state can be produced with a fidelity of 84% in cold Rb atoms.
Spectral line width broadening in Rydberg gases, a phenomenon previously attributed to the many-body effect, was observed experimentally almost a decade ago. The observed line width was typically 80-100 times larger than the average interaction strength predicted from a binary interaction. The interpretation of such a phenomenon is usually based on the so-called diffusion model, where the line width broadening mostly originates from the diffusion of excitations. In this paper, we present a model calculation to show that diffusion is not the main mechanism to the line width broadening. We find …
We analyze the effects of Zeeman degeneracies on the long-range interactions between like Rydberg atoms, with particular emphasis on applications to quantum information processing using van der Waals blockade. We present a general analysis of how degeneracies affect the primary error sources in blockade experiments, emphasizing that blockade errors are sensitive primarily to the weakest possible atom-atom interactions between the degenerate states, not the mean interaction strength. We present explicit calculations of the van der Waals potentials in the limit where the fine-structure interacti…
We obtain ab initio the Hubbard parameters for Rydberg-dressed atoms in an one-dimensional sinusoidal optical lattice in the basis of maximally localized wannier states. Finite range, soft-core inter-atomic interactions become the trait of Rydberg admixed atoms, which can be extended over many neighbouring lattice sites. On contrary to dipolar gases, where the interactions follow an inverse cubic law, the key feature of Rydberg-dressed interactions being the possibility of making neighbouring couplings to the same magnitude as that of the onsite ones. The maximally localized Wannier functions …
Using the reactance matrix approach, we systematically develop new multichannel quantum defect theory models for the singlet and triplet S, P, D and F states of strontium based on improved energy level measurements. The new models reveal additional insights into the character of doubly excited perturber states, and the improved energy level measurements for certain series allow fine structure to be resolved for those series' perturbers. Comparison between the predictions of the new models and those of previous empirical and \emph{ab initio} studies reveals good agreement with most series, howe…
In this paper, we report on numerical calculations of the spontaneous emission rates and Lamb shifts of a $^{87}\text{Rb}$ atom in a Rydberg-excited state $\left(n\leq30\right)$ located close to a silica optical nanofiber. We investigate how these quantities depend on the fiber's radius, the distance of the atom to the fiber, the direction of the atomic angular momentum polarization as well as the different atomic quantum numbers. We also study the contribution of quadrupolar transitions, which may be substantial for highly polarizable Rydberg states. Our calculations are performed in the macr…
We present a detailed error analysis of a Rydberg blockade mediated controlled-NOT quantum gate between two neutral atoms as demonstrated recently in Phys. Rev. Lett. 104, 010503 (2010) and Phys. Rev. A 82, 030306 (2010). Numerical solutions of a master equation for the gate dynamics, including all known sources of technical error, are shown to be in good agreement with experiments. The primary sources of gate error are identified and suggestions given for future improvements. We also present numerical simulations of quantum process tomography to find the intrinsic fidelity, neglecting technic…
We have recently shown that Alkali atoms contained in a vapor cell can serve as a highly accurate standard for microwave electric field strength as well as polarization using the principles of Rydberg atom electromagnetically induced transparency. Here, we show, for the first time, that Rydberg atom electromagnetically induced transparency can be used to image microwave electric fields with unprecedented precision. The spatial resolution of the method is far into the sub-wavelength regime. The electric field resolutions are similar to those we have demonstrated in our prior experiments. Our ex…
Transfer mechanism of orbital angular moment(OAM) of light to trapped ground-state atoms under paraxial approximation is well known. Here we show how optical OAM of a Laguerre-Gaussian(LG) beam under paraxial approximation can be transferred to trapped Rydberg atoms. Optical OAM is shown to be transferable to a Rydberg electronic state in dipole transition. The Gaussian part of the profile of the LG beam, which is generally neglected , is found to have an important effect on the OAM transfer to the Rydberg atoms. Numerical calculations are calculated based on this theory for Rubidium Rydberg a…
The strong interaction between Rydberg atoms can be used to control the strength and character of the interatomic interaction in ultracold gases by weakly dressing the atoms with a Rydberg state. Elaborate theoretical proposals for the realization of various complex phases and applications in quantum simulation exist. Also a simple model has been already developed that describes the basic idea of Rydberg dressing in a two-atom basis. However, an experimental realization has been elusive so far. We present a model describing the ground state of a Bose-Einstein condensate dressed with a Rydberg …
Non dispersive electronic Rydberg wave packets may be created in atoms illuminated by a microwave field of circular polarization. We discuss the spontaneous emission from such states and show that the elastic incoherent component (occuring at the frequency of the driving field) dominates the spectrum in the semiclassical limit, contrary to earlier predictions. We calculate the frequencies of single photon emissions and the associated rates in the "harmonic approximation", i.e. when the wave packet has approximately a Gaussian shape. The results agree well with exact quantum mechanical calculat…
We study a system of atoms that are laser-driven to $nD_{3/2}$ Rydberg states and assess how accurately they can be mapped onto spin-$1/2$ particles for the quantum simulation of anisotropic Ising magnets. Using non-perturbative calculations of the pair interaction potentials between two atoms in the presence of both electric and magnetic fields, we emphasize the importance of a careful selection of the experimental parameters in order to maintain the Rydberg blockade and avoid excitation of unwanted Rydberg states. We then benchmark these theoretical observations against experiments using two…
We report on numerical calculations of the spontaneous emission rate of a Rydberg-excited sodium atom in the vicinity of an optical nanofibre. In particular, we study how this rate varies with the distance of the atom to the fibre, the fibre's radius, the symmetry s or p of the Rydberg state as well as its principal quantum number. We find that a fraction of the spontaneously emitted light can be captured and guided along the fibre. This suggests that such a setup could be used for networking atomic ensembles, manipulated in a collective way due to the Rydberg blockade phenomenon.
We report on the direct measurement of the van der Waals interaction between two isolated, single Rydberg atoms separated by a controlled distance of a few micrometers. By working in a regime where the single-atom Rabi frequency of the laser used for excitation to the Rydberg state is comparable to the interaction energy, we observe a \emph{partial} Rydberg blockade, whereby the time-dependent populations of the various two-atom states exhibit coherent oscillations with several frequencies. A quantitative comparison of the data with a simple model based on the optical Bloch equations allows us…
We analyze the zero-temperature phases of an array of neutral atoms on the kagome lattice, interacting via laser excitation to atomic Rydberg states. Density-matrix renormalization group calculations reveal the presence of a wide variety of complex solid phases with broken lattice symmetries. In addition, we identify a novel regime with dense Rydberg excitations that has a large entanglement entropy and no local order parameter associated with lattice symmetries. From a mapping to the triangular lattice quantum dimer model, and theories of quantum phase transitions out of the proximate solid p…
We have observed resonant energy transfer between cold Rydberg atoms in spatially separated cylinders. Resonant dipole-dipole coupling excites the 49s atoms in one cylinder to the 49p state while the 41d atoms in the second cylinder are transferred down to the 42p state. We have measured the production of the 49p state as a function of separation of the cylinders (0 - 80 um) and the interaction time (0 - 25 us). In addition we measured the width of the electric field resonances. A full many-body quantum calculation reproduces the main features of the experiments.
Analytical arguments and numerical simulations suggest that the shapes of 3D microwave ionization curves measured by Koch and collaborators (see P. M. Koch and K. A. H. van Leeuwen, Phys. Rep. {\bf 255}, 289 (1995)) depend only weakly on the angular momentum of the atoms in the initial microcanonical ensemble, but strongly on the principal quantum number and the magnetic quantum number. Based on this insight, coupled with the computational power of a high-end 60-node Beowulf PC cluster, we present the first 3D quantum calculations of microwave ionization curves in the experimentally relevant p…
We study the Landau-Zener (LZ) dynamics in a setup of two Rydberg atoms with time-dependent detuning, both linear and periodic, using both the exact numerical calculations as well as the method of adiabatic impulse approximation (AIA). By varying the Rydberg-Rydberg interaction strengths, the system can emulate different three-level LZ models, for instance, bow-tie and triangular LZ models. The LZ dynamics exhibits non-trivial dependence on the initial state, the quench rate, and the interaction strengths. For large interaction strengths, the dynamics is well captured by AIA. In the end, we an…
We calculate the bound state spectrum of the highly excited valence electron in the heavy alkali atoms solving the radial Schrödinger eigenvalue problem numerically with an accurate spectral collocation algorithm that applies also for a large principal quantum number $n\gg1$. As an effective single-particle potential we favor the reputable potential of Marinescu \emph{et al}., {[}Phys. Rev.A \textbf{49}, 982(1994){]}. Recent quasiclassical calculations of the quantum defect of the valence electron agree for orbital angular momentum $l=0,1,2,...$ overall remarkably well with the results of the …
We calculate properties of dipolar interacting ultracold molecules or Rydberg atoms in a semi-synthetic three-dimensional configuration -- one synthetic dimension plus a two-dimensional real space optical lattice or periodic microtrap array -- using the stochastic Green function Quantum Monte Carlo method. Through a calculation of thermodynamic quantities and appropriate correlation functions, along with their finite size scalings, we show that there is a second order transition to a low temperature phase in which two-dimensional `sheets' form in the synthetic dimension of internal rotational …
