Wigner crystallization of single photons in cold Rydberg ensembles

Quantum Light from Lossy Semiconductor Rydberg Excitons

The emergence of photonic quantum correlations is typically associated with emitters strongly coupled to a photonic mode. Here, we show that semiconductor Rydberg excitons, which are only weakly coupled to a free-space light mode can produce strongly antibunched fields, i.e., quantum light. This effect is fueled by a micron-scale excitation blockade between Rydberg excitons inducing pair-wise polariton scattering events. Photons incident on an exciton resonance are scattered into blue- and red-detuned pairs, which enjoy relative protection from absorption and thus dominate the transmitted light. We demonstrate that this effect persists in the presence of additional phonon coupling, strong nonradiative decay, and across a wide range of experimental parameters. Our results pave the way for the observation of quantum statistics from weakly coupled semiconductor excitons.

Quantum vortices of strongly interacting photons

Vortices are topologically nontrivial defects that generally originate from nonlinear field dynamics. All-optical generation of photonic vortices-phase singularities of the electromagnetic field-requires sufficiently strong nonlinearity that is typically achieved in the classical optics regime. We report on the realization of quantum vortices of photons that result from a strong photon-photon interaction in a quantum nonlinear optical medium. The interaction causes faster phase accumulation for copropagating photons, producing a quantum vortex-antivortex pair within the two-photon wave function. For three photons, the formation of vortex lines and a central vortex ring confirms the existence of a genuine three-photon interaction. The wave function topology, governed by two- and three-photon bound states, imposes a conditional phase shift of π per photon, a potential resource for deterministic quantum logic operations.

Light propagation in a three-dimensional Rydberg gas with a nonlocal optical response

We theoretically investigate the linear susceptibility and propagation of light in a three-dimensional (3-D) Rydberg gas under conditions of electromagnetically induced transparency. Rydberg atoms with two relevant S states are coupled via exchange interactions. When the gas is initially prepared in an entangled spin-wave state, this coupling induces a strong, nonlocal susceptibility whereby the photon field at one point of the medium acts as a source at a distant position. The nonlocal propagation occurs not only in the propagation direction but also in the paraxial direction. We discuss the absorption features and numerically simulate the 3-D propagation of probe laser light. Combined with the long-range exchange interaction, we show that the 3-D Rydberg gas is an ideal medium for studying nonlocal wave phenomena, in which the strength, range, and sign of the nonlocal interaction kernel can be widely tuned.

Quantum Many-Body Dynamics of Driven-Dissipative Rydberg Polaritons

We study the propagation of strongly interacting Rydberg polaritons through an atomic medium in a one-dimensional optical lattice. We derive an effective single-band Hubbard model to describe the dynamics of the dark-state polaritons under realistic assumptions. Within this model, we analyze the driven-dissipative transport of polaritons through the system by considering a coherent drive on one side and by including the spontaneous emission of the metastable Rydberg state. Using a variational approach to solve the many-body problem, we find strong antibunching of the outgoing photons despite the losses from the Rydberg state decay.

Exotic Photonic Molecules via Lennard-Jones-like Potentials

Ultracold systems offer an unprecedented level of control of interactions between atoms. An important challenge is to achieve a similar level of control of the interactions between photons. Towards this goal, we propose a realization of a novel Lennard-Jones-like potential between photons coupled to the Rydberg states via electromagnetically induced transparency (EIT). This potential is achieved by tuning Rydberg states to a Förster resonance with other Rydberg states. We consider few-body problems in 1D and 2D geometries and show the existence of self-bound clusters ("molecules") of photons. We demonstrate that for a few-body problem, the multibody interactions have a significant impact on the geometry of the molecular ground state. This leads to phenomena without counterparts in conventional systems: For example, three photons in two dimensions preferentially arrange themselves in a line configuration rather than in an equilateral-triangle configuration. Our result opens a new avenue for studies of many-body phenomena with strongly interacting photons.

Multistability of Driven-Dissipative Quantum Spins

We study the dynamics of lattice models of quantum spins one-half, driven by a coherent drive and subject to dissipation. Generically the mean-field limit of these models manifests multistable parameter regions of coexisting steady states with different magnetizations. We introduce an efficient scheme accounting for the corrections to mean field by correlations at leading order, and benchmark this scheme using high-precision numerics based on matrix-product operators in one- and two-dimensional lattices. Correlations are shown to wash the mean-field bistability in dimension one, leading to a unique steady state. In dimension two and higher, we find that multistability is again possible, provided the thermodynamic limit of an infinitely large lattice is taken first with respect to the longtime limit. Variation of the system parameters results in jumps between the different steady states, each showing a critical slowing down in the convergence of perturbations towards the steady state. Experiments with trapped ions can realize the model and possibly answer open questions in the nonequilibrium many-body dynamics of these quantum systems, beyond the system sizes accessible to present numerics.

Photon propagation through dissipative Rydberg media at large input rates

We study the dissipative propagation of quantized light in interacting Rydberg media under the conditions of electromagnetically induced transparency. Rydberg blockade physics in optically dense atomic media leads to strong dissipative interactions between single photons. The regime of high incoming photon flux constitutes a challenging many-body dissipative problem. We experimentally study in detail the pulse shapes and the second-order correlation function of the outgoing field and compare our data with simulations based on two novel theoretical approaches well-suited to treat this many-photon limit. At low incoming flux, we report good agreement between both theories and the experiment. For higher input flux, the intensity of the outgoing light is lower than that obtained from theoretical predictions. We explain this discrepancy using a simple phenomenological model taking into account pollutants, which are nearly stationary Rydberg excitations coming from the reabsorption of scattered probe photons. At high incoming photon rates, the blockade physics results in unconventional shapes of measured correlation functions.

Efimov States of Strongly Interacting Photons

We demonstrate the emergence of universal Efimov physics for interacting photons in cold gases of Rydberg atoms. We consider the behavior of three photons injected into the gas in their propagating frame, where a paraxial approximation allows us to consider them as massive particles. In contrast to atoms and nuclei, the photons have a large anisotropy between their longitudinal mass, arising from dispersion, and their transverse mass, arising from diffraction. Nevertheless, we show that, in suitably rescaled coordinates, the effective interactions become dominated by s-wave scattering near threshold and, as a result, give rise to an Efimov effect near unitarity. We show that the three-body loss of these Efimov trimers can be strongly suppressed and determine conditions under which these states are observable in current experiments. These effects can be naturally extended to probe few-body universality beyond three bodies, as well as the role of Efimov physics in the nonequilibrium, many-body regime.

Quantum Critical Behavior of One-Dimensional Soft Bosons in the Continuum

We consider a zero-temperature one-dimensional system of bosons interacting via the soft-shoulder potential in the continuum, typical of dressed Rydberg gases. We employ quantum Monte Carlo simulations, which allow for the exact calculation of imaginary-time correlations, and a stochastic analytic continuation method, to extract the dynamical structure factor. At finite densities, in the weakly interacting homogeneous regime, a rotonic spectrum marks the tendency to clustering. With strong interactions, we indeed observe cluster liquid phases emerging, characterized by the spectrum of a composite harmonic chain. Luttinger theory has to be adapted by changing the reference lattice density field. In both the liquid and cluster liquid phases, we find convincing evidence of a secondary mode, which becomes gapless only at the transition. In that region, we also measure the central charge and observe its increase towards c=3/2, as recently evaluated in a related extended Bose-Hubbard model, and we note a fast reduction of the Luttinger parameter. For two-particle clusters, we then interpret such observations in terms of the compresence of a Luttinger liquid and a critical transverse Ising model, related to the instability of the reference lattice density field towards coalescence of sites, typical of potentials which are flat at short distances. Even in the absence of a true lattice, we are able to evaluate the spatial correlation function of a suitable pseudospin operator, which manifests ferromagnetic order in the cluster liquid phase, exponential decay in the liquid phase, and algebraic order at criticality.

Simulating quantum light propagation through atomic ensembles using matrix product states

A powerful method to interface quantum light with matter is to propagate the light through an ensemble of atoms. Recently, a number of such interfaces have emerged, most prominently Rydberg ensembles, that enable strong nonlinear interactions between propagating photons. A largely open problem is whether these systems produce exotic many-body states of light and developing new tools to study propagation in the large photon number limit is highly desirable. Here we provide a method based on a “spin model” that maps quasi one-dimensional (1D) light propagation to the dynamics of an open 1D interacting spin system, where all photon correlations are obtained from those of the spins. The spin dynamics in turn are numerically solved using the toolbox of matrix product states. We apply this formalism to investigate vacuum induced transparency, wherein the different photon number components of a pulse propagate with number-dependent group velocity and separate at output.

Numerical simulation of light propagation through 1D atomic systems in the many-body limit rapidly saturates hardware capabilities. Here, the authors tackle the problem by mapping the dynamics to an open 1D interacting spin system and solving it using the matrix product state ansatz.

Induced Cavities for Photonic Quantum Gates

Effective cavities can be optically induced in atomic media and employed to strengthen optical nonlinearities. Here we study the integration of induced cavities with a photonic quantum gate based on Rydberg blockade. Accounting for loss in the atomic medium, we calculate the corresponding finesse and gate infidelity. Our analysis shows that the conventional limits imposed by the blockade optical depth are mitigated by the induced cavity in long media, thus establishing the total optical depth of the medium as a complementary resource.

Quantum simulations and many-body physics with light

In this review we discuss the works in the area of quantum simulation and many-body physics with light, from the early proposals on equilibrium models to the more recent works in driven dissipative platforms. We start by describing the founding works on Jaynes-Cummings-Hubbard model and the corresponding photon-blockade induced Mott transitions and continue by discussing the proposals to simulate effective spin models and fractional quantum Hall states in coupled resonator arrays (CRAs). We also analyse the recent efforts to study out-of-equilibrium many-body effects using driven CRAs, including the predictions for photon fermionisation and crystallisation in driven rings of CRAs as well as other dynamical and transient phenomena. We try to summarise some of the relatively recent results predicting exotic phases such as super-solidity and Majorana like modes and then shift our attention to developments involving 1D nonlinear slow light setups. There the simulation of strongly correlated phases characterising Tonks-Girardeau gases, Luttinger liquids, and interacting relativistic fermionic models is described. We review the major theory results and also briefly outline recent developments in ongoing experimental efforts involving different platforms in circuit QED, photonic crystals and nanophotonic fibres interfaced with cold atoms.

Effective Field Theory for Rydberg Polaritons

We develop an effective field theory (EFT) to describe the few- and many-body propagation of one-dimensional Rydberg polaritons. We show that the photonic transmission through the Rydberg medium can be found by mapping the propagation problem to a nonequilibrium quench, where the role of time and space are reversed. We include effective range corrections in the EFT and show that they dominate the dynamics near scattering resonances in the presence of deep bound states. Finally, we show how the long-range nature of the Rydberg-Rydberg interactions induces strong effective N-body interactions between Rydberg polaritons. These results pave the way towards studying nonperturbative effects in quantum field theories using Rydberg polaritons.

Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances

Mapping the strong interaction between Rydberg atoms onto single photons via electromagnetically induced transparency enables manipulation of light at the single-photon level and few-photon devices such as all-optical switches and transistors operated by individual photons. Here we demonstrate experimentally that Stark-tuned Förster resonances can substantially increase this effective interaction between individual photons. This technique boosts the gain of a single-photon transistor to over 100, enhances the non-destructive detection of single Rydberg atoms to a fidelity beyond 0.8, and enables high-precision spectroscopy on Rydberg pair states. On top, we achieve a gain larger than 2 with gate photon read-out after the transistor operation. Theory models for Rydberg polariton propagation on Förster resonance and for the projection of the stored spin-wave yield excellent agreement to our data and successfully identify the main decoherence mechanism of the Rydberg transistor, paving the way towards photonic quantum gates.

Single photon level of light control is possible by using the effective interaction between single photons and Rydberg atoms. Here the authors utilized such interaction of Stark-tuned Forster resonances to boost the gain of a Rydberg single-photon transistor and perform high precision spectroscopy.

Three-Body Interaction of Rydberg Slow-Light Polaritons

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.

Broadband photon-photon interactions mediated by cold atoms in a photonic crystal fiber

We demonstrate theoretically that photon-photon attraction can be engineered in the continuum of scattering states for pairs of photons propagating in a hollow-core photonic crystal fiber filled with cold atoms. The atoms are regularly spaced in an optical lattice configuration and the photons are resonantly tuned to an internal atomic transition. We show that the hard-core repulsion resulting from saturation of the atomic transitions induces bunching in the photonic component of the collective atom-photon modes (polaritons). Bunching is obtained in a frequency range as large as tens of GHz, and can be controlled by the inter-atomic separation. We provide a fully analytical explanation for this phenomenon by proving that correlations result from a mismatch of the quantization volumes for atomic excitations and photons in the continuum. Even stronger correlations can be observed for in-gap two-polariton bound states. Our theoretical results use parameters relevant for current experiments and suggest a simple and feasible way to induce interactions between photons.

Exotic Attractors of the Nonequilibrium Rabi-Hubbard Model

We explore the phase diagram of the dissipative Rabi-Hubbard model, as could be realized by a Raman-pumping scheme applied to a coupled cavity array. There exist various exotic attractors, including ferroelectric, antiferroelectric, and incommensurate fixed points, as well as regions of persistent oscillations. Many of these features can be understood analytically by truncating to the two lowest lying states of the Rabi model on each site. We also show that these features survive beyond mean field, using matrix product operator simulations.

Tunable Polarons of Slow-Light Polaritons in a Two-Dimensional Bose-Einstein Condensate

When an impurity interacts with a bath of phonons it forms a polaron. For increasing interaction strengths the mass of the polaron increases and it can become self-trapped. For impurity atoms inside an atomic Bose-Einstein condensate (BEC) the nature of this transition is not understood. While Feynman's variational approach to the Fröhlich model predicts a sharp transition for light impurities, renormalization group studies always predict an extended intermediate-coupling region characterized by large phonon correlations. To investigate this intricate regime and to test polaron physics beyond the validity of the Fröhlich model we suggest a versatile experimental setup that allows us to tune both the mass of the impurity and its interactions with the BEC. The impurity is realized as a dark-state polariton (DSP) inside a quasi-two-dimensional BEC. We show that its interactions with the Bogoliubov phonons lead to photonic polarons, described by the Bogoliubov-Fröhlich Hamiltonian, and make theoretical predictions using an extension of a recently introduced renormalization group approach to Fröhlich polarons.

Many-body decoherence dynamics and optimized operation of a single-photon switch

We develop a theoretical framework to characterize the decoherence dynamics due to multi-photon scattering in an all-optical switch based on Rydberg atom induced nonlinearities. By incorporating the knowledge of this decoherence process into optimal photon storage and retrieval strategies, we establish optimized switching protocols for experimentally relevant conditions, and evaluate the corresponding limits in the achievable fidelities. Based on these results we work out a simplified description that reproduces recent experiments (Nat. Commun. 7 12480) and provides a new interpretation in terms of many-body decoherence involving multiple incident photons and multiple gate excitations forming the switch. Aside from offering insights into the operational capacity of realistic photon switching capabilities, our work provides a complete description of spin wave decoherence in a Rydberg quantum optics setting, and has immediate relevance to a number of further applications employing photon storage in Rydberg media.

Nonequilibrium many-body steady states via Keldysh formalism

Many-body systems with both coherent dynamics and dissipation constitute a rich class of models which are nevertheless much less explored than their dissipationless counterparts. The advent of numerous experimental platforms that simulate such dynamics poses an immediate challenge to systematically understand and classify these models. In particular, nontrivial many-body states emerge as steady states under nonequilibrium dynamics. While these states and their phase transitions have been studied extensively with mean-field theory, the validity of the mean-field approximation has not been systematically investigated. In this paper, we employ a field-theoretic approach based on the Keldysh formalism to study nonequilibrium phases and phase transitions in a variety of models. In all cases, a complete description via the Keldysh formalism indicates a partial or complete failure of the mean-field analysis. Furthermore, we find that an effective temperature emerges as a result of dissipation, and the universal behavior including the dynamics near the steady state is generically described by a thermodynamic universality class.

Incompressible Polaritons in a Flat Band

We study the interplay of geometric frustration and interactions in a nonequilibrium photonic lattice system exhibiting a polariton flat band as described by a variant of the Jaynes-Cummings-Hubbard model. We show how to engineer strong photonic correlations in such a driven, dissipative system by quenching the kinetic energy through frustration. This produces an incompressible state of photons characterized by short-ranged crystalline order with period doubling. The latter manifests itself in strong spatial correlations, i.e., on-site and nearest-neighbor antibunching combined with extended density-wave oscillations at larger distances. We propose a state-of-the-art circuit QED realization of our system, which is tunable in situ.

Coulomb Bound States of Strongly Interacting Photons

We show that two photons coupled to Rydberg states via electromagnetically induced transparency can interact via an effective Coulomb potential. This interaction gives rise to a continuum of two-body bound states. Within the continuum, metastable bound states are distinguished in analogy with quasibound states tunneling through a potential barrier. We find multiple branches of metastable bound states whose energy spectrum is governed by the Coulomb potential, thus obtaining a photonic analogue of the hydrogen atom. Under certain conditions, the wave function resembles that of a diatomic molecule in which the two polaritons are separated by a finite "bond length." These states propagate with a negative group velocity in the medium, allowing for a simple preparation and detection scheme, before they slowly decay to pairs of bound Rydberg atoms.

Dipolar Dephasing of Rydberg D-State Polaritons

We experimentally study the effects of the anisotropic Rydberg interaction on D-state Rydberg polaritons slowly propagating through a cold atomic sample. 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 wave function through our system with nonperturbative calculations of the anisotropic Rydberg interaction to show that the observed effect can be attributed to pairwise interaction of individual Rydberg polaritons at distances larger than the Rydberg blockade.

Dissipative preparation of spatial order in Rydberg-dressed Bose-Einstein condensates

We propose a technique for engineering momentum-dependent dissipation in Bose-Einstein condensates with nonlocal interactions. The scheme relies on the use of momentum-dependent dark states in close analogy to velocity-selective coherent population trapping. During the short-time dissipative dynamics, the system is driven into a particular finite-momentum phonon mode, which in real space corresponds to an ordered structure with nonlocal density-density correlations. Dissipation-induced ordering can be observed and studied in present-day experiments using cold atoms with dipole-dipole or off-resonant Rydberg interactions. Because of its dissipative nature, the ordering does not require artificial breaking of translational symmetry by an optical lattice or harmonic trap. This opens up a perspective of direct cooling of quantum gases into strongly interacting phases.

Electromagnetically induced transparency in an entangled medium

We theoretically investigate light propagation and electromagnetically induced transparency in a quasi-one-dimensional gas in which atoms interact strongly via exchange interactions. We focus on the case in which the gas is initially prepared in a many-body state that contains a single excitation and conduct a detailed study of the absorptive and dispersive properties of such a medium. This scenario is achieved in interacting gases of Rydberg atoms with two relevant S states that are coupled through exchange. Of particular interest is the case in which the medium is prepared in an entangled spin-wave state. This, in conjunction with the exchange interaction, gives rise to a nonlocal susceptibility that--in comparison to conventional Rydberg electromagnetically induced transparency--qualitatively alters the absorption and propagation of weak probe light, leading to nonlocal propagation and enhanced absorption.

Two-photon dynamics in coherent Rydberg atomic ensemble

We study the interaction of two photons in a Rydberg atomic ensemble under the condition of electromagnetically induced transparency, combining a semiclassical approach for pulse propagation and a complete quantum treatment for quantum state evolution. We find that the blockade regime is not suitable for implementing photon-photon cross-phase modulation due to pulse absorption and dispersion. However, approximately ideal cross-phase modulation can be realized based on relatively weak interactions, with counterpropagating and transversely separated pulses.