Casimir energy of a dilute dispersive dielectric ball: realistic microscopic model
Valery N. Marachevsky
备注：
Latex 2e, 4 pages, 1 figure, to appear in Int. J. Mod. Phys. A [a talk given at the Fifth Leipzig Workshop on Quantum Field Theory under the Influence of External Conditions, September 10-14, 2001]
The Casimir energy of a dilute homogeneous nonmagnetic dielectric ball at zero temperature is derived analytically within a microscopic realistic model of dielectrics for an arbitrary physically possible frequency dispersion of dielectric permittivity. Divergences are absent in calculations, a minimum interatomic distance is a physical cut-off. Casimir surface force is proved to be attractive. A physical definition of the Casimir energy is discussed.
The Casimir energy, free energy and Casimir force are evaluated, at arbitrary finite temperature, for a dilute dielectric ball with uniform velocity of light inside the ball and in the surrounding medium. In particular, we investigate the classical limit at high temperature. The Casimir force found is repulsive, as in previous calculations.
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.
The finite temperature Casimir free energy is calculated for a dielectric ball of radius $a$ embedded in an infinite medium. The condition $εμ=1$ is assumed for the inside/outside regions. Both the Green function method and the mode summation method are considered, and found to be equivalent. For a dilute medium we find, assuming a simple "square" dispersion relation with an abrupt cutoff at imaginary frequency $\hat ω= ω_0$, the high temperature Casimir free energy to be negative and proportional to $x_0 \equiv ω_0 a$. Also, a physically more realistic dispersion relation involving spatial di…
The microscopic theory of the Casimir effect in the dielectric is studied in the framework when absorption is realized via a reservoir modeled by a set of oscillators with continuously distributed frquencies with the aim to see if the effects depend on the form of interaction with the reservoir. A simple case of the one-dimensional dielectric between two metallic plates is considered. Two possible models are investigated, the direct interaction of the electromagnetic field with the reservoir and indirect interaction via an intermediate oscillator imitating the atom. It is found that with the s…
Using perturbation theory the first order dispersive correction to the Casimir energy between two plates separated by a dielectric material is calculated. It falls off with the plate separation as 1/L^6. The result is derived both from evaluation of the zero-point energy and within the Lifshitz formulation. It is pointed out that a possible surface term can be more important, varying like 1/L^5.
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 paper we sum over the spherical modes appearing in the expression for the Casimir energy of a conducting sphere and of a dielectric ball (assuming the same speed of light inside and outside), before doing the frequency integration. We derive closed integral expressions that allow the calculations to be done to all orders, without the use of regularization procedures. The technique of mode summation using a contour integral is critically examined.
We consider a dielectric medium with an ultraviolet behavior as it follows from the Drude model. Compared with dilute models, this has the advantage that, for large frequencies, two different media behave the same way. As a result one expects the Casimir energy to contain less divergencies than for the dilute media approximation. We show that the Casimir energy of a spherical dielectric ball contains just one divergent term, a volume one, which can be renormalized by introducing a contact term analogous to the volume energy counterterm needed in bag models. PACS: 12.20.Ds, 03.70.+k, 77.22.Ch
We consider the vacuum energy of the electromagnetic field in the background of spherically symmetric dielectrics, subject to a cut-off frequency in the dispersion relations. The effect of this frequency dependent boundary condition between media is described in terms of the {\it incomplete} $ζ$-functions of the problem. The use of the Debye asymptotic expansion for Bessel functions allows to determine the dominant (volume, area, ...) terms in the Casimir energy. The application of these expressions to the case of a gas bubble immersed in water is discussed, and results consistent with Schwing…
Assuming the conventional Casimir setting with two thick parallel perfectly conducting plates of large extent with a homogeneous and isotropic medium between them, we discuss the physical meaning of the electromagnetic field energy $W_{\rm disp}$ when the intervening medium is weakly dispersive but nondissipative. The presence of dispersion means that the energy density contains terms of the form $d[ωε(ω)] /dω$ and $d[ωμ(ω)] /dω$. We find that, as $W_{\rm disp}$ refers thermodynamically to a non-closed physical system, it is {\it not} to be identified with the internal thermodynamic energy $U$…
In this paper, we theoretically propose and investigate a feasible experimental scheme for realizing the dynamical Casimir effect (DCE) of phonons in an optomechanical setup formed by a ground-state precooled mechanical oscillator (MO) inside a Fabry-P{é}rot cavity, which is driven by an amplitude-modulated classical laser field in the dispersive (far-detuned) regime. The time modulation of the driving field leads to the parametric amplification of the mechanical vacuum fluctuations of the MO, which results in the generation of Casimir phonons over time scales longer than the cavity lifetime. …
We develop a unified scattering approach to dynamical Casimir problems which can be applied to both accelerating boundaries, as well as dispersive objects in relative motion. A general (trace) formula is derived for the radiation from accelerating boundaries. Applications are provided for objects with different shapes in various dimensions, and undergoing rotational or linear motion. Within this framework, photon generation is discussed in the context of a modulated optical mirror. For dispersive objects, we find general results solely in terms of the scattering matrix. Specifically, we discus…
We calculate a force due to zero-temperature quantum fluctuations on a stationary object in a moving superfluid flow. We model the object by a localized potential varying only in the flow direction and model the flow by a three-dimensional weakly interacting Bose-Einstein condensate at zero temperature. We show that this force exists for any arbitrarily small flow velocity and discuss the implications for the stability of superfluid flow.
We investigate the Casimir-Polder interaction between two atoms one of which is excited. We show that the perturbation theory results in divergence of integrals for the interaction between an excited atom and a media of dilute gas. We considered the interaction between two atoms embedded in a dielectric medium. The non-perturbative method used in this paper shows that the interaction between the atoms is suppressed due to absorption of photons by the medium. Now the integrals are divergent no more. Interaction between two media of dilute gases is considered for the case of high temperatures.
Casimir-Polder interactions are considered in an inhomogeneous, dispersive background. We consider both the interaction between a polarizable atom and a perfectly conducting wall, and between such an atom and a plane interface between two different inhomogeneous dielectric media. Renormalization is achieved by subtracting the interaction with the local inhomogeneous medium by itself. The results are expressed in general form, and generalize the Dzyaloshinskii-Lifshitz-Pitaevskii interaction between a spatially homogeneous dielectric interface and a polarizable atom.
We use the mode summation method together with zeta-function regularization to compute the Casimir energy of a dilute dielectric cylinder. The method is very transparent, and sheds light on the reason the resulting energy vanishes.
The Casimir energy of a dilute dielectric cylinder, with the same light-velocity as in its surrounding medium, is evaluated exactly to first order in $ξ^2$ and numerically to higher orders in $ξ^2$. The first part is carried out using addition formulas for Bessel functions, and no Debye expansions are required.
We show that the Casimir, or zero-point, energy of a dilute dielectric ball, or of a spherical bubble in a dielectric medium, coincides with the sum of the van der Waals energies between the molecules that make up the medium. That energy, which is finite and repulsive when self-energy and surface effects are removed, may be unambiguously calculated by either dimensional continuation or by zeta function regularization. This physical interpretation of the Casimir energy seems unambiguous evidence that the bulk self-energy cannot be relevant to sonoluminescence.
In this Letter we study the entanglement of a quantum radiation field interacting with a dielectric medium. In particular, we describe the quantum mixed state of a field interacting with a dielectric through plasma and Drude models and show that these generate very different entanglement behavior, as manifested in the entanglement entropy of the field. We also present a formula for a "Casimir" entanglement entropy, i.e., the distance dependence of the field entropy. Finally, we study a toy model of the interaction between two plates. In this model, the field entanglement entropy is divergent; …
The retarded dispersion interaction (Casimir interaction) between two dilute dielectric media at high temperatures is considered. The excited atoms are taken into account. It is shown that the perturbation technique can not be applied to this problem due to the divergence of integrals. A non-perturbative approach based on kinetic Green functions is implemented. We consider interaction between two atoms (one of them is excited0 embedded in an absorbing dielectric medium. We take into account possible absorption of photons in the medium, which solves the problem of divergence. The force between …
An expression for the Casimir stress on arbitrary dispersive and lossy linear magnetodielectric matter at finite temperature, including left-handed material, is derived and applied to spherical systems. To cast the relevant part of the scattering Green tensor for a general magnetodielectric sphere in a convenient form, classical Mie scattering is reformulated.
A general, exact formula is derived for the expectation value of the electromagnetic energy density of an inhomogeneous absorbing and dispersive dielectric medium in thermal equilibrium, assuming that the medium is well approximated as a continuum. From this formula we obtain the formal expression for the Casimir force density. Unlike most previous approaches to Casimir effects in which absorption is either ignored or admitted implicitly through the required analytic properties of the permittivity, we include dissipation explicitly via the coupling of each dipole oscillator of the medium to a …
The Casimir effect in a dispersive and absorbing multilayered system is considered adopting the (net) vacuum-field pressure point of view to the Casimir force. Using the properties of the macroscopic field operators appropriate for absorbing systems and a convenient compact form of the Green function for a multilayer, a straightforward and transparent derivation of the Casimir force in a lossless layer of an otherwise absorbing multilayer is presented. The resulting expression in terms of the reflection coefficients of the surrounding stacks of layers is of the same form as that obtained by Zh…
We propose a method to probe dispersive atom-surface interactions by measuring via two-photon Bragg spectroscopy the dynamic structure factor of a Bose-Einstein condensate above corrugated surfaces. This method takes advantage of the condensate coherence to reveal the spatial Fourier components of the lateral Casimir-Polder interaction energy.
We investigate the dispersive Casimir-Polder interaction between a Rubidium atom and a suspended graphene sheet subjected to an external magnetic field B. We demonstrate that this concrete physical system allows for an unprecedented control of dispersive interactions at micro and nanoscales. Indeed, we show that the application of an external magnetic field can induce a 80% reduction of the Casimir-Polder energy relative to its value without the field. We also show that sharp discontinuities emerge in the Casimir-Polder interaction energy for certain values of the applied magnetic field at low…
It has always been conventionally understood that, in the dilute limit, the Casimir energy of interaction between bodies or the Casimir self-energy of a dielectric body could be identified with the sum of the van der Waals or Casimir-Polder energies of the constituents of the bodies. Recently, this proposition for self-energies has been challenged by Avni and Leonhardt [Ann.\ Phys.\ {\bf 395}, 326 (2018)], who find that the energy or self-stress of a homogeneous dielectric ball with permittivity $\varepsilon$ begins with a term of order $\varepsilon-1$. Here we demonstrate that this cannot be …
Finite temperature Casimir theory of the Dirichlet scalar field is developed, assuming that there is a conventional Casimir setup in physical space with two infinitely large plates separated by a gap R and in addition an arbitrary number q of extra compacified dimensions. As a generalization of earlier theory, we assume in the first part of the paper that there is a scalar 'refractive index' N filling the whole of the physical space region. After presenting general expressions for free energy and Casimir forces we focus on the low temperature case, as this is of main physical interest both for…
We develop the scattering approach to calculate the exact dispersive Casimir-Polder potential between a ground-state atom and a rectangular grating. Our formalism allows, in principle, for arbitrary values of the grating amplitude and period, and of the atom-grating distance. We compute numerically the potential for a Rb atom on top of a Si grating and compare the results with the potential for a flat surface taken at the local atom-surface distance (proximity force approximation). Except for very short separation distances, the potential is nearly sinusoidal along the direction transverse to …
Despite suggestions to the contrary, we show in this paper that the usual dispersive form of the electromagnetic energy must be used to derive the Lifshitz force between parallel dielectric media. This conclusion follows from the general form of the quantum vacuum energy, which is the basis of the multiple-scattering formalism. As an illustration, we explicitly derive the Lifshitz formula for the interaction between parallel dielectric semispaces, including dispersion, starting from the expression for the total energy of the system. The issues of constancy of the energy between parallel plates…
We present an explicit analytic calculation of the energy-level shift of an atom in front of a non-dispersive and non-dissipative dielectric slab. We work with the fully quantized electromagnetic field, taking retardation into account. We give the shift as a two-dimensional integral and use asymptotic analysis to find expressions for it in various retarded and non-retarded limiting cases. The results can be used to estimate the energy shift of an atom close to layered microstructures.
The Casimir-Polder potential for interaction between an excited atom and a ground-state one in the retarded case obtained with the help of perturbation technique drops as R^-2 with the distance between the atoms [E.A. Power, T.Thirunamachandran, Phys. Rev. A, 47, 2539 (1993)]. It results in diverdent integrals for interaction between an excited atom and a dilute gas medium. We investigate interaction between two atoms embedded in a dielectric medium with the help of non-perturbative approach. We take into account absorption of photons in the medium. This approach solves the problem of divergen…
The Casimir energy for a compact dielectric sphere is considered in a novel way, using the quantum statistical method introduced by H\oye - Stell and others. Dilute media are assumed. It turns out that this method is a very powerful one: we are actually able to derive an expression for the Casimir energy that contains also the negative part resulting from the attractive van der Waals forces between the molecules. It is precisely this part of the Casimir energy that has turned out to be so difficult to extract from the formalism when using the conventional field theoretical methods for a contin…
The dielectric sphere has been an important test case for understanding and calculating the vacuum force of a dielectric body onto itself. Here we develop a method for computing this force in homogeneous spheres of arbitrary dielectric properties embedded in arbitrary homogeneous backgrounds, assuming only that both materials are isotropic and dispersionless. Our results agree with known special cases; most notably we reproduce the prediction of Boyer and Schwinger et al. of a repulsive Casimir force of a perfectly reflecting shell. Our results disagree with the literature in the dilute limit.…
The canonical quantization of macroscopic electromagnetism was recently presented in New J. Phys. 12 (2010) 123008. This theory is here used to derive the Casimir effect, by considering the special case of thermal and zero-point fields. The stress-energy-momentum tensor of the canonical theory follows from Noether's theorem, and its electromagnetic part in thermal equilibrium gives the Casimir energy density and stress tensor. The results hold for arbitrary inhomogeneous magnetodielectrics and are obtained from a rigorous quantization of electromagnetism in dispersive, dissipative media. Conti…
Quantum electrodynamics near a boundary is investigated by considering the inertial mass shift of an electron near a dielectric or conducting surface. We show that in all tractable cases the shift can be written in terms of integrals over the TE and TM reflection coefficients associated with the surface, in analogy to the Lifshitz formula for the Casimir effect. We discuss the applications and potential limitations of this formula, and provide exact results for several models of the surface.
We present a study of atom-wall interactions in non-relativistic quantum electrodynamics by functional integral methods. The Feynman-Kac path integral representation is generalized to the case when the particle interacts with a radiation field, providing an additional effective potential that contains all the interactions induced by the field. We show how one can retrieve the standard van der Waals, Casimir-Polder and classical Lifshiftz forces in this formalism for an atom in its ground state. Moreover, when electrostatic interactions are screened in the medium, we find low temperature correc…
The finite temperature Casimir free energy, entropy, and internal energy are considered anew for a conventional parallel-plate configuration, in the light of current discussions in the literature. In the case of an "ideal" metal, characterized by a refractive index equal to infinity for all frequencies, we recover, via a somewhat unconventional method, conventional results for the temperature dependence, meaning that the zero-frequency transverse electric mode contributes the same as the transverse magnetic mode. For a real metal, however, approximately obeying the Drude dispersive model at lo…
We find the action that describes the electromagnetic field in a spatially dispersive, homogeneous medium. This theory is quantized and the Hamiltonian is diagonalized in terms of a continuum of normal modes. It is found that the introduction of nonlocal response in the medium automatically regulates some previously divergent results, and we calculate a finite value for the intensity of the electromagnetic field at a fixed frequency within a homogeneous medium. To conclude we discuss the potential importance of spatial dispersion in taming the divergences that arise in calculations of Casimir-…
We compute the expectations of the squares of the electric and magnetic fields in the vacuum region outside a half-space filled with a uniform dispersive dielectric. We find a positive energy density of the electromagnetic field which diverges at the interface despite the inclusion of dispersion in the calculation. We also investigate the mean squared fields and the energy density in the vacuum region between two parallel half-spaces. Of particular interest is the sign of the energy density. We find that the energy density is described by two terms: a negative position independent (Casimir) te…
The Casimir interaction between two thick parallel plates, one made of metal and the other of dielectric, is investigated at nonzero temperature. It is shown that in some temperature intervals the Casimir pressure and the free energy of a fluctuating field are the nonmonotonous functions of temperature and the respective Casimir entropy can be negative. The physical interpretation for these conclusions is given. At the same time we demonstrate that the entropy vanishes when the temperature goes to zero, i.e., in the Casimir interaction between metal and dielectric the Nernst heat theorem is sa…
We study the consequences of dissipation in homogeneous media when the system is subject to a sudden change, thereby producing pairs of correlated quasi-particles with opposite momenta. We compute both the modifications of the spectrum, and those of the correlations. In particular, we compute the final coherence level, and identify the regimes where the state is non-separable. To isolate the role of dissipation, we first consider dispersive media and study the competition between the intensity of the jump which induces some coherence, and the temperature which reduces it. The contributions of …
We calculate the Casimir energy at spherical cavities within a host made up of an arbitrary material described by a possibly dispersive and lossy dielectric response. To that end, we add to the coherent optical response a contribution that takes account of the incoherent radiation emitted by the host in order to guarantee the detailed balance required to keep the system at thermodynamic equilibrium in the presence of dissipation. The resulting boundary conditions allow a conventional quantum mechanical treatment of the radiation within the cavity from which we obtain the contribution of the ca…
From the beginning of the subject, calculations of quantum vacuum energies or Casimir energies have been plagued with two types of divergences: The total energy, which may be thought of as some sort of regularization of the zero-point energy, $\sum\frac12\hbarω$, seems manifestly divergent. And local energy densities, obtained from the vacuum expectation value of the energy-momentum tensor, $\langle T_{00}\rangle$, typically diverge near boundaries. The energy of interaction between distinct rigid bodies of whatever type is finite, corresponding to observable forces and torques between the bod…
The Casimir and van der Waals interaction between two dissimilar thick dielectric plates is reconsidered on the basis of thermal quantum field theory in Matsubara formulation. We briefly review two main derivations of the Lifshitz formula in the framework of thermal quantum field theory without use of the fluctuation-dissipation theorem. A set of special conditions is formulated under which these derivations remain valid in the presence of dissipation. The low-temperature behavior of the Casimir and van der Waals interactions between dissimilar dielectrics is found analytically from the Lifshi…
We study the interaction between a neutral atom or molecule and a conductor-patched dielectric surface. We model this system by a perfectly reflecting disc lying atop of a non-dispersive dielectric half-space, both interacting with the neutral atom or molecule. We assume the interaction to be non-retarded and at zero temperature. We find an exact solution to this problem. In addition we generate a number of other useful results. For the case of no substrate we obtain the exact formula for the van der Waals interaction energy of an atom near a perfectly conducting disc. We show that the Casimir…
Electromagnetic Casimir stresses are of relevance to many technologies based on mesoscopic devices such as MEMS embedded in dielectric media, Casimir induced friction in nano-machinery, micro-fluidics and molecular electronics. Computation of such stresses based on cavity QED generally require numerical analysis based on a regularization process. A new scheme is described that has the potential for wide applicability to systems involving realistic inhomogeneous media. From a knowledge of the spectrum of the stationary modes of the electromagnetic field the scheme is illustrated by estimating n…
The derivation of Casimir forces between dielectrics can be simplified by ignoring absorption, calculating energy changes due to displacements of the dielectrics, and only then admitting absorption by allowing permittivities to be complex. As a first step towards a better understanding of this situation we consider in this paper the model of a dielectric as a collection of oscillators, each of which is coupled to a reservoir giving rise to damping and Langevin forces on the oscillators and a noise polarization acting as a source of a fluctuating electromagnetic (EM) field in the dielectric. Th…
The stress tensor for the quantized electromagnetic field is calculated in the region between a pair of dispersive, dielectric half-spaces. This generalizes the stress tensor for the Casimir energy to the case where the boundaries have finite reflectivity. We also include the effects of finite temperature. This allows us to discuss the circumstances under which the weak energy condition and the null energy condition can be violated in the presence of finite reflectivity and finite temperature.
Quantum Optomechanics with Imperfect Mirrors and Casimir-Polder Effect of Atoms with Different Dynamic Polarizabilities: A Unified Treatment via a Microscopic Model
University of Maryland, College Park（马里兰大学帕克分校）；Joint Quantum Institute and Maryland Center for Fundamental Physics, University of Maryland, College Park（马里兰大学联合量子研究所与马里兰基础物理中心）
We determine the energy-level shift experienced by a neutral atom due the quantum electromagnetic interaction with a layered dielectric body. We use the technique of normal-mode expansion to quantize the electromagnetic field in the presence of a layered, non-dispersive and non-absorptive dielectric. We explicitly calculate the equal-time commutation relations between the electric field and vector potential operators. We show that the commutator can be expressed in terms of a generalized transverse delta-function and that this is a consequence of using the generalized Coulomb gauge to quantize…
