Recent Experiments on the Casimir Effect: Description and Analysis

The Casimir force between rough metallic plates

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.

An intuitive picture of the Casimir effect

The Casimir effect, which predicts the emergence of an attractive force between two parallel, highly reflecting plates in vacuum, plays a vital role in various fields of physics, from quantum field theory and cosmology to nanophotonics and condensed matter physics. Nevertheless, Casimir forces still lack an intuitive explanation and current derivations rely on regularisation procedures to remove infinities. Starting from special relativity and treating space and time coordinates equivalently, this paper overcomes no-go theorems of quantum electrodynamics and obtains a local relativistic quantu…

Thermal Casimir Effect in the Plane-Sphere Geometry

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.

Dynamical Casimir effect via modulated Kerr or higher nonlinearities

We show two examples in which the dynamical Casimir effect can be achieved by modulating the Kerr or higher order nonlinearities. In the first case the cavity field is coupled to an arbitrary number of qubits or an harmonic oscillator via the dipole interaction. In the second case, the modulation of the nonlinearities is accompanied by the off-resonance modulation of the cavity frequency. We present the analytic description of the phenomenon and supplement it with numeric simulations, demonstrating that photons can be created from vacuum and the resulting hyper-Poissonian photon statistics is …

A Particle Theory of the Casimir Effect

In previous works Suppes and de Barros used a pure particle model to derive interference effects, where individual photons have well-defined trajectories, and hence no wave properties. In the present paper we extend that description to account for the Casimir effect. We consider that the linear momentum $\sum\frac{1}{2}\hbar {\bf k}$ of the vacuum state in quantum electrodynamics corresponds to the linear momentum of virtual photons. The Casimir effect, in the cases of two parallel plates and the solid ball, is explained in terms of the pressure caused by the photons. Contrary to quantum elect…

Recent measurements of the Casimir force: Comparison between experiment and theory

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.

Dynamical Casimir effect via four- and five-photon transitions using a strongly detuned atom

The scenario of a single-mode cavity with harmonically modulated frequency is revisited in the presence of strongly detuned qubit or cyclic qutrit. It is found that when the qubit frequency is close to $3ν$ there is a peak in the photon generation rate via four-photon transitions for the modulation frequency $4ν$, where $ν$ is the average cavity frequency. Effective five-photon processes can occur for the modulation frequency $5ν$ in the presence of a cyclic qutrit, and the corresponding transition rates exhibit series of peaks. Closed analytical description is derived for the unitary evolutio…

Complete roughness and conductivity corrections for the recent Casimir force measurement

We consider detailed roughness and conductivity corrections to the Casimir force in the recent Casimir force measurement employing an Atomic Force Microscope. The roughness of the test bodies-a metal plate and a sphere- was investigated with the Atomic Force Microscope and the Scanning Electron Microscope respectively. It consists of separate crystals of different heights and a stochastic background. The amplitude of roughness relative to the zero roughness level was determined and the corrections to the Casimir force were calculated up to the fourth order in a small parameter (which is this a…

一维双δ函数势标量场中的有限温度卡西米尔效应

Finite temperature Casimir effect in one-dimensional scalar field with double delta-function potentials

本文研究一维双δ函数势标量场的有限温度卡西米尔效应，采用正则量子化方法计算力与熵，并与Lifshitz理论对比，发现正则量子化给出热力学自洽的正熵。

Casimir effect for Dirac lattices

We consider polarizable sheets, which recently received some attention, especially in context of the dispersion interaction of thin sheets like graphene. These sheets are modeled by a collection of delta function potentials and resemble "zero range potentials", known in quantum mechanics. We develop a theoretical description and apply the so-called "TGTG"-formula to calculate the interaction of two such lattices. Thereby we make use of the formulation of the scattering of waves off such sheets provided earlier. We consider all limiting cases, providing link to earlier results. Also, we discuss…

Nonlocal impedances and the Casimir entropy at low temperatures

The problem with the temperature dependence of the Casimir force is investigated. Specifically, the entropy behavior in the low temperature limit, which caused debates in the literature, is analyzed. It is stressed that the behavior of the relaxation frequency in the $T\to0$ limit does not play a physical role since the anomalous skin effect dominates in this range. In contrast with the previous works, where the approximate Leontovich impedance was used for analysis of nonlocal effects, we give description of the problem in terms of exact nonlocal impedances. It is found that the Casimir entro…

Interplay between optomechanics and the dynamical Casimir effect

We develop a model of a quantum field confined within a cavity with a movable wall where the position of the wall is quantized. We obtain a full description of the dynamics of both the quantum field and the confining wall depending on the initial state of the whole system. Both the reaction and back-reaction of the field on the wall, and the wall on the field, can be taken into account, as well as external driving forces on both the cavity and the wall. The model exactly reproduces the resonant cavity mode stimulation due to the periodic motion of the mirror (dynamical Casimir effect), as well…

Temperature correction to the Casimir force in cryogenic range and anomalous skin effect

Temperature correction to the Casimir force is considered for real metals at low temperatures. With the temperature decrease the mean free path for electrons becomes larger than the field penetration depth. In this condition description of metals with the impedance of anomalous skin effect is shown to be more appropriate than with the permittivity. The effect is crucial for the temperature correction. It is demonstrated that in the zero frequency limit the reflection coefficients should coincide with those of ideal metal if we demand the entropy to be zero at T=0. All the other prescriptions d…

Casimir Friction Force for Moving Harmonic Oscillators

Casimir friction is analyzed for a pair of dielectric particles in relative motion. We first adopt a microscopic model for harmonically oscillating particles at finite temperature T moving non-relativistically with constant velocity. We use a statistical-mechanical description where time-dependent correlations are involved. This description is physical and direct, and, in spite of its simplicity, is able to elucidate the essentials of the problem. This treatment elaborates upon, and extends, an earlier theory of ours back in 1992. The energy change Delta E turns out to be finite in general, co…

Comment on "Thermal Effects on the Casimir Force in the 0.1-5 micrometer Range"

In a recent paper (M. Bostrom and Bo E. Sernelius, Phys. Rev. Lett. 84, 4757 (2000)) the combined effect of finite conductivity and finite temperature on the Casimir force is analyzed, and significant deviations from other theoretical results and a recent experiment are obtained. In this Comment, I show that the extrapolation to zero frequency is incorrect because the authors have neglected that the wavenumber and frequency of the electromagentic mode must simultaneously appraoch zero

Comment on ``Lateral Casimir force beyond the proximity-force approximation''

We demonstrate that the approach by Rodrigues et al. (PRL, v.96, 100402, 2006) to the description of the lateral Casimir force between corrugated surfaces is in disagreement with a path-integral theory by Emig et al. (PRA, v.67, 022114, 2003), and the approximations used are not appropriate for comparison to experiment.

Contribution of drifting carriers to the Casimir-Lifshitz and Casimir-Polder interactions with semiconductor materials

We develop a new theory for Casimir-Lifshitz and Casimir-Polder interactions with semiconductor surfaces that takes into account charge drift in the bulk material. The corresponding frequency-dependent dispersion relations describe a crossover between Lifshitz results for dielectrics and for good conductors. In the quasi-static limit, our calculated reflection amplitudes coincide with those recently computed to account for Debye screening in the thermal Lifshitz force with conducting surfaces with small density of carriers.

Experiment, theory and the Casimir effect

Several problems at the interface between the field-theoretical description of the Casimir effect and experiments on measuring the Casimir force are discussed. One of these problems is connected with the definition of the Casimir free energy in ideal metal rectangular boxes satisfying the general physical requirements. It is shown that the consideration of rectangular boxes with a partition (piston) does not negate the previously known results obtained for boxes without a piston. Both sets of results are found to be in mutual agreement. Another problem is related to the use of the proximity fo…

Reply to "Comment on 'Precision measurement of the Casimir-Lifshitz force in a fluid'"

We have reviewed the Comment of Geyer et al. [arXiv:0708.1548] concerning our recent work [Phys. Rev. A 75, 060102 (R) (2007)], and while we disagree with their criticisms, we acknowledge them for giving us the opportunity to add interesting addition material and a more detailed description of our experiment. We describe further our calculation and explain why a more sophisticated model is not warranted. We also present detailed experiments on the effects of electrostatic forces in our measurements and show that the contribution due to work function differences is small and that the residual e…

Comment on "Contribution of drifting carriers to the Casimir-Lifshitz and Casimir-Polder interactions with semiconductor materials"

Recently D. A. R. Dalvit and S. K. Lamoreaux arXiv:0805.1676 [Phys. Rev. Lett. v.101, 163203 (2008)] suggested the modification of the reflection coefficients in the Lifshitz theory taking into account the screening effects and diffusion currents. We demonstrate that the suggested modification is thermodynamically and experimentally inconsistent.

Casimir Momentum of a Chiral Molecule in a Magnetic Field

In a classical description, a neutral, polarizable object acquires a kinetic momentum when exposed to crossed electric and magnetic fields. In the presence of only a magnetic field no such momentum exists classically, although it is symmetry-allowed for an object lacking mirror symmetry. We perform a full QED calculation to show that the quantum vacuum coupled to a chiral molecule provides a kinetic "Casimir" momentum directed along the magnetic field, proportional to its rotatory power and the fine structure constant.

Experimental Verifications of the Casimir Attractive Force between Solid Bodies

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

New approach to the thermal Casimir force between real metals

The new approach to the theoretical description of the thermal Casimir force between real metals is presented. It uses the plasma-like dielectric permittivity that takes into account the interband transitions of core electrons. This permittivity precisely satisfies the Kramers-Kronig relations. The respective Casimir entropy is positive and vanishes at zero temperature in accordance with the Nernst heat theorem. The physical reasons why the Drude dielectric function, when substituted in the Lifshitz formula, is inconsistent with electrodynamics are elucidated. The proposed approach is the sing…

Three-body Casimir-Polder interactions

As part of our program to develop the description of three-body effects in quantum vacuum phenomena, we study the three-body interaction of two anisotropically polarizable atoms with a perfect electrically conducting plate, a generalization of earlier work. Three- and four-scattering effects are important, and lead to nonmonotonic behavior.

Resonant cavity photon creation via the dynamical Casimir effect

Motivated by a recent proposal for an experimental verification of the dynamical Casimir effect, the macroscopic electromagnetic field within a perfect cavity containing a thin slab with a time-dependent dielectric permittivity is quantized in terms of the dual potentials. For the resonance case, the number of photons created out of the vacuum due to the dynamical Casimir effect is calculated for both polarizations (TE and TM). PACS: 42.50.Lc, 03.70.+k, 42.50.Dv, 42.60.Da.

Casimir forces between spheres and loop integrals

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.

Casimir energy in spherical cavities

We calculate the Casimir energy at spherical cavities within a host made up of an arbitrary material described by a possibly dispersive and lossy dielectric response. To that end, we add to the coherent optical response a contribution that takes account of the incoherent radiation emitted by the host in order to guarantee the detailed balance required to keep the system at thermodynamic equilibrium in the presence of dissipation. The resulting boundary conditions allow a conventional quantum mechanical treatment of the radiation within the cavity from which we obtain the contribution of the ca…

An Optical Approach to the Dynamical Casimir Effect

We recently proposed a new approach to analyze the parametric resonance in a vibrating cavity based on the analysis of classical optical paths. This approach is used to examine various models of cavities with moving walls. We prove that our method is useful to extract easily basic physical outcome.

Comment on Repulsive Casimir Forces

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…

介电介质中的多重散射展开：Casimir效应

Multiple scattering expansion for dielectric media: Casimir effect

本文提出了一种用于任意形状和材料组成物体间Casimir相互作用的多重散射描述方法，该方法将其展开为物体间和物体内的波散射序列，无需散射振幅的先验知识即可在实验分辨率内计算复杂几何形状的相互作用。

Casimir effect between dissimilar materials: a test for the proximity theorem

We calculate exactly the Casimir force between a spherical particle and a plane, both with arbitrary dielectric properties, in the non-retarded limit. Using a Spectral Representation formalism, we show that the Casimir force of a sphere made of a material A and a plane made of a material B, differ from the case when the sphere is made of B, and the plane is made of A. The differences in energy and force show the importance of the geometry, and make evident the necessity of realistic descriptions of the sphere-plane system beyond the Proximity Theorem approximation.

探寻卡西米尔能量

Seeking the Casimir Energy

本研究围绕卡西米尔能量展开，通过实验探究卡西米尔腔对超导体临界温度$T_c$的影响，未检测到大于12微开尔文的$T_c$变化，理论预期变化量级为0.025微开尔文，低于当前实验灵敏度。

Relation between the thermodynamic Casimir effect in Bose-gas slabs and critical Casimir forces

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.

Reply to comment on Casimir energy for spherical boundaries

It is shown that the recent criticism of Brevik et al. (hep-th/0004041) is in error.

Recent Developments in the Casimir Effect

In this talk I review various developments in the past year concerning quantum vacuum energy, the Casimir effect. In particular, there has been continuing controversy surrounding the temperature correction to the Lifshitz formula for the Casimir force between real materials, be they metals or semiconductors. Consensus has emerged as to how Casimir energy accelerates in a weak gravitational field; quantum vacuum energy, including the divergent parts which renormalize the masses of the Casimir plates, accelerates indeed according to the equivalence principle. Significant development has been for…

Progress in Experimental Measurements of the Surface-Surface Casimir Force: Electrostatic Calibrations and Limitations to Accuracy

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.

The Casimir effect: Recent controversies and progress

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…

Experimental and theoretical investigation of angular dependence of the Casimir force between sinusoidally corrugated surfaces

In the current work we present the complete results for the measurement of normal Casimir force between shallow and smooth sinusoidally corrugated gold coated sphere and a plate at various angles between the corrugations using an atomic force microscope. All measured data were compared with the theoretical approach using the proximity force approximation and theory based on derivative expansion. In both cases real material properties of the surfaces and non-zero temperature were taken into account. Special attention is paid to the description of electrostatic interactions between corrugated su…

Non-perturbative Dynamical Casimir Effect in Optomechanical Systems: Vacuum Casimir-Rabi Splittings

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 …

Improved constraints on the coupling constants of axion-like particles to nucleons from recent Casimir-less experiment

We obtain improved constraints on the coupling constants of axion-like particles to nucleons from a recently performed Casimir-less experiment. For this purpose, the differential force between a Au-coated sphere and either Au or Si sectors of a rotating disc, arising due to two-axion exchange, is calculated. Over a wide region of axion masses from 1.7 meV to 0.9 eV the obtained constraints are stronger up to a factor of 60 than the previously known ones following from the Cavendish-type experiment and measurements of the effective Casimir pressure.

Casimir effect for the sphere revisited

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.

Reply to "On the heat transfer across a vacuum gap mediated by Casimir force"

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.

Thermal and non-thermal signatures of the Unruh effect in Casimir-Polder forces

We show that Casimir-Polder forces between two relativistic uniformly accelerated atoms exhibit a transition from the short distance thermal-like behavior predicted by the Unruh effect, to a long distance non-thermal behavior, associated with the breakdown of a local inertial description of the system. This phenomenology extends the Unruh thermal response detected by a single accelerated observer to an accelerated spatially extended system of two particles, and we identify the characteristic length scale for this crossover with the inverse of the proper acceleration of the two atoms. Our resul…

Casimir Effect Associated with Fractional Klein-Gordon Field

This paper gives a brief review on the recent work on fractional Klein-Gordon fields, in particular on the Casimir effect associated to fractional Klein-Gordon fields in various geometries and boundary conditions. New results on Casimir piston due to a fractional Klein-Gordon massive field are given.

Normal and lateral Casimir force: Advances and prospects

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…

Casimir Effects: an Optical Approach II. Local Observables and Thermal Corrections

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.

Recent Results on Thermal Casimir Force between Dielectrics and Related Problems

We review recent results obtained in the physics of the thermal Casimir force acting between two dielectrics, dielectric and metal, and between metal and semiconductor. The detailed derivation for the low-temperature behavior of the Casimir free energy, pressure and entropy in the configuration of two real dielectric plates is presented. For dielectrics with finite static dielectric permittivity it is shown that the Nernst heat theorem is satisfied. Hence, the Lifshitz theory of the van der Waals and Casimir forces is demonstrated to be consistent with thermodynamics. The nonzero dc conductivi…

Refractive index perturbations -- Unruh effect, Hawking radiation or dynamical Casimir effect?

Motivated by recent experimental progress to manipulate the refractive index of dielectric materials by strong laser beams, we study some aspects of the quantum radiation created by such refractive index perturbations.

金属对低频 s 偏振倏逝波异常介电响应的测量及其对 Casimir 效应的启示

Measurement of the Unusual Dielectric Response to Low-Frequency s-Polarized Evanescent Waves in Metals with {\break} Implications for the Casimir Effect

通过精确测量铜板反射的振荡磁场横向分量，发现实验数据排除了基于 Drude 模型的经典电动力学预测，证明该模型无法完整描述金属对 s 偏振倏逝波的电磁响应，对 Casimir 效应等领域具有重要启示。

The thermal Casimir effect for conducting plates and the Bohr-van Leeuwen theorem

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.

The Casimir Effect in Relativistic Quantum Field Theories

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…

New constraints on Yukawa-type interactions from the Casimir effect

Measurements of the Casimir force are used to obtain stronger constraints on the parameters of hypothetical interactions predicted in different unification schemes beyond the Standard Model. We review new strong constraints on the Yukawa-type interactions derived during the last two years from recent experiments on measuring the lateral Casimir force, Casimir force in configurations with corrugated boundaries and the Casimir-Polder force. Specifically, from measurements of the lateral Casimir force compared with the exact theory the strengthening of constraints up to a factor of 24 millions wa…

What is the Temperature Dependence of the Casimir Effect?

There has been recent criticism of our approach to the Casimir force between real metallic surfaces at finite temperature, saying it is in conflict with the third law of thermodynamics and in contradiction with experiment. We show that these claims are unwarranted, and that our approach has strong theoretical support, while the experimental situation is still unclear.

Stronger constraints on non-Newtonian gravity from the Casimir effect

We review new constraints on the Yukawa-type corrections to Newtonian gravity obtained recently from gravitational experiments and from the measurements of the Casimir force. Special attention is paid to the constraints following from the most precise dynamic determination of the Casimir pressure between the two parallel plates by means of a micromechanical torsional oscillator. The possibility of setting limits on the predictions of chameleon field theories using the results of gravitational experiments and Casimir force measurements is discussed.