Unconventional Distance Scaling of Casimir-Polder Force between Atomic Arrays

TL;DR

The paper shows that Casimir-Polder forces between intrinsically discrete atomic arrays can violate the usual retardation-induced faster decay seen in continuous bodies, due to the lattice spacing $a$ introducing an extra length scale. A microscopic scattering Green's-function framework is developed for two parallel 2D atomic arrays, with Weyl-decomposition and k-space methods to treat Bravais lattices and multi-sublattice geometries, and is extended to Rydberg arrays. The force on ground-state arrays crosses from a short-distance $F_{CP}\propto -1/h^7$ to a retarded-domain $F_{CP}\propto -1/h^6$, while Rydberg arrays can exhibit even stronger deviations (e.g., nonretarded $F_{CP}\propto -1/(a^2 h^5)$). The authors also propose a realistic measurement scheme using a single Rydberg atom near a 2D Rydberg array to directly observe this unconventional scaling, highlighting a new route to explore dispersion forces beyond the continuum limit.

Abstract

Conventionally, dispersion forces mediated by quantum vacuum fluctuations are known to exhibit universal distance scalings, with retardation typically leading to a faster decay of the interaction. Here, we show that this expectation fails for intrinsically discrete systems. Using the microscopic scattering approach, we study the Casimir-Polder interaction between two atomic arrays, and uncover an unconventional distance scaling in which the force crosses over from a faster decay at short separations to a slower decay in the retarded regime. This behavior originates from the discrete lattice structure and can be consistently understood within the scattering picture. Extending our analysis to Rydberg atomic arrays, we predict an even stronger deviation from conventional scaling and propose an experimentally feasible scheme for direct measurement. Our results provide a new platform for exploring dispersion forces beyond the continuum limit.

Figures (3)

• Figure 1: Illustration of the Casimir-Polder force between two 2D atomic arrays. The inset illustrates a quantum fluctuation that excites a two-level atom marked by the red circle in Array 1 and emits a virtual photon (dashed wavy arrow), which can be scattered by atoms in the arrays. Solid colored wavy lines indicate the virtual photons scattering between atomic arrays.
• Figure 2: (a) CP or Casimir force per atom between various objects. White dash-dot line: between two conducting plate (rescaled by $10^{-4}$). Black dashed line: between two atoms. The solid lines from dark to light purple: between two ground-state Rb atomic arrays with lattice constant $a = 0.1\lambda_0$, $0.3\lambda_0$, $0.5\lambda_0$, and $0.9\lambda_0$ where $\lambda_0 = 780.2\,\mathrm{nm}$ for transition from $\mathrm{5S_{1/2}}$ to $\mathrm{5P_{3/2}}$. Numbers next to each curve indicate the power-law scaling exponent in the corresponding distance regime. Inset: the CP force between two arrays of Rydberg atoms, computed for the $53D_{3/2}\rightarrow 52F_{5/2}$ transition with the corresponding wavelength $\lambda_0 = 1.913\times10^{-2}\,\mathrm{m}$ and $\mu = 1.491\times10^{-26}\,\mathrm{C\cdot m}$. The horizontal axis and the vertical axis are both omitted since they are identical to that in Fig. 2(a). The lattice constant is $a = 7000\,\mathrm{nm}$. (b) Scaling diagram of the CP force between atomic arrays.
• Figure 3: (a) Sketch of atom-array system. Gray beams indicate optical trapping potentials, while red beams represent the additional modulated potential $U(t) = U_0(1+\epsilon \cos \Omega t)$. (b) Rydberg-atom excitation process, with the $70D_{5/2} \rightarrow 71P_{5/2}$ transition shown as a gray dashed line representing the virtual-photon excitation process. (c) Simulated parametric resonance of a single Rydberg atom positioned above a 2D atomic array. The plotted quantity is the atomic displacement under a periodically modulated trap; the peak amplitude indicates the resonance position. (d) Measured resonance peak shifts (left axis) and the corresponding Casimir force (right axis) as a function of atom–array distance $h$. The peak shifts reflect trap curvature changes induced by the Casimir interaction, from which the force is inferred.