Barash has calculated the Casimir forces between parallel birefringent plates with optical axes parallel to the plate boundaries [Izv. Vyssh. Uchebn.\ Zaved., Radiofiz., {\bf 12}, 1637 (1978)]. The interesting new feature of the solution compared to the case of isotropic plates is the existence of a Casimir torque which acts to line up the optical axes if they are not parallel or perpendicular. The forces were found from a calculation of the Helmholtz free energy of the electromagnetic field. Given the length of the calculations in this problem and hopes of an experimental measurement of the torque, it is important to check the results for the Casimir forces by a different method. We provide this check by calculating the electromagnetic stress tensor between the plates and showing that the resulting forces are in agreement with those found by Barash.
We present detailed numerical calculations of the mechanical torque induced by quantum fluctuations on two parallel birefringent plates with in plane optical anisotropy, separated by either vacuum or a liquid (ethanol). The torque is found to vary as $\sin(2θ)$, where $θ$ represents the angle between the two optical axes, and its magnitude rapidly increases with decreasing plate separation $d$. For a 40 $μ$m diameter disk, made out of either quartz or calcite, kept parallel to a Barium Titanate plate at $d\simeq 100$ nm, the maximum torque (at $θ={π\over 4}$) is of the order of $\simeq 10^{-19…
The linear momentum and angular momentum of virtual photons of quantum vacuum fluctuations can induce the Casimir force and the Casimir torque, respectively. While the Casimir force has been measured extensively, the Casimir torque has not been observed experimentally though it was predicted over forty years ago. Here we propose to detect the Casimir torque with an optically levitated nanorod near a birefringent plate in vacuum. The axis of the nanorod tends to align with the polarization direction of the linearly polarized optical tweezer. When its axis is not parallel or perpendicular to the…
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 describe theoretically the main characteristics of the steady state regime of a type II Optical Parametric Oscillator (OPO) containing a birefringent plate. In such a device the signal and idler waves are at the same time linearly coupled by the plate and nonlinearly coupled by the $χ^{(2)}$ crystal. This mixed coupling allows, in some well-defined range of the control parameters, a frequency degenerate operation as well as phase locking between the signal and idler modes. We describe here a complete model taking into account all possible effects in the system, \emph{i.e.} arbitrary rotatio…
We consider vacuum fluctuations of the quantum electromagnetic field in the presence of an infinite and perfectly conducting plate. We evaluate how the change of vacuum fluctuations due to the plate modifies the Casimir-Polder potential between two atoms placed near the plate. We use two different methods to evaluate the Casimir-Polder potential in the presence of the plate. They also give new insights on the role of boundary conditions in the Casimir-Polder interatomic potential, as well as indications for possible generalizations to more complicated boundary conditions.
The Casimir force between two parallel plates separated by anisotropic media is investigated. We theoretically calculate the Casimir force between two parallel plates when the interspace between the plates is filled with anisotropic media. Our result shows that the anisotropy of the material between the plates can significantly affect the Casimir force, especially the direction of the force. If ignoring the anisotropy of the in-between material makes the force to be repulsive (attractive), by contrast, taking the anisotropy into account may produce an extra attractive (repulsive) force. The ph…
We make remarks on Hoodbhoy's paper [J. Phys. A: Math. Gen. 38 (2005) 10253-10256] by pointing out the single plate considered could be elastic and there will be deformations once the net (one-sided) force being upon the plate which will change the position-dependent potential imposed by Hoodbhoy.
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…
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 …
We investigate in detail the Casimir torque induced by quantum vacuum fluctuations between two nanostructured plates. Our calculations are based on the scattering approach and take into account the coupling between different modes induced by the shape of the surface which are neglected in any sort of proximity approximation or effective medium approach. We then present an experimental setup aiming at measuring this torque.
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 apply the generalized zeta function method to compute the Casimir energy and pressure between an unusual pair of parallel plates at finite temperature, namely: a perfectly conducting plate and an infinitely permeable one. The high and low temperature limits of these quantities are discussed; relationships between high and low temperature limits are estabkished by means of a modified version of the temperature inversion symmetry.
We study the Casimir interaction between perfectly conducting sphere and plate in the classical limit of high temperatures. By taking the small-distance expansion of the exact scattering formula, we compute the leading correction to the Casimir energy beyond the commonly employed proximity force approximation. We find that for a sphere of radius $R$ at distance $d$ from the plate the correction is of the form $\ln^2 (d/R)$, in agreement with indications from recent large-scale numerical computations. We develop a fast-converging numerical scheme for computing the Casimir interaction to high pr…
We consider two parallel corrugated plates and show that a Casimir torque arises when the corrugation directions are not aligned. We follow the scattering approach and calculate the Casimir energy up to second order in the corrugation amplitudes, taking into account nonspecular reflections, polarization mixing and the finite conductivity of the metals. We compare our results with the proximity force approximation, which overestimates the torque by a factor 2 when taking the conditions that optimize the effect. We argue that the Casimir torque could be measured for separation distances as large…
The Casimir force between two metallic plates is affected by their roughness state. This effect is usually calculated through the so-called `proximity force approximation' which is only valid for small enough wavevectors in the spectrum of the roughness profile. We introduce here a more general description with a wavevector-dependent roughness sensitivity of the Casimir effect. Since the proximity force approximation underestimates the effect, a measurement of the roughness spectrum is needed to achieve the desired level of accuracy in the theory-experiment comparison.
We present a calculation of the Casimir torque acting on a liquid crystal near a birefringent crystal. In this system, a liquid crystal bulk is uniformly aligned at one surface and is twisted at the other surface by a birefringent crystal, e.g. barium titanate. The liquid crystal is separated from the solid crystal by an isotropic, transparent material such as SiO$_2$. By varying the thickness of the deposited layer, we can observe the effect of retardation on the torque (which differentiates it from the close-range van der Waals torque). We find that a barium titanate slab would cause 5CB (4-…
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 propose a new theory of thermal Casimir effect, holding for the experimentally important case of metallic surfaces with a roughness having a spatial scale smaller than the skin depth. The theory is based on a simple phenomenological model for a rough conductor, that explicitly takes account of the fact that ohmic conduction in the immediate vicinity of the surface of a conductor is much impeded by surface roughness, if the amplitude of roughness is smaller than the skin depth. As a result of the new model, we find that surface roughness strongly influences the magnitude of the thermal corre…
We calculate the Casimir force between two parallel ideal metal plates when there is an intervening chiral medium present. Making use of methods of quantum statistical mechanics we show how the force can be found in a simple and compact way. The expression for the force is in agreement with that obtained recently by Q.-D. Jiang and F. Wilczek [Phys. Rev. B {\bf 99}, 125403 (2019)], in their case with the use of Green function methods.
Using a recently developed theory of the Casimir force (Raabe C and Welsch D-G 2005 Phys. Rev. A 71 013814), we calculate the force that acts on a plate in front of a planar wall and the force that acts on the plate in the case where the plate is part of matter that fills the space in front of the wall. We show that in the limit of a dielectric plate whose permittivity is close to unity, the force obtained in the former case reduces to the ordinary, i.e., unscreened Casimir-Polder force acting on isolated atoms. In the latter case, the theory yields the Casimir-Polder force that is screened by…
We investigate the Casimir force between a microfabricated elliptic cylinder (cylindrical lens) and a plate made of real materials. After a brief discussion of the fabrication procedure, which typically results in elliptic rather than circular cylinders, the Lifshitz-type formulas for the Casimir force and for its gradient are derived. In the specific case of equal semiaxes, the resulting formulas coincide with those derived previously for circular cylinders. The nanofabrication procedure may also result in asymmetric cylindrical lenses obtained from parts of two different cylinders, or rotate…
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…
We describe theoretically the quantum properties of atype-II Optical Parametric Oscillator containing a birefringent plate which induces a linear coupling between the orthogonally polarized signal and idler beams and results in phase locking between these two beams. As in a classical OPO, the signal and idler waves show large quantum correlations which can be measured experimentally due to the phase locking between the two beams. We study the influence of the waveplate on the various criteria characterizing quantum correlations. We show in particular that the quantum correlations can be maximi…
Casimir-Polder interactions cause energy and momentum exchange between microscopic and macroscopic bodies, a process mediated by quantum fluctuations in the coupled matter-electromagnetic field system. The dynamics of such effects are yet to be experimentally investigated due to the dominance of static effects at currently attainable atomic velocities. However, Y. Guo and Z. Jacob [\textit{Opt. Express}, 22:26193-26202, 2014] have proposed a non-static two-plate set-up where quantum fluctuation mediated effects have a strong velocity-dependent resonance, leading to a giant friction force on th…
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 …
We propose a test on the role of relaxation properties of conduction electrons in the Casimir pressure between two parallel metal-coated plates kept at different temperatures. It is shown that for sufficiently thick metallic coatings the Casimir pressure and pressure gradient are determined by the mean of the equilibrium contributions calculated at temperatures of the two plates and by the term independent on separation. Numerical computations of the nonequilibrium pressures are performed for two parallel Au plates of finite thickness as a function of separation and temperature of one of the p…
The Casimir effect is a physical manifestation of zero point energy of quantum vacuum. In a relativistic quantum field theory, Poincaré symmetry of the theory seems, at first sight, to imply that non-zero vacuum energy is inconsistent with translational invariance of the vacuum. In the setting of two uniform boundary plates at rest, quantum fields outside the plates have (1+2)-dimensional Poincaré symmtry. Taking a massless scalar field as an example, we have examined the consistency between the Poincaré symmetry and the existence of the vacuum enegy. We note that, in quantum theory, symmetrie…
We calculate the roughness correction to the Casimir effect in the parallel plates geometry for metallic plates described by the plasma model. The calculation is perturbative in the roughness amplitude with arbitrary values for the plasma wavelength, the plate separation and the roughness correlation length. The correction is found to be always larger than the result obtained in the Proximity Force Approximation.
A recent theoretical calculation shows that the Casimir force between two parallel plates can be repulsive for plates with nontrivial magnetic properties (O. Kenneth et al., Phys. Rev. Lett. 89, 033001 (2002)). According to the authors, the effect may be observed with known materials, such as ferrites and garnets, and it might be possible to engineer micro- or nanoelectromechanical systems (MEMS or NEMS) that could take advantage of a short range repulsive force. Here we show that on the contrary the Casimir force between two parallel plates in vacuum at micron and submicron distance is always…
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 investigate Casimir effect as well as thermal Casimir effect for a pair of parallel perfectly plates placed in general stationary space-time background. It is found that the Casimir energy is influenced by the 00-component of metric and the corresponding quantity in dragging frame. We give a scheme to renormalize thermal correction to free energy in curved space-time. It is shown that the thermal corrections to Casimir thermodynamic quantities not only depend on the proper temperature and proper geometrical parameters of the plates, but also on the determinant of space-time metric.
This paper extends our recent study on Casimir friction forces for dielectric plates moving parallel to each other [J. S. Høye and I. Brevik, Eur. Phys. J. D {\bf 68}, 61 (2014)], to the case where the plates are no longer restricted to rectilinear motion. Part of the mathematical formalism thereby becomes more cumbersome, but reduces in the end to the form that we could expect to be the natural one in advance. As an example, we calculate the Casimir torque on a planar disc rotating with constant angular velocity around its vertical symmetry axis next to another plate.
We study the Casimir effect at finite temperature for a massless scalar field in the parallel plates geometry in N spatial dimensions, under various combinations of Dirichlet and Neumann boundary conditions on the plates. We show that in all these cases the entropy, in the limit where energy equipartitioning applies, is a geometrical factor whose sign determines the sign of the Casimir force.
In this paper we calculate the Casimir-Polder force density (force per unit area acting on the elements of the surface) on a metallic plate placed in front of a neutral atom. To obtain the force density we use the quantum operator associated to the electromagnetic stress tensor. We explicitly show that the integral of this force density over the plate reproduces the total force acting on the plate. This result shows that, although the force is obtained as a sum of surface element-atom contributions, the stress-tensor method includes also nonadditive components of Casimir-Polder forces in the e…
The temperature dependence of the Casimir effect for the radiation field confined between two conducting plates is analysed. The Casimir energy is shown to decline exponentially with temperature while the Casimir entropy which is defined in the text is shown to approach a limit which depends only on the geometry of the constraining plates. The results are discussed in terms of the relation between the Bose distribution function and the equipartition theorem - a relation based on a study by Einstein and Stern circa 1913.
Using perturbation theory the first order dispersive correction to the Casimir energy between two plates separated by a dielectric material is calculated. It falls off with the plate separation as 1/L^6. The result is derived both from evaluation of the zero-point energy and within the Lifshitz formulation. It is pointed out that a possible surface term can be more important, varying like 1/L^5.
We find the lateral projection of the Casimir force for a configuration of a sphere above a corrugated plate. This force tends to change the sphere position in the direction of a nearest corrugation maximum. The probability distribution describing different positions of a sphere above a corrugated plate is suggested which is fitted well with experimental data demonstrating the nontrivial boundary dependence of the Casimir force.
Birefringent materials or nanostructures that introduce phase differences between two linear polarizations underpin the operation of wave plates for polarization control of light. Here we develop metasurfaces realizing a distinct class of complex-birefringent wave plates, which combine polarization transformation with a judiciously tailored polarization-dependent phase retardance and amplitude filtering via diffraction. We prove that the presence of loss enables the mapping from any chosen generally non-orthogonal pair of polarizations to any other pair at the output. We establish an optimal t…
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 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.
For the first time, we have studied the impact of magnetic polarizability of a neutral metallic particle on the dynamical Casimir-Polder force, when the particle is uniformly moving with relativistic velocity parallel to the surface of an ideally conducting plate. The material properties of particle are taken into account within the Drude model approach.
We examine recent prescriptions for estimating the thermal Casimir force between two metallic plates from the point of view of their consistency with the Bohr-van Leeuwen theorem of classical statistical physics. We find that prescriptions including the effect of ohmic dissipation satisfy the theorem, while prescriptions that neglect ohmic dissipation do not.
Electromagnetic waves propagation on either rotating or anisotropic spacetime backgrounds (such as Kerr and Gödel metrics, or Bianchi-I metric) produce a reduction of the magnitude of Casimir forces between plates. These curved spacetimes behave as chiral or birefringent materials producing dispersion of electromagnetic waves, in such a way that right- and left-circularly polarized light waves propagate with different phase velocities. Results are explicitly calculated for discussed cases. The difference on the wavevectors of the two polarized electromagnetic waves produces an abatement of a C…
We combine linear response theory and dimensional regularization in order to derive the dynamical Casimir force in the low frequency regime. We consider two parallel plates moving along the normal direction in $D-$dimensional space. We assume the free-space values for the mass of each plate to be known, and obtain finite, separation-dependent mass corrections resulting from the combined effect of the two plates. The global mass correction is proportional to the static Casimir energy, in agreement with Einstein's law of equivalence between mass and energy for stressed rigid bodies.
We study the Casimir pressure between two similar plates of finite thickness kept at different temperatures in the case when the dielectric permittivity of the plates depends on temperature. It is suggested to consider the dielectric permittivity at two different temperatures as the permittivities of two dissimilar bodies, thus allowing to apply the theory of Casimir forces out of thermal equilibrium developed earlier in the literature. Following this approach, we show that, in addition to the equilibrium contribution to the nonequilibrium Casimir pressure, a proper nonequilibrium contribution…
In the presence of an external field, the imposition of specific boundary conditions can lead to interesting new manifestations of the Casimir effect. In particular, it is shown here that even a single conducting plate may experience a non-zero force due to vacuum fluctuations. The origins of this force lie in the change induced by the external potential in the density of available quantum states.
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…
We present boundary conditions for the electromagnetic fields on a δ-function plate, having both electric and magnetic properties, sandwiched between two magneto-electric semi-infinite half spaces. The optical properties for an isolated δ-function plate are shown to be independent of the longitudinal material properties of the plate. The Casimir-Polder energy between an isotropically polarizable atom and a magneto-electric δ-function plate is attractive for a purely electric δ-function plate, repulsive for a purely magnetic δ-function plate, and vanishes for the simultaneous perfect conductor …
In this work, we consider the Casimir effect due to massless fermionic fields in the presence of long cylinders. More precisely, we consider the interaction between a cylinder parallel to a plate, between two parallel cylinders outside each other, and between a cylinder lying parallelly inside another cylinder. We derive the explicit formulas for the Casimir interaction energies and compute the leading and the next-to-leading order terms of the small separation asymptotic expansions. As expected, the leading order terms coincide with the proximity force approximations. We compare the results o…
We study the Casimir energy of a massless scalar field that obeys Dirichlet boundary conditions on a hyperboloid facing a plate. We use the optical approximation including the first six reflections and compare the results with the predictions of the proximity force approximation and the semi-classical method. We also consider finite size effects by contrasting the infinite with a finite plate. We find sizable and qualitative differences between the new optical method and the more traditional approaches.
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…
The Casimir energy is constructed for a system consisting of two semi-infinite slabs of anisotropic material. Each of them is characterized by bulk complex dielectric permittivity tensor and surface conductivity on the free boundary. We found general form of the scattering matrix and Fresnel coefficients for each part of the system by solving Maxwell equations in the anisotropic media.
We discuss possibility to obtain stronger constraints on non-Newtonian gravity from measuring the gradient of the Casimir force between a cylinder and a plate. Exact analytical expression for the Yukawa-type force in a cylinder-plate configuration is obtained, as well as its asymptotic expansions. The gravitational force is compared with the Casimir force acting between a cylinder and a plate. Numerical computations for the prospective constraints on non-Newtonian gravity are performed for recently proposed experiment using a microfabricated cylinder attached to a micromachined oscillator. Spe…
