Using cold atoms to simulate strongly interacting quantum systems is an exciting frontier of physics. However, because atoms are nominally neutral point particles, this limits the types of interaction that can be produced. We propose to use the powerful new platform of cold atoms trapped near nanophotonic systems to extend these limits, enabling a novel quantum material in which atomic spin degrees of freedom, motion and photons strongly couple over long distances. In this system, an atom trapped near a photonic crystal seeds a localized, tunable cavity mode around the atomic position. We find that this effective cavity facilitates interactions with other atoms within the cavity length, in a way that can be made robust against realistic imperfections. Finally, we show that such phenomena should be accessible using onedimensional photonic crystal waveguides in which coupling to atoms has already been experimentally demonstrated.T rapped ultracold atoms are a rich resource for physicists. Isolated from the environment and routinely manipulated, they can act as a quantum simulator for a wide variety of physical models 1 . However, although short-range interactions between atoms can be adjusted by Feshbach resonance, these systems typically lack the long-range interactions required to produce some of the most interesting condensed-matter phenomena. For example, exotic phases such as supersolids are predicted in systems with long-range interactions 2 , as well as Wigner crystallization 3 and topological states 4 . Long-range interactions can also lead to the breakdown of concepts such as additivity in statistical mechanics 5,6 and the violation of speed limits (Lieb-Robinson bounds) for the propagation of information [7][8][9] . As a result, there are active efforts to achieve long-range interactions using specific properties of the atoms 10 , such as their magnetic moment 11,12 , Rydberg excitation 13 or by using polar molecules 14 .In this Article we investigate another paradigm, where, instead of relying on atomic properties, we design the medium through which the atoms interact-specifically, by coupling the atoms via the photon modes of a photonic crystal. Our proposal is inspired by demonstrations of strong coupling of photons in nanophotonic systems with individual solid-state emitters 15 and, more recently, with cold atoms [16][17][18][19] . For example, systems of ∼10 3 atoms have been trapped by and coupled to the evanescent guided modes of nanofibres 16,17 , and single atoms have been coupled to photonic crystal cavities 18 and waveguides 19. One aim of these efforts is to utilize strong, controlled light-matter interactions for quantum information processing and networks 20. Here, we show that atoms interfaced with photonic crystals can also have remarkable consequences for the exploration of quantum many-body physics [21][22][23] .A photonic crystal is a periodic dielectric structure that controls the propagation of light 24 . By introducing a defect into this regular structure, it is possible to induce c...

This Colloquium describes a new paradigm for creating strong quantum interactions of light and matter by way of single atoms and photons in nanoscopic lattices. Beyond the possibilities for quantitative improvements for familiar phenomena in atomic physics and quantum optics, there is a growing research community that is exploring novel quantum phases and phenomena that arise from atom-photon interactions in one-and two-dimensional nanophotonic lattices. Nanophotonic structures offer the intriguing possibility to control atom-photon interactions by engineering the medium properties through which they interact. An important aspect of these new research lines is that they have become possible only by pushing the state-of-the-art capabilities in nanophotonic device fabrication and by the integration of these capabilities into the realm of ultracold atoms. This Colloquium attempts to inform a broad physics community of the emerging opportunities in this new field on both theoretical and experimental fronts. The research is inherently multidisciplinary, spanning the fields of nanophotonics, atomic physics, quantum optics, and condensed matter physics.

There has been rapid development of systems that yield strong interactions between freely propagating photons in one-dimension via controlled coupling to quantum emitters. This raises interesting possibilities such as quantum information processing with photons or quantum many-body states of light, but treating such systems generally remains a difficult task theoretically. Here, we describe a novel technique in which the dynamics and correlations of a few photons can be exactly calculated, based upon knowledge of the initial photonic state and the solution of the reduced effective dynamics of the quantum emitters alone. We show that this generalized 'input-output' formalism allows for a straightforward numerical implementation regardless of system details, such as emitter positions, external driving, and level structure. As a specific example, we apply our technique to show how atomic systems with infinite-range interactions and under conditions of electromagnetically induced transparency enable the selective transmission of correlated multi-photon states.from other emitters appears to be a difficult task. An exception is the weak excitation limit, in which atoms can be treated as linear scatterers and the powerful transfer matrix method of linear optics can be employed [18,19].The full quantum case has been solved exactly in a limited number of situations in which nonlinear systems are coupled to 1D waveguides [20][21][22][23][24][25][26][27][28][29][30]. The formalism employed in [21] is particularly elegant, because it establishes an input-output relation to determine the nonlinear scattering from a two-level atom. Here, we show that this technique can be efficiently generalized to many atoms, chiral or bi-directional waveguides, and arbitrary atomic configurations, providing a powerful tool to investigate nonlinear optical dynamics in all systems of interest.This paper is organized in the following way: first, we present a generalized input-output formalism to treat few-photon propagation in waveguides coupled to many atoms. We show that the infinite degrees of freedom associated with the photonic modes can be effectively integrated out, yielding an open, interacting 'spin' model that involves only the internal degrees of freedom of the atoms. This open system can be solved using a number of conventional, quantum optical techniques. Then, we show that the solution of the spin problem can be used to re-construct the optical fields. In particular, we provide a prescription to map spin correlations to S-matrix elements, which contain full information about the photon dynamics, and give explicit closed-form expressions for the one-and two-photon cases. Importantly, in analogy with the cavity QED case, our technique enables analytical solutions under some scenarios, but in general allows for simple numerical implementation under a wide variety of circumstances of interest, such as different level structures, external driving, atomic positions, atomic motion, etc. Finally, to illustrate the ease of usage, we apply ...

Realizing systems that support robust, controlled interactions between individual photons is an exciting frontier of nonlinear optics. To this end, one approach that has emerged recently is to leverage atomic interactions to create strong and spatially non-local interactions between photons. In particular, effective interactions have been successfully created via interactions between atoms excited to Rydberg levels. Here, we investigate an alternative approach, in which atomic interactions arise via their common coupling to photonic crystal waveguides. This technique takes advantage of the ability to separately tailor the strength and range of interactions via the dispersion engineering of the structure itself, which can lead to qualitatively new types of phenomena. As an example, we discuss the formation of correlated transparency windows, in which photonic states of a certain number and shape selectively propagate through the system. Through this technique, we show in particular that one can create molecular-like potentials that lead to molecular bound states of photon pairs.

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.

There have been concerted efforts in recent years to realize the next generation of clocks using alkaline earth atoms in an optical lattice. Assuming that the atoms are independent, such a clock would benefit from a √ N enhancement in its stability, associated with the improved signal-tonoise ratio of a large atom number N . An interesting question, however, is what type of atomic interactions might affect the clock dynamics, and whether these interactions are deleterious or could even be beneficial. In this work, we investigate the effect of dipole-dipole interactions, in which atoms excited during the clock protocol emit and re-absorb photons. Taking a simple system consisting of a 1D atomic array, we find that dipole-dipole interactions in fact result in an open quantum system exhibiting critical dynamics, as a set of collective excitations acquires a decay rate approaching zero in the thermodynamic limit due to subradiance. A first consequence is that the decay of atomic excited population at long times exhibits a slow power-law behavior, instead of the exponential expected for non-interacting atoms. We also find that excitations among the atoms exhibit fermionic spatial correlations at long times, due to the microscopic properties of the multi-excitation subradiant states. Interestingly, these properties cannot be captured by mean-field dynamics, suggesting the strongly interacting nature of this system. We finally characterize the time-dependent frequency shift in the atomic frequency measurement, and find that it is dominated by the interaction energy of subradiant states at long times. Furthermore, we show that the decay of the clock signal displays at long times a non-exponential behavior, which might be useful to improve the uncertainty limit with which the atomic frequency can be resolved. We attribute the lack of robust power-law dynamics for the clock signal to an effective many-body dephasing caused by purely coherent interactions.

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