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Statistical properties of interacting Rydberg gases

D. Breyel

Published Jan 1, 2014• 0 citations
Publisher

Abstract

The present thesis treats the interacting ultracold Rydberg gas with special emphasison the statistical footprint of the phase transition between unordered and crystalline phase, which can be understood as a consequence of the dipole blockade effect. After mapping the system onto an effective spin-1/2-model, exact diagonalization of the effective Hamiltonian is used to obtain the many-body ground state. Repeated application of this procedure on random realizations reveals the underlying probability distribution of the number of Rydberg atoms, allowing to calculate its statistical moments. In the regimes of weak and strong interaction these observables have power law character. The critical interaction strength is estimated by extrapolating these power laws up to their intersection point. The same procedure is applied to interacting excitons in bilayer heterostructures uncovering a phase transition here as well. Furthermore, new methods are introduced to handle the effects of finite detection efficiency and parameter fluctuations to establish a better connection between experimental and theoretical results. Finally, new models are introduced to include dynamics or additional Rydberg states. The last model is of purely statistical nature and its results may be used as a tool for detecting a potential clustering of Rydberg atoms.

DOI
10.11588/heidok.00016035

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The present thesis treats the interacting ultracold Rydberg gas with special emphasison the statistical footprint of the phase transition between unordered and crystalline phase, which can be understood as a consequence of the dipole blockade effect. After mapping the system onto an effective spin-1/2-model, exact diagonalization of the effective Hamiltonian is used to obtain the many-body ground state. Repeated application of this procedure on random realizations reveals the underlying probability distribution of the number of Rydberg atoms, allowing to calculate its statistical moments. In the regimes of weak and strong interaction these observables have power law character. The critical interaction strength is estimated by extrapolating these power laws up to their intersection point. The same procedure is applied to interacting excitons in bilayer heterostructures uncovering a phase transition here as well. Furthermore, new methods are introduced to handle the effects of finite detection efficiency and parameter fluctuations to establish a better connection between experimental and theoretical results. Finally, new models are introduced to include dynamics or additional Rydberg states. The last model is of purely statistical nature and its results may be used as a tool for detecting a potential clustering of Rydberg atoms.
In this thesis I present several theoretical methods to explore the excitation dynamics of Ultracold Rydberg systems in various regimes. For some applications the details of interactions between Rydberg atoms have to be carefully examined, while for others, the many-body aspect of Rydberg excitation may be crucial. In both cases, the basic prerequisite is to know the interactions sufficiently well. I will review the basic concepts how to evaluate molecular potentials in various Hund's cases and approximations. Long-range Rydberg-Rydberg interactions between Rydberg atoms induce ℓ-mixing which in certain situations gives rise to molecular resonances. For these resonances, we calculate long-range potentials in Hund's case (c) by diagonalization of an interaction matrix. The excitation dynamics of the resonances is always modeled as pair excitation. At high principal quantum number n, the interactions between Rydberg atoms can blockade the excitation of many surrounding atoms in the range of few μm. The atoms within this range are strongly correlated so that many-body treatments are, in general, needed. To describe this blockading effect and other manifestations of the collective behavior of Rydberg excitation, such as the many-body Rabi oscillations, spatial correlations between atoms and the fluctuations of the number of excited atoms, I have used three different approaches. In the mean-field approach the interactions between different atoms is modeled by a distribution of mean-field shifts for which a distribution of probabilities is calculated, A good agreement between the theoretical model and experimental measurements is found. To study the correlations between atoms the many-body wavefunction is numerically computed. The possibility of observing the many-body Rabi oscillations of Rydberg excitation and other aspects of many-body dynamics is analyzed. Although strong interactions cannot be treated perturbatively other expansions may be feasible. I will show that the expansion in powers of the single atom Rabi frequency can be evaluated in the interaction and Heisenberg picture. The expansion is expected to behave well for arbitrary strong interactions. ^
Recent achievements in ultra-cold experiments have made quantum simulation of interacting many-body systems possible in a well controllable environment. Of many candidates as quantum simulators, Rydberg atoms have been extensively utilised due to its exaggerated and fascinating atomic properties. Example includes high susceptibility to electric fields and relatively long life time in comparison to atoms in low-lying states. The tunable interaction between Rydberg atoms have made them even more versatile in simulating quantum many-body systems, e.g. interacting spin-1/2 particles. We will start the thesis by reviewing these properties of Rydberg atoms and explain how they lead to the Rydberg lattice gases that of interest. Following the review of the essential knowledge of Rydberg atoms, we first study the ground states of interacting Rydberg lattice gases in both one-dimension and two-dimensions. The many-body system we are interested in is initially prepared in a Mott-insulator state, with each lattice site containing one atom that is laser coupled to its highly excited Rydberg state. The extremely huge van der Waals interactions between Rydberg atoms at close distance leads to an interesting Rydberg blockade effect. As we shall show, these strong interactions lead to rich phases and critical behaviours in the ground states of the many-body Hamiltonians that describes the systems. The aim of the first three chapters is to analyse these ground states in detail. Having investigated the static properties, we then move on to study the dynamical behaviour of a class of generic spin models which can in principal be realised by Rydberg lattice gases with tunable blockade radius. By deriving an effective master equation, and comparing it to the exact calculation, we will demonstrate how different pure initial states eventually evolve to the same equilibrium state and analysed in detail the time evolution and the steady state.
The spectral structure underlying excitonic energy transfer in ultracold Rydberg gases is studied numerically, in the framework of random matrix theory, and via self-consistent diagrammatic techniques. Rydberg gases are made up of randomly distributed, highly polarizable atoms that interact via strong dipolar forces. Dynamics in such a system is fundamentally different from cases in which the interactions are of short range, and is ultimately determined by the spectral and eigenvector structure. In the energy levels' spacing statistics, we find evidence for a critical energy that separates delocalized eigenstates from states that are localized at pairs or clusters of atoms separated by less than the typical nearest-neighbor distance. We argue that the dipole blockade effect in Rydberg gases can be leveraged to manipulate this transition across a wide range: As the blockade radius increases, the relative weight of localized states is reduced. At the same time, the spectral statistics, in particular, the density of states and the nearest-neighbor level-spacing statistics, exhibits a transition from approximately a 1-stable L\'evy to a Gaussian orthogonal ensemble. Deviations from random matrix statistics are shown to stem from correlations between interatomic interaction strengths that lead to an asymmetry of the spectral density and profoundly affect localization properties. We discuss approximations to the self-consistent Matsubara-Toyozawa locator expansion that incorporate these effects.
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 level based on the Rydberg blockade. This approach provides an intuitive understanding of the transition from pure two-body interaction to a regime of collective interactions. Furthermore it enables us to calculate the deformation of a three-dimensional sample under realistic experimental conditions in mean-field approximation. We compare full three-dimensional numerical calculations of the ground state to an analytic expression obtained within Thomas–Fermi approximation. Finally we discuss limitations and problems arising in an experimental realization of Rydberg dressing based on our experimental results and point out possible solutions for future approaches. Our work enables the reader to straight forwardly estimate the experimental feasibility of Rydberg dressing in realistic three-dimensional atomic samples.
The dipole blockade of Rydberg excitations is a hallmark of the strong interactions between atoms in these high-lying quantum states [M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod. Phys. 82, 2313 (2010); D. Comparat and P. Pillet, J. Opt. Soc. Am. B 27, A208 (2010)]. One of the consequences of the dipole blockade is the suppression of fluctuations in the counting statistics of Rydberg excitations, of which some evidence has been found in previous experiments. Here we present experimental results on the dynamics and the counting statistics of Rydberg excitations of ultracold rubidium atoms both on and off resonance, which exhibit sub- and super-Poissonian counting statistics, respectively. We compare our results with numerical simulations using a novel theoretical model based on Dicke states of Rydberg atoms including dipole-dipole interactions, finding good agreement between experiment and theory.
Ultracold atomic gases excited to strongly interacting Rydberg states are a promising system for quantum simulations of many-body systems. The dipole blockade of Rydberg excitations is a hallmark of the strong interactions between atoms in these high-lying quantum states. We have measured the Rydberg excitation for rubidium ultracold atoms in magneto-optical traps and for Bose-Einstein condensates loaded into quasi one-dimensional traps. One of the consequences of the dipole blockade is the suppression of fluctuations in the counting statistics of Rydberg excitations. We have obtained experimental results on the dynamics and the counting statistics of Rydberg excitations of ultra-cold Rubidium atoms both on and off resonance, which exhibit sub- and super-Poissonian counting statistics, respectively. We have found strongly bimodal counting distributions in the offresonant regime.