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Abstract
Manipulation of light using atoms plays a fundamental and important role in emerging technologies such as integrated photonics, information storage, and quantum sensors. Specifically, there have been intense theoretical efforts involving large samples of cold neutral atoms for coherent control of light. Here we present a theoretical scheme that enables efficient computation of collective optical responses of mono- and bi-layer planar square lattices of dense, cold two-level atoms using classical electrodynamics of coupled dipoles in the limit of low laser intensity. The steady-state transmissivity and reflectivity are obtained at a field point far away from the atomic lattices in the regime with no Bragg reflection. While our earlier method was based on exact solution of the electrodynamics for a small-scale lattice, here we calculate the dipole moments assuming that they are the same at all lattice sites, as for an infinite lattice. Atomic lattices with effectively over one hundred times more sites than in our earlier exact computations can then be simulated numerically with fewer computational resources. We have implemented an automatic selection of the number of sites under the given convergence criteria. We compare the numerical results from both computational schemes. We also find similarities and differences of a stack of two atomic lattices from a two-atom sample. Such aspects may be exploited to engineer a stack for potential applications.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Due mainly to the modification of spontaneous emission of resonant emitters [1,2], the optical response of an assembly of emitters in close proximity is either collectively reduced (subradiant) [3–9] or enhanced (superradiant) [10,11]. The geometry of the emitters [12] is essential to engineering collective optical responses [13–17] and dipole-dipole interactions. In particular, a sample of $N$ identical cold atoms in a regular array is an ideal platform for engineering optical effects, for example, collective reflection and transmission of light through the sample. Such arrays of atoms can be experimentally realized in optical lattices [18–21], dipole traps [22], two-dimensional (2D) atomic structures [23,24], one-dimensional (1D) array of optical tweezers [25], photonic crystal waveguides [11], and holographic microtraps [26]. Recent theoretical analysis has revealed that 2D dipolar arrays can be exploited as an almost perfect mirror [14,27–32]. Other applications include controlled light manipulation [7,18,29,33], cooperative optical properties [18,30,34–36] and antennae [37], topological quantum optics [32,38–40], quantum information processing [4,41], testing condensed-matter models [42], and the emulator [31] of regularly spaced atoms in a lossless 1D waveguide.
Theoretical treatments [30–32,34,43,44] where coupling of dense cold atoms is fully taken into account are scarce, whereas different approximations to dilute cold atom samples [45] are widely employed. Likewise, direct simulation results for large atomic lattices are not many because of computational issues. As a continuation of the earlier effort [30,34], here we discuss a faster numerical scheme that can obtain dipole moments in larger-scale lattices.
We study the collective optical response of cold atoms using classical electrodynamics of coupled dipoles. The atoms are confined either to a single planar square lattice with the lattice spacing $a$, or to a stack of two lattices separated by $\Delta z$ along the direction of light propagation. In the limit of low light intensity such a classical approach is valid even for the simulation of quantum systems, and, while we do not go into this, could properly include the quantum fluctuations of atomic positions [32,44]. However, the classical electrodynamics approach cannot simulate a sample at high atomic densities (small lattice spacings) with saturation light intensity as discussed in [32]. Assuming a homogeneous infinite lattice [34], we obtain the expression of the identical dipole moments induced by a circularly polarized incident light at all sites, and numerically simulate optical depth, transmissivity, and reflectivity of the atomic lattices at a far-field point. We mainly consider the regime of the lattice constant $a$ where Bragg scattering is absent, and the lattice-to-lattice distance $\Delta z$ is large compared to $a$.
The main improvements of the present numerical scheme, initially introduced in [31], are faster computation, larger total number of sites $N$ in a lattice plane, automatic selection of $N$, and algorithm for computing the relevant sums that incorporates Richardson extrapolation [46]. We compare the present numerical results with the ones from exact electrodynamics obtained for a finite-size lattice; specifically, the optical density versus detuning of the light from resonance. Both methods are in good agreement about collective shifts of the resonance line, but the line shapes display substantial deviations.
We also obtain an analytical expression of the transmissivity of a stack of two lattices. We have developed an analogy [30] of a single atomic lattice with a single superatom acting as a mirror. By extension, a stack of two atomic lattices acts as if it were a pair of superatoms. The stack provides a rudimentary cooperative system of atoms in a one-dimensional channel of light, but with the advantage that the reflection and transmission characteristics of each atom can be engineered by varying the lattice constant.
2. Light-driven strongly coupled atoms in lattices
We apply an exact classical electrodynamics formalism of coupled dipoles [30,31,34,43,47] to a lattice of a large number of identical atoms in order to understand the collective optical features. For the time being we discuss the new computational methods, and proceed as if the lattice were infinite. The present scheme, different from our previous ones [30,34], is adapted from Ref. [31], so we only enumerate some key points without derivations.
The atoms are confined to a 2D lattice with the lattice constant $a$. We neglect position fluctuations and motion of the atoms, light pressure, and photon recoil effects. The driving light comes to the lattice at normal incidence. We take a particular atomic transition from $J_g = 0$ to $J_e = 1$ in the limit of low light intensity, when saturation of the excited state population may be neglected. Thus the optical response of an induced dipole in the sample is linear and isotropic. All of our analysis is for the steady-state of the system, which prevails a long time after the light has been turned on.
Time-dependent quantities such as electric field and dipole moment are written in terms of the slowly-varying positive-frequency quantities, without the explicit oscillations $e^{- i\omega t}$ at the frequency $\omega$ of the driving light. The dimensionless detuning between the light frequency and the atomic transition frequency $\omega _0$ is defined as $\delta = (\omega -\omega _0)/\gamma$, given the HWHM linewidth $\gamma$ of the optical transition. We use custom units in which the numerical values for the wave number of the incident light and for certain natural constants are
The unit of length is $k^{-1}$, and the units of quantities related with length are determined accordingly. The polarizability for an isotropic two-level atom in our units is2.1 Dipolar field
The planar square lattice, aligned with the $x$ and $y$ axes, contains one atom per site, and is driven by a circularly polarized incident plane-wave of light
2.2 Scalar dipolar field in a single planar square lattice
Due to the circularly polarized dipole moments with the amplitude $d$ in the plane of the lattice, the dipolar field is also circularly polarized, and its expression at all sites can be written in a scalar form
where the self-sum is defined as [31]In our earlier computations [34] the exact solution for the electrodynamics of the atoms was found site by site under the approximation that there is a finite number $N$ of lattice sites, instead of the ideal infinite lattice. Here the approximation is to compute the sum (10) for finite values of $M$ and $\eta$, instead of the limits $M\rightarrow \infty$ and $\eta \rightarrow 0+$. The values of the parameters are determined using a Richardson extrapolation [46] method [31] in such a way that the relative error from the idealized limits is less than a specified bound, typically $10^{-3}$. We occasionally cite the number of the lattice sites with $0 < |\mathbf {n}|\le M$ as a representative of the number $N$ of lattice sites involved in the new scheme.
2.3 Scalar dipolar field in a stack of two lattices
As the first step to obtain the expression of the dipolar field emitted by a single lattice as a whole and detected at a field point outside the lattice, we consider a stack of two identical lattice planes separated by a distance $\Delta z$. By virtue of the translation invariance of the lattices, the dipolar field of the first lattice is the same at each site of the second lattice, and circularly polarized. We can therefore express the dipolar field in a scalar form
Here a transfer sum initially introduced in [31] reads3. Analytical calculations
General expressions of reflectivity and transmissivity of light for two model systems will now be developed by applying the theoretical treatment described in Sec. 2 in the limit of far field, or small lattice constant.
3.1 Collective optical response of a single lattice
Let us first consider the response of a single infinite lattice to an incident plane wave of light propagating perpendicularly to the lattice plane. All atoms have identical induced electric dipole moments with the amplitude $d=\alpha E_{on}$, where $E_{on}$ is the electric field acting on each atom. The center of the lattice at the origin serves as the representative point for the calculations.
The self-consistent total field acting on an atom is the sum of the incident field and the dipolar fields of the other atoms,
Substituting the dipole moment strength $d = \alpha E_{on}$ and Eq. (2) into Eq. (14), we obtain the expression of the dipole moment where the effective collective detuning and HWHM transition linewidth are These collective quantities characterize the cooperative optical response of the lattice. They depend on the geometry of the lattice, which determines the light-mediated dipole-dipole interactions between the atoms. Equation (16) implies that the presence of the lattice causes the transition frequency of the atoms to shift by $-\frac {3}{2}\Re [S_S(a)]$.Let us consider the total electric field detected at a field point $z$ far downstream from the lattice plane. In such a regime of far field $z \gg a$, the electric field reads
We may obtain an explicit expression of the transmitted field by inserting Eq. (15) into Eq. (13) and Eq. (18). We find the field reflected from the lattice analogously, and finally the corresponding amplitudes for light transmission and reflection:As has been argued in [31], at the detunings
for which $\Delta = 0$, and at any lattice constant $a \in (0, \lambda )$, no light gets transmitted through the lattice. Figure 1 demonstrates a reason why the transmission coefficient becomes zero in the regime of $a \in (0, \lambda )$ at the detuning expressed in Eq. (21) by plotting the function This is the imaginary part of the numerator of the transmission coefficient Eq. (19). In the regime of $a \in (0, \lambda )$, $Imt$ equals zero, implying $\Gamma = -\frac {3}{2}i s_T(a)$, resulting in $t_f(\delta ,a)=0$ and $r_f(\delta ,a)=-1$ for the detunings given in Eq. (21). The materially same observation has been made elsewhere [28].
Fig. 1. Imaginary part of the numerator $Imt$ of the transmission coefficient $t_f$ in the far-field regime for a single square lattice as a function of the lattice constant $a$. The plot exhibits $Imt$ equal to zero for $a \in (0,\lambda )$, demonstrating a regime of $a$ where light can get completely reflected from the lattice.
The spikes in Fig. 1, divergences although truncated by numerical resolution, occur at the lattice spacings when a new order of Bragg scattering set in. To avoid Bragg scattering, we restrict the interatomic separation to the interval $a \in (0, \lambda )$ unless otherwise indicated. Thus, at the particular detuning given in Eq. (21), an incident beams gets completely reflected, and no light is transmitted through the lattice. Furthermore, with $a \in (0, \lambda )$, since $Imt$ equals zero for a lattice with any lattice constant, the general expressions for the reflection and transmission coefficients of a single lattice have the same forms as those of a single atom in a one-dimensional channel of light, such as an idealized optical fiber that leaks no light. The expressions are
Next, the respective general expressions of the transmissivity, reflectivity, and optical depth valid for any lattice constant $a$ in the far-field regime, are
3.2 Collective optical response of a stack of two lattices
We next consider two identical square lattices with the lattice constant $a$, displaced from one another by the distance $\Delta z$ in the direction perpendicular to the planes of the lattices. We label the lattices by 1 and 2. The lattices are driven by a circularly polarized plane wave of light at normal incidence. All atoms in each lattice will acquire the same circularly polarized dipole moment. We will denote the scalar amplitudes of the dipole moments by $d_1$ and $d_2$. The electric field on the atoms is also the same at all sites of each lattice. We label the amplitudes by $E_1$ and $E_2$.
The amplitude of the dipolar field at the position $z_n$ ($n = 1$ or 2) of the lattice $n$ from the dipole moment $d_m$ ($m = 2$ or 1) of the lattice $m$ at $z_m$ is
One can enumerate all electric fields (driving light, self-fields of each lattice, transmitted dipolar fields between the lattices) and dipole moments along these lines, and the relation (2) leads to a unique solution for all unknown quantities. To discuss the results, we introduce the reflection coefficient valid for an arbitrary distance $\Delta z$ from one lattice The transmissivity of the stack (at a field point far behind the stack) is correspondinglyLet us consider the limit of a small lattice constant $a \ll |\Delta z|$. The transfer sum $S_T(a,\Delta z)$ approximately equals its small lattice constant form $s_T(a)$, so that the reflection coefficient $r_l(\delta , a, \Delta z)$ almost equals its far-field form $r_f(\delta , a)$ in Eq. (20). Then the first three terms in the numerator of Eq. (29) can be factored into $(1+r_f)^2$ that equals $t^2_f(\delta , a)$, and the last term becomes zero. The transmissivity reduces to the far-field and small lattice constant form
Summarizing, in the limits of far field ($z \gg a$) and a small lattice constant ($\Delta z \gg a$), two geometrically identical lattices in the stack can be considered as optically identical “superatoms.” However, the parameters of the superatom governing the optical response, transition frequency and linewidth, depend cooperatively on the lattice constant.
4. Numerical simulations
Here we compute collective transmissivities of monolayer and bilayer lattices of atoms. The objective is to approach the limit of a large lattice. In our earlier method the response of each individual atom was solved from exact electrodynamics for a lattice as large as practicable [30,34], and the radiation from the lattice(s) was analyzed using an approximation that speeds up the convergence to the large-sample limit [52]. The present method [31] is based on the assumption that the dipole moments are the same at each lattice site, as they would be in an infinite lattice. The self-sum and the transfer sum for large enough upper limit $M$ approximate the infinite sums, enabling numerical simulations of an infinite homogeneous lattice.
4.1 Comparison of the theoretical schemes
We compare the two theoretical schemes by computing the optical depth $OD(\delta ,a)$ of a monolayer square lattice, as observed at a far-away field point, as a function of a dimensionless detuning $\delta$. See Fig. 2. Results from the current and earlier schemes are given as dashed and solid lines, respectively. The earlier method takes a lattice with $N = 35\times 35 = 1225$ sites, at the respective lattice constants $a = 0.5 \lambda$ (blue lines) and $0.803 \lambda$ (red lines), while the current one invokes lattices with the effective numbers of atoms $N = 344978$ for $a = 0.5 \lambda$ and $N = 134028$ for $a = 0.803 \lambda$. These numbers are automatically determined during the computation. While the detunings giving the maximum absorption are nearly the same for both computational methods, the optical depth obtained for the small-scale lattice with $1225$ sites deviates significantly from the results of the new scheme. The proximate reason is the small size of the lattice, limited by the computational resources at our disposal. The new scheme took into account two orders of magnitude more lattice sites, and ran for less than 1/10 of the CPU time. Thus, the current theoretical scheme is robust, and beneficial for analyzing the optical properties of large-scale atomic lattices.
Fig. 2. Optical depth $OD$ in monolayer lattices of lattice constants $a = 0.5 \lambda$ (blue lines) and $0.803 \lambda$ (red lines) as a function of the dimensionless detuning $\delta$. Dashed lines display the results from the model of a homogeneous infinite lattice, and solid lines the exact classical electrodynamics results for a finite-size lattice with $N = 1225$ atoms.
4.2 Simulations for a stack of two lattices
We first simulate numerically bilayer lattices in the regimes of far field and small lattice constant. Figure 3 shows the transmissivities, Eq. (30), as a function of the detuning $\delta$, for a stack of two lattices with a fixed interlattice separation $\Delta z = 0.075 \lambda$, for different lattice constants $a$. The separation obviously is not large compared to the lattice constants. However, in the present limiting case the optical properties of the stack are periodic in the separation with the period $\pi$, which is the value of $\lambda /2$ in our units. In a physical realization the separation could be $\Delta z = 0.075 \lambda + n \lambda /2$ with a large enough $n$ that the asymptotic limit $\Delta z\gg a$ is reached, and the results of the calculations for $\Delta z = 0.075 \lambda$ should be accurate.
Fig. 3. Transmissivity $T$ as a function of the detuning $\delta$ for a stack of two planar square lattices with the same interlattice separation $\Delta z = 0.075 \lambda$, and the lattice constants $a = 0.202 \lambda$ (blue line), $0.803 \lambda$ (red line), and $0.425\lambda$ (dash-dotted blue line).
At $a = 0.202 \lambda$ (blue line) and $0.803 \lambda$ (red line), the transmissivities are zero for on-resonance excitation, $\delta = 0$. This is as expected, because at these lattice constants there is no cooperative resonance shift even for a single lattice. In contrast, for $a = 0.425 \lambda$ (blue dash-dotted line) the zero transmission is shifted from $\delta =0$. Fano resonances, which give a window with unit (as far as we can tell) maximum transmission, are distinctly notable in all curves. Such lineshapes have been pointed out [30] as signatures of the cooperative response of the lattices to the driving light. In fact, similar plots are shown for a two-atom case [44], revealing the connection of light propagation in a single atom and a monolayer lattice of atoms.
Cooperative features are found even for atomic area density $\sigma \, (=a^{-2}) \ll 1$ for each lattice; for $a=0.802\lambda$ we have $\sigma =0.04$. Broadening of the resonance line is inversely proportional to the square of the lattice constant $a$, as predicted in Eq. (16). It is evident that the lineshapes of transmissivity (and reflectivity) vary depending on the lattice geometry. However, when the optical lineshapes are plotted as a function of the scaled parameter $\Delta /\Gamma$, the cooperative effects within each lattice are scaled away, and all three graphs in Fig. 3 fold into the same curve [31]. In fact, it can be verified explicitly that Eq. (30) depends on the parameters $\Delta$ and $\Gamma$ only through the combination $\Delta /\Gamma$.
We next proceed to the case when the separation between the two lattices varies while the lattice constant is fixed. In an earlier study [30] we have analytically calculated the transmissivity and optical depth for a sample of two point-like atoms in an effectively one-dimensional channel for light. Here we explore numerically optical analogies and differences between a two-atom and a two-lattice sample further.
In Fig. 4 we compute the the transmissivity $T(\delta , a, \Delta z)$ of the lattice stack as a function of the detuning $\delta$ from Eq. (29) valid for an arbitrary lattice separation (solid lines). The lattice constant is $a = 0.803 \lambda$, and the separation of the lattices has two values, $\Delta z = 0.500 \lambda$ (solid red line) and $\Delta z = 0.450 \lambda$ (solid blue line). The similar transmissivities in Fig. 1(c) of [44] are obtained in two-atom waveguides with the interatomic separations $x_{12} = 0.500 \lambda$ and $x_{12} = 0.450 \lambda$, respectively. Such a feature is also the evidence for the similarity of two-atom and two-lattice samples in light propagation. For this lattice constant there is no shift of the resonance, but the cooperative linewidth is 0.245 times the linewidth of an individual atom. In the computations of the two-atom data we therefore use the linewidth $\Gamma = 0.245$ for the dimensionless detunings of the two atoms, by dividing $\delta$ with $\Gamma = 0.245$. The transmissivity $T^{(2)}$ of two atoms obtained in this way is the same as the one computed from Eq. (30). It is also plotted for the separations $\Delta z = 0.500 \lambda$ (dashed red line) and $\Delta z = 0.450 \lambda$ (dashed blue line).
Fig. 4. Solid lines: Transmissivities $T$ as a function of the detuning $\delta$ for stacks of two lattices with the same lattice constant $a = 0.803 \lambda$, but separated by different distances $\Delta z = 0.450 \lambda$ (solid blue line) and $0.500 \lambda$ (solid red line). Dashed lines: Transmissivities $T^{(2)}$ as a function of $\delta$ for two-atom samples whose atomic separations are $\Delta z = 0.450 \lambda$ (dashed blue line) and $0.500 \lambda$ (dashed red line).
We know from earlier work on two-atom and two-lattice systems [30] that for a distance $\Delta z$ that is close to, but not exactly, an integer multiple of half of the wavelength, there tends to be a sharp Fano resonance in the line shape. This feature is seen again in Fig. 4, in that there is a Fano resonance in the blue curves for $\Delta z = 0.450 \lambda$, but not in the red curves for $\Delta z = 0.500 \lambda$. However, while in the limit of a large interlattice separation the two-atom and two-lattice models would give identical results (same for adjacent solid and dashed lines), it is not the case in Fig. 4. The difference is particularly notable in the position and shape of the Fano resonance in the blue curves for $\Delta z = 0.450 \lambda$. The distance between the lattices is a parameter that can be used to tune the optical response of the lattice stack above and beyond what can be done with two atoms. The analogies and differences between a stack and a two-atom sample may be exploited to engineer stacks of two lattices for applications such as quantum information processing [4], emulator of atoms in a lossless 1D waveguide [31], and nanometer-sized 2D atom laser [36].
5. Conclusion
We have derived the expressions of transmissivity and reflectivity of low-intensity plane wave of light propagating at perpendicular incidence through a stack of two identical planar square lattices of two-level atoms. Our theoretical scheme is developed from a classical treatment of the light that mediates a strong interaction between the induced dipoles. The solution of the coupled equations for the dipole moments at the sites of an assumedly infinite lattices involves two sums determined by the geometry of the lattices, and may in part be characterized in terms of an effective collective detuning and linewidth. Mostly in the limits of far field and large interlattice separation (or small lattice constant), we have obtained numerical values of light transmissivity by computing such sums [31].
One of the advantages of the present numerical scheme over our previous one [34] is the automated selection of the total number of lattice atoms from a preset convergence criterion. By comparing with earlier finite-size lattice calculations, we have gained insights into the effect of the finite size of the lattice on the results. In the case when the separation between the lattices is large compared to the lattice constant, the new simulation results quantitatively confirm our proposal that a stack of two planar atomic lattices may be modeled as two effective superatoms [34]. When the distance is not large, the lattice system offers additional flexibility to engineer the optical response.
The realization of artificial systems of large-scale quantum bits that can be individually addressed is one of the great challenges for applications in quantum computation. Trapped systems of atoms [53] and ions [54] are such systems, along with those involving superconducting circuits [55] or quantum dots [56]. Our classical electrodynamics formalism may provide a theoretical method to understand another candidate systems for quantum technologies with modest computational effort.
Funding
National Research Foundation of Korea (2019R1F1A1043770); Hongik University (2016 Research Fund).
Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)(2019R1F1A1043770) and 2016 Hongik University Research Fund. We thank the Korea Institute for Advanced Study for providing computing resources (KIAS Center for Advanced Computation Abacus4) for this work.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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