Recent measurements of Casimir forces have provided evidence of an intricate modification of quantum fluctuations of the electromagnetic field in complex geometries. Here we introduce a multiple scattering description for Casimir interactions between bodies of arbitrary shape and material composition, admitting an expansion as a sequence of inter- and intra-body wave scatterings. Interactions in complex geometries can be computed within the current experimental resolution from typically a few wave scatterings, notably without any a priori knowledge of the scattering amplitudes of the bodies. Some first applications demonstrate the power of the approach.
The dielectric sphere has been an important test case for understanding and calculating the vacuum force of a dielectric body onto itself. Here we develop a method for computing this force in homogeneous spheres of arbitrary dielectric properties embedded in arbitrary homogeneous backgrounds, assuming only that both materials are isotropic and dispersionless. Our results agree with known special cases; most notably we reproduce the prediction of Boyer and Schwinger et al. of a repulsive Casimir force of a perfectly reflecting shell. Our results disagree with the literature in the dilute limit.…
The Casimir energy, free energy and Casimir force are evaluated, at arbitrary finite temperature, for a dilute dielectric ball with uniform velocity of light inside the ball and in the surrounding medium. In particular, we investigate the classical limit at high temperature. The Casimir force found is repulsive, as in previous calculations.
We present a full analysis of the mode spectrum in a cavity formed by two parallel plates, one of which is a magneto-dielectric, e.g. a metamaterial, while the other one is metallic, and obtain dispersion relations in closed form. The optical properties of the cavity walls are described in terms of realistic models for the effective permittivity and the permeability. Surface polaritons, i.e. electromagnetic modes that have at least partly an evanescent character, are shown to dominate the Casimir interaction at small separations. We analyze in detail the s-polarized polaritons, which are a cha…
The Casimir energy of a dilute dielectric cylinder, with the same light-velocity as in its surrounding medium, is evaluated exactly to first order in $ξ^2$ and numerically to higher orders in $ξ^2$. The first part is carried out using addition formulas for Bessel functions, and no Debye expansions are required.
We derive an exact solution for the Casimir force between two arbitrary periodic dielectric gratings and illustrate our method by applying it to two nanostructured silicon gratings. We also reproduce the Casimir force gradient measured recently [1] between a silicon grating and a gold sphere taking into account the material dependence of the force. We find good agreement between our theoretical results and the measured values both in absolute force values and the ratios between the exact force and PFA predictions.
We have used density functional theory to calculate the anisotropic dielectric functions for ultrathin gold sheets (composed of 1, 3, 6, and 15 atomic layers). Such films are important components in nano-electromechanical systems. When using correct dielectric functions rather than bulk gold dielectric functions we predict an enhanced attractive Casimir-Lifshitz force (at most around 20%) between two atomically thin gold sheets. For thicker sheets the dielectric properties and the corresponding Casimir forces approach those of gold half-spaces. The magnitude of the corrections that we predict …
We review recent results obtained in the physics of the thermal Casimir force acting between two dielectrics, dielectric and metal, and between metal and semiconductor. The detailed derivation for the low-temperature behavior of the Casimir free energy, pressure and entropy in the configuration of two real dielectric plates is presented. For dielectrics with finite static dielectric permittivity it is shown that the Nernst heat theorem is satisfied. Hence, the Lifshitz theory of the van der Waals and Casimir forces is demonstrated to be consistent with thermodynamics. The nonzero dc conductivi…
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.
We derive an expression for the Casimir force between slabs with arbitrary dielectric properties characterized by their reflection coefficients. The formalism presented here is applicable to media with a local or a non-local dielectric response, an infinite or a finite width, inhomogeneous dissipative, etc. Our results reduce to the Lifshitz formula for the force between semi-infinite dielectric slabs by replacing the reflection coefficients by the Fresnel amplitudes.
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.
A diffraction problem for a flat Chern-Simons layer at plane boundary of a dielectric half space is solved. The Casimir energy of two dielectric half spaces with Chern-Simons layers at plane-parallel boundaries separated by a vacuum slit is derived. Crossing from the repulsive to the attractive Casimir force is analyzed for two Au and two Si half spaces with boundary Chern-Simons layers. Boundary quantum Hall layers in external magnetic field lead to Casimir repulsion at nanoscales. We discuss features that make systems with boundary quantum Hall layers unique for force measurements and search…
We introduce the concept of Casimir friction, i.e. friction due to quantum fluctuations. In this first article we describe the calculation of a constant torque, arising from the scattering of quantum fluctuations, on a dielectric rotating in an electromagnetic vacuum.
The Casimir interaction between two thick parallel plates, one made of metal and the other of dielectric, is investigated at nonzero temperature. It is shown that in some temperature intervals the Casimir pressure and the free energy of a fluctuating field are the nonmonotonous functions of temperature and the respective Casimir entropy can be negative. The physical interpretation for these conclusions is given. At the same time we demonstrate that the entropy vanishes when the temperature goes to zero, i.e., in the Casimir interaction between metal and dielectric the Nernst heat theorem is sa…
The Casimir energy of a dilute homogeneous nonmagnetic dielectric ball at zero temperature is derived analytically within a microscopic realistic model of dielectrics for an arbitrary physically possible frequency dispersion of dielectric permittivity. Divergences are absent in calculations, a minimum interatomic distance is a physical cut-off. Casimir surface force is proved to be attractive. A physical definition of the Casimir energy is discussed.
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…
In 1948 H.B.G.Casimir predicted that an attractive force between two perfectly conducting neutral plates exists due to changes in the electromagnetic vacuum energy caused by the influence of the plates. In 1956 E.M. Lifshitz derived an extension of Casimir's expression applicable to finite temperatures and arbitrary dielectric constants for the two half-spaces and the gap in between them. It is shown in this brief report that, while the Lifshitz formula predicts an attractive force for the case of identical dielectric constants for the two half-spaces, a repulsive force results in some circums…
Starting from a Lagrangian, electromagnetic field in the presence of a nonlinear dielectric medium is quantized using path-integral techniques and correlation functions of different fields are calculated. The susceptibilities of the nonlinear medium are obtained and their relation to coupling functions are determined. Finally, the Casimir energy and force in the presence of a nonlinear medium at finite temperature is calculated.
We use the mode summation method together with zeta-function regularization to compute the Casimir energy of a dilute dielectric cylinder. The method is very transparent, and sheds light on the reason the resulting energy vanishes.
The statistical mechanical approach to Casimir problems for dielectrics separated by a vacuum gap turns out to be compact and effective. A central ingredient of this method is the effect of interacting fluctuating dipole moments of the polarizable particles. At arbitrary temperature the path integral formulation of quantized particles, developed by Høye-Stell and others, is needed. At high temperature - the limit considered in the present paper - the classical theory is however sufficient. Our present theory is related to an idea put forward earlier by Jancovici and {Š}amaj (2004), namely to e…
We study the lateral Casimir force between corrugated dielectric plates. We use the dielectric contrast perturbation theory [R. Golestanian, {\it Phys. Rev. Lett.} {\bf 95}, 230601, (2005)], which allows us to consider arbitrary deformations with large amplitudes. We consider sinusoidal, rectangular, and sawtooth corrugations, for samples made of silicon and gold. We use the plasma and Drude-Lorentz models for the permittivity of gold and silicon, respectively. For these geometries and materials, the lateral Casimir force is {\it not} a sinusoidal function of the relative lateral displacement …
Electromagnetic field quantization in the presence of two semi-infinite dielectrics with moving interface is investigated in $1+1$-dimensional space-time. The moving interface is modeled for small displacements and the field equation is solved perturbatively. Input-output relations and spectral distribution of emitted photons are obtained and the effect of small transitions trough the interface discussed.
We study the Casimir pressure for a dielectric-diamagnetic cylinder subject to light velocity conservation and with a dispersion law analogous to Sellmeir's rule. Similarities to and differences from the spherical case are pointed out.
The Casimir effect in an inhomogeneous dielectric is investigated using Lifshitz's theory of electromagnetic vacuum energy. A permittivity function that depends continuously on one Cartesian coordinate is chosen, bounded on each side by homogeneous dielectrics. The result for the Casimir stress is infinite everywhere inside the inhomogeneous region, a divergence that does not occur for piece-wise homogeneous dielectrics with planar boundaries. A Casimir force per unit volume can be extracted from the infinite stress but it diverges on the boundaries between the inhomogeneous medium and the hom…
We find the low-temperature behavior of the Casimir-Polder free energy for a polarizable and magnetizable atom interacting with a plate made of ferromagnetic dielectric material. It is shown that the corresponding Casimir-Polder entropy goes to zero with vanishing temperature, i.e., the Nernst heat theorem is satisfied, if the dc conductivity of the plate material is disregarded in calculations. If the dc conductivity is taken into account, the Nernst theorem is violated. These results are discussed in light of recent experiments.
We analyze some consequences of the Casimir-type zero-point radiation pressure. These include macroscopic "vacuum" forces on a metallic layer in-between a dielectric medium and an inert ($ε(ω) = 1$) one. Ways to control the sign of these forces, based on dielectric properties of the media, are thus suggested. Finally, the large positive Casimir pressure, due to surface plasmons on thin metallic layers, is evaluated and discussed.
Based on calculations involving an idealized boundary condition, it has long been assumed that the stress on a spherical conducting shell is repulsive. We use the more realistic case of a Drude dielectric to show that the stress is attractive, matching the generic behavior of Casimir forces in electromagnetism. We trace the discrepancy between these two cases to interactions between the electromagnetic quantum fluctuations and the dielectric material.
The low-temperature behavior of the Casimir-Polder free energy and entropy for an atom near a dielectric plate are found on the basis of the Lifshitz theory. The obtained results are shown to be thermodynamically consistent if the dc conductivity of the plate material is disregarded. With inclusion of dc conductivity, both the standard Lifshitz theory (for all dielectrics) and its generalization taking into account screening effects (for a wide range of dielectrics) violate the Nernst heat theorem. The inclusion of the screening effects is also shown to be inconsistent with experimental data o…
Casimir forces on suspended thin dielectric membranes covered with graphene
The Hong Kong University of Science and Technology（香港科技大学）；Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier（夏尔·库隆实验室（L2C），CNRS-蒙彼利埃大学联合研究单位5221）
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…
The application of the Lifshitz theory to describe the thermal Casimir force between dielectrics and semiconductors is considered. It is shown that for all true dielectrics (i.e., for all materials having zero conductivity at zero temperature) the inclusion of a nonzero conductivity arising at nonzero temperature into the model of dielectric response leads to the violation of the Nernst heat theorem. This result refers equally to simple insulators, intrinsic semiconductors, Mott-Hubbard dielectrics and doped semiconductors with doping concentration below a critical value. We demonstrate that i…
We study numerically the Casimir interaction between dielectrics in both two and three dimensions. We demonstrate how sparse matrix factorizations enable one to study torsional interactions in three dimensions. In two dimensions we study the full cross-over between non-retarded and retarded interactions as a function of separation. We use constrained factorizations in order to measure the interaction of a particle with a rough dielectric surface and compare with a scaling argument.
The Casimir free energy of peptide films on a silicon substrate: Impact of dielectric-to-metal transition in silicon and nanoparticles in peptide
Central Astronomical Observatory at Pulkovo of the Russian Academy of Sciences（俄罗斯科学院普尔科沃中央天文台）；Peter the Great Saint Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
The interaction between drifting carriers and traveling electromagnetic waves is considered within the context of the classical Boltzmann transport equation to compute the Casimir-Lifshitz force between media with small density of charge carriers, including dielectrics and intrinsic semiconductors. We expand upon our previous work [Phys. Rev. Lett. {\bf 101}, 163203 (2008)] and derive in some detail the frequency-dependent reflection amplitudes in this theory and compute the corresponding Casimir free energy for a parallel plate configuration. We critically discuss the the issue of verificatio…
We study the Casimir interaction between a dielectric nanosphere and a metallic plane, using the multiple scattering theory. Exact results are obtained with the dielectric described by a Sellmeier model and the metal by a Drude model. Asymptotic forms are discussed for small spheres, large or small distances. The well-known Casimir-Polder formula is recovered at the limit of vanishingly small spheres, while an expression better behaved at small distances is found for any finite value of the radius. The exact results are of particular interest for the study of quantum states of nanospheres in t…
The Casimir effect is considered for a wedge with opening angle $α$, with perfectly conducting walls, when the interior region is filled with an isotropic and nondispersive medium with permittivity $ε$ and permeability $μ$. The electromagnetic energy-momentum tensor in the bulk is calculated, together with the surface stress on the walls. A discussion is given on the possibilities for measuring the influence of the medium, via the Casimir-Polder force.
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.
The analytic asymptotic expressions for the Casimir free energy, pressure and entropy at low temperature in the configuration of one metal and one dielectric plate are obtained. For this purpose we develop the perturbation theory in a small parameter proportional to the product of the separation between the plates and the temperature. This is done using both the simplified model of an ideal metal and of a dielectric with constant dielectric permittivity and for the realistic case of the metal and dielectric with frequency-dependent dielectric permittivities. The analytic expressions for all re…
Surface Scattering Expansion of the Casimir-Polder Interaction for Magneto-dielectric Bodies: Convergence Properties for Insulators, Conductors and Semiconductors
We demonstrate that the Casimir-Polder force for a molecule near the surface of a real dielectric substrate out of thermal equilibrium displays distinctive behaviors as compared to that at thermal equilibrium. In particular, when the molecule-substrate separation is much less than the molecular transition wave-length, the CP force in the high temperature limit can be dramatically manipulated by varying the relative magnitude of the temperatures of the substrate and the environment so that the attractive-to-repulsive transition can occur beyond certain a threshold temperature of either the subs…
We present an explicit analytic calculation of the energy-level shift of an atom in front of a non-dispersive and non-dissipative dielectric slab. We work with the fully quantized electromagnetic field, taking retardation into account. We give the shift as a two-dimensional integral and use asymptotic analysis to find expressions for it in various retarded and non-retarded limiting cases. The results can be used to estimate the energy shift of an atom close to layered microstructures.
In this paper we sum over the spherical modes appearing in the expression for the Casimir energy of a conducting sphere and of a dielectric ball (assuming the same speed of light inside and outside), before doing the frequency integration. We derive closed integral expressions that allow the calculations to be done to all orders, without the use of regularization procedures. The technique of mode summation using a contour integral is critically examined.
Recently the influence of dielectric and geometrical properties on the Casimir force between dispersing and absorbing multilayered plates in the zero-temperature limit has been studied within a 1D quantization scheme for the electromagnetic field in the presence of causal media [R. Esquivel-Sirvent, C. Villarreal, and G.H. Cocoletzi, Phys. Rev. Lett. 64, 052108 (2001)]. In the present paper a rigorous 3D analysis is given, which shows that for complex heterostructures the 1D theory only roughly reflects the dependence of the Casimir force on the plate separation in general. Further, an extensi…
We develop a general numerical method aimed at studying particle production from vacuum states in a variety of settings. As a first example we look at particle production in a simple cosmological model. We apply the same approach to the dynamical Casimir effect, with special focus on the case of an oscillating mirror. We confirm previous estimates and obtain long-time production rates and particle spectra for both resonant and off-resonant frequencies. Finally, we simulate a system with space and time-dependent optical properties, analogous to a one-dimensional expanding dielectric bubble. We …
In this paper we calculate the Casimir energy for a dielectric-diamagnetic cylinder with the speed of light differing on the inside and outside. Although the result is in general divergent, special cases are meaningful. The well-known results for a uniform speed of light are reproduced. The self-stress on a purely dielectric cylinder is shown to vanish through second order in the deviation of the permittivity from its vacuum value, in agreement with the result calculated from the sum of van der Waals forces.These results are unambiguously separated from divergent terms.
A general, exact formula is derived for the expectation value of the electromagnetic energy density of an inhomogeneous absorbing and dispersive dielectric medium in thermal equilibrium, assuming that the medium is well approximated as a continuum. From this formula we obtain the formal expression for the Casimir force density. Unlike most previous approaches to Casimir effects in which absorption is either ignored or admitted implicitly through the required analytic properties of the permittivity, we include dissipation explicitly via the coupling of each dipole oscillator of the medium to a …
The possibility of repulsive Casimir forces between small metal spheres and a dielectric half-space is discussed. We treat a model in which the spheres have a dielectric function given by the Drude model, and the radius of the sphere is small compared to the corresponding plasma wavelength. The half-space is also described by the same model, but with a different plasma frequency. We find that in the retarded limit, the force is quasi-oscillatory. This leads to the prediction of stable equilibrium points at which the sphere could levitate in the Earth's gravitational field. This seems to lead t…
Exploiting methods of Quantum Field Theory we compute the bulk polarization tensor and bulk dielectric functions for Dirac materials in the presence of a mass gap, chemical potential, and finite temperature. Using these results (and neglecting eventual boundary effects), we study the Casimir interaction of Dirac materials. We describe in detail the characteristic features of the dielectric functions and their influence on the Casimir pressure.
It has always been conventionally understood that, in the dilute limit, the Casimir energy of interaction between bodies or the Casimir self-energy of a dielectric body could be identified with the sum of the van der Waals or Casimir-Polder energies of the constituents of the bodies. Recently, this proposition for self-energies has been challenged by Avni and Leonhardt [Ann.\ Phys.\ {\bf 395}, 326 (2018)], who find that the energy or self-stress of a homogeneous dielectric ball with permittivity $\varepsilon$ begins with a term of order $\varepsilon-1$. Here we demonstrate that this cannot be …
The Casimir and van der Waals interaction between two dissimilar thick dielectric plates is reconsidered on the basis of thermal quantum field theory in Matsubara formulation. We briefly review two main derivations of the Lifshitz formula in the framework of thermal quantum field theory without use of the fluctuation-dissipation theorem. A set of special conditions is formulated under which these derivations remain valid in the presence of dissipation. The low-temperature behavior of the Casimir and van der Waals interactions between dissimilar dielectrics is found analytically from the Lifshi…
Reflection of electromagnetic waves from a periodic grating can be described in terms of a discrete coupled multichannel scattering problem. By modeling the grating as a space- and frequency-dependent dielectric, it is possible to use a variable phase method, applied to a generalized Helmholtz equation incorporating both transverse and longitudinal modes, to efficiently compute the scattering $S$-matrix. The projection onto transverse modes of this result, evaluated for imaginary wave vector, provides the information necessary for a Casimir energy calculation. This approach is of particular in…
We consider the vacuum energy of the electromagnetic field in the background of spherically symmetric dielectrics, subject to a cut-off frequency in the dispersion relations. The effect of this frequency dependent boundary condition between media is described in terms of the {\it incomplete} $ζ$-functions of the problem. The use of the Debye asymptotic expansion for Bessel functions allows to determine the dominant (volume, area, ...) terms in the Casimir energy. The application of these expressions to the case of a gas bubble immersed in water is discussed, and results consistent with Schwing…
We discuss repulsive Casimir forces between dielectric materials with non trivial magnetic susceptibility. It is shown that considerations based on naive pair-wise summation of Van der Waals and Casimir Polder forces may not only give an incorrect estimate of the magnitude of the total Casimir force, but even the wrong sign of the force when materials with high dielectric and magnetic response are involved. Indeed repulsive Casimir forces may be found in a large range of parameters, and we suggest that the effect may be realized in known materials. The phenomenon of repulsive Casimir forces ma…
