We give a rigorous analytical derivation of low-temperature behavior of the Casimir entropy in the framework of the Lifshitz formula combined with the Drude dielectric function. An earlier result that the Casimir entropy at zero temperature is not equal to zero and depends on the parameters of the system is confirmed, i.e. the third law of thermodynamics (the Nernst heat theorem) is violated. We illustrate the resolution of this thermodynamical puzzle in the context of the surface impedance approach by several calculations of the thermal Casimir force and entropy for both real metals and dielectrics. Different representations for the impedances, which are equivalent for real photons, are discussed. Finally, we argue in favor of the Leontovich boundary condition which leads to results for the thermal Casimir force that are consistent with thermodynamics.
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.
The Casimir force is calculated between Au thin films that are described by a Drude model with a frequency dependent damping function. The model parameters are obtained from available experimental data for Au thin films. Two cases are considered; annealed and nonannealed films that have a different damping function. Compared with the calculations using a Drude model with a constant damping parameter, we observe changes in the Casimir force of a few percent. This behavior is only observed in films of no more than 300 $Å$ thick.
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.
In view of the current discussion on the subject, an effort is made to show very accurately both analytically and numerically how the Drude dispersive model, assuming the relaxation is nonzero at zero temperature (which is the case when impurities are present), gives consistent results for the Casimir free energy at low temperatures. Specifically, we find that the free energy consists essentially of two terms, one leading term proportional to T^2, and a next term proportional to T^{5/2}. Both these terms give rise to zero Casimir entropy as T -> 0, thus in accordance with Nernst's theorem.
We calculate the Casimir force between a Ge plate and a Ge spherical lens with neglected and included free charge carriers. Computations with included carriers are performed using the Drude, plasma, and diffusion models. It is shown that the plasma and the Drude models lead to almost identical computational results, while the results obtained using the diffusion model are sandwiched between those obtained with neglected charge carriers and with charge carriers described by the plasma or Drude models.
We find analytic asymptotic expressions at low temperature for the Casimir free energy, entropy and pressure of two parallel graphene sheets in the framework of the Lifshitz theory. The reflection coefficients of electromagnetic waves on graphene are described on the basis of first principles of quantum electrodynamics at nonzero temperature using the polarization tensor in (2+1)-dimensional space-time. The leading contributions to the Casimir entropy and to the thermal corrections to the Casimir energy and pressure are given by the thermal correction to the polarization tensor at nonzero Mats…
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 find the low-temperature behavior of the Casimir-Polder free energy and entropy for an atom interacting with real graphene sheet possessing nonzero energy gap and chemical potential. Employing the formalism of the polarization tensor, it is shown that the Casimir-Polder entropy goes to zero by the power law with vanishing temperature, i.e., the Nernst heat theorem is satisfied. This result is discussed in connection with the problems connected with account of free charge carriers in the Lifshitz theory.
We derive an analytic behavior of the Casimir free energy, entropy and pressure of metallic films in vacuum at low temperature. It is shown that this behavior differs significantly depending on whether the plasma or the Drude model is used to describe the dielectric properties of film metal. For metallic films described by the lossless plasma model the thermal corrections to the Casimir energy and pressure drop to zero exponentially fast with increasing film thickness. There is no classical limit in this case. The Casimir entropy satisfies the Nernst heat theorem. For metallic films with perfe…
We 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 investigate the Casimir free energy of a metallic film either sandwiched between two dielectric plates or in vacuum. It is shown that even for a thin film of several tens of nanometer thickness the Casimir free energy and pressure calculated with the Lifshitz theory using the Drude model and the plasma model approaches take significantly different values and can be easily discriminated. According to our results, the classical limit is already achieved for films of about 100\,nm thickness if the Drude model approach is used in calculations. In this case the classical expressions for the Casi…
We discuss theoretical predictions for the thermal Casimir force and compare them with available experimental data. Special attention is paid to the recent claim of the observation of that effect, as predicted by the Drude model approach. We show that this claim is in contradiction with a number of experiments reported so far. We suggest that the experimental errors, as reported in support of the observation of the thermal Casimir force, are significantly underestimated. Furthermore, the experimental data at separations above $3\,μ$m are shown to be in agreement not with the Drude model approa…
We calculate the Casimir force and its gradient between a spherical and a planar gold surface. Significant numerical improvements allow us to extend the range of accessible parameters into the experimental regime. We compare our numerically exact results with those obtained within the proximity force approximation (PFA) employed in the analysis of all Casimir force experiments reported in the literature so far. Special attention is paid to the difference between the Drude model and the dissipationless plasma model at zero frequency. It is found that the correction to PFA is too small to explai…
Recently, Høye, Brevik, Ellingsen and Aarseth (quant-ph/0703174) claimed that the use of the Drude dielectric function leads to zero Casimir entropy at zero temperature in accordance with Nernst's theorem. We demonstrate that their proof is not applicable to metals with perfect crystal lattices having no impurities. Thus there is no any contradiction with previous results in the literature proving that the Drude dielectric function violates the Nernst theorem for the Casimir entropy in the case of perfect crystal lattices. We also indicate mistakes in the coefficients of their asymptotic expre…
The situation with the temperature corrections to the Casimir force between real metals of finite conductivity is reported. It is shown that the plasma dielectric function is well adapted to the Lifshitz formula and leads to reasonable results for real conductors. The Drude dielectric function which describes media with dissipation is found not to belong to the application range of the Lifshitz formula at nonzero temperature. For Drude metals the special modification of the zero-frequency term of this formula is suggested. The contradictory results on the subject in recent literature are analy…
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…
Recently J.S. Hoye, I.Brevik, J.B. Aarseth, and K.A. Milton [Phys. Rev. E v.67, 056116 (2003); quant-ph/0212125] proposed that if the Lifshitz formula is combined with the Drude model, the transverse electric zero mode does not contribute to the result for real metals and there arises a linear temperature correction to the Casimir force at small temperatures. The authors claim that in spite of the fact that the Casimir entropy in their approach is negative, the Nernst heat theorem is satisfied. In the present Comment we show that the authors' conclusion regarding the Nernst heat theorem is in …
An extended Drude model, termed as the Gurzhi model, which takes into account the electron-phonon and electron-electron interactions, is applied to calculate the Casimir force between two metallic plates. It is shown that although the dielectric permittivity of the Gurzhi model has a first order pole in the upper half-plane of complex frequencies and, thus, violates the causality principle, it can be used in a restricted frequency interval in combination with the experimental permittivity determined by the optical data for the complex index of refraction. The imaginary part of the Gurzhi diele…
The Lifshitz theory and its modifications are discussed with respect to the Nernst heat theorem and the experimental data of several recent experiments. An analysis of all available information leads to the conclusion that some concepts of statistical physics might need reconsideration.
It is well known that the Casimir force calculated at large separations using the Lifshitz theory differs by a factor of 2 for metals described by the Drude or plasma models. We argue that this difference is entirely determined by the contribution of transverse electric (s) evanescent waves. Taking into account that there is a lack of experimental information on the electromagnetic response of metals to low-frequency evanescent waves, we propose an experiment on measuring the magnetic field of an oscillating magnetic dipole spaced in vacuum above a thick metallic plate. According to our result…
We perform detailed comparison of the experimental data of the experiment on the determination of the Casimir pressure between two parallel Au plates with the theoretical values computed using the Lifshitz formula at zero temperature. Computations are done using the optical data for the complex index of refraction of Au extrapolated to low frequencies by means of the Drude model with both most often used and other suggested Drude parameters. It is shown that the experimental data exclude the Lifshitz formula at zero temperature at a 70% confidence level if the Drude model with most often used …
The analytic expressions for the free energy and entropy of the Casimir-Polder interaction between a polarizable and magnetizable atom and a graphene sheet are found in the limiting case of low temperature. In so doing, the response of graphene to electromagnetic fluctuations is described in the framework of the Dirac model by means of the polarization tensor in (2+1)-dimensional space-time. It is shown that the dominant contribution to the low-temperature behavior is given by an explicit dependence of the polarization tensor on temperature as a parameter. We demonstrate that the Lifshitz theo…
We extend our approach to the Casimir effect between absorbing dielectric multilayers [M. S. Tomas, Phys. Rev. A 66, 052103 (2002)] to magnetodielectric systems. The resulting expression for the force is used to numerically explore the effect of the medium dispersion on the attractive/repulsive force in a metal-magnetodielectric system described by the Drude-Lorentz permittivities and permeabilities.
In this Reply to the preceding Comment of Klimchitskaya and Mostepanenko (cf. also quant-ph/0703214), we summarize and maintain our position that the Drude dispersion relation when inserted in the Lifshitz formula gives a thermodynamically satisfactory description of the Casimir force, also in the limiting case when the relaxation frequency goes to zero (perfect crystals).
The Casimir interactions in the solid-liquid-solid systems as a function of separation distance have been studied by the Lifshitz theory. The dielectric permittivity functions for a wide range of materials are described by Drude, Drude-Lorentz and oscillator models. We find that the Casimir forces between gold and silica or MgO materials are both the repulsive and attractive. We also find the stable forms for the systems. Our studies would provide a good guidance for the future experimental studies on the dispersion interactions.
We investigate the thermodynamical aspects of the Casimir effect in the case of plane parallel plates made of real metals. The thermal corrections to the Casimir force between real metals were recently computed by several authors using different approaches based on the Lifshitz formula with diverse results. Both the Drude and plasma models were used to describe a real metal. We calculate the entropy density of photons between metallic plates as a function of the surface separation and temperature. Some of these approaches are demonstrated to lead to negative values of entropy and to nonzero en…
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…
In view of the current discussion on the subject, an effort is made to show very accurately both analytically and numerically how the Drude dispersion model gives consistent results for the Casimir free energy at low temperatures. Specifically, for the free energy near T=0 we find the leading term to be proportional to T^2 and the next-to-leading term proportional to T^{5/2}. These terms give rise to zero Casimir entropy as T approaches zero, and is thus in accordance with Nernst's theorem.
Based on a generalization of the Lifshiftz theory we calculate Casimir forces involving magnetodielectric and possibly anisotropic metamaterials, focusing on the possibility of repulsive forces. It is found that Casimir repulsion decreases with magnetic dissipation, and even a small Drude background in metallic-based metamaterials acts to make attractive a Casimir force that would otherwise be predicted to be repulsive. The sign of the force also depends sensitively on the degree of optical anisotropy of the metamaterial, and on the form of the frequency-dependency of the magnetic response.
We derive analytically the asymptotic behavior of the Casimir interaction between a sphere and a plate when the distance between them, $d$, is much smaller than the radius of the sphere, $R$. The leading order and next-to-leading order terms are derived from the exact formula for the Casimir interaction energy. They are found to depend nontrivially on the dielectric functions of the objects. As expected, the leading order term coincides with that derived using the proximity force approximation. The result on the next-to-leading order term complements that found by Bimonte, Emig and Kardar [App…
We find the analytic expressions for the Casimir free energy, entropy and pressure at low temperature in the configuration of two parallel plates made of magnetodielectic material. The cases of constant and frequency-dependent dielectic permittivity and magnetic permeability of the plates are considered. Special attention is paid to the account of dc conductivity. It is shown that in the case of finite static dielectric permittivity and magnetic permeability the Nernst heat theorem for the Casimir entropy is satisfied. If the dc conductivity is taken into account, the Casimir entropy goes to a…
The recently suggested modification of the transverse electric contribution to the Lifshitz formula (S. K. Lamoreaux, arXiv:0801.1283) is discussed. We show that this modification is inconsistent with the data of two precise experiments, and violates the Nernst heat theorem. The preprint's suggestion concerning the resolution of the "apparent violation of the Third Law of Thermodynamics" is shown to be incorrect.
We present theoretical calculations of the Casimir force for Au thin films near the insulator-conductor transition that has been observed experimentally. The dielectric function of the Au thin films is described by the Drude-Smith model. The parameters needed to model the dielectric function such as the relaxation time, plasma frequency and the backscattering constant depend on the thickness of the film. The Casimir force decreases as the film thickness decreases until it reaches a minimum after which the force increases again. The minimum of the force coincides with the critical film thicknes…
We carefully re-examine the conditions of validity for the consistent derivation of the Lifshitz-Matsubara sum formula for the Casimir pressure between metallic plane mirrors. We recover the usual expression for the lossy Drude model, but not for the lossless plasma model. We give an interpretation of this new result in terms of the modes associated with the Foucault currents which play a role in the limit of vanishing losses, in contrast to common expectations.
We derive the low-temperature behavior of the Casimir-Polder free energy for a polarizable atom interacting with graphene sheet which possesses the nonzero energy gap $Δ$ and chemical potential $μ$. The response of graphene to the electromagnetic field is described by means of the polarization tensor in the framework of Dirac model on the basis of first principles of thermal quantum field theory in the Matsubara formulation. It is shown that the thermal correction to the Casimir-Polder energy consists of three contributions. The first of them is determined by the Matsubara summation using the …
We investigate the Casimir force acting between real metals at nonzero temperature. It is shown that the zero-frequency term of Lifshitz formula has interpretation problem in the case of real metal described by Drude model. It happens because the scattering theory underlying Lifshitz formula is not well formulated when the dielectric permittivity takes account of dissipation. To give the zeroth term of Lifshitz formula the definite meaning different prescriptions were used recently by different authors with diversed results. These results are shown to be improper and in disagreement with exper…
We propose a definitive test of whether plates involved in Casimir experiments should be modeled with ballistic or diffusive electrons--a prominent controversy highlighted by a number of conflicting experiments. The unambiguous test we propose is a measurement of the Casimir force between a disordered quasi-2D metallic plate and a three-dimensional metallic system at low temperatures, in which disorder-induced weak localization effects modify the well-known Drude result in an experimentally tunable way. We calculate the weak localization correction to the Casimir force as a function of magneti…
We examine the Casimir free energy and pressure of magnetic metal films, which are free-standing in vacuum, sandwiched between two dielectric plates, and deposited on either nonmagnetic or magnetic metallic plates. All calculations are performed using both the Drude and plasma model approaches to the Lifshitz theory. According to our results, the Casimir free energies and pressures calculated using both theoretical approaches are significantly different in the magnitude and sign even for thin films of several tens of nanometers thickness. Thus, for the Ni film of 47\,nm thickness deposited on …
We derive perturbation analytic expressions for the Casimir free energy and entropy between two dissimilar ferromagnetic plates which are applicale at arbitrarily low temperature. The dielectric properties of metals are described using either the nondissipative plasma model or the Drude model taking into account the dissipation of free charge carriers. Both cases of constant and frequency-dependent magnetic permeability are considered. It is shown that for ferromagnetic metals described by the plasma model the Casimir entropy goes to zero when the temperature vanishes, i.e., the Nernst heat th…
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.
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 …
It was recently found that thermodynamic anomalies which arise in the Casimir effect between metals described by the Drude model can be attributed to the interaction of fluctuating Foucault (or eddy) currents [Phys. Rev. Lett. 103, 130405 (2009)]. We show explicitly that the two leading terms of the low-temperature correction to the Casimir free energy of interaction between two plates, are identical to those pertaining to the Foucault current interaction alone, up to a correction which is very small for good metals. Moreover, a mode density along real frequencies is introduced, showing that t…
The nonlocal response functions to quantum fluctuations are used to find asymptotic expressions for the Casimir free energy and entropy at arbitrarily low temperature in the configuration of two parallel metallic plates. It is shown that by introducing an alternative nonlocal response to the off-the-mass-shell fluctuations the Lifshitz theory is brought into agreement with the requirements of thermodynamics. According to our results, the Casimir entropy calculated using the nonlocal response functions, which take into account dissipation of conduction electrons, remains positive and monotonous…
We consider Casimir-Polder and Casimir forces with finite dissipation by coupling heat baths to the dipoles introducing, this way, dissipation from 'first principles'. We derive a representation of the free energy as an integral over real frequencies, which can be viewd as an generalization of the 'remarkable formula' introduced by Ford et. al. 1985. For instance, we obtain a nonperturbative representation for the atom-atom and atom-wall interactions. We investigate several limiting cases. From the limit $T\to0$ we show that the third law of thermodynamics cannot be violated within the given a…
Differential force measurements between spheres coated with either nickel or gold and rotating disks with periodic distributions of nickel and gold are reported. The rotating samples are covered by a thin layer of titanium and a layer of gold. While titanium is used for fabrication purposes, the gold layer (nominal thicknesses of 21, 37, 47 and 87 nm) provides an isoelectronic environment, and is used to nullify the electrostatic contribution but allow the passage of long wavelength Casimir photons. A direct comparison between the experimental results and predictions from Drude and plasma mode…
The interaction between drifting carriers and traveling electromagnetic waves is considered within the context of the classical Boltzmann transport equation to compute the Casimir-Lifshitz force between media with small density of charge carriers, including dielectrics and intrinsic semiconductors. We expand upon our previous work [Phys. Rev. Lett. {\bf 101}, 163203 (2008)] and derive in some detail the frequency-dependent reflection amplitudes in this theory and compute the corresponding Casimir free energy for a parallel plate configuration. We critically discuss the the issue of verificatio…
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…
Recently V. A. Yampol'skii, S. Savel'ev, Z. A. Mayselis, S. S. Apostolov, and F. Nori [Phys. Rev. Lett., v.101, 096803 (2008),arXiv:0712.1395] claimed that the Casimir force between a thin metal film described by the Drude model and an ideal metal semispace can decrease with temperature whereas for bulk samples the Casimir force increases with temperature. On this basis the experimental observation of the decreasing temperature dependence of the Casimir force magnitude between a film and a bulk ideal metal was proposed. We demonstrate that, contrary to what is claimed by the authors, the magni…
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 investigate the Casimir force between two dissimilar plane mirrors the material properties of which are described by Drude or Lorentz models. We calculate analytically the short and long distance asymptote of the force and relate its behavior to the influence of interacting surface plasmons. In particular we discuss conditions under which Casimir repulsion could be achieved.
We review complicated problems in the Lifshitz theory describing the Casimir force between real material plates made of metals and dielectrics including different approaches to their resolution. It has been shown that both for metallic plates with perfect crystal lattices and for any dielectric plates the Casimir entropy calculated in the framework of the Lifshitz theory violates the Nernst heat theorem when the well approved dielectric functions are used in computations. The respective theoretical Casimir forces are excluded by the measurement data of numerous precision experiments. In the li…
The controversy concerning the temperature correction to the Casimir force has been ongoing for almost a decade with no view to a solution and has recently been extended to include semiconducting materials. We review some theoretical aspects of formal violations of Nernst's heat theorem in the context of Casimir Lifshitz thermodynamics and the role of the exponent of the leading term of the dielectric permittivity with respect to imaginary frequency. A general formalism for calculating the temperature corrections to free energy at low temperatures is developed for systems which do not exhibit …
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.
Recently Guerout et al. [Phys. Rev. E, v.93, 022108 (2016)] advocated that the lossless plasma model has to be redefined as the limit of the Drude model when the relaxation parameter goes to zero. It was claimed that the previously used plasma model cannot correctly describe the Casimir pressure between two plates made of both nonmagnetic and magnetic metals and has to be replaced with the redefined one. We show that the suggested redefinition does not satisfy necessary physical requirements imposed on the dielectric permittivity. We also present a plausible explanation to the fact that the lo…
