Casimir effect is the attractive force which acts between two plane parallel, closely spaced, uncharged, metallic plates in vacuum. This phenomenon was predicted theoretically in 1948 and reliably investigated experimentally only in recent years. In fact, the Casimir force is similar to the familiar van der Waals force in the case of relatively large separations when the relativistic effects come into play. We review the most important experiments on measuring the Casimir force by means of torsion pendulum, atomic force microscope and micromechanical torsional oscillator. Special attention is paid to the puzzle of the thermal Casimir force, i.e., to the apparent violation of the third law of thermodynamics when the Lifshitz theory of dispersion forces is applied to real metals. Thereafter we discuss the role of the Casimir force in nanosystems including the stiction phenomenon, actuators, and interaction of hydrogen atoms with carbon nanotubes. The applications of the Casimir effect for constraining predictions of extra-dimensional unification schemes and other physics beyond the standard model are also considered.
Theory of the Casimir effect is presented in several examples. Casimir - Polder type formulas, Lifshitz theory and theory of the Casimir effect for two gratings separated by a vacuum slit are derived. Equations for the electromagnetic field in the presence of a medium and dispersion are discussed. Casimir effect for systems with a layer of 2+1 fermions is studied.
Zero-point energy is generally known to be unphysical. Casimir effect, however, is often presented as a counterexample, giving rise to a conceptual confusion. To resolve the confusion we study foundational aspects of Casimir effect at a qualitative level, but also at a quantitative level within a simple toy model with only 3 degrees of freedom. In particular, we point out that Casimir vacuum is not a state without photons, and not a ground state for a Hamiltonian that can describe Casimir force. Instead, Casimir vacuum can be related to the photon vacuum by a non-trivial Bogoliubov transformat…
The Casimir effect for spherical geometry is calculated using generalized Thermofield Dynamics for the case of scalar field. Casimir force and Casimir pressure are presented. It is found that for high temperatures the Casimir force does change sign.
The experimentally demonstration of Casimir force transition from attraction to repulsion is still challenging. Herein, the Casimir forces for a sphere above a plate immersed in different liquids were precisely measured using Atomic force microscope, and the long-range repulsive Casimir force in the gold-cyclohexane-PTFE system is observed for the first time. The experimental data are consistent with the calculation by Lifshitz theory, which offers the direct evidence for the system of ε1<ε3<ε2. It further verifies the reasonability of van Zwol et al. dielectric model to describe the interveni…
We analyze the Casimir interaction of doped graphene. To this end we derive a simple expression for the finite temperature polarization tensor with a chemical potential. It is found that doping leads to a strong enhancement of the Casimir force reaching almost $60\%$ in quite realistic situations. This result should be important for planning and interpreting the Casimir measurements, especially taking into account that the Casimir interaction of undoped graphene is rather weak.
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 consider the Casimir effect of a massive vector field between two semi-infinite dielectric slabs. We first derive the generalization of the Lifshitz formula that gives the Casimir interaction energy of two magnetodielectric slabs separated by a magnetodielectric medium due to the vacuum fluctuations of a massive vector field. We then discuss the asymptotic behaviors of the Casimir energy and the Casimir force in various limits, such as low temperature, high temperature, small mass, large mass, up to the first order in the finite conductivity correction, for two real metal semispaces whose d…
Several new experiments have extended studies of the Casimir force into new and interesting regimes. This recent work will be briefly reviewed. With this recent progress, new issues with background electrostatic effects have been uncovered. The myriad of problems associated with both patch potentials and electrostatic calibrations are discussed and the remaining open questions are brought forward.
We give a short review on the static and dynamical Casimir effects, recalling their historical prediction, as well as their more recent experimental verification. We emphasise on the central role played by so-called {\it dynamical boundary conditions} (for which the boundary condition depends on a second time derivative of the field) in the experimental verification of the dynamical Casimir effect by Wilson et al. We then go on to review our previous work on the static Casimir effect with dynamical boundary conditions, providing an overview on how to compute the so-called local Casimir energy,…
We have measured the normalized gradient of the Casimir force between Au-coated surfaces of the sphere and the plate and equivalent Casimir pressure between two parallel Au plates at T=77K. These measurements have been performed by means of dynamic force microscope adapted for operating at low temperatures in the frequency shift technique. It was shown that the measurement results at T=77K are in a very good agreement with those at T=300K and with computations at T=77K using both theoretical approaches to the thermal Casimir force proposed in the literature. No thermal effect in the Casimir pr…
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 physical origin of the Casimir force is connected with the existence of zero-point and thermal fluctuations. The Casimir effect is very general and finds applications in various fields of physics. This review is limited to the rapid progress at the intersection of experiment and theory that has been achieved in the last few years. It includes a critical assessment of the proposed approaches to the resolution of the puzzles arising in the applications of the Lifshitz theory of the van der Waals and Casimir forces to real materials. All the primary experiments on the measurement of the Casim…
Like Casimir's original force between conducting plates in vacuum, Casimir forces are usually attractive. But repulsive Casimir forces can be achieved in special circumstances. These might prove useful in nanotechnology. We give examples of when repulsive quantum vacuum forces can arise with conducting materials.
We discuss corrections to the Casimir effect at finite temperature and effective field theory. Recently, it has been shown that effective field theories can reproduce radiative corrections to the Casimir energy calculated in full QED. We apply effective field theory methods at finite temperature and reproduce the Casimir free energy. We show that the system undergoes dimensional reduction at high temperature and that it can be described by an effective three-dimensional field theory.
We recently proposed a new approach to the Casimir effect based on classical ray optics (the "optical approximation"). In this paper we show how to use it to calculate the local observables of the field theory. In particular we study the energy-momentum tensor and the Casimir pressure. We work three examples in detail: parallel plates, the Casimir pendulum and a sphere opposite a plate. We also show how to calculate thermal corrections, proving that the high temperature `classical limit' is indeed valid for any smooth geometry.
We propose a theory of the thermal Casimir interaction for multilayered test bodies coated with a graphene sheet. The reflection coefficients on such structures are expressed in terms of the components of the polarization tensor and the dielectric permittivities of material layers. The developed theory is applied to calculate the gradient of the Casimir force between an Au-coated sphere and a graphene sheet deposited on a SiO${}_2$ film covering a Si plate, which is the configuration of a recent experiment performed by means of a dynamic atomic force microscope. The theoretical results are fou…
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.
Experimental methods and procedures required for precision measurements of the Casimir force are presented. In particular, the best practices for obtaining stable cantilevers, calibration of the cantilever, correction of thermal and mechanical drift, measuring the contact separation, sphere radius and the roughness are discussed.
In his autobiography Casimir barely mentioned the Casimir effect, but remarked that it is "of some theortical significance." We will describe some aspects of Casimir effects that appear to be of particular significance now, more than half a century after Casimir's famous paper.
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.
The Casimir effect is investigated in cylindrical and toroidal carbon nanotubes within the framework of the Dirac-like model for the electronic states. The topological Casimir energy is positive for metallic cylindrical nanotubes and is negative for semiconducting ones. The toroidal compactification of a cylindrical nanotube along its axis increases the Casimir energy for metallic-type (periodic) boundary conditions along its axis and decreases the Casimir energy for the semiconducting-type compactifications. For finite length metallic nanotubes the Casimir forces acting on the tube edges are …
In a recent letter, Martin and Zagrebnov [Europhys. Lett., 73 (2006) 1] discussed the thermodynamic Casimir effect for the ideal Bose gas confined in a thin film. We point out that their findings can be expressed in terms of previous general results for the Casimir effect induced by confined critical fluctuations. This highlights the links between the Casimir effect in the contexts of critical phenomena and Bose-Einstein condensation.
The Casimir effect is one of the most direct manifestations of the existence of the vacuum quantum fluctuations, discovered by H. B Casimir in 1948. On the other hand, Lorentz invariance is one of the main and basic concepts in special relativity, which states that, the laws of physics are invariant under Lorentz transformation. In this work, we calculate the corrections imposed by LIV on Casimir effect (force). This may provide a direct probe to test LIV in nature.
We present an approach to studying the Casimir effects by means of the effective theory. An essential point of our approach is replacing the mirror separation into the size of space S^1 in the adiabatic approximation. It is natural to identify the size of space S^1 with the scale factor of the Robertson-Walker-type metric. This replacement simplifies the construction of a class of effective models to study the Casimir effects. To check the validity of this replacement we construct a model for a scalar field coupling to the two-dimensional gravity and calculate the Casimir effects by the effect…
This article is divided in three sections. In the first section we briefly review some high precision experiments on the Casimir force, underlying an important aspect of the analysis of the data. In the second section we discuss our recent results in the measurement of the Casimir force using non-trivial materials. In the third section we present some original ideas for experiments on new phenomena related to the Casimir effects.
We investigate the influence of a dark photon on the Casimir effect and calculate the corresponding leading contribution to the Casimir energy. For expected magnitudes of the photon - dark photon mixing parameter, the influence turns out to be negligible. The plasmon dispersion relation is also not noticeably modified by the presence of a dark photon.
We study the Topological Casimir effect, in which extra vacuum energy emerges as a result of the topological features of the theory, rather than due to the conventional fluctuations of the physical propagating degrees of freedom. We compute the corresponding topological term in quantum Maxwell theory defined on a compact manifold. Numerically, the topological effect is much smaller than the conventional Casimir effect. However, we argue that the Topological Casimir Effect is highly sensitive to an external magnetic field, which may help to discriminate it from the conventional Casimir effect. …
We study the dynamical Casimir effect using a fully quantum-mechanical description of both the cavity field and the oscillating mirror. We do not linearize the dynamics, nor do we adopt any parametric or perturbative approximation. By numerically diagonalizing the full optomechanical Hamiltonian, we show that the resonant generation of photons from the vacuum is determined by a ladder of mirror-field {\em vacuum Rabi splittings}. We find that vacuum emission can originate from the free evolution of an initial pure mechanical excited state, in analogy with the spontaneous emission from excited …
We compute Casimir forces in open geometries with edges, involving parallel as well as perpendicular semi-infinite plates. We focus on Casimir configurations which are governed by a unique dimensional scaling law with a universal coefficient. With the aid of worldline numerics, we determine this coefficient for various geometries for the case of scalar-field fluctuations with Dirichlet boundary conditions. Our results facilitate an estimate of the systematic error induced by the edges of finite plates, for instance, in a standard parallel-plate experiment. The Casimir edge effects for this cas…
A brief review of the recent experimental verifications of the Casimir force between extended bodies is presented. With modern techniques, it now appears feasible to test the force law with 1% precision; I will address the issues relating to the interpretation of experiments at this level of accuracy
We study the Casimir effect in axion electrodynamics. A finite $θ$-term affects the energy dispersion relation of photon if $θ$ is time and/or space dependent. We focus on a special case with linearly inhomogeneous $θ$ along the $z$-axis. Then we demonstrate that the Casimir force between two parallel plates perpendicular to the $z$-axis can be either attractive or repulsive, dependent on the gradient of $θ$. We call this repulsive component in the Casimir force induced by inhomogeneous $θ$ the anomalous Casimir effect.
We review recent developments in the Casimir effect which arises in quantization volumes restricted by material boundaries and in spaces with non-Euclidean topology. The starting point of our discussion is the novel exact solution for the electromagnetic Casimir force in the configuration of a cylinder above a plate. The related work for the scalar Casimir effect in sphere-plate configuration is also considered, and the application region of the proximity force theorem is discussed. Next we consider new experiments on the measurement of the Casimir force between metals and between metal and se…
The phenomena implied by the existence of quantum vacuum fluctuations, grouped under the title of the Casimir effect, are reviewed, with emphasis on new results discovered in the past four years. The Casimir force between parallel plates is rederived as the strong-coupling limit of $δ$-function potential planes. The role of surface divergences is clarified. A summary of effects relevant to measurements of the Casimir force between real materials is given, starting from a geometrical optics derivation of the Lifshitz formula, and including a rederivation of the Casimir-Polder forces. A great de…
In discussions of the cosmological constant, the Casimir effect is often invoked as decisive evidence that the zero point energies of quantum fields are "real''. On the contrary, Casimir effects can be formulated and Casimir forces can be computed without reference to zero point energies. They are relativistic, quantum forces between charges and currents. The Casimir force (per unit area) between parallel plates vanishes as α, the fine structure constant, goes to zero, and the standard result, which appears to be independent of α, corresponds to the α\to\infty limit.
We address several different Casimir experiments where theory and experiment disagree. First out is the classical Casimir force measurement between two metal half spaces; here both in the form of the torsion pendulum experiment by Lamoreaux and in the form of the Casimir pressure measurement between a gold sphere and a gold plate as performed by Decca et al.; theory predicts a large negative thermal correction, absent in the high precision experiments. The third experiment is the measurement of the Casimir force between a metal plate and a laser irradiated semiconductor membrane as performed b…
We consider experiments on measuring the Casimir interaction which have been performed in the last four years. The emphasis is made on measuring differences in the Casimir pressures under a transition of the plate metal from normal to superconducting state and on the Casimir metrology platform using a commercial micromechanical sensor. In both cases several problems in the comparison between experiment and theory are discussed.
A number of experimental measurements of the Casimir force have observed a logarithmic distance variation of the voltage that minimizes electrostatic force between the plates in a sphere-plane geometry. We show that this variation can be simply understood from a geometric averaging of surface potential patches together with the Proximity Force Approximation.
We describe experimental and related theoretical work on the measurement of the Casimir force using semiconductor test bodies. This field of research started in 2005 and several important and interesting results have already been obtained. Specifically, the Casimir force or its gradient were measured in the configuration of an Au-coated sphere and different semiconductor surfaces. It was found that the force magnitude depends significantly on the replacement of the metal with a semiconductor and on the concentration of charge carriers in the semiconductor material. Special attention is paid to…
The conventional Casimir effect has been studied in the continuous spacetime, but to elucidate its counterpart in the lattice space is an important subject. Here, we discuss various types of Casimir effects for quantum fields on the lattice. By using a definition of the Casimir energy on the lattice, we show that the Casimir effect for the Wilson fermion is similar to that for the continuous Dirac fermion. We apply our definition to an effective Hamiltonian describing Dirac semimetals, such as Cd$_{3}$As$_{2}$ and Na$_{3}$Bi, and find an oscillatory behavior of the Casimir energy as a function…
The Casimir effects for one-dimensional fractal networks, so-called quantum graphs is studied. Based on the Green function approach for quantum graphs zero-point energy for some simplest topologies is written explicitly.
After reviewing some essential features of the Casimir effect and, specifically, of its regularization by zeta function and Hadamard methods, we consider the dynamical Casimir effect (or Fulling-Davis theory), where related regularization problems appear, with a view to an experimental verification of this theory. We finish with a discussion of the possible contribution of vacuum fluctuations to dark energy, in a Casimir like fashion, that might involve the dynamical version.
An experimental set-up for the measurement of the Casimir effect at separations larger than a few microns is presented. The apparatus is based on a mechanical resonator and uses a homodyne detection technique to sense the Casimir force in the plane-parallel configuration. First measurements in the 3-10 micrometer range show an unexpected large force probably due to patch effects.
The Casimir energy is calculated in one-, two-, and three-dimensional spaces for the field with generalized coordinates and momenta satisfying the deformed Poisson brackets leading to the minimal length.
Quantum fluctuations are the key concepts of quantum mechanics. Quantum fluctuations of quantum fields induce a zero-point energy shift under spatial boundary conditions. This quantum phenomenon, called the Casimir effect, has been attracting much attention beyond the hierarchy of energy scales, ranging from elementary particle physics to condensed matter physics together with photonics. However, the application of the Casimir effect to spintronics has not yet been investigated enough, particularly to ferrimagnetic thin films, although yttrium iron garnet (YIG) is one of the best platforms for…
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.
We address three different problematic Casimir experiments in this work. The first is the classical Casimir force measured between two metal half spaces; here in the form of the Casimir pressure measurement between a gold sphere and a gold plate as performed by Decca et al. [Phys. Rev. D 75, 077101 (2007)]; theory predicts a large negative thermal correction, absent in the high precision experiment. The second experiment is the measurement of the Casimir force between a metal plate and a laser irradiated semiconductor membrane as performed by Chen et al. [Phys. Rev. B 76, 035338 (2007)]; the c…
The Casimir effect is a crucial prediction of Quantum Field Theory which has fascinating connections with open questions in fundamental physics. The ideal formula written by Casimir does not describe real experiments and it has to be generalized by taking into account the effects of imperfect reflection, thermal fluctuations, geometry as well as the corrections coming from surface physics. We discuss these developments in Casimir physics and give the current status in the comparison between theory and experiment after years of improvements in measurements as well as theory.
We propose a new approach to the Casimir effect based on classical ray optics. We define and compute the contribution of classical optical paths to the Casimir force between rigid bodies. We reproduce the standard result for parallel plates and agree over a wide range of parameters with a recent numerical treatment of the sphere and plate with Dirichlet boundary conditions. Our approach improves upon proximity force approximation. It can be generalized easily to other geometries, other boundary conditions, to the computation of Casimir energy densities and to many other situations.
The geometrical phase is shown to be integral of motion. Deformation of particle distribution function corresponding to nonstationary Casimir effect is expressed in terms of multivariable Hermite polynomials. Correction to Planck distribution due to q--nonlinearity is discussed.
Advanced Science Research Center, Japan Atomic Energy Agency (JAEA)（日本原子力机构先进科学研究中心）；Research Center for Nuclear Physics, Osaka University（大阪大学核物理研究中心）；RIKEN Center for Computational Science（理化学研究所计算科学中心）
The Casimir effect is a well-known macroscopic consequence of quantum vacuum fluctuations, but whereas the static effect (Casimir force) has long been observed experimentally, the dynamic Casimir effect is up to now undetected. From an experimental viewpoint a possible detection would imply the vibration of a mirror at gigahertz frequencies. Mechanical motions at such frequencies turn out to be technically unfeasible. Here we present a different experimental scheme where mechanical motions are avoided, and the results of laboratory tests showing that the scheme is practically feasible. We thin…
We present explicitly another example of a temperature inversion symmetry in the Casimir effect for a nonsymmetric boundary condition. We also give an interpretation for our result.
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.
