Non-retarded Regime Research Articles

Requirements for probing chiral Casimir-Polder forces in a molecular Talbot-Lau interferometer

We theoretically investigate the influence of chiral Casimir-Polder (CP) forces in Talbot-Lau interferometry, based on three nanomechanical gratings. We study scenarios where the second grating is either directly written into a chiral material or where the nanomask is coated with chiral substances. We show requirements for probing enantiospecific effects in matter-wave interferometry in the transmission signal and the interference fringe visibility, which depend on the de Broglie wavelength and the molecular chirality. The proposed setup is particularly sensitive to CP forces in the nonretarded regime where chiral effects can be comparable in magnitude to their electric and magnetic counterparts. While the first and third gratings do not change the phase of the matter wave, applying a coating of chiral substances to them enhances the instrument's chiral selectivity. Published by the American Physical Society 2024

Casimir-Polder interactions of S -state Rydberg atoms with graphene

We investigate the thermal Casimir-Polder (CP) potential of $^{87}\mathrm{Rb}$ atoms in Rydberg $nS$-states near single- and double-layer graphene, and briefly look into the lifetimes near graphene-hexagonal boron nitride (hBN) multilayered structures. The dependence of the CP potential on parameters such as atom-surface distance, temperature, principal quantum number $n$, and graphene Fermi energy are explored. Through large-scale numerical simulations, we show that, in the nonretarded regime, the CP potential is dominated by the nonresonant and evanescent-wave terms which are monotonic, and that, in the retarded regime, the CP potential exhibits spatial oscillations. We identify that the most important contributions to the resonant component of the CP potential come from the $nS\text{\ensuremath{-}}nP$ and $nS\text{\ensuremath{-}}(n\ensuremath{-}1)P$ transitions. Scaling of the CP potential as a function of the principal quantum number and temperature is obtained. A heterostructure comprising hexagonal boron nitride layers sandwiched between two graphene layers is also studied. When the boron nitride layer is sufficiently thin, the CP potential can be weakened by changing the Fermi energy of the top graphene layer. Our study provides insights for understanding and controlling CP potentials experienced by Rydberg atoms near single- and multilayer graphene-based van der Waals heterostructures.

Edge plasmon-polaritons on isotropic semi-infinite conducting sheets

From a three-dimensional boundary value problem for the time harmonic classical Maxwell equations, we derive the dispersion relation for a surface wave, the edge plasmon-polariton (EP), which is localized near and propagates along the straight edge of a planar, semi-infinite sheet with a spatially homogeneous, scalar conductivity. The sheet lies in a uniform and isotropic medium and serves as a model for some two-dimensional (2D) conducting materials such as the doped monolayer graphene. We formulate a homogeneous system of integral equations for the electric field tangential to the plane of the sheet. By the Wiener–Hopf method, we convert this system to coupled functional equations on the real line for the Fourier transforms of the fields in the surface coordinate normal to the edge and solve these equations exactly. The derived EP dispersion relation smoothly connects two regimes: a low-frequency regime, where the EP wave number, q, can be comparable to the propagation constant, k0, of the ambient medium, and the nonretarded frequency regime in which |q| ≫ |k0|. Our analysis indicates two types of 2D surface plasmon-polaritons on the sheet away from the edge. We extend the formalism to the geometry of two coplanar sheets.

Nonretarded edge plasmon-polaritons in anisotropic two-dimensional materials

By an integral equation approach to the time-harmonic classical Maxwell equations, we describe the dispersion in the nonretarded frequency regime of the edge plasmon-polariton (EPP) on a semi-infinite flat sheet. The sheet has an arbitrary, physically admissible, tensor valued and spatially homogeneous conductivity, and serves as a model for a family of two-dimensional conducting materials. We formulate a system of integral equations for the electric field tangential to the sheet in a homogeneous and isotropic ambient medium. We show how this system is simplified via a length scale separation. This view entails the quasi-electrostatic approximation, by which the tangential electric field is replaced by the gradient of a scalar potential, . By the Wiener–Hopf method, we solve an integral equation for in some generality. The EPP dispersion relation comes from the elimination of a divergent limiting Fourier integral for at the edge. We connect the existence, or lack thereof, of the EPP dispersion relation to the index for Wiener–Hopf integral equations, an integer of topological character. We indicate that the values of this index may express an asymmetry due to the material anisotropy in the number of wave modes propagating on the sheet away from the edge with respect to the EPP direction of propagation. We discuss extensions such as the setting of two semi-infinite, coplanar sheets. Our theory forms a generalization of the treatment by Volkov and Mikhailov (1988 Sov. Phys.—JETP 67 1639).

Nonlinear optical response in graphene nanoribbons: The critical role of electron scattering

Nonlinear nanophotonics has many potential applications, such as in mode locking, frequency-comb generation, and all-optical switching. The development of materials with large nonlinear susceptibility is key to realizing nonlinear nanophotonics. Nanostructured graphene systems, such as graphene nanoribbons and nanoislands, have been predicted to have a strong plasmon-enhanced nonlinear optical behavior in the nonretarded regime. Plasmons concentrate the light field down to subwavelength scales and can enhance the nonlinear optical effects; however, plasmon resonances are narrowband and sensitive to the nanostructure geometry. Here we show that graphene nanoribbons, particularly armchair graphene nanoribbons, have a remarkably strong nonlinear optical response in the long-wavelength regime and over a broad frequency range, from terahertz to the near infrared. We use a quantum-mechanical master equation with a detailed treatment of scattering and show that, in the retarded regime, electron scattering has a critical effect on the optical nonlinearity of graphene nanoribbons, which cannot be captured via the commonly used relaxation-time approximation. At terahertz frequencies, where intraband optical transitions dominate, the strong nonlinearity (in particular, third-order Kerr nonlinearity) stems from the jagged shape of the electron energy distribution, caused by the interband electron scattering mechanisms along with the intraband inelastic scattering mechanisms. We show that the relaxation-time approximation fails to capture this quantum-mechanical phenomenon and results in a significant underestimation of the intraband nonlinearity. At the midinfrared to near infrared frequencies, where interband optical transitions dominate, the Kerr nonlinearity is significantly overestimated within the relaxation-time approximation. These findings unveil the critical effect of electron scattering on the optical nonlinearity of nanostructured graphene, and also underscore the capability of this class of materials for nonlinear nanophotonic applications.

Velocity-dependent dipole forces on an excited atom

We present a time-dependent calculation of the velocity-dependent forces which act on an excited atomic dipole in relative motion with respect to ground state atoms of a different kind. Both, its interaction with a single atom and with a dilute atomic plate are evaluated. In either case, the total force consists of a conservative van der Waals component and a non-conservative Rontgen component. On physical grounds, the former corresponds to the velocity-dependent recoil experienced by the excited atom in the processes of absorption and emission of the photons that it exchanges with the ground state atoms on a periodic basis. The latter corresponds to the time-variation of the Rontgen momentum, which is also mediated by the periodic exchange of quasi-resonant photons. We find that, at leading order, all these interactions are linear in the velocity. In the non-retarded regime the van der Waals force dominates, being antiparallel to the velocity. On the contrary, in the retarded regime the velocity-dependent forces oscillate in space, van der Waals and Rontgen forces are of the same order in the atom-atom interaction, and the Rontgen component dominates in the atom-surface interaction.

Influence of a surface in the nonretarded interaction between two atoms

In this work we obtain analytical expressions for the non-additivity effects in the dispersive interaction between two atoms and perfectly conducting surface of arbitrary shape in the non-retarded regime. We show that this three bodies quantum-mechanical problem can be solved by mapping it into a two-bodies electrostatic one. We apply the general formulas developed in this paper in several examples. Firstly we re-derive the London interaction as a particular case of our formalism. Then we treat two atoms in the presence of a plane, re-obtained the result displayed in the literature. After we add some new examples. A particularly interesting one is two atoms inside a plate capacitor, a situation where non-additivity is very manifest since the plates lead to the exponentially suppression of the interaction of the atoms, provided the atoms are separated by a distance of the order of the distance between the plates or greater. Our results holds even in the presence of other atoms inside the plate capacitor. As a last example we deal with two atoms in the presence of a sphere, both grounded and isolated. We show that for realistic experimental parameters the non-additivity may be relevant for the force in each atom.

Quantum electrodynamics near anisotropic polarizable materials: Casimir-Polder shifts near multilayers of graphene

In a recent paper we have formulated a theory of non-relativistic quantum electrodynamics in the presence of an inhomogeneous Huttner-Barnett dielectric. Here we generalize the formalism to anisotropic materials and show how it may be modified to include conducting surfaces. We start with the derivation of the photon propagator for a slab of material and use it to work out the energy-level shift near a medium whose conductivity in the direction parallel to the surface far exceeds that in the direction perpendicular to the surface. We investigate the influence of the anisotropy of the material's electromagnetic response on the Casimir-Polder shifts, both analytically and numerically, and show that it may have a significant impact on the atom-surface interaction, especially in the non-retarded regime, i.e. for small atom-surface separations. Our results for the energy shift may be used to estimate the Casimir-Polder force acting on quantum objects close to multilayers of graphene or graphite. They are particularly important for the case of trapped cold molecules whose dispersive interactions with surfaces often fall within the non-retarded regime where the anisotropy of the material strongly influences the Casimir-Polder force. We also give a formula for the change in the spontaneous decay rate of an excited atom or molecule near an anisotropically conducting surface.

Casimir-Polder interaction of fullerene molecules with surfaces

We calculate the thermal Casimir-Polder potential of ${\mathrm{C}}_{60}$ and ${\mathrm{C}}_{70}$ fullerene molecules near gold and silicon nitride surfaces, motivated by their relevance for molecular matter-wave interference experiments. We obtain the coefficients governing the asymptotic power laws of the interaction in the thermal, retarded, and nonretarded distance regimes and evaluate the full potential numerically. The interaction is found to be dominated by electronic transitions and hence independent of the internal temperature of the molecules. The contributions from phonon transitions, which are affected by the molecular temperature, give rise to only a small correction. Moreover, we find that the sizable molecular line widths of thermal fullerenes may modify the nonretarded interaction, depending on the model used. Detailed measurements of the nonretarded potential of fullerene thus allow one to distinguish between different theories of incorporating damping.

Tangential and Quantum Frictional Forces on a Neutral Particle Moving Near a Surface

We calculate tangential forces applied to a ground state atom (nanoparticle) moving with nonrelativistic velocity parallel to the surface of Drude -modelled or Lorentz -modelled half -space using the formalism of fluctuation electrodynamics.The obtained formulae assume the nonretarded regime and arbitrary temperatures of the particle and medium. The quantum frictional force is realized at zero temperature of the particle and medium and can reach about 1/3 of the static value of the Casimir -Polder force in the case of ground state atom. In the case of nanoparticles and nonzero temperature, the tangential forces appears to be both frictional and accelerating. Numerical examples are given for atoms of Na above the surface of Au and MgO nanoparticles above the surface of SiC.

Repulsive Casimir–Polder forces from cosmic strings

We investigate the Casimir-Polder force acting on a polarizable microparticle in the geometry of a straight cosmic string. In order to develop this analysis we evaluate the electromagnetic field Green tensor on the imaginary frequency axis. The expression for the Casimir-Polder force is derived in the general case of anisotropic polarizability. In dependence of the eigenvalues for the polarizability tensor and of the orientation of its principal axes, the Casimir-Polder force can be either repulsive or attractive. Moreover, there are situations where the force changes the sign with separation. We show that for an isotropic polarizability tensor the force is always repulsive. At large separations between the microparticle and the string, the force varies inversely as the fifth power of the distance. In the non-retarded regime, corresponding to separations smaller than the relevant transition wavelengths, the force decays as the inverse fourth power of the distance. In the case of anisotropic polarizability, the dependence of the Casimir-Polder potential on the orientation of the polarizability tensor principal axes also leads to the moment of force acting on the particle.

Erratum: Thermal Casimir-Polder shifts in Rydberg atoms near metallic surfaces [Phys. Rev. A82, 010901(R) (2010)

The present thesis focuses on several topics within three separate but related branches of the overall field of dispersion forces. The three branches are: temperature corrections to the Casimir force between real materials (Part 1), explicit calculation of Casimir energy in wedge geometries (Part 2), and Casimir-Polder forces on particles out of thermal equilibrium (Part 3). Part 1 deals primarily with analysis of a previously purported thermodynamic inconsistency in the Casimir-Lifshitz free energy of the interaction of two plane mirrors – violation of the third law of thermodynamics – when the latter’s dielectric response is described with dissipative models. It is shown analytically and numerically that the Casimir entropy of the interaction between two metallic mirrors described by the Drude model does tend to zero at zero temperature, provided electronic relaxation does not vanish. The leading order terms at low temperature are found. A similar calculation is carried out for the interaction of semiconductors with small but non-zero DC conductivity. In a generalisation, it is shown that a violation of the third law can only occur for permittivities whose low-frequency behaviour is temperature dependent near zero temperature. A calculation using path integral methods shows that the low temperature behaviour of the interaction of fluctuating Foucault currents in two mirrors of Drude metal is identical to that of the full Casimir-Lifshitz free energy, reasserting a previous finding by Intravaia and Henkel that such fluctuating bulk currents are the physical reason for the anomalous entropy behaviour. In a related effort, an analysis of the frequency dependence of the Casimir force by Ford is generalised to imperfectly reflecting mirrors. A paradox is pointed out, in that the effects of a perturbation of the reflecting properties of the mirrors in a finite frequency window can be calculated in two ways giving different results. It is concluded that optimistic conclusions reached by Ford based on one of these methods, which seems to allow radically changing and tailoring the Casimir force with engineered materials, can not be realised. Part 2 presents several explicit calculations of the Casimir energy of different wedge and cylinder geometries. The Casimir energy of a perfectly conducting wedge intercut by a circularly cylindrical arc, either perfectly conducting or (magneto)dielectric, is calculated. The energy is found to include a singular and non-regularisable term due to the corners where the arc meets the wedge, whereas the finite part is an immediate generalisation of the previously known results for a circular cylinder. The energy of a magnetodielectric wedge obeying a criterion of isorefractivity (spatially uniform speed of light) superimposed coaxially on a perfectly conducting cylindrical shell is calculated. This is the first expression for the energy of a wedge which is not perfectly reflecting. Finally, the energy of the perfectly conducting wedge and arc (and, as a special case, cylinder) is extended to the case of non-zero temperatures. After a regularisation procedure making use of the Chowla-Selberg formula an analytical expression for the temperature-dependent energy at all temperatures is derived, and showed to coincide with previously calculated high-temperature asymptotics by Bordag, Nesterenko and Pirozhenko. Part 3 considers numerical and analytical studies of the Casimir-Polder forces acting on particles prepared in a given eigenstate (or superposition of such) in an environment which is otherwise at thermal equilibrium. We first consider cold polar molecules outside a metallic halfspace. It is found that the force in the near-zone (non-retarded regime) is much weaker than what would result from a naive perturbative calculation, and that in the far-zone (retarded regime) the force becomes spatially oscillatory. It is demonstrated how these spatial oscillations may be enhanced in a resonating planar cavity, although for polar molecules the resulting amplitude is still insufficient for observation. A cylindrical cavity, however, can achieve a better enhancement factor. The Casimir-Polder forces on Rydberg atoms near a surface are calculated; because of the very large transition dipole moments of Rydberg transitions, the force is enormous on an atomic scale. We show that the oscillating force on Rydberg atoms can be enhanced into the observable regime by use of a fine-tuned cylindrical cavity. A particle in an eigenstate which is in the non-retarded regime with respect to all its dominant transitions is shown to feel a Casimir-Polder force which is virtually independent of temperature from zero to room temperature and beyond. Both for cold polar molecules and Rydberg atoms, the temperature-independent regime extends to a few and hundreds of micrometers, respectively, and includes the separations generally accessed in experiments

Optically anisotropic infinite cylinder above an optically anisotropic half space: Dispersion interaction of a single-walled carbon nanotube with a substrate.

A complete form of the van der Waals dispersion interaction between an infinitely long anisotropic semiconducting/insulating thin cylinder and an anisotropic half space is derived for all separations between the cylinder and the half space. The derivation proceeds from the theory of dispersion interactions between two anisotropic infinite half spaces as formulated in Phys. Rev. A 71, 042102 (2005). The approach is valid in the retarded as well as nonretarded regimes of the interaction and is coupled with the recently evaluated ab initio dielectric response functions of various semiconducting/insulating single wall carbon nanotubes, enables the authors to evaluate the strength of the van der Waals dispersion interaction for all orientation angles and separations between a thin cylindrical nanotube and the half space. The possibility of repulsive and/or nonmonotonic dispersion interactions is examined in detail.

Resonant Casimir–Polder forces in planar meta-materials

We study the resonant Casimir–Polder potential of an excited atom near a half-space containing magneto-electric meta-material of various kinds on the basis of macroscopic quantum electrodynamics. Analytical results are obtained in the nonretarded and retarded distance regimes and numerical examples are given. We compare our findings with the potential of an excited atom near a left-handed superlens.

Modifying atom-surface interactions with optical fields

The ability to control matter on the nanometer scale is greatly influenced by the van der Waals vdW interaction. Therefore, understanding and manipulating the vdW interaction is of interest to the fields of nanotechnology and atom optics. We show that near-resonant light can significantly modify atom-surface vdW interactions in the nonretarded regime. A theory based on quantized electromagnetic fields is used to calculate 1 the ordinary vdW interaction, 2 corrections to the ordinary vdW interaction due to thermal radiation, and 3 modifications to the ordinary vdW interaction that result from monochromatic laser radiation. Near- resonant laser light with an intensity of 5 W /cm 2 is predicted to double the vdW interaction strength for sodium atoms, and possible experiments to detect this effect are discussed. The strength of the van der Waals vdW interaction be- tween matter is usually determined by the atomic or bulk material properties of a system. Therefore, the vdW force can typically only be altered by changing the atomic constituents of the matter involved in the interaction, limiting the ability to control nanometer scale forces. This circumstance reso- nates with the point of view in which the vdW interaction can be thought of as being caused by the fluctuating dipole moment of a quantum-mechanical atom 1. These fluctuat- ing dipoles can then interact with each other, leading to the vdW force between the atoms. However, there exists an al- ternative perspective in which the origin of the vdW interac- tion is due to electromagnetic-field fluctuations in the vacuum, which in turn induce a fluctuating dipole moment in the atoms 2. This suggests that the vdW interaction can be affected by the radiation environment that the matter resides in, opening the exciting possibility of using light to control or even inhibit the vdW interaction. One way to change the radiation environment of matter is through temperature, which exposes the atoms to a thermal electromagnetic field. In principle, this should modify the vdW or Casimir-Polder interaction. Just such an effect has recently been observed for an atom and a nearby surface in the retarded regime 3, where the atom-surface distance z is much larger than the principle transition wavelength of the atom. However, thermal modifications are predicted to be negligible in the nonretarded regime 4,5, where distances are such that z. This is unfortunate since the ability to change the force between atoms and a surface would have an impact on atom chips, atom optics, and quantum reflection experiments.

Dispersion forces between an atom and a perfectly conducting wedge

We consider the interaction between an electrically polarizable atom in its fundamental state and a wedge constituted by two semi-infinite perfectly conducting plates. Using a formalism based on a master equation, we compute the dispersion force on the atom for both retarded and non-retarded regimes.

Efficient calculation of van der Waals dispersion coefficients with time-dependent density functional theory in real time: Application to polycyclic aromatic hydrocarbons

The van der Waals dispersion coefficients of a set of polycyclic aromatic hydrocarbons, ranging in size from the single-cycle benzene to circumovalene (C(66)H(20)), are calculated with a real-time propagation approach to time-dependent density functional theory (TDDFT). In the nonretarded regime, the Casimir-Polder integral is employed to obtain C(6), once the dynamic polarizabilities have been computed at imaginary frequencies with TDDFT. On the other hand, the numerical coefficient that characterizes the fully retarded regime is obtained from the static polarizabilities. This ab initio strategy has favorable scaling with the size of the system--as demonstrated by the size of the reported molecules--and can be easily extended to obtain higher order van der Waals coefficients.

Van der Waals interactions between atoms and dispersive surfaces at finite temperature

The long-range interactions between an atomic system in an arbitrary energy level and dispersive surfaces in thermal equilibrium at non-zero temperature are revisited within the framework of the quantum-mechanical linear response theory, using generalized susceptibilities for both atom and electromagnetic field. After defining the observables of interest, one presents a general analysis of the atomic level shift valid for any number and form of dielectric surfaces. It is shown that, at zero temperature, one recovers well-known results previously obtained in the linear response regime. The case of a plane dispersive surface is elaborated on in the non-retarded regime. Calculations are given in detail for a dielectric surface exhibiting a single polariton resonance. Theoretical predictions are presented within a physical viewpoint allowing one to discriminate between the various interaction processes: on one hand, the level shift induced by non-resonant quantum fluctuations, on the other hand, two potentially resonant atom-surface couplings. The first resonant process appears for excited-state atoms and originates in an atomic de-excitation channel resonantly coupled to the surface polariton mode. It exists also at zero temperature, and has been studied and observed previously. The second physical process, which exists at non-zero temperature only, corresponds to the reverse process in which a thermal quantum excitation of a surface polariton resonantly couples to an atomic absorption channel. This novel phenomenon is predicted as well for a ground state atom, and can turn the ordinary long-range van der Waals attraction of atoms into a surface repulsion at increasing temperatures. This opens the way to the control and engineering of the sign and amplitude of van der Waals forces via surface temperature adjustment.

Casimir–Polder forces from density matrix formalism

We use the density matrix formalism in order to calculate the energy level shifts, in second order on interaction, of an atom in the presence of a perfectly conducting wall in the dipole approximation. The thermal corrections are also examined when $\hbar \omega_0/k_B T = k_0 \lambda_T \gg 1$, where ${$\omega_0=k_0 c$}$ is the dominant transition frequency of the atom and $\lambda_T$ is the thermal length. When the distance $z$ between the atom and the wall is larger than $\lambda_T$ we find the well known result obtained from Lifshitz's formula, whose leading term is proportional to temperature and is independent of $c$, $\hbar$ and $k_0$. In the short distance limit, when $z\ll\lambda_T$, only very small corrections to the leading vacuum term occur. We also show, for all distance regimes, that the main thermal corrections are independent of $k_0$ (dispersion is not important) and dependent of $c$, which means that there is not a non-retarded regime for the thermal contributions.

Retarded and nonretarded van der Waals interactions between a cluster and a second cluster or a conducting surface

In some respects, a cluster consisting of many atoms may be regarded as a single large atom. Knowing the dielectric properties of such a cluster permits one to evaluate the form of the van der Waals (dispersion) interactions between two clusters or between one cluster and a surface. In this paper, we derive these interactions in two extreme opposite regimes of separation: fully retarded and nonretarded. In the fully retarded regime (very large separation), the magnitude of the interaction is determined by just the static polarizability of the cluster(s). In the nonretarded regime (small separation), we employ a single resonant frequency model of thecluster polarizability to derive expressions for the interactions' coefficients. Numerical examples are presented to demonstrate that many-body screening of these interactions can be significant. The results represent the corrections to the commonly used approximation of pairwise additivity of interatomic interactions.