Cooperative optical response of 2D dense lattices with strongly correlated dipoles

Sung-Mi Yoo; Sun Mok Paik

doi:10.1364/OE.24.002156

1. Introduction

When a coherent light interacts with a group of many identical atoms in close proximity or in regular spacing, the system reveals modified radiative and optical properties that do not occur in single-atom systems. Examples of such many-body behavior include a large shift of the emitted radiation from resonance [1, 2], strong modification of spatial configuration [3], superradiance (subradiance) [4], highly directional scattering [5], dipole-induced electromagnetic transparency [6], quantum phase transitions [7, 8], and localized excitation [9, 10]. One of the underlying mechanisms for these phenomena is the interference between incident field and scattered field from dipoles in the sample. Potential applications range from nano-optics [11] to quantum information processing with cold atoms [12]. Recent experiments on light interaction with a system of nanometric size both at room [13] and sub-microkelvin [14] temperatures reach the regime of high atomic density with ρk³ ≥ 1, where ρ and k denote the atomic number density and the wave number of resonant light, respectively. Access to small samples and promising applications prompt theoretical studies about frequency shifts of a resonance line in this density regime. One of pronounced many-atom frequency shifts is the Lorentz-Lorenz (LL) shift [15] and collective Lamb shift (CLS) [16–18] of the resonance line. In particular, the LL shift are consequences of the mean-field (MF) theory, which assumes that the local field at a dipole is given by the sum of the average macroscopic field and the correction term, and serves as the frequency scale for density-dependent quantities such as polarization, susceptibility, absorption or transmission, and optical thickness.

Until recent work by Keaveney et al. [13], there has been no experimental confirmation of theories [1, 2, 18] on LL shift or CLS in a small sample with sufficiently high density because of the difficulties with realizing an optically thick sample. If the sample is subject to inhomogeneous broadening, the response of the sample is collective, and predictions [1, 2, 16, 18–20] based on MF approach agree with experiments on frequency shifts in a many-body system [13]. Conversely, if the sample is homogeneously broadened and confined in a volume of a small size R < λ, where λ is the resonant light wavelength, the response is called to be cooperative. This is the case when the sample is in the high density regime and strongly correlated, and MF based electrodynamics theory cannot explain precisely the cooperative phenomena accessible in recent years. In the previous simulations [21] we had studied the cooperative response of a subwavelength-thickness slab with randomly distributed atom positions at high density. However, we are concerned with exploring how lateral structure and spatial regularity interplay for the cooperative optical behaviors so that we focus on 2D systems with regular spacing.

Here we continue our theoretical analysis of dense cold atoms in two-dimensional square and kagome lattice systems in terms of basically exact numerical simulations to find out how different geometries of dipolar emitters affect their cooperative optical response, as suggested in [19]. In the limit of low laser intensity, we believe that while using classical simulation, we are in effect solving the quantum problem exactly. Thus classical-electrodynamics approaches [21–23] taken previously is used for computing various optical responses such as optical thickness and absorption or transmission. We model square and kagome lattices with many point-like atoms in a monolayer and calculate the stationary dipole moments in the low light-intensity limit by solving a coupled equation [21, 24] for light and atoms. As cooperative behaviors emerge with increasing atomic number density or decreasing lattice spacing, our 2D lattice systems exhibit frequency shifts of resonance line which is proportional to the atomic density. We derive shifts of resonance within square lattices illuminated by a linearly polarized plane-wave beam using a 2D isotropic infinite lattice model. Our simulations are confirmed by showing that numerical results of shifts are in good agreement with experimental data [13] as well as our derived expressions. For the kagome lattice quantitatively different resonance features are displayed, as shown in [24].

2. Models and methods

We consider 2D square and kagome lattices in the xy plane consisting of N identical two-level atoms with a J = 0 → J′ = 1 transition. Laser beams impinging on the atoms in our models are plane-wave and Gaussian beams. The former is ideal for driving the atomic dipoles in a small-size lattice, the latter for a large-size one. Both beams are monochromatic, linearly polarized along the y direction, and propagating in the z direction perpendicular to a lattice plane. The plane-wave beam and Gaussian beam with waist w₀ are respectively expressed by

E_{0} (r, t) = E_{0} {\hat{e}}_{y} \exp [i (k z - ω t)],

E_{0} (r, t) = E_{0} {\hat{e}}_{y} \exp (- \frac{r^{2} - z^{2}}{w_{0}^{2}}) \exp [i (k z - ω t)] .

Here E₀ is the real amplitude of incoming electric field that determines the Rabi frequency $Ω = D E_{0} / \bar{h}$ of the atom with the dipole moment matrix element $D$ , ê_y the Cartesian unit vector in the y direction, and ω the light frequency. A two-level atom tends to align in a preferred direction for polarization, however, J = 0 → J′ = 1 transition scheme makes the dipole moment d of an atom to be parallel with the polarization of the incident light. The detuning between the light frequency ω and the atomic transition frequency ω₀ is defined as

Δ = ω - ω_{0}

and γ denotes the HWHM linewidth of the atomic transition. We use the limit of low light intensity in which saturation of the excited state is neglected, and the optical response of the sample is linear. The response of lattice atoms is isotropic for a J = 0 → J′ = 1 transition.

At the macroscopic level a dipole experiences the average field E_a. However, at the microscopic level it is clear that the electric field experienced by the dipole varies with position so that actual dipole within the lattices sees different electric field. Because of simplicity while making a reliable connection between macroscopic measurements and microscopic model, using local-field E_local = E_a +E_LOR has been a common practice in the mean-field description of a system of atoms interacting with light field. Here E_LOR is so called Lorentz field that results from polarization charge on the surface of a fictitious spherical cavity within a 3D material. However, there exists no Lorentz field E_LOR for materials in the lower dimensions, it is the case with our 2D lattices. It is obvious that such a mean-field approach cannot describe cooperative optical responses of a dense cold atom lattice because all positions between atoms are strongly correlated.

We aim to present numerically solutions to the atom and light interaction following the quantum electrodynamics formalism [10, 25, 26]. In the limit of low light intensity, exact quantum mechanical solutions to the problem are obtained, and they are equivalent with classical simulation results as already demonstrated in our previous work [21]. We have a lattice sample with N atoms at positions r_n(n = 1,…,N), where atoms are considered as damped oscillators weakly driven by the total external electric field

E (r_{n}) = E_{0} (r_{n}) + E^{S} (r_{n})

inducing dipole moment

d (r_{n}) = α E (r_{n})

for the isotropic polarizability α. Here E₀(r_n) is the incoming field and

E^{S} (r_{n}) = \sum_{m \neq n} G (r_{n} - r_{m}) d (r_{m})

the dipolar field from all the N − 1 atoms, where G is the monochromatic dipole field propagator that is a 3 × 3 matrix and describes the propagation of light from one atom to another. The familiar expression [22] for the components of G reads

G_{k l} (r) = \frac{1}{4 π ε_{0}} {\hat{e}}_{k} \cdot {k^{2} (\hat{r} \times {\hat{e}}_{l}) \times \hat{r} + [3 \hat{r} (\hat{r} \cdot {\hat{e}}_{l}) - {\hat{e}}_{l}] (\frac{1}{r^{2}} - \frac{i k}{r})} \frac{e^{i k r}}{r} .

Here ê_k and ê_l are the Cartesian unit vectors for k,l ∈ x,y,z and $\hat{r} = r / r$ is the unit vector pointing from the source point r_m to the field point r_n for the separation r = |r_n − r_m|. We have the explicit near fields (∝ 1/r² and ∝ 1/r³) in addition to the far-field radiation (∝ 1/r). Cooperative responses in our model originate from propagator matrix G, where microscopic near- and far-fields are included.

The present simulation focuses on stationary two-level atoms within 2D lattices. We do not consider oscillation and tunnelling into the other sites. This is valid if the atoms do not move substantially during the characteristic lifetime of the excited state and if the photon recoils are negligible. The total electric field driving the dipole at r_n is then described as

E (r_{n}) = E_{0} (r_{n}) + α \sum_{m \neq n} G (r_{n} - r_{m}) E (r_{m}) .

Note Eq. (7) is equivalent in quantum-mechanical and classical formalisms as shown in [27]. This makes a closed set of 3N × 3N linear equations for the Cartesian components of the field amplitude driving the dipoles at the positions r_n. In the codes, we consider atoms in the square and kagome lattices centered at the origin. We first assign N positions of the atoms at the fixed area density σ = 1/a² for square lattice and $0.5 \sqrt{3} / a^{2}$ for kagome lattice, respectively. Square lattices used in our numerical experiments have the same number of sites along the x and y directions. Because of different geometries of two lattices as depicted in Fig. 1, area density for square sample is greater than for kagome one at the same lattice constant a. Next, we modify the LU decomposition algorithms from Numerical Recipes [28] for the complex coefficients and solve Eq. (7) for the total amplitude E(r_n). Given the solutions E(r_n), we have dipole moment d(r_n) from Eq. (5) and the electric field amplitude at the field position r in the form

E (r) = E_{0} (r) + α \sum_{i} G (r - r_{i}) E (r_{i}) .

Fig. 1 Spatial configurations for a square lattice of N = 25 dipoles and a kagome lattice of N = 15 dipoles with lattice constant a.

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For a two-level atom the polarizability is

α = - \frac{D^{2}}{\bar{h} (Δ + i γ)} .

We use relation between the dipole matrix element $D$ and the transition linewidth γ,

γ = \frac{D^{2} ω_{0}^{3}}{6 π \bar{h} ε_{0} c^{3}} ≃ \frac{D^{2} k^{3}}{6 π \bar{h} ε_{0}},

where we take the resonant wavenumber k₀ and the actual wavenumber of light k to be the same

k_{0} = ω_{0} / c ≃ k

as it is true in the usual spectroscopy experiments.

To validate our numerical scheme, we derive shifts of the resonance line for isotropic infinite square lattice model. The electric susceptibility is calculated explicitly and reads

\frac{ε (ω)}{ε_{0}} - 1 = \frac{η_{k} (a)}{Δ + s_{k} (a) + i γ_{k} (a)} .

Here η_k(a) = [ig_k(a) − f_k(a)]γ, the shift of resonance Δ_shift = −s_k(a) is expressed as

Δ_{s h i f t} = - {Re [f_{k} (a)] + Im [g_{k} (a)]} γ,

and γ_k(a) = γ+ {Im[f_k(a)] − Re[g_k(a)]}γ where the structure functions are written in terms of infinite series as follows:

f_{k} (a) = \frac{3}{2} \sum_{n, m = - \infty \neq (0, 0)}^{\infty} [\frac{m^{2}}{{(n^{2} + m^{2})}^{3 / 2}} \frac{1}{k a} + \frac{2 n^{2} - m^{2}}{{(n^{2} + m^{2})}^{5 / 2}} \frac{1}{{(k a)}^{3}}] \exp (i k a \sqrt{n^{2} + m^{2}}),

g_{k} (a) = \frac{3}{2} \sum_{n, m = - \infty \neq (0, 0)}^{\infty} \frac{(2 n^{2} - m^{2}) \exp (i k a \sqrt{n^{2} + m^{2}})}{{(k a)}^{2} (n^{2} + m^{2})} .

The units we use in the simulation are

k = c = \bar{h} = \frac{1}{4 π ε_{0}} = 1.

In particular, the unit of length is expressed in terms of the free-space wavelength of the radiation λ as k⁻¹ = λ/2π. From Eq. (10), it follows that the numerical values of the dipole moment matrix element $D$ and the linewidth γ are related by

D = \sqrt{\frac{3 γ}{2}} .

When a monochromatic incoming light propagates through a medium whose absorption length is long compared to the resonance radiation wavelength, the optical thickness is expressed as

D = - \ln T

for the transmission coefficient of light T. Even if optical thickness has already been presented in [20] for different purposes, we have derived optical thickness D from a different perspective and used in our previous simulation [21]. Here we describe methods briefly. We take an observation point in the far field with x = 0,y = 0,z = ξ ≥ 1 and also assume that area of lattice plane A >> ξ². The average field for a dipole at the observation point is derived as

\bar{E} (ξ {\hat{e}}_{z}) = 2 π i \frac{d}{A} \exp (i ξ)

in the limit of far field ξ » 1. We derive the amplitude of the transmitted light E_T as a sum over dipolar fields at the position of dipole r_n given below

E_{T} = E_{0} + \frac{2 π i}{A} \sum_{n} [d (r_{n}) - {\hat{e}}_{z} \cdot d (r_{n}) {\hat{e}}_{z}] \exp (- i z_{n}) .

For our model with dipoles in the xy plane, transmission coefficient is

T = \frac{{| E_{0} + \frac{2 π i}{A} \sum_{n} d (r_{n}) |}^{2}}{{| E_{0} |}^{2}}

and the coefficient of absorption is A = 1 − T.

3. Numerical experiments

In this section, we describe setup of our numerical experiments and demonstrate main results. Samples with lattice constant a have a monolayer of area A perpendicular to the incoming light. We deposit different numbers of dipoles on square and kagome lattices. For experiments at constant densities, we pick the area densities in the range of σ = 0.5k² and 9.0k². These area densities in the real units are of the order of 10⁹ cm⁻²~10¹⁰ cm⁻² for the wavelength λ₀ = 780nm of the D2 resonance in Rb atoms. Conversion from area to volume number density ρ = N/V for a cubic lattice with the same edge lengths gives the order of 10¹⁴ cm⁻³~10¹⁵ cm⁻³, which can be achieved in the experiments [13, 14, 29, 30] for ground-state atoms these days. We express simulation results in terms of dimensionless units, which are detuning $\bar{δ}$ shift of, defined as resonance $\bar{s}$ at k = 1, lattice spacing ā, and area density $\bar{σ}$ , define as

\bar{δ} \equiv Δ / γ, \bar{s} \equiv Δ_{s h i f t} / γ, \bar{a} \equiv a k, \bar{σ} \equiv σ k^{- 2} .

Our main results to be presented are shifts of resonance line, broadenings of absorption line, and comparison of numerical shifts with analytical ones, which has been derived in Eq. (12), and experimental results [13]. The first two have to do with cooperative response, and the last for validating the present simulations. We first demonstrate shifts of a resonance line by computing optical thickness D of the samples illuminated by a Gaussian beam with a waist w = 4πk⁻¹. For both lattices the resonance lines shift toward lower frequencies in the high area density regime whereas the shifts of a resonance line are approximately zero in the low density regime. Figure 2 presents optical thickness D of a square lattice with N = 441 dipoles as a function of dimensionless detuning $\bar{δ}$ at the high area densities $\bar{σ}$ = 2.0, 3.0, 4.0, and 5.0 with the corresponding dimensionless lattice spacings ā ≈ 0.71, 0.58, 0.50, and 0.45. As cooperative effects set in with increased area density $\bar{σ}$ (decreased lattice spacing ā), resonance frequencies shift to red detuning. We did numerical experiments with N = 441 dipoles in kagome lattices at the same area densities with the corresponding spacings ā ≈ 0.66, 0.54, 0.47, and 0.42. Cooperative optical responses are conspicuously different from square lattices as shown in Fig. 3: The splitting in absorption lines are observed with high area densities at the fixed number of dipoles. The similar features have been demonstrated for a 2D kagome lattice with N = 47 dipoles at $\bar{σ}$ ≈ 0.13 [24]. We find the detunings at which the resonance lines occur depends on the lattice constant ā for the different number of dipoles. At the fixed lattice constant, we also found that the separation of two resonance lines are approximately the same, and they nearly coincide at the different number of dipoles. Further investigations are under way about kagome lattices at the high density regime. We also find that the magnitude of frequency shifts in square lattices, which have greater lattice constant ā than kagome lattices, is less than in kagome lattices. This suggests greater lattice spacing leads to less shifts of resonance line. Comparison of Figs. 2 and 3 demonstrates that the spatial arrangement of sample dipoles significantly affects the absorption lines. Difference in shifts and cooperative behaviors on both lattices are also true at the other high area densities used in our numerical experiments. These differences are explained by the separation dependence of the dipolar field given in Eq. (6). Due to the different spatial configurations of square and kagome lattices as shown in Fig. 1, each dipole in both lattices is driven by the different total amplitude of the external electric field, which equals the sum of the incoming field and emitted field from all the N − 1 dipoles. Therefore not only the number density(or lattice constant) but the arrangement of atoms and incoming fields make a difference in cooperative optical responses of the sample.

Fig. 2 Optical thickness D as a function of detuning $\bar{δ}$ in a square lattice sample with N = 441 dipoles for the dimensionless area densities $\bar{σ}$ = 2.0 (dashed line), 3.0 (dotted line), 4.0 (dot-dashed line), and 5.0 (solid line).

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Fig. 3 Optical thickness D as a function of detuning $\bar{δ}$ in a kagome lattice sample with N = 441 dipoles for the dimensionless area densities $\bar{σ}$ = 2.0 (dashed line), 3.0 (dotted line), 4.0 (dot-dashed line), and 5.0 (solid line).

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Figure 4 deals with the other aspect of cooperativity. Here the area density was fixed at $\bar{σ}$ = 2.0. We used the number of dipoles N = 121 (solid line), N = 441 (dashed line), N = 961 (gray line) for square lattice on the Left, and N = 125 (solid line), N = 434 (dashed line), N = 940 (gray line) for kagome lattice on the Right. Kagome lattice used for this numerical experiment has a rhombus-like shape, which differs from kagome lattice used in Fig. 3. The line shape broadens as dipole number increases in both lattices, while broadening is not substantial with increased density at the fixed number of dipoles as already depicted in Figs. 2 and 3. For the same lattice, shifts of a resonance line are approximately the same at different dipole numbers, which insinuates the frequency shifts depend more on the area density than on the number of dipoles.

Fig. 4 Optical thickness D at fixed number density $\bar{σ}$ = 2.0 as a function of detuning $\bar{δ}$ in (Left) a square lattice sample for N = 121, 441, and 961 dipoles and (Right) a kagome lattice sample for N = 125, 434, and 940 dipoles from top to bottom.

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We plot finally the shifts within square lattices in the high density regime as demonstrated in Fig. 5. The shifts are numerically calculated and plotted for samples with N = 64(Δ), 256(⊗), 441(⋄), and 784(×) dipoles at the lattice constant in the range of ā = 0.3 and 1.0. The prediction derived in Eq. (12) are plotted as dashed line. We also plot shifts extracted from experimental results [13] for slabs at the thickness of 90nm and 250nm as filled and empty circles, respectively. Considering the collisional shift is included in the experiment and the sample is not a real 2D system, our results are in good agreement with the experimental data. As the thickness of the slab is decreased, the experimental values approach to our numerical data and prediction. Simulation results also agree with the analytical ones. When light propagates through a dense low-temperature sample, dipoles are so strongly correlated that we include all orders of dipole-dipole correlations in this simulation. In our previous simulation [21] with 3D thin slabs of cold atomic vapors, LL shift, which is the MF prediction of a many-body system, was not observed. This states average fields cannot explain new phenomena on dense cold ensembles that are revealed in recent experiments [13, 14, 31].

Fig. 5 Numerical values for shift of the resonance line $\bar{s}$ in square lattices with N = 64(Δ), 256(⊗), 441(⋄), and 784(×) dipoles as a function of dimensionless lattice constant ā. The prediction of Eq. (12) are also plotted as dashed line and experimental data from [13] for slabs at the thickness of 90nm and 250nm as filled and empty circles.

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To validate the present simulation scheme further, we also found that the line shapes of optical thickness on square and kagome lattice at small area densities $\bar{σ}$ < 1 with many dipoles greater than N = 430 were exactly Lorentzian. Numerical experiments were repeated with a plane-wave beam, where cooperative features did not make a big difference with a Gaussian beam except that the absorption lines were narrower.

4. Conclusion

We have obtained numerical solution to light propagation in homogeneously broadened dense 2D lattices using classical microscopic electrodynamics. We emphasize that shifts of the resonance line in the lattice atoms are observed when atoms are in close proximity so that they reach sufficiently high number density. Even though dipole-dipole interactions induce strong correlations, because dipoles in this simulation are in regular spacing, the isotropic infinite lattice model allowed us to determine analytically the resonance shifts within a square lattice. Good agreement of the numerical data with the prediction confirms validity of numerical solutions for the cooperative optical response in the present and previous [21] simulations. We have also found different lattice geometries take an essential role in the absorption lines.

Recent experiments with dense and low-temperature samples allow us to access knowledge on cooperative behavior of atoms beyond the level at which we have acquired from the classical electrodynamics textbooks. We hope that our study provides a theoretical scheme for understanding not only system of cold atomic vapors but also other systems such as Rydberg atoms [32] and trapped ions [31].

Acknowledgments

This work is supported by the Hongik University new faculty research support fund. The authors are indebted to Prof. Juha Javanainen for his comprehensive guidance, discussion, and support. Sun Mok Paik is supported by Kangwon National University. We thank the Korea Institute for Advanced Study for providing computing resources (KIAS Center for Advanced Computation Abacus4) for this work.

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Cooperative optical response of 2D dense lattices with strongly correlated dipoles

Abstract

1. Introduction

2. Models and methods

3. Numerical experiments

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (21)

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