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arXiv:2010.13728 · Physical Review A

Modeling Rydberg gases using random sequential adsorption on random graphs

Daan Rutten, Jaron Sanders

Published Mar 3, 2021• 0 citations
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Abstract

The statistics of strongly interacting, ultracold Rydberg gases are governed by the interplay of two factors: geometrical restrictions induced by blockade effects and quantum mechanical effects. To shed light on their relative roles in the statistics of Rydberg gases, we compare three models in this paper: a quantum mechanical model describing the excitation dynamics within a Rydberg gas, a random sequential adsorption (RSA) process on a random geometric graph (RGG), and a RSA process on a decomposed random intersection graph (DRIG). The last model refers to choosing a particular subgraph of a mixture of two other random graphs. Contrary to the first two models, it lends itself for a rigorous mathematical analysis, and it is built specifically to have particular structural properties of a RGG. We establish for it a fluid limit describing the time evolution of the number of Rydberg atoms and show numerically that the expression remains accurate across a wider range of particle densities than an earlier approach based on an RSA process on an Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi random graph (ERRG). Finally, we also develop a heuristic using random graphs that gives a recursion to describe a normalized pair-correlation function of a Rydberg gas. Our results suggest that even without dissipation, on long timescales the statistics are affected most by the geometrical restrictions induced by blockade effects, while on short timescales the statistics are affected most by quantum mechanical effects.

DOI
10.1103/physreva.103.033302

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The statistics of strongly interacting, ultracold Rydberg gases are governed by the interplay of two factors: geometrical restrictions induced by blockade effects and quantum mechanical effects. To shed light on their relative roles in the statistics of Rydberg gases, we compare three models in this paper: a quantum mechanical model describing the excitation dynamics within a Rydberg gas, a random sequential adsorption (RSA) process on a random geometric graph (RGG), and a RSA process on a decomposed random intersection graph (DRIG). The last model refers to choosing a particular subgraph of a mixture of two other random graphs. Contrary to the first two models, it lends itself for a rigorous mathematical analysis, and it is built specifically to have particular structural properties of a RGG. We establish for it a fluid limit describing the time evolution of the number of Rydberg atoms and show numerically that the expression remains accurate across a wider range of particle densities than an earlier approach based on an RSA process on an Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R'enyi random graph (ERRG). Finally, we also develop a heuristic using random graphs that gives a recursion to describe a normalized pair-correlation function of a Rydberg gas. Our results suggest that even without dissipation, on long timescales the statistics are affected most by the geometrical restrictions induced by blockade effects, while on short timescales the statistics are affected most by quantum mechanical effects.
Several recent experiments have established by measuring the Mandel Q parameter that the number of Rydberg excitations in ultracold gases exhibits sub-Poissonian statistics. This effect is attributed to the Rydberg blockade that occurs due to the strong interatomic interactions between highly excited atoms. Because of this blockade effect, the system can end up in a state in which all particles are either excited or blocked: a jamming limit. We analyze appropriately constructed random-graph models that capture the blockade effect, and derive formulae for the mean and variance of the number of Rydberg excitations in jamming limits. This yields an explicit relationship between the Mandel Q parameter and the blockade effect, and comparison to measurement data shows strong agreement between theory and experiment.
Several recent experiments have established by measuring the Mandel Q parameter that the number of Rydberg excitations in ultracold gases exhibits sub-Poissonian statistics. This effect is attributed to the Rydberg blockade that occurs due to the strong interatomic interactions between highly-excited atoms. Because of this blockade effect, the system can end up in a state in which all particles are either excited or blocked: a jamming limit. We analyze appropriately constructed random-graph models that capture the blockade effect, and derive formulae for the mean and variance of the number of Rydberg excitations in jamming limits. This yields an explicit relationship between the Mandel Q parameter and the blockade effect, and comparison to measurement data shows strong agreement between theory and experiment.
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 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.
Rydberg atom arrays are powerful platforms for studying quantum many-body systems. We consider the Rydberg-Ising Hamiltonian in a ring geometry and numerically study ensembles of states generated by random global pulse sequences subject to hardware constraints and fixed evolution times. We compare the statistical properties of such states with those of Haar-random states within the relevant lattice symmetry sector. In the strong-interaction regime (short interatomic distance), the dynamics is governed by an effective blockade that restricts Hilbert-space exploration and limits entanglement growth. In this regime, level-spacing statistics of the entanglement spectrum are close to random-matrix predictions, while the distribution of measurement probabilities deviates from Porter-Thomas behavior. For weaker interactions (larger interatomic distance), the system approaches Haar-like statistics at long times, as reflected in entanglement entropy, entanglement-spectrum statistics, and the distribution of measurement probabilities. At intermediate interactions, this behavior is observed on experimentally relevant timescales. Motivated by this observation, we investigate whether individual target states can be efficiently prepared using quantum optimal control in this regime. Employing target states drawn from an ensemble with a broad entropy distribution, we observe high fidelities (infidelities between 10^-5 and 3 x 10^-2 for nine spins). Within this ensemble, the fidelity decreases with the entanglement entropy of the target state, indicating that highly entangled targets are more difficult to prepare under the considered finite-time control constraints.
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. ^