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# On the attraction between two perfectly conducting plates

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... If the mean value of a nonzero random variable vanishes, its variance nonetheless differs from zero. This simple mathematical fact leads to nontrivial physical consequences such as the occurrence of the so-called Casimir force [1]. In 1948 [1], after a discussion with Niels Bohr, the Dutch physicist H. B. G. Casimir realized that the zero-point fluctuations of the electromagnetic field in vacuum lead to the remarkable mechanical effect of the appearance of a long-ranged attractive force between two perfectly conducting, uncharged, parallel plates at a distance L from each other, and he calculated this force. ...
... This simple mathematical fact leads to nontrivial physical consequences such as the occurrence of the so-called Casimir force [1]. In 1948 [1], after a discussion with Niels Bohr, the Dutch physicist H. B. G. Casimir realized that the zero-point fluctuations of the electromagnetic field in vacuum lead to the remarkable mechanical effect of the appearance of a long-ranged attractive force between two perfectly conducting, uncharged, parallel plates at a distance L from each other, and he calculated this force. In the absence of charges on the plates the mean value of both the electric field E and the magnetic field B between the plates vanishes, i.e., E = 0 and B = 0, but E 2 0 and B 2 0, so that the expectation value of the energy due to the electromagnetic field in the volume between the plates, i.e., H ...
... Thus, upon changing the distance L between them, a force emerges which acts on the plates in normal direction. Nowadays this phenomenon is known as the quantum-mechanical Casimir effect [1] and currently there is a vast number of publications devoted to this effect (see the few corresponding reviews [2][3][4][5][6]). ...
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If a material body is immersed into a fluctuating medium, its shape and the properties of its constituents modify the fluctuations in the surrounding medium. If in the same medium there is a second body, modifications of the fluctuation due to the first one influence the modifications due to the second one. This mutual influence results in a force between these bodies. If the fluctuating medium consists of the confined electromagnetic field in vacuum, one speaks of the quantum mechanical Casimir effect. In the case that the order parameter of material fields fluctuates - such as differences of number densities or concentrations - and that the corresponding fluctuations of the order parameter are long-ranged, one speaks of the critical Casimir effect. This holds, e.g., in the case of systems which undergo a second-order phase transition and which are thermodynamically located near the corresponding critical point, or for systems with a continuous symmetry exhibiting Goldstone mode excitations. Here we review the currently available exact results concerning the critical Casimir effect in systems encompassing the one-dimensional Ising, XY, and Heisenberg models, the two-dimensional Ising model, the Gaussian and the spherical models, as well as the mean field results for the Ising and the XY model. Special attention is paid to the influence of the boundary conditions on the behavior of the Casimir force.
... Upon the inclusion of the Coulomb interaction, the mass renormalization leads also to a shrinking of the Wigner-Seitz radius, which implies a localization effect for the electrons. From the energy density of the photon field in the cavity we compute the corresponding Casimir force [99,100] (pressure) and we find that due to the interaction of the cavity field with the 2DEG, the Casimir force is repulsive [101]. Furthermore, we are able to describe consistently and from first principles dissipation and absorption processes without the need of any artificial damping parameter [3,93]. ...
... Having defined and constructed the effective field theory for the continuum of modes, we want to proceed by computing the zero point energy of the electromagnetic field. The zero point energy of the electromagnetic field it is known to be responsible for forces like the interatomic van der Waals forces, the Casimir-Polder forces between an atom and a body 5 [100,122], and the Casimir force between parallel conducting plates [99]. Since we consider a 2D material in a cavity we fall in the third category and the macroscopic forces in the system should be Casimir forces. ...
... In this model the photon field shrinks the Wigner-Seitz radius which implies a localization effect on the electrons. Moreover, the energy density of the photon field makes itself manifest by producing a Casimir force [99,100,122,123] between the mirrors of the cavity, which is repulsive due to the lightmatter coupling. Then, we performed linear response in the effective field theory and we showed that due to the continuum of photon modes we are able to describe dissipation and absorption processes without the need of any artificial damping parameter or having to introduce an environment for the system. ...
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Cavity modification of material properties and phenomena is a novel research field largely motivated by the advances in strong light-matter interactions. Despite this progress, exact solutions for extended systems strongly coupled to the photon field are not available, and both theory and experiments rely mainly on finite-system models. Therefore, a paradigmatic example of an exactly solvable extended system in a cavity becomes highly desirable. To fill this gap we revisit Sommerfeld's theory of the free electron gas in cavity quantum electrodynamics. We solve this system analytically in the long-wavelength limit for an arbitrary number of noninteracting electrons, and we demonstrate that the electron-photon ground state is a Fermi liquid which contains virtual photons. In contrast to models of finite systems, no ground state exists if the diamagentic A2 term is omitted. Further, by performing linear response we show that the cavity field induces plasmon-polariton excitations and modifies the optical and the DC conductivity of the electron gas. Our exact solution allows us to consider the thermodynamic limit for both electrons and photons by constructing an effective quantum field theory. The continuum of modes leads to a many-body renormalization of the electron mass, which modifies the fermionic quasiparticle excitations of the Fermi liquid and the Wigner-Seitz radius of the interacting electron gas. Last, we show how the matter-modified photon field leads to a repulsive Casimir force and how the continuum of modes introduces dissipation into the light-matter system. Several of the presented findings should be experimentally accessible.
... By contrast, the second zero-point (ZP) contribution is associated with the zero-point energy, it is a quantum property of the vacuum, and by definition it involves all the photons in vacuum, i.e. inside and outside the physical system. Accordingly, its value diverges with frequency, it does not vanish at T = 0, it has never been measured directly but only through its effects, and in particular the Casimir effect that evidences an attractive or repulsive force acting between finite parts of the physical system [15], as predicted by Casimir in 1948 [8] for the simple case of two parallel perfect conducting plates. ( For an updated bibliography on the Casimir effect the reader can refer to J. Babb, bibliography on the Casimir Effect web site, https://www.cfa.harvard.edu/ ...
... The sum gives a divergent energy contribution when evaluated all over the space, which leads to the so-called vacuum catastrophe [9]. Otherwise, as observed by Casimir [8], real measurements are performed on finite-size systems where manifestations of zero-point energy are directly observable [16]. In particular, the different content of EM energy inside and outside an assigned region produces a finite value of the zero-point total energy. ...
... In general, Eq. (18) is solved by using specific boundary conditions related to: (i) the shape of the physical system, and (ii) the material used to construct the physical system. Calculations of the Casimir force are in general not easy to be performed [17][18][19], and here we report the simple but significant case considered by Casimir [8] and further confirmed by more detailed mathematical approaches [20]: ...
... Since each statistical field in the hierarchy (Fig.1) has a root-mean-square speed r v  and usually a much faster "atomic" speed u  , in view of (39) one may associate a "wave" and a "particle" speed with each statistical field [43,52] that is the speed of typical detonation wave [58]. At the scale LKD ( Fig. 1) physical space is identified as Casimir vacuum [45] and is considered to be a compressible fluid, Planck compressible ether [59] as discussed in [60]. Lorentz perceptions about the medium of space as Aristotle or Huygens ether [60] is further described in the following quotation by Verhulst [61] from Lorentz 1915 lecture at the Royal Academy of Sciences in Amsterdam: ...
... As discussed in Section 2, absolute thermodynamic temperature identified as Wien wavelength w T  = of thermal oscillations [32] additional compactified dimensions could be associated with 3-rotational and 3vibrational internal degrees of freedom in Eq. (20). Therefore, the total number of dimensions required for the description of each statistical field ( Fig. 1) including the physical space or Casimir [45] vacuum will be 4 + 6 = 10 in harmony with models of superstring theories [71]. By the equation of state pT R     = at constant pressure, density is also inversely related to absolute temperature. ...
... Because pressure can be viewed as volumetric potential energy density of the field [32], negative values of pressure, often assumed in inflationary models of cosmology, are expected to be nonphysical. In view of finite value of Casimir [45] zero-point energy, it is reasonable to anticipate a finite positive pressure of Casimir vacuum in accordance with modified van der Waals equation of state [72] 0 p p p p p =     =  is shown in Fig. 7. Therefore, if one introduces the concept of space "scalar curvature" as deviation of space density from the density of Casimir [45] vacuum v    −  , one finds that ( 0 , 0 , 0)    =   could be respectively associated with ( , , ) Riemannian Euclidian Lobachevskian space. The hydrodynamic model of chromodynamics shock waves in compressible space (Fig. 7) is in harmony with the perceptions of 't Hooft [73] concerning quantum gravity as a dissipative deterministic dynamic system. ...
Chapter
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An invariant model of Boltzmann statistical mechanics is applied to derive invariant Schrödinger equation of quantum mechanics from invariant Bernoulli equation of hydrodynamics. The results suggest new perspectives regarding quantum mechanics wave function and its collapse, stationary versus propagating wave functions, and wave-particle duality. The invariant hydrodynamic model also leads to the definition of generalized shock waves in “supersonic” flows at molecular-, electro-, and chromo-dynamic scales with (Mach, Lorentz, and Michelson) numbers exceeding unity. The invariant internal hydro-thermo-diffusive structure of such generalized “shock” waves are described.
... Although not that often introduced in the context of thermal physics, the Casimir effect is another example of counter-intuitive phenomena [2], which fits nicely in the framework of statistical mechanics [3,4,5,6,7,8,9]. The usual setup consists of two large perfect conductor neutral plates facing each other at a distance d. ...
... (1), the "Casimir pressure" in eq. (2) does not vanish at the absolute zero of temperature. Still more surprising, the non vanishing eq. ...
... Moreover, the negativeness of eq. (2) indicates that the plates are attracted to each other, and that seems to be a very unusual behavior for the walls of a cavity containing blackbody radiation. When temperature is taken to be arbitrarily large (T → ∞), within the same framework that leads to eq. (2), now it results as leading contribution the ordinary blackbody radiation pressure, namely, ...
Preprint
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This paper is a study of the electromagnetic radiation at temperature $T$ in a thin slab whose walls are made of a perfect conductor. The two large parallel walls of area $A$ are apart by a distance $d\ll \sqrt{A}$. We take $T$, $A$, and $d$ as thermodynamic parameters, obtaining the free energy from a procedure that involves the integration over the slab of the ensemble average of the stress-energy-momentum tensor calculated long ago by Brown and Maclay. Both thermodynamic regimes $kTd/\hbar c\gg 1$ and $kTd/\hbar c\ll 1$ are fully addressed. We show that certain thermodynamic quantities which are notoriously ill defined (or trivial) in ordinary blackbody thermodynamics are now well defined (or nontrivial) due to presence of boundary conditions at the walls of the slab ("Casimir's effect"). The relationships among such quantities are fully explored and it is speculated that they may be experimentally checked. Stability is addressed, showing that electromagnetic radiation in the slab is thermally stable; but mechanically unstable. We investigate thermodynamic processes where temperature, internal energy, entropy and enthalpy are each taken to be constant, revealing rather atypical behaviors. For example, in sharp contrast with what one would expect from a gas, when $kTd/\hbar c\ll 1$, "free expansion" gives place to "free contraction'' in accordance with the second law of thermodynamics. As a check of consistency of the formulae we remark that various Carnot cycles have been examined and verified that they correctly lead to Carnot's efficiency.
... Casimir effect is a fascinating phenomenon (Pauli initially dismissed it as "absolute nonsense" [13]) in which two electrically neutral, parallel metal plates, spaced a short distance from each other, experience a force of mutual attraction. After its first discovery by Hendrik Casimir [14], there is still debate about whether the Casimir effect is a manifestation of the reality of zero-point quantum fluctuations of the electromagnetic vacuum or is it just the relativistic, retarded van der Waals force between the metal plates [15]. Casimir himself, when asked the same question twice with an interval of eleven years, whether the Casimir effect is the result quantum fluctuations of the electromagnetic field, or is it caused by van der Waals forces between molecules in two objects, replied that he has not made up his mind (see foreword by I.H. ...
... The picture of a fluctuating quantum vacuum, altered by the presence of an object, has proven to be very useful for predicting new and interesting effects such as the quantum atmosphere effect [18] and the dissipationless rotation-induced axial Casimir force that emerges when a particle rotates above a plate that exhibits either time-reversal symmetry breaking or parity-symmetry breaking [19]. Therefore, we consider it appropriate to remind the spirit of the Casimir's original derivation [14]. ...
... where the notation ∞ n=(0)1 means that the term n = 0 in the sum must be multiplied by 1/2 [14]. When calculating the zero-point energy of the fluctuating electromagnetic field outside the conducting plates (but inside the quantization volume L 3 ) W L−R , we should specify the boundary conditions in the x-direction. ...
Article
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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.
... Why another paper on Casimir forces? Remarkably, even though the seminal works are from the nineteen fifties [1,2], the topic of Casimir forces is still very much active and outstanding questions remain so far unanswered (see e.g. [3] for a review). ...
... Such divergences are not removable using the usual introduction of local counterterms. The presence of such divergences has led to various partial solutions, of both conceptual and technical nature, and sometimes an in-depth questioning of standard QFT calculations [3,[10][11][12][13]. 2 In this work we claim that we bring a new and satisfactory answer to the puzzles raised by the existence of these divergences, using only conventional quantum field theory techniques. ...
... The quantum work in the limiting case of thinshell geometry is further evaluated. Section 4 specialises to two rigid bodies, showing that Casimir and Casimir-Polder forces are asymptotically recovered as limits of our unifying 2 Our calculations bear some technical similarities with those presented in these references. For example the source we consider has finite density, and the Dirichlet limit is recovered by sending the density to infinity. ...
Preprint
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We introduce a unified description of Casimir and Casimir-Polder forces between classical objects in quantum field theory (QFT). We focus on interactions mediated by a scalar field. We first show that the quantum work felt by an arbitrary (either rigid or deformable) classical body is finite upon requiring conservation of matter. Using our formulation, we explicitly show how the complete QFT prediction for the quantum pressure inside the Dirichlet sphere is finite, thereby solving a long-standing problem. We then show that our general result interpolates between Casimir and Casimir-Polder forces for arbitrary rigid bodies. We provide the expressions for the generalised Casimir force in the simple cases of plate-plate and plate-point geometries. In the latter geometry, we show how to compute phase shifts observable in atomic interferometry, induced by the generalized quantum force.
... One interesting and intriguing phenomenon which is of pure quantum nature is the Casimir effect. It was first predicted by H. Casimir in (1948) [1] who noticed that the effect consists in a force of attraction that arises between two neutral parallel and perfectly conducting plates, placed in vacuum within a very short distance from each other. This force of attraction is described in the framework of a quantum electromagnetic field and it is the result of modifications on the vacuum fluctuations. ...
... One interesting and intriguing phenomenon which is of pure quantum nature is the Casimir effect. It was first predicted by H. Casimir in (1948) [1] who noticed that the effect consists in a force of attraction that arises between two neutral parallel and perfectly conducting plates, placed in vacuum within a very short distance from each other. This force of attraction is described in the framework of a quantum electromagnetic field and it is the result of modifications on the vacuum fluctuations. ...
... [9,[30][31][32], using other methods. Additionally, we also calculate the second loop correction to the effective potential, which translates in being the correction to the Casimir energy density at order λ due to the self-interaction considered in Eq. (1). The generation of a correction, of order λ, to the mass m of the field is also obtained. ...
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In this paper the effective potential approach in quantum field theory is used in order to investigate self-interaction loop correction to the Casimir energy density and generation of topological mass for both massless and massive real scalar fields. It is assumed that the scalar field obeys a helix boundary condition. In addition, it is also considered a CPT-even aether-type violation of the Lorentz symmetry. In the absence of the Lorentz violation we obtain analytical expressions for the loop correction to the Casimir energy density and to the mass of the scalar field. The same expressions are also obtained assuming the Lorentz violation in each of the spacetime directions. We also show some graphs that exhibit how the loop correction and the Lorentz violation affect the the Casimir energy density and the mass of the scalar field.
... By contrast, the second zero-point (ZP) contribution is associated with the zero-point energy, it is a quantum property of the vacuum, and by definition it involves all the photons in vacuum, i.e. inside and outside the physical system. Accordingly, its value diverges with frequency, it does not vanish at T = 0, it has never been measured directly but only through its effects, and in particular the Casimir effect that evidences an attractive or repulsive force acting between finite parts of the physical system [13], as predicted by Casimir in 1948 [6] for the simple case of two parallel perfect conducting plates. ...
... The sum gives a divergent energy contribution when evaluated all over the space, which leads to the so-called vacuum catastrophe [7]. Otherwise, as observed by Casimir [6], real measurements are performed on finite-size systems where manifestations of zero-point energy are directly observable [14]. In particular, the different content of EM energy inside and outside an assigned region produces a finite value of the zero-point total energy. ...
... In general, Eq. (18) is solved by using specific boundary conditions related to: (i) the shape of the physical system, and (ii) the material used to construct the physical system. Calculations are not easy to be performed [15][16][17], and here we report the simple but significant case considered by Casimir [6] and further confirmed by more detailed mathematical approaches [18]: ...
Preprint
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The Stefan-Boltzmann (SB) law relates the emissivity q,to the fourth power of the absolute temperature T, the proportionaly factor, $\sigma$ firstly estimated by Stefan to within 11 per cent of the actual value. The law is a pillar of modern physics since its microscopic derivation implies the quantization of the energy related to the electromagnetic field. Somewhat astonishing, Boltzmann presented his derivation in 1878 making use only of electrodynamic and thermodynamic classical concepts, apparently without introducing any quantum hypothesis (here called first Boltzmann paradox). By using Planck (1901) quantization of the radiation field in terms of a gas of photons, the SB law received a microscopic interpretation providing also the value of the SB constant on the basis of a set of universal constants including the quantum action constant h. However, the successive consideration by Planck (1912) of the zero-point energy contribution was found to be responsible of another divergence of the radiation energy-density for the single photon mode at high frequencies. This divergence is of pure quantum origin and is responsible for a vacuum-catastrophe, to keep the analogy with the well-known ultraviolet catastrophe of the classical black-body radiation spectrum, given by the Rayleigh-Jeans law in 1900. As a consequence, from a rigorous quantum-mechanical derivation we expect the divergence of the SB law (here called second Boltzmann paradox). In this paper we revisit the SB law by accounting for genuine quantum effects associated with Planck energy quantization and Casimir size quantization thus resolving both Boltzmann paradoxes.
... The Casimir effect is the attraction of two neutral interfaces induced by the zero-point fluctuation (i.e. the uncertainty principle) of electromagnetic field, which is initially discovered by Casimir in 1948 [38] motivated by his previous work with Polder [39]. The van der Waals effect is closely related to the Casimir effect. ...
... In his work [38], Casimir discussed a pair of planar mirrors separated by a vacuum gap with a width d shown in Figure 3.1. Since he assumed that the mirrors are perfect ones, they can be represented by the Dirichlet boundary conditions. ...
... The sum representation in Eq. (3.3) is called the Casimir sum. Casimir calculated the derivative of the Casimir energy with respect to the gap width [38], ...
Preprint
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In this thesis, I analyse the electromagnetic properties of dynamical metasurfaces and find two critical phenomena. The first is the Casimir-induced instability of a deformable metallic film. In general, two charge-neutral interfaces attract with or repel each other due to the contribution from the zero-point fluctuation of the electromagnetic field between them, namely, the Casimir effect. The effects of perturbative interface corrugation on the Casimir energy in the film system is studied by the proximity force approximation with dispersion correction. If the corrugation period exceeds a critical value, the Casimir effect dominates the surface tension and brings about structural instability. The second is \v{C}erebkov radiation in the vacuum from a time-varying, corrugated surface. Travelling faster than light brings about electromagnetic shock waves, \v{C}erenkov radiation. Since light is the fastest object in a vacuum, it has been considered that \v{C}erenkov radiation is emitted only in the presence of some refractive index. Here, I propose mimicking a series of particles travelling faster than light in a vacuum by dynamical surface corrugation to find \v{C}erenkov radiation in a vacuum from the surface. The dynamical corrugation induces an effective current source on the surface with an external electrostatic field applied. When the corrugation profile is of travelling wave type, the source can be regarded as a series of dipoles virtually travelling along the surface. If the phase velocity of the travelling wave profile exceeds the speed of light, and so do the dipoles, they emit \v{C}erenkov radiation in a vacuum.
... However, If one is interested in the change of energy of the electromagnetic field due to a pair of perfectly conducting metallic plates, the zero-point energy needs to be handled properly. The zero-point energy in this case leads to the emergence of macroscopic Casimir [88] and Casimir-Polder forces [89] which have measurable effects. We will look into these forces more closely in chapter 7. ...
... The zero-point energy of the photon field (in the continuum) is known to be responsible for the emergence of forces like the interatomic van der Waals forces, the Casimir-Polder forces between an atom and a macroscopic body [89,83], and the Casimir force between two parallel conducting plates [88]. Here, we are considering a 2D material inside a cavity and consequently we fall in the third category, which means that the macroscopic forces in our system will be Casimir forces. ...
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Condensed matter physics and quantum electrodynamics (QED) have been long considered as distinct disciplines. This situation is changing by the progress in cavity QED materials. Motivated by these advances we aim to bridge these fields by merging fundamental concepts coming from both sides. In the first part of the thesis we present how non-relativistic QED can be constructed and we discuss the light-matter interaction in different gauges and that neglecting particular quadratic terms can lead to instabilities. In the second part, we revisit the Sommerfeld model of the free electron gas in cavity QED and provide the analytic solution for this paradigmatic system coupled to the cavity. We show that the cavity field modifies the optical conductivity of the electron gas and suppresses its Drude peak. Further, by constructing an effective field theory in the continuum of photon modes we show how the photon field leads to a many-body renormalization of the electron mass, which modifies the fermionic quasiparticle excitations of the Fermi liquid. In the last part, we show that translational symmetry for periodic materials in homogeneous magnetic fields can be restored by embedding the problem into QED. This leads to a generalization of Bloch's theory for electron-photon systems, that we named as QED-Bloch theory, which can be applied for the description of periodic materials in homogeneous magnetic fields and strongly coupled to the quantized cavity field. As a first application we consider Landau levels coupled to a cavity and we show that quasiparticle excitations between Landau levels and photons appear, called Landau polaritons. Further, for periodic materials in such setups, QED-Bloch theory predicts the emergence of novel fractal polaritonic energy spectra, which we name as fractal polaritons. The fractal polaritons are a polaritonic, QED analogue of the Hofstadter butterfly.
... The Casimir effect is a phenomenon that arises due to fluctuations in the quantum vacuum. In its simplest version, uncharged metal plates that are spaced much closer than a micron freeze out the long wavelength modes of these fluctuations [1][2][3]. The lower density of modes between the plates in comparison with the density of modes outside the plates gives rise to a net attractive force between the plates. ...
... As ħ goes to zero, so does the force. The other thing to note in Eq. (1) is the (1) Force/area = −ℏc 2 ∕240d 4 dependence on the fourth power of the separation, 1/d 4 . This is in contrast to gravitational or electrostatic forces which scale as 1/d 2 . ...
Article
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In this paper, we discuss using the Casimir force in conjunction with a MEMS parametric amplifier to construct a quantum displacement amplifier. Such a mechanical amplifier converts DC displacements into much larger AC oscillations via the quantum gain of the system which, in some cases, can be a factor of a million or more. This would allow one to build chip scale metrology systems with zeptometer positional resolution. This approach leverages quantum fluctuations to build a device with a sensitivity that can’t be obtained with classical systems.
... Here we present a reasoning, which follows the lines of eqs. (6) and clarifies the meaning of R. It invokes analytic continuation, which is the only valid justification. Let us consider |z| < 1, and the limit z → −1, for ...
... The Casimir effect was first predicted in Ref. [5], but here we follow the point of view which Hendrik Casimir (1909Casimir ( -2000 expressed a little later [6], after a discussion with Niels Bohr. For comprehensive overviews, we refer to Refs. ...
Preprint
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Srinivasa Ramanujan was a great self-taught Indian mathematician, who died a century ago, at the age of only 32, one year after returning from England. Among his numerous achievements is the assignment of sensible, finite values to divergent series, which correspond to Riemann's $\zeta$-function with negative integer arguments. He hardly left any explanation about it, but following the few hints that he gave, we construct a direct justification for the best known example, based on analytic continuation. As a physical application of Ramanujan summation we discuss the Casimir effect, where this way of removing a divergent term corresponds to the renormalization of the vacuum energy density, in particular of the photon field. This leads to the prediction of the Casimir force between conducting plates, which has now been accurately confirmed by experiments. Finally we review the discussion about the meaning and interpretation of the Casimir effect. This takes us to the mystery surrounding the magnitude of Dark Energy.
... показав [240], що паралельнi провiднi пластини, розмiщенi у вакуумi, повиннi притягатися завдяки iснуванню нульових вакуумних флуктуацiй електромагнiтного поля. Казимир розрахував енергiю взаємодiї δE(L) мiж цими поверхнями на одиницю площi i отримав результат ...
... Ця функцiя застосовується також у багатьох фiзичних задачах. Одним з важливих прикладiв, який має безпосереднє вiдношення до наших дослiджень, є дослiдження ефекту Казимира [240] у квантовiй теорiї поля i його аналогiв у статистичнiй фiзицi i температурнiй теорiї поля при високих температурах, див. напр. ...
Thesis
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The habilitation thesis in Ukrainian. Contains an abstract in English on p. 6-10.
... Here we present a reasoning, which follows the lines of Eqs. (6) and clarifies the meaning of R. It invokes analytic continuation, which is the only valid justification. Let us consider |z| < 1, and the limit z → −1, for ...
... The Casimir effect was first predicted in Ref. [5], but here we follow the point of view which Hendrik Casimir (1909Casimir ( -2000 expressed a little later [6], after a discussion with Niels Bohr. For comprehensive overviews, we refer to Refs. ...
Article
Full-text available
Srinivasa Ramanujan was a great self-taught Indian mathematician, who died a century ago, at the age of only 32, one year after returning from England. Among his numerous achievements is the assignment of sensible, finite values to divergent series, which correspond to Riemann's $\zeta$-function with negative integer arguments. He hardly left any explanation about it, but following the few hints that he gave, we construct a direct justification for the best known example, based on analytic continuation. As a physical application of Ramanujan summation we discuss the Casimir effect, where this way of removing a divergent term corresponds to the renormalization of the vacuum energy density, in particular of the photon field. This leads to the prediction of the Casimir force between conducting plates, which has now been accurately confirmed by experiments. Finally we review the discussion about the meaning and interpretation of the Casimir effect. This takes us to the mystery surrounding the magnitude of Dark Energy.
... As predicted by Casimir [1], two parallel uncharged ideal metal planes separated by a distance a at zero temperature should attract each other by the force ...
... Here, k B is the Boltzmann constant, k ⊥ = (k 2 x + k 2 y ) 1/2 is the magnitude of the wave vector projection on the plane of plates, ξ l = 2πk B T l/ are the Matsubara frequencies, q l = (k 2 ⊥ + ξ 2 l /c 2 ) 1/2 , and the prime on the summation sign divides by 2 the term of the first sum with l = 0. The reflection coefficients of electromagnetic waves with the transverse magnetic (α = TM) and transverse electric (α = TE) polarizations on the first and second plates are r (1) α (iξ l , k ⊥ ) and r (2) α (iξ l , k ⊥ ), respectively. In the original version of the Lifshitz theory [2,3], it is assumed that the plate materials possess only the temporal dispersion, i.e., their dielectric permittivities, ε n (ω), and magnetic permeability, µ n (ω), where n = 1, 2 for the first and second plates, depend on the frequency ω (a dependence on T , e.g., for metals is also allowed). ...
Preprint
It has been known that the Lifshitz theory of the Casimir force comes into conflict with the measurement data if the response of conduction electrons in metals to electromagnetic fluctuations is described by the well tested dissipative Drude model. The same theory is in a very good agreement with measurements of the Casimir force from graphene whose spatially nonlocal electromagnetic response is derived from the first principles of quantum electrodynamics. Here, we propose the spatially nonlocal phenomenological dielectric functions of metals which lead to nearly the same response, as the standard Drude model, to the propagating waves, but to a different response in the case of evanescent waves. Unlike some previous suggestions of this type, the response functions considered here depend on all components of the wave vector as is most natural in the formalism of specular reflection used. It is shown that these response functions satisfy the Kramers-Kronig relations. We derive respective expressions for the surface impedances and reflection coefficients. The obtained results are used to compute the effective Casimir pressure between two parallel plates, the Casimir force between a sphere and a plate, and its gradient in configurations of the most precise experiments performed with both nonmagnetic (Au) and magnetic (Ni) test bodies. It is shown that in all cases (Au-Au, Au-Ni, and Ni-Ni test bodies) the predictions of the Lifshitz theory found by using the dissipative nonlocal response functions are in as good agreement with the measurement data, as was reached previously with the dissipationless plasma model. Possible developments and applications of these results are discussed.
... While past studies have mostly focused on the behavior of tracer particles passively carried by a fluctuating solvent, in recent years increasing attention has been paid to cases in which the particle and the solvent affect each other dynamically [3][4][5][6][7][8][9][10]. Particularly interesting is the case in which the medium is a fluid near a critical point, thus displaying long-range spatial correlations and long relaxation times. Objects immersed in near-critical fluids are known to experience fluctuation-induced forces [11][12][13] such as the critical Casimir force, i.e., the thermal analog of the celebrated effect in quantum electrodynamics [14]. While equilibrium field-mediated effects have long since been explored, the dynamical behavior of these systems has rarely been addressed in the literature. ...
... At O λ 0 , Eq. (14) is solved by the Ornstein-Uhlenbeck process, recalled in Appendix A. The higher-order corrections X (n) can be formally expressed as ...
... Toward the other extreme, at the Planck scale, dimensional reduction to effective 2D behavior [25] and the breaking up of space into a large number of causally disconnected regions [26]. For the more tangible, laboratory-accessible phenomena, the simplest example of a vacuum polarization effect is perhaps the Casimir effect [27][28][29][30], e.g., an attractive force between two conducting plates, or a repulsive force in a conducting sphere. When the plates are moved rapidly, particles are created from the vacuum. ...
... As n 0 = 0, we have the zero temperature limit here. The result actually corresponds to the quantum Casimir effect [27][28][29][30] due to the spatial circle. This is consistent with the usual derivation of the Casimir energy. ...
Article
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An important yet perplexing result from work in the 1990s and 2000s is the near-unity value of the ratio of fluctuations in the vacuum energy density of quantum fields to the mean in a collection of generic spacetimes. This was carried out by way of calculating the noise kernels which are the correlators of the stress-energy tensor of quantum fields. In this paper, we revisit this issue via a quantum thermodynamics approach, by calculating two quintessential thermodynamic quantities: the heat capacity and the quantum compressibility of some model geometries filled with a quantum field at high and low temperatures. This is because heat capacity at constant volume gives a measure of the fluctuations of the energy density to the mean. When this ratio approaches or exceeds unity, the validity of the canonical distribution is called into question. Likewise, a system’s compressibility at constant pressure is a criterion for the validity of grand canonical ensemble. We derive the free energy density and, from it, obtain the expressions for these two thermodynamic quantities for thermal and quantum fields in 2d Casimir space, 2d Einstein cylinder and 4d (S1×S3 ) Einstein universe. To examine the dependence on the dimensionality of space, for completeness, we have also derived these thermodynamic quantities for the Einstein universes with even-spatial dimensions: S1×S2 and S1×S4. With this array of spacetimes we can investigate the thermodynamic stability of quantum matter fields in them and make some qualitative observations on the compatibility condition for the co-existence between quantum fields and spacetimes, a fundamental issue in the quantum and gravitation conundrum.
... Objects immersed in a fluctuating medium experience induced interactions due to the constraints they impose on its fluctuating modes. Among these interactions [1][2][3][4][5][6][7] are the critical Casimir forces [8][9][10][11][12] observed in classical systems close to the critical point of a second-order phase transition: they are the thermal and classical counterpart of the well-known Casimir effect in quantum electrodynamics [1]. Even when fluctuations are negligible, particles deforming a correlated elastic medium still experience field-mediated interactions [13,14]. ...
... Objects immersed in a fluctuating medium experience induced interactions due to the constraints they impose on its fluctuating modes. Among these interactions [1][2][3][4][5][6][7] are the critical Casimir forces [8][9][10][11][12] observed in classical systems close to the critical point of a second-order phase transition: they are the thermal and classical counterpart of the well-known Casimir effect in quantum electrodynamics [1]. Even when fluctuations are negligible, particles deforming a correlated elastic medium still experience field-mediated interactions [13,14]. ...
Preprint
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We study the non-equilibrium dynamics of two particles confined in two spatially separated harmonic potentials and linearly coupled to the same thermally fluctuating scalar field, a cartoon for optically trapped colloids in contact with a medium close to a continuous phase transition. When an external periodic driving is applied to one of these particles, a non-equilibrium periodic state is eventually reached in which their motion synchronizes thanks to the field-mediated effective interaction, a phenomenon already observed in experiments. We fully characterize the nonlinear response of the second particle as a function of the driving frequency, and in particular far from the adiabatic regime in which the field can be assumed to relax instantaneously. We compare the perturbative, analytic solution to its adiabatic approximation, thus determining the limits of validity of the latter, and we qualitatively test our predictions against numerical simulations.
... In the following, we will subsume all these environment-assisted quantum-vacuum effects under a generalised notion of the Purcell effect, which was originally conceived for the cavity-induced modification of spontaneous decay [3]. Besides its influence on atoms and molecules the changes of the quantum vacuum induced by macroscopic objects can also have an impact onto the macroscopic objects themselves; they lead e.g. to the Casimir force [8]. ...
Article
Full-text available
The effect of cavities or plates upon the electromagnetic quantum vacuum are considered in the context of electro-optic sampling, revealing how they can be directly studied. These modifications are at the heart of e.g. the Casimir force or the Purcell effect such that a link between electro-optic sampling of the quantum vacuum and environment-induced vacuum effects is forged. Furthermore, we discuss the microscopic processes underlying electro-optic sampling of quantum-vacuum fluctuations, leading to an interpretation of these experiments in terms of exchange of virtual photons. With this in mind it is shown how one can reveal the dynamics of vacuum fluctuations by resolving them in the frequency and time domains using electro-optic sampling experiments.
... It is interesting to recall here that quantum-optical fluctuation-induced phenomena can also be found beyond atomic systems. One closely related example is the counterpart of the Casimir-Polder force, the so-called Casimir force which acts between macroscopic bodies 34 . Even though both effects were theoretically predicted at about the same time, they come with rather distinct experimental challenges. ...
Preprint
When two or more objects move relative to one another in vacuum, they experience a drag force which, at zero temperature, usually goes under the name of quantum friction. This contactless non-conservative interaction is mediated by the fluctuations of the material-modified quantum electrodynamic vacuum and, hence, is purely quantum in nature. Numerous investigations have revealed the richness of the mechanisms at work, thereby stimulating novel theoretical and experimental approaches and identifying challenges as well as opportunities. In this article, we provide an overview of the physics surrounding quantum friction and a perspective on recent developments.
... Intermolecular forces such as van der Waals forces and Casimir force have been considered in the modeling of nanostructures [1,2]. Although these intermolecular forces describe the same physical phenomena at different nano levels [3], the authors modeled the intermolecular forces in nanosystems separately or simultaneously. ...
Article
The aim of the present research is to understand the bouncing dynamic behavior of nano electromechanical (NEM) switches in order to improve the switch performance and reliability. It is well known that bouncing can dramatically degrade the switch performance and life; hence, in the present study, the bouncing dynamics of a cantilever-based NEM switch has been studied in detail. To this end, the repulsive van der Waals force is incorporated into a nano-switch model to capture the contact dynamics. Intermolecular forces, surface effects, and gas rarefication effects were also included in the proposed model. The Euler-Bernoulli beam theory and an approximate approach based on Galerkin’s method have been employed to predict transient dynamic responses. In the present study, performance parameters such as initial contact time, permanent contact time, major bounce height, and the number of bounces, were quantified in the presence of interactive system nonlinearities. The performance parameters were used to investigate the influence of surface effects and rarefication effects on the performance of an electrostatically actuated switch. Recommended operating conditions are suggested to avoid excessive bouncing for these types of NEM switches.
... Although provocative and intriguing, the Steinhardt-Turok model and related theories , like Einstein's oscillating universe, have been rejected by most scientists for a variety of reasons which will not be detailed here, except in passing. For example, if these parallel universes are encircled by membranes, then rather than the hypothetical "dark energy" it is more likely that the voids between each membrane encapsulated bubble Universe would be filled with negative energy (Joseph 2014), thereby drawing these universes together--similar to a Casimir vacuum (Casimir 1948)--and never letting them part. ...
Article
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A "Big Bang" creation event which begins as a subatomic singularity leads to the question: where did the singularity come from? As detailed here, the evidence indicates that this (observable) "universe" recycles itself by expanding outward and collapsing back into a singularity which explodes outward again. The cosmos, however, may be infinite, and consist of innumerable "universes" all of which eventually collapse into a singularity which then mushrooms outward giving rise to new universes (including our own) and thus explaining why our universe behaves and is organized contrary to Big Bang theory. These theories of cyclic, oscillating universes and of repeated episodes of expansion, contraction, and colliding universes, have failed to generate widespread support, and are based on the beliefs that: A) this universe is expanding-when, it may already be accelerating toward a collapse-and that B) a singularity explodes outward. Quantum physics and relativity, however, predict that a singularity-which has shrunk to smaller than a Planck Length (1.61619926 x 10-33 cm), will blow an imploding hole through the fabric of the space-time quantum continuum, forming an Einstein-Rosen bridge and creating a mirror universe on the other side. The mirror of a positive-matter universe, is an antimatter universe. If cycles of creation alternate from antimatter to matter, and if a collapsing/imploding antimatter universe gave birth to our own, there is no violation of the second law of thermodynamics, entropy ceases to be a limiting factor, and the conservation laws of energy and mass are maintained. As predicted by Einstein's theory of relativity, and when coupled with quantum physics, it appears we may be dwelling in a Mirror Universe which formed from the remnants of a collapsing antimatter universe which upon shrinking to a singularity smaller than a Planck length, blew a hole in the quantum continuum thereby leading to the creation of this universe on the other side. Further, our Mirror Universe is not be expanding, but already collapsing and accelerating to its doom. If so, this collapse may account for the clumping and formation of great galactic walls separated by vast voids, colliding galaxies, and phenomenon attributed to the purely hypothetical "dark energy" which may not exist at all.
... Casimir interaction was initially studied as an attractive force between two perfectly conducting metal plates [1]. Casimir interpreted that this interaction is dependent only on the distance between the plates, and electromagnetic fluctuations whose wavelength are comparable with the distance between the plates would be contributing to the Casimir effect. ...
Preprint
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The Casimir interaction energy for a class of discrete self-similar configuration of parallel plates is evaluated using existing methods. The similarities to characteristics of an attractive Casimir force is deduced only at infinite range of configuration. Further, the emergence of Casimir-like energy is qualitatively described for a Gaussian model of Landau-Ginzburg scalar field. Its relevance to self-similarity in the statistical field is shown at infinite range of fluctuations.
... In the late 1940s Casimir and Polder set forth to calculate the force excerted between two polarisable atoms and between an atom and a neutral conducting plate [1]. Soon after Casimir understood that a force would also be present between two neutral conducting plates [2]. The latter effect is a particular instance of what is known today as the (static) Casimir effect. ...
Preprint
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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, the total Casimir energy and the Casimir force. We give as a future perspective the direction in which this work should be generalised to put the theoretical predictions of the dynamical Casimir effect experiments on a rigorous footing.
... In 1948, Dutch physicist Hendrik Casimir discovered a physical phenomenon known as Casimir effect [1][2][3][4][5]; It has been widely valued and studied for a long time [6][7][8][9][10]. This is because it provides a direct demonstration and proof that "quantum vacuum is a physical reality". ...
Article
The Casimir effect is one observable of the existence of the vacuum energy, i.e. the existence of vacuum electromagnetic field. The meaning of this word "vacuum" is physical vacuum, not technology vacuum. Then, we say that the change in the vacuum structure enforced by the plates. There are two kinds of vacuum, one is usual vacua or free vacua (outside the plates). Another is the negative energy vacua (inside the plates), and the refraction index less than 1(n<1). That cause a change in the light speed for electromagnetic waves propagating perpendicular to the plates: △c/c1.6×10-60d-4, and d is the plate distance. When d=10-9m(1nm), △c=10-24c. Then, a two-loop QED effect cause the phase and group velocities of an electromagnetic wave to slightly exceed c. Though the difference are very small, that raise interesting matters of principle. The focus of this paper is to improve the understanding of the nature of quantum vacuum. In the past, to say that "vacuum is not empty" was already a criticism and subversion of classical physics. Now it seems doubly strange to say that there is a negative energy vacuum that is "empty"than the normal physical vacuum. But these theories are rigorously justified; Casimir effect can create an environment with refractive index less than 1(n<1) and lead to the appearance of superluminality, which is one of the representations of "quantum superluminality". These advances in basic science will certainly open up new fields of application, In short, it is not the Casimir structure that creates the quantum vacuum, but the structure that makes the quantum vacuum "emerge"in a clever way as a perceptible physical reality. This is truly a scientific achievement.
... The Casimir energy may be thought-of as the energy difference due to the distortion of the vacuum; (see e.g [15] and [23]). The energy difference gives rise to what is known as the Casimir force. ...
Thesis
The spectral zeta function, introduced by Minakshisundaram and Pleijel encodes important spectral information for the Laplacian on Riemannian manifolds. For instance, the important notions of the determinant of the Laplacian and Casimir energy are defined via the spectral zeta function. On homogeneous manifolds, it is known that the spectral zeta function is critical with respect to conformal metric perturbations, (see e.g works of Richardson and Okikiolu), cited in the work. In this thesis, we compute a second variation formula of the spectral zeta function on closed homogeneous Riemannian manifolds under conformal metric perturbations. It is well known that the quadratic form corresponding to this second variation is given by a certain pseudodifferential operator that depends meromorphically on the argument, s. The symbol of this operator was analysed by Okikiolu. We analyse it in more detail on homogeneous spaces, in particular on the n-dimensional spheres. The case n=3 is treated in great detail. In order to describe the second variation we introduce a certain distributional integral kernel, analyse its meromorphic properties and the pole structure. The Casimir energy defined as the finite part of spectral zeta function at s=- \frac{1}{2} on the n-sphere and other points of zeta function are used to illustrate our results. The techniques employed are heat kernel asymptotics on Riemannian manifolds, the associated meromorphic continuation of the zeta function, harmonic analysis on spheres, and asymptotic analysis.
... Quantum field theory predicts that quantum fields, even in the vacuum state, can fluctuate and thus may have rich content. Theory and experiment have adequately shown that vacuum fluctuations can give rise to some unusual physical consequences, such as spontaneous emission [1], Lamb shift [2,3], Casimir effect [4,5], and Casimir-Polder effect [6]. These effects are related to the interaction of matter (such as atoms) with quantum fields. ...
Article
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We investigate the influence of atomic uniform motion on radiative energy shifts of a multilevel atom when it interacts with black-body radiation. Our analysis reveals that the atomic energy shifts depend crucially on three factors: the temperature of black-body thermal radiation, atomic velocity, and atomic polarizability. In the low-temperature limit, the presence of atomic uniform motion always enhances the effect of the thermal field on the atomic energy shifts. However, in the high-temperature limit, the atomic uniform motion enhances the effect of the thermal field for an atom polarizable perpendicular to the atomic velocity but weakens it for an atom polarizable parallel to the atomic velocity. Our work indicates that the physical properties of atom–field coupling systems can in principle be regulated and controlled by the combined action of the thermal field and the atomic uniform motion.
... The creation of real particles by the excitation of the quantum vacuum due to a moving mirror was predicted by Moore [1] and investigated in other pioneering works in the 1970s [2][3][4][5]. Nowadays, this effect is commonly called the dynamical Casimir effect (DCE), a name adopted by Yablonovitch [6] and Schwinger [7], motivated by a certain similarity with another quantum vacuum effect involving mirrors, the so-called Casimir effect [8]. On the DCE, there are some excellent reviews [9,10]. ...
Article
In the present paper, we show that a partially reflecting static mirror with time-dependent properties can produce, via dynamical Casimir effect in the context of a massless scalar field in 1+1 dimensions, a larger number of particles than a perfectly reflecting one. As particular limits, our results recover those found in the literature for a perfect static mirror imposing a generalized or a usual time-dependent Robin boundary condition.
... Experimental investigations for normal metals have been carried out by Shushkov et al. 15 in this distance scale. Also, we included the equilibrium Casimir pressure P eq obtained by setting T 1 = T 2 = 300 K in Eq. (1). In order to gain insight on the magnitude of the predicted forces, we compare our results with those expected in the case of two gold plates in thermal equilibrium at 300 K. ...
Article
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We present a comprehensive analysis of the out-of-equilibrium Casimir pressure between two high-Tc superconducting plates, each kept at a different temperature. Two interaction regimes can be distinguished. While the zero-point energy dominates in the near field, thermal effects become important at large interplate separations causing a drop in the force’s magnitude compared with the usual thermal-equilibrium case. Our detailed calculations highlight the competing role played by propagating and evanescent modes. Moreover, as one of the plates undergoes the superconducting transition, we predict an abrupt change in the force for any plate distance, which has not been previously observed in other systems. The sensitivity of the dielectric function of the high-Tc superconductors makes them ideal systems for a possible direct measurement of the out-of-equilibrium Casimir pressure.
... The electromagnetic Casimir effect [1][2][3] and the socalled critical Casimir effect [4][5][6][7][8] are two examples of long-range forces appearing when fluctuations are confined within walls [9]. The former is often considered as associated to quantum field fluctuations and the latter to classical thermal fluctuations in matter. ...
Article
We study the Casimir interaction between two dielectric spheres immersed in a salted solution at distances larger than the Debye screening length. The long distance behavior is dominated by the nonscreened interaction due to low-frequency transverse magnetic thermal fluctuations. It shows universality properties in its dependence on geometric dimensions and independence of dielectric functions of the particles, with these properties related to approximate conformal invariance. The universal interaction overtakes nonuniversal contributions at distances of the order of or larger than 0.1 μm, with a magnitude of the order of the thermal scale kBT such as to make it important for the modeling of colloids and biological interfaces.
... The generation of such force depends on the way that the fluctuations of the medium are modified by the presence of the walls [1]. The seminal work of Casimir describes the emergence of an attractive force between two conducting plates confining the quantum fluctuations of the electromagnetic field [2]. Analog effects to the Casimir force, also called fluctuation-induced forces, have been observed in numerous systems spanning from the quantum to macroscopic scale, and showing fluctuations of very diverse nature in equilibrium or in non-equilibrium conditions [3][4][5][6][7][8][9]. ...
Article
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We present experimental results on a fluctuation-induced force observed in Faraday wave- driven turbulence. As recently reported, a long-range attraction force arises between two walls that confine the wave-driven turbulent flow. In the Faraday waves system, the turbulent fluid motion is coupled with the disordered wave motion. This study describes the emergence of the fluctuation-induced force from the viewpoint of the wave dynamics. The wave amplitude is unaffected by the confinement while the wave erratic motion is. As the wall spacing decreases, the wave motion becomes less energetic and more anisotropic in the cavity formed by the walls, giving rise to a stronger attraction. These results clarify why the modelling of the attraction force in this system cannot be based on the wave amplitude but has to be built upon the wave-fluid motion coupling. When the wall spacing is comparable to the wavelength, an intermittent wave resonance is observed, and it leads to a complex short-range interaction. These results contribute to the study of aggregation processes in the presence of turbulence and its related problems such as the accumulation of plastic debris in coastal marine ecosystems or the modelling of planetary formation.
... As a result of Heisenberg's uncertainty relation, the quantum vacuum field obtains non-vanishing eigen-energy, which is associated to the zero-point fluctuation of infinite harmonic oscillators of different frequencies [1]. There are several observable effects to support the existence of vacuum fluctuation, such as the Casimir effect [2]. In fact, being a fluctuating medium, the quantum vacuum can exhibit astonishing effects under external perturbations. ...
Article
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The vacuum radiation of a massive scalar field is studied by means of a single moving mirror. The field equation with an arbitrary-shaped mirror moving in (d+1) dimensions is given perturbatively in the non-relativistic limit. Explicit results are obtained for a flat mirror moving in (1+1) dimensions and (3+1) dimensions. The vacuum radiation power and vacuum friction force on the mirror are given in (1+1) dimensions. The intrinsic mass of the field is found to suppress the vacuum radiation. In (3+1) dimensions, the modification of the frequency spectra and angular spectra of emitted particles due to the intrinsic mass are obtained. In the limit of m→0, we recover the results of the massless field.
... While there have been remarkable advances in our understanding of collisional aggregation processes between microscopic particles carried by turbulence [4,7], relatively little is known about the ability of turbulent fluctuations to mediate attractive interaction forces between larger objects or particle aggregates. This lack is striking when compared to the continued research efforts devoted to fluctuationinduced forces and Casimir-like effects across various disciplines [16][17][18][19][20][21][22]. ...
Article
Full-text available
Fluctuation-induced forces are observed in numerous physical systems spanning from quantum to macroscopic scale. However, there is as yet no experimental report of their existence in hydrodynamic turbulence. Here, we present evidence of an attraction force mediated via turbulent fluctuations by using two walls locally confining 2D turbulence. This long-range interaction is a function of the wall separation and the energy injection rate in the turbulent flow. As the wall spacing decreases, the confined flow becomes less energetic and more anisotropic in the bounded domain, producing stronger attraction. The mechanism of force generation is rooted in a nontrivial fluid-wall coupling where coherent flow structures are guided by the cavity walls. For the narrowest cavities studied, a resonance phenomenon at the flow forcing scale leads to a complex short-range interaction. The results could be relevant to problems encountered in a range of fields from industrial multiphase flows to modeling of planetary formation.
... The interaction between three objects give rise to many fascinating phenomena such as chaos of astronomical objects [1], Efimov bound states of ultracold atoms [2], and frustrated states of quantum spin systems [3]. It is intriguing to consider the potential of three-body interactions arising solely from quantum vacuum fluctuations (virtual photons) [4][5][6]. The Casimir effect due to virtual photons can provide a new approach to couple mechanical resonators [7]. ...
Preprint
The dynamics of three interacting objects has been investigated extensively in Newtonian gravitational physics (often termed the three-body problem), and is important for many quantum systems, including nuclei, Efimov states, and frustrated spin systems. However, the dynamics of three macroscopic objects interacting through quantum vacuum fluctuations (virtual photons) is still an unexplored frontier. Here, we report the first observation of Casimir interactions between three isolated macroscopic objects. We propose and demonstrate a three terminal switchable architecture exploiting opto-mechanical Casimir interactions that can lay the foundations of a Casimir transistor. Beyond the paradigm of Casimir forces between two objects in different geometries, our Casimir transistor represents an important development for control of three-body virtual photon interactions and will have potential applications in sensing and information processing with the Casimir effect.
... While past studies have mostly focused on the behavior of tracer particles passively carried by a fluctuating solvent, in recent years increasing attention has been paid to cases in which the particle and the solvent affect each other dynamically [3][4][5][6][7][8][9][10]. Particularly interesting is the case in which the medium is a fluid near a critical point, thus displaying long-range spatial correlations and long relaxation times. Objects immersed in near-critical fluids are known to experience fluctuation-induced forces [11][12][13] such as the critical Casimir force, i.e., the thermal analog of the celebrated effect in quantum electrodynamics [14]. While equilibrium field-mediated effects have long since been explored, the dynamical behavior of these systems has rarely been addressed in the literature. ...
Preprint
We study the non-equilibrium relaxational dynamics of a probe particle linearly coupled to a thermally fluctuating scalar field and subject to a harmonic potential, which provides a cartoon for an optically trapped colloid immersed in a fluid close to its bulk critical point. The average position of the particle initially displaced from the position of mechanical equilibrium is shown to feature long-time algebraic tails as the critical point of the field is approached, the universal exponents of which are determined in arbitrary spatial dimensions. As expected, this behavior cannot be captured by adiabatic approaches which assume fast field relaxation. The predictions of the analytic, perturbative approach are qualitatively confirmed by numerical simulations.
... The Casimir effect, predicted in 1948 by Hendrick Casimir, is a fundamental aspect of study in quantum electrodynamics for past seven decades [1]. This phenomenon portrays an origin of attractive force between parallelly placed conducting plates in vacuum. ...
Preprint
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In this work we have studied the plausibility of studying Casimir effect in spherically symmetric sonoluminescence bubble. The suggestions in this report are based on literature and experimental evidences which has been discussed in detail. Schwinger calculations were mainly used to predict the time of bubble expansion under irradiated shock wave. The contradictions for not relating sonoluminescence (SL) by Casimir effect is also being discussed. It is found that SL is very sensitive to measure experimentally and depends upon, the nature of gas used to make bubble, temperature of the fluid and gas, surface tension, density, pressure and radius. The final conclusion is drawn on the basis of suggesting modifications in different gases specially in different temperatures to measure SL using Casimir effects of force and energy. We expect that this study will assist in accelerating the further investigations on measuring SL bubble accurately.
... Dispersion forces (DFs) acting between uncharged bodies originate from quantum fluctuations of electromagnetic field. At distances of the order of 1 nm they are called van der Waals (vdW) forces, but at larger separations, when the retardation of the electromagnetic signal becomes important, they are called Casimir [1] (or retarded vdW) forces. Since the physical origin of the forces is the same, a general name DFs can be used. ...
Article
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The average distance h0 separating two contacting bodies is a critical parameter for many problems, for example, for control of unwanted stiction in micro/nanoelectromechanical systems. Here this problem is analysed in relation to precise determination of the dispersion forces in a difficult for measurement range of 5-30 nm. The unloaded contact between two deposited rough films characterized by a relatively large number of high asperities is considered. The equilibrium distance h0 can be found from the balance of attractive dispersion forces and repulsive forces acting in the spots of real contact. A simple columnar model associated with AFM images of rough surfaces is used to describe the balance. The numerical analysis, which treats the high asperities as elastoplastic semi-spheroids, demonstrates that the columnar model describes the contact adequately. It is shown that in contrast with the value of h0 the adhesion energy between the surfaces is nearly entirely defined by the dispersion interaction, but the effects of contact interaction and plastic deformations can be neglected. This property is proposed to use for more precise determination of the equilibrium distance.
... In the present author's first paper [1] a model to naively study chemisorption was proposed, which, with the appropriate modifications, happens that can be applied to study, in a naïve way too, the Casimir effect [2]. ...
Article
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The Casimir effect is discussed via an HMO treatment. At this schematic theoretical level, the Casimir effect might be considered as the result of the general quantum mechanical interaction behavior of two sets of particles.
... One of the remarkable consequences of quantization of fields is the existence of their vacuum fluctuations as a result of the Heisenberg uncertainty principle, which has profoundly changed our usual understanding of vacuum. The fluctuations of quantum fields in vacuum can result in various physical phenomena, such as the spontaneous emission [1], the Lamb shift [2], and the Casimir effect [3] and so on. When a thermal bath rather than a vacuum is considered, thermal fluctuations of a bath of thermal photons at nonzero temperature occur, from which the thermal corrections to vacuum-fluctuation-induced effects then arise. ...
Preprint
We study the thermal corrections induced by fluctuations of thermal quantum fields to the van der Waals and Casimir-Polder interactions between two ground-state atoms. We discover a particular region, i.e., $\sqrt[4]{\lambda^3\beta}\ll L\ll \lambda$ with $L$, $\beta$ and $\lambda$ denoting the interatomic separation, the wavelength of thermal photons and the transition wavelength of the atoms respectively, where the thermal corrections remarkably render the van der Waals force, which is usually attractive, repulsive, leading to an interesting crossover phenomenon of the interatomic interaction from attractive to repulsive as the temperature increases. We also find that the thermal corrections cause significant changes to the Casimir-Polder force when the temperature is sufficiently high, resulting in an attractive force proportional to $TL^{-3}$ in the $\lambda\ll\beta\ll L$ region, with $T$ being the temperature, and a force which can be either attractive or repulsive and even vanishing in the $\beta\ll\lambda\ll L$ region depending on the interatomic separation.
... The Casimir force depends significantly on the optical properties of objects [3,4]. The Casimir force between various materials, such as perfectly conductive plates [5], metals [2,6], ferromagnetic materials [7,8], and metamaterials [9] has been investigated. In addition, superconductors are important materials for understanding the Casimir effect more effectively because they exhibit marked changes in conductivity and cause variations in the Casimir energy. ...
Article
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The Casimir effect between type-II superconducting plates in the coexisting phase of a superconducting phase and a normal phase is investigated. The dependence of the optical conductivity of the superconducting plates on the external magnetic field is described in terms of the penetration depth of the incident electromagnetic field, and the permittivity along the imaginary axis is represented by a linear combination of the permittivities for the plasma model and Drude models. The characteristic frequency in each model is determined using the force parameters for the motion of the magnetic field vortices. The Casimir force between parallel YBCO plates in the mixed state is calculated, and the dependence on the applied magnetic field and temperature is considered.
Article
In a microscopic model of the photoelectric effect, it becomes clear that the conservation of energy is exclusively determined by Doppler shift processes, i.e. , the whole energy of the photon vanishes by means of Doppler redshifts. Accordingly, if a photon is generated, the energy is won by Doppler blueshifts. This is supposed to be valid for all processes with energy conservation. An experiment is carried out to make this Doppler energy flow visible by means of interactions with probes. The result of this experiment is that a weak force is measurable in the vicinity of processes with energy conservation. With the aid of a twisted rubber-driven low-power device ( P ˜ = 10 W \tilde{P}=10\hspace{.5em}\text{W} ), periodic accelerations and decelerations of about 10 ⁻⁶ m/s ² are measurable. In the close vicinity of the device, accelerations with values up to 10 ⁻³ m/s ² can be concluded. The consequences that result from this force are discussed.
Chapter
Dirac-Gleichung und Maxwell-Gleichungen liefern gemeinsam eine Theorie des Elektromagnetismus. Dabei müssen jedoch die Quantenpostulate berücksichtigt – d. h. das Feld quantisiert – werden; die daraus resultierende Quantenfeldtheorie des Elektromagnetismus heißt Quantenelektrodynamik, QED. Die in diesem Kapitel dargestellte kurze Einführung in Prinzipien der Quantenfeldtheorien und speziell der QED soll einen ersten Einblick in Fragestellungen und Methoden geben. Im physikalischen Bild der QED kann auch ein ruhendes Elektron als Folge der Energie-Zeit-Unschärferelation ein virtuelles Photon aussenden, sofern dieses schnell genug wieder absorbiert wird. Durch die resultierenden quantenelektrodynamischen Effekte wird der g-Faktor des Elektrons in perfekter Übereinstimmung von Theorie und Experiment im Promillebereich über den Dirac-Wert $$g=2$$ vergrößert, die Lamb-Shift lässt sich präzise berechnen und auch der Casimir-Effekt kann durch den Einfluss der virtuellen Photonen erklärt werden. Als einfachstes Verfahren zur Feldquantisierung wird zunächst die kanonische Quantisierung behandelt, die wie die erste Quantisierung wieder das Korrespondenzprinzip nutzt. Die Feldoperatoren lassen sich in der Besetzungszahldarstellung mit Erzeugungs- und Vernichtungsoperatoren darstellen. Sie erfüllen für Bose- und Fermifelder jeweils unterschiedliche Vertauschungs- bzw. Antivertauschungsrelationen. Als Folge der Auszeichnung der Zeit ist die kanonische Quantisierung jedoch nicht Lorentz-invariant. Zur Quantisierung des elektromagnetischen Strahlungsfeldes wird deshalb ein Verfahren eingesetzt, das Raum- und Zeit-Koordinaten in gleicher Weise behandelt und deshalb Lorentz-invariant ist. Durch die (zusätzliche) Lorenz-Bedingung werden die dabei auftretenden unphysikalischen Polarisationszustände mit dem Gupta-Bleuler-Verfahren eliminiert und gleichzeitig werden auf diese Weise Zustände positiver Norm erzeugt. Unter Berücksichtigung der Quantenpostulate durch die Vertauschungsrelationen werden dann Erwartungswerte physikalischer Größen wie der elektrischen und magnetischen Feldstärken berechnet, sowie von Energie und Impuls. Dabei lassen sich divergente Resultate durch Subtraktion der Vakuumenergie bzw. Einführung des normalgeordneten Produkts bei Erzeugungs- und Vernichtungsoperatoren vermeiden.
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We present a comprehensive analysis of the out-of-equilibrium Casimir pressure between two high-T c superconducting plates, each kept at a different temperature. Two interaction regimes can be distinguished. While the zero-point energy dominates in the near field, thermal effects become important at large interplate separations causing a drop in the force’s magnitude compared with the usual thermal-equilibrium case. Our detailed calculations highlight the competing role played by propagating and evanescent modes. Moreover, as one of the plates undergoes the superconducting transition, we predict a sudden discontinuity in the force for any plate distance, which has not been previously observed in other systems. The sensitivity of the dielectric function of the high-T c superconductors makes them ideal systems for a possible direct measurement of the out-of-equilibrium Casimir pressure.
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Wettability is of pivotal importance in many areas of science and technology, ranging from the extractive industry to development of advanced functional materials and biomedicine problems. An increasing interest to wetting-related phenomena stimulates impetuous growth of research activity in this field. The presented review is aimed at the cumulative coverage of issues related to wettability and its investigation. It outlines basic concepts of wetting as a physical phenomenon, methods for its characterisation (with the emphasis on sessile drop techniques), and performances of contemporary instrumentation for wettability measurements. In the first section, physics of wettability is considered. The intermolecular interactions related to wetting are classified as dependent on their nature. Thus, discussion of interactions involving polar molecules covers permanent dipole - permanent dipole interactions and freely rotating permanent dipoles. Consideration of interactions resulting from the polarization of molecules includes interactions between ions and uncharged molecules, Debye interactions, and London dispersion interactions. Hydrogen bonds are discussed separately. The second section deals with the issues related to surface tension and its effect on shaping the surface of a liquid brought in contact with a solid body. The relationship between the surface tension and the contact angle as well as equations that quantify this relationship are discussed. The Young–Laplace equation governing the shape of the drop resting on the surface is analysed. The third section is devoted to the experimental characterization of surface wettability and the underlying theoretical analysis. Particular attention is paid to the method known as the Axisymmetric Drop Shape Analysis (ADSA). Principles of automated determination of relevant physical values from experimental data are briefly discussed. Basics of numerical techniques intended for analysing the digitized image of the drop and extracting information on surface tension and contact angle are outlined. In the fourth section, an overview of commercially available instrumentation for studying wettability and the contact angle measurements is presented. The prototype contact angle analyser designed and manufactured at the ISP NASU is introduced.
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We consider systems consisting of multiple plates, where the dynamics of the positions and the temperatures of the plates is determined by the non-equilibrium Casimir forces and the heat transfer between them, respectively. We numerically and theoretically show that in these systems, the temporal dependence of the position of a plate can be made to conform to a prescribed trajectory through a feedback control scheme. We illustrate by simulations on both two-plate and three-plate systems. Our results indicate the significant potentials of using feedback control schemes to engineer the behaviors of mechanical systems governed by quantum and thermal fluctuations.
Article
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One of the fundamental predictions of quantum mechanics is the occurrence of random fluctuations in a vacuum caused by the zero-point energy. Remarkably, quantum electromagnetic fluctuations can induce a measurable force between neutral objects, known as the Casimir effect¹, and it has been studied both theoretically2,3 and experimentally4,5,6,7,8,9. The Casimir effect can dominate the interaction between microstructures at small separations and is essential for micro- and nanotechnologies10,11. It has been utilized to realize nonlinear oscillation¹², quantum trapping¹³, phonon transfer14,15 and dissipation dilution¹⁶. However, a non-reciprocal device based on quantum vacuum fluctuations remains an unexplored frontier. Here we report quantum-vacuum-mediated non-reciprocal energy transfer between two micromechanical oscillators. We parametrically modulate the Casimir interaction to realize a strong coupling between the two oscillators with different resonant frequencies. We engineer the system’s spectrum such that it possesses an exceptional point17,18,19,20 in the parameter space and explore the asymmetric topological structure in its vicinity. By dynamically changing the parameters near the exceptional point and utilizing the non-adiabaticity of the process, we achieve non-reciprocal energy transfer between the two oscillators with high contrast. Our work demonstrates a scheme that employs quantum vacuum fluctuations to regulate energy transfer at the nanoscale and may enable functional Casimir devices in the future.
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The time-varying resonance interaction between two identical two-level atoms near a conducting plane is theoretically studied by quantum electrodynamics. Initially one of the atoms is excited, the other is in ground state and the light field is in vacuum state. Dynamical analysis method is employed to study the temporal evolution of the resonance interaction, and the full variation process of the resonance interaction from its occurrence to decay is represented. The analytical and numerical results of our calculation show that the characteristics of resonance interaction, including its occurrence, temporal dependence, energy transferring, energy level shifts and damping rates, will be sufficiently affected when the atoms are very close to the conducting plane. This effect has potential in developing quantum manipulation of inter-atom interaction.
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