Abstract

Nonlinear optical phenomena are typically local. Here, we predict the possibility of highly nonlocal optical nonlinearities for light propagating in atomic media trapped near a nano-waveguide, where long-range interactions between the atoms can be tailored. When the atoms are in an electromagnetically induced transparency configuration, the atomic interactions are translated to long-range interactions between photons and thus to highly nonlocal optical nonlinearities. We derive and analyze the governing nonlinear propagation equation, finding a roton-like excitation spectrum for light and the emergence of order in its output intensity. These predictions open the door to studies of unexplored wave dynamics and many-body physics with highly nonlocal interactions of optical fields in one dimension.

© 2016 Optical Society of America

1. INTRODUCTION

Optical nonlinearities are commonly described by local nonlinear responses of the material to the optical field, resulting in the dependence of the refractive index at point z on the field at the same point, E(z) [1]. Recently however, there has been a growing interest in nonlocal nonlinear optics, namely, in mechanisms whereby the refractive index at z depends on the field intensity at different points z in the material [2]. Mechanisms that give rise to such nonlocal nonlinearities include heat diffusion [3,4], molecular reorientation in liquid crystals [5], and atomic diffusion [6,7]. This paper discusses a new regime of extremely nonlocal nonlinearities affecting both the frequency and the quadratic dispersion of optical waves [see Eq. (1) below]. The physical mechanism that leads to this new regime is very different than those explored previously [27]: it relies on an atomic medium prepared in an electromagnetically induced transparency (EIT) configuration whose optical nonlinearity is controlled by shaping the nonlocal dipolar interactions between the atoms.

EIT is associated with the lossless and slow propagation of light pulses in resonant atomic media subject to the coherent driving of an auxiliary atomic transition [8,9]. Since the early days of EIT, it has been explored as a means of enhancing optical nonlinearities [8,1012]. A particularly effective mechanism for giant optical nonlinearities is provided by dipolar interactions between atoms that form the medium. Since EIT can be described by the propagation of the so-called dark-state polariton [13], which is a superposition of the light field and an atomic spin wave, the inherently nonlocal dipolar interactions between atoms are translated to nonlocal nonlinearities in polariton propagation. In the case of dipolar interactions between Rydberg atoms in free space, most theoretical [1422] and experimental [2327] studies have focused on their remarkable strength, a useful feature in quantum information. Exploration of the nonlocal aspect of this nonlinearity, on the other hand, has received less attention, and was restricted to the nonlocality dictated by the 1/r6 spatial dependence of the van der Waals potential between Rydberg atoms [28].

The present work rests on two recently explored mechanisms: that of EIT polaritons and that of modified long-range dipolar interactions in confining geometries, such as fibers, waveguides, photonic band structures, and transmission lines, which are currently attracting considerable interest [2938]. Yet, we show that the combined effect of these mechanisms may allow for new and unfamiliar possibilities of nonlocal nonlinear optics. More specifically, we show how dispersive laser-induced dipolar interactions between atoms coupled to a nano-waveguide with a grating, which can be designed to extend over hundreds of optical wavelengths [32,33], are translated via EIT into extremely nonlocal optical nonlinearities. We then analyze light propagation in this medium along the waveguide and find a roton-like excitation spectrum for light and the emergence of spatial self-order in its output intensity.

2. SUMMARY AND SCOPE OF THE RESULTS

The main general result of this work is the derivation of a nonlinear propagation equation for the (possibly quantum) EIT polariton field Ψ^ comprised of light guided by the waveguide, and an atomic medium tightly trapped along the waveguide axis z with an effectively one-dimensional (1D) inter-atomic potential U(zz),

(t+vz)Ψ^(z)=iΔ^cαΨ^(z)iΔ^cCv2z2Ψ^(z),Δ^c=δc+δ^NL,δ^NL=αLdzU(zz)Ψ^(z)Ψ^(z).
The left-hand side of the equation describes an envelope of a wave traveling with a group velocity v, whereas the first and second terms of the right-hand side present its frequency shift Δ^c (multiplied by a coefficient α) and its quadratic dispersion with a coefficient proportional to the detuning Δ^c (and to a constant C), respectively. nonlinearity comes about by noting the detuning Δ^c: it contains a linear component δc, which is controlled by the EIT configuration [the so-called coupling-field detuning, see Fig. 1(b)], and a nonlocal nonlinear detuning δ^NL, which depends on field intensities integrated over the medium with the interaction kernel U(z). The appearance of a nonlocal nonlinearity not only in the frequency shift but also in the dispersion coefficient gives rise to a new regime of nonlocal nonlinear optics. The physical system and reasoning that lead to Eq. (1), as well as its derivation, are discussed in Section 3 below. In principle, we assume sufficient conditions for a lossless EIT propagation (coefficients v, α, and C are real), whereas loss and decoherence mechanisms due to imperfections and scattering are analyzed in Section 6.

 figure: Fig. 1.

Fig. 1. (a) Setup: atoms (black dots), illuminated by the EIT fields [see (b)] E^ and Ω (thin blue arrow), are trapped at a distance ra from a nano-waveguide (gray cylinder) along z, from z=0 to z=L. A far-detuned laser ΩL (thick orange arrow), tilted by an angle θL from the z-axis, induces long-range interactions between the atoms, mediated by the waveguide modes [see (c)]. (b) EIT atomic configuration: the probe field E^ is resonantly coupled to the |g|e transition, whereas the coupling field Ω is coupled to the |d|e transition with detuning δc. Interaction between the atoms in the |d-level [see (c)] induces its energy shift δNL, which is effectively added to the detuning δc. (c) Laser-induced dipolar interactions: the laser with Rabi frequency ΩL and detuning δL operates on the |d|s transition of all atoms, |s being an additional level, thus inducing a dipolar potential U(z) between pairs of atoms (z apart) populating the state |d [59].

Download Full Size | PPT Slide | PDF

The second part of this work (Sections 4 and 5) is dedicated to study some first implications of wave propagation in such a medium, and specifically to the analysis of wave excitations on top of a continuous-wave (CW) background. General results for the excitation spectrum (dispersion relation) and the intensity correlations at the output are presented in Section 4, Eqs. (8) and (11), respectively. These are followed in Section 5 by specific results for a medium of atoms trapped near a waveguide grating that supports extremely long-range interactions U(z), as per Eq. (12). The results for the excitation spectrum exhibit a roton-like narrow-band shape, which may be probed by a homodyne detection scheme (Fig. 3). The roton-like spectrum signifies the tendency of light in this regime to exhibit spatial self-order, namely, crystal-like correlations; these can be revealed by measuring the photon intensity at the waveguide’s output (Fig. 4).

 figure: Fig. 2.

Fig. 2. Linear susceptibility of the EIT atomic medium to the probe field as a function of its detuning Δp [8,9] (Δp in units of level width γ and Ω=2.5γ). (a) For total detuning of the coupling field Δc=0, the absorption Imχ (red dashed line) and dispersion Reχ (blue solid line) are symmetrical and antisymmetric, respectively, with respect to Δp, so that no (real) quadratic dispersion exists for the probe envelope centered around Δp=0. (b) For Δc0, Reχ is not antisymmetric, so that quadratic dispersion exists, giving rise to the term ΔcCv2z2 in Eq. (1), with Δc=δc+δNL [Fig. 1(b)].

Download Full Size | PPT Slide | PDF

 figure: Fig. 3.

Fig. 3. Dispersion relations for EIT polaritons with waveguide-grating mediated atomic interactions (k values presented in all plots are within the EIT transparency window; values of physical parameters used here are given in Supplement 1, Section 4). (a) Roton-like excitation spectrum (dispersion relation) ωk for the potential of Eq. (12) in the anomalous dispersion case (opposite signs of ωk0 and Uk). The analytical results from Eq. (8) (blue solid line) agree well with those of direct numerical simulations of Eq. (1) (gray dots). Compared with the spectrum of a local interaction (Uk independent of k, dashed red line), the roton-like spectrum exhibits a dip in a narrow band of k-values around kR=kLzkB. (b) Anti-roton peak of the spectrum ωk in a narrow band around kR in the normal dispersion case (identical signs of ωk0 and Uk). (c) Possible homodyne detection scheme: the input probe field consists of a CW field + perturbation at wavenumber k and frequency ω(0)(k). The field is split before entering the EIT medium (z=0) so that a local oscillator of ω(0)(k) is formed (lower arm) by filtering out the CW component. Then, mixing the output signal (z=L) with the local oscillator reveals their phase difference, from which ωk can be inferred (see text).

Download Full Size | PPT Slide | PDF

 figure: Fig. 4.

Fig. 4. Instability and self-ordering (values of physical parameters are given in Supplement 1, Section 4). (a) γk=Imωk is the exponential growth rate of unstable perturbations at spatial frequency k (anomalous dispersion case). It exhibits a narrow peak around kR (blue dotted line), compared to the broadband instability of the local interaction case (red dotted line). The instability is accompanied by the generation of quantum entanglement, characterized by a narrow-band squeezing spectrum Gk (blue solid line; Gk<1 quantifies entanglement between ±k photon modes), in contrast to the broadband spectrum for a local interaction (horizontal red solid line). (b) Dynamics of spatial spectrum Nk(t) of the field intensity for different propagation times t inside the medium (t=L/v at the output). The emergence of a large peak around kR (and kR) out of the initial Gaussian perturbation is clearly seen. Results of both the linearized theory (solid lines) and numerical simulations of Eq. (1) (dots) are shown: slight differences at later t-values are attributed to nonlinear corrections. (c) Self-ordering of the field intensity: considering the narrow peaks of Nk(t) around ±kR, fluctuations at these k-values become dominant, resulting in ordered intensity correlations g(2) which grow with propagation time t (t=L/v at the output). Excellent agreement between the theory, Eq. (11) (solid lines), and numerical simulations of Eq. (1) (dots) is observed. The correlations oscillate with a period 2π/kR6.1mm and a range of a few l2.7mm (where L=2.68cm and v=4340ms1), determined by the long-range interaction U(z). This is in contrast to the local interaction case, where the correlations vanish (g(2)=1) after a short distance π/qtr1.75mm, which is determined by the bandwidth qtr of the initial fluctuations [see inset: local (red thin line) and nonlocal (blue thick line) cases for the output field at t=L/v].

Download Full Size | PPT Slide | PDF

The predicted self-order of light constitutes a new, hitherto unexplored, optical “phase,” analogous to the spatial structure of cold atomic media subject to light-induced dipolar interactions [32,3944]. We discuss important aspects and prospects of this work in Section 7.

3. EIT POLARITONS WITH NONLOCAL INTERACTIONS

A. System

Consider a medium of identical atoms in an EIT configuration as in Fig. 1: the atoms are trapped at a distance ra from a nano-waveguide along its longitudinal z-axis [31,4554] [Fig. 1(a)]. A strong (external) coupling field with a constant and uniform Rabi frequency Ω drives the |d|e atomic transition with detuning δc and wavenumber kc, whereas a weak (possibly quantized) probe field with carrier frequency ω0, wavenumber k0, and envelope E^ is resonantly coupled to the |g|e transition (ω0=ωeg) [Fig. 1(b)]. Under tight transverse trapping (around ra) with respect to the transition wavelength, the atomic positions can be characterized by their longitudinal component z [see Supplement 1, Section 1.A] [32,45]. We assume the existence of a dipolar interaction U(z) between atoms that occupy the state |d [Fig. 1(c) and Section 5.A below]. Then, the energy of level |d of an atom at z is shifted by δNLnadzU(zz)Pd(z), na being the atomic density (per unit length) and Pd(z) the occupation of state |d in an atom at z.

We may now explain the physical reasoning that leads to Eq. (1). In Fig. 2, we plot the complex linear susceptibility χ of the EIT medium to the probe field E^ as a function of its detuning Δp in the presence of a coupling field detuned by Δc [8], which in our case is given by Δc=δc+δNL [Fig. 1(b)]. When Δc=0 [Fig. 2(a)], the absorption coefficient Imχ is symmetric with respect to Δp, whereas the dispersion Reχ is antisymmetric, so no (real) quadratic dispersion exists for the probe envelope centered around Δp=0. By contrast, when Δc0 is introduced [Fig. 2(b)], Reχ is no longer antisymmetric and quadratic dispersion exists, which explains the term ΔcCv2z2 in Eq. (1). However, this comes at the price of non-vanishing losses at Δp=0 (see also Section 6). For this reason, we choose to work in the so-called Autler–Townes regime, Ωγ, Δc [8], where γ is the width of the level |e. Then, for Δc smaller than the single-atom transparency window, ΔcΩ2/γ,Ω, but still larger than γ, the absorption per atom can become negligible while dispersion is still significant, as illustrated in Fig. 2(b) (see also Supplement 1, Section 1). This explains the lossless propagation described by Eq. (1) with real parameters α, v, C. As long as the absorption, associated with dissipation due to spontaneous emission at rate γ, is negligible, so are the noise effects of vacuum fluctuations; Eq. (1) then holds in operator form without additional Langevin quantum noise operators.

B. Derivation of Eq. (1)

The formal derivation of Eq. (1) goes as follows (see Supplement 1, Section 1 for more details). The field envelope E^(z)=ka^keikz/L, with commutation relations [a^k,a^k]=δkk and hence [E^(z),E^(z)]=δ(zz), is assumed to be spectrally narrow and is guided by a transverse mode of the waveguide (later taken to be the HE11 mode of a fiber) with effective area A at the atomic position ra and polarization vector e0. The Hamiltonian in the interaction picture is

HAF=nadz[igE^(z)eik0zσ^eg(z)+h.c.],HAC=nadz[iΩeiδcteikczσ^ed(z)+h.c.],HDD=12na2dzdzU(zz)σ^dd(z)σ^dd(z)
and HF=kcka^ka^k, where g=ω0/(2ε0A)d·e0, d being the dipole matrix element of the |g|e transition, and σ^ij(z)=|ij| for an atom at z, with i, j representing the states {g,d,e}. By writing the Heisenberg equations for the atom and field operators and assuming a sufficiently weak probe field such that the atomic |g|e transition is far from saturation, we obtain coupled equations for the |g|d spin wave and the field [Supplement 1, Section 1.B],
σ¯gd(z)=gΩE^(z)1Ω[(t+γ)(1Ω*)×(t+iδc+iS^(z))σ¯gd(z)F^],(t+cz)E^(z)=nag*Ω*[t+iδc+iS^(z)]σ¯gd(z),
where σ¯gd(z)=σ^gd(z)ei(kck0)zeiδct. Here, γ and F^ are the spontaneous emission rate and corresponding Langevin noise operator, respectively, due to the coupling of the |g|e transition to the reservoir formed by photon modes other than those guided by the waveguide. The effect of interaction is a nonlinear detuning for the coupling field,
S^(z)=naLdzU(zz)σ¯gd(z)σ¯gd(z).

Moving to the polariton picture of EIT [13], we define the dark and bright polaritons, Ψ^ and Φ^, respectively,

(Ψ^(z)Φ^(z))=(cosθNsinθsinθNcosθ)(E^(z)σ¯gd(z)/L),
with tan2θ=nag2/|Ω|2, and transform Eqs. (3) into equations of motion for the polaritons,
(t+ccos2θz)Ψ^(z)=sinθcosθczΦ^(z)isinθ(δc+S^(z))[sinθΨ^(z)cosθΦ^(z)],Φ^(z)=cosθ|Ω|2(t+γ)(t+iδc+iS^(z))×[sinθΨ^(z)cosθΦ^(z)]+nacosθΩF^.

In the adiabatic regime, where the probe field is a CW and in the absence of detunings (δc, U=0), the bright polariton Φ^ vanishes [13]. Here, we take the first non-adiabatic correction [Supplement 1, Section 1.C] by inserting the equation for Φ^ into that of Ψ^, and we assume all detunings to be smaller than the EIT transparency window δtr=Ω2/(γOD), with OD=(na/A)Lσa and σa the cross section of the |g|e transition, finally arriving at Eq. (1) with α=sin2θ, v=ccos2θ, and C=sin2θ(23sin2θ)/|Ω|2.

4. COLLECTIVE EXCITATIONS AND PROPAGATION IN A CW BACKGROUND

A. Excitation Spectrum of Polariton Waves

With Eq. (1) in hand, we now turn to the analysis of the polariton wave propagation it predicts. Specifically, we consider the CW polariton solution and find the dispersion relation of wave excitations around this CW background, analogous to the Bogoliubov spectrum of excitations in a Bose–Einstein condensate (BEC) [55,56]. The CW solution of Eq. (1) is ψ(t)=ψ0ei(αδc+npU0)t, with ψ0=|ψ0|eiφ, np=α2|ψ0|2 being an effective polariton density per unit length and U0 the k=0 component of the spatial Fourier transform of the potential Uk=dzU(z)eikz. Here we have neglected edge effects by assuming l<z<Ll, l being the range of the potential U(z). The dispersion relation of small fluctuations ϕ(z,t) around the large average CW field Ψ=ψ(t) are found as usual upon inserting Ψ=ψ(t)+ϕ(z,t) into Eq. (1) and linearizing it by keeping the fluctuations ϕ to linear order. Then, introducing the ansatz [55],

ϕ(z,t)=ei[φ(αδc+npU0)t][ukeikzei(ωk+kv)tvk*eikzei(ωk+kv)t]
into the linearized equation for ϕ, uk and vk being c-number (Bogoliubov) coefficients, and using standard procedures, we find the modified Bogoliubov spectrum [Supplement 1, Section 2.A],
ωk=ωk0(ωk0+2npUk),ωk0=(npU0/α+δc)Cv2k2.

This means that a polariton wave distortion (about the CW solution) with a wavenumber k relative to the carrier wavenumber (inside the EIT medium) oscillates at a frequency (relative to the carrier frequency ω0)

ω(k)=αδc+npU0±(vk+ωk),
where the ± sign is for positive/negative k, respectively. The dispersion relation [Eq. (9)] is composed of the detuning αδc due to the coupling field, the self-phase modulation of the CW component npU0 (analogous to the chemical potential in a BEC [55]), the linear dispersion vk, and the modified Bogoliubov excitation spectrum ωk. The spectrum ωk is determined by the interplay between the interaction Fourier transform Uk and the “kinetic-energy” quadratic dispersion ωk0, which is affected by both the detuning δc and by the k=0 component of Uk. This interplay is further discussed below for U(z) resulting from laser-induced interactions near a waveguide grating.

B. Generation of Two-Mode Correlations

The parametric process described by the foregoing modified Bogoliubov theory also entails the dynamic generation of two-mode squeezing, i.e., pairs of entangled polaritons with wavenumbers ±k. The analysis is similar to that of propagation in fibers with local Kerr nonlinearity [57,58]. Upon inserting the expansion of small quantum fluctuations in the longitudinal wavenumber modes k, ϕ^(z)=ka^keikz/L, into the linearized equation for ϕ(z,t), we obtain coupled first-order differential equations (in time) for a^k(t) and a^k(t), whose solution is a dynamic Bogoliubov transformation [Supplement 1, Section 2.B],

a^k(t)=ei(npU0+αδc+kv)t[μk(t)a^k(0)+ei2φνk(t)a^k(0)],μk(t)=cos(ωkt)inpUk+ωk0ωksin(ωkt),νk(t)=inpUkωksin(ωkt).
The number of entangled pairs generated after propagation time t at wavenumbers ±k can be quantified by the so-called squeezing spectrum, whose optimum is given by Gk=(|μk||νk|)2 [57] (Gk<1 signifies entanglement) or by the normalized second-order (intensity) correlation function g(2)(z,z,t)=[Ψ^(z)Ψ^(z)Ψ^(z)Ψ^(z)]/[Ψ^(z)Ψ^(z)Ψ^(z)Ψ^(z)] (all fields measured at the waveguide’s output after propagation time t=L/v), where the averaging is performed with respect to the initial probability distribution, e.g., the quantum state, of the polariton (probe) field. For initial zero-mean fluctuations (around the CW solution) with average polariton occupation at mode k, a^k(0)a^k(0)=Nkδkk, and vanishing anomalous correlations a^k(0)a^k(0)=0, we find [Supplement 1, Section 2.C]
g(2)(z,z)1+2α2πnp0dk[|μk|2Nk+|νk|2(Nk+1)+(2Nk+1)|μk||νk|cosφk]cos[k(zz)],
where φk=arg(μkνk), and we used (1/L)kdk/(2π). As we shall see below, these correlations may reveal the ordering of a nonlocal system caused by pair generation at preferred k-values.

5. HIGHLY NONLOCAL LASER-INDUCED INTERACTION VIA WAVEGUIDE GRATING

Our analysis up to this point was kept general, without specifying the interaction potential U(z). Let us now turn to a particularly interesting case of an extremely long-range interaction, where novel nonlinear optical effects can be illustrated.

A. Shaping the Interaction Potential

The illumination of atoms by an off-resonant laser virtually excites the atoms and allows them to resonantly interact via virtual photons. The spatial dependence of the resulting interaction potential U(z) then follows the spatial structure of the mediating photon modes. This is the essence of the laser-induced interaction potential we wish to employ [59]. Specifically, consider another laser with Rabi frequency ΩL, which is detuned by |δL|ΩL from the transition |d|s, |s being a fourth atomic level [Fig. 1(c)], and assume that this transition is distinct and separated from the transitions used for EIT [Fig. 1(b)], either spectrally or by polarization. Then, the extended waveguide modes can mediate long-range interactions between the trapped atoms [Fig. 1(a)] [32]. As a specific example, we consider a nano-waveguide that incorporates a grating, i.e., periodic perturbation of the refractive index with period Λπ/kB, so that the photon modes exhibit a bandgap (see, e.g., Refs. [48,60]). Then, for a laser frequency ωL inside the gap (the probe field’s carrier frequency ω0 being outside the gap), the laser-induced interaction potential becomes [32] (Supplement 1, Section 3)

U(z)=UL12cos(kLzz)cos(kBz)e|z|/l.
Here, kLz=kLcosθL with kL as the laser wavenumber and θL its orientation with respect to the waveguide axis z [Fig. 1(c)], and UL depends on the laser parameters, atomic transition, and effective mode area at the transverse position ra. The spatial dependence of this potential exhibits a sinusoidal behavior at kLz due to the laser ΩL beaten with another sinusoidal due to the Bragg grating at kB. The decaying envelope with a decay length l1/ωuωL, ωu being the frequency of the upper band edge, is due to the evanescent nature of the mediating waveguide modes inside the gap (ωL<ωu), where the length l can extend to hundreds of wavelengths for laser frequency ωL close to the band edge ωu [32,33] (see also Supplement 1, Section 3).

The resulting spatial Fourier transform Uk then consists of four Lorentzian peaks of width 1/l centered around the spatial beating frequencies ±(kLzkB) and ±(kLz+kB). We note that the laser ΩL and the grating are unrelated to the linear propagation of the probe field and their sole role is to induce the long-range dipolar interaction Eq. (12) between atoms, which is in turn translated via EIT to interaction between polaritons as in Eq. (1) (see also Discussion, Section 7.B).

B. Roton and Anti-Roton Spectra

Let us focus on the peak of Uk around kRkLzkB and its effect on the dispersion relation (spectrum), shown in Eq. (8). We first consider the case of anomalous dispersion, where the signs of ωk0 and Uk are opposite. Analogous to BEC, this describes the case of an attractive potential Uk that competes with the “kinetic energy” ωk0. Then, for k-values satisfying |ωk0|>2np|Uk|, ωk is real and exhibits a dip around kR, in contrast to the case of a local potential, for which Uk is independent of k (standard Bogoliubov spectrum) and this feature is absent. This is seen in Fig. 3(a), where both the analytical results of Eq. (8) and numerical simulations of the nonlinear Eq. (1) (Supplement 1, Section 5) are plotted and shown to agree very well. The narrow-band “dip” of this ωk spectrum is in analogy with the roton minimum in He II [61]. It reflects the fact that wave distortions about the CW field with spatial frequencies around kR cost less energy and are hence favorable. This feature implies that the intensity of the polariton field in its ground state would tend to self-order with typical wavenumber kR [61].

Turning to the case of normal dispersion, where the signs of ωk0 and Uk are identical, ωk exhibits an “anti-roton” peak around kR [Fig. 3(b)]. This means that distortions of spatial frequencies around kR are costly, so the system prefers to avoid these spatial variations. This behavior again manifests the tendency of the system to order, since it indicates the spatial distortions that the system is unlikely to be found in should it be in its ground state.

In order to measure the roton and anti-roton spectra, we first recall the meaning of the dispersion relation ω(k) from Eq. (9): without an interaction, the frequency associated with a wave envelope at wavenumber k traveling inside the EIT medium is given by ω(0)(k)=αδc+vk+δcCv2k2=ω(k)npU0(ωkδcCv2k2), so that ωk (together with npU0) expresses a frequency, or phase velocity, shift due to the nonlinearity. Namely, the wavenumber k describes a spatial eigenmode of propagation, both with and without interaction, with eigenfrequencies ω(k) and ω(0)(k), respectively. Now, suppose we let a weak quasi-CW pulse of length Lp<L and frequency ωp=ω(0)(k) enter the medium (on top of the strong CW) when the laser and hence the interaction Uk are turned off. Since upon entering the medium, the field does not change its frequency ωp, we deduce from the dispersion relation in the absence of interaction ω(0)(k) that the field inside the medium exhibits a perturbation ϕ(z) at spatial frequency k on top of the strong CW. Subsequently, when the entire pulse is in the medium, we immediately (non-adiabatically) turn on the laser ΩL and hence the interaction Uk, so that the temporal frequency of the perturbation ϕ(z) at wavenumber k becomes ω(k)=ω(0)(k)+npU0+(ωkδcCv2k2). The frequency shift npU0+(ωkδcCv2k2) can be thought of as an extra energy acquired by the mode k due to the interaction energy. Therefore, the spectrum ωk can be inferred from the frequency shift, measurable by homodyne detection of the pulse ϕ(z) that exits the EIT medium [Fig. 3(c)]. A similar procedure was proposed for measuring the tachyon-like spectrum of polaritons in inverted media [62].

C. Dynamical Instability: Pair Generation

In the anomalous dispersion case, consider now a sufficiently strong interaction such that for a narrow band of k-values around kR, where Uk is peaked, the condition 2np|Uk|>|ωk0| is satisfied and ωk becomes imaginary. Then, field perturbations around kR become exponentially unstable [Fig. 4(a)], resulting in parametric amplification and the generation of entangled photon pairs in this narrow band of unstable k-values. The strength of the amplified perturbation and generated field is characterized by the magnitude of the coefficients μk(t) and νk(t) from Eq. (10), which grow exponentially with propagation time t and are largest for the narrow peak around kR. The resulting squeezing spectrum Gk at the output t=L/v [Fig. 4(a)] then exhibits stronger squeezing at a narrow bandwidth around kR, which may be measured by homodyne detection [57].

D. Dynamical Instability: Emergence of Self-Order

An interesting implication of the extremely nonlocal potential of Eq. (12) is the dynamic formation of order in the system. Consider that at t=0, there exist fluctuations around the CW, with a spatial spectrum Nk=N0e(k/qtr)2, i.e., “δ-correlated” noise limited by the EIT transparency window of width δtr=vqtr. Then, fluctuations at k-values around the peak kR will be parametrically amplified as they propagate through the medium, as verified in Fig. 4(b), where the spatial spectrum of the polariton field, Nk(t)=a^k(t)a^k(t), at different propagation times t (t=L/v describing the output field) is calculated both analytically and via direct (classical) numerical simulations of Eq. (1). This suggests that the system becomes spatially ordered with a period 2π/kR, which may be revealed by measuring the correlation function g(2) between the intensities of the output field that arrive at a detector at the waveguide’s end (z=L) and time difference (zz)/v [scheme from Fig. 3(c) without the lower local oscillator arm]. We obtain the corresponding g(2) by numerically integrating over k in the classical limit of Eq. (11), where the vacuum-fluctuation contributions are neglected. This calculation is compared to g(2) measured via direct numerical simulations of Eq. (1), yielding excellent agreement. Figure 4(c) reveals the emergence of order by presenting the intensity correlations g(2) at different propagation times t through the medium (t=L/v at the output). The resulting g(2) exhibits oscillations with a period zz2π/(kR), which persist over a few l. Considering that the interaction range l can reach hundreds or even thousands of optical wavelengths (l3027λL and L10l0.026m in our example, see Supplement 1, Section 4), the light intensity clearly becomes ordered due to the long-range interaction U(z), as can also be seen by the comparison to the local interaction case [inset of Fig. 4(c)].

6. SCATTERING AND IMPERFECTIONS

So far, we have considered a purely coherent evolution of the polariton field. Here we address three main sources of scattering and loss of field excitations; we estimate the decoherence rate each of them imposes on the polariton field and its possible effect on the observability of the self-ordering effects discussed above (see Supplement 1, Section 6, for details).

First, since the interaction U(z) from Fig. 1(b) is induced by the illumination of the atoms by an off-resonant laser, it is accompanied by an incoherent process of scattering of laser photons ΩL from the |d|s transition to non-guided modes at rate Rfs [59]. This process limits the coherence time of the σ¯gd spin wave and hence that of the polariton to be below Rfs1, which in the examples of Figs. 3 and 4 is nevertheless much longer than the experiment time L/v (Supplement 1, Section 4).

Second, consider material imperfections in the waveguide grating structure (e.g., defects and surface roughness) that give rise to the scattering of photons off the guided modes. This leads to a decay rate (width) κ for each of the longitudinal (guided) photon modes k (see Supplement 1 Section 6.C and [33,49]). Then, since atoms at level |d are coupled via ΩL to these lossy waveguide grating modes, the σ¯gd spin wave and hence the polariton are decohered at a rate Rim(1)UL/Im1iκ/(2δu), δu being the detuning of the laser ΩL from the upper band edge of the grating. However, the effect of Rim(1) on the spectra ωk and the observables mentioned above can in principle be made arbitrary small by noting that ωk depends on npUL, whereas Rim(1) depends solely on UL, reflecting the fact that Rim(1) is a single-polariton loss mechanism, whereas ωk is a cooperative effect. Then, decreasing |UL| while keeping npUL constant (by increasing the CW power np) reduces Rim(1) but keeps ωk unchanged.

Nevertheless, a cooperative decoherence can in fact result from κ represented by an imaginary part acquired by U(z); namely, a pair of atoms at a distance z apart can jointly scatter photons at a rate ImU(z), since the virtual photons that mediate their interaction U are now lossy and enable excitation decay to non-guided modes. Then, for a single polariton at z, the scattering induced by the entire atomic medium becomes Rim(2)npdzImU(z)np|U0|κΔu/(κ2/4+Δu2), with ΔuωL(kR/kB)2/(n¯Δn). In Supplement 1, Section 6.B, we show that in cases where this dephasing channel limits the observability of ωk,γk, its effect can be reduced by increasing the EIT coupling laser Ω.

Finally, we recall that EIT losses were neglected considering all detunings and the probe bandwidth to be within the transparency width δtr. The EIT losses analysis of Supplement 1, Section 6.C goes beyond these simple considerations and treats the EIT-induced decoherence rate REIT using the full EIT loss spectrum. We find that the observability of the anti-roton peak is indeed not affected much by the EIT decoherence, whereas a similar conclusion is reached in the case of the instability by possibly detuning the carrier frequency of the probe field to match that of the coupling field (two-photon resonance). EIT losses may, however, pose a significant limitation to the observability of the more sensitive roton “dip.”

In principle, all loss terms discussed above should be accompanied by corresponding quantum noise terms, which are not fully considered here, restricting this discussion to the classical field case (for which all of the above results apply, excluding the squeezing spectrum Gk).

7. DISCUSSION

This study predicts a new and hitherto unexplored regime of nonlinear optics; namely, that of highly nonlocal interactions between photons in 1D. These nonlocal optical nonlinearities arise for light propagation inside driven atomic media in the vicinity of a waveguide and affect both the frequency and quadratic dispersion of the light field. We have derived the nonlinear equation that governs light propagation in this regime and have analyzed it around its CW solution, finding a narrow roton-like dispersion relation [Figs. 3(a) and 3(b)] and squeezing (entanglement) spectrum [Fig. 4(a)] and the emergence of order in the field intensity [Figs. 4(b) and 4(c)], all of which reflect the tendency of the system to self-organize, which in turn results from the long-range interactions between photons. Tremendous experimental progress in both key ingredients of our scheme has been recently reported and constantly being pursued, namely, that of electromagnetically induced transparency in waveguide systems [51,63] and the ability to design dipolar interactions therein [64]. In the following, we wish to discuss some important aspects of this work.

A. Structure and Generality of Eq. (1)

It is important to note the nonlocal and nonlinear dispersion term appearing in Eq. (1), absent in previous works on nonlocal nonlinear optics [2,47] and EIT-based nonlinear optics [1418,20,22,28]. We identify the physical conditions under which this term becomes significant and may lead to new phenomena, namely, in the Autler–Townes regime of EIT and for sufficiently large nonlinear detuning.

Another interesting point concerns the generality of this work in relation to different confining geometries. The derivation of Eq. (1) does not depend on the specific form of the potential U(z), which naturally opens the way to the exploration of nonlocal nonlinear optics due to laser-induced dipolar potentials mediated by confined photon modes of geometries other than the waveguide grating considered here. For example, the laser-induced potential in a cavity along z is in general not translational-invariant and depends on the z-positions of both interacting atoms (rather than just their difference), a case which is qualitatively different from the one analyzed here.

A different possibility is to consider modified laser-induced dipolar interactions shaped by controlling the spectrum of the exciting laser ΩL, rather than by working in a confined geometry [65,66]. Using our approach, this may allow exploring the generalization of Eq. (1) to a three-dimensional (free-space) environment with a designed dipolar potential.

B. Origin and Length Scale of Self-Ordering

We stress that the length scale associated with order kR=kLcosθLkB originates from the interaction potential of the light field with itself, Eq. (12); hence, ordering spontaneously occurs in this optical system, much like in other condensed-mater systems and crystals. This is in contrast to, e.g., the order of atoms in a deep-potential optical lattice, where the atoms are situated at lattice sites determined by the potential imposed by an external laser rather than by their mutual interactions. In our case, the role of the grating is not to trap the propagating polariton, but merely to create dispersive and long-range dipolar interactions between atoms U(z) (induced by an external laser ΩL that is unrelated to the light component of the polariton), which underlies the nonlinearity in the polariton propagation.

Moreover, it is important to note that the length scale kR is a signature of the specific spatial shape of the potential U(z) [sinusoidal in the case of Eq. (12)], as also revealed by the excitation and instability spectra of Figs. 3 and 4(a). This is in contrast to, e.g., the polariton crystallization process in a short-range potential described in Ref. [19] using Luttinger’s liquid theory, where the specific shape of the potential is irrelevant. In this respect, our results are more related to the modulational instability discussed in Ref. [28] for Rydberg-atom EIT, though there the spatial dependence of the potential is not tunable and is restricted to a power law (no length scale). However, the considered system is three-dimensional and the quadratic dispersion is linear and emerges simply due to diffraction. Other recent works where photon spatial correlations follow those of the interaction potential include Refs. [24] and [67,68], which, however, treat a probe field with only a few photons (typically two), in contrast with the largely collective behavior discussed in the present work.

C. Prospects

To conclude, this work opens the way to experimental and theoretical investigations of new nonlinear wave phenomena, especially by venturing beyond the linearized regime and exploring the role of the nonlocal and nonlinear dispersion term δ^NLCv2z2. Specific directions of further research may include: nonlocal nonlinear optics in 1D, concerning the study of solitons, where the lensing effect created by the nonlinear refractive index change is now highly nonlocal, following U(z); thermalization in 1D, which has been considered both experimentally [69] and theoretically [70] for isolated BEC in 1D, may become qualitatively different here due to the nonlocal character of the designed interactions U(z) and the nonlinear dispersion (“mass”); and effects of non-additivity of systems with long-range interactions [71] may be studied here for quantum/classical optical fields.

Funding

Austrian Science Fund (FWF) [P25329-N27, SFB F41 (VICOM), I830-N13 (LODIQUAS)]; Israel Science Foundation (ISF); United States–Israel Binational Science Foundation (BSF).

Acknowledgment

We acknowledge discussions with Ofer Firstenberg, Arno Rauschenbeutel, Philipp Schneeweiß, Darrick Chang, Nir Davidson, Tommaso Caneva, and Boris Malomed.

 

See Supplement 1 for supporting content.

REFERENCES

1. R. B. Boyd, Nonlinear Optics (Academic, 2008).

2. W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004). [CrossRef]  

3. C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005). [CrossRef]  

4. C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769–774 (2006). [CrossRef]  

5. C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004). [CrossRef]  

6. D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583–4587 (1993). [CrossRef]  

7. S. Skupin, M. Saffman, and W. Królikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007). [CrossRef]  

8. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]  

9. P. Lambropoulos and D. Petrosyan, Fundamentals of Quantum Optics and Quantum Information (Springer, 2007).

10. H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996). [CrossRef]  

11. S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611–3614 (1998). [CrossRef]  

12. M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000). [CrossRef]  

13. M. Fleischhauer and M. D. Lukin, “Quantum memory for photons: dark-state polaritons,” Phys. Rev. A 65, 022314 (2002). [CrossRef]  

14. I. Friedler, D. Petrosyan, M. L. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slow single-photon pulses,” Phys. Rev. A 72, 043803 (2005). [CrossRef]  

15. E. Shahmoon, G. Kurizki, M. Fleischhauer, and D. Petrosyan, “Strongly interacting photons in hollow-core waveguides,” Phys. Rev. A 83, 033806 (2011). [CrossRef]  

16. A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011). [CrossRef]  

17. B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011). [CrossRef]  

18. D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg atoms,” Phys. Rev. Lett. 107, 213601 (2011). [CrossRef]  

19. J. Otterbach, M. Moos, D. Muth, and M. Fleischhauer, “Wigner crystallization of single photons in cold Rydberg ensembles,” Phys. Rev. Lett. 111, 113001 (2013). [CrossRef]  

20. B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014). [CrossRef]  

21. A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014). [CrossRef]  

22. P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014). [CrossRef]  

23. T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012). [CrossRef]  

24. O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013). [CrossRef]  

25. D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013). [CrossRef]  

26. H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014). [CrossRef]  

27. D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, “Single-photon transistor using a Förster resonance,” Phys. Rev. Lett. 113, 053602 (2014). [CrossRef]  

28. S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal nonlinear optics in cold Rydberg gases,” Phys. Rev. Lett. 107, 153001 (2011). [CrossRef]  

29. E. Shahmoon and G. Kurizki, “Nonradiative interaction and entanglement between distant atoms,” Phys. Rev. A 87, 033831 (2013). [CrossRef]  

30. E. Shahmoon, I. Mazets, and G. Kurizki, “Giant vacuum forces via transmission lines,” Proc. Natl. Acad. Sci. USA 111, 10485–10490 (2014). [CrossRef]  

31. A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” arXiv:1503.04503 (2015).

32. E. Shahmoon, I. Mazets, and G. Kurizki, “Non-additivity in laser-illuminated many-atom systems,” Opt. Lett. 39, 3674–3677 (2014). [CrossRef]  

33. J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015). [CrossRef]  

34. A. González-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9, 320–325 (2015). [CrossRef]  

35. G. Kurizki, “Two-atom resonant radiative coupling in photonic band structures,” Phys. Rev. A 42, 2915–2924 (1990). [CrossRef]  

36. A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-mediated interactions between distant artificial atoms,” Science 342, 1494–1496 (2013). [CrossRef]  

37. J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014). [CrossRef]  

38. G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. USA 112, 3866–3873 (2015). [CrossRef]  

39. S. Giovanazzi, D. O’Dell, and G. Kurizki, “Density modulations of Bose–Einstein condensates via laser-induced interactions,” Phys. Rev. Lett. 88, 130402 (2002). [CrossRef]  

40. D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki, “Rotons in gaseous Bose–Einstein condensates irradiated by a laser,” Phys. Rev. Lett. 90, 110402 (2002). [CrossRef]  

41. R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions,” Science 336, 1570–1573 (2012). [CrossRef]  

42. D. E. Chang, J. I. Cirac, and H. J. Kimble, “Self-organization of atoms along a nanophotonic waveguide,” Phys. Rev. Lett. 110, 113606 (2013). [CrossRef]  

43. T. Grießer and H. Ritsch, “Light-Induced crystallization of cold atoms in a 1D optical trap,” Phys. Rev. Lett. 111, 055702 (2013). [CrossRef]  

44. P. Domokos and H. Ritsch, “Collective cooling and self-organization of atoms in a cavity,” Phys. Rev. Lett. 89, 253003 (2002). [CrossRef]  

45. E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010). [CrossRef]  

46. D. Reitz, C. Sayrin, R. Mitsch, P. Schneeweiss, and A. Rauschenbeutel, “Coherence properties of nanofiber-trapped cesium atoms,” Phys. Rev. Lett. 110, 243603 (2013). [CrossRef]  

47. A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012). [CrossRef]  

48. A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014). [CrossRef]  

49. J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013). [CrossRef]  

50. M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014). [CrossRef]  

51. B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a memory for tightly guided light in an optical nanofiber,” Phys. Rev. Lett. 114, 180503 (2015). [CrossRef]  

52. J. Lee, J. A. Grover, J. E. Hoffman, L. A. Orozco, and S. L. Rolston, “Inhomogeneous broadening of optical transitions of Rb87 atoms in an optical nanofiber trap,” J. Phys. B 48, 165004 (2015). [CrossRef]  

53. M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014). [CrossRef]  

54. V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2013). [CrossRef]  

55. C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases (Cambridge University, 2002).

56. L. P. Pitaevskii and S. Stringari, Bose–Einstein Condensation (Clarendon, 2003).

57. M. J. Potasek and B. Yurke, “Squeezed-light generation in a medium governed by the nonlinear Schrödinger equation,” Phys. Rev. A 35, 3974(R) (1987). [CrossRef]  

58. P. D. Drummond and Z. Ficek, eds., Quantum Squeezing (Springer-Verlag, 2004).

59. E. Shahmoon and G. Kurizki, “Nonlinear theory of laser-induced dipolar interactions in arbitrary geometry,” Phys. Rev. A 89, 043419 (2014). [CrossRef]  

60. K. P. Nayak and K. Hakuta, “Photonic crystal formation on optical nanofibers using femtosecond laser ablation technique,” Opt. Express 21, 2480–2490 (2013). [CrossRef]  

61. P. Nozieres and D. Pines, Theory of Quantum Liquids: Superfluid Bose Liquids (Addison-Wesley, 1990).

62. R. Y. Chiao, A. E. Kozhekin, and G. Kurizki, “Tachyonlike excitations in inverted two-level media,” Phys. Rev. Lett. 77, 1254–1257 (1996). [CrossRef]  

63. C. Sayrin, C. Clausen, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Storage of fiber-guided light in a nanofiber-trapped ensemble of cold atoms,” Optica 2, 353–356 (2015). [CrossRef]  

64. J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom–atom interactions around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).

65. J. Rodrguez and D. L. Andrews, “Inter-particle interaction induced by broadband radiation,” Opt. Commun. 282, 2267–2269 (2009).

66. E. Shahmoon, “van der Waals and Casimir-Polder dispersion forces,” in Forces of the Quantum Vacuum: An Introduction to Casimir Physics, W. Simpson and U. Leonhardt, eds. (World Scientific, 2015), Chap. 2.

67. T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015). [CrossRef]  

68. J. S. Douglas, T. Caneva, and D. E. Chang, “Photon molecules in atomic gases trapped near photonic crystal waveguides,” arXiv: 1511.00816 (2015).

69. S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,” Nature 449, 324–327 (2007). [CrossRef]  

70. P. Grisins and I. E. Mazets, “Thermalization in a one-dimensional integrable system,” Phys. Rev. A 84, 053635 (2011). [CrossRef]  

71. A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. R. B. Boyd, Nonlinear Optics (Academic, 2008).
  2. W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
    [Crossref]
  3. C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
    [Crossref]
  4. C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769–774 (2006).
    [Crossref]
  5. C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
    [Crossref]
  6. D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583–4587 (1993).
    [Crossref]
  7. S. Skupin, M. Saffman, and W. Królikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007).
    [Crossref]
  8. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
    [Crossref]
  9. P. Lambropoulos and D. Petrosyan, Fundamentals of Quantum Optics and Quantum Information (Springer, 2007).
  10. H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996).
    [Crossref]
  11. S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611–3614 (1998).
    [Crossref]
  12. M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000).
    [Crossref]
  13. M. Fleischhauer and M. D. Lukin, “Quantum memory for photons: dark-state polaritons,” Phys. Rev. A 65, 022314 (2002).
    [Crossref]
  14. I. Friedler, D. Petrosyan, M. L. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slow single-photon pulses,” Phys. Rev. A 72, 043803 (2005).
    [Crossref]
  15. E. Shahmoon, G. Kurizki, M. Fleischhauer, and D. Petrosyan, “Strongly interacting photons in hollow-core waveguides,” Phys. Rev. A 83, 033806 (2011).
    [Crossref]
  16. A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
    [Crossref]
  17. B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011).
    [Crossref]
  18. D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg atoms,” Phys. Rev. Lett. 107, 213601 (2011).
    [Crossref]
  19. J. Otterbach, M. Moos, D. Muth, and M. Fleischhauer, “Wigner crystallization of single photons in cold Rydberg ensembles,” Phys. Rev. Lett. 111, 113001 (2013).
    [Crossref]
  20. B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
    [Crossref]
  21. A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
    [Crossref]
  22. P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
    [Crossref]
  23. T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
    [Crossref]
  24. O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
    [Crossref]
  25. D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
    [Crossref]
  26. H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
    [Crossref]
  27. D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, “Single-photon transistor using a Förster resonance,” Phys. Rev. Lett. 113, 053602 (2014).
    [Crossref]
  28. S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal nonlinear optics in cold Rydberg gases,” Phys. Rev. Lett. 107, 153001 (2011).
    [Crossref]
  29. E. Shahmoon and G. Kurizki, “Nonradiative interaction and entanglement between distant atoms,” Phys. Rev. A 87, 033831 (2013).
    [Crossref]
  30. E. Shahmoon, I. Mazets, and G. Kurizki, “Giant vacuum forces via transmission lines,” Proc. Natl. Acad. Sci. USA 111, 10485–10490 (2014).
    [Crossref]
  31. A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” arXiv:1503.04503 (2015).
  32. E. Shahmoon, I. Mazets, and G. Kurizki, “Non-additivity in laser-illuminated many-atom systems,” Opt. Lett. 39, 3674–3677 (2014).
    [Crossref]
  33. J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
    [Crossref]
  34. A. González-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9, 320–325 (2015).
    [Crossref]
  35. G. Kurizki, “Two-atom resonant radiative coupling in photonic band structures,” Phys. Rev. A 42, 2915–2924 (1990).
    [Crossref]
  36. A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-mediated interactions between distant artificial atoms,” Science 342, 1494–1496 (2013).
    [Crossref]
  37. J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014).
    [Crossref]
  38. G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. USA 112, 3866–3873 (2015).
    [Crossref]
  39. S. Giovanazzi, D. O’Dell, and G. Kurizki, “Density modulations of Bose–Einstein condensates via laser-induced interactions,” Phys. Rev. Lett. 88, 130402 (2002).
    [Crossref]
  40. D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki, “Rotons in gaseous Bose–Einstein condensates irradiated by a laser,” Phys. Rev. Lett. 90, 110402 (2002).
    [Crossref]
  41. R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions,” Science 336, 1570–1573 (2012).
    [Crossref]
  42. D. E. Chang, J. I. Cirac, and H. J. Kimble, “Self-organization of atoms along a nanophotonic waveguide,” Phys. Rev. Lett. 110, 113606 (2013).
    [Crossref]
  43. T. Grießer and H. Ritsch, “Light-Induced crystallization of cold atoms in a 1D optical trap,” Phys. Rev. Lett. 111, 055702 (2013).
    [Crossref]
  44. P. Domokos and H. Ritsch, “Collective cooling and self-organization of atoms in a cavity,” Phys. Rev. Lett. 89, 253003 (2002).
    [Crossref]
  45. E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
    [Crossref]
  46. D. Reitz, C. Sayrin, R. Mitsch, P. Schneeweiss, and A. Rauschenbeutel, “Coherence properties of nanofiber-trapped cesium atoms,” Phys. Rev. Lett. 110, 243603 (2013).
    [Crossref]
  47. A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
    [Crossref]
  48. A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
    [Crossref]
  49. J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
    [Crossref]
  50. M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
    [Crossref]
  51. B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a memory for tightly guided light in an optical nanofiber,” Phys. Rev. Lett. 114, 180503 (2015).
    [Crossref]
  52. J. Lee, J. A. Grover, J. E. Hoffman, L. A. Orozco, and S. L. Rolston, “Inhomogeneous broadening of optical transitions of Rb87 atoms in an optical nanofiber trap,” J. Phys. B 48, 165004 (2015).
    [Crossref]
  53. M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
    [Crossref]
  54. V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2013).
    [Crossref]
  55. C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases (Cambridge University, 2002).
  56. L. P. Pitaevskii and S. Stringari, Bose–Einstein Condensation (Clarendon, 2003).
  57. M. J. Potasek and B. Yurke, “Squeezed-light generation in a medium governed by the nonlinear Schrödinger equation,” Phys. Rev. A 35, 3974(R) (1987).
    [Crossref]
  58. P. D. Drummond and Z. Ficek, eds., Quantum Squeezing (Springer-Verlag, 2004).
  59. E. Shahmoon and G. Kurizki, “Nonlinear theory of laser-induced dipolar interactions in arbitrary geometry,” Phys. Rev. A 89, 043419 (2014).
    [Crossref]
  60. K. P. Nayak and K. Hakuta, “Photonic crystal formation on optical nanofibers using femtosecond laser ablation technique,” Opt. Express 21, 2480–2490 (2013).
    [Crossref]
  61. P. Nozieres and D. Pines, Theory of Quantum Liquids: Superfluid Bose Liquids (Addison-Wesley, 1990).
  62. R. Y. Chiao, A. E. Kozhekin, and G. Kurizki, “Tachyonlike excitations in inverted two-level media,” Phys. Rev. Lett. 77, 1254–1257 (1996).
    [Crossref]
  63. C. Sayrin, C. Clausen, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Storage of fiber-guided light in a nanofiber-trapped ensemble of cold atoms,” Optica 2, 353–356 (2015).
    [Crossref]
  64. J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom–atom interactions around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).
  65. J. Rodrguez and D. L. Andrews, “Inter-particle interaction induced by broadband radiation,” Opt. Commun. 282, 2267–2269 (2009).
  66. E. Shahmoon, “van der Waals and Casimir-Polder dispersion forces,” in Forces of the Quantum Vacuum: An Introduction to Casimir Physics, W. Simpson and U. Leonhardt, eds. (World Scientific, 2015), Chap. 2.
  67. T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
    [Crossref]
  68. J. S. Douglas, T. Caneva, and D. E. Chang, “Photon molecules in atomic gases trapped near photonic crystal waveguides,” arXiv: 1511.00816 (2015).
  69. S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,” Nature 449, 324–327 (2007).
    [Crossref]
  70. P. Grisins and I. E. Mazets, “Thermalization in a one-dimensional integrable system,” Phys. Rev. A 84, 053635 (2011).
    [Crossref]
  71. A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
    [Crossref]

2015 (7)

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

A. González-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9, 320–325 (2015).
[Crossref]

G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. USA 112, 3866–3873 (2015).
[Crossref]

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a memory for tightly guided light in an optical nanofiber,” Phys. Rev. Lett. 114, 180503 (2015).
[Crossref]

J. Lee, J. A. Grover, J. E. Hoffman, L. A. Orozco, and S. L. Rolston, “Inhomogeneous broadening of optical transitions of Rb87 atoms in an optical nanofiber trap,” J. Phys. B 48, 165004 (2015).
[Crossref]

C. Sayrin, C. Clausen, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Storage of fiber-guided light in a nanofiber-trapped ensemble of cold atoms,” Optica 2, 353–356 (2015).
[Crossref]

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

2014 (12)

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

E. Shahmoon and G. Kurizki, “Nonlinear theory of laser-induced dipolar interactions in arbitrary geometry,” Phys. Rev. A 89, 043419 (2014).
[Crossref]

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
[Crossref]

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014).
[Crossref]

E. Shahmoon, I. Mazets, and G. Kurizki, “Giant vacuum forces via transmission lines,” Proc. Natl. Acad. Sci. USA 111, 10485–10490 (2014).
[Crossref]

E. Shahmoon, I. Mazets, and G. Kurizki, “Non-additivity in laser-illuminated many-atom systems,” Opt. Lett. 39, 3674–3677 (2014).
[Crossref]

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, “Single-photon transistor using a Förster resonance,” Phys. Rev. Lett. 113, 053602 (2014).
[Crossref]

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
[Crossref]

2013 (11)

J. Otterbach, M. Moos, D. Muth, and M. Fleischhauer, “Wigner crystallization of single photons in cold Rydberg ensembles,” Phys. Rev. Lett. 111, 113001 (2013).
[Crossref]

E. Shahmoon and G. Kurizki, “Nonradiative interaction and entanglement between distant atoms,” Phys. Rev. A 87, 033831 (2013).
[Crossref]

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-mediated interactions between distant artificial atoms,” Science 342, 1494–1496 (2013).
[Crossref]

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref]

D. Reitz, C. Sayrin, R. Mitsch, P. Schneeweiss, and A. Rauschenbeutel, “Coherence properties of nanofiber-trapped cesium atoms,” Phys. Rev. Lett. 110, 243603 (2013).
[Crossref]

D. E. Chang, J. I. Cirac, and H. J. Kimble, “Self-organization of atoms along a nanophotonic waveguide,” Phys. Rev. Lett. 110, 113606 (2013).
[Crossref]

T. Grießer and H. Ritsch, “Light-Induced crystallization of cold atoms in a 1D optical trap,” Phys. Rev. Lett. 111, 055702 (2013).
[Crossref]

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2013).
[Crossref]

K. P. Nayak and K. Hakuta, “Photonic crystal formation on optical nanofibers using femtosecond laser ablation technique,” Opt. Express 21, 2480–2490 (2013).
[Crossref]

2012 (3)

R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions,” Science 336, 1570–1573 (2012).
[Crossref]

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
[Crossref]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref]

2011 (6)

E. Shahmoon, G. Kurizki, M. Fleischhauer, and D. Petrosyan, “Strongly interacting photons in hollow-core waveguides,” Phys. Rev. A 83, 033806 (2011).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011).
[Crossref]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref]

S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal nonlinear optics in cold Rydberg gases,” Phys. Rev. Lett. 107, 153001 (2011).
[Crossref]

P. Grisins and I. E. Mazets, “Thermalization in a one-dimensional integrable system,” Phys. Rev. A 84, 053635 (2011).
[Crossref]

2010 (1)

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
[Crossref]

2009 (2)

J. Rodrguez and D. L. Andrews, “Inter-particle interaction induced by broadband radiation,” Opt. Commun. 282, 2267–2269 (2009).

A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
[Crossref]

2007 (2)

S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,” Nature 449, 324–327 (2007).
[Crossref]

S. Skupin, M. Saffman, and W. Królikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007).
[Crossref]

2006 (1)

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769–774 (2006).
[Crossref]

2005 (3)

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

I. Friedler, D. Petrosyan, M. L. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slow single-photon pulses,” Phys. Rev. A 72, 043803 (2005).
[Crossref]

2004 (2)

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[Crossref]

2002 (4)

M. Fleischhauer and M. D. Lukin, “Quantum memory for photons: dark-state polaritons,” Phys. Rev. A 65, 022314 (2002).
[Crossref]

S. Giovanazzi, D. O’Dell, and G. Kurizki, “Density modulations of Bose–Einstein condensates via laser-induced interactions,” Phys. Rev. Lett. 88, 130402 (2002).
[Crossref]

D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki, “Rotons in gaseous Bose–Einstein condensates irradiated by a laser,” Phys. Rev. Lett. 90, 110402 (2002).
[Crossref]

P. Domokos and H. Ritsch, “Collective cooling and self-organization of atoms in a cavity,” Phys. Rev. Lett. 89, 253003 (2002).
[Crossref]

2000 (1)

M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000).
[Crossref]

1998 (1)

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611–3614 (1998).
[Crossref]

1996 (2)

H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996).
[Crossref]

R. Y. Chiao, A. E. Kozhekin, and G. Kurizki, “Tachyonlike excitations in inverted two-level media,” Phys. Rev. Lett. 77, 1254–1257 (1996).
[Crossref]

1993 (1)

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583–4587 (1993).
[Crossref]

1990 (1)

G. Kurizki, “Two-atom resonant radiative coupling in photonic band structures,” Phys. Rev. A 42, 2915–2924 (1990).
[Crossref]

1987 (1)

M. J. Potasek and B. Yurke, “Squeezed-light generation in a medium governed by the nonlinear Schrödinger equation,” Phys. Rev. A 35, 3974(R) (1987).
[Crossref]

Abdolvand, A.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
[Crossref]

Abdumalikov, A. A.

J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014).
[Crossref]

Adams, C. S.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Akimov, A. V.

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref]

Albrecht, B.

Alfassi, B.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769–774 (2006).
[Crossref]

Alton, D. J.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
[Crossref]

Andrews, D. L.

J. Rodrguez and D. L. Andrews, “Inter-particle interaction induced by broadband radiation,” Opt. Commun. 282, 2267–2269 (2009).

Arcari, M.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

Asenjo-Garcia, A.

J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom–atom interactions around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).

Assanto, G.

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[Crossref]

Ates, C.

S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal nonlinear optics in cold Rydberg gases,” Phys. Rev. Lett. 107, 153001 (2011).
[Crossref]

Bang, O.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Baumann, K.

R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions,” Science 336, 1570–1573 (2012).
[Crossref]

Baur, S.

D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, “Single-photon transistor using a Förster resonance,” Phys. Rev. Lett. 113, 053602 (2014).
[Crossref]

Bertet, P.

G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. USA 112, 3866–3873 (2015).
[Crossref]

Bienias, P.

P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
[Crossref]

Bimbard, E.

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

Blais, A.

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-mediated interactions between distant artificial atoms,” Science 342, 1494–1496 (2013).
[Crossref]

Blasberg, T.

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583–4587 (1993).
[Crossref]

Boddeda, R.

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

Boyd, R. B.

R. B. Boyd, Nonlinear Optics (Academic, 2008).

Brennecke, F.

R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions,” Science 336, 1570–1573 (2012).
[Crossref]

Brion, E.

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

Büchler, H. P.

P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
[Crossref]

Busche, H.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Campa, A.

A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
[Crossref]

Caneva, T.

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

J. S. Douglas, T. Caneva, and D. E. Chang, “Photon molecules in atomic gases trapped near photonic crystal waveguides,” arXiv: 1511.00816 (2015).

Carmon, T.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref]

Champion, T. F. M.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
[Crossref]

Chang, D. E.

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

A. González-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9, 320–325 (2015).
[Crossref]

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

D. E. Chang, J. I. Cirac, and H. J. Kimble, “Self-organization of atoms along a nanophotonic waveguide,” Phys. Rev. Lett. 110, 113606 (2013).
[Crossref]

J. S. Douglas, T. Caneva, and D. E. Chang, “Photon molecules in atomic gases trapped near photonic crystal waveguides,” arXiv: 1511.00816 (2015).

J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom–atom interactions around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).

Chiao, R. Y.

R. Y. Chiao, A. E. Kozhekin, and G. Kurizki, “Tachyonlike excitations in inverted two-level media,” Phys. Rev. Lett. 77, 1254–1257 (1996).
[Crossref]

Choi, K. S.

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
[Crossref]

Choi, S.

P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
[Crossref]

Cirac, I.

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

Cirac, J. I.

A. González-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9, 320–325 (2015).
[Crossref]

D. E. Chang, J. I. Cirac, and H. J. Kimble, “Self-organization of atoms along a nanophotonic waveguide,” Phys. Rev. Lett. 110, 113606 (2013).
[Crossref]

Clausen, C.

Cohen, O.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769–774 (2006).
[Crossref]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref]

Conti, C.

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[Crossref]

Dauxois, T.

A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
[Crossref]

Dawkins, S. T.

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
[Crossref]

de Leon, N. P.

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref]

Ding, D.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
[Crossref]

Domokos, P.

P. Domokos and H. Ritsch, “Collective cooling and self-organization of atoms in a cavity,” Phys. Rev. Lett. 89, 253003 (2002).
[Crossref]

Donner, T.

R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions,” Science 336, 1570–1573 (2012).
[Crossref]

Douglas, J. S.

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

J. S. Douglas, T. Caneva, and D. E. Chang, “Photon molecules in atomic gases trapped near photonic crystal waveguides,” arXiv: 1511.00816 (2015).

Dürr, S.

D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, “Single-photon transistor using a Förster resonance,” Phys. Rev. Lett. 113, 053602 (2014).
[Crossref]

Edmundson, D.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Eichler, C.

J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014).
[Crossref]

England, D. G.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
[Crossref]

Esslinger, T.

R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions,” Science 336, 1570–1573 (2012).
[Crossref]

Fedder, H.

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

Fedorov, A.

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-mediated interactions between distant artificial atoms,” Science 342, 1494–1496 (2013).
[Crossref]

Feist, J.

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref]

Firstenberg, O.

P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
[Crossref]

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref]

Fischer, B.

S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,” Nature 449, 324–327 (2007).
[Crossref]

Fleischhauer, M.

J. Otterbach, M. Moos, D. Muth, and M. Fleischhauer, “Wigner crystallization of single photons in cold Rydberg ensembles,” Phys. Rev. Lett. 111, 113001 (2013).
[Crossref]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref]

E. Shahmoon, G. Kurizki, M. Fleischhauer, and D. Petrosyan, “Strongly interacting photons in hollow-core waveguides,” Phys. Rev. A 83, 033806 (2011).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

M. Fleischhauer and M. D. Lukin, “Quantum memory for photons: dark-state polaritons,” Phys. Rev. A 65, 022314 (2002).
[Crossref]

Fleischhauer, M. L.

I. Friedler, D. Petrosyan, M. L. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slow single-photon pulses,” Phys. Rev. A 72, 043803 (2005).
[Crossref]

Friedler, I.

I. Friedler, D. Petrosyan, M. L. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slow single-photon pulses,” Phys. Rev. A 72, 043803 (2005).
[Crossref]

Gaeta, A. L.

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2013).
[Crossref]

Gauguet, A.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Giovanazzi, S.

S. Giovanazzi, D. O’Dell, and G. Kurizki, “Density modulations of Bose–Einstein condensates via laser-induced interactions,” Phys. Rev. Lett. 88, 130402 (2002).
[Crossref]

D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki, “Rotons in gaseous Bose–Einstein condensates irradiated by a laser,” Phys. Rev. Lett. 90, 110402 (2002).
[Crossref]

Goban, A.

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
[Crossref]

J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom–atom interactions around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).

A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” arXiv:1503.04503 (2015).

González-Tudela, A.

A. González-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9, 320–325 (2015).
[Crossref]

Gorniaczyk, H.

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

Gorshkov, A. V.

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
[Crossref]

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

Gouraud, B.

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a memory for tightly guided light in an optical nanofiber,” Phys. Rev. Lett. 114, 180503 (2015).
[Crossref]

Grangier, P.

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

Grankin, A.

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

Grießer, T.

T. Grießer and H. Ritsch, “Light-Induced crystallization of cold atoms in a 1D optical trap,” Phys. Rev. Lett. 111, 055702 (2013).
[Crossref]

Grisins, P.

P. Grisins and I. E. Mazets, “Thermalization in a one-dimensional integrable system,” Phys. Rev. A 84, 053635 (2011).
[Crossref]

Grover, J. A.

J. Lee, J. A. Grover, J. E. Hoffman, L. A. Orozco, and S. L. Rolston, “Inhomogeneous broadening of optical transitions of Rb87 atoms in an optical nanofiber trap,” J. Phys. B 48, 165004 (2015).
[Crossref]

Gullans, M.

P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
[Crossref]

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref]

Habibian, H.

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

Hakuta, K.

Harris, S. E.

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611–3614 (1998).
[Crossref]

He, B.

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref]

B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011).
[Crossref]

Henkel, N.

S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal nonlinear optics in cold Rydberg gases,” Phys. Rev. Lett. 107, 153001 (2011).
[Crossref]

Hofferberth, S.

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref]

S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,” Nature 449, 324–327 (2007).
[Crossref]

Hoffman, J. E.

J. Lee, J. A. Grover, J. E. Hoffman, L. A. Orozco, and S. L. Rolston, “Inhomogeneous broadening of optical transitions of Rb87 atoms in an optical nanofiber trap,” J. Phys. B 48, 165004 (2015).
[Crossref]

Hood, J. D.

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom–atom interactions around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).

A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” arXiv:1503.04503 (2015).

Hung, C.-L.

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

A. González-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9, 320–325 (2015).
[Crossref]

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” arXiv:1503.04503 (2015).

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000).
[Crossref]

H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996).
[Crossref]

Javadi, A.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

Jin, X.-M.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
[Crossref]

Jones, M. P. A.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Kimble, H. J.

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

A. González-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9, 320–325 (2015).
[Crossref]

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

D. E. Chang, J. I. Cirac, and H. J. Kimble, “Self-organization of atoms along a nanophotonic waveguide,” Phys. Rev. Lett. 110, 113606 (2013).
[Crossref]

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
[Crossref]

J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom–atom interactions around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).

A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” arXiv:1503.04503 (2015).

Kolthammer, W. S.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
[Crossref]

Kozhekin, A. E.

R. Y. Chiao, A. E. Kozhekin, and G. Kurizki, “Tachyonlike excitations in inverted two-level media,” Phys. Rev. Lett. 77, 1254–1257 (1996).
[Crossref]

Królikowski, W.

S. Skupin, M. Saffman, and W. Królikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007).
[Crossref]

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Kubo, Y.

G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. USA 112, 3866–3873 (2015).
[Crossref]

Kurizki, G.

G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. USA 112, 3866–3873 (2015).
[Crossref]

E. Shahmoon, I. Mazets, and G. Kurizki, “Non-additivity in laser-illuminated many-atom systems,” Opt. Lett. 39, 3674–3677 (2014).
[Crossref]

E. Shahmoon and G. Kurizki, “Nonlinear theory of laser-induced dipolar interactions in arbitrary geometry,” Phys. Rev. A 89, 043419 (2014).
[Crossref]

E. Shahmoon, I. Mazets, and G. Kurizki, “Giant vacuum forces via transmission lines,” Proc. Natl. Acad. Sci. USA 111, 10485–10490 (2014).
[Crossref]

E. Shahmoon and G. Kurizki, “Nonradiative interaction and entanglement between distant atoms,” Phys. Rev. A 87, 033831 (2013).
[Crossref]

E. Shahmoon, G. Kurizki, M. Fleischhauer, and D. Petrosyan, “Strongly interacting photons in hollow-core waveguides,” Phys. Rev. A 83, 033806 (2011).
[Crossref]

I. Friedler, D. Petrosyan, M. L. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slow single-photon pulses,” Phys. Rev. A 72, 043803 (2005).
[Crossref]

S. Giovanazzi, D. O’Dell, and G. Kurizki, “Density modulations of Bose–Einstein condensates via laser-induced interactions,” Phys. Rev. Lett. 88, 130402 (2002).
[Crossref]

D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki, “Rotons in gaseous Bose–Einstein condensates irradiated by a laser,” Phys. Rev. Lett. 90, 110402 (2002).
[Crossref]

R. Y. Chiao, A. E. Kozhekin, and G. Kurizki, “Tachyonlike excitations in inverted two-level media,” Phys. Rev. Lett. 77, 1254–1257 (1996).
[Crossref]

G. Kurizki, “Two-atom resonant radiative coupling in photonic band structures,” Phys. Rev. A 42, 2915–2924 (1990).
[Crossref]

Lacroute, C.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
[Crossref]

Lalumière, K.

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-mediated interactions between distant artificial atoms,” Science 342, 1494–1496 (2013).
[Crossref]

Lambropoulos, P.

P. Lambropoulos and D. Petrosyan, Fundamentals of Quantum Optics and Quantum Information (Springer, 2007).

Landig, R.

R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions,” Science 336, 1570–1573 (2012).
[Crossref]

Laurat, J.

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a memory for tightly guided light in an optical nanofiber,” Phys. Rev. Lett. 114, 180503 (2015).
[Crossref]

Lee, E. H.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

Lee, J.

J. Lee, J. A. Grover, J. E. Hoffman, L. A. Orozco, and S. L. Rolston, “Inhomogeneous broadening of optical transitions of Rb87 atoms in an optical nanofiber trap,” J. Phys. B 48, 165004 (2015).
[Crossref]

Lee, J. H.

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

Lesanovsky, I.

S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,” Nature 449, 324–327 (2007).
[Crossref]

Liang, Q.-Y.

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref]

Lin, Q.

B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011).
[Crossref]

Lindskov Hansen, S.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

Liu, J.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

Lodahl, P.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

Lu, M.

J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom–atom interactions around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).

Lukin, M. D.

P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
[Crossref]

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref]

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

M. Fleischhauer and M. D. Lukin, “Quantum memory for photons: dark-state polaritons,” Phys. Rev. A 65, 022314 (2002).
[Crossref]

M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000).
[Crossref]

Maghrebi, M. F.

P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
[Crossref]

Mahmoodian, S.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

Manela, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref]

Manzoni, M. T.

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Martin, M. J.

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

Maxein, D.

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a memory for tightly guided light in an optical nanofiber,” Phys. Rev. Lett. 114, 180503 (2015).
[Crossref]

Maxwell, D.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Mazets, I.

E. Shahmoon, I. Mazets, and G. Kurizki, “Non-additivity in laser-illuminated many-atom systems,” Opt. Lett. 39, 3674–3677 (2014).
[Crossref]

E. Shahmoon, I. Mazets, and G. Kurizki, “Giant vacuum forces via transmission lines,” Proc. Natl. Acad. Sci. USA 111, 10485–10490 (2014).
[Crossref]

Mazets, I. E.

P. Grisins and I. E. Mazets, “Thermalization in a one-dimensional integrable system,” Phys. Rev. A 84, 053635 (2011).
[Crossref]

McClung, A. C.

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

Michelberger, P. S.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
[Crossref]

Mitsch, R.

D. Reitz, C. Sayrin, R. Mitsch, P. Schneeweiss, and A. Rauschenbeutel, “Coherence properties of nanofiber-trapped cesium atoms,” Phys. Rev. Lett. 110, 243603 (2013).
[Crossref]

Mlynek, J. A.

J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014).
[Crossref]

Mølmer, K.

G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. USA 112, 3866–3873 (2015).
[Crossref]

Moos, M.

J. Otterbach, M. Moos, D. Muth, and M. Fleischhauer, “Wigner crystallization of single photons in cold Rydberg ensembles,” Phys. Rev. Lett. 111, 113001 (2013).
[Crossref]

Morin, O.

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a memory for tightly guided light in an optical nanofiber,” Phys. Rev. Lett. 114, 180503 (2015).
[Crossref]

Mottl, R.

R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions,” Science 336, 1570–1573 (2012).
[Crossref]

Muniz, J. A.

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” arXiv:1503.04503 (2015).

Muth, D.

J. Otterbach, M. Moos, D. Muth, and M. Fleischhauer, “Wigner crystallization of single photons in cold Rydberg ensembles,” Phys. Rev. Lett. 111, 113001 (2013).
[Crossref]

Nayak, K. P.

Neshev, D.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Nicolas, A.

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a memory for tightly guided light in an optical nanofiber,” Phys. Rev. Lett. 114, 180503 (2015).
[Crossref]

Nikolov, N. I.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Nozieres, P.

P. Nozieres and D. Pines, Theory of Quantum Liquids: Superfluid Bose Liquids (Addison-Wesley, 1990).

Nunn, J.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
[Crossref]

O’Dell, D.

S. Giovanazzi, D. O’Dell, and G. Kurizki, “Density modulations of Bose–Einstein condensates via laser-induced interactions,” Phys. Rev. Lett. 88, 130402 (2002).
[Crossref]

O’Dell, D. H. J.

D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki, “Rotons in gaseous Bose–Einstein condensates irradiated by a laser,” Phys. Rev. Lett. 90, 110402 (2002).
[Crossref]

Orozco, L. A.

J. Lee, J. A. Grover, J. E. Hoffman, L. A. Orozco, and S. L. Rolston, “Inhomogeneous broadening of optical transitions of Rb87 atoms in an optical nanofiber trap,” J. Phys. B 48, 165004 (2015).
[Crossref]

Otterbach, J.

J. Otterbach, M. Moos, D. Muth, and M. Fleischhauer, “Wigner crystallization of single photons in cold Rydberg ensembles,” Phys. Rev. Lett. 111, 113001 (2013).
[Crossref]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

Ourjoumtsev, A.

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

Painter, O.

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” arXiv:1503.04503 (2015).

Paredes-Barato, D.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Peccianti, M.

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[Crossref]

Pethick, C. J.

C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases (Cambridge University, 2002).

Petrosyan, D.

G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. USA 112, 3866–3873 (2015).
[Crossref]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref]

E. Shahmoon, G. Kurizki, M. Fleischhauer, and D. Petrosyan, “Strongly interacting photons in hollow-core waveguides,” Phys. Rev. A 83, 033806 (2011).
[Crossref]

I. Friedler, D. Petrosyan, M. L. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slow single-photon pulses,” Phys. Rev. A 72, 043803 (2005).
[Crossref]

P. Lambropoulos and D. Petrosyan, Fundamentals of Quantum Optics and Quantum Information (Springer, 2007).

Peyronel, T.

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref]

Pines, D.

P. Nozieres and D. Pines, Theory of Quantum Liquids: Superfluid Bose Liquids (Addison-Wesley, 1990).

Pitaevskii, L. P.

L. P. Pitaevskii and S. Stringari, Bose–Einstein Condensation (Clarendon, 2003).

Pohl, T.

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref]

S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal nonlinear optics in cold Rydberg gases,” Phys. Rev. Lett. 107, 153001 (2011).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

Potasek, M. J.

M. J. Potasek and B. Yurke, “Squeezed-light generation in a medium governed by the nonlinear Schrödinger equation,” Phys. Rev. A 35, 3974(R) (1987).
[Crossref]

Pototschnig, M.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
[Crossref]

Pritchard, J. D.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Rabl, P.

G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. USA 112, 3866–3873 (2015).
[Crossref]

Rasmussen, J. J.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Rauschenbeutel, A.

C. Sayrin, C. Clausen, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Storage of fiber-guided light in a nanofiber-trapped ensemble of cold atoms,” Optica 2, 353–356 (2015).
[Crossref]

D. Reitz, C. Sayrin, R. Mitsch, P. Schneeweiss, and A. Rauschenbeutel, “Coherence properties of nanofiber-trapped cesium atoms,” Phys. Rev. Lett. 110, 243603 (2013).
[Crossref]

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
[Crossref]

Reitz, D.

D. Reitz, C. Sayrin, R. Mitsch, P. Schneeweiss, and A. Rauschenbeutel, “Coherence properties of nanofiber-trapped cesium atoms,” Phys. Rev. Lett. 110, 243603 (2013).
[Crossref]

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
[Crossref]

Rempe, G.

D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, “Single-photon transistor using a Förster resonance,” Phys. Rev. Lett. 113, 053602 (2014).
[Crossref]

Ritsch, H.

T. Grießer and H. Ritsch, “Light-Induced crystallization of cold atoms in a 1D optical trap,” Phys. Rev. Lett. 111, 055702 (2013).
[Crossref]

P. Domokos and H. Ritsch, “Collective cooling and self-organization of atoms in a cavity,” Phys. Rev. Lett. 89, 253003 (2002).
[Crossref]

Rodrguez, J.

J. Rodrguez and D. L. Andrews, “Inter-particle interaction induced by broadband radiation,” Opt. Commun. 282, 2267–2269 (2009).

Rolston, S. L.

J. Lee, J. A. Grover, J. E. Hoffman, L. A. Orozco, and S. L. Rolston, “Inhomogeneous broadening of optical transitions of Rb87 atoms in an optical nanofiber trap,” J. Phys. B 48, 165004 (2015).
[Crossref]

Rotschild, C.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769–774 (2006).
[Crossref]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref]

Ruffo, S.

A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
[Crossref]

Russell, P. St. J.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
[Crossref]

Saffman, M.

S. Skupin, M. Saffman, and W. Królikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007).
[Crossref]

Sagué, G.

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
[Crossref]

Saha, K.

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2013).
[Crossref]

Sanders, B. C.

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-mediated interactions between distant artificial atoms,” Science 342, 1494–1496 (2013).
[Crossref]

Sayrin, C.

C. Sayrin, C. Clausen, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Storage of fiber-guided light in a nanofiber-trapped ensemble of cold atoms,” Optica 2, 353–356 (2015).
[Crossref]

D. Reitz, C. Sayrin, R. Mitsch, P. Schneeweiss, and A. Rauschenbeutel, “Coherence properties of nanofiber-trapped cesium atoms,” Phys. Rev. Lett. 110, 243603 (2013).
[Crossref]

Schmidt, H.

Schmidt, J.

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

Schmidt, R.

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
[Crossref]

Schmiedmayer, J.

G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. USA 112, 3866–3873 (2015).
[Crossref]

S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,” Nature 449, 324–327 (2007).
[Crossref]

Schneeweiss, P.

C. Sayrin, C. Clausen, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Storage of fiber-guided light in a nanofiber-trapped ensemble of cold atoms,” Optica 2, 353–356 (2015).
[Crossref]

D. Reitz, C. Sayrin, R. Mitsch, P. Schneeweiss, and A. Rauschenbeutel, “Coherence properties of nanofiber-trapped cesium atoms,” Phys. Rev. Lett. 110, 243603 (2013).
[Crossref]

Schneider, K.

D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, “Single-photon transistor using a Förster resonance,” Phys. Rev. Lett. 113, 053602 (2014).
[Crossref]

Schumm, T.

S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,” Nature 449, 324–327 (2007).
[Crossref]

Segev, M.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769–774 (2006).
[Crossref]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref]

Sevinçli, S.

S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal nonlinear optics in cold Rydberg gases,” Phys. Rev. Lett. 107, 153001 (2011).
[Crossref]

Shahmoon, E.

E. Shahmoon, I. Mazets, and G. Kurizki, “Giant vacuum forces via transmission lines,” Proc. Natl. Acad. Sci. USA 111, 10485–10490 (2014).
[Crossref]

E. Shahmoon, I. Mazets, and G. Kurizki, “Non-additivity in laser-illuminated many-atom systems,” Opt. Lett. 39, 3674–3677 (2014).
[Crossref]

E. Shahmoon and G. Kurizki, “Nonlinear theory of laser-induced dipolar interactions in arbitrary geometry,” Phys. Rev. A 89, 043419 (2014).
[Crossref]

E. Shahmoon and G. Kurizki, “Nonradiative interaction and entanglement between distant atoms,” Phys. Rev. A 87, 033831 (2013).
[Crossref]

E. Shahmoon, G. Kurizki, M. Fleischhauer, and D. Petrosyan, “Strongly interacting photons in hollow-core waveguides,” Phys. Rev. A 83, 033806 (2011).
[Crossref]

E. Shahmoon, “van der Waals and Casimir-Polder dispersion forces,” in Forces of the Quantum Vacuum: An Introduction to Casimir Physics, W. Simpson and U. Leonhardt, eds. (World Scientific, 2015), Chap. 2.

Sharypov, A. V.

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref]

Sheng, J.

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref]

Shi, T.

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

Simon, C.

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref]

B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011).
[Crossref]

Skupin, S.

S. Skupin, M. Saffman, and W. Królikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007).
[Crossref]

Smith, H.

C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases (Cambridge University, 2002).

Söllner, I.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

Song, J. D.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

Sprague, M. R.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
[Crossref]

Stern, N. P.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
[Crossref]

Stobbe, S.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

Stringari, S.

L. P. Pitaevskii and S. Stringari, Bose–Einstein Condensation (Clarendon, 2003).

Suter, D.

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583–4587 (1993).
[Crossref]

Szwer, D. J.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Thiele, T.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
[Crossref]

Thompson, J. D.

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref]

Thyrrestrup, H.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

Tiarks, D.

D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, “Single-photon transistor using a Förster resonance,” Phys. Rev. Lett. 113, 053602 (2014).
[Crossref]

Tiecke, T. G.

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref]

Tresp, C.

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

Usmani, I.

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

van Loo, A. F.

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-mediated interactions between distant artificial atoms,” Science 342, 1494–1496 (2013).
[Crossref]

Venkataraman, V.

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2013).
[Crossref]

Vetsch, E.

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
[Crossref]

Vuletic, V.

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref]

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref]

Wallraff, A.

J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014).
[Crossref]

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-mediated interactions between distant artificial atoms,” Science 342, 1494–1496 (2013).
[Crossref]

Walmsley, I. A.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
[Crossref]

Weatherill, K. J.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

Wyller, J.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Xiao, M.

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref]

Yamamoto, Y.

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611–3614 (1998).
[Crossref]

Yu, S.-P.

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom–atom interactions around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).

A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” arXiv:1503.04503 (2015).

Yurke, B.

M. J. Potasek and B. Yurke, “Squeezed-light generation in a medium governed by the nonlinear Schrödinger equation,” Phys. Rev. A 35, 3974(R) (1987).
[Crossref]

Zibrov, A. S.

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref]

J. Opt. B (1)

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

J. Phys. B (1)

J. Lee, J. A. Grover, J. E. Hoffman, L. A. Orozco, and S. L. Rolston, “Inhomogeneous broadening of optical transitions of Rb87 atoms in an optical nanofiber trap,” J. Phys. B 48, 165004 (2015).
[Crossref]

Nat. Commun. (2)

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014).
[Crossref]

Nat. Photonics (4)

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
[Crossref]

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2013).
[Crossref]

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

A. González-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9, 320–325 (2015).
[Crossref]

Nat. Phys. (1)

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769–774 (2006).
[Crossref]

Nature (3)

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref]

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref]

S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,” Nature 449, 324–327 (2007).
[Crossref]

New J. Phys. (2)

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

Opt. Commun. (1)

J. Rodrguez and D. L. Andrews, “Inter-particle interaction induced by broadband radiation,” Opt. Commun. 282, 2267–2269 (2009).

Opt. Express (1)

Opt. Lett. (2)

Optica (1)

Phys. Rep. (1)

A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
[Crossref]

Phys. Rev. A (11)

P. Grisins and I. E. Mazets, “Thermalization in a one-dimensional integrable system,” Phys. Rev. A 84, 053635 (2011).
[Crossref]

E. Shahmoon and G. Kurizki, “Nonlinear theory of laser-induced dipolar interactions in arbitrary geometry,” Phys. Rev. A 89, 043419 (2014).
[Crossref]

M. J. Potasek and B. Yurke, “Squeezed-light generation in a medium governed by the nonlinear Schrödinger equation,” Phys. Rev. A 35, 3974(R) (1987).
[Crossref]

E. Shahmoon and G. Kurizki, “Nonradiative interaction and entanglement between distant atoms,” Phys. Rev. A 87, 033831 (2013).
[Crossref]

G. Kurizki, “Two-atom resonant radiative coupling in photonic band structures,” Phys. Rev. A 42, 2915–2924 (1990).
[Crossref]

P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
[Crossref]

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583–4587 (1993).
[Crossref]

M. Fleischhauer and M. D. Lukin, “Quantum memory for photons: dark-state polaritons,” Phys. Rev. A 65, 022314 (2002).
[Crossref]

I. Friedler, D. Petrosyan, M. L. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slow single-photon pulses,” Phys. Rev. A 72, 043803 (2005).
[Crossref]

E. Shahmoon, G. Kurizki, M. Fleischhauer, and D. Petrosyan, “Strongly interacting photons in hollow-core waveguides,” Phys. Rev. A 83, 033806 (2011).
[Crossref]

B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011).
[Crossref]

Phys. Rev. Lett. (24)

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref]

J. Otterbach, M. Moos, D. Muth, and M. Fleischhauer, “Wigner crystallization of single photons in cold Rydberg ensembles,” Phys. Rev. Lett. 111, 113001 (2013).
[Crossref]

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

S. Skupin, M. Saffman, and W. Królikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007).
[Crossref]

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[Crossref]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref]

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611–3614 (1998).
[Crossref]

M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, “Single-photon transistor using a Förster resonance,” Phys. Rev. Lett. 113, 053602 (2014).
[Crossref]

S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal nonlinear optics in cold Rydberg gases,” Phys. Rev. Lett. 107, 153001 (2011).
[Crossref]

R. Y. Chiao, A. E. Kozhekin, and G. Kurizki, “Tachyonlike excitations in inverted two-level media,” Phys. Rev. Lett. 77, 1254–1257 (1996).
[Crossref]

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a memory for tightly guided light in an optical nanofiber,” Phys. Rev. Lett. 114, 180503 (2015).
[Crossref]

S. Giovanazzi, D. O’Dell, and G. Kurizki, “Density modulations of Bose–Einstein condensates via laser-induced interactions,” Phys. Rev. Lett. 88, 130402 (2002).
[Crossref]

D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki, “Rotons in gaseous Bose–Einstein condensates irradiated by a laser,” Phys. Rev. Lett. 90, 110402 (2002).
[Crossref]

D. E. Chang, J. I. Cirac, and H. J. Kimble, “Self-organization of atoms along a nanophotonic waveguide,” Phys. Rev. Lett. 110, 113606 (2013).
[Crossref]

T. Grießer and H. Ritsch, “Light-Induced crystallization of cold atoms in a 1D optical trap,” Phys. Rev. Lett. 111, 055702 (2013).
[Crossref]

P. Domokos and H. Ritsch, “Collective cooling and self-organization of atoms in a cavity,” Phys. Rev. Lett. 89, 253003 (2002).
[Crossref]

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
[Crossref]

D. Reitz, C. Sayrin, R. Mitsch, P. Schneeweiss, and A. Rauschenbeutel, “Coherence properties of nanofiber-trapped cesium atoms,” Phys. Rev. Lett. 110, 243603 (2013).
[Crossref]

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
[Crossref]

Proc. Natl. Acad. Sci. USA (2)

G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. USA 112, 3866–3873 (2015).
[Crossref]

E. Shahmoon, I. Mazets, and G. Kurizki, “Giant vacuum forces via transmission lines,” Proc. Natl. Acad. Sci. USA 111, 10485–10490 (2014).
[Crossref]

Rev. Mod. Phys. (1)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Science (3)

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-mediated interactions between distant artificial atoms,” Science 342, 1494–1496 (2013).
[Crossref]

R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions,” Science 336, 1570–1573 (2012).
[Crossref]

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref]

Other (10)

C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases (Cambridge University, 2002).

L. P. Pitaevskii and S. Stringari, Bose–Einstein Condensation (Clarendon, 2003).

J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom–atom interactions around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).

P. D. Drummond and Z. Ficek, eds., Quantum Squeezing (Springer-Verlag, 2004).

P. Nozieres and D. Pines, Theory of Quantum Liquids: Superfluid Bose Liquids (Addison-Wesley, 1990).

E. Shahmoon, “van der Waals and Casimir-Polder dispersion forces,” in Forces of the Quantum Vacuum: An Introduction to Casimir Physics, W. Simpson and U. Leonhardt, eds. (World Scientific, 2015), Chap. 2.

J. S. Douglas, T. Caneva, and D. E. Chang, “Photon molecules in atomic gases trapped near photonic crystal waveguides,” arXiv: 1511.00816 (2015).

A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” arXiv:1503.04503 (2015).

R. B. Boyd, Nonlinear Optics (Academic, 2008).

P. Lambropoulos and D. Petrosyan, Fundamentals of Quantum Optics and Quantum Information (Springer, 2007).

Supplementary Material (1)

NameDescription
» Supplement 1: PDF (2741 KB)      Supplemental Document

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1. (a) Setup: atoms (black dots), illuminated by the EIT fields [see (b)] E ^ and Ω (thin blue arrow), are trapped at a distance r a from a nano-waveguide (gray cylinder) along z , from z = 0 to z = L . A far-detuned laser Ω L (thick orange arrow), tilted by an angle θ L from the z -axis, induces long-range interactions between the atoms, mediated by the waveguide modes [see (c)]. (b) EIT atomic configuration: the probe field E ^ is resonantly coupled to the | g | e transition, whereas the coupling field Ω is coupled to the | d | e transition with detuning δ c . Interaction between the atoms in the | d -level [see (c)] induces its energy shift δ NL , which is effectively added to the detuning δ c . (c) Laser-induced dipolar interactions: the laser with Rabi frequency Ω L and detuning δ L operates on the | d | s transition of all atoms, | s being an additional level, thus inducing a dipolar potential U ( z ) between pairs of atoms ( z apart) populating the state | d [59].
Fig. 2.
Fig. 2. Linear susceptibility of the EIT atomic medium to the probe field as a function of its detuning Δ p [8,9] ( Δ p in units of level width γ and Ω = 2.5 γ ). (a) For total detuning of the coupling field Δ c = 0 , the absorption Im χ (red dashed line) and dispersion Re χ (blue solid line) are symmetrical and antisymmetric, respectively, with respect to Δ p , so that no (real) quadratic dispersion exists for the probe envelope centered around Δ p = 0 . (b) For Δ c 0 , Re χ is not antisymmetric, so that quadratic dispersion exists, giving rise to the term Δ c C v 2 z 2 in Eq. (1), with Δ c = δ c + δ NL [Fig. 1(b)].
Fig. 3.
Fig. 3. Dispersion relations for EIT polaritons with waveguide-grating mediated atomic interactions ( k values presented in all plots are within the EIT transparency window; values of physical parameters used here are given in Supplement 1, Section 4). (a) Roton-like excitation spectrum (dispersion relation) ω k for the potential of Eq. (12) in the anomalous dispersion case (opposite signs of ω k 0 and U k ). The analytical results from Eq. (8) (blue solid line) agree well with those of direct numerical simulations of Eq. (1) (gray dots). Compared with the spectrum of a local interaction ( U k independent of k , dashed red line), the roton-like spectrum exhibits a dip in a narrow band of k -values around k R = k L z k B . (b) Anti-roton peak of the spectrum ω k in a narrow band around k R in the normal dispersion case (identical signs of ω k 0 and U k ). (c) Possible homodyne detection scheme: the input probe field consists of a CW field + perturbation at wavenumber k and frequency ω ( 0 ) ( k ) . The field is split before entering the EIT medium ( z = 0 ) so that a local oscillator of ω ( 0 ) ( k ) is formed (lower arm) by filtering out the CW component. Then, mixing the output signal ( z = L ) with the local oscillator reveals their phase difference, from which ω k can be inferred (see text).
Fig. 4.
Fig. 4. Instability and self-ordering (values of physical parameters are given in Supplement 1, Section 4). (a)  γ k = Im ω k is the exponential growth rate of unstable perturbations at spatial frequency k (anomalous dispersion case). It exhibits a narrow peak around k R (blue dotted line), compared to the broadband instability of the local interaction case (red dotted line). The instability is accompanied by the generation of quantum entanglement, characterized by a narrow-band squeezing spectrum G k (blue solid line; G k < 1 quantifies entanglement between ± k photon modes), in contrast to the broadband spectrum for a local interaction (horizontal red solid line). (b) Dynamics of spatial spectrum N k ( t ) of the field intensity for different propagation times t inside the medium ( t = L / v at the output). The emergence of a large peak around k R (and k R ) out of the initial Gaussian perturbation is clearly seen. Results of both the linearized theory (solid lines) and numerical simulations of Eq. (1) (dots) are shown: slight differences at later t -values are attributed to nonlinear corrections. (c) Self-ordering of the field intensity: considering the narrow peaks of N k ( t ) around ± k R , fluctuations at these k -values become dominant, resulting in ordered intensity correlations g ( 2 ) which grow with propagation time t ( t = L / v at the output). Excellent agreement between the theory, Eq. (11) (solid lines), and numerical simulations of Eq. (1) (dots) is observed. The correlations oscillate with a period 2 π / k R 6.1 mm and a range of a few l 2.7 mm (where L = 2.68 cm and v = 4340 ms 1 ), determined by the long-range interaction U ( z ) . This is in contrast to the local interaction case, where the correlations vanish ( g ( 2 ) = 1 ) after a short distance π / q tr 1.75 mm , which is determined by the bandwidth q tr of the initial fluctuations [see inset: local (red thin line) and nonlocal (blue thick line) cases for the output field at t = L / v ].

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

( t + v z ) Ψ ^ ( z ) = i Δ ^ c α Ψ ^ ( z ) i Δ ^ c C v 2 z 2 Ψ ^ ( z ) , Δ ^ c = δ c + δ ^ NL , δ ^ NL = α L d z U ( z z ) Ψ ^ ( z ) Ψ ^ ( z ) .
H A F = n a d z [ i g E ^ ( z ) e i k 0 z σ ^ e g ( z ) + h.c. ] , H A C = n a d z [ i Ω e i δ c t e i k c z σ ^ e d ( z ) + h.c. ] , H D D = 1 2 n a 2 d z d z U ( z z ) σ ^ d d ( z ) σ ^ d d ( z )
σ ¯ g d ( z ) = g Ω E ^ ( z ) 1 Ω [ ( t + γ ) ( 1 Ω * ) × ( t + i δ c + i S ^ ( z ) ) σ ¯ g d ( z ) F ^ ] , ( t + c z ) E ^ ( z ) = n a g * Ω * [ t + i δ c + i S ^ ( z ) ] σ ¯ g d ( z ) ,
S ^ ( z ) = n a L d z U ( z z ) σ ¯ g d ( z ) σ ¯ g d ( z ) .
( Ψ ^ ( z ) Φ ^ ( z ) ) = ( cos θ N sin θ sin θ N cos θ ) ( E ^ ( z ) σ ¯ g d ( z ) / L ) ,
( t + c cos 2 θ z ) Ψ ^ ( z ) = sin θ cos θ c z Φ ^ ( z ) i sin θ ( δ c + S ^ ( z ) ) [ sin θ Ψ ^ ( z ) cos θ Φ ^ ( z ) ] , Φ ^ ( z ) = cos θ | Ω | 2 ( t + γ ) ( t + i δ c + i S ^ ( z ) ) × [ sin θ Ψ ^ ( z ) cos θ Φ ^ ( z ) ] + n a cos θ Ω F ^ .
ϕ ( z , t ) = e i [ φ ( α δ c + n p U 0 ) t ] [ u k e i k z e i ( ω k + k v ) t v k * e i k z e i ( ω k + k v ) t ]
ω k = ω k 0 ( ω k 0 + 2 n p U k ) , ω k 0 = ( n p U 0 / α + δ c ) C v 2 k 2 .
ω ( k ) = α δ c + n p U 0 ± ( v k + ω k ) ,
a ^ k ( t ) = e i ( n p U 0 + α δ c + k v ) t [ μ k ( t ) a ^ k ( 0 ) + e i 2 φ ν k ( t ) a ^ k ( 0 ) ] , μ k ( t ) = cos ( ω k t ) i n p U k + ω k 0 ω k sin ( ω k t ) , ν k ( t ) = i n p U k ω k sin ( ω k t ) .
g ( 2 ) ( z , z ) 1 + 2 α 2 π n p 0 d k [ | μ k | 2 N k + | ν k | 2 ( N k + 1 ) + ( 2 N k + 1 ) | μ k | | ν k | cos φ k ] cos [ k ( z z ) ] ,
U ( z ) = U L 1 2 cos ( k L z z ) cos ( k B z ) e | z | / l .

Metrics