We calculate the Casimir force on an isolated dielectric sphere in an ensemble of $N$ spheres due to multiple mutual interactions of the collection of spheres. In particular we consider dielectric spheres immersed in some other background dielectric. As an example, the Casimir force between two and three spheres at zero and finite temperature is evaluated. For a very large number of spheres, we consider a large-$N$ scaling limit of the Casimir force.
We discuss the calculation of Casimir forces between a collection of $N$-dielectric spheres. This is done by evaluating directly the force on a sphere constructed from a stress tensor, rather than an interaction energy. Two and three body forces between the spheres are evaluated for setups of two and three sphere systems respectively. An approximate large-$N$ limit is also obtained for the functional dependence on the number of spheres.
An expression for the Casimir stress on arbitrary dispersive and lossy linear magnetodielectric matter at finite temperature, including left-handed material, is derived and applied to spherical systems. To cast the relevant part of the scattering Green tensor for a general magnetodielectric sphere in a convenient form, classical Mie scattering is reformulated.
Exploiting conformal symmetry, we derive a simple exact formula for the classical electromagnetic Casimir interaction of two perfectly conducting three-spheres, including the sphere-plate geometry as a special case, in four euclidean dimensions. We verify that the short distance expansion of the Casimir energy agrees to leading order with the Proximity Force Approximation (PFA), while the next-to-leading-order is in agreement with a recently proposed derivative expansion of the Casimir energy. At the next-to-next-to-leading order we find a non-analytic correction to PFA, which for a sphere-pla…
A summary of recent calculations of Casimir forces between a collection of $N$-dielectric spheres is presented. This is done by evaluating directly the force on a sphere constructed from a stress tensor, rather than an interaction energy. A loop integral formulation is also discussed where we rewrite the expressions for the force in terms of loop integrals for the effective classical propagation of the electric and magnetic fields.
A recent experiment [J.L. Garrett et al., Phys. Rev. Lett {\bf 120}, 040401 (2018)] measured for the first time the gradient of the Casimir force between two gold spheres at room temperature. The theoretical analysis of the data was carried out using the standard Proximity Force Approximation (PFA). A fit of the data, using a parametrization of the force valid for the sphere-plate geometry, was used by the authors to place a bound on deviations from PFA. Motivated by this work, we compute the Casimir force between two gold spheres at finite temperature. The semi-analytic formula for the Casimi…
We study the Casimir interaction between perfectly conducting sphere and plate in the classical limit of high temperatures. By taking the small-distance expansion of the exact scattering formula, we compute the leading correction to the Casimir energy beyond the commonly employed proximity force approximation. We find that for a sphere of radius $R$ at distance $d$ from the plate the correction is of the form $\ln^2 (d/R)$, in agreement with indications from recent large-scale numerical computations. We develop a fast-converging numerical scheme for computing the Casimir interaction to high pr…
We compute the Casimir interaction between a plane and a sphere, the configuration employed in the most precise experiments. The scattering formula is developed by taking a suitably chosen plane-wave and multipole basis and is valid for arbitrary values of the sphere radius, inter-plate distance, temperature and arbitrary dielectric functions for both sphere and plate. Our analytical and numerical results for metallic surfaces show a non-trivial interplay between the effects of curvature, temperature, finite conductivity and dissipation.
We calculate the Casimir force in the non-retarded limit between a spherical nanoparticle and a substrate, and we found that high-multipolar contributions are very important when the sphere is very close to the substrate. We show that the highly inhomegenous electromagnetic field induced by the presence of the substrate, can enhance the Casimir force by orders of magnitude, compared with the classical dipolar approximation.
We derive the explicit expression for the Casimir energy between a sphere and a 1D grating, in terms of the sphere and grating reflection matrices, and valid for arbitrary materials, sphere radius, and grating geometric parameters. We then numerically calculate the Casimir energy between a metallic (gold) sphere and a dielectric (fused silica) lamellar grating at room temperature, and explore its dependence on the sphere radius, grating-sphere separation, and lateral displacement. We quantitatively investigate the geometrical dependence of the interaction, which is sensitive to the grating hei…
Quantum fluctuations of the electromagnetic field in the medium surrounding two discharged macroscopic polarizable bodies induce a force between the two bodies, the so called Casimir force. In the last two decades many experiments have accurately measured this force, and significant efforts are made to harness it in the actuation of micro and nano machines. The inherent many body character of the Casimir force makes its computation very difficult in non-planar geometries, like the standard experimental sphere-plate configuration. Here we derive an approximate semi-analytic formula for the sphe…
The Casimir energy is computed in the geometry of interest for the most precise experiments, a plane and a sphere in electromagnetic vacuum. The scattering formula is developed on adapted plane-waves and multipole basis, leading to an expression valid for arbitrary relative values of the sphere radius and inter-plate distance. In the limiting case of perfect reflection, the electromagnetic result is found to depart from the commonly used proximity-force approximation (PFA) significantly more rapidly than expected from scalar computations.
We report measurements of the Casimir force between a gold sphere and a silicon surface with an array of nanoscale, rectangular corrugations using a micromechanical torsional oscillator. At distance between 150 nm and 500 nm, the measured force shows significant deviations from the pairwise additive formulism, demonstrating the strong dependence of the Casimir force on the shape of the interacting bodies. The observed deviation, however, is smaller than the calculated values for perfectly conducting surfaces, possibly due to the interplay between finite conductivity and geometry effects.
We critically discuss the common belief that the thermal Casimir-Polder force can always be represented as a Matsubara sum, as it is true for the Casimir force. On the contrary, an exact treatment of the atom-field coupling reveals that for a ground-state atom, terms associated with virtual-photon absorption contribute to the force which are identified as a signature of non-equilibrium dynamics. Even the equilibrium force on a thermalized atom or molecule may be overestimated when using the ground-state polarizability instead of its thermal counterpart.
We develop a spectral representation formalism to calculate the Casimir force in the non-retarded limit, between a spherical particle and a substrate, both with arbitrary local dielectric properties. This spectral formalism allows one to do a systematic study of the force as a function of the geometrical variables separately from the dielectric properties. We found that the force does not follow a simple power-law as a function of the separation between the sphere and substrate. As a consequence, the non-retarded Casimir force is enhanced by several orders of magnitude as the sphere approaches…
The Casimir force for charge-neutral, perfect conductors of non-planar geometric configurations have been investigated. The configurations are: (1) the plate-hemisphere, (2) the hemisphere-hemisphere and (3) the spherical shell. The resulting Casimir forces for these physical arrangements have been found to be attractive. The repulsive Casimir force found by Boyer for a spherical shell is a special case requiring stringent material property of the sphere, as well as the specific boundary conditions for the wave modes inside and outside of the sphere. The necessary criteria in detecting Boyer's…
Using a recently developed theory of the Casimir force (Raabe C and Welsch D-G 2005 Phys. Rev. A 71 013814), we calculate the force that acts on a plate in front of a planar wall and the force that acts on the plate in the case where the plate is part of matter that fills the space in front of the wall. We show that in the limit of a dielectric plate whose permittivity is close to unity, the force obtained in the former case reduces to the ordinary, i.e., unscreened Casimir-Polder force acting on isolated atoms. In the latter case, the theory yields the Casimir-Polder force that is screened by…
On the basis of macroscopic quantum electrodynamics and point-scattering techniques, we derive a closed expression for the Casimir-Polder force between a ground-state atom and a small magnetodielectric sphere in an arbitrary environment. In order to allow for the presence of both bodies and media, local-field corrections are taken into account. Our results are compared with the known van der Waals force between two ground-state atoms. To continuously interpolate between the two extreme cases of a single atom and a macroscopic sphere, we also derive the force between an atom and a sphere of var…
We find the lateral projection of the Casimir force for a configuration of a sphere above a corrugated plate. This force tends to change the sphere position in the direction of a nearest corrugation maximum. The probability distribution describing different positions of a sphere above a corrugated plate is suggested which is fitted well with experimental data demonstrating the nontrivial boundary dependence of the Casimir force.
The thermal Casimir force between two metallic plates is known to depend on the description of material properties. For large separations the dissipative Drude model leads to a force a factor of 2 smaller than the lossless plasma model. Here we show that the plane-sphere geometry, in which current experiment are performed, decreases this ratio to a factor of 3/2, as revealed by exact numerical and large distance analytical calculations. For perfect reflectors, we find a repulsive contribution of thermal photons to the force and negative entropy values at intermediate distances.
Dissipative electromagnetic response and scattering geometry are potential sources for the appearance of a negative Casimir entropy. We show that the dissipative contribution familiar from the plane-plane geometry appears also in the plane-sphere and the sphere-sphere geometries and adds to the negative Casimir entropy known to exist in these geometries even for perfectly reflecting objects. Taking the sphere-sphere geometry as an example, we carry out a scattering-channel analysis which allows to distinguish between the contributions of different polarizations. We demonstrate that dissipation…
A 3D finite-element numerical simulation was developed to investigate Casimir forces in arbitrary geometries. The code was verified comparing it with results obtained from analytical equations. Appling the simulation to previously not assessed configurations, new Casimir properties were found such as repulsive Casimir forces in groove like structures.
We calculated the force of the quantum vacuum, the Casimir force, in a spherically symmetric medium, Maxwell's fish eye, surrounded by a perfect mirror and derived an exact analytic solution. Our solution questions the idea that the Casimir force of a spherical mirror is repulsive - we found an attractive force that diverges at the mirror.
We show that the Casimir force between mirrors with arbitrary frequency dependent reflectivities obeys bounds due to causality and passivity properties. The force is always smaller than the Casimir force between two perfectly reflecting mirrors. For narrow-band mirrors in particular, the force is found to decrease with the mirrors bandwidth.
The Casimir-Polder and van der Waals interaction energy of an atom with infinitely thin sphere with finite conductivity is investigated in the framework of the hydrodynamic approach. We put the sphere into spherical cavity inside the infinite dielectric media, then calculate the energy of vacuum fluctuations in the context of the zeta-function approach. The energy for a single atom is obtained by rarefying media. The Casimir-Polder expression for an atom and plate is recovered in the limit of the infinite radius of the sphere. Assuming a finite radius of the sphere, the interaction energy of a…
By using silicon oxide based aerogels we show numerically that the Casimir force can be reduced several orders of magnitude, making its effect negligible in nanodevices. This decrease in the Casimir force is also present even when the aerogels are deposited on metallic substrates. To calculate the Casimir force we model the dielectric function of silicon oxide aerogels using an effective medium dielectric function such as the Clausius-Mossotti approximation. The results show that both the porosity of the aerogel and its thickness can be use as control parameters to reduce the magnitude of the …
We investigate repulsive Casimir force between slabs containing left-handed materials with controllable electromagnetic properties. The sign of Casimir force is determined by the electric and magnetic properties of the materials, and it is shown that the formation of the repulsive force is related to the wave impedances of two slabs. The sign change of the Casimir force as a function of the distance is studied. Special emphasis is put on the restoring Casimir force which may be found to exist between perfectly conducting material and metamaterial slabs. This restoring force is a natural power …
We study the nonequilibrium Casimir-Polder force on an atom prepared in an incoherent superposition of internal energy-eigenstates, which is placed in a magnetoelectric environment of nonuniform temperature. After solving the coupled atom--field dynamics within the framework of macroscopic quantum electrodynamics, we derive a general expression for the thermal Casimir-Polder force.
We derive the Casimir force expression from Maxwell's stress tensor by means of original quantum-electro-dynamical cavity modes. In contrast with similar calculations, our method is straightforward and does not rely on intricate mathematical extrapolation relations.
Linear in temperature correction to the Casimir force is discussed. The correction is important for small separations between bodies tested in the recent experiments and disappears in the case of perfect conductors.
In this paper we review the role of magneto plasmon polaritons in the Casimir force calculations. By applying an external constant magnetic field a strong optical anisotropy is induced on two parallel slabs reducing the reflectivity and thus the Casimir force. As the external magnetic field increases, the Casimir force decreases. Thus, with an an external magnetic field the Casimir force can be controlled.The calculations are done in the Voigt configuration where the magnetic field is parallel to the slabs. In this configuration the reflection coefficients for TE and TM modes do not show mode …
We study the electromagnetic Casimir interaction of a compact object contained inside a closed cavity of another compact object. We express the interaction energy in terms of the objects' scattering matrices and translation matrices that relate the coordinate systems appropriate to each object. When the enclosing object is an otherwise empty metallic spherical shell, much larger than the internal object, and the two are sufficiently separated, the Casimir force can be expressed in terms of the static electric and magnetic multipole polarizabilities of the internal object, which is analogous to…
It is well known that residual electrostatic forces create significant difficulties to precise measurements of the Casimir force and to wide use of Casimir-operated microdevices. We experimentally demonstrate that with the help of Ar-ion cleaning of the surfaces it is possible to make electrostatic effects negligibly small as compared to the Casimir interaction. Our experimental setup consists of the dynamic atomic force microscope supplemented with an Ar-ion gun and argon reservoir. The residual potential difference between the Au-coated surfaces of a sphere and a plate was measured both befo…
The normal and tangential Casimir force for the rack gear is calculated numerically in the case of ideal boundary conditions for the electromagnetic field - perfect reflection on the boundaries. The resulting tangential force appears to be essentially shape dependent. Relatively small shape variations lead to the essential changes in tangential force, whereas normal force remains almost unchanged.
Recent progress in the simulation of Casimir forces between various objects has allowed traditional computational electromagnetic solvers to be used to find Casimir forces in arbitrary three-dimensional objects. The underlying theory to these approaches requires knowledge and manipulation of quantum field theory and statistical physics. We present a calculation of the Casimir force using the method of moments via the argument principle. This simplified derivation allows greater freedom in the moment matrix where the argument principle can be used to calculate Casimir forces for arbitrary geome…
We present theoretical and numerical results for the screened Casimir effect between perfect metal surfaces in a plasma. We show how the Casimir effect in an electron-positron plasma can provide an important contribution to nuclear interactions. Our results suggest that there is a connection between Casimir forces and nucleon forces mediated by mesons. Correct nuclear energies and meson masses appear to emerge naturally from the screened Casimir-Lifshitz effect.
Analytic expressions that describe Casimir interactions over the entire range of separations have been limited to planar surfaces. Here we derive analytic expressions for the classical or high-temperature limit of Casimir interactions between two spheres (interior and exterior configurations), including the sphere-plane geometry as a special case, using bispherical coordinates. We consider both Dirichlet boundary conditions and metallic boundary conditions described by the Drude model. At short distances, closed-form expansions are derived from the exact result, displaying an intricate structu…
In this article we investigate the Casimir effect in the presence of a medium by quantizing the Electromagnetic (EM) field in the presence of a magnetodielectric medium by using the path integral formalism. For a given medium with definite electric and magnetic susceptibilities, explicit expressions for the Casimir force are obtained which are in agree with the original Casimir force between two conducting parallel plates immersed in the quantum electromagnetic vacuum.
The possibility to modify the strength of the Casimir effect by tailoring the dielectric functions of the interacting surfaces is regarded as a unique opportunity in the development of Micro- and NanoElectroMechanical Systems. In air, however, one expects that, unless noble metals are used, the electrostatic force arising from trapped charges overcomes the Casimir attraction, leaving no room for exploitation of Casimir force engineering at ambient conditions. Here we show that, in the presence of a conductive oxide, the Casimir force can be the dominant interaction even in air, and that the us…
The possibility in principle is shown that the noncompensated Casimir force can exist in nanosized open metal cavities. The force shows up as time-constant expulsion of open cavities toward their least opening. The optimal parameters of the angles of the opening, of "generating lines" of cavities and their lengths are found at which the expulsive force is maximal. The theory is created for trapezoid configurations, in particular for parallel mirrors which experience both the transverse Casimir pressure and longitudinal compression at zero general expulsive force.
We show that exact results are obtained for the calculation of Casimir forces between arbitrary materials using the concept of surface impedances, obtaining in a trivial way the force in the limit of perfect conductors and also Lifshitz formula in the limit of semi-infinite media. As an example we present a full and rigorous calculation of the Casimir force between two metallic half-spaces described by a hydrodynamic nonlocal dielectric response.
We consider the Casimir interaction energy between a plane and a sphere of radius $R$ at finite temperature $T$ as a function of the distance of closest approach $L$. Typical experimental conditions are such that the thermal wavelength $λ_T=\hbar c/k_\mathrm{B}T$ satisfies the condition $L\ll λ_T\ll R$. We derive the leading correction to the proximity-force approximation valid for such intermediate temperatures by developing the scattering formula in the plane-wave basis. Our analytical result captures the joint effect of the spherical geometry and temperature and is written as a sum of tempe…
The Casimir force between two parallel plates separated by anisotropic media is investigated. We theoretically calculate the Casimir force between two parallel plates when the interspace between the plates is filled with anisotropic media. Our result shows that the anisotropy of the material between the plates can significantly affect the Casimir force, especially the direction of the force. If ignoring the anisotropy of the in-between material makes the force to be repulsive (attractive), by contrast, taking the anisotropy into account may produce an extra attractive (repulsive) force. The ph…
A general analysis of thermal noise in torsion pendulums is presented. The specific case where the torsion angle is kept fixed by electronic feedback is analyzed. This analysis is applied to a recent experiment that employed a torsion pendulum to measure the Casimir force. The ultimate limit to the distance at which the Casimir force can be measured to high accuracy is discussed, and in particular the prospects for measuring the thermal correction are elaborated upon.
The difference of the thermal Casimir forces at different temperatures between real metals is shown to increase with a decrease of the separation distance. This opens new opportunities for the demonstration of the thermal dependence of the Casimir force. Both configurations of two parallel plates and a sphere above a plate are considered. Different approaches to the theoretical description of the thermal Casimir force are shown to lead to different measurable predictions.
We present a modal approach to calculate finite temperature Casimir interactions between two periodically modulated surfaces. The scattering formula is used and the reflection matrices of the patterned surfaces are calculated decomposing the electromagnetic field into the natural modes of the structures. The Casimir force gradient from a deeply etched silicon grating is evaluated using the modal approach and compared to experiment for validation. The Casimir force from a two dimensional periodic structure is computed and deviations from the proximity force approximation examined.
We discuss a proposal to measure the Casimir force in the parallel plate configuration in the $1-10μ$m range via a high-sensitivity torsional balance. This will allow to measure the thermal contribution to the Casimir force therefore discriminating between the various approaches discussed so far. The accurate control of the Casimir force in this range of distances is also required to improve the limits to the existence of non-Newtonian forces in the micrometer range predicted by unification models of fundamental interactions.
A number of experimental measurements of the Casimir force have observed a logarithmic distance variation of the voltage that minimizes electrostatic force between the plates in a sphere-plane geometry. We show that this variation can be simply understood from a geometric averaging of surface potential patches together with the Proximity Force Approximation.
We discuss the Casimir effect for boundary conditions involving perfect electromagnetic conductors (PEMCs). Based on the corresponding reciprocal Green's tensor we construct the Green's tensor for two perfectly reflecting plates with magnetoelectric coupling (non-reciprocal media) within the framework of macroscopic quantum electrodynamics. We calculate the Casimir force between two PEMC plates in terms of the PEMC parameter M and the duality transformation angle $θ$ resulting in a universal analytic expression that connects the attractive Casimir force with the repulsive Boyer force. We relat…
After a short recall of our previous standing wave approach to the Casimir force problem, we consider Lifshitz's temperature Green's function method and its virtues from a physical point of view. Using his formula, specialized for perfectly reflecting mirrors, we present a quantitative discussion of the temperature effect on the attractive force.
We study the effect of two non-interacting impurity atoms near by a one-dimensional nanowire, which is modeled as a tight-binding hopping model. The virtual single-electron hopping between two impurities will induce an additional energy depending on the distance of two impurities, which gives a electronic Casimir-Polder effect. We find that the Casimir-Polder force between the two impurities decreases with the impurity-impurity distance exponentially. And the effects of nanowire and finite temperature on the Casimir-Polder force are also discussed in detail, respectively.
Using a template-stripping method, macroscopic gold surfaces with root-mean-square (rms) roughness less than 0.4 nm have been prepared, making them useful for studies of surface interactions in the nanometer range. The utility of such substrates is demonstrated by measurements of the Casimir force at surface separations between 20 and 100 nm, resulting in good agreement with theory. The significance and quantification of this agreement is addressed, as well as some methodological aspects regarding the measurement of the Casimir force with high accuracy.
We propose the use of a tunneling electromechanical transducer to dynamically detect Casimir forces between two conducting surfaces. The maximum distance for which Casimir forces should be detectable with our method is around $1 μ$m, while the lower limit is given by the ability to approach the surfaces. This technique should permit to study gravitational forces on the same range of distances, as well as the vacuum friction provided that very low dissipation mechanical resonators are used.
We develop a computational study of Casimir forces between three dimensional (3D) finite objects with an internal granular structure. The objects in the model consist of a finite arrangement of nanometer sized spherical particles having a dipolar interaction. In this model system one can both study the basic properties of the Casimir forces, and the effects from changing the parameters of the nano-structured materials that constitute the particles; this last type of study leads to a form of control of the Casimir force and an insight into possible technological applications. We present example…
In this paper we calculate the Casimir-Polder force density (force per unit area acting on the elements of the surface) on a metallic plate placed in front of a neutral atom. To obtain the force density we use the quantum operator associated to the electromagnetic stress tensor. We explicitly show that the integral of this force density over the plate reproduces the total force acting on the plate. This result shows that, although the force is obtained as a sum of surface element-atom contributions, the stress-tensor method includes also nonadditive components of Casimir-Polder forces in the e…
We consider the quantum fluctuations of the Casimir-Polder force between a neutral atom and a perfectly conducting wall in the ground state of the system. In order to obtain the atom-wall force fluctuation we first define an operator directly associated to the force experienced by the atom considered as a polarizable body in an electromagnetic field, and we use a time-averaged force operator in order to avoid ultraviolet divergences appearing in the fluctuation of the force. This time-averaged force operator takes into account that any measurement involves a finite time. We also calculate the …
We consider the three-body Casimir-Polder interaction between three atoms during their dynamical self-dressing. We show that the time-dependent three-body Casimir-Polder interaction energy displays nonlocal features related to quantum properties of the electromagnetic field and to the nonlocality of spatial field correlations. We discuss the measurability of this intriguing phenomenon and its relation with the usual concept of stationary three-body forces.
The finite conductivity corrections to the Casimir force in two configurations are calculated in the third and fourth orders in relative penetration depth of electromagnetic zero oscillations into the metal. The obtained analytical perturbation results are compared with recent computations. Applications to the modern experiments are discussed.
We present systematic measurements of the Casimir force between a gold-coated plate and a sphere coated with a Hydrogen Switchable Mirror (HSM). HSMs are shiny metals that can become transparent by hydrogenation. In spite of such a dramatic change of the optical properties of the sphere, we did not observe any significant decrease of the Casimir force after filling the experimental apparatus with hydrogen. This counterintuitive result can be explained by the Lifshitz theory that describes the Casimir attraction between metallic and dielectric materials.
