Universal Approach for Quantum Interfaces with Atomic Arrays

We develop a general framework for the analysis of two-sided quantum interfaces, composed of collections of atoms interacting with paraxial light. Accounting for photon-mediated dipole-dipole interactions, our approach is based on the mapping of collective atom-photon interfaces onto a generic one-dimensional model of light scattering, characterized by a reflectivity parameter $r_0$. This entails two key practical advantages: (i) the efficiency of the quantum interface in performing various quantum tasks, such as quantum memory or entanglement generation, is universally given by $r_0$ and is hence reduced to a measurement or classical calculation of a reflectivity; (ii) the efficiency can be greatly enhanced by a properly designed photon mode that spatially matches a collective-dipole eigenmode of the atoms. We demonstrate our approach for realistic cases of finite-size atomic arrays, partially filled arrays, and circular arrays. This provides a unified approach for treating collective light-matter coupling in various platforms, such as optical lattices and optical tweezers.

Popular Summary

The ability to efficiently couple propagating photons to atomic systems is crucial for a variety of quantum applications. Of particular relevance is the creation of an interface between quantum light and ordered arrays of atoms, which recently emerged as prominent quantum platforms. This work provides a universal approach for the analysis of such atom-array interfaces by mapping them to a simple one-dimensional (1D) problem. The latter is fully characterized by a reflectivity parameter, and it is proven that this parameter determines the efficiency of quantum tasks such as quantum memory and photonic entanglement generation. Therefore, efficiencies of quantum applications are reduced by this approach to the classical calculation of a reflectivity.

While typical light-matter interfaces require the use of bulk or macroscopic systems to confine or absorb the light, it was recently realized that mesoscopic atomic arrays may also efficiently couple to light: for dozens of atoms in free space, their collective response can lead to strong reflection of light. The current work underscores the fundamental role of this array reflectivity by revealing that it forms the figure of merit of a variety of quantum applications. This idea is illustrated by our establishing the mapping of realistic cases of two-dimensional and three-dimensional arrays to the 1D model and deriving their reflectivity.

This provides a unified framework for the analysis of light-matter phenomena in timely systems such as optical lattices and tweezer arrays. Efficiencies of quantum protocols are estimated within the simple 1D model in terms of the reflectivity parameter, while the connection to realistic systems is established by the evaluation of their actual reflectivity, for example, calculated classically with accounting for various imperfections.

Figure 1

Generic 1D model of a two-sided light-matter interface [Eq. (1)] onto which various systems are mapped. (a) A collective dipole $\hat{P}$ of the atomic system is coupled to a spatially matched, target photon mode $\hat{\mathcal{E}}$ at an emission rate $\mathrm{\Gamma }$, while scattering to other modes (loss) at rate ${\gamma }_{\mathrm{loss}}$. The atom-photon coupling is taken to be symmetric on both sides [Eq. (4)]. (b) Level structure of the atoms in the three-level variant of the model, with a tunable coupling field $\mathrm{\Omega }(t)$ [Eq. (6)]. (c) Ladder-configuration version of three-level atom scheme.

Figure 2

2D ordered array. A finite-size array with lattice spacing $a$ is situated on the $x$-$y$ plane ($z=0$). The target-mode input consists of the symmetric combination of fields from both sides, ${\hat{\mathcal{E}}}_{0,\pm }$. The full free-space multimode problem is mapped to the generic model in Fig. 1, with the relevant field and collective dipole modes projected to the target-mode profile [Eqs. (16) and (19)] and with the effective parameters $\mathrm{\Gamma }$ and ${\gamma }_{\mathrm{loss}}$ from Table 1.

Figure 3

Sources of imperfection and loss in an ordered atom array. (a) Disorder $\delta r$ in atomic positions around the perfect lattice points leads to an effectively individual-atom scattering to undesired modes at rate ${\gamma }_{\mathrm{dis}}$. (b) The finite size of the array (linear size $L_a=\sqrt{N}a$) leads to its nonideal overlap with the target mode, $\eta ={\mathrm{erf}}^2(L_a/\sqrt{2}w)<1$, and hence to a fraction ($1-\eta$) of the collective emission that spills over to undesired modes. The error scalings induced by these loss mechanisms are summarized in Table 2.

Figure 4

Error (inefficiency) $1-r_0$ in an ordered 2D array interface coupled to a Gaussian-beam target mode, estimated from independent numerical calculations of both the scattering (reflectivity) and quantum memory efficiency (see Sec. 4c). Agreement with the predicted analytical scaling in Table 2 is exhibited, in correspondence to the dominant source of imperfection. (a) Effect of finite array size: $1-r_0$ as a function of the atom number $N=L_a^2/a^2$ for a perfectly ordered array. The error $1-r_0$ decreases exponentially with $N$, as predicted analytically from the error-function behavior of the overlap $\eta$ between the array and the Gaussian target mode. Here the waist and lattice spacing were fixed to $w/\lambda =4.8$ and $a/\lambda =0.6$, and the reflectivity was extracted from scattered fields calculated at distance $z=5\lambda$ from the array. The corresponding quantum memory error is obtained from the calculation of the retrieval fidelity [31] with an identical initial Gaussian profile. (b) Disorder imperfection: the error $1-r_0=1/(C+1)\approx 1/C$ plotted as a function of the standard deviation $\delta r$ around perfect 2D array positions (averaged over $50$ disorder realizations). The linear fit to the log-log plot in both the reflectivity and the memory calculations verifies the scaling $1/C={\gamma }_{\mathrm{dis}}/{\mathrm{\Gamma }}_0\sim (2\pi \delta r/\lambda {)}^2$ predicted analytically for a large enough array with respect to the incident-beam waist, $L_a=\sqrt{N}a>w$ (see Table 2 and the main text). Numerical parameters: $w/\lambda =4.8$, $a/\lambda =0.6$, and atom number $N=30\times 30$. Slight deviations between the reflectivity and memory errors arise mainly from the $z$ component of the position disorder, which breaks the symmetry between left-scattered and right-scattered light.

Figure 5

Paraxial dipole eigenmodes, Eq. (21), of a realization of a partially filled 2D array with $30\times 30$ lattice sites, lattice constant $a/\lambda =0.6$, and filling fraction (a) $0.85$ and (b) 0.95. The top panel displays the most paraxial eigenmode $v_{0,n}$ represented on the discrete array sites. The middle panel shows the corresponding continuous function $u({\mathbf{r}}_{\perp })$ constructed from the interpolation of the eigenmode, and chosen as the target photon mode. The Fourier transform of the mode $u({\mathbf{r}}_{\perp })$ is shown in the bottom panel: we note its paraxial character evident by its small bandwidth around ${\mathbf{k}}_{\perp }=(k_x,k_y)=(0,0)$.

Figure 6

Boosting the efficiency using dipole-eigenmode target modes. Efficiency (reflectivity) $r_0$ of a partially filled 2D array as a function of the filling fraction for a randomly filled square lattice of $30\times 30$ sites and lattice constant $a=0.6\lambda$ averaged over ten realizations. The theoretically expected value $r_0=\mathrm{\Gamma }/(\mathrm{\Gamma }+{\gamma }_{\mathrm{loss}})$ (solid red line), determined from Eqs. (28) and (30) for each realization, exhibits excellent agreement with direct numerical calculations of both the scattering reflectivity (blue circles) and the quantum memory efficiency (black crosses): in the numerical calculations the target mode is taken as the most paraxial eigenmode constructed as in Fig. 5. High efficiencies $r_0>0.99$ and $r_0>0.94$ are observed for filling fractions as low as $0.85$ and $0.70$, respectively. In comparison, we plot the numerically calculated reflectivity and quantum memory efficiency for a Gaussian-beam target mode (with a waist optimized to the perfectly filled array), which are seen to reach much lower values, demonstrating the superiority of the eigenmode approach suggested by the theory.

Figure 7

Two examples (a) and (b) of paraxial eigenmodes in the circular array described in Sec. 6. The top panel presents the eigenmodes $v_{l,n}$ on the array sites, the middle panel shows the corresponding continuous functions $u_l({\mathbf{r}}_{\perp })$ constructed from the eigenmodes by interpolation, and the bottom panel displays the Fourier transform of these functions. The paraxial character of the modes is exhibited by their narrow bandwidth in Fourier space around ${\mathbf{k}}_{\perp }=(k_x,k_y)=(0,0)$.

Figure 8

(a) A 3D ordered atomic array is described as consisting of $N_z$ layers, each being a 2D array (here $N_z=3$). (b) Cooperativity $C=\mathrm{\Gamma }/{\gamma }_{\mathrm{loss}}$ and error $1-r_0=1/(1+C)$ versus the number of layers $N_z$ of a phase-matched array [$a_z=(\lambda /2)\times \mathbb{N}$]. For the parameters chosen ($a/\lambda =0.6$, $N_{\mathrm{\perp }}=30$, $a_z/\lambda =1$, ${\gamma }_s=\gamma$, and $w/\lambda =6$), the individual decay dominates the loss, ${\gamma }_{\mathrm{loss}}\approx {\gamma }_s$, so $C$ increases linearly with $N_z$ (see Table 1). Excellent agreement between the analytical theory [Eq. (41)] and numerical scattering calculations is exhibited.

Figure 9

Multilayer system represented as a series of building blocks (optical elements), each comprising a single layer followed by a propagation over distance $a_z$. The input and output are divided into forward and backward propagation on the left-hand and right-hand sides of the multilayer system.

Figure 10

Intensity transmittance ($T$) and reflectance ($R$) as a function of the number of layers $N_z$ for interlayer spacing (a) $a_z/\lambda =0.5$ and (b) $a_z/\lambda =0.25$. Solid and dotted lines represent transfer-matrix numerical calculations and theoretical analytical predictions, respectively (see the text). In both plots the ratio between individual losses and collective emission to the target mode of each layer is taken to be ${\gamma }_s/{\mathrm{\Gamma }}_{1\mathrm{D}}=10$.

Figure 11

Intensity reflection coefficient $R$ of a 3D multilayered array as a function of detuning ${\delta }_p-\mathrm{\Delta }$ and the ratio $a/\lambda$ for (a) $a_z=\lambda$, and (b) $a_z=\lambda /2$. The detuning correction ${\mathrm{\Delta }}^{\prime }$ (dashed line) fits the maximal reflectivity. The reflectivity was calculated for an array with $N_z=10$, $\eta =1$, and ${\gamma }_{\text{loss}}=0.05{\mathrm{\Gamma }}_0$.