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.
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.
We extend our approach to the Casimir effect between absorbing dielectric multilayers [M. S. Tomas, Phys. Rev. A 66, 052103 (2002)] to magnetodielectric systems. The resulting expression for the force is used to numerically explore the effect of the medium dispersion on the attractive/repulsive force in a metal-magnetodielectric system described by the Drude-Lorentz permittivities and permeabilities.
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.
A path-integral approach to the quantization of the electromagnetic field in a linearly amplifying magnetodielectric medium is presented. Two continua of inverted harmonic oscillators are used to describe the polarizability and magnetizability of the amplifying medium. The causal susceptibilities of the amplifying medium, with negative imaginary parts in finite frequency intervals, are identified and their relation to microscopic coupling functions are determined. By carefully relating the two-point functions of the field theory to the optical Green functions, we calculate the Casimir energy a…
We extend our previous work [Phys. Rev. Lett. {\bf 100}, 183602 (2008)] on the generalization of the Casimir-Lifshitz theory to treat anisotropic magnetodielectric media, focusing on the forces between metals and magnetodielectric metamaterials and on the possibility of inferring magnetic effects by measurements of these forces. We present results for metamaterials including structures with uniaxial and biaxial magnetodielectric anisotropies, as well as for structures with isolated metallic or dielectric properties that we describe in terms of filling factors and a Maxwell Garnett approximatio…
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…
Based on macroscopic QED in linear, causal media, we present a consistent theory for the Casimir-Polder force acting on an atom positioned near dispersing and absorbing magnetodielectric bodies. The perturbative result for the van-der-Waals energy is shown to exhibit interesting new features in the presence of magnetodielectric bodies. To go beyond perturbation theory, we start with the center-of-mass equation of motion and derive a dynamical expression for the Casimir-Polder force acting on an atom prepared in an arbitrary electronic state. For a non-driven atom in the weak coupling regime, t…
We present a worldline method for the calculation of Casimir energies for scalar fields coupled to magnetodielectric media. The scalar model we consider may be applied in arbitrary geometries, and it corresponds exactly to one polarization of the electromagnetic field in planar layered media. Starting from the field theory for electromagnetism, we work with the two decoupled polarizations in planar media and develop worldline path integrals, which represent the two polarizations separately, for computing both Casimir and Casimir-Polder potentials. We then show analytically that the path integr…
A general theory of the Casimir-Polder interaction of single atoms with dispersing and absorbing magnetodielectric bodies is presented, which is based on QED in linear, causal media. Both ground-state and excited atoms are considered. Whereas the Casimir-Polder force acting on a ground-state atom can conveniently be derived from a perturbative calculation of the atom-field coupling energy, an atom in an excited state is subject to transient force components that can only be fully understood by a dynamical treatment based on the body-assisted vacuum Lorentz force. The results show that the Casi…
A powerful procedure is presented for calculating the Casimir attraction between plane parallel multilayers made up of homogeneous regions with arbitrary magnetic and dielectric properties by use of the Minkowski energy-momentum tensor. The theory is applied to numerous geometries and shown to reproduce a number of results obtained by other authors. Although the various pieces of theory drawn upon are well known, the relative ease with which the Casimir force density in even complex planar structures may be calculated, appears not to be widely appreciated, and no single paper to the author's k…
We find the analytic expressions for the Casimir free energy, entropy and pressure at low temperature in the configuration of two parallel plates made of magnetodielectic material. The cases of constant and frequency-dependent dielectic permittivity and magnetic permeability of the plates are considered. Special attention is paid to the account of dc conductivity. It is shown that in the case of finite static dielectric permittivity and magnetic permeability the Nernst heat theorem for the Casimir entropy is satisfied. If the dc conductivity is taken into account, the Casimir entropy goes to a…
We argue that the results for the vacuum forces on a slab and on an atom embedded in a magnetodielectric medium near a mirror, obtained using a recently suggested Lorentz-force approach to the Casimir effect, are equivalent to the corresponding results obtained in a traditional way. We also derive a general expression for the atom-atom force in a medium and extend a few classical results concerning this force in vacuum and dielectrics to magnetodielectric systems. This, for example, reveals that the (repulsive) interaction between atoms of different polarizability type is at small distances un…
We demonstrate that, according to a recently suggested Lorentz-force approach to the Casimir effect, the vacuum force on an atom embedded in a material cavity differs substantially from the force on an atom of the cavity medium. The force on an embedded atom is of the familiar (van der Waals and Casimir-Polder) type, however, more strongly modified by the cavity medium than usually considered. The force on an atom of the cavity medium is of the medium-assisted force type with rather unusual properties, as demonstrated very recently [M. S. Tomas, Phys. Rev. A 71, 060101(R) (2005)]. This implies…
The van der Waals potential of two atoms in the presence of an arbitrary arrangement of dispersing and absorbing magnetodielectric bodies is studied. Starting from a polarizable atom placed within a given geometry, its interaction with a second polarizable/magnetizable atom is deduced from its Casimir-Polder interaction with a weakly polarizable/magnetizable test body. The general expressions for the van der Waals potential hence obtained are illustrated by considering first the case of two atoms in free space, with special emphasis on the interaction between (i) two polarizable atoms and (ii)…
Boyer showed that a perfect electrically conducting slab repels a perfect magnetically conducting slab, in contrast to the attractive Casimir force between two identical perfect electrically/magnetically conducting slabs. To gain insight for the difference between the Boyer force and the Casimir force, we present the derivation of the Boyer force using the stress tensor method and then using the method of variation in dielectric. The Green dyadic, in terms of electric and magnetic Green's functions, is presented for an electric medium filling half of space and another magnetic medium filling a…
We investigate the Casimir pressure between two parallel plates made of magnetic materials at nonzero temperature. It is shown that for real magnetodielectric materials only the magnetic properties of ferromagnets can influence the Casimir pressure. This influence is accomplished through the contribution of the zero-frequency term of the Lifshitz formula. The possibility of the Casimir repulsion through the vacuum gap is analyzed depending on the model used for the description of the dielectric properties of the metal plates.
Based on a generalization of the Lifshiftz theory we calculate Casimir forces involving magnetodielectric and possibly anisotropic metamaterials, focusing on the possibility of repulsive forces. It is found that Casimir repulsion decreases with magnetic dissipation, and even a small Drude background in metallic-based metamaterials acts to make attractive a Casimir force that would otherwise be predicted to be repulsive. The sign of the force also depends sensitively on the degree of optical anisotropy of the metamaterial, and on the form of the frequency-dependency of the magnetic response.
We consider the interaction pressure acting on the surface of a dielectric sphere enclosed within a magnetodielectric cavity. We determine the sign of this quantity regardless of the geometry of the cavity for systems at thermal equilibrium, extending the Dzyaloshinskii-Lifshitz-Pitaevskii result for homogeneous slabs. As in previous theorems regarding Casimir-Lifshitz forces, the result is based on the scattering formalism. In this case the proof follows from the variable phase approach of electromagnetic scattering. With this, we present configurations in which both the interaction and the s…
We develop a dynamical approach to study the Casimir-Polder force between a initially bare molecule and a magnetodielectric body at finite temperature. Switching on the interaction between the molecule and the field at a particular time, we study the resulting temporal evolution of the Casimir-Polder interaction. The dynamical self-dressing of the molecule and its population-induced dynamics are accounted for and discussed. In particular, we find that the Casimir-Polder force between a chiral molecule and a perfect mirror oscillates in time with a frequency related to the molecular transition …
We consider the Casimir energy in a geometry of an infinite magnetodielectric wedge closed by a circularly cylindrical, perfectly conducting arc embedded in another magnetodielectric medium, under the condition that the speed of light be the same in both media. An expression for the Casimir energy corresponding to the arc is obtained and it is found that in the limit where the reflectivity of the wedge boundaries tends to unity the finite part of the Casimir energy of a perfectly conducting wedge-shaped sheet closed by a circular cylinder is regained. The energy of the latter geometry possesse…
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
This is the abstract of an invited contribution to be presented at the 9th International Conference on Squeezed States and Uncertainty Relations, ICSSUR '05, Besancon, France, May 2-6, 2005.
We study the radiative heat transfer and the Casimir-Lifshitz force occurring between two bodies in a system out of thermal equilibrium. We consider bodies of arbitrary shape and dielectric properties, held at two different temperatures, and immersed in a environmental radiation at a third different temperature. We derive explicit closed-form analytic expressions for the correlations of the electromagnetic field, and for the heat transfer and Casimir-Lifshitz force, in terms of the bodies scattering matrices. We then consider some particular cases which we investigate in detail: the atom-surfa…
Within leading-order perturbation theory, the Casimir-Polder potential of a ground-state atom placed within an arbitrary arrangement of dispersing and absorbing linear bodies can be expressed in terms of the polarizability of the atom and the scattering Green tensor of the body-assisted electromagnetic field. Based on a Born series of the Green tensor, a systematic expansion of the Casimir-Polder potential in powers of the susceptibilities of the bodies is presented. The Born expansion is used to show how and under which conditions the Casimir-Polder force can be related to microscopic many-at…
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…
The canonical quantization of macroscopic electromagnetism was recently presented in New J. Phys. 12 (2010) 123008. This theory is here used to derive the Casimir effect, by considering the special case of thermal and zero-point fields. The stress-energy-momentum tensor of the canonical theory follows from Noether's theorem, and its electromagnetic part in thermal equilibrium gives the Casimir energy density and stress tensor. The results hold for arbitrary inhomogeneous magnetodielectrics and are obtained from a rigorous quantization of electromagnetism in dispersive, dissipative media. Conti…
We calculate the Casimir force between a Ge plate and a Ge spherical lens with neglected and included free charge carriers. Computations with included carriers are performed using the Drude, plasma, and diffusion models. It is shown that the plasma and the Drude models lead to almost identical computational results, while the results obtained using the diffusion model are sandwiched between those obtained with neglected charge carriers and with charge carriers described by the plasma or Drude models.
We establish a general relation between dispersion forces. First, based on QED in causal media, leading-order perturbation theory is used to express both the single-atom Casimir-Polder and the two-atom van der Waals potentials in terms of the atomic polarizabilities and the Green tensor for the body-assisted electromagnetic field. Endowed with this geometry-independent framework, we then employ the Born expansion of the Green tensor together with the Clausius-Mosotti relation to prove that the macroscopic Casimir-Polder potential of an atom in the presence of dielectric bodies is due to an inf…
The Casimir force acting between two test bodies made of different metals is considered. The finiteness of the conductivity of the metals is taken into account perturbatively up to the fourth order of the relative penetration depths of electromagnetic zero-point oscillations into the metals. The influence of nonzero temperature is computed explicitly for separate orders of perturbation and found to be important in the zeroth and first orders only. The configurations of two parallel plates and a sphere (spherical lens) above a plate are considered made of $Au$ and $Cr$. The obtained results can…
In a recent paper [C. Raabe and D.-G. Welsch, Phys. Rev. A 71, 013814 (2005)] an electromagnetic energy-momentum tensor is suggested as an alternative to the Abraham-Minkowski tensor and is applied to calculations of Casimir forces in planar geometries. We argue that the universality of the suggested tensor is doubtful; application of the Raabe-Welsch theory to a simple example in classical electrodynamics shows that their proposed tensor is unable to describe the situation in a simple way. We also show that modified Casimir forces acting on the cavity medium as prescribed by these authors suf…
The repulsive nature of the static Casimir force between two half-spaces, one perfectly dielectric and the other purely magnetic, has been known since Boyer's work [T. H. Boyer, Phys. Rev. A {\bf 9}, 2078 (1974)]. We here analyze the corresponding friction force in the magnetodielectric case. Our main method is that of quantum mechanical statistical mechanics. The basic model we introduce is a harmonic oscillator model: an electric dipole oscillating in the $x$ direction and a magnetic one oscillating in the $y$ direction, while their separation is in the $z$ direction. This is then extended t…
We study the Casimir-Lifshitz force and the radiative heat transfer in a system consisting of three bodies held at three independent temperatures and immersed in a thermal environment, the whole system being in a stationary configuration out of thermal equilibrium. The theory we develop is valid for arbitrary bodies, i.e. for any set of temperatures, dielectric and geometrical properties, and describes each body by means of its scattering operators. For the three-body system we provide a closed-form unified expression of the radiative heat transfer and of the Casimir-Lifshitz force (both in an…
We discuss the temperature correction to the Casimir force between nonideal metallic bodies which caused disagreement in the literature. A general method to find the troubling term is proposed that does not require a direct reference to the Lifshitz formula. The linear in temperature correction is shown to survive for nonideal metals. It is important for small separations between bodies tested in the recent experiments.
The Casimir-Lifshitz interaction between metamaterials is studied using a model that takes into account the structural heterogeneity of the dielectric and magnetic properties of the bodies. A recently developed perturbation theory for the Casimir-Lifshitz interaction between arbitrary material bodies is generalized to include non-uniform magnetic permeability profiles, and used to study the interaction between the magneto-dielectric heterostructures within the leading order. The metamaterials are modeled as two dimensional arrays of domains with varying permittivity and permeability. In the ca…
The Casimir interaction energy due to the vacuum fluctuations of a massive vector field between two perfectly conducting concentric spherical bodies is computed. The TE contribution to the Casimir interaction energy is a direct generalization of the massless case but the TM contribution is much more complicated. Each TM mode is a linear combination of a transverse mode which is the generalization of a TM mode in the massless case and a longitudinal mode that does not appear in the massless case. In contrast to the case of two parallel perfectly conducting plates, the are no TM discrete modes t…
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.
Various applications of the multiple scattering technique to calculating Casimir energy are described. These include the interaction between dilute bodies of various sizes and shapes, temperature dependence, interactions with multilayered and corrugated bodies, and new examples of exactly solvable separable bodies.
In earlier papers we have applied multiple scattering techniques to calculate Casimir forces due to scalar fields between different bodies described by delta function potentials. When the coupling to the potentials became weak, closed-form results were obtained. We simplify this weak-coupling technique and apply it to the case of tenuous dielectric bodies, in which case the method involves the summation of van der Waals (Casimir-Polder) interactions. Once again exact results for finite bodies can be obtained. We present closed formulas describing the interaction between spheres and between cyl…
Within the frame of macroscopic QED in linear, causal media, we study the radiation force of Casimir-Polder type acting on an atom which is positioned near dispersing and absorbing magnetodielectric bodies and initially prepared in an arbitrary electronic state. It is shown that minimal and multipolar coupling lead to essentially the same lowest-order perturbative result for the force acting on an atom in an energy eigenstate. To go beyond perturbation theory, the calculations are based on the exact center-of-mass equation of motion. For a nondriven atom in the weak-coupling regime, the force …
We extend our previous work on the electromagnetic Casimir-Lifshitz interaction between two bodies when one is contained within the other. We focus on the fluctuation-induced pressure acting on the cavity wall, which is assumed to be spherical. This pressure can be positive or negative depending on the response functions describing the bodies and the medium filling the cavity. However, we find that, under general hypotheses, the sign is independent of the geometry of the configuration. This result is based on the representation of the Casimir-Lifshitz energy in terms of transition operators. I…
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…
It is shown that the criticism by Raabe and Welsch of the Dzyaloshinskii-Lifshitz-Pitaevskii theory of the van der Waals-Casimir forces inside a medium is based on misunderstandings. It is explained why and at which conditions one can use the ''Minkowski-like '' stress tensor for calculations of the forces. The reason, why approach of Raabe and Welsch is incorrect, is discussed.
The measurement of the Casimir force between a large gold coated sphere and single crystal silicon plate is performed with an atomic force microscope. A rigorous statistical comparison of data with theory is done, without use of the concept of root-mean-square deviation, and excellent agreement is obtained. The Casimir force between metal and semiconductor is demonstrated to be significantly different than between two similar or dissimilar metals.
We consider detailed roughness and conductivity corrections to the Casimir force in the recent Casimir force measurement employing an Atomic Force Microscope. The roughness of the test bodies-a metal plate and a sphere- was investigated with the Atomic Force Microscope and the Scanning Electron Microscope respectively. It consists of separate crystals of different heights and a stochastic background. The amplitude of roughness relative to the zero roughness level was determined and the corrections to the Casimir force were calculated up to the fourth order in a small parameter (which is this a…
Surface Scattering Expansion of the Casimir-Polder Interaction for Magneto-dielectric Bodies: Convergence Properties for Insulators, Conductors and Semiconductors
We consider the Casimir-Polder interaction energy between a metallic nanoparticle and a metallic plate, as well as the Casimir interaction energy between two macroscopic metal plates, in terms of the many-body dispersion interactions between their constituents. Expressions for two- and three-body dispersion interactions between the microscopic parts of a real metal are first obtained, both in the retarded and non-retarded limits. These expressions are then used to evaluate, a compare each other, the overall two- and three-body contributions to the macroscopic Casimir-Polder and Casimir force, …
Vacuum fluctuations provide a fundamental source of dissipation for systems coupled to quantum fields by radiation pressure. In the dynamical Casimir effect, accelerating neutral bodies in free space give rise to the emission of real photons while experiencing a damping force which plays the role of a radiation reaction force. Analog models where non-stationary conditions for the electromagnetic field simulate the presence of moving plates are currently under experimental investigation. A dissipative force might also appear in the case of uniform relative motion between two bodies, thus leadin…
We study the Casimir-Lifshitz force and the radiative heat transfer occurring between two arbitrary bodies, each one held at a given temperature, surrounded by environmental radiation at a third temperature. The system, in stationary configuration out of thermal equilibrium, is characterized by a force and a heat transfer depending on the three temperatures, and explicitly expressed in terms of the scattering operators of each body. We find a closed-form analytic expression valid for bodies of any geometry and dielectric properties. As an example, the force between two parallel slabs of finite…
Current theory is able to predict the Casimir force between macroscopic bodies with an accuracy only limited by the knowledge of the material parameters. Yet surprisingly after almost 70 years of research, while theory describes the Casimir force between bodies, the force inside macroscopic bodies is poorly understood. In this paper, we develop from a physically intuitive picture the macroscopic theory of the Casimir stress inside inhomogeneous planar media. Our result agrees with a previous ad hoc procedure [T. G. Philbin et al., Ann. Phys. (New York) 325, 579 (2010)], giving both mathematica…
The Casimir and van der Waals forces acting between two metallic plates or a sphere (lens) above a plate are calculated accounting for the finite conductivity of the metals. The simple formalism of surface modes is briefly presented which allows the possibility to obtain the generalization of Lifshitz results for the case of two semi-spaces covered by the thin layers. Additional clarifications of the regularization procedure provides the means to obtain reliable results not only for the force but also for the energy density. This, in turn, leads to the value of the force for the configuration …
The Casimir effect, the dispersion force attracting neutral objects to each other, may be understood in terms of multiple scattering of light between the interacting bodies. We explore the simple model in which the bodies are assumed to possess reflection coefficients independent of the energy and angle of incidence of an impinging field and show how a multitude of information can be extracted within the geometry of two parallel plates. The full thermal behaviour of the model is found and we discuss how non-analytic behaviour emerges in the combined limits of zero temperature and perfect refle…
We compare theoretical expectations for the Casimir force with the results of precise measurements. The force is calculated at finite temperature for multilayered covering of the bodies using the Lifshitz theory. We argue that the dielectric function of the metallization has to be directly measured to reach the necessary precision in the force calculation. Without knowledge of this function one can establish a well defined upper limit on the force using parameters of perfect single-crystal materials. The force measured in the torsion pendulum experiment does not contradict to the upper limit. …
The Casimir force between macroscopic bodies is well understood, but not the Casimir stress inside bodies. Suppose empty space or a uniform medium meets a soft wall where the refractive index is continuous but its derivative jumps. For this situation we predict a characteristic power law for the stress inside the soft wall and close to its edges. Our result shows that such edges are not tolerated in the aggregation of liquids at surfaces, regardless whether the liquid is attracted or repelled.
