Free-space quantum interface of a single atomic tweezer array with light

TL;DR

This work tackles the limited interface efficiency between light and two-dimensional atomic tweezer arrays caused by the array’s multi-diffraction-order radiation. It introduces a multi-beam target mode that is a unique superposition of beams directed at all radiative diffraction orders, engineered from a single Gaussian input via spatial light modulators and a standard objective, turning diffraction losses into coupling and achieving near-unity efficiency. Analytical and numerical results show that for triangular lattices with $a/\lambda \lesssim 2$ and modest NA (e.g., $\mathrm{NA}=0.7$), the interface efficiency $r_0$ can exceed $0.99$ for $N=149$ and approach $0.9999$ for $N\sim 10^3$, with the inefficiency scaling as $1/N$ in the large-$N$ limit. The approach is robust to finite-size effects and atomic-position errors, suggesting practical, high-fidelity light–matter interfaces for current and future tweezer-array quantum technologies, including quantum memories and nonlinear optics with Rydberg states.

Abstract

We present a practical approach for interfacing light with a two-dimensional atomic tweezer array. Typical paraxial fields are poorly matched to the array's multi-diffraction-order radiation pattern, thus severely limiting the interface coupling efficiency. Instead, we propose to design a field mode that naturally couples to the array: it consists of a unique superposition of multiple beams corresponding to the array's diffraction orders. This composite mode can be generated from a single Gaussian beam using standard free-space optics, including spatial light modulators and a single objective lens. For a triangular array with lattice spacing about twice the wavelength, all diffraction angles remain below 35 degrees, making the scheme compatible with standard objectives of numerical aperture NA <= 0.7. Our analytical theory and scattering simulations reveal that the interface efficiency r0 for quantum information tasks scales favorably with the array atom number N: reaching >0.99 (>0.9999) for N = 149 (N approximately 1000) and scaling as 1 - r0 scales as 1/N for large N. The scheme is robust to optical imperfections and atomic-position errors, offering a viable path for quantum light-matter applications and state readout in current tweezer-array platforms.

Figures (7)

• Figure 1: Coupling light to a 2D tweezer array: triangular lattice. (a) For lattice spacings $a$ exceeding the wavelength $\lambda$, the uniform collective excitation of the array couples to multiple radiative diffraction orders ${\mathbf{m}}=(m_1,m_2)$ with reciprocal wavevectors $\mathbf{q}_{\mathbf{m}}$ (radiative orders for $2/\sqrt{3}<a/\lambda<2$ are shown). For coupling to a normal-incident field, only the order $\mathbf{m} = 0$ contributes (coupling rate $\Gamma_{0,0}\equiv\Gamma_{0}$), while the rest of the orders appear as losses (rates $\Gamma_{\mathbf{m}\neq 0}$). (b) The multi-beam target mode is composed of beams corresponding to all radiative diffraction orders, which now contribute to the coupling, thus yielding high coupling efficiencies. A beam component corresponding to order $\mathbf{m}$ is directed at an angle $\theta_{\mathbf{m}}$ with respect to the $z$ axis [Eq. (\ref{['Gm']})]. The transverse profile in the beam reference frame is an elliptical Gaussian with waists $w$ and $w\cos \theta_{\mathbf{m}}$ such that all beams form a single Gaussian of waist $w$ on the array plane. Here 3 beams out of the 7 orders from (a) are shown (with $\theta_{\mathbf{m}}\equiv \theta$ for $\mathbf{m}\neq 0$). (c) Setup for generating (collecting) the multi-beam target mode from (into) a single Gaussian beam at the input (output). Spatial light modulators (SLMs) convert the Gaussian beam into a set of beams, which are then directed through an objective lens to the array at the required angles $\theta_{\mathbf{m}}$. For a triangular lattice, a single standard objective of NA = 0.7 suffices.
• Figure 2: Infinite array theory: interface efficiency $r_{0}$ as a function of lattice spacing $a/\lambda$ for triangular (a) and square (b) arrays, considering a target mode comprising beams corresponding to the first diffraction orders $R_1$ in addition to the zeroth order $\mathbf{m} = 0$. When only this set is radiative (left of the vertical dashed line), the target mode perfectly overlaps with the array's radiation pattern yielding efficiency $r_0=1$, Eq. (\ref{['G']}). For larger $a/\lambda$, more radiative orders beyond $\mathbf{m}\in\{R_1,0\}$ emerge, and the efficiency drops to $r_0=\Gamma/(\Gamma+\gamma_{\mathrm{loss}})<1$, with $\Gamma=\sum_{\mathbf{m}\in R_1,0}\Gamma_{\mathbf{m}}$, $\gamma_{\text{loss}}=\sum_{\mathbf{m}\in R,0}\Gamma_{\mathbf{m}}-\Gamma$, and $\Gamma_{\mathbf{m}}$ from Eq. (\ref{['Gm']}) (text). (c,d) diffraction angle $\theta_{\mathbf{m}}$ of the first set of radiative orders, approaching $35^{\circ}$ ($45^{\circ}$) for the triangular (square) lattice at $a/\lambda \rightarrow 2$ ($a/\lambda \rightarrow \sqrt{2}$).
• Figure 3: Interface efficiency $r_{0}$ as a function of lattice spacing $a/\lambda$ for triangular (a) and square (b) finite-size arrays (atom number $N$). In both cases, we plot the region of $a/\lambda$ where only the set of the first diffraction orders and the zeroth order $\mathbf{m} = 0$ are radiative, taking the target mode containing all corresponding finite-waist beams. $r_{0}$ is evaluated numerically from scattering calculations of reflectivity (solid lines) and theoretically from Eq. (\ref{['r0']}) with (\ref{['Gu']}) (dotted lines). For each data point, the reflectivity is optimized with respect to the beam waist $w$ (Appendix B).
• Figure 4: Interface efficiency $r_{0}$ for different numerical apertures (NAs), plotted as a function of lattice spacing $a/\lambda$ of a triangular array with $N=149$ atoms. $r_0$ is evaluated numerically from the scattering reflectivity with additional NA filtering (text), and is optimized over the waist $w$ for each NA and lattice spacing $a$.
• Figure 5: Coupling inefficiency $1-r_{0}$ vs. the number of atoms $N$ in a triangular array, shown for NA = 1 (blue squares) and NA = 0.7 (red triangles). At each point, both the lattice spacing $a$ and the waist $w$ are optimized to maximize $r_{0}$ (evaluated numerically from scattering reflectivity). For NA = 1, the results are consistent with the favorable scaling $1/N$ as indicated by the fit performed for $N\geq 203$ (gray line). Similar scaling is observed for NA = 0.7 at large enough $N$.