The energy of fluctuating electromagnetic field is investigated for the thermal Casimir force acting between parallel plates made of real metal. It is proved that for nondissipative media with temperature independent dielectric permittivity the energy at nonzero temperature comprises of the (renormalized) energies of the zero-point and thermal photons. In this manner photons can be considered as collective elementary excitations of the matter of plates and electromagnetic field. If the dielectric permittivity depends on temperature the energy contains additional terms proportional to the derivatives of dielectric permittivity with respect to temperature, and the quasiparticle interpretation of the fluctuating field is not possible. The correlation between energy and free energy is considered. Previous calculations of the Casimir energy in the framework of the Lifshitz formula at zero temperature and optical tabulated data supplemented by the Drude model at room temperature are analysed. It is demonstrated that this quantity is not a good approximation either for the free energy or the energy. A physical interpretation of this hybrid quantity is suggested. The contradictory results in the recent literature on whether the zero-frequency term of the Lifshitz formula for the perpendicular polarized modes has any effective contribution to the physical quantities are discussed. Four main approaches to the resolution of this problem are specified. The precise expressions for entropy of the fluctuating field between plates made of real metal are obtained, which helps to decide between the different approaches. The conclusion is that the Lifshitz formula supplemented by the plasma model and the surface impedance approach are best suited to describe the thermal Casimir force between real metals.
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.
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…
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.
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 …
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…
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 …
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.
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.
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 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 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.
This article is divided in three sections. In the first section we briefly review some high precision experiments on the Casimir force, underlying an important aspect of the analysis of the data. In the second section we discuss our recent results in the measurement of the Casimir force using non-trivial materials. In the third section we present some original ideas for experiments on new phenomena related to the Casimir effects.
We calculate the roughness correction to the Casimir effect in the parallel plates geometry for metallic plates described by the plasma model. The calculation is perturbative in the roughness amplitude with arbitrary values for the plasma wavelength, the plate separation and the roughness correlation length. The correction is found to be always larger than the result obtained in the Proximity Force Approximation.
A brief review of the recent experimental verifications of the Casimir force between extended bodies is presented. With modern techniques, it now appears feasible to test the force law with 1% precision; I will address the issues relating to the interpretation of experiments at this level of accuracy
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.
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.
We demonstrate that a very recently obtained formula for the force on a slab in a material planar cavity based on the calculation of the vacuum Lorentz force [C. Raabe and D.-G. Welsch, Phys. Rev. A 71, 013814 (2005)] describes a (medium) screened Casimir force and, in addition to it, a medium-assisted force. The latter force also describes the force on the cavity medium. For dilute media, it implies the atom-mirror interaction of the Casimir-Polder type at large and of the Coulomb type at small atom-mirror distances of which the sign is insensitive to the polarizability type (electric or magn…
We show that in order to account for the repulsive Casimir effect in the parallel plate geometry in terms of the quantum version of the Lorentz force, virtual surface densities of magnetic charges and currents must be introduced. The quantum version of the Lorentz force expressed in terms of the correlators of the electric and magnetic fields for planar geometries yields then correctly the Casimir pressure.
Casimir forces of massive fermionic Dirac fields are calculated for parallel plates geometry in spatial space with dimension d and imposing bag model boundary conditions.It is shown that in the range of ma>>1 where m is mass of fields quanta and a is the separation distance of the plates,it is equal to massive bosonic fields Casimir force for each degree of freedom.We argue this equality exists for any massive anyonic field in two-dimensional spatial space.Also the ratio of massless fermionic field Casimir force to its bosonic correspondent in d-dimentional spatial space is $(1-1/2^d)$.
We present theoretical calculations of the Casimir force for Au thin films near the insulator-conductor transition that has been observed experimentally. The dielectric function of the Au thin films is described by the Drude-Smith model. The parameters needed to model the dielectric function such as the relaxation time, plasma frequency and the backscattering constant depend on the thickness of the film. The Casimir force decreases as the film thickness decreases until it reaches a minimum after which the force increases again. The minimum of the force coincides with the critical film thicknes…
Within the framework of macroscopic quantum electrodynamics, general expressions for the Casimir force acting on linearly and causally responding magnetodielectric bodies that can be embedded in another linear and causal magnetodielectric medium are derived. Consistency with microscopic harmonic-oscillator models of the matter is shown. The theory is applied to planar structures and proper generalizations of Casimir's and Lifshitz-type formulas are given.
The general perturbative expression for the lateral Casimir force between two plates covered by longitudinal corrugations of arbitrary shape is obtained. This expression is applicable for corrugation periods larger than the separation distance. The cases of asymmetric corrugations are considered, which allow to increase the maximum to minimum force ratio and affect the character of equilibrium points. This opens new opportunities to control the lateral Casimir forces for use in microelectromechanical devices based entirely on the vacuum fluctuation properties.
It has recently been argued that Casimir-Lifshitz forces depend in detail on the microphysics of a system; calculations of the Casimir force in inhomogeneous media yield results that are cutoff-dependent. This result has been shown to hold generally. But suppose we introduce an inhomogeneous metamaterial into a cavity that effectively implements a simple distortion of the coordinate system. Considered in its 'virtual space', the optical properties of such a material are homogeneous and consequently free from the cutoff-dependency associated with inhomogeneous media. This conclusion should be r…
We establish strict upper limits for the Casimir interaction between multilayered structures of arbitrary dielectric or diamagnetic materials. We discuss the appearance of different power laws due to frequency-dependent material constants. Simple analytical expressions are in good agreement with numerical calculations based on Lifshitz theory. We discuss the improvements required for current (meta) materials to achieve a repulsive Casimir force.
We report an improved precision measurement of the Casimir force. The force is measured between a large Al coated sphere and flat plate using an Atomic Force Microscope. The primary experimental improvements include the use of smoother metal coatings, reduced noise, lower systematic errors and independent measurement of surface separations. Also the complete dielectric spectrum of the metal is used in the theory. The average statistical precision remains at the same 1% of the forces measured at the closest separation.
he Casimir force is calculated between plates with thin metallic coating. Thin films are described with spatially dispersive (nonlocal) dielectric functions. For thin films the nonlocal effects are more relevant than for half-spaces. However, it is shown that even for film thickness smaller than the mean free path for electrons, the difference between local and nonlocal calculations of the Casimir force is of the order of a few tenths of a percent. Thus the local description of thin metallic films is adequate within the current experimental precision and range of separations.
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.
Based on a unified approach to macroscopic QED that allows for the inclusion of amplification in a limited space and frequency range, we study the Casimir force as a Lorentz force on an arbitrary partially amplifying system of linearly locally responding (isotropic) magnetoelectric bodies. We demonstrate that the force on a weakly polarisable/magnetisable amplifying object in the presence of a purely absorbing environment can be expressed as a sum over the Casimir--Polder forces on the excited atoms inside the body. As an example, the resonant force between a plate consisting of a dilute gas o…
A recent theoretical calculation shows that the Casimir force between two parallel plates can be repulsive for plates with nontrivial magnetic properties (O. Kenneth et al., Phys. Rev. Lett. 89, 033001 (2002)). According to the authors, the effect may be observed with known materials, such as ferrites and garnets, and it might be possible to engineer micro- or nanoelectromechanical systems (MEMS or NEMS) that could take advantage of a short range repulsive force. Here we show that on the contrary the Casimir force between two parallel plates in vacuum at micron and submicron distance is always…
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…
