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arXiv:2601.22640

Unconventional Distance Scaling of Casimir-Polder Force between Atomic Arrays

Qihang Ye, Qihang Ye, Bing Miao, Lei Ying

quant-phphysics.atom-phSubmitted Jan 30, 2026• Announced Feb 2, 2026• 0 citations
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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.

Comments
Submitted to Physical Review Letters

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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.
The Casimir-Polder (CP) effect -- the force between a neutral atom and an uncharged conducting plate in empty space -- is an intriguing consequence of quantum vacuum fluctuations. The typically attractive CP potential crosses over from a scaling of z−3z^{-3}z−3 at short separations to z−4z^{-4}z−4 at long distances, where retardation effects due to the finite speed of light become important. At intermediate distances, where the atom--surface separation is of the order of the wavelength of the dominant atomic transition, experiments have so far relied on indirect methods, such as diffraction or quantum reflection, to observe the CP effect. Here, we directly reveal the CP force between strontium atoms and a dielectric surface via the induced shifts in the atomic energy levels in the intermediate regime. We spectroscopically probe the CP-induced kHz-frequency shift of ultracold atoms confined by a magic-wavelength optical lattice at 189(2)~nm from the surface -- on the scale of the dominant 461-nm transition. Our measurements agree well with QED calculations and differ from the short-range approximation, while excluding the long-distance one. This paves the way for studying the CP effect across various surface properties and geometries, as well as exploring the tensor nature of the atom-surface potential -- all important for the development of hybrid atomic optical-magnetic quantum devices.
We present the first direct measurements of Casimir-Polder forces between solid surfaces and atomic gases in the transition regime between the electrostatic short-distance and the retarded long-distance limit. The experimental method is based on ultracold ground-state Rb atoms that are reflected from evanescent wave barriers at the surface of a dielectric glass prism. Our novel approach does not require assumptions about the potential shape. The experimental data are compared to the theoretical predictions valid in the different regimes. They agree best with a full QED calculation.
We develop a microscopic description of the fluctuation-mediated Casimir-Polder (CP) shifts on a 'test' two-level atom placed near a two-dimensional atomic array of two-level atoms. We derive the resonant and off-resonant CP potentials experienced by the excited test atom using fourth-order perturbation theory, under the assumption that the test atom resonance is far detuned from those of the array atoms. The total potential on the test atom can be described as the sum of the pairwise resonant and off-resonant potentials resulting from its interaction with the individual atoms of the array. We analyze the asymptotic scaling of CP shifts as a function of the test atom-array separation, and its dependence on various system parameters: array spacing and size, and dipole orientation of the array atoms. Our results bridge the description of CP potential across two distinct regimes: (i) from a single-atom limit where we recover the well-known two-atom Van der Waals potential, (ii) to a macroscopic boundary limit, where we demonstrate new asymptotic scaling laws. We demonstrate that these scaling laws can be tuned via the microscopic parameters of the atomic array, establishing atomically-controlled arrays as a versatile platform for tailoring fluctuation-induced QED phenomena.
The distance dependence of the anisotropic atom-wall interaction is studied. The central result is the 1/z61/{z}^{6}1/z6 quadrupolar anisotropy decay in the retarded Casimir-Polder regime. Analysis of the transition region between nonretarded van der Waals regime (in 1/z31/{z}^{3}1/z3) and Casimir-Polder regime shows that the anisotropy crossover occurs at very short distances from the surface, on the order of $0.03\ensuremath{\lambda}$, where $\ensuremath{\lambda}$ is the atom characteristic wavelength. Possible experimental verifications of this distance dependence using surface-induced inelastic transitions are discussed.
At nanometer separation, the dominant interaction between an atom and a material surface is the fluctuation-induced Casimir-Polder potential. We demonstrate that slow atoms crossing a silicon nitride transmission nanograting are a remarkably sensitive probe for that potential. A 15% difference between nonretarded (van der Waals) and retarded Casimir-Polder potentials is discernible at distances smaller than 51 nm. We discuss the relative influence of various theoretical and experimental parameters on the potential in detail. Our work paves the way to high-precision measurement of the Casimir-Polder potential as a prerequisite for understanding fundamental physics and its relevance to applications in quantum-enhanced sensing.
Interactions between objects can be classified as fundamental or emergent. Fundamental interactions are either extremely short-range or decay inversely with the separation distance, such as the Coulomb potential between charges or the gravitational attraction between masses. In contrast, emergent quantum van der Waals (vdW) and Casimir interactions decay considerably faster (R−6R^{-6}R−6 or R−7R^{-7}R−7) with distance RRR. Here we apply perturbative quantum electrodynamics (QED) to a many-body (MB) system of atoms modeled as charged harmonic oscillators, and reveal a persistent inverse-distance MB-QED interaction stemming from the coupling between virtual photons and molecular plasmons in the non-retarded regime. This interaction, scaling with the third power of the fine-structure constant, is reminiscent of the Lamb shift for a single atom. Although weaker than vdW forces, this MB-QED R−1R^{-1}R−1 interaction may substantially surpass gravitational attraction in future experiments probing quantum gravity at microscopic scales.