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arXiv:1501.03269 · Physical Review Letters

Spatial Patterns in Rydberg Excitations from Logarithmic Pair Interactions

Wolfgang Lechner, P. Zoller

Published Sep 14, 2015• 6 citations
arXivHTMLPDF

Abstract

The collective excitations in ensembles of dissipative, laser driven ultracold atoms exhibit crystal-like patterns, a many-body effect of the Rydberg blockade mechanism. These crystalline structures are revealed in an experiment from a postselection of configurations with fixed numbers of excitations. Here, we show that these subensembles can be well represented by ensembles of effective particles that interact via logarithmic pair potentials. This allows one to study the emergent patterns with a small number of effective particles to determine the phases of Rydberg crystals and to systematically study contributions from N-body terms.

DOI
10.1103/physrevlett.115.125301

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The collective excitations in ensembles of dissipative, laser driven ultracold atoms exhibit crystal-like patterns, a many-body effect of the Rydberg blockade mechanism. These crystalline structures are revealed in an experiment from a postselection of configurations with fixed numbers of excitations. Here, we show that these subensembles can be well represented by ensembles of effective particles that interact via logarithmic pair potentials. This allows one to study the emergent patterns with a small number of effective particles to determine the phases of Rydberg crystals and to systematically study contributions from N-body terms.
Crystalline structures of highly excited Rydberg atoms can be a model for dilute metallic solids with tunable parameters, and provide access to the regime of strongly coupled systems. In practice, crystal creation is made difficult by the 'Rydberg blockade', where one atom in the Rydberg state shifts the energy levels of its neighbours out of resonance with the excitation laser. In our experiment, we can spatially detect Rydberg atoms by a combination of field-ionization, and subsequent acceleration of the remaining ions onto a multi-channel plate. In this way, we were able to determine the blockade radius from measuring second-order correlations. We have also made small ordered Rydberg structures by optically imprinting local excitations in a disordered atomic gas using a spatial light modulator. With this diffractive optical element it is possible to control the intensity of light in the far ¿eld and create almost any desired intensity pattern. We found that a Rydberg system with a dissipative mechanism can be mapped onto a set of rate equations that describe a wireless computer network. In these networks neighbouring transmitters have a similar blockade effect on each other. For the Rydberg system, we used mathematical tools developed for wireless networks, and applied this to optically imprinted square lattices.
Laser-driven Rydberg gases are many-body quantum systems that exhibit a pronounced collective excitation behavior due to the strong interaction between atoms in high-lying electronic states. Atoms located within a so-called blockade volume share a single Rydberg excitation, which is dynamically created and annihilated. For sufficiently long times, this driven system approaches a steady state, which lends its properties from a maximum entropy state of a Tonks gas. Using this connection, we show that spatial correlations between Rydberg atoms are controlled by the number of atoms contained within a blockade volume. For a small number, the system favors a disordered arrangement of Rydberg atoms, whereas in the opposite limit, Rydberg atoms tend to arrange in an increasingly ordered configuration. We argue that this is an entropic effect which is observable in current experiments. Our work demonstrates how ordered structures can spontaneously emerge in a strongly interacting, closed many-body system out of equilibrium.
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. ^
We study many-body correlations in the stationary state of a continuously driven, strongly interacting dissipative system. Specifically, we examine resonant optical excitations of Rydberg states of atoms interacting via long-range dipole-dipole and van der Waals potentials employing numerical and analytical techniques. Collection of atoms within a blockade distance form a ``superatom'' that can accommodate at most one Rydberg excitation. The superatom excitation probability saturates to 12\frac{1}{2}21​ for coherently driven atoms but is significantly higher in the presence of dephasing, approaching unity as the number of atoms increases. Using the exact numerical solution of the density-matrix equations for a small system, we demonstrate that strong dephasing of the optically driven dipoles renders the many-body problem amenable to semiclassical Monte Carlo simulations. We employ the Monte Carlo algorithm for a large number of atoms and find that in the steady state of a uniformly driven, extended one-dimensional system, the saturation of superatoms leads to quasicrystallization of Rydberg excitations whose correlations exhibit damped spatial oscillations. We show that the behavior of the system under the van der Waals interaction potential can be approximated by a rate-equations model based on a ``hard-rod'' interatomic potential, and by solving it we obtain the period and correlation length for the density wave of Rydberg excitations.
We investigate the spectral properties of a finite laser-driven lattice of ultracold Rydberg atoms exploiting the dipole blockade effect in the frozen Rydberg gas regime. Uniform one-dimensional lattices as well as lattices with variable spacings are considered. In the case of a weak laser coupling, we find a multitude of many-body Rydberg states with well-defined excitation properties which are adiabatically accessible starting from the ground state. A comprehensive analysis of the degeneracies of the spectrum as well as of the single- and pair-excitation numbers of the eigenstates is performed. In the strong laser regime, analytical solutions for the pseudo-fermionic eigenmodes are derived. Perturbative energy corrections for this approximative approach are provided.