We propose a method of generating a vortex ring in a BoseEinstein condensate by means of electromagnetically-induced atomic transitions. Some remnant population of atoms in a second internal state remains within the toroidal trap formed by the mean field repulsion of the vortex ring. This population can be removed, or it can be made to flow around the torus (i.e. within the vortex ring). If this flow has unit topological winding number, the entire structure formed by the two condensates is an example of a three-dimensional skyrmion texture. 03.75.Fi,05.30.Jp With the recent controlled creation of vortices [1,2] in trapped atomic Bose-Einstein condensates (BECs), and evidence for vortex nucleation above a critical flow velocity [3], the dynamics of topological structures in weakly interacting superfluids has become an active subject of experimental research. In this paper we propose methods of preparing a vortex ring within a BEC, allowing the controlled study of a vortex structure away from surface effects [4]. The vortex ring is created by means of driving electromagnetic (em) fields inducing transitions between internal atomic levels. The em field couples to the relative phases between the levels, in such a way as to transfer angular momentum to the atoms. With an appropriately chosen superposition of em fields the local axes of this effective rotation form a closed circular loop, producing a vortex ring in the BEC. We also consider the stability of the vortex ring, and identify a way to use a second atomic component of the BEC to increase the stability of the ring against collapsing under its own string tension. Conversely, the vortex ring acts as a trap for the second component, providing purely atomoptical confinement of cold atoms. Finally, the combined two-component structure may be identified as a threedimensional (3D) skyrmion [5], shown in Fig. 1. This nonsingular texture is of interest in its own right, as a topological structure beyond the simple vortex.Topological defects such as vortices and monopoles possess cores at which the order parameter is singular, and indeed such defects are characterized by winding numbers defined on paths or surfaces that enclose the singular cores. The simplest topological defect relevant to superfluid physics is that of a current flowing around a closed 1D path parametrized by the angular coordinate φ. The simple mapping α(φ) = φ, from physical space into order parameter space, where φ is defined on a 1D circle, has 'winding number' one, which will be unchanged if α(φ) is continuously deformed. Topologically nontrivial configurations are also possible without singular cores: Textures are defined by the way in which the compact order parameter space is 'stretched over' physical space. An SU(2) order parameter, for instance, takes values on a sphere. One can 'puncture' this sphere, stretch the resulting hole to infinity, and so 'spread' the order parameter space over a 2D plane. Such a mapping, or any continuous deformation of it, is known as a 2D (or 'baby') skyrmion...

We study the emergence of collective scattering in the presence of dipole-dipole interactions when we illuminate a cold cloud of rubidium atoms with a near-resonant and weak intensity laser. The size of the atomic sample is comparable to the wavelength of light. When we gradually increase the number of atoms from 1 to ∼450, we observe a broadening of the line, a small redshift and, consistently with these, a strong suppression of the scattered light with respect to the noninteracting atom case. We compare our data to numerical simulations of the optical response, which include the internal level structure of the atoms. DOI: 10.1103/PhysRevLett.113.133602 PACS numbers: 42.50.Ct, 03.65.Nk, 32.80.Qk, 42.50.Nn When resonant emitters, such as atoms, molecules, quantum dots, or metamaterial circuits, with a transition at a wavelength λ, are confined inside a volume smaller than λ 3 , they are coupled via strong dipole-dipole interactions. In this situation, the response of the ensemble to near-resonant light is collective and originates from the excitation of collective eigenstates of the system, such as super-and subradiant modes [1][2][3]. Dipole-dipole interactions affect the response of the system and the collective scattering of near-resonant light differs from the case of an assembly of noninteracting emitters [4]. It has even been predicted to be suppressed for a dense gas of cold twolevel atoms [5].Following the recent measurement of the collective Lamb shift [6] in a Fe layer [7], in a hot thermal vapor [8], and in arrays of trapped ions [9], it was pointed out [10] that the collective response of interacting emitters is different between ensembles exhibiting inhomogeneous broadening, such as solid state systems or thermal vapors, and those free of it, such as cold-atom clouds. In particular, inhomogeneous broadening suppresses the correlations induced by the interactions between dipoles, leading to the textbook theory of the optical response of continuous media [10,11]. In the absence of broadening, however, this theory fails and should be revisited to include the lightinduced correlations [12][13][14][15][16][17][18][19]. Several recent experiments aiming at studying collective scattering with identical emitters used large and optically thick ensembles of cold atoms [20][21][22][23]. However, the case of a cold-atom ensemble with a size comparable to the optical wavelength has not been studied experimentally, nor has the transition between the well-understood case of scattering by an individual atom [24] to collective scattering. In particular, the suppression of light scattering when the number of atoms increases in a regime of collective scattering has never been directly observed.Here, we study-both experimentally and theoreticallythe emergence of collective effects in the optical response of a cold-atom sample due to dipole-dipole interactions, as we gradually increase the number of atoms. To do so, we send low-intensity near-resonant laser light onto a cloud containing from 1 to ∼450 cold 87 Rb ato...

We study the interactions of a possibly dense and/or quantum degenerate gas with driving light. Both the atoms and the electromagnetic fields are represented by quantum fields throughout the analysis. We introduce a field theory version of Markov and Born approximations for the interactions of light with matter, and devise a procedure whereby certain types of products of atom and light fields may be put to a desired, essentially normal, order. In the limit of low light intensity we find a hierarchy of equations of motion for correlation functions that contain one excited-atom field and one, two, three, etc., ground state atom fields. It is conjectured that the entire linear hierarchy may be solved by solving numerically the classical equations for the coupled system of electromagnetic fields and charged harmonic oscillators. We discuss the emergence of resonant dipole-dipole interactions and collective linewidths, and delineate the limits of validity of the column density approach in terms of non-cooperative atoms by presenting a mathematical example in which this approach is exact.Comment: 35 pages, RevTe

We show how strong light-mediated resonant dipole-dipole interactions between atoms can be utilized in a control and storage of light. The method is based on a high-fidelity preparation of a collective atomic excitation in a single correlated subradiant eigenmode in a lattice. We demonstrate how a simple phenomenological model captures the qualitative features of the dynamics and sharp transmission resonances that may find applications in sensing. DOI: 10.1103/PhysRevLett.117.243601 Resonant emitters play a key role in optical devices for classical and quantum technologies. Atoms have particular advantages because of an excellent isolation from environmental noise with well-specified resonance frequencies and no absorption due to nonradiative losses. At high densities, however, they exhibit strong light-mediated resonant dipoledipole (DD) interactions that can lead to uncontrolled and unwanted phenomena, such as resonance broadening, shifts, and dephasing. According to common wisdom, these are considered as a design limitation in quantum and classical light technologies, e.g., in quantum metrology [1,2], sensing [3], information processing [4], in the storage of light, and in the implementations of quantum memories [5][6][7][8]. DD interactions also receive significant attention, e.g., in Rydberg gases [9][10][11][12][13]. Here, we show how strong radiative interactions can be harnessed in engineering long living collective excitations that open up avenues for utilizing resonant DD interactions in the control and storage of light, and in sensing. Our protocol is based on controlled preparation of large, many-atom subradiant excitations, where the light-mediated interactions between the atoms strongly suppress radiative losses.Superradiance [14], where the emission of light is coherently enhanced in an ensemble of emitters has continued to attract considerable interest [15] with the recent experiments focusing on light in confined geometries [16], weak excitation regime [17][18][19], and the related shifts of the resonance frequencies [20][21][22][23][24]. Its counterpart, subradiance, describes coherently suppressed emission due to a weak coupling to the radiative vacuum. Because of the weak coupling, subradiant states are challenging to excite and have experimentally proved elusive. In atomic and molecular systems subradiance has been observed in pairs of trapped ions [25] and molecules [26], as well as in weakly bound ultracold molecular states [27,28]. In a large atom cloud, a subradiant decay was recently observed in the long tails of a radiative decay distribution [29] that indicated a small fraction of the atoms exhibiting a suppressed emission.In our model, an incident light excites a collective atomic state that exhibits a significant radiative vacuum coupling.The excitation is then transferred to a radiatively isolated cooperative state. The cold atoms that store the light excitation are confined in a planar lattice, providing a protection against nonradiative losses, which typically are a common hin...

We study the collective response of a dense atomic sample to light essentially exactly using classicalelectrodynamics simulations. In a homogeneously broadened atomic sample there is no overt LorentzLorenz local field shift of the resonance, nor a collective Lamb shift. However, the addition of inhomogeneous broadening restores the usual mean-field phenomenology. DOI: 10.1103/PhysRevLett.112.113603 PACS numbers: 42.50.Nn, 32.70.Jz, 42.25.Bs Textbook arguments [1,2] tell us that in a dielectric medium the local electric field E l seen by an atom (molecule) is different from the macroscopic electric field E by an amount proportional to the polarization P of the medium, E l ¼ E þ P=3ϵ 0 . This is the origin of the localfield corrections in electrodynamics embodied in the Clausius-Mossotti and Lorentz-Lorenz relations. As a result, the frequency dependence of the microscopic polarizability and the macroscopic susceptibility are different. If the polarizability has a Lorentzian line shape then so does the susceptibility, but the resonance is shifted by what is known as the Lorentz-Lorenz (LL) shift [3]. The LL shift serves as the generic frequency scale for other density dependent phenomena in an atomic sample such as collisional self-broadening of absorption lines [4,5] and collective Lamb shift (CLS) [6][7][8][9][10].Local-field corrections are a standard workhorse in solid and liquid media. On the other hand, in a resonant atomic gas a density conducive to LL shift and CLS results in an optically thick sample, which might explain the sparsity of laser spectroscopy era experiments. There are careful experiments on related phenomenology that agree with the respective theory [8,9,[11][12][13], but except for the nuclear-physics experiment of Ref.[9] the published experiments we know of deal with inhomogeneously broadened samples with a substantial line broadening due to the motion of the atoms. Atomic-physics experiments with cold and dense clouds such as those in Ref. [14] are presently underway [15]. Optically thick samples are needed for a good quantum interface between photons and matter [16], so that local-field effects, and, more generally, cooperative response of matter to light, are likely to become issues in the quest toward quantum technologies.Here we study the cooperative response of a dense atomic sample to light in the limit of low excitation essentially exactly [17] using classical-electrodynamics simulations [18][19][20][21][22][23][24][25] in a slab geometry analogously to theory [6] and experiments [8] on CLS. In these simulations with an unprecedentedly large scale, we have discovered that a homogeneously broadened sample with fixed atomic positions in fact does not exhibit the expected LorentzLorenz or collective Lamb shifts. However, when we add inhomogeneous broadening [24] to the atomic samples, the traditional phenomenology of local-field corrections together with density-dependent collective effects reemerges. Basically, in a homogeneously broadened sample the correlations between nearby ...

We study interactions of light with a sample of two-level atoms, with full inclusion of angular momentum degeneracy, at temperatures and densities such that the quantum statistics of the atoms may have an effect.Coupled propagation equations are given for light and matter fields, and plausible general simplifications are enumerated. In particular, the motion of the atoms during the excited-state lifetime may often be ignored. The propagation equations of light and matter fields are decoupled within the assumption that the detuning of the driving light from atomic resonance is large, and the spectrum of scattered light is studied for ideal Bose and Fermi gases. In addition to the expected image of the velocity distribution, the spectra contain qualitatively distinct features that depend on the statistics of the atoms. This is because for bosons (fermions), those scattering events in which an atom recoils to an already occupied state are enhanced (inhibited). PACS number(s): 42.50.Vk, 03.75.Fi, 05.30.Jp ( gm),~em')). Here g stands for the "ground" and e for the "excited" level, the corresponding angular momenta being jg and j,, and m, m' denote the z components of angular momentum. We use n as the generic energy level label, so that n = g or e. The frequency of the optical transition between the levels g and e is denoted by coo, and d stands for the dipole moment operator of the transition.

We measure the coherent scattering of light by a cloud of laser-cooled atoms with a size comparable to the wavelength of light. By interfering a laser beam tuned near an atomic resonance with the field scattered by the atoms, we observe a resonance with a redshift, a broadening, and a saturation of the extinction for increasing atom numbers. We attribute these features to enhanced light-induced dipole-dipole interactions in a cold, dense atomic ensemble that result in a failure of standard predictions such as the "cooperative Lamb shift". The description of the atomic cloud by a mean-field model based on the Lorentz-Lorenz formula that ignores scattering events where light is scattered recurrently by the same atom and by a microscopic discrete dipole model that incorporates these effects lead to progressively closer agreement with the observations, despite remaining differences. DOI: 10.1103/PhysRevLett.116.233601 The understanding of light propagation in dense media relies traditionally on a continuous description of the sample characterized by macroscopic quantities such as susceptibility or refractive index [1,2]. Their derivation from a microscopic theory is in general challenging owing to the interactions between the light-induced dipoles that can be large when the light is tuned near an atomic resonance. In dilute media, their role can be analyzed using the perturbative approach of Friedberg, Hartmann, and Manassah (FHM) [3], which predicts in particular a "cooperative Lamb shift" measured recently in inhomogeneously broadened media [4,5] and cold dilute atomic gases [6]. For an atom slab, the FHM approach was shown to correspond to the low-density limit of the local-field model introduced by Lorentz [7], which replaces the action of all the atoms of the medium on a particular one by an average effective field [1,2], thus ignoring correlations between the light-induced dipoles. This mean-field approach leads to the Lorentz-Lorenz formula, which allows calculating the index of refraction of many dense media with an excellent accuracy [1,8]. However, it was pointed out [7,9] that in the absence of inhomogeneous broadening, such as in cold atomic ensembles, the mean-field response may not be valid due to recurrent scattering where the field radiated by one atom can be scattered back by another atom [10,11]. Recurrent scattering should become important when the incident light (wavelength λ ¼ 2π=k) is tuned near an atomic resonance, and the atomic density approaches k 3 . This calls for an experiment operating in this regime, where a comparison between the standard mean-field theories of light scattering and a microscopic approach, including recurrent scattering, can be performed.Here, we perform this comparison. To do so, we need to access a quantity relevant to both the macroscopic and the microscopic approaches. The coherent electric field hE sc i scattered by the cloud fulfills this condition: it is obtained by averaging the scattered field E sc over many realizations of the spatial random distribution o...

We study propagation of low-intensity light in a medium within a one-dimensional ͑1D͒ model electrodynamics. It is shown that the coupled theory for light and matter fields may be solved, in principle exactly, by means of stochastic simulations that account for both collective linewidths and line shifts, and for quantum statistical position correlations of the atoms. Such simulations require that one synthesize atomic positions that have correlation functions appropriate for the given type of atomic sample. We demonstrate how one may simulate both a Bose-Einstein condensate ͑BEC͒ and a zero-temperature noninteracting Fermi-Dirac gas. Results of simulations of light propagation in such quantum degenerate gases are compared with analytical density expansions obtained by adapting the approach of Morice, Castin, and Dalibard ͓Phys. Rev. A 51, 3896 ͑1995͔͒ to the 1D electrodynamics. A BEC exhibits an optical resonance that narrows and stays somewhat below the atomic resonance frequency as collective effects set in with increasing atom density. The first two terms in the analytical density expansion are in excellent agreement with numerical results for a condensate. While fermions display a similar narrowing and shift of the resonance with increasing density, already in the limit of very dilute gas the linewidth is only half of the resonance linewidth of an isolated atom. We attribute this to the regular spacing between the atoms, which is enforced by the Pauli exclusion principle. The analytical density expansion successfully predicts the narrowing, and also gives the next term in the density expansion of the optical response in semiquantitative agreement with numerical simulations.