Fundamental limits to attractive and repulsive Casimir-Polder forces

Prashanth S. Venkataram, Sean Molesky, Pengning Chao, and Alejandro W. Rodriguez

Phys. Rev. A 101, 052115 – Published 21 May 2020

Abstract

We derive upper and lower bounds on the Casimir-Polder force between an anisotropic dipolar body and a macroscopic body, separated by vacuum, via algebraic properties of Maxwell's equations. These bounds require only a coarse characterization of the system—the material composition of the macroscopic object, the polarizability of the dipole, and any convenient partition between the two objects—to encompass all structuring possibilities. We find that the attractive Casimir-Polder force between a polarizable dipole and a uniform planar semi-infinite bulk medium always comes within 10% of the lower bound, implying that nanostructuring is of limited use for increasing attraction. In contrast, the possibility of repulsion is observed even for isotropic dipoles, and is routinely found to be several orders of magnitude larger than any known design, including recently predicted geometries involving conductors with sharp edges. These results may have ramifications for the design of surfaces to trap, suspend, or adsorb ultracold gases.

Received 25 November 2019
Revised 25 April 2020
Accepted 27 April 2020

DOI:https://doi.org/10.1103/PhysRevA.101.052115

Physics Subject Headings (PhySH)

Casimir effect & related phenomena Optimization problems

Classical electromagnetism

General PhysicsStatistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Prashanth S. Venkataram, Sean Molesky, Pengning Chao , and Alejandro W. Rodriguez

Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA

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Vol. 101, Iss. 5 — May 2020

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Figure 1
Schematic of investigation. We derive shape-independent upper and lower bounds on the Casimir-Polder force between a polarizable dipolar body of parallel (perpendicular) polarizability $α_{∥}$ ( $α_{⊥}$ ) above any nanostructured medium of susceptibility $χ$ within a given domain $Ω$ .
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Figure 2
Material and anisotropy dependence of bounds on Casimir-Polder forces. (a) Upper and lower bounds to the CP force (blue and red lines, respectively, in the upper or lower parts of the plot) on a nondispersive anisotropic dipole of parallel and perpendicular polarizabilities $α_{∥}$ and $α_{⊥}$ , above a planar semi-infinite half-space domain, along with the actual force above a planar semi-infinite bulk (gray lines, nearly overlapping with red lines in the lower part of the plot), normalized to $ℏ c α_{∥} / 8 π^{2} d^{5}$ for separation $d$ , as a function of $α_{⊥} / α_{∥}$ . The macroscopic susceptibility $χ_{0}$ is nondispersive and increases logarithmically from $10^{- 2}$ to $10^{6}$ (going from lighter to darker shades upward in the upper part of the plot or downward in the lower part of the plot). (b) Same as panel (a) but plotted against $χ_{0}$ for the isotropic case $α_{⊥} = α_{∥}$ .
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Figure 3
Distance dependence of bounds on Casimir-Polder forces for gold nanostructures. Repulsive and attractive bounds on CP forces as in Fig. 2 but with the macroscopic susceptibility $χ$ corresponding to that of gold and $α_{⊥} / α_{∥}$ increasing logarithmically from $10^{- 2}$ to $10^{2}$ (going from lighter to darker shades upward in the upper part of the plot or downward in the lower part of the plot). Also shown is the CP force on a gold needle above a gold plate with a circular aperture from Ref. [49] (dark blue star), corresponding to a static anisotropic polarizability ratio $α_{⊥} / α_{∥} \approx 51.1$ at $d = 200 nm$ ; bounds for $α_{⊥} / α_{∥} = 50$ are marked in dashed lines.
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Figure 4
Distance dependence of bounds on Casimir-Polder forces for silicon nanostructures. Repulsive and attractive bounds on CP forces (blue and red lines, respectively, in the upper or lower part of the plot) and actual force (gray lines, nearly overlapping with red lines in the lower part of the plot) for the macroscopic susceptibility $χ$ corresponding to that of silicon and $α_{⊥} / α_{∥}$ increasing logarithmically from $10^{- 2}$ to $10^{2}$ (going from lighter to darker shades upward in the upper part of the plot or downward in the lower part of the plot).
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Figure 5
Distance dependence of bounds on Casimir-Polder forces for gold nanostructures: dependence on model and temperature. Repulsive and attractive bounds on CP forces (blue and red lines, respectively in the upper or lower part of the plots) and actual force (gray lines, nearly overlapping with red lines in the lower part of the plots) with the macroscopic susceptibility $χ$ corresponding to that of gold and $α_{⊥} / α_{∥}$ increasing logarithmically from $10^{- 2}$ to $10^{2}$ (going from lighter to darker shades upward in the upper part of the plots or downward in the lower part of the plots), for the Drude model at zero (a) or room (b) temperatures. [(c), (d)] Same as panels (a) and (b), respectively, using the plasma model instead.
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Figure 6
Ratios of bounds on Casimir-Polder forces for gold nanostructures. Ratios of corresponding repulsive and attractive bounds on CP forces (blue and red lines) and actual forces (gray lines, nearly overlapping with red lines), with the macroscopic susceptibility $χ$ corresponding to that of gold and $α_{⊥} / α_{∥}$ increasing logarithmically from $10^{- 2}$ to $10^{2}$ (lighter to darker shades). For panels (a) and (b), ratios are of $\frac{F (300 K)}{F (0)}$ for the Drude (a) or plasma (b) models: For both plots, from top to bottom, the top-most dark blue curve is the repulsive bound ratio for $α_{⊥} / α_{∥} = 10^{2}$ , the next dark red curve is the attractive bound ratio for $α_{⊥} / α_{∥} = 10^{2}$ , the following light red curve is the attractive bound ratio for $α_{⊥} / α_{∥} = 10^{- 2}$ , and the bottom-most light blue curve is the repulsive bound ratio for $α_{⊥} / α_{∥} = 10^{- 2}$ . For panels (c) and (d), ratios are of $\frac{F_{Au, plasma}}{F_{Au, Drude}}$ , for zero (c) or room (d) temperatures: For both plots, from top to bottom, the top-most light red curve is the attractive bound ratio for $α_{⊥} / α_{∥} = 10^{- 2}$ , the next light blue curve is the repulsive bound ratio for $α_{⊥} / α_{∥} = 10^{- 2}$ , the following dark red curve is the attractive bound ratio for $α_{⊥} / α_{∥} = 10^{2}$ , and the bottom-most dark blue curve is the repulsive bound ratio for $α_{⊥} / α_{∥} = 10^{2}$ .
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