Unconventional Distance Scaling of Casimir-Polder Force between Atomic Arrays

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.

💡 Research Summary

The paper investigates how Casimir‑Polder (CP) forces behave when the interacting bodies are not continuous media but discrete atomic arrays. Using a microscopic scattering formalism, the authors calculate the CP interaction between two parallel, infinite two‑dimensional (2D) square lattices of identical two‑level atoms separated by a distance h. Each atom is modeled as an isotropic electric dipole with a polarizability α(ω) that includes the resonant transition frequency ω₀, decay rate γ, and dipole moment d₀. The total Hamiltonian comprises the atomic, field, and atom‑field interaction parts; because the coupling is weak, a single‑photon perturbative treatment suffices.

The CP potential is expressed as an integral over imaginary frequencies (iξ) of the product of the atomic polarizability and the induced dipole moments, which themselves are determined by the full Green’s tensor of the combined atom‑array system. By Fourier‑transforming the Green’s tensor for an infinite lattice and applying the Weyl decomposition, the problem becomes semi‑analytical, allowing efficient evaluation of the lattice sums that encode the discrete geometry.

A key result is the lattice‑sum function
S(h; ξ)=∑{m,n} exp(−2R{mn}ξ/c)/R_{mn}⁶,
where R_{mn}=√(h²+(ma)²+(na)²) is the distance from a probe atom to each lattice site. The asymptotic behavior of S determines the distance scaling of the CP force. Three distinct regimes emerge, governed by the relative magnitudes of the lattice constant a, the resonant wavelength λ₀=c/ω₀, and the separation h:

1. Short‑range, non‑retarded (h ≪ a ≪ λ₀): The nearest lattice site dominates, giving S ∝ h⁻⁶ and a force F ∝ h⁻⁷, identical to the familiar atom‑atom CP law.

2. Intermediate non‑retarded (a ≲ h ≪ λ₀): Many lattice atoms contribute; the sum can be approximated by a continuum integral over the plane, yielding S ∝ a⁻² h⁻⁴ and therefore F ∝ a⁻² h⁻⁵. The force now decays more slowly because the areal density (∝ a⁻²) multiplies the distance dependence.

3. Retarded (h ≫ λ₀): The exponential factor from the Green’s functions suppresses contributions at large ξ, but the lattice geometry still leads to S ∝ a⁻² h⁻⁵, giving F ∝ a⁻² h⁻⁶. This is markedly slower than the classic h⁻⁴ force between perfect conducting plates (which becomes h⁻⁵ after differentiation).

The authors emphasize that the extra intrinsic length scale a breaks the universal scaling seen in continuous systems. In particular, for Rydberg atoms the resonant wavelength can be centimeters (e.g., λ₀≈1.9 cm for the 53 D₃/₂→52 F₅/₂ transition), while the lattice spacing can be a few micrometers. Consequently, the condition a ≪ λ₀ holds across the entire experimentally relevant range, and the force follows the slower h⁻⁵ scaling even in the non‑retarded regime. This “distance‑slow‑down” is a distinctive signature of discrete arrays and cannot be captured by Lifshitz‑type continuum theories.

To demonstrate experimental feasibility, the paper proposes measuring the CP‑induced shift of the motional trapping frequency of a single Rydberg atom placed near a 2D Rydberg array. The CP potential adds a curvature term to the harmonic trap, modifying the effective frequency as
ω_eff ≈ ω₀ + (1/2mω₀)∂²U_CP/∂h².
By modulating the trap depth (parametric excitation) and detecting the resulting frequency shift, one can extract the distance dependence of the force. The suggested parameters—Rb atoms excited to the 70 D₅/₂ state, lattice spacing a≈6 µm, separations h ranging from 10 µm to 100 µm, and trap frequencies in the kHz range—are within current experimental capabilities.

In summary, the work reveals that Casimir‑Polder forces between discrete atomic lattices do not obey the conventional distance scaling derived for continuous media. The presence of the lattice constant introduces new scaling regimes: h⁻⁷ (single‑atom dominated), h⁻⁵ (many‑atom non‑retarded), and h⁻⁶ (retarded). For Rydberg arrays the h⁻⁵ regime dominates, offering a clear experimental target. These findings open a pathway to engineer dispersion forces by tailoring lattice geometry and atomic resonances, with potential applications in quantum simulation, metamaterial design, and precision force metrology at the nanoscale.