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 of excited atoms and a perfect mirror is calculated.
On the basis of macroscopic quantum electrodynamics, a theory of Casimir forces in the presence of on linearly amplifying bodies is presented which provides a consistent framework for studying the effect of, e.g., amplifying left-handed metamaterials on dispersion forces. A general formula for the Casimir force acting on an amplifying body is derived. It is shown that the force can be given in terms of the classical Green tensor and that it can be decomposed into a resonant component associated with emission processes and an off-resonant Lifshitz-type component looking formally the same as for…
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…
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.
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…
The dynamics of three interacting objects has been investigated extensively in Newtonian gravitational physics (often termed the three-body problem), and is important for many quantum systems, including nuclei, Efimov states, and frustrated spin systems. However, the dynamics of three macroscopic objects interacting through quantum vacuum fluctuations (virtual photons) is still an unexplored frontier. Here, we report the first observation of Casimir interactions between three isolated macroscopic objects. We propose and demonstrate a three terminal switchable architecture exploiting opto-mecha…
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…
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.
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…
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 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…
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…
Recently, we have developed a unified approach to QED in arbitrary linearly responding media in equilibrium--media that give rise to absorption [Phys. Rev. A \textbf{75}, (2007) 053813]. In the present paper we show that, under appropriate conditions, the theory can be quite naturally generalized to amplifying media the effect of which is described within the framework of linear response theory. We discuss the limits of validity of the generalized theory and make contact with earlier quantization schemes suggested for the case of linearly and locally responding amplifying dielectric-type media…
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…
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…
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.
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.
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.
Surface Scattering Expansion of the Casimir-Polder Interaction for Magneto-dielectric Bodies: Convergence Properties for Insulators, Conductors and Semiconductors
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…
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, …
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…
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…
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…
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 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.
It has been known that the Lifshitz theory of the Casimir force comes into conflict with the measurement data if the response of conduction electrons in metals to electromagnetic fluctuations is described by the well tested dissipative Drude model. The same theory is in a very good agreement with measurements of the Casimir force from graphene whose spatially nonlocal electromagnetic response is derived from the first principles of quantum electrodynamics. Here, we propose the spatially nonlocal phenomenological dielectric functions of metals which lead to nearly the same response, as the stan…
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…
A multiple scattering formulation is used to calculate the force, arising from fluctuating scalar fields, between distinct bodies described by $δ$-function potentials, so-called semitransparent bodies. (In the limit of strong coupling, a semitransparent boundary becomes a Dirichlet one.) We obtain expressions for the Casimir energies between disjoint parallel semitransparent cylinders and between disjoint semitransparent spheres. In the limit of weak coupling, we derive power series expansions for the energy, which can be exactly summed, so that explicit, very simple, closed-form expressions a…
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. …
We consider the Casimir interaction between (non-magnetic) dielectric bodies or conductors. Our main result is a proof that the Casimir force between two bodies related by reflection is always attractive, independent of the exact form of the bodies or dielectric properties. Apart from being a fundamental property of fields, the theorem and its corollaries also rule out a class of suggestions to obtain repulsive forces, such as the two hemisphere repulsion suggestion and its relatives.
Casimir and Casimir-Polder repulsion have been known for more than 50 years. The general "Lifshitz" configuration of parallel semi-infinite dielectric slabs permits repulsion if they are separated by a dielectric fluid that has a value of permittivity that is intermediate between those of the dielectric slabs. This was indirectly confirmed in the 1970s, and more directly by Capasso's group recently. It has also been known for many years that electrically and magnetically polarizable bodies can experience a repulsive quantum vacuum force. More amenable to practical application are situations wh…
Based on the photon-exciton Hamiltonian a microscopic theory of the Casimir problem for dielectrics is developed. Using well-known many-body techniques we derive a perturbation expansion for the energy which is free from divergences. In the continuum limit we turn off the interaction at a distance smaller than a cut-off distance $a$ to keep the energy finite. We will show that the macroscopic theory of the Casimir effect with hard boundary conditions is not well defined because it ignores the finite distance between the atoms, hence is including infinite self-energy contributions. Nevertheless…
We describe experimental and related theoretical work on the measurement of the Casimir force using semiconductor test bodies. This field of research started in 2005 and several important and interesting results have already been obtained. Specifically, the Casimir force or its gradient were measured in the configuration of an Au-coated sphere and different semiconductor surfaces. It was found that the force magnitude depends significantly on the replacement of the metal with a semiconductor and on the concentration of charge carriers in the semiconductor material. Special attention is paid to…
The impedance boundary condition is used to calculate the Casimir force in configurations of two parallel plates and a shpere (spherical lens) above a plate at both zero and nonzero temperature. The impedance approach allows one to find the Casimir force between the realistic test bodies regardless of the electromagnetic fluctuations inside the media. Although this approach is an approximate one, it has wider areas of application than the Lifshitz theory of the Casimir force. The general formulas of the impedance approach to the theory of the Casimir force are given and the formal substitution…
The thermal Casimir-Lifshitz force among two bodies held at different temperatures displays striking features that are absent in systems in thermal equilibrium. The manifestation of this force has been observed so far only in Bose-Einstein condensates close to a heated substrate, but never between two macroscopic bodies. Observation of the thermal Casimir-Lifhitz force out of thermal equilibrium with conventional Casimir setups is very difficult, because for experimentally accessible separations the thermal force is small compared to the zero-temperature quantum Casimir force, unless prohibiti…
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.
Casimir forces are of fundamental interest because they originate from quantum fluctuations of the electromagnetic field. Apart from controlling the Casimir force via the optical properties of the materials, a number of novel geometries have been proposed to generate repulsive and/or non-monotonic Casimir forces between bodies separated by vacuum gaps. Experimental realization of these geometries, however, is hindered by the difficulties in alignment when the bodies are brought into close proximity. Here, using an on-chip platform with integrated force sensors and actuators, we circumvent the …
We analyse the result of precise measurement of the Casimir force between bodies covered with gold. The values of the parameters used to extrapolate the gold dielectric function to low frequencies are very important and discussed in detail. The finite temperature effect is shown to exceed considerably the experimental errors. The upper limit on the force is found which is smaller than the measured force. Many experimental and theoretical uncertainties were excluded with gold covering and we conclude that, possibly, a new force has been detected at small separations between bodies.
Although Casimir, or quantum vacuum, forces between distinct bodies, or self-stresses of individual bodies, have been calculated by a variety of different methods since 1948, they have always been plagued by divergences. Some of these divergences are associated with the volume, and so may be more or less unambiguously removed, while other divergences are associated with the surface. The interpretation of these has been quite controversial. Particularly mysterious is the contradiction between finite total self-energies and surface divergences in the local energy density. In this paper we clarif…
We present the foundations of a new approach to the Casimir effect based on classical ray optics. We show that a very useful approximation to the Casimir force between arbitrarily shaped smooth conductors can be obtained from knowledge of the paths of light rays that originate at points between these bodies and close on themselves. Although an approximation, the optical method is exact for flat bodies, and is surprisingly accurate and versatile. In this paper we present a self-contained derivation of our approximation, discuss its range of validity and possible improvements, and work out three…
For more than 35 years theorists have studied quantum or Casimir friction, which occurs when two smooth bodies move transversely to each other, experiencing a frictional dissipative force due to quantum electromagnetic fluctuations, which break time-reversal symmetry. These forces are typically very small, unless the bodies are nearly touching, and consequently such effects have never been observed, although lateral Casimir forces have been seen for corrugated surfaces. Partly because of the lack of contact with phenomena, theoretical predictions for the frictional force between parallel plate…
Recently Yampol'skii et al. [Phys. Rev. A v.82, 032511 (2010)] advocated that Lifshitz theory is not applicable when the characteristic wavelength of the fluctuating electromagnetic field, responsible for the thermal correction to the Casimir force, is larger than the size of the metal test bodies. It was claimed that this is the case in experiments which exclude Lifshitz theory combined with the Drude model. We calculate the wavelengths of the evanescent waves giving the dominant contribution to the thermal correction and we find that they are much smaller than the sizes of the test bodies. T…
We propose an experiment for measuring the effective Casimir pressure between two parallel SiC plates with aligned nuclear spins. The prospective constraints on an axion-neutron coupling constant for both hadronic and GUT axions are calculated using the process of one-axion exchange. For this purpose, a general expression for the additional pressure arising between two polarized plates due to the exchange of one axion between their constituent fermions is derived. We demonstrate that only the polarization component perpendicular to the plates contribute to the pressure. The obtained pressure c…
Dynamical Casimir effect (DCE) is the name assigned to the process of generating quanta from vacuum due to an accelerated motion of macroscopic neutral bodies (mirrors) or time-modulation of cavity material properties, as well as the simulation of such processes. Here I review the theoretical results on the detection of DCE using intracavity quantum detectors, such as multi-level atoms, atomic networks and harmonic oscillators. I also stress the mathematical equivalence of this problem to the physics of optical parametric oscillators interacting with atoms or quantum wells, studied in Quantum …
