The Casimir energy is computed in the geometry of interest for the most precise experiments, a plane and a sphere in electromagnetic vacuum. The scattering formula is developed on adapted plane-waves and multipole basis, leading to an expression valid for arbitrary relative values of the sphere radius and inter-plate distance. In the limiting case of perfect reflection, the electromagnetic result is found to depart from the commonly used proximity-force approximation (PFA) significantly more rapidly than expected from scalar computations.
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.
We discuss the calculation of Casimir forces between a collection of $N$-dielectric spheres. This is done by evaluating directly the force on a sphere constructed from a stress tensor, rather than an interaction energy. Two and three body forces between the spheres are evaluated for setups of two and three sphere systems respectively. An approximate large-$N$ limit is also obtained for the functional dependence on the number of spheres.
An experimentally feasible realization of testing quantum-vacuum geometric phases of photons by using a gyrotropic-medium optical fibre via Casimir's effect is proposed.
An expression for the Casimir stress on arbitrary dispersive and lossy linear magnetodielectric matter at finite temperature, including left-handed material, is derived and applied to spherical systems. To cast the relevant part of the scattering Green tensor for a general magnetodielectric sphere in a convenient form, classical Mie scattering is reformulated.
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…
A summary of recent calculations of Casimir forces between a collection of $N$-dielectric spheres is presented. This is done by evaluating directly the force on a sphere constructed from a stress tensor, rather than an interaction energy. A loop integral formulation is also discussed where we rewrite the expressions for the force in terms of loop integrals for the effective classical propagation of the electric and magnetic fields.
A recent experiment [J.L. Garrett et al., Phys. Rev. Lett {\bf 120}, 040401 (2018)] measured for the first time the gradient of the Casimir force between two gold spheres at room temperature. The theoretical analysis of the data was carried out using the standard Proximity Force Approximation (PFA). A fit of the data, using a parametrization of the force valid for the sphere-plate geometry, was used by the authors to place a bound on deviations from PFA. Motivated by this work, we compute the Casimir force between two gold spheres at finite temperature. The semi-analytic formula for the Casimi…
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 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 in the non-retarded limit between a spherical nanoparticle and a substrate, and we found that high-multipolar contributions are very important when the sphere is very close to the substrate. We show that the highly inhomegenous electromagnetic field induced by the presence of the substrate, can enhance the Casimir force by orders of magnitude, compared with the classical dipolar approximation.
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 three-body Casimir-Polder potential between one excited and two ground-state atoms is evaluated. A physical model based on the dressed field correlations of vacuum fluctuations is used, generalizing a model previously introduced for three ground-state atoms. Although the three-body potential with one excited atom is already known in the literature, our model gives new insights on the nature of non-additive Casimir-Polder forces with one or more excited atoms.
We derive the explicit expression for the Casimir energy between a sphere and a 1D grating, in terms of the sphere and grating reflection matrices, and valid for arbitrary materials, sphere radius, and grating geometric parameters. We then numerically calculate the Casimir energy between a metallic (gold) sphere and a dielectric (fused silica) lamellar grating at room temperature, and explore its dependence on the sphere radius, grating-sphere separation, and lateral displacement. We quantitatively investigate the geometrical dependence of the interaction, which is sensitive to the grating hei…
The electromagnetic vacuum is known to have energy. It has been recently argued that the quantum vacuum can possess momentum, that adds up to the momentum of matter. This ``Casimir momentum'' is closely related to the Casimir effect, in which case energy is exchanged. In previous theory it was treated semi-classically. We present a non-relativistic quantum theory for the linear momentum of electromagnetic zero-point fluctuations, considering an harmonic oscillator subject to crossed, quasi-static magnetic and electric and coupled to the quantum vacuum. We derive a contribution of the quantum v…
We consider the angular momentum of a charge q rotating in a homogeneous magnetic field and study the role of the electromagnetic quantum vacuum. Its orbital angular momentum is caused by the recoil of energetic vacuum photons that grows as n^2, i.e. faster than the kinetic angular momentum -2n\hbar of a Landau level.
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 report measurements of the Casimir force between a gold sphere and a silicon surface with an array of nanoscale, rectangular corrugations using a micromechanical torsional oscillator. At distance between 150 nm and 500 nm, the measured force shows significant deviations from the pairwise additive formulism, demonstrating the strong dependence of the Casimir force on the shape of the interacting bodies. The observed deviation, however, is smaller than the calculated values for perfectly conducting surfaces, possibly due to the interplay between finite conductivity and geometry effects.
We develop a spectral representation formalism to calculate the Casimir force in the non-retarded limit, between a spherical particle and a substrate, both with arbitrary local dielectric properties. This spectral formalism allows one to do a systematic study of the force as a function of the geometrical variables separately from the dielectric properties. We found that the force does not follow a simple power-law as a function of the separation between the sphere and substrate. As a consequence, the non-retarded Casimir force is enhanced by several orders of magnitude as the sphere approaches…
The Casimir force for charge-neutral, perfect conductors of non-planar geometric configurations have been investigated. The configurations are: (1) the plate-hemisphere, (2) the hemisphere-hemisphere and (3) the spherical shell. The resulting Casimir forces for these physical arrangements have been found to be attractive. The repulsive Casimir force found by Boyer for a spherical shell is a special case requiring stringent material property of the sphere, as well as the specific boundary conditions for the wave modes inside and outside of the sphere. The necessary criteria in detecting Boyer's…
Electromagnetic vacuum fluctuations have observable consequences, like the Casimir force between mirrors in vacuum. This force is now measured with good accuracy and agreement with theory when the effect of imperfect reflection of mirrors is properly taken into account. We discuss the simple case of bulk metallic mirrors described by a plasma model and show that simple scaling laws are obtained at the limits of long and short distances. The crossover between the short and long-distance laws is quite similar to the crossover between Van der Waals and Casimir-Polder forces for two atoms in vacuu…
On the basis of macroscopic quantum electrodynamics and point-scattering techniques, we derive a closed expression for the Casimir-Polder force between a ground-state atom and a small magnetodielectric sphere in an arbitrary environment. In order to allow for the presence of both bodies and media, local-field corrections are taken into account. Our results are compared with the known van der Waals force between two ground-state atoms. To continuously interpolate between the two extreme cases of a single atom and a macroscopic sphere, we also derive the force between an atom and a sphere of var…
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.
We calculate the Casimir-Polder intermolecular potential using an effective Hamiltonian recently introduced. We show that the potential can be expressed in terms of the dynamical polarizabilities of the two atoms and the equal-time spatial correlation of the electric field in the vacuum state. This gives support to an interesting physical model recently proposed in the literature, where the potential is obtained from the classical interaction between the instantaneous atomic dipoles induced and correlated by the vacuum fluctuations. Also, the results obtained suggest a more general validity of…
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.
Dissipative electromagnetic response and scattering geometry are potential sources for the appearance of a negative Casimir entropy. We show that the dissipative contribution familiar from the plane-plane geometry appears also in the plane-sphere and the sphere-sphere geometries and adds to the negative Casimir entropy known to exist in these geometries even for perfectly reflecting objects. Taking the sphere-sphere geometry as an example, we carry out a scattering-channel analysis which allows to distinguish between the contributions of different polarizations. We demonstrate that dissipation…
We discuss the possibility of verifying the equivalence principle for the zero-point energy of quantum electrodynamics, by evaluating the force, produced by vacuum fluctuations, acting on a rigid Casimir cavity in a weak gravitational field. The resulting force has opposite direction with respect to the gravitational acceleration; the order of magnitude for a multi-layer cavity configuration is derived and experimental feasibility is discussed, taking into account current technological resources.
The Casimir-Polder and van der Waals interaction energy of an atom with infinitely thin sphere with finite conductivity is investigated in the framework of the hydrodynamic approach. We put the sphere into spherical cavity inside the infinite dielectric media, then calculate the energy of vacuum fluctuations in the context of the zeta-function approach. The energy for a single atom is obtained by rarefying media. The Casimir-Polder expression for an atom and plate is recovered in the limit of the infinite radius of the sphere. Assuming a finite radius of the sphere, the interaction energy of a…
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.
In the work, the thermal and vacuum fluctuation is predicted capable of generating a Casimir thrust force on a rotating chiral particle, which will push or pull the particle along the rotation axis. The Casimir thrust force comes from two origins: i) the rotation-induced symmetry-breaking in the vacuum and thermal fluctuation and ii) the chiral cross-coupling between electric and magnetic fields and dipoles, which can convert the vacuum spin angular momentum (SAM) to the vacuum force. Using the fluctuation dissipation theorem (FDT), we derive the analytical expressions for the vacuum thrust fo…
We study the electromagnetic Casimir interaction of a compact object contained inside a closed cavity of another compact object. We express the interaction energy in terms of the objects' scattering matrices and translation matrices that relate the coordinate systems appropriate to each object. When the enclosing object is an otherwise empty metallic spherical shell, much larger than the internal object, and the two are sufficiently separated, the Casimir force can be expressed in terms of the static electric and magnetic multipole polarizabilities of the internal object, which is analogous to…
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 consider the Casimir interaction energy between a plane and a sphere of radius $R$ at finite temperature $T$ as a function of the distance of closest approach $L$. Typical experimental conditions are such that the thermal wavelength $λ_T=\hbar c/k_\mathrm{B}T$ satisfies the condition $L\ll λ_T\ll R$. We derive the leading correction to the proximity-force approximation valid for such intermediate temperatures by developing the scattering formula in the plane-wave basis. Our analytical result captures the joint effect of the spherical geometry and temperature and is written as a sum of tempe…
We study the Casimir force between a nanoparticle and a substrate. We consider the interaction of metal nanoparticles with different substrates within the dipolar approximation. We study the force as a function of the distance for gold and potassium spheres, which are over a substrate of titanium dioxide, sapphire and a perfect conductor. We show that Casimir force is important in systems at the nanometer scale. We study the force as a function of the material properties, radii of the spheres, and the distance between the sphere and the substrate.
In a recent work Brevik \emph{et al.} have offered formal proofs of two results which figure prominently in calculations of the Casimir pressure on a sphere. It is shown by means of simple counterexamples that each of those proofs is necessarily incorrect.
We calculate the Casimir energy and entropy for two spheres described by the perfect metal model, plasma model, and Drude model in the large separation limit. We obtain nonmonotonic behavior of the Helmholtz free energy with separation and temperature for the perfect metal and plasma models, leading to parameter ranges with negative entropy, and also nonmonotonic behavior of the entropy with temperature and the separation between the spheres. This nonmonotonic behavior has not been found for Drude model. The appearance of this anomalous behavior of the entropy is discussed as well as its therm…
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.
The attractive Casimir force acting on a micrometer-sphere suspended in a spherical dip, close to the wall, is discussed. This setup is in principle directly accessible to experiment. The sphere and the substrate are assumed to be made of the same perfectly conducting material.
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 present systematic measurements of the Casimir force between a gold-coated plate and a sphere coated with a Hydrogen Switchable Mirror (HSM). HSMs are shiny metals that can become transparent by hydrogenation. In spite of such a dramatic change of the optical properties of the sphere, we did not observe any significant decrease of the Casimir force after filling the experimental apparatus with hydrogen. This counterintuitive result can be explained by the Lifshitz theory that describes the Casimir attraction between metallic and dielectric materials.
The van der Waals and Casimir-Polder interaction energy of an atom with an infinitely thin sphere with finite conductivity is investigated in the framework of the hydrodynamic approach at finite temperature. This configuration models the real interaction of an atom with fullerene. The Lifshitz approach is used to find the free energy. We find the explicit expression for the free energy and perform the analysis of it for i) high and low temperatures, ii) large radii of sphere and ii) short separation between an atom and sphere. At low temperatures the thermal part of the free energy approaches …
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.…
We develop an exact method for computing the Casimir energy between arbitrary compact objects, both with boundary conditions for a scalar field and dielectrics or perfect conductors for the electromagnetic field. The energy is obtained as an interaction between multipoles, generated by quantum source or current fluctuations. The objects' shape and composition enter only through their scattering matrices. The result is exact when all multipoles are included, and converges rapidly. A low frequency expansion yields the energy as a series in the ratio of the objects' size to their separation. As e…
The Casimir effect results from the optomechanical coupling between field fluctuations and mirrors in quantum vacuum. This contribution to the 20th International Conference on Laser Spectroscopy (ICOLS 2011) discusses the current status in the comparison between theory and experiments.
Negative values of the Casimir entropy occur quite frequently at low temperatures in arrangements of metallic objects. The physical reason lies either in the dissipative nature of the metals as is the case for the plane-plane geometry or in the geometric form of the objects involved. Examples for the latter are the sphere-plane and the sphere-sphere geometry, where negative Casimir entropies can occur already for perfect metal objects. After appropriately scaling out the size of the objects, negative Casimir entropies of geometric origin are particularly pronounced in the limit of large distan…
The electromagnetic vacuum energy is considered in the presence of a perfectly conducting plane and a ball with dielectric permittivity $\varepsilon$ and magnetic permeability $μ$, $μ\ne1$. The attention is focused on the Casimir repulsion in this system caused by magnetic permeability of the sphere. In the case of perfectly permeable sphere, $μ=\infty$, the vacuum energy is estimated numerically. The short and long distance asymptotes corresponding to the repulsive force and respective low temperature corrections and high temperature limits are found for a wide range of $μ$. The constraints o…
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 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 consider the Casimir interaction between two spheres at zero and finite temperature, for both scalar fields and electromagnetic fields. Of particular interest is the asymptotic expansions of the Casimir free energy when the distance between the spheres is small. The scenario where one sphere is inside the other is discussed in detail. At zero temperature, we compute analytically the leading and the next-to-leading order terms from the functional determinant representation of the Casimir energy. As expected, the leading order term agrees with the proximity force approximation. The results fo…
The Casimir and van der Waals forces acting between two metallic plates or a sphere (lens) above a plate are calculated accounting for the finite conductivity of the metals. The simple formalism of surface modes is briefly presented which allows the possibility to obtain the generalization of Lifshitz results for the case of two semi-spaces covered by the thin layers. Additional clarifications of the regularization procedure provides the means to obtain reliable results not only for the force but also for the energy density. This, in turn, leads to the value of the force for the configuration …
The Casimir force, originating from vacuum zero-point energy, is one of the most intriguing purely quantum effects. It has attracted renewed interests in current field of nanomechanics, due to the rapid size decrease of on-chip devices. Here we study the optomechanically-induced transparency (OMIT) with a tunable Casimir force. We find that the optical output rate can be significantly altered by the vacuum force, even terminated and then restored, indicating a highly-controlled optical switch. Our result addresses the possibility of designing exotic optical nano-devices by harnessing the power…
Vacuum fluctuations have observable consequences, like the Casimir force appearing between two mirrors in vacuum. This force is now measured with good accuracy and agreement with theory. We discuss the meaning and consequences of these statements by emphasizing their relation with the problem of vacuum energy, one of the main unsolved problems at the interface between gravitational and quantum theory.
In a recent paper (arXiv:2003.00760) and a related article (Z. Naturforsch. A 75, 803-807), S.-A. Biehs et al. present erroneous interpretations of our work on the heat transfer driven by Casimir force. Here, we clarify several key concepts and refute their claims.
The regularized total Casimir energy in spacetimes with boundaries is not in general equal to the integral of the regularized energy density. This paradoxical phenomenon is most transparently analyzed in the simple example of a massless scalar field in 1+1 dimensions confined to a line element of length $L$ and obeying Dirichlet boundary conditions.
We calculate the Casimir energy and entropy for two perfect metal spheres in the large and short separation limit. We obtain nonmonotonic behavior of the Helmholtz free energy with separation and temperature, leading to parameter ranges with negative entropy, and also nonmonotonic behavior of the entropy with temperature and with the separation between the spheres. The appearance of this anomalous behavior of the entropy is discussed as well as its thermodynamic consequences.
We study the Casimir interaction between a sphere and a cylinder both subjected to Dirichlet, Neumann or perfectly conducting boundary conditions. Generalizing the operator approach developed by Wittman [IEEE Trans. Antennas Propag. 36, 1078 (1988)], we compute the scalar and vector translation matrices between a sphere and a cylinder, and thus write down explicitly the exact TGTG formula for the Casimir interaction energy. In the scalar case, the formula shows manifestly that the Casimir interaction force is attractive at all separations. Large separation leading term of the Casimir interacti…
The Casimir effect giving rise to an attractive force between the closely spaced two concentric spheres that confine the massless scalar field is calculated by using a direct mode summation with contour integration in the complex plane of eigenfrequencies. We devoleped a new approach appropriate for the calculation of the Casimir energy for spherical boundary conditions. The Casimir energy for a massless scalar field between the closely spaced two concentric spheres coincides with the Casimir energy of the parallel plates for a massless scalar field in the limit when the dimensionless paramete…
The recent discovery that silicon nitride membranes can be used as extremely high Q mechanical resonators makes possible a number of novel experiments, which include improved long range vacuum Casimir force measurements, and measurments of the properties of liquid helium below the lambda point. It is noted that in the thermal correction to the Casimir force, the phase velocity of the excitations does not enter, with the force per unit area between parallel plates depending only on the temperature and distance between the plates. Thus it appears as possible to measure the phonon analog of the f…
Here we reply to the comment of Li et al. (arXiv:2009.09377) on the preprint (arXiv:2003.00760) of our publication (Z. Naturforsch. A 75, 803-807; DOI:10.1515/zna-2020-0204).
A recent experiment [J. L. Garrett, D. A. T. Somers, and J. N. Munday, Phys. Rev. Lett {\bf 120}, 040401 (2018)] measured for the first time the gradient of the Casimir force between two gold spheres in vacuum at room temperature, and placed a bound on the magnitude of the deviation of the measured force from the proximity force approximation (PFA). The present work extends a previous theoretical analysis of this experiment [G. Bimonte, Phys. Rev. D {\bf 97}, 085011 (2018)], by analyzing in detail how the magnitude of the deviation from PFA is affected by the inclusion or neglect of ohmic diss…
