We derive perturbation analytic expressions for the Casimir free energy and entropy between two dissimilar ferromagnetic plates which are applicale at arbitrarily low temperature. The dielectric properties of metals are described using either the nondissipative plasma model or the Drude model taking into account the dissipation of free charge carriers. Both cases of constant and frequency-dependent magnetic permeability are considered. It is shown that for ferromagnetic metals described by the plasma model the Casimir entropy goes to zero when the temperature vanishes, i.e., the Nernst heat theorem is satisfied. For ferromagnetic metals with perfect crystal lattices described by the Drude model the Casimir entropy goes to a nonzero constant depending on the parameters of a system with vanishing temperature, i.e., the Nernst heat theorem is violated. This constant can be positive which is quite different from the earlier investigated case of two nonmagnetic metals.
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 demonstrate the Casimir interaction between two ferromagnetic boundary surfaces using the dynamic atomic force microscope. The experimental data are found to be in excellent agreement with the predictions of the Lifshitz theory for magnetic boundary surfaces combined with the plasma model approach. It is shown that for magnetic materials the role of hypothetical patch potentials is opposite to that required for reconciliation of the data with the Drude model.
We measured the gradient of the Casimir force between an Au sphere and a plate made of ferromagnetic metal (Ni). It is demonstrated that the magnetic properties influence the force magnitude. This opens prospective opportunities for the control of the Casimir force in nanotechnology and for obtaining Casimir repulsion by using ferromagnetic dielectrics.
We consider the Casimir interaction between a ferromagnetic and a non-magnetic mirror, and show how the Casimir effect gives rise to a magnetic anisotropy in the ferromagnetic layer. The anisotropy is out-of-plane if the non-magnetic plate is optically isotropic. If the non-magnetic plate shows a uniaxial optical anisotropy (with optical axis in the plate plane), we find an in-plane magnetic anisotropy. In both cases, the energetically most favorable magnetization orientation is given by the competition between polar, longitudinal and transverse contributions to the magneto-optical Kerr effect…
Analytic expressions that describe Casimir interactions over the entire range of separations have been limited to planar surfaces. Here we derive analytic expressions for the classical or high-temperature limit of Casimir interactions between two spheres (interior and exterior configurations), including the sphere-plane geometry as a special case, using bispherical coordinates. We consider both Dirichlet boundary conditions and metallic boundary conditions described by the Drude model. At short distances, closed-form expansions are derived from the exact result, displaying an intricate structu…
We discuss recent experimental and theoretical results on the Casimir force between real material bodies made of different materials. Special attention is paid to calculations of the normal Casimir force acting perpendicular to the surface with the help of the Lifshitz theory taking into account the role of free charge carriers. Theoretical results for the thermal Casimir force acting between metallic, dielectric and semiconductor materials are presented and compared with available experimental data. Main attention is concentrated on the possibility to control the magnitude and sign of the Cas…
We derive an exact analytic expression for the high-temperature limit of the Casimir interaction between two Drude spheres of arbitrary radii. Specifically, we determine the Casimir free energy by using the scattering approach in the plane-wave basis. Within a round-trip expansion, we are led to consider the combinatorics of certain partitions of the round trips. The relation between the Casimir free energy and the capacitance matrix of two spheres is discussed. Previously known results for the special cases of a sphere-plane geometry as well as two spheres of equal radii are recovered. An asy…
We develop a diagrammatic representation of the Casimir energy of a multibody configuration. The diagrams represent multiple reflections between the objects and can be organized by a few simple rules. The lowest-order diagrams (or reflections) give the main contribution to the Casimir interaction which proves the usefulness of this expansion. Among some applications of this, we find analytical formulae describing the interaction between "edges", i.e. semi-infinite plates, where we also give a first example of blocking in the context of the Casimir energy. We also find the interaction of edges …
We present an almost fully analytical technique for computing Casimir interactions between periodic lamellar gratings based on a modal approach. Our method improves on previous work on Casimir modal approaches for nanostructures by using the exact form of the eigenvectors of such structures, and computing eigenvalues by solving numerically a simple transcendental equation. In some cases eigenvalues can be solved for exactly, such as the zero frequency limit of gratings modeled by a Drude permittivity. Our technique also allows us to predict analytically the behavior of the Casimir interaction …
The Casimir interaction between one-dimensional metallic objects (cylinders, wires) displays unconventional features. Here we study the orientation dependence of this interaction by computing the Casimir energy between two inclined cylinders over a wide range of separations. We consider Dirichlet, Neumann and perfect metal boundary conditions, both at zero temperature and in the classical high temperature limit. For all types of boundary conditions, we find that at large distances the interaction decays slowly with distance, similarly to the case of parallel cylinders, and at small distances s…
We investigate the thermal Casimir effect of a massless scalar field for two parallel plates moving in the equatorial orbit in Kerr space-time. Under the assumption that the typical cavity size is much smaller than the orbital radius, proposed by Sorge, we deduce the analytical expression of the renormalized free energy in this curved space-time. We also get the analytical representation for the renormalized internal energy, and find that there is a thermal correction to the Casimir energy, which depends on the proper temperature and the proper geometrical parameters of the plates. The asympto…
In view of the current discussion on the subject, an effort is made to show very accurately both analytically and numerically how the Drude dispersive model, assuming the relaxation is nonzero at zero temperature (which is the case when impurities are present), gives consistent results for the Casimir free energy at low temperatures. Specifically, we find that the free energy consists essentially of two terms, one leading term proportional to T^2, and a next term proportional to T^{5/2}. Both these terms give rise to zero Casimir entropy as T -> 0, thus in accordance with Nernst's theorem.
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…
We study the geometrical corrections to the simple Proximity Force Approximation for the non-retarded Casimir force. We present analytical results for the force between objects of various shapes and substrates, and between pairs of objects. We compare the results to those from more exact numerical calculations. We treat spheres, spheroids, cylinders, cubes, cones, and wings; the analytical PFA results together with the geometrical correction factors are summarized in a table.
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 establish strict upper limits for the Casimir interaction between multilayered structures of arbitrary dielectric or diamagnetic materials. We discuss the appearance of different power laws due to frequency-dependent material constants. Simple analytical expressions are in good agreement with numerical calculations based on Lifshitz theory. We discuss the improvements required for current (meta) materials to achieve a repulsive Casimir force.
We obtain constraints on parameters of the Yukawa-type corrections to Newton's gravitational law from measurements of the gradient of the Casimir force between surfaces coated with ferromagnetic metal Ni and from measurements of the Casimir force between Au-coated sinusoidally corrugated surfaces at various angles between corrugations. It is shown that constraints following from the experiment with magnetic surfaces are slightly weaker than currently available strongest constraints, but benefit from increased reliability and independence of systematic effects.The constraints derived from the e…
We investigate the nontrivial interplay between geometry and temperature in the Casimir effect for the sphere-plate and cylinder-plate configurations. At low temperature, the thermal contribution to the Casimir force is dominated by this interplay, implying that standard approximation techniques such as the PFA are inapplicable even in the limit of small surface separation. Thermal fluctuations on scales of the thermal wavelength lead to a delocalization of the thermal force density at low temperatures. As a consequence, the temperature dependence strongly differs from naive expectations. Most…
In this paper we treat the Law's effective Hamiltonian of the Dynamical Casimir Effect in a cavity and construct an analytic approximate solution of the time-dependent Schr{ö}dinger equation under the general setting through a kind of rotating wave approximation (RWA). To the best of our knowledge this is the finest analytic approximate solution.
We derive analytically the asymptotic behavior of the Casimir interaction between a sphere and a plate when the distance between them, $d$, is much smaller than the radius of the sphere, $R$. The leading order and next-to-leading order terms are derived from the exact formula for the Casimir interaction energy. They are found to depend nontrivially on the dielectric functions of the objects. As expected, the leading order term coincides with that derived using the proximity force approximation. The result on the next-to-leading order term complements that found by Bimonte, Emig and Kardar [App…
Using numerical analytic continuation, we compute the Zeta function for the Casimir Effect for circular geometries in 2+1 dimensions. After subtraction of the simple pole of the zeta function, essentially MS renormalization, we find the Casimir force is repulsive. Implications for the stability of 2+1 dimensional domain walls and Axion membranes in 3+1 dimensions are discussed, especially in the context of strongly coupled underlying physics.
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…
The Casimir force is calculated analytically for configurations of two parallel plates and a spherical lens (sphere) above a plate with account of nonzero temperature, finite conductivity of the boundary metal and surface roughness. The permittivity of the metal is described by the plasma model. It is proved that in case of the plasma model the scattering formalism of quantum field theory in Matsubara formulation underlying Lifshitz formula is well defined and no modifications are needed concerning the zero-frequency contribution. The temperature correction to the Casimir force is found comple…
We provide further evidence for the nontrivial interplay between geometry and temperature in the Casimir effect. We investigate the temperature dependence of the Casimir force between an inclined semi-infinite plate above an infinite plate in D dimensions using the worldline formalism. Whereas the high-temperature behavior is always found to be linear in T in accordance with dimensional-reduction arguments, different power-law behaviors at small temperatures emerge. Unlike the case of infinite parallel plates, which shows the well-known T^D behavior of the force, we find a T^{D-1} behavior for…
We investigate the Casimir force between two dissimilar plane mirrors the material properties of which are described by Drude or Lorentz models. We calculate analytically the short and long distance asymptote of the force and relate its behavior to the influence of interacting surface plasmons. In particular we discuss conditions under which Casimir repulsion could be achieved.
In this paper the Casimir energy of two parallel plates made by materials of different penetration depth and no medium in between is derived. We study the Casimir force density and derive analytical constraints on the two penetration depths which are sufficient conditions to ensure repulsion. Compared to other methods our approach needs no specific model for dielectric or magnetic material properties and constitutes a complementary analysis.
We consider the small separation asymptotic expansions of the Casimir interaction energy and the Casimir interaction force between two parallel cylinders. The leading order terms and the next-to-leading order terms are computed analytically. Four combinations of boundary conditions are considered, which are Dirichlet-Dirichlet (DD), Neumann-Neumann (NN), Dirichlet-Neumann (DN) and Neumann-Dirichlet (ND). For the case where one cylinder is inside another cylinder, the computations are shown in detail. In this case, we restrict our attention to the situation where the cylinders are strictly ecce…
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…
We compute the Casimir interaction between a plane and a sphere, the configuration employed in the most precise experiments. The scattering formula is developed by taking a suitably chosen plane-wave and multipole basis and is valid for arbitrary values of the sphere radius, inter-plate distance, temperature and arbitrary dielectric functions for both sphere and plate. Our analytical and numerical results for metallic surfaces show a non-trivial interplay between the effects of curvature, temperature, finite conductivity and dissipation.
We calculate the Casimir force and free energy for plane metallic mirrors at non-zero temperature. Numerical evaluations are given with temperature and conductivity effects treated simultaneously. The results are compared with the approximation where both effects are treated independently and the corrections simply multiplied. The deviation between the exact and approximated results takes the form of a temperature dependent function for which an analytical expression is given. The knowledge of this function allows simple and accurate estimations at the % level.
We study the geometry dependence of the Casimir energy for deformed metal plates by a path integral quantization of the electromagnetic field. For the first time, we give a complete analytical result for the deformation induced change in Casimir energy δ\cal E in an experimentally testable, nontrivial geometry, consisting of a flat and a corrugated plate. Our results show an interesting crossover for δ\cal E as a function of the ratio of the mean plate distance H, to the corrugation length λ: For λ\ll H we find a {\em slower} decay \sim H^{-4}, compared to the H^{-5} behavior predicted by the …
The time-dependent Casimir-Polder force arising during the time evolution of an initially bare two-level atom, interacting with the radiation field and placed near a perfectly conducting wall, is considered. Initially the electromagnetic field is supposed to be in the vacuum state and the atom in its ground state. The analytical expression of the force as a function of time and atom-wall distance, is evaluated from the the time-dependent atom-field interaction energy. Physical features and limits of validity of the results are discussed in detail.
Exploiting conformal symmetry, we derive a simple exact formula for the classical electromagnetic Casimir interaction of two perfectly conducting three-spheres, including the sphere-plate geometry as a special case, in four euclidean dimensions. We verify that the short distance expansion of the Casimir energy agrees to leading order with the Proximity Force Approximation (PFA), while the next-to-leading-order is in agreement with a recently proposed derivative expansion of the Casimir energy. At the next-to-next-to-leading order we find a non-analytic correction to PFA, which for a sphere-pla…
We compute the Casimir Energy of a spherical region using a Surface Impedance approach. We characterize the Surface Impedance of the boundary using plasma model. Exact analytical formulae are obtained by means of the zeta function regularization method and all the divergencies are explicitly computed. We find that it is possible to have negative Casimir energy for some range of values of the relevant parameter. Limits of applicability of the model are discussed.
A new solution for an analytic spectrum of particle creation by an accelerating mirror (dynamical Casimir effect) is given. It is the first model to simultaneously radiate thermally and emit a finite number of particles.
We demonstrate that Casimir forces associated with zero-point fluctuations of quantum vacuum may be substantially affected by the presence of dynamical topological defects. In order to illustrate this nonperturbative effect we study the Casimir interactions between dielectric wires in a compact formulation of Abelian gauge theory in two spatial dimensions. The model possesses topological defects, instanton-like monopoles, which are known to be responsible for nonperturbative generation of a mass gap and for a linear confinement of electrically charged probes. Despite the model has no matter fi…
Using recently developed techniques based on scattering theory, we find the electromagnetic Casimir energy for geometries involving semi-infinite planes, a case that is of particular interest in the design of microelectromechanical devices. We obtain both approximate analytic formulae and exact results requiring only modest numerical computation. Using these results, we analyze the effects of edges and orientation on the Casimir energy. We also demonstrate the accuracy, simplicity, and utility of our approximation scheme, which is based on a multiple reflection expansion.
We numerically evaluate the Casimir interaction energy for configurations involving two perfectly conducting eccentric cylinders and a cylinder in front of a plane. We consider in detail several special cases. For quasi-concentric cylinders, we analyze the convergence of a perturbative evaluation based on sparse matrices. For concentric cylinders, we obtain analytically the corrections to the proximity force approximation up to second order, and we present an improved numerical procedure to evaluate the interaction energy at very small distances. Finally, we consider the configuration of a cyl…
Using the recently derived representation for the polarization tensor in (2+1)-dimensional space-time allowing an analytic continuation to the entire plane of complex frequencies, we obtain simple analytic expressions for the reflection coefficients of graphene at nonzero Matsubara frequencies. In the framework of the Lifshitz theory, these coefficients are shown to lead to nearly exact results for the Casimir free energy and pressure between two graphene sheets. The constituent parts of large thermal effect, arising in the Casimir interaction at short separations due to an explicit parametric…
In this paper we study the role of surface plasmon modes in the Casimir effect. The Casimir energy can be written as a sum over the modes of a real cavity and one may identify two sorts of modes, two evanescent surface plasmon modes and propagative modes. As one of the surface plasmon modes becomes propagative for some choice of parameters we adopt an adiabatic mode definition where we follow this mode into the propagative sector and count it together with the surface plasmon contribution, calling this contribution "plasmonic". We evaluate analytically the contribution of the plasmonic modes t…
We evaluate the Casimir-Polder potential between two atoms in the presence of an infinite perfectly conducting plate and at nonzero temperature. In order to calculate the potential, we use a method based on equal-time spatial correlations of the electric field, already used to evaluate the effect of boundary conditions on interatomic potentials. This method gives also a transparent physical picture of the role of a finite temperature and boundary conditions on the Casimir-Polder potential. We obtain an analytical expression of the potential both in the near and far zones, and consider several …
The thermal Casimir force between two metallic plates is known to depend on the description of material properties. For large separations the dissipative Drude model leads to a force a factor of 2 smaller than the lossless plasma model. Here we show that the plane-sphere geometry, in which current experiment are performed, decreases this ratio to a factor of 3/2, as revealed by exact numerical and large distance analytical calculations. For perfect reflectors, we find a repulsive contribution of thermal photons to the force and negative entropy values at intermediate distances.
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.
We investigate the electromagnetic Casimir interactions of an object contained within an otherwise empty, perfectly conducting spherical shell. For a small object we present analytical calculations of the force, which is directed away from the center of the cavity, and the torque, which tends to align the object opposite to the preferred alignment outside the cavity. For a perfectly conducting sphere as the interior object, we compute the corrections to the proximity force approximation (PFA) numerically. In both cases the results for the interior configuration match smoothly onto those for th…
We show how to compute Casimir forces at nonzero temperatures with time-domain electromagnetic simulations, for example using a finite-difference time-domain (FDTD) method. Compared to our previous zero-temperature time-domain method, only a small modification is required, but we explain that some care is required to properly capture the zero-frequency contribution. We validate the method against analytical and numerical frequency-domain calculations, and show a surprising high-temperature disappearance of a non-monotonic behavior previously demonstrated in a piston-like geometry.
We review a simple technique for evaluating the vacuum energy stemming from non-trivial boundary conditions and review results for the Casimir energy of a massive fermionic field confined in a d+1-dimensional slab-bag and the effect of a uniform magnetic field on the vacuum energy of confined massive bosonic and fermionic fields. New results concerning the Casimir energy and the evaluation of the rate of creation of quanta in kappa-deformed theories are presented.
Casimir pistons are models in which finite Casimir forces can be calculated without any suspect renormalizations. It has been suggested that such forces are always attractive, but we present several counterexamples, notably a simple type of quantum graph in which the sign of the force depends upon the number of edges. We also show that Casimir forces in quantum graphs can be reliably computed by summing over the classical orbits, and study the rate of convergence of the periodic orbit expansion. In generic situations where no analytic expression is available, the sign and approximate magnitude…
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.
We give a rigorous analytical derivation of low-temperature behavior of the Casimir entropy in the framework of the Lifshitz formula combined with the Drude dielectric function. An earlier result that the Casimir entropy at zero temperature is not equal to zero and depends on the parameters of the system is confirmed, i.e. the third law of thermodynamics (the Nernst heat theorem) is violated. We illustrate the resolution of this thermodynamical puzzle in the context of the surface impedance approach by several calculations of the thermal Casimir force and entropy for both real metals and diele…
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 derive an analytic behavior of the Casimir free energy, entropy and pressure of metallic films in vacuum at low temperature. It is shown that this behavior differs significantly depending on whether the plasma or the Drude model is used to describe the dielectric properties of film metal. For metallic films described by the lossless plasma model the thermal corrections to the Casimir energy and pressure drop to zero exponentially fast with increasing film thickness. There is no classical limit in this case. The Casimir entropy satisfies the Nernst heat theorem. For metallic films with perfe…
We discuss the strongest constraints on the Yukawa-type corrections to Newton's gravitational law within a submicrometer interaction range following from measurements of the Casimir force. In this connection the complicated problems arising when comparing the measurement data with the Lifshitz theory are analyzed. Special attention is paid to the results of two recent experiments on measuring the Casimir interaction between ferromagnetic surfaces and sinusoidally corrugated surfaces at various angles between corrugations.
We develop the worldline formalism for computations of composite operators such as the fluctuation induced energy-momentum tensor. As an example, we use a fluctuating real scalar field subject to Dirichlet boundary conditions. The resulting worldline representation can be evaluated by worldline Monte-Carlo methods in continuous spacetime. We benchmark this worldline numerical algorithm with the aid of analytically accessible single-plate and parallel-plate Casimir configurations, providing a detailed analysis of statistical and systematic errors. The method generalizes straightforwardly to arb…
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 vacuum energy of a bosonic field interacting locally with objects is decomposed into irreducible $N$-body parts. The irreducible $N$-body contribution to the vacuum energy is finite if the common intersection $O_1\cap O_2...\cap O_N$ of all $N$ objects $O_i,i=1,..., N$ is empty. I prove that the perturbative expansion of the corresponding irreducible $N$-body spectral function $\tphi^{(N)}(β)$ for $β\sim 0$ vanishes to all orders even if some of the objects intersect. These irreducible spectral functions and their associated Casimir energies in principle can be computed numerically or appr…
Recently, Høye, Brevik, Ellingsen and Aarseth (quant-ph/0703174) claimed that the use of the Drude dielectric function leads to zero Casimir entropy at zero temperature in accordance with Nernst's theorem. We demonstrate that their proof is not applicable to metals with perfect crystal lattices having no impurities. Thus there is no any contradiction with previous results in the literature proving that the Drude dielectric function violates the Nernst theorem for the Casimir entropy in the case of perfect crystal lattices. We also indicate mistakes in the coefficients of their asymptotic expre…
