Abstract

We theoretically explore cooperative effects of equally spaced multiemitters in a 1D dense array driven by a low-intensity probe field propagating through a 1D waveguide by modeling the emitters as point-like coupled electric dipoles. We calculate the collective optical spectra of a number of 1D emitter arrays with any radiation-retention coefficient η using both exact classical-electrodynamics and mean-field-theory formalisms. We illustrate cooperative effects of lossless 1D emitter arrays with η = 1 at the emitter spacings, which are displayed by steep edges accompanied by a deep minimum and Fano resonances in the plots of transmissivities as a function of the detuning of the incident light from the emitter resonance. Numerical simulation of the full width of such optical bandgaps reveals that cooperativity between emitters is greater in a small array of size N ≤ 8 than in a larger one of size N > 8. For a lossy 1D emitter array in which the radiation retention coefficient is equal to or less than 0.1 the transmissivity obtained by exact-electrodynamics scheme exhibits no bandgap structures, being in good agreement with the mean-field-theory result. We propose that a 1D multiemitter array may work as a nanoscale filter blocking transmission of light with a frequency in the range of optical bandgaps.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Rapid advances in techniques have enabled trapping light in a system of many resonant emitters and controlling the state of individual emitters [14]. Such advances offer fascinating possibilities [59] for manipulating light and tuning dipole-dipole interactions for novel devices with unique functionalities [10,11]. Theories on light interaction with multiemitters delve into topics regarding storage [9] and improvement [6] of quantum information, storage of light [7,8,12], engineering of optical spectra [13,12,1416], quantum simulations [1720], and to name a few.

One of the simplest yet illuminating platforms for light and emitter engineering is a 1D array of cold atoms [21], transmons [22], plasmon polaritons [23], and quantum dots [18], which are coupled to a common nanofiber [2] or photonic-crystal [24] waveguide, electric circuit [25], and optical lattice [19]. Particularly, in a 1D dense array of $N$ identical emitters with equal spacing of order the incident light’s wavelength or smaller, cooperation between emitters [12,26,27] plays a key role in light propagation. Underlying mechanism of optical cooperative effect is multiple scattering in which light radiated by a single emitter in spontaneous emission may be absorbed by one of its neighbors and be reemitted a little later. Such multiple scatterings continue until the light escapes from the array. A sample where all radiations by spontaneous emission are retained, which is what we will refer to as either “lossless” or “ideal” sample, shows cooperativity in a measurement such as peak transmitted intensity of light scaled as $N^2$. In addition, the interference of reflected lights from $N - 1$ emitters at the position of a specific emitter contributes to different amounts of light each emitter experiences. These observations are why a mean-field theory may not accurately explain collective optical responses in an ideal 1D array without radiation loss. However, for a lossy 1D emitter array, where the ratio $\eta$ of trapped radiation energy to total radiation energy in spontaneous emission is small, effect of multiple scattering might be less dominant than for a lossless 1D array.

In parallel with the experimental realizations of 1D multiemitter arrays [2,3,9,18,24,25,28,29], a variety of theories on electromagnetic radiation interacting with a 1D array of multiemitters [22,3045] have emerged. Among the most recent theoretical works reporting optical bandgap structures are those on real atomic ensembles [32] in a waveguide, chains of qubits [22], and transmons [46]. Nearly zero transmission profile around the emitter resonance is referred to as optical bandgap. This bandgap may be used for inhibiting spontaneous emission by emitters and has potential applications [22,47,48]. The emergence of optical bandgap with steep edges is an evidence of cooperativity in a 1D multiemitter array so that its shape can be engineered by varying interemitter spacing and array size. How the radiation retention coefficient affects cooperative optical responses of a 1D emitter array has not been discussed so far. The present interests are on which quantities could represent optical cooperative effects of emitters, what array size is needed for observing such effects, and how to engineer optical bandgap. We consider a 1D dense emitter array with an arbitrary radiation retention coefficient $\eta$ to answer these questions.

We extend our earlier theoretical approaches [14,17,49] for light-driven strongly coupled cold atoms in a 2D planar dense lattice to a 1D dense array of $N$ emitters at deterministic fixed positions, which may be side coupled to a 1D waveguide. In particular, the 1D array occupies a single emitter at a fixed position in the spatial periodicity, inducing optical bandgap. Emitters in the 1D array interacting with the low-intensity light are modeled as coupled point-like electric dipoles [13] for which steady states are considered. We solve a set of simultaneous equations for the excitation amplitudes of electric dipoles using classical electrodynamics [4952] that has been derived by the mean-field approximation where light-mediated correlations are not included. Because emitters under consideration are tightly confined at fixed positions without spatial fluctuations, the mean-field approximation can be used. Collective optical spectra of the 1D array are then obtained in terms of the Rabi frequency at a position past the last emitter light travels through. For a 1D dense emitter array of the emitter spacing equals a half of the resonant light wavelength, the transmissivity is analytically calculated as a function of the detuning of the incident light, array size, and radiation retention coefficient. The present classical electrodynamics framework, however, cannot analyze a 1D dense emitter array in the saturation intensity of light [53] because an isolated emitter can scatter light quantum-mechanically.

Major finding from our numerical simulations is that with a lossless 1D multiemitter array of $\eta = 1$ the cooperativity, which is represented by the plots of the full width of bandgap in the transmissivity versus $N^{-2}$, is greater for an array of size $N \le 8$ than for that of size $N > 8$. Despite fixed positions of emitters without correlation, our theoretical scheme, derived from the mean-field approximation, enables the measurement of cooperative feature of light-mediated interaction between emitters. We also resolve a computation issue at the edges of optical bandgap [47], which is distinct with increasing array size, by minimizing the round-off errors and increasing a machine precision and data points for the dimensionless detuning up to 3000. Furthermore, it is found that with a lossy 1D array of $\eta \le 0.1$ the lineshapes of transmissivity are in good agreement with those from the mean-field-theory which assumes equal amount of light on each of emitters light passes through. Such a result implies that in the case of a low radiation retention, cooperative effect tends to be weakened despite the small interemitter spacing.

2. Coupled-dipole model

We consider a 1D array of $N$ emitters placed at the fixed positions $x_n (n=1,\ldots, N)$ and coupled to a driving field traveling along the axis of a 1D waveguide. The driving field is a plane wave of coherent light linearly polarized in the $y$ direction and traveling along the $x$ axis of the form

$$\mathbf{E}_\textrm{inc}(x)=\mathcal{E}_0\hat{\mathbf{e}}_{y}\exp{(ikx)}.$$

Here $\mathcal {E}_0$, ${\hat {\mathbf {e}}}_{y}$, and $k$ are the amplitude, unit polarization vector, and wave number of the light field. Electric field of the driving light with the angular frequency $\omega$ is expressed in terms of the slowly-varying positive-frequency quantities without the explicit oscillations $e^{-i\omega t}$. The detuning of the driving light is defined as $\Delta = \omega -\omega _0$ for the emitter resonance $\omega _{0}$, and $\gamma$ denotes the HWHM linewidth of the optical transition of a single emitter $J=0 \rightarrow J'=1$. Figure 1 illustrates the schematics of the case of $N = 8$ emitters coupled to the 1D waveguide. In the present theoretical scheme light intensity is low enough such that optical response of emitters are linear. In addition, inelastically scattered radiation is not included. Thus we can model emitters in a 1D array, coupled to the electromagnetic field by the dipole-dipole interaction, as classical linear point-dipole oscillators in a two-level approximation. We also assume collective emission where emitters interact through the guided modes of the coupled 1D waveguide and three-dimensional (3D) free-space electromagnetic modes. The linewidth $\gamma$ of the optical transition then equals the sum of the 1D linewidth $\gamma _w$ and the 3D linewidth $\gamma _l$. For the current purposes, we neglect position fluctuation of the emitters around the equilibrium position in the assumption of tight confinement of emitters at the fixed points. In a lossless 1D array of multiemitters, all of the light an emitter captures would be delivered back to the original mode that they left without loss, and their propagation phase is determined by the wave number $k$. We consider a 1D multiemitter array with an arbitrary radiation retention coefficient $\eta = \gamma _w/\gamma$, where light propagates with attenuation due to radiation into channels other than the 1D emitter array or the 1D waveguide. An array of emitters in the evanescent wave of a nanofiber [6] is illustrated in Fig. 1. Experiments on the sample of emitters placed within a 1D photonic crystal waveguide [54] have been realized as well.

 figure: Fig. 1.

Fig. 1. A candidate system of 1D emitter array coupled to a 1D waveguide is illustrated for the case of $N = 8$ emitters at the positions $x_n (n=1,\ldots,8)$ separated by the equal spacing of $a \leq \lambda$. Incident plane-wave probe field of the wave vector $\vec {k}$ with the linear polarization in the $y$ direction propagates along the axis of the waveguide, the $x$-direction. The emitter at $x_4$ driven by the sum of the incident light and scattered light by the other seven emitters radiates light with rates $\gamma _{w}$ into the 1D waveguide and $\gamma _{l}$ into the 3D free space. Light transmits through the 1D array with the wave number $k$, inducing the identical electric dipole moments on the emitters in the $y$ direction.

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In the low light intensity limit and a two-level approximation, the saturation of the excited state is neglected. Additional simplification arising from collective emission enables to use an effective classical electrodynamics [12,32,55] for the collective dynamics of the sample. We describe a modeled 1D emitter array using the time evolution of the excitation amplitude $\mathcal {P}_n$ of $n$th emitter as follows:

$$\frac{d\mathcal{P}_n}{dt}=(i\Delta-\gamma)\mathcal{P}_n + i\kappa_{0}(x_n)-\gamma\eta\sum_{l \neq n}\exp{(ik|x_n-x_l|)}\mathcal{P}_l.$$

Here the Rabi frequency of the driving light with electric field amplitude $\mathbf {E}_\textrm {inc}(x)$ at the position $x$ is defined as

$$\kappa_0(x) = \frac{\mathcal{D}}{\hbar}\mathbf{E}_\textrm{inc}(x)\cdot\hat{\mathbf{e}}_y \equiv K_{0}\exp(ikx)$$
in terms of the dipole transition matrix element $\mathcal {D}$ for the $J=0 \rightarrow J'=1$ transition, unit polarization vector $\hat {\mathbf {e}}_y$ of the electric dipoles, and a constant amplitude $K_0$. Detailed derivation of Eq. (2) can be found in the Supplement 1. In terms of column matrices $\mathbf {P}$ and $\mathbf {K}$ consisting of the excitation amplitudes $\mathcal {P}_n$ and Rabi frequencies $\kappa _{0}(x_n)$, the equations for the emitters in the steady state can be written below:
$$\mathbf{H}\mathbf{P}=i\mathcal{\mathbf{K}}.$$

The $N$ by $N$ matrix $\mathbf {H}$ represents dissipative response of the emitters in the 1D array whose element is expressed as

$$H_{nl}=(\gamma-i\Delta)\delta_{nl}+\gamma\eta(1-\delta_{nl})\exp(ik|x_n-x_l|)$$
for $n, l =1,\ldots, N$, and Kronecker delta $\delta _{nl} = 1$ for $n = l$ and 0 for $n \neq l$. Solving Eq. (4) for $\mathbf {P}$ of the steady-state emitters, we find the Rabi frequency at an arbitrary position $x > \max {(x_1,\ldots,x_N)}$ past the last emitter through which light passes:
$$\kappa_T(x)=\kappa_{0}(x)+i\gamma\eta \sum_{n}\exp{(ik|x-x_{n}|)\mathcal{P}_n}.$$

This can be rewritten as $\kappa _{T}(x) = K_{T} \exp {(ikx)}$ with an amplitude $K_{T}$. In a study on optical cooperation between many emitters by the use of the coupled-dipole model, we take a 1D array in which $N$ emitters at the positions $x_n$ are apart from one another by the same emitter spacing $a$, homogeneous, and isotropic. We use the dimensionless detuning defined as $\delta = \Delta /\gamma$ for the calculations. The primary quantity to calculate is the transmissivity of a 1D $N$-emitter array with the same spacing $a$, which is defined as

$$T^{(N)}(\delta,\eta, a) = \left |\frac{\kappa_T(x)}{\kappa_0(x)}\right|^2 = \left | \frac{K_T}{K_0}\right |^2$$
for $K_T = K_T(\delta,\eta, a)$ and a constant $K_0$. The related quantity is the optical thickness
$$O^{(N)}(\delta,\eta, a)={-}\ln{T^{(N)}}(\delta,\eta, a).$$

3. Examples

Calculation of cooperative effects of multiemitters to driving light can be often performed analytically for particular cases: specific-emitter-spacing and small-size cases. Here we begin with presenting the analytical expressions of the transmissivity of plane-wave light confining a 1D $N$-emitter array with the emitter separation $a = \lambda /2$ and a two-emitter array with any spacing $a$. We proceed to use the transmissivity of the two-emitter array when modeling a four-emitter array as a pair of two-emitter arrays apart by a long distance so that the transmissivity in the four-emitter array can be approximated by the mean-field theory. We solve a set of linear equations in Eq. (4) for the collective transmissivity, whereas we used the transfer-matrix method in the earlier work on cold atoms [14]. When using an analysis model of coupled dipole, both methods obtain exact solutions of classical electrodynamics for a 1D multiemitter array, producing the same results.

3.1 $N$-emitter array with $\lambda /2$ spacing

The transmissivity in a 1D dense array of $N$ emitters can be exactly calculated for the emitter separation $a = \lambda /2$ at any $\eta$ as

$$T^{(N)}(\delta,\eta, \lambda/2) =\frac{\delta^2+(\eta-1)^2}{ \delta^2+\left[(N-1)\eta+1\right]^2}.$$

The same result as Eq. (9) was presented for a lossless 1D array of $N$ qubits [22] with $\eta = 1$. This indicates that the collective linewidth $\gamma _N = (N-1)\eta + 1$ can be tuned by varying both $N$ and $\eta$, leading to the lifetime scaled as the inverse of $\gamma _N$. For example, a $16$-emitter array with $\eta = 1$ exhibits almost the same lineshape as that of a $128$-emitter one with $\eta = 15/127$. The dependence of lifetime on $N^{-1}$ indicates a phenomena similar to superradiance is observed at the emitter spacing $a = \lambda /2$, even if this is not cooperative effect scaled as $N^2$. Specially, in case of $\eta = 1$ where radiative energy is conserved such that $R^{(N)}+T^{(N)} = 1$ is satisfied, the reflectivity becomes

$$R^{(N)} (\delta, 1, \lambda/2) = \frac{N^2}{ \delta^2+N^2}.$$

Eq. (10) implies that in case of $\eta = 1$ collective HWHM linewidths in both transmissivity and reflectivity equal $N$; their lineshapes are Lorentzian. Even if available techniques implementing $\eta = 1$ case has not been known even in experiments on nanometer optical fibers, advances in techniques hopefully may allow to count total number of emitters by measurement of the optical spectra in a 1D array of equally-spaced emitters. Furthermore, it is quite interesting to notice that the transmissivity becomes greater than one in a single-emitter sample at optical resonance ($\delta = 0$) for $\eta > 2$. This means that light passing through a single emitter will have stronger intensity than at a position $x < x_1$ before entering the emitter. The increase in light intensity is related with the probability that a photon of light is amplified as it travels in a one-emitter sample. Consequently, the optical thickness of the one-emitter sample can be negative. Such $\eta$ dependence indicates the possibility of engineering the optical spectra by the control of external parameter $\eta$ in the sample.

Because transmissivity has periodicity in the emitter spacing $a = \lambda /2$, the same optical spectra as Eq. (9) are obtained for any spacing $a = \lambda /2 + m\lambda /2$ ($m = 1,2,3,\ldots$). This rule of thumb applies to any other emitter spacing $a_\textrm {given}$: $T^{(N)}(\delta,\eta, a_\textrm {given}) = T^{(N)}(\delta,\eta, a_\textrm {given} + m\lambda /2)$. Additionally, we refer to [22,32,34] for the analytical transmission amplitudes in a 1D multiemitter array.

3.2 Four-emitter array

The transmissivity of a coherent light interacting with a two-emitter array is

$$T^{(2)}(\delta, \eta, a) = \frac{\left [\delta^2 +(\eta-1)^2\right ]^2}{|(\delta+i)^2+\eta^2 \exp(2 i a)|^2}.$$

For a particular value of $\eta = 1$, the same result as Eq. (11) was derived in our earlier work on cold atoms [14] using transfer-matrix method from Fresnel equations [56]. It has been reported the reflectivity of two individually controlled Rydberg atoms [57] in a nanophotonic cavity was measured. We proceed to obtain the analytical expression for the transmissivity in a 1D four-emitter array. For the mean-field-theory calculation, the transmissivity $T^{(4)}_\textrm {m}(\delta, \eta, a)$ can be approximated by $[T^{(2)}(\delta, \eta, a)]^2$ and $[T^{(1)}(\delta, \eta )]^4$ for the one-emitter transmissivity

$$T^{(1)}(\delta, \eta) =\frac{\delta^2+(\eta-1)^2}{\delta^2+1}.$$

For a lossless 1D emitter array with the energy retention ratio $\eta = 1$, cooperative feature in the exact solutions using classical electrodynamics is manifest even for a small-size array [47]. As an example, Fig. 2 displays the transmissivities of a four-emitter array as a function of detuning $\delta$ for the emitter spacings $a = 0.250 \lambda$ and $0.425 \lambda$, respectively. The curves in blue solid line represent the exact numerical results from Eq. (7), while red and black dashed lines demonstrate the mean-field-theory results $[T^{(2)}]^2$ from Eq. (11) and $[T^{(1)}]^4$ from Eq. (12). Substantial deviations of the mean-field-theory results from the exact ones are manifest except in a range $|\delta | \le 1.0$ for a 1D emitter array of $a = 0.250 \lambda$. Furthermore, the transmissivity in the exact calculation shows steep edges at $\delta _F = \pm 1.41$ where Fano resonances occur revealing cooperative responses of emitters, and narrower lineshapes than in the mean-field-theory transmissivity. Figure 2(b) demonstrates the transmissivities for a 1D emitter array with $a = 0.425 \lambda$. The exact transmissivity exhibits two narrow peaks at $\delta _\textrm {F} = 0.284$, 0.510 and one broad peak at $\delta _\textrm {F} = 2.47$, whereas the mean-field-theory result from $[T^{(2)}]^2$ shows a single narrow peak at $\delta _\textrm {F} = 0.510$. Figure 2 evidently reveals dependence of Fano resonance lineshapes on the emitter spacing. Quantitative analysis for the narrow and broad peaks in Fano resonance can be performed calculating collective eigenmodes. Here we address the origin of such peaks as highly subradiant and superradiant eigenmodes of low light intensity [53].

Multiple scattering of light with each of optically thick emitters is unavoidable in a 1D emitter array along which driving light propagates if there exists no other channel of light than the one through emitters in sequence. Such multiple scattering causes cooperation among the emitters mediated by light, which is manifest in the transmissivity even for a lossless small-size array like that with $N = 4$. Narrow bandgap with steep edge in Fig. 2 might be explained by suppression spontaneous emission of collective emitter excitation caused by strong interference in emission between emitters [6]. In fact, dynamics of emitters, which is derived from the mean-field approximation, exhibits correlated behavior of emitters arising from repeated absorptions and emissions of light of the same emitter.

 figure: Fig. 2.

Fig. 2. Transmissivities $T^{(4)}$ of a lossless 1D four-emitter array with the energy retention ratio $\eta = 1$ as a function of detuning $\delta$ at two different emitter spacings of $a = 0.250 \lambda$ and $0.425 \lambda$. The mean-field-theory results approximated by $[T^{(2)}]^2$ (red dashed) and $[T^{(1)}]^4$ (black dashed) are compared with the exact results (blue solid).

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We illustrate the transmissivity of a four-emitter array with the energy retention ratio $\eta = 0.1$ in Fig. 3. The mean-field-theory transmissivities $T_\textrm {m} ^{(4)} = [T^{(2)}]^2$ (red dashed) and $T_\textrm {m} ^{(4)} = [T^{(1)}]^4$ (black dashed) are compared with the exact-electrodynamics results (blue solid line) at the emitter spacings $a = 0.250 \lambda$ and $0.425 \lambda$. It is found that cooperative features exhibited as a form of Fano resonances diminish with decreasing $\eta$, displaying almost no deviation of the mean-field-theory results from the exact-electrodynamics ones with $\eta = 0.1$. For a 1D emitter array of $a = 0.250 \lambda$ deviation is displayed only in a near-resonance region, and no deviation of exact result from the mean-field-theory transmissivities $[T^{(2)}]^2$ and $[T^{(1)}]^4$ in the other region. Thus, in the limit of small $\eta$, the mean-field-theory formalism may accurately predict some aspects of collective optical spectra like lineshape, whereas shifts of resonance cannot be predicted accurately enough with the mean-field theory as shown in Fig. 3(b). Notable feature with the $\eta = 0.1$ case for the four-emitter array is that a bandgap does not exist. In contrast, ideal 1D emitter arrays with $\eta = 1$, as illustrated in Fig. 2, have optical bandgap in the range of $\delta \in [−0.5, 0.5]$ for $a = 0.250 \lambda$ and $\delta \in [−1.0, 0.25]$ for $a = 0.425 \lambda$, respectively. Furthermore, because 90$\%$ of spontaneously emitted photons are lost in the $\eta = 0.1$ case, the transmissivity at a detection point is in a narrower range of values than in a lossless 1D emitter array with $\eta = 1$. The exact transmissivities in the array with $a = 0.250 \lambda$ and $0.425 \lambda$ are respectively in the ranges of $[0.414, 0.958]$ and $[0.440, 0.956]$.

 figure: Fig. 3.

Fig. 3. Transmissivities $T^{(4)}$ of four-emitter arrays with $\eta = 0.1$ as a function of detuning $\delta$ at two different emitter spacings of $a = 0.250 \lambda$ and $0.425 \lambda$. Mean-field-theory results using $[T^{(2)}]^2$ (red dashed) and $[T^{(1)}]^4$ (black dashed line) are compared with the exact results (blue solid lines).

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4. Numerical simulations

Using Mathematica to solve a set of linear equations in Eq. (4) symbolically for a 1D multiemitter array, we obtain exact results of collective optical responses. Major computation issue with the array of the radiation retention coefficient $\eta = 1$ is to avoid round-off errors so that we could plot a curve at steep edges of optical bandstops. Adapting a numerical scheme in [47], we resolve such an issue by computing internally above 40 significant figures exceeding precision which any analog computing device can have.

Figure 4 illustrates the numerically simulated transmissivities $T^{(4)}, T^{(8)}, T^{(16)}$, and $T^{(32)}$ (top to bottom) as a function of detuning $\delta$ at an emitter spacing $a = 0.125 \lambda$ and for $\eta = 1$ from the exact-classical-electrodynamics (Fig. 4(b)) and mean-field-theory (Fig. 4(b)) schemes. As addressed in [47], Fig. 4 from the exact electrodynamics exhibits that a flat bandgap with steep edges, which is antisymmetric around the emitter resonance frequency. Note that the transmissivity converges to 1 as the detuning increases. Fano resonances at $\delta _{F} \approx −0.5$ is a signature of optical cooperative response of emitters: line shift of Fano resonance has smaller value in the red detunings, and its linewidth becomes narrower with the increased number of emitters. An array with $N = 32$ emitters exhibits the steepest edges at $\delta _{F} \approx −0.5, 2.5$ and greatest full width of bandgap, being a potential system for a nanoscale filter [47] that blocks transmission of light with a frequency in the range of $\delta \in [−0.5, 2.5]$. Cooperation between emitters by multiple scattering of light with the same emitter produce the different amount of radiation energy transmitted to each emitter. This is what differentiates exact classical-electrodynamics approach and mean-field one where each of emitters has the same excitation amplitude. In Fig. 4(b) the mean-field-theory transmissivities in which $T_\textrm {m}^{(4)} = [T^{(1)}]^4, T_\textrm {m}^{(8)} = [T^{(1)}]^8, T_\textrm {m}^{(16)} = [T^{(1)}]^{16}$, and $T_\textrm {m}^{(32)} = [T^{(1)}]^{32}$ (top to bottom) are used reveal substantial deviations from the exact transmissivities shown in Fig. 4. Moreover, Fano resonances and optical bandgaps with the steep edges are not displayed in the mean-field-theory curves.

 figure: Fig. 4.

Fig. 4. Transmissivities $T^{(4)}, T^{(8)}, T^{(16)}$, and $T^{(32)}$ (top to down) as a function of detuning $\delta$ at the emitter spacing $a = 0.125 \lambda$ and for $\eta = 1$ using both the (a) exact-electrodynamics and (b) mean-field-theory calculations. An offset of 1 separates the plots in vertical direction, and unit transmissivity are represented by horizontal lines above each of four graphs.

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Let’s now explore the effects of the radiation retention coefficient $\eta$ on the cooperative optical responses of 1D large-size emitter arrays. Transmissivities $T^{(64)}$ of a $N = 64$ array with an emitter spacing $a = 0.250 \lambda$ are demonstrated in Fig. 5 for $\eta = 1$, 0.55, and 0.1 (bottom to top). We compare the exact-electrodynamics $T^{(64)}$ (Fig. 5) and mean-field-theory calculations (Fig. 5(b)) in which $T_\textrm {m}^{(64)} = [T^{(1)}]^{64}$ is used. At the coefficient $\eta = 0.1$ the mean-field-theory transmissivities are in good agreement with the exact ones as it is true with a 1D small-size array like the four-emitter array in Fig. 3. In the limit of small $\eta \le 0.1$ it is thus confirmed that the mean-field-theory scheme accurately describes lineshapes of collective optical spectra of a 1D large-size emitter array at $a = 0.250 \lambda$. While the mean-field-theory transmissivities display broadened bandgaps with increasing $\eta$, the exact-electrodynamics ones display the broadest bandgap almost at $\eta = 0.55$. For any radiation retention coefficient $\eta \le 1$, however, edges of bandgaps are not as steep as for $\eta = 1$. We can account for the physics on these findings: The more number of emitters are involved in spontaneous emission of light, the more subradiant cooperative eigenmodes with smaller linewidths are produced, leading to the steeper edges of optical bandgap. Note that many Fano resonances are displayed in $T^{(64)}$ of the array with $\eta = 1$. The narrow Fano resonances in light transmission originate from the interference between the broad superradiant and narrow subradiant cooperative eigenmodes. The transmissivities of $N = 64$ emitter array with $\eta = 0.1$ have flat bandgap structures, whereas the four-emitter array does not as shown in Fig. 3. Increasing width of bandgap with $N$ manifests cooperative effect of emitters.

 figure: Fig. 5.

Fig. 5. Transmissivity $T^{(64)}$ as a function of detuning $\delta$ at the emitter spacing $a = 0.250 \lambda$ and for the energy retention coefficients $\eta = 1$, 0.55, and 0.1 (bottom to top) using both the (a) exact-electrodynamics and (b) mean-field-theory calculations. An offset of 1 separates the plots in vertical direction, and unit transmissivity are represented by horizontal lines above each of three graphs.

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The total number of emitters, their spacing, and radiation retention coefficient in the array make a difference in the scattering of coherent incident light into the 1D waveguide, leading to the different cooperative effects. The $N^2$(or $N^{-2}$) dependence of a simulated quantity is referred to as cooperativity, and quantity manifesting cooperativity can be optical spectra such as transmissivity and optical thickness. Our choice here is the full width of bandgaps in the transmissivity denoted by $BW$, which we define as smallest difference between the two detunings at which the transmissivities are unity. As examples, $N = 8$ emitter array of $a = 0.125\lambda$ shown in Fig. 4(a), $BW$ of the bandgap is nearly 3.696, and for $N = 4$ array of $a = 0.250 \lambda$ in Fig. 2(a), $BW$ is 2.828. Figure 6 demonstrates the full optical bandgap width as a function of $N^{-2}$. Figure 6 is for a 1D array with smaller number of emitters $N \in [4, 8]$, and Fig. 6(b) for that with larger $N \in [9,14]$. We consider 1D arrays with emitter spacings $a = 0.125\lambda$ (blue triangles) and $0.250\lambda$ (red circles). Cooperative effects characterized by the $N^{-2}$ dependence of the full optical bandgap width $BW$ is shown for the larger array, whereas the $N^{-4}$ dependence is shown for the smaller one. This finding indicates two regimes of cooperativity, e.g., linear regime with $N^{-2}$ valid for the large-size array and nonlinear one with $N^{-2}$ for the small-size array. Such a finding suggests that a 1D dense emitter array tightly confined by a 1D waveguide can be a potential system for studying collective nonlinear optics [58]. In the array with smaller emitter spacing $a = 0.125 \lambda$ we obtain greater full optical bandgap widths as demonstrated in Fig. 6, and converges to a specific value with increasing number of emitters. For example, in the linear regime the full width of optical bandgap $BW$ converges to 2.05 for $a = 0.250\lambda$ array and to 3.06 for $a = 0.125\lambda$ array as shown in Fig. 6(b). For a 1D emitter array with $N \ge 15$ rapid change in the transmissivity interrupts numerical simulation of the full width of bandgaps beyond $N = 15$ case.

 figure: Fig. 6.

Fig. 6. Full width of bandgap in the transmissivity as a function of $N^{-2}$ for perfect 1D arrays with the emitter spacings $a = 0.125 \lambda$ (blue markers) and $0.250 \lambda$ (red markers). (a) Small array of size $N \in [4, 8]$, (b) Large array of size $N \in [9, 14]$.

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5. Conclusion

We calculated the spectrum of transmitted light through a 1D waveguide confining a 1D dense array of equally spaced two-level emitters modeled as coupled point-like electric dipoles at steady states. In the limit of low-intensity light we analyzed transmissivities of a 1D emitter array with an arbitrary radiation retention coefficient $\eta$ that can be controlled in practice by geometry of waveguide and index of refraction of a dielectric filling the waveguide. Both exact classical-electrodynamics and mean-field-theory approaches are used as theoretical schemes. Difference of the present exact simulation scheme from our earlier one for cold atoms [14,16,48] is a method to obtain collective optical spectra: a set of simultaneous equations for the excitation amplitudes of electric dipoles are solved to calculate the Rabi frequency at a detection point.

For a $N$-emitter array of interemitter distance $a$ equals half of the wavelength of light we obtain the analytical expression of transmissivity whose linewidth is proportional to $N$ and $\eta$, displaying collective effect rather than cooperative one. However, numerical simulations for lossless arrays with $\eta = 1$ at three different emitter spacings manifest flat bandgaps with steep edges in the transmissivity. Analysis of the full widths of such bandgap structures has revealed cooperative effects scaled as $N^{-2}$ in the exact electrodynamics approach even for a 1D small-size emitter array of $\eta = 1$. Furthermore, we have addressed that a small array of size $N \le 8$ shows greater cooperativity between emitters than an array of size $N > 8$, on which we do not account for the physics. For lossy arrays of $\eta \le 0.1$, however, the exact-electrodynamics transmissivity has displayed no flat bandgap structures with steep edges, and has been in good agreement with the mean-field-theory one. The absence of optical bandgap indicates that multiple scattering of light with the same emitter, which induces cooperative effect, is not underlying mechanism for the $\eta \le 1$ case. We also point out Fano resonance shown in the exact transmissivity as a signature of the cooperativity between emitters, which stems from strong light-mediated dipole-dipole interaction.

We speculate that an engineered 1D multiemitter array with steepest edges accompanied by a deep minimum in the transmissivity might be exploited as a nanoscale filter blocking transmission of light with a frequency in the range of bandgap. Similar idea has been addressed in [47] for a 1D waveguide of atoms. If experimental techniques enabled the radiation retention coefficient of a single-emitter sample to be greater than two, a 1D multiemitter array with transmissivity greater than one (or negative optical thickness) would be realized. One possibility to make a sample with $\eta > 1$ may be to exploit total reflection of light through a solid-state material enclosing the atom [59]. Because cooperative effects of multiemitter underlie fundamental principles upon which top-notch technologies in nano-optics are based, experimental demonstration of cooperative effects [21,60] of radiation from a 1D multiemitter array comes up rapidly. Analyzing experimental data, one may extend the classical electrodynamics approach to various 1D arrays involving artificial atoms such as quantum dots, excitons, and superconducting qubits [61,62]. Classical electrodynamics framework in the limit beyond the low-intensity light [63] may be developed as well.

Funding

Hongik University (2020 Research Fund); National Research Foundation of Korea (2019R1F1A1043770).

Acknowledgements

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2019R1F1A1043770) and 2020 Hongik University Research Fund. We appreciate the Korea Institute for Advanced Study for providing a space for this work.

Disclosures

The author declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

References

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References

  • View by:

  1. S. Casulleras, C. Gonzalez-Ballestero, P. Maurer, J. J. García-Ripoll, and O. Romero-Isart, “Remote individual addressing of quantum emitters with chirped pulses,” Phys. Rev. Lett. 126(10), 103602 (2021).
    [Crossref]
  2. P. Solano, P. Barberis-Blostein, F. K. Fatemi, L. A. Orozco, and S. L. Rolston, “Super-radiance reveals infinite-range dipole interactions through a nanofiber,” Nature Communications 8(1), 1857 (2017).
    [Crossref]
  3. 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,” Nature Communications 5(1), 3808 (2014).
    [Crossref]
  4. J. Pellegrino, R. Bourgain, S. Jennewein, Y. R. P. Sortais, A. Browaeys, S. D. Jenkins, and J. Ruostekoski, “Observation of suppression of light scattering induced by dipole-dipole interactions in a cold-atom ensemble,” Phys. Rev. Lett. 113(13), 133602 (2014).
    [Crossref]
  5. M. O. Scully, “Single photon subradiance: Quantum control of spontaneous emission and ultrafast readout,” Phys. Rev. Lett. 115(24), 243602 (2015).
    [Crossref]
  6. A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. J. Kimble, and D. E. Chang, “Exponential improvement in photon storage fidelities using subradiance and “selective radiance” in atomic arrays,” Phys. Rev. X 7(3), 031024 (2017).
    [Crossref]
  7. G. Facchinetti, S. D. Jenkins, and J. Ruostekoski, “Storing light with subradiant correlations in arrays of atoms,” Phys. Rev. Lett. 117(24), 243601 (2016).
    [Crossref]
  8. S. D. Jenkins, J. Ruostekoski, N. Papasimakis, S. Savo, and N. I. Zheludev, “Many-body subradiant excitations in metamaterial arrays: Experiment and theory,” Phys. Rev. Lett. 119(5), 053901 (2017).
    [Crossref]
  9. P.-O. Guimond, A. Grankin, D. V. Vasilyev, B. Vermersch, and P. Zoller, “Subradiant bell states in distant atomic arrays,” Phys. Rev. Lett. 122(9), 093601 (2019).
    [Crossref]
  10. X. Gu, A. F. Kockum, A. Miranowicz, Y. xi Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Physics Reports 718-719, 1–102 (2017).Microwave photonics with superconducting quantum circuits
    [Crossref]
  11. A. J. Sternbach, S. H. Chae, S. Latini, A. A. Rikhter, Y. Shao, B. Li, D. Rhodes, B. Kim, P. J. Schuck, X. Xu, X.-Y. Zhu, R. D. Averitt, J. Hone, M. M. Fogler, A. Rubio, and D. N. Basov, “Programmable hyperbolic polaritons in van der waals semiconductors,” Science 371(6529), 617–620 (2021).
    [Crossref]
  12. S. D. Jenkins and J. Ruostekoski, “Controlled manipulation of light by cooperative response of atoms in an optical lattice,” Phys. Rev. A 86(3), 031602 (2012).
    [Crossref]
  13. L. Chomaz, L. Corman, T. Yefsah, R. Desbuquois, and J. Dalibard, “Absorption imaging of a quasi-two-dimensional gas: a multiple scattering analysis,” New J. Phys. 14(5), 055001 (2012).
    [Crossref]
  14. S.-M. Yoo, “Strongly coupled cold atoms in bilayer dense lattices,” New J. Phys. 20(8), 083012 (2018).
    [Crossref]
  15. J. Javanainen and R. Rajapakse, “Light propogation in systems involving two-dimensional atomic lattices,” Phys. Rev. A 100(1), 013616 (2019).
    [Crossref]
  16. S.-M. Yoo and J. Javanainen, “Light reflection and transmission in planar lattices of cold atoms,” Opt. Express 28(7), 9764–9776 (2020).
    [Crossref]
  17. J. Gao, L.-F. Qiao, X.-F. Lin, Z.-Q. Jiao, Z. Feng, Z. Zhou, Z.-W. Gao, X.-Y. Xu, Y. Chen, H. Tang, and X.-M. Jin, “Non-classical photon correlation in a two-dimensional photonic lattice,” Opt. Express 24(12), 12607–12616 (2016).
    [Crossref]
  18. T. Hensgens, T. Fujita, L. Janssen, X. Li, C. J. Van Diepen, C. Reichl, W. Wegscheider, S. Das Sarma, and L. M. K. Vandersypen, “Quantum simulation of a fermi–hubbard model using a semiconductor quantum dot array,” Nature 548(7665), 70–73 (2017).
    [Crossref]
  19. C. Gross and I. Bloch, “Quantum simulations with ultracold atoms in optical lattices,” Science 357(6355), 995–1001 (2017).
    [Crossref]
  20. A. Browaeys and T. Lahaye, “Many-body physics with individually controlled rydberg atoms,” Nat. Phys. 16(2), 132–142 (2020).
    [Crossref]
  21. A. Glicenstein, G. Ferioli, N. Šibalić, L. Brossard, I. Ferrier-Barbut, and A. Browaeys, “Collective shift in resonant light scattering by a one-dimensional atomic chain,” Phys. Rev. Lett. 124(25), 253602 (2020).
    [Crossref]
  22. Y. S. Greenberg, A. A. Shtygashev, and A. G. Moiseev, “Waveguide band-gap n-qubit array with a tunable transparency resonance,” Phys. Rev. A 103(2), 023508 (2021).
    [Crossref]
  23. V. Smirnov, S. Stephan, M. Westphal, D. Emmrich, A. Beyer, A. Gölzhäuser, C. Lienau, and M. Silies, “Transmitting surface plasmon polaritons across nanometer-sized gaps by optical near-field coupling,” ACS Photonics 8(3), 832–840 (2021).
    [Crossref]
  24. A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007).
    [Crossref]
  25. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
    [Crossref]
  26. W. Guerin, M. Rouabah, and R. Kaiser, “Light interacting with atomic ensembles: collective, cooperative and mesoscopic effects,” Journal of Modern Optics 64(9), 895–907 (2017).
    [Crossref]
  27. K. E. Ballantine and J. Ruostekoski, “Optical magnetism and huygens’ surfaces in arrays of atoms induced by cooperative responses,” Phys. Rev. Lett. 125(14), 143604 (2020).
    [Crossref]
  28. A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109(3), 033603 (2012).
    [Crossref]
  29. J. Rui, D. Wei, A. Rubio-Abadal, S. Hollerith, J. Zeiher, D. M. Stamper-Kurn, C. Gross, and I. Bloch, “A subradiant optical mirror formed by a single structured atomic layer,” Nature 583(7816), 369–374 (2020).
    [Crossref]
  30. I.-C. Hoi, C. M. Wilson, G. Johansson, J. Lindkvist, B. Peropadre, T. Palomaki, and P. Delsing, “Microwave quantum optics with an artificial atom in one-dimensional open space,” New J. Phys. 15(2), 025011 (2013).
    [Crossref]
  31. Z. Liao, X. Zeng, S.-Y. Zhu, and M. S. Zubairy, “Single-photon transport through an atomic chain coupled to a one-dimensional nanophotonic waveguide,” Phys. Rev. A 92(2), 023806 (2015).
    [Crossref]
  32. J. Ruostekoski and J. Javanainen, “Emergence of correlated optics in one-dimensional waveguides for classical and quantum atomic gases,” Phys. Rev. Lett. 117(14), 143602 (2016).
    [Crossref]
  33. R. J. Bettles, S. A. Gardiner, and C. S. Adams, “Cooperative eigenmodes and scattering in one-dimensional atomic arrays,” Phys. Rev. A 94(4), 043844 (2016).
    [Crossref]
  34. J. Ruostekoski and J. Javanainen, “Arrays of strongly coupled atoms in a one-dimensional waveguide,” Phys. Rev. A 96(3), 033857 (2017).
    [Crossref]
  35. G.-Z. Song, E. Munro, W. Nie, F.-G. Deng, G.-J. Yang, and L.-C. Kwek, “Photon scattering by an atomic ensemble coupled to a one-dimensional nanophotonic waveguide,” Phys. Rev. A 96(4), 043872 (2017).
    [Crossref]
  36. M. Moreno-Cardoner, D. Plankensteiner, L. Ostermann, D. E. Chang, and H. Ritsch, “Subradiance-enhanced excitation transfer between dipole-coupled nanorings of quantum emitters,” Phys. Rev. A 100(2), 023806 (2019).
    [Crossref]
  37. D. Mukhopadhyay and G. S. Agarwal, “Multiple fano interferences due to waveguide-mediated phase coupling between atoms,” Phys. Rev. A 100(1), 013812 (2019).
    [Crossref]
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2021 (8)

S. Casulleras, C. Gonzalez-Ballestero, P. Maurer, J. J. García-Ripoll, and O. Romero-Isart, “Remote individual addressing of quantum emitters with chirped pulses,” Phys. Rev. Lett. 126(10), 103602 (2021).
[Crossref]

A. J. Sternbach, S. H. Chae, S. Latini, A. A. Rikhter, Y. Shao, B. Li, D. Rhodes, B. Kim, P. J. Schuck, X. Xu, X.-Y. Zhu, R. D. Averitt, J. Hone, M. M. Fogler, A. Rubio, and D. N. Basov, “Programmable hyperbolic polaritons in van der waals semiconductors,” Science 371(6529), 617–620 (2021).
[Crossref]

Y. S. Greenberg, A. A. Shtygashev, and A. G. Moiseev, “Waveguide band-gap n-qubit array with a tunable transparency resonance,” Phys. Rev. A 103(2), 023508 (2021).
[Crossref]

V. Smirnov, S. Stephan, M. Westphal, D. Emmrich, A. Beyer, A. Gölzhäuser, C. Lienau, and M. Silies, “Transmitting surface plasmon polaritons across nanometer-sized gaps by optical near-field coupling,” ACS Photonics 8(3), 832–840 (2021).
[Crossref]

F. Robicheaux and D. A. Suresh, “Beyond lowest order mean-field theory for light interacting with atom arrays,” Phys. Rev. A 104(2), 023702 (2021).
[Crossref]

J. D. Brehm, A. N. Poddubny, A. Stehli, T. Wolz, H. Rotzinger, and A. V. Ustinov, “Waveguide bandgap engineering with an array of superconducting qubits,” npj Quantum Mater. 6(1), 10 (2021).
[Crossref]

C. D. Parmee and J. Ruostekoski, “Bistable optical transmission through arrays of atoms in free space,” Phys. Rev. A 103(3), 033706 (2021).
[Crossref]

G. Karpat, I. Yalçinkaya, B. Çakmak, G. L. Giorgi, and R. Zambrini, “Synchronization and non-markovianity in open quantum systems,” Phys. Rev. A 103(6), 062217 (2021).
[Crossref]

2020 (16)

L. A. Williamson, M. O. Borgh, and J. Ruostekoski, “Superatom picture of collective nonclassical light emission and dipole blockade in atom arrays,” Phys. Rev. Lett. 125(7), 073602 (2020).
[Crossref]

A. Cidrim, T. S. do Espirito Santo, J. Schachenmayer, R. Kaiser, and R. Bachelard, “Photon blockade with ground-state neutral atoms,” Phys. Rev. Lett. 125(7), 073601 (2020).
[Crossref]

Z. Wang, H. Li, W. Feng, X. Song, C. Song, W. Liu, Q. Guo, X. Zhang, H. Dong, D. Zheng, H. Wang, and D.-W. Wang, “Controllable switching between superradiant and subradiant states in a 10-qubit superconducting circuit,” Phys. Rev. Lett. 124(1), 013601 (2020).
[Crossref]

R. J. Bettles, M. D. Lee, S. A. Gardiner, and J. Ruostekoski, “Quantum and nonlinear effects in light transmitted through planar atomic arrays,” Commun Phys 3(1), 141–149 (2020).
[Crossref]

P. Samutpraphoot, T. Ðorđević, P. L. Ocola, H. Bernien, C. Senko, V. Vuletić, and M. D. Lukin, “Strong coupling of two individually controlled atoms via a nanophotonic cavity,” Phys. Rev. Lett. 124(6), 063602 (2020).
[Crossref]

J. Javanainen, “Cooperative band-stop filters,” Optik 216, 164792 (2020).
[Crossref]

Y.-X. Zhang, C. Yu, and K. Mølmer, “Subradiant bound dimer excited states of emitter chains coupled to a one dimensional waveguide,” Phys. Rev. Research 2(1), 013173 (2020).
[Crossref]

R. Jones, G. Buonaiuto, B. Lang, I. Lesanovsky, and B. Olmos, “Collectively enhanced chiral photon emission from an atomic array near a nanofiber,” Phys. Rev. Lett. 124(9), 093601 (2020).
[Crossref]

B. Olmos, G. Buonaiuto, P. Schneeweiss, and I. Lesanovsky, “Interaction signatures and non-gaussian photon states from a strongly driven atomic ensemble coupled to a nanophotonic waveguide,” Phys. Rev. A 102(4), 043711 (2020).
[Crossref]

V. A. Pivovarov, A. S. Sheremet, L. V. Gerasimov, J. Laurat, and D. V. Kupriyanov, “Quantum interface between light and a one-dimensional atomic system,” Phys. Rev. A 101(5), 053858 (2020).
[Crossref]

L. A. Williamson and J. Ruostekoski, “Optical response of atom chains beyond the limit of low light intensity: The validity of the linear classical oscillator model,” Phys. Rev. Research 2(2), 023273 (2020).
[Crossref]

A. Browaeys and T. Lahaye, “Many-body physics with individually controlled rydberg atoms,” Nat. Phys. 16(2), 132–142 (2020).
[Crossref]

A. Glicenstein, G. Ferioli, N. Šibalić, L. Brossard, I. Ferrier-Barbut, and A. Browaeys, “Collective shift in resonant light scattering by a one-dimensional atomic chain,” Phys. Rev. Lett. 124(25), 253602 (2020).
[Crossref]

K. E. Ballantine and J. Ruostekoski, “Optical magnetism and huygens’ surfaces in arrays of atoms induced by cooperative responses,” Phys. Rev. Lett. 125(14), 143604 (2020).
[Crossref]

J. Rui, D. Wei, A. Rubio-Abadal, S. Hollerith, J. Zeiher, D. M. Stamper-Kurn, C. Gross, and I. Bloch, “A subradiant optical mirror formed by a single structured atomic layer,” Nature 583(7816), 369–374 (2020).
[Crossref]

S.-M. Yoo and J. Javanainen, “Light reflection and transmission in planar lattices of cold atoms,” Opt. Express 28(7), 9764–9776 (2020).
[Crossref]

2019 (6)

J. Javanainen and R. Rajapakse, “Light propogation in systems involving two-dimensional atomic lattices,” Phys. Rev. A 100(1), 013616 (2019).
[Crossref]

P.-O. Guimond, A. Grankin, D. V. Vasilyev, B. Vermersch, and P. Zoller, “Subradiant bell states in distant atomic arrays,” Phys. Rev. Lett. 122(9), 093601 (2019).
[Crossref]

M. Moreno-Cardoner, D. Plankensteiner, L. Ostermann, D. E. Chang, and H. Ritsch, “Subradiance-enhanced excitation transfer between dipole-coupled nanorings of quantum emitters,” Phys. Rev. A 100(2), 023806 (2019).
[Crossref]

D. Mukhopadhyay and G. S. Agarwal, “Multiple fano interferences due to waveguide-mediated phase coupling between atoms,” Phys. Rev. A 100(1), 013812 (2019).
[Crossref]

Y.-X. Zhang and K. Mølmer, “Theory of subradiant states of a one-dimensional two-level atom chain,” Phys. Rev. Lett. 122(20), 203605 (2019).
[Crossref]

L. Ostermann, C. Meignant, C. Genes, and H. Ritsch, “Super- and subradiance of clock atoms in multimode optical waveguides,” New Journal of Physics 21(2), 025004 (2019).
[Crossref]

2018 (2)

P. Back, S. Zeytinoglu, A. Ijaz, M. Kroner, and A. Imamoğlu, “Realization of an electrically tunable narrow-bandwidth atomically thin mirror using monolayer mose2,” Phys. Rev. Lett. 120(3), 037401 (2018).
[Crossref]

S.-M. Yoo, “Strongly coupled cold atoms in bilayer dense lattices,” New J. Phys. 20(8), 083012 (2018).
[Crossref]

2017 (9)

X. Gu, A. F. Kockum, A. Miranowicz, Y. xi Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Physics Reports 718-719, 1–102 (2017).Microwave photonics with superconducting quantum circuits
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P. Solano, P. Barberis-Blostein, F. K. Fatemi, L. A. Orozco, and S. L. Rolston, “Super-radiance reveals infinite-range dipole interactions through a nanofiber,” Nature Communications 8(1), 1857 (2017).
[Crossref]

A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. J. Kimble, and D. E. Chang, “Exponential improvement in photon storage fidelities using subradiance and “selective radiance” in atomic arrays,” Phys. Rev. X 7(3), 031024 (2017).
[Crossref]

S. D. Jenkins, J. Ruostekoski, N. Papasimakis, S. Savo, and N. I. Zheludev, “Many-body subradiant excitations in metamaterial arrays: Experiment and theory,” Phys. Rev. Lett. 119(5), 053901 (2017).
[Crossref]

W. Guerin, M. Rouabah, and R. Kaiser, “Light interacting with atomic ensembles: collective, cooperative and mesoscopic effects,” Journal of Modern Optics 64(9), 895–907 (2017).
[Crossref]

T. Hensgens, T. Fujita, L. Janssen, X. Li, C. J. Van Diepen, C. Reichl, W. Wegscheider, S. Das Sarma, and L. M. K. Vandersypen, “Quantum simulation of a fermi–hubbard model using a semiconductor quantum dot array,” Nature 548(7665), 70–73 (2017).
[Crossref]

C. Gross and I. Bloch, “Quantum simulations with ultracold atoms in optical lattices,” Science 357(6355), 995–1001 (2017).
[Crossref]

J. Ruostekoski and J. Javanainen, “Arrays of strongly coupled atoms in a one-dimensional waveguide,” Phys. Rev. A 96(3), 033857 (2017).
[Crossref]

G.-Z. Song, E. Munro, W. Nie, F.-G. Deng, G.-J. Yang, and L.-C. Kwek, “Photon scattering by an atomic ensemble coupled to a one-dimensional nanophotonic waveguide,” Phys. Rev. A 96(4), 043872 (2017).
[Crossref]

2016 (8)

J. Javanainen and J. Ruostekoski, “Light propagation beyond the mean-field theory of standard optics,” Opt. Express 24(2), 993–1001 (2016).
[Crossref]

S.-M. Yoo and S. M. Paik, “Cooperative optical response of 2d dense lattices with strongly correlated dipoles,” Opt. Express 24(3), 2156–2165 (2016).
[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,” Proc Natl Acad Sci USA 113(38), 10507–10512 (2016).
[Crossref]

M. D. Lee, S. D. Jenkins, and J. Ruostekoski, “Stochastic methods for light propagation and recurrent scattering in saturated and nonsaturated atomic ensembles,” Phys. Rev. A 93(6), 063803 (2016).
[Crossref]

J. Ruostekoski and J. Javanainen, “Emergence of correlated optics in one-dimensional waveguides for classical and quantum atomic gases,” Phys. Rev. Lett. 117(14), 143602 (2016).
[Crossref]

R. J. Bettles, S. A. Gardiner, and C. S. Adams, “Cooperative eigenmodes and scattering in one-dimensional atomic arrays,” Phys. Rev. A 94(4), 043844 (2016).
[Crossref]

G. Facchinetti, S. D. Jenkins, and J. Ruostekoski, “Storing light with subradiant correlations in arrays of atoms,” Phys. Rev. Lett. 117(24), 243601 (2016).
[Crossref]

J. Gao, L.-F. Qiao, X.-F. Lin, Z.-Q. Jiao, Z. Feng, Z. Zhou, Z.-W. Gao, X.-Y. Xu, Y. Chen, H. Tang, and X.-M. Jin, “Non-classical photon correlation in a two-dimensional photonic lattice,” Opt. Express 24(12), 12607–12616 (2016).
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2015 (2)

M. O. Scully, “Single photon subradiance: Quantum control of spontaneous emission and ultrafast readout,” Phys. Rev. Lett. 115(24), 243602 (2015).
[Crossref]

Z. Liao, X. Zeng, S.-Y. Zhu, and M. S. Zubairy, “Single-photon transport through an atomic chain coupled to a one-dimensional nanophotonic waveguide,” Phys. Rev. A 92(2), 023806 (2015).
[Crossref]

2014 (3)

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,” Nature Communications 5(1), 3808 (2014).
[Crossref]

J. Pellegrino, R. Bourgain, S. Jennewein, Y. R. P. Sortais, A. Browaeys, S. D. Jenkins, and J. Ruostekoski, “Observation of suppression of light scattering induced by dipole-dipole interactions in a cold-atom ensemble,” Phys. Rev. Lett. 113(13), 133602 (2014).
[Crossref]

J. Javanainen, J. Ruostekoski, Y. Li, and S.-M. Yoo, “Shifts of a resonance line in a dense atomic sample,” Phys. Rev. Lett. 112(11), 113603 (2014).
[Crossref]

2013 (1)

I.-C. Hoi, C. M. Wilson, G. Johansson, J. Lindkvist, B. Peropadre, T. Palomaki, and P. Delsing, “Microwave quantum optics with an artificial atom in one-dimensional open space,” New J. Phys. 15(2), 025011 (2013).
[Crossref]

2012 (3)

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109(3), 033603 (2012).
[Crossref]

S. D. Jenkins and J. Ruostekoski, “Controlled manipulation of light by cooperative response of atoms in an optical lattice,” Phys. Rev. A 86(3), 031602 (2012).
[Crossref]

L. Chomaz, L. Corman, T. Yefsah, R. Desbuquois, and J. Dalibard, “Absorption imaging of a quasi-two-dimensional gas: a multiple scattering analysis,” New J. Phys. 14(5), 055001 (2012).
[Crossref]

2007 (1)

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007).
[Crossref]

2004 (1)

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref]

1999 (1)

J. Javanainen, J. Ruostekoski, B. Vestergaard, and M. R. Francis, “One-dimensional modeling of light propagation in dense and degenerate samples,” Phys. Rev. A 59(1), 649–666 (1999).
[Crossref]

Adams, C. S.

R. J. Bettles, S. A. Gardiner, and C. S. Adams, “Cooperative eigenmodes and scattering in one-dimensional atomic arrays,” Phys. Rev. A 94(4), 043844 (2016).
[Crossref]

Agarwal, G. S.

D. Mukhopadhyay and G. S. Agarwal, “Multiple fano interferences due to waveguide-mediated phase coupling between atoms,” Phys. Rev. A 100(1), 013812 (2019).
[Crossref]

Akimov, A. V.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007).
[Crossref]

Albrecht, A.

A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. J. Kimble, and D. E. Chang, “Exponential improvement in photon storage fidelities using subradiance and “selective radiance” in atomic arrays,” Phys. Rev. X 7(3), 031024 (2017).
[Crossref]

Alton, D. J.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109(3), 033603 (2012).
[Crossref]

Asenjo-Garcia, A.

A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. J. Kimble, and D. E. Chang, “Exponential improvement in photon storage fidelities using subradiance and “selective radiance” in atomic arrays,” Phys. Rev. X 7(3), 031024 (2017).
[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,” Proc Natl Acad Sci USA 113(38), 10507–10512 (2016).
[Crossref]

Averitt, R. D.

A. J. Sternbach, S. H. Chae, S. Latini, A. A. Rikhter, Y. Shao, B. Li, D. Rhodes, B. Kim, P. J. Schuck, X. Xu, X.-Y. Zhu, R. D. Averitt, J. Hone, M. M. Fogler, A. Rubio, and D. N. Basov, “Programmable hyperbolic polaritons in van der waals semiconductors,” Science 371(6529), 617–620 (2021).
[Crossref]

Bachelard, R.

A. Cidrim, T. S. do Espirito Santo, J. Schachenmayer, R. Kaiser, and R. Bachelard, “Photon blockade with ground-state neutral atoms,” Phys. Rev. Lett. 125(7), 073601 (2020).
[Crossref]

Back, P.

P. Back, S. Zeytinoglu, A. Ijaz, M. Kroner, and A. Imamoğlu, “Realization of an electrically tunable narrow-bandwidth atomically thin mirror using monolayer mose2,” Phys. Rev. Lett. 120(3), 037401 (2018).
[Crossref]

Ballantine, K. E.

K. E. Ballantine and J. Ruostekoski, “Optical magnetism and huygens’ surfaces in arrays of atoms induced by cooperative responses,” Phys. Rev. Lett. 125(14), 143604 (2020).
[Crossref]

Barberis-Blostein, P.

P. Solano, P. Barberis-Blostein, F. K. Fatemi, L. A. Orozco, and S. L. Rolston, “Super-radiance reveals infinite-range dipole interactions through a nanofiber,” Nature Communications 8(1), 1857 (2017).
[Crossref]

Basov, D. N.

A. J. Sternbach, S. H. Chae, S. Latini, A. A. Rikhter, Y. Shao, B. Li, D. Rhodes, B. Kim, P. J. Schuck, X. Xu, X.-Y. Zhu, R. D. Averitt, J. Hone, M. M. Fogler, A. Rubio, and D. N. Basov, “Programmable hyperbolic polaritons in van der waals semiconductors,” Science 371(6529), 617–620 (2021).
[Crossref]

Bernien, H.

P. Samutpraphoot, T. Ðorđević, P. L. Ocola, H. Bernien, C. Senko, V. Vuletić, and M. D. Lukin, “Strong coupling of two individually controlled atoms via a nanophotonic cavity,” Phys. Rev. Lett. 124(6), 063602 (2020).
[Crossref]

Bettles, R. J.

R. J. Bettles, M. D. Lee, S. A. Gardiner, and J. Ruostekoski, “Quantum and nonlinear effects in light transmitted through planar atomic arrays,” Commun Phys 3(1), 141–149 (2020).
[Crossref]

R. J. Bettles, S. A. Gardiner, and C. S. Adams, “Cooperative eigenmodes and scattering in one-dimensional atomic arrays,” Phys. Rev. A 94(4), 043844 (2016).
[Crossref]

Beyer, A.

V. Smirnov, S. Stephan, M. Westphal, D. Emmrich, A. Beyer, A. Gölzhäuser, C. Lienau, and M. Silies, “Transmitting surface plasmon polaritons across nanometer-sized gaps by optical near-field coupling,” ACS Photonics 8(3), 832–840 (2021).
[Crossref]

Blais, A.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref]

Bloch, I.

J. Rui, D. Wei, A. Rubio-Abadal, S. Hollerith, J. Zeiher, D. M. Stamper-Kurn, C. Gross, and I. Bloch, “A subradiant optical mirror formed by a single structured atomic layer,” Nature 583(7816), 369–374 (2020).
[Crossref]

C. Gross and I. Bloch, “Quantum simulations with ultracold atoms in optical lattices,” Science 357(6355), 995–1001 (2017).
[Crossref]

Borgh, M. O.

L. A. Williamson, M. O. Borgh, and J. Ruostekoski, “Superatom picture of collective nonclassical light emission and dipole blockade in atom arrays,” Phys. Rev. Lett. 125(7), 073602 (2020).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed (Cambridge University Press, Cambridge, 1999).

Bourgain, R.

J. Pellegrino, R. Bourgain, S. Jennewein, Y. R. P. Sortais, A. Browaeys, S. D. Jenkins, and J. Ruostekoski, “Observation of suppression of light scattering induced by dipole-dipole interactions in a cold-atom ensemble,” Phys. Rev. Lett. 113(13), 133602 (2014).
[Crossref]

Brehm, J. D.

J. D. Brehm, A. N. Poddubny, A. Stehli, T. Wolz, H. Rotzinger, and A. V. Ustinov, “Waveguide bandgap engineering with an array of superconducting qubits,” npj Quantum Mater. 6(1), 10 (2021).
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Supplementary Material (1)

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Supplement 1       Supplemental document ver 2

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)

Fig. 1.
Fig. 1. A candidate system of 1D emitter array coupled to a 1D waveguide is illustrated for the case of $N = 8$ emitters at the positions $x_n (n=1,\ldots,8)$ separated by the equal spacing of $a \leq \lambda$. Incident plane-wave probe field of the wave vector $\vec {k}$ with the linear polarization in the $y$ direction propagates along the axis of the waveguide, the $x$-direction. The emitter at $x_4$ driven by the sum of the incident light and scattered light by the other seven emitters radiates light with rates $\gamma _{w}$ into the 1D waveguide and $\gamma _{l}$ into the 3D free space. Light transmits through the 1D array with the wave number $k$, inducing the identical electric dipole moments on the emitters in the $y$ direction.
Fig. 2.
Fig. 2. Transmissivities $T^{(4)}$ of a lossless 1D four-emitter array with the energy retention ratio $\eta = 1$ as a function of detuning $\delta$ at two different emitter spacings of $a = 0.250 \lambda$ and $0.425 \lambda$. The mean-field-theory results approximated by $[T^{(2)}]^2$ (red dashed) and $[T^{(1)}]^4$ (black dashed) are compared with the exact results (blue solid).
Fig. 3.
Fig. 3. Transmissivities $T^{(4)}$ of four-emitter arrays with $\eta = 0.1$ as a function of detuning $\delta$ at two different emitter spacings of $a = 0.250 \lambda$ and $0.425 \lambda$. Mean-field-theory results using $[T^{(2)}]^2$ (red dashed) and $[T^{(1)}]^4$ (black dashed line) are compared with the exact results (blue solid lines).
Fig. 4.
Fig. 4. Transmissivities $T^{(4)}, T^{(8)}, T^{(16)}$, and $T^{(32)}$ (top to down) as a function of detuning $\delta$ at the emitter spacing $a = 0.125 \lambda$ and for $\eta = 1$ using both the (a) exact-electrodynamics and (b) mean-field-theory calculations. An offset of 1 separates the plots in vertical direction, and unit transmissivity are represented by horizontal lines above each of four graphs.
Fig. 5.
Fig. 5. Transmissivity $T^{(64)}$ as a function of detuning $\delta$ at the emitter spacing $a = 0.250 \lambda$ and for the energy retention coefficients $\eta = 1$, 0.55, and 0.1 (bottom to top) using both the (a) exact-electrodynamics and (b) mean-field-theory calculations. An offset of 1 separates the plots in vertical direction, and unit transmissivity are represented by horizontal lines above each of three graphs.
Fig. 6.
Fig. 6. Full width of bandgap in the transmissivity as a function of $N^{-2}$ for perfect 1D arrays with the emitter spacings $a = 0.125 \lambda$ (blue markers) and $0.250 \lambda$ (red markers). (a) Small array of size $N \in [4, 8]$, (b) Large array of size $N \in [9, 14]$.

Equations (12)

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E inc ( x ) = E 0 e ^ y exp ( i k x ) .
d P n d t = ( i Δ γ ) P n + i κ 0 ( x n ) γ η l n exp ( i k | x n x l | ) P l .
κ 0 ( x ) = D E inc ( x ) e ^ y K 0 exp ( i k x )
H P = i K .
H n l = ( γ i Δ ) δ n l + γ η ( 1 δ n l ) exp ( i k | x n x l | )
κ T ( x ) = κ 0 ( x ) + i γ η n exp ( i k | x x n | ) P n .
T ( N ) ( δ , η , a ) = | κ T ( x ) κ 0 ( x ) | 2 = | K T K 0 | 2
O ( N ) ( δ , η , a ) = ln T ( N ) ( δ , η , a ) .
T ( N ) ( δ , η , λ / 2 ) = δ 2 + ( η 1 ) 2 δ 2 + [ ( N 1 ) η + 1 ] 2 .
R ( N ) ( δ , 1 , λ / 2 ) = N 2 δ 2 + N 2 .
T ( 2 ) ( δ , η , a ) = [ δ 2 + ( η 1 ) 2 ] 2 | ( δ + i ) 2 + η 2 exp ( 2 i a ) | 2 .
T ( 1 ) ( δ , η ) = δ 2 + ( η 1 ) 2 δ 2 + 1 .