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arXiv:1304.8096 · Physical Review Letters

Wigner Crystallization of Single Photons in Cold Rydberg Ensembles

Johannes Otterbach, Matthias Moos, Dominik Muth, Michael Fleischhauer

Published Sep 9, 2013• 92 citations
arXivHTMLPDF

Abstract

The coupling of weak light fields to Rydberg states of atoms under conditions of electromagnetically induced transparency leads to the formation of Rydberg polaritons which are quasiparticles with tunable effective mass and nonlocal interactions. Confined to one spatial dimension their low energy physics is that of a moving-frame Luttinger liquid which, due to the nonlocal character of the repulsive interaction, can form a Wigner crystal of individual photons. We calculate the Luttinger K parameter using density-matrix renormalization group simulations and find that under typical slow-light conditions kinetic energy contributions are too strong for crystal formation. However, adiabatically increasing the polariton mass by turning a light pulse into stationary spin excitations allows us to generate true crystalline order over a finite length. The dynamics of this process and asymptotic correlations are analyzed in terms of a time-dependent Luttinger theory.

DOI
10.1103/physrevlett.111.113001

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The coupling of weak light fields to Rydberg states of atoms under conditions of electromagnetically induced transparency leads to the formation of Rydberg polaritons which are quasiparticles with tunable effective mass and nonlocal interactions. Confined to one spatial dimension their low energy physics is that of a moving-frame Luttinger liquid which, due to the nonlocal character of the repulsive interaction, can form a Wigner crystal of individual photons. We calculate the Luttinger K parameter using density-matrix renormalization group simulations and find that under typical slow-light conditions kinetic energy contributions are too strong for crystal formation. However, adiabatically increasing the polariton mass by turning a light pulse into stationary spin excitations allows us to generate true crystalline order over a finite length. The dynamics of this process and asymptotic correlations are analyzed in terms of a time-dependent Luttinger theory.
Coupling light to Rydberg states of atoms under conditions of electromagnetically induced transparency (EIT) leads to the formation of interacting quasiparticles, termed Rydberg polaritons. We derive a one-dimensional model describing the time evolution of these polaritons under paraxial propagation conditions. Specifically, we obtain a master equation governing the dynamics of Rydberg polaritons and identify conditions when it can essentially be described by an effective Hamiltonian of a single-species polariton. We verify this Hamiltonian by numerical two-excitation simulations. Under typical stationary EIT conditions it is impossible to reach the strongly interacting regime where long-range density-density correlations emerge. In contrast, by employing the time dependence of the control field the regime of strong interactions can be reached where the polaritons attain quasicrystalline order. We provide a physical explanation for the differences between stationary and time-dependent schemes.
Electromagnetically-induced transparency (EIT) with Rydberg atoms enables strong interactions between atoms to be controlled and measured by pulses of light. Consequently, Rydberg dark polaritons are a promising platform for quantum simulations. Of particular interest are simulations of one-dimensional systems, known to exhibit unique properties compared to their higher-dimensional counterparts. One limitation of standard EIT is that reducing the polariton group velocity to bring the dark polaritons to rest also implies activating strong Rydberg interactions. To achieve independent control of polariton speed and interaction strength, we propose a stationary light-scheme to bring Rydberg polaritons to a stand-still. This allows them to remain primarily photonic and, therefore, non-interacting, before Rydberg interactions are turned on. Once activated, in the dilute regime and in the presence of strong transverse confinement, the strong polarizabilities of Rydberg atoms give rise to an effective contact interaction between polaritons. By tuning the various parameters of the stationary-light scheme, the effective one-dimensional scattering length may be adjusted to yield various kinds of physics, including the regime of a Tonks-Girardeau gas, when polaritons exhibit fermionic behavior. We outline a protocol to observe the resulting spatial correlations between neighboring polaritons.
Abstract Photonic materials are an emerging platform to explore quantum matter (Carusotto and Ciuti 2013 Rev. Mod. Phys. 85 299; Sommer et al 2015 arXiv: 1506.00341 ) and quantum dynamics (Peyronel et al 2012 Nature 488 57). The development of Rydberg electromagnetically induced transparency (Weatherill et al 2008 J. Phys. B: At. Mol. Opt. Phys. 41 201002; Petrosyan et al 2011 Phys. Rev. Lett. 107 213601) provided a clear route to strong interactions between individual optical photons. In conjunction with carefully designed optical resonators, it is now possible to achieve extraordinary control of the properties of individual photons, introducing tunable gauge fields (Schine et al 2016 Nature 534 671) whilst imbuing the photons with mass and embedding them on curved spatial manifolds (Sommer et al 2016 New J. Phys. 18 035008). Building on work formalizing Rydberg mediated interactions between propagating photons (Gorshkov et al 2011 Phys. Rev. Lett. 107 133602; Gullans et al 2016 Phys. Rev. Lett. 117 113601), we develop a theory of interacting Rydberg polaritons in multimode optical resonators, where the strong interactions are married with tunable single particle properties to build and probe exotic matter. In the presence of strong coupling between the resonator field and a Rydberg-dressed atomic ensemble, a quasiparticle called the ‘cavity Rydberg polariton’ emerges. We investigate its properties, finding that it inherits both the fast dynamics of its photonic constituents and the strong interactions of its atomic constituents. We develop tools to properly renormalize the interactions when polaritons approach each other, and investigate the impact of atomic motion on the coherence of multi-mode polaritons, showing that most channels for atom-polariton cross-thermalization are strongly suppressed. Finally, we propose to harness the repeated diffraction and refocusing of the optical resonator to realize interactions which are local in momentum-space. This work points the way to efficient modeling of polaritonic quantum materials in properly renormalized strongly interacting effective theories, thereby enabling experimental studies of photonic fractional quantum Hall fluids and crystals (Sommer et al 2015 arXiv: 1506.00341 ; Umucalılar et al 2014 Phys. Rev. A 89 023803; Grusdt and Fleischhauer 2013 Phys. Rev. A 87 043628), plus photonic quantum information processors and repeaters (Jaksch et al 2000 Phys. Rev. Lett. 85 2208; Saffman et al 2010 Rev. Mod. Phys. 82 2313; Han et al 2010 Phys. Rev. A 81 , 052311).
Abstract The propagation of light fields through a quasi one-dimensional cold atomic gas, exciting atomic Rydberg levels of large principal quantum number under conditions of electromagnetically induced transparency, can lead to a stable two-mode Luttinger liquid system. Atomic van der Waals interactions induce a coupling of bosonic field modes that display both photonic and atomic character, the Rydberg dark-state polaritons (RDPs). It is shown that by tunable control of the van der Waals coupling, the RDP may decouple into independent ‘spin’ and ‘charge’ fields which propagate at different speeds, analogous to spin–charge separation of electrons in a one-dimensional metal.
In recent years, the development of various platforms for quantum simulation made it possible to study many-body systems under controlled conditions. Of particular interest are one-dimensional systems, known to exhibit some unique properties compared to their higher-dimensional counterparts. One of the current promising platforms for simulations of one-dimensional systems are Rydberg dark polaritons, quasiparticles of a photon-matter hybrid character emerging in the context of electromagnetically induced transparency (EIT) coupled with Rydberg atoms. These polaritons are long-lived (owing to the properties of EIT) and inherit interaction properties from the huge polarizability of Rydberg atoms, both attractive features in the context of simulations. We herein construct an experimental proposal for realizing one-dimensional physics through Rydberg dark polaritons. By using a stationary-light scheme, a sequence of steps allows a laser pulse incident on a transversally trapped gas of Rydberg atoms to yield a stationary pulse of weakly interacting polaritons. Within the stationarity regime, the lasers are further tuned to adjust the interaction effective one-dimensional scattering length through a Feshbach-like resonance to create a Tonks-Girardeau-type gas of polaritons. Signatures of non-trivial correlations can then experimentally probed by retrieving the photons through a subsequent adjustment of the laser strengths. Realizing this proposal would be a landmark achievement in the fields of many-body simulations and Rydberg physics.
We study a system of three photons in an atomic medium coupled to Rydberg states near the conditions of electromagnetically induced transparency. Based on the analytical analysis of the microscopic set of equations in the far-detuned regime, the effective three-body interaction for these Rydberg polaritons is derived. For slow light polaritons, we find a strong three-body repulsion with the remarkable property that three polaritons can become essentially noninteracting at short distances. This analysis allows us to derive the influence of the three-body repulsion on bound states and correlation functions of photons propagating through a one-dimensional atomic cloud.