We perform detailed comparison of the experimental data of the experiment on the determination of the Casimir pressure between two parallel Au plates with the theoretical values computed using the Lifshitz formula at zero temperature. Computations are done using the optical data for the complex index of refraction of Au extrapolated to low frequencies by means of the Drude model with both most often used and other suggested Drude parameters. It is shown that the experimental data exclude the Lifshitz formula at zero temperature at a 70% confidence level if the Drude model with most often used values of the parameters is employed. If other values of the Drude parameters are used, the Lifshitz formula at zero frequency is experimentally excluded at a 95% confidence level. The Lifshitz formula at zero temperature combined with the generalized plasma-like model with most often used value of the plasma frequency is shown to be experimentally consistent. We propose a decisive experiment which will shed additional light on the role of relaxation properties of conduction electrons in the Casimir effect.
More than forty years ago, Barash published a calculation of the full retarded Casimir-Lifshitz torque for planar birefringent media with arbitrary degrees of anisotropy. An independent theoretical confirmation has been lacking since. We report a systematic and transparent derivation of the torque between two media with both electric and magnetic birefringence. Our approach, based on an eigenmode decomposition of Maxwell's equations, generalizes Barash's result for electrically birefringent materials, and can be generalized to a wide range of anisotropic materials and finite thickness effects.
We propose the Casimir effect as a general method to observe Lifshitz transitions in electron systems. The concept is demonstrated with a planar spin-orbit coupled semiconductor in a magnetic field. We calculate the Casimir force between two such semiconductors and between the semiconductor and a metal as a function of the Zeeman splitting in the semiconductor. The Zeeman field causes a Fermi pocket in the semiconductor to form or collapse by tuning the system through a topological Lifshitz transition. We find that the Casimir force experiences a kink at the transition point and noticeably dif…
We develop a new theory for Casimir-Lifshitz and Casimir-Polder interactions with semiconductor surfaces that takes into account charge drift in the bulk material. The corresponding frequency-dependent dispersion relations describe a crossover between Lifshitz results for dielectrics and for good conductors. In the quasi-static limit, our calculated reflection amplitudes coincide with those recently computed to account for Debye screening in the thermal Lifshitz force with conducting surfaces with small density of carriers.
Recently D. A. R. Dalvit and S. K. Lamoreaux arXiv:0805.1676 [Phys. Rev. Lett. v.101, 163203 (2008)] suggested the modification of the reflection coefficients in the Lifshitz theory taking into account the screening effects and diffusion currents. We demonstrate that the suggested modification is thermodynamically and experimentally inconsistent.
We reply to the "Comment on 'The Lifshitz-Matsubara sum formula for the Casimir pressure between magnetic metallic mirrors'" [arXiv:1604.06432]. We believe the comment misrepresents our papers, and fails to provide a plausible resolution to the conflict between theory and experiment.
A new model for calculating the Casimir-Lifshitz force per unit length for two dielectric rods is proposed, based on the Green function method of classical electrodynamics and the Lorentz model for permittivity.
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 interaction between arbitrary dielectric heterostructures is studied within the framework of a recently developed dielectric contrast perturbation theory. It is shown that periodically patterned dielectric or metallic structures lead to oscillatory lateral Casimir-Lifshitz forces, as well as modulations in the normal force as they are displaced with respect to one another. The strength of these oscillatory contributions increases with decreasing gap size and increasing contrast in the dielectric properties of the materials used in the heterostructures.
In this paper we correct a previously obtained expression for the Casimir-Lifshitz attraction force (Dedkov, Kyasov, Surf. Sci., 2010) in a system of two parallel plates in relative nonrelativistic motion, assuming the total thermal equilibrium of the system.
We show that the claims expressed in the Comment arXiv:0810.3244v1 by R.S. Decca et al against our paper, D.A.R. Dalvit and S.K. Lamoreaux, Phys. Rev. Lett. {\bf 101}, 163203 (2008), are wrong and manifestly inconsistent with basic principles of statistical physics.
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 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…
The zero-temperature Casimir-Lifshitz force between two plates moving parallel to each other at arbitrary constant speed was found in [New J. Phys. 11, 033035 (2009)]. The solution is here generalized to the case where the plates are at different temperatures. The Casimir-Lifshitz force is obtained by calculating the electromagnetic stress tensor, using the method employed by Antezza et al. [Phys. Rev. A 77, 022901 (2008)] for non-moving plates at different temperatures. The perpendicular force on the plates has contributions from the quantum vacuum and from the thermal radiation; both of thes…
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…
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 examine the conditions of validity for the Lifshitz-Matsubara sum formula for the Casimir pressure between magnetic metallic plane mirrors. As in the previously studied case of non-magnetic materials (Guerout et al, Phys. Rev. E 90 042125), we recover the usual expression for the lossy model of optical response, but not for the lossless plasma model. We also show that the modes associated with the Foucault currents play a crucial role in the limit of vanishing losses, in contrast to expectations.
We carefully re-examine the conditions of validity for the consistent derivation of the Lifshitz-Matsubara sum formula for the Casimir pressure between metallic plane mirrors. We recover the usual expression for the lossy Drude model, but not for the lossless plasma model. We give an interpretation of this new result in terms of the modes associated with the Foucault currents which play a role in the limit of vanishing losses, in contrast to common expectations.
In this work, we present detailed measurements of the Casimir-Lifshitz force between two gold surfaces (a sphere and a plate) immersed in ethanol and study the effect of residual electrostatic forces, which are dominated by static fields within the apparatus and can be reduced with proper shielding. Electrostatic forces are further reduced by Debye screening through the addition of salt ions to the liquid. Additionally, the salt leads to a reduction of the Casimir-Lifshitz force by screening the zero-frequency contribution to the force; however, the effect is small between gold surfaces at the…
We study the singularities that the Casimir energy of a scalar field in spacetimes with Lifshitz dimensions exhibits, and provide expressions of the energy in terms of multidimensional zeta functions for the massless case. Using the zeta-regularization method, we found that when the 4-dimensional spacetime has Lifshitz dimensions, then for specific values of the critical exponents, the Casimir energy is singular, in contrast to the non-Lifshitz case. Particularly we found that when the value of the critical exponent is $z=2$, the Casimir energy is singular, while for $z\geq 3$ the Casimir ener…
The applicability of the Lifshitz formula is discussed to the case of two thick parallel plates made of real metal. The usual description of the zero-point vacuum oscillations on the background of the frequency-dependent dielectric permittivity is shown to be in contradiction with thermodynamics. Instead, the Lifshitz formula for the Casimir free energy should be reformulated in terms of the reflection coefficients containing the surface impedance instead of the dielectric permittivity. This approach is presently confirmed experimentally by precision measurements of the van der Waals and Casim…
The general theory of electromagnetic--fluctuation--induced interactions in dielectric bodies as formulated by Dzyaloshinskii, Lifshitz, and Pitaevskii is rewritten as a perturbation theory in terms of the spatial contrast in (imaginary) frequency dependent dielectric function. The formulation can be used to calculate the Casimir-Lifshitz forces for dielectric objects of arbitrary geometry, as a perturbative expansion in the dielectric contrast, and could thus complement the existing theories that use perturbation in geometrical features. We find that expansion in dielectric contrast recasts t…
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…
The Casimir-Lifshitz torque between two biaxially polarizable anisotropic planar slabs is shown to exhibit a non-trivial sign-reversal in its rotational sense. The critical distance $a_c$ between the slabs that marks this reversal is characterized by the frequency $ω_c\!\sim \!c/2a_c$ at which the in-planar polarizabilities along the two principal axes are equal. The two materials seek to align their principal axes of polarizabilities in one direction below $a_c$, while above $a_c$ their axes try to align rotated perpendicular relative to their previous minimum energy orientation. The sign-rev…
The controversy concerning the temperature correction to the Casimir force has been ongoing for almost a decade with no view to a solution and has recently been extended to include semiconducting materials. We review some theoretical aspects of formal violations of Nernst's heat theorem in the context of Casimir Lifshitz thermodynamics and the role of the exponent of the leading term of the dielectric permittivity with respect to imaginary frequency. A general formalism for calculating the temperature corrections to free energy at low temperatures is developed for systems which do not exhibit …
We provide a Quantum Field Theory derivation of Lifshitz formula for the Casimir force due to a fluctuating real scalar field in $d+1$ dimensions. The field is coupled to two imperfect, thick, plane mirrors, which are modeled by background potentials localized on their positions. The derivation proceeds from the calculation of the vacuum energy in the Euclidean version of the system, reducing the problem to the evaluation of a functional determinant. The latter is written, via Gelfand-Yaglom's formula, in terms of functions depending on the structure of the potential describing each mirror; th…
A classification of the electromagnetic modes on open and closed spatial domains containing media with piecewise homogeneous permittivities is used to facilitate the derivation of quantum induced Casimir stresses in dielectrics. By directly exploiting the complex analytic properties of solutions of the macroscopic Maxwell equations for open systems it is shown how regular expressions for such stresses can be expressed in terms of double integrals involving either real or pure imaginary frequencies associated with harmonic modes in conformity with the Lifshitz theory for separated planar dielec…
A general theory of photon-mediated energy and momentum transfer in N-body planar systems out of thermal equilibrium is introduced. It is based on the combination of the scattering theory and the fluctuational-electrodynamics approach in many-body systems. By making a Landauer-like formulation of the heat transfer problem, explicit formulas for the energy transmission coefficients between two distinct slabs as well as the self-coupling coefficients are derived and expressed in terms of the reflection and transmission coefficients of the single bodies. We also show how to calculate local equili…
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…
Recently Guerout et al. [Phys. Rev. E, v.93, 022108 (2016)] advocated that the lossless plasma model has to be redefined as the limit of the Drude model when the relaxation parameter goes to zero. It was claimed that the previously used plasma model cannot correctly describe the Casimir pressure between two plates made of both nonmagnetic and magnetic metals and has to be replaced with the redefined one. We show that the suggested redefinition does not satisfy necessary physical requirements imposed on the dielectric permittivity. We also present a plausible explanation to the fact that the lo…
The Casimir force and free energy at low temperatures has been the subject of focus for some time. We calculate the temperature correction to the Casimir-Lifshitz free energy between two parallel plates made of dielectric material possessing a constant conductivity at low temperatures, described through a Drude-type dielectric function. For the transverse magnetic (TM) mode such a calculation is new. A further calculation for the case of the TE mode is thereafter presented which extends and generalizes previous work for metals. A numerical study is undertaken to verify the correctness of the a…
Despite suggestions to the contrary, we show in this paper that the usual dispersive form of the electromagnetic energy must be used to derive the Lifshitz force between parallel dielectric media. This conclusion follows from the general form of the quantum vacuum energy, which is the basis of the multiple-scattering formalism. As an illustration, we explicitly derive the Lifshitz formula for the interaction between parallel dielectric semispaces, including dispersion, starting from the expression for the total energy of the system. The issues of constancy of the energy between parallel plates…
We have reviewed the Comment of Geyer et al. [arXiv:0708.1548] concerning our recent work [Phys. Rev. A 75, 060102 (R) (2007)], and while we disagree with their criticisms, we acknowledge them for giving us the opportunity to add interesting addition material and a more detailed description of our experiment. We describe further our calculation and explain why a more sophisticated model is not warranted. We also present detailed experiments on the effects of electrostatic forces in our measurements and show that the contribution due to work function differences is small and that the residual e…
The problem of thermal Casimir force, which consists in disagreement of theoretical predictions of the fundamental Lifshitz theory with the measurement data of high precision experiments and some peculiar properties of the Casimir entropy, is reviewed. We discuss different approaches to the resolution of this problem proposed in the literature during the last twenty years. Particular attention is given to the role of the effects of spatial dispersion. The recently suggested nonlocal Drude-like permittivities which take proper account of the dissipation of conduction electrons and bring the pre…
We consider the interaction between a ZnO nanorod and a SiO2 nanorod in bromobenzene. Using optical data for the interacting objects and ambient we calculate the force - from short-range attractive van der Waals force to intermediate range repulsive Casimir-Lifshitz force to long range entropically driven attraction. The nonretarded van der Waals interaction is attractive at all separations. We demonstrate a retardation driven repulsion at intermediate separations. At short separations (in the nonretarded limit) and at large separations (in the classical limit) the interaction is attractive. T…
Based on the theory of macroscopic quantum electrodynamics, we generalize the expression of the Casimir force for nonreciprocal media. The essential ingredient of this result is the Green's tensor between two nonreciprocal semi-infinite slabs including a reflexion matrix with four coefficients that mixes optical polarizations. This Green's tensor does not obey Lorentz's reciprocity and thus violates time-reversal symmetry. The general result for the Casimir force is analyzed in the retarded and nonretarded limits, concentrating on the influences arising from reflections with or without change …
We investigate the force acting between two parallel plates held at different temperatures. The force reproduces, as limiting cases, the well known Casimir-Lifshitz surface-surface force at thermal equilibrium and the surface-atom force out of thermal equilibrium recently derived by M. Antezza \emph{et al.}, Phys. Rev. Lett. {\bf 95}, 113202 (2005). The asymptotic behavior of the force at large distances is explicitly discussed. In particular when one of the two bodies is a rarefied gas the force is not additive, being proportional to the square root of the density. Nontrivial cross-over regio…
We study the Casimir-Lifshitz interaction out of thermal equilibrium, with particular attention devoted to the surface-surface and surface-atom configurations. A systematic investigation of the contributions to the force coming from the propagating and evanescent components of the electromagnetic radiation is performed. The large distance behaviors of such interactions is discussed, and both analytical and numerical results are compared with the equilibrium ones. A detailed analysis of the crossing between the surface-surface and the surface-rarefied body, and finally the surface-atom force is…
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…
The thermal free energy and pressure of dispersion interaction between two graphene sheets described by the Dirac model are calculated using the Lifshitz formula with reflection coefficients expressed via the polarization tensor. The obtained results for a pristine graphene are found to be in agreement with computations using Coulomb coupling between density fluctuations. For a graphene with nonzero mass gap parameter a qualitatively different behavior for the free energy and pressure is obtained. The Lifshitz formula with reflection coefficients expressed via the polarization tensor is used a…
We study the field theoretical model of a scalar field in presence of spacial inhomogeneities in form of one and two finite width mirrors (material slabs). The interaction of the scalar field with the defect is described with position-dependent mass term. Within this model we derive the interaction of two finite width mirrors, establish the correspondence of the model to the Lifshitz formula and construct limiting procedure to obtain finite self-energy of a single mirror without any normalization condition.
Lifshitz-type formulas are obtained for the van der Waals and Casimir interaction between graphene and a material plate, graphene and an atom or a molecule, and between a single-wall carbon nanotube and a plate. The reflection properties of electromagnetic oscillations on graphene are governed by the specific boundary conditions imposed on the infinitely thin positively charged plasma sheet, carrying a continuous fluid with some mass and charge density. The obtained formulas are applied to graphene interacting with Au and Si plates, to hydrogen atoms and molecules interacting with graphene, an…
Mainly on nano-scale, but maybe not exclusively, it can be imagined a spontaneous charge disjunction inside certain media due to the fluctuations, collisions, wall effects, radiation or/and other presently unknown interactions. The repulsive Casimir-Lifshitz force may be also a good candidate as a source of this kind of phenomenon. To cover this assumption mathematically an additional term should appear in the Maxwell equations. As a consequence of this term, similar Klein-Gordon equations with a negative mass term will be obtained for the electric and magnetic fields. These equations may have…
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…
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…
