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This article explores a number of issues associated with the problem of calculating and detecting electromagnetic quantum induced energy and stress in a stationary dielectric material with a smooth inhomogeneous polarizability. By concentrating on a particular system composed of an ENZ-type (epsilon-near-zero) meta-material, chosen to have a particular anisotropic and smooth inhomogeneous permittivity, confined in an infinitely long perfectly conducting open rectangular waveguide, we are able to deduce analytically from the source-free Maxwell’s equations and their boundary conditions a complete set of bounded harmonic electromagnetic evanescent eigen-modes and their associated eigen-frequencies. Since these solutions prohibit the existence of asymptotic scattering states in the guide, the application of the conventional Lifshitz approach to the Casimir stress problem becomes uncertain. An alternative approach is adopted based upon the spectral properties of the system and a regularization scheme constructed with direct applicability to more general systems composed of dielectrics with smooth inhomogeneous permittivities and open systems that may only admit evanescent modes. This more general scheme enables one, for the first time, to prescribe precise criteria for the extraction of finite quantum expectation values from regularized mode sums together with error bounds on these values, and is used to derive analytic or numeric results for regularized electromagnetic ground state expectation values in the guide.

Delving into spring-like helical configurations, such as DNA within our cells, motivates physicists to inquire about the effects of such structures in the realm of quantum field theory, specifically unraveling their manifestation of effectiveness in Casimir energy. To explore this, we initiated our investigation with the Casimir effect and proceeded to calculate the Casimir energy, along with its thermal and radiative corrections, for massive and massless Lorentz-violating scalar fields across (d+1)-dimensional spacetime. Adhering to the principle that the renormalization program should be consistent with the boundary conditions applied to the quantum fields was paramount. Accordingly, one-loop correction to the Casimir energy for both massive and massless scalar fields under helical boundary conditions by employing a position-dependent counterterm was performed. Our results, spanning across spatial dimensions, exhibited convergence and coherence with established physical grounds. Additionally, we presented the Casimir energy density using graphical plots for systems featuring time-like and space-like Lorentz violations across all spatial dimensions, followed by a comprehensive discussion of the obtained results.

The Casimir force between inhomogeneous slabs that exhibit a band-like structure is calculated. The slabs are made of basic unit cells each made of two layers of different materials. As the number of unit cells increases the Casimir force between the slabs changes, since the reflectivity develops a band-like structure characterized by frequency regions of high reflectivity. This is also evident in the difference of the local density of states between free and boundary distorted vacuum, that becomes maximum at frequencies corresponding to the band gaps. The calculations are restricted to vacuum modes with wave vectors perpendicular to the slabs.

By engineering the boundary conditions of electromagnetic fields between material interfaces, one can dramatically change the Casimir-Lifshitz force between surfaces as a result of the modified zero-point energy density of the system. Repulsive interactions between macroscopic bodies occur when their dielectric responses obey a particular inequality, as pointed out by Dzyaloshinskii, Lifshitz, and Pitaevskii. We discuss experimental verification of this behavior as well as a description of how this can be used to develop a scheme for quantum levitation. Based on these concepts, we discuss the possible development of a new class of devices based on ultra-low static friction and the ability to sort objects based on their dielectric functions.

This article addresses the discrepancies between theoretical and experimental results obtained for the thermal Casimir effect. Here we test the possibility that saturation effects may be the root of the problems. We present graphs that describe the numerical derivations in great detail.

In the study of quantum vacuum energy and the Casimir effect, it is desirable to model the conductor by a potential of the form $V(z)=z^{\alpha }$. This “soft wall” model was proposed so as to avoid the violation of the principle of virtual work under ultraviolet regularization that occurs for the standard Dirichlet wall. The model was formalized for a massless scalar field, and the expectation value of the stress tensor has been expressed in terms of the reduced Green function of the equation of motion. In the limit of interest, $\alpha \gg 1$, which approximates a Dirichlet wall, a closed-form expression for the reduced Green function cannot be found, so piecewise approximations incorporating the perturbative and WKB expansions of the Green function, along with interpolating splines in the region where neither expansion is valid, have been developed. After reviewing this program, in this paper, we apply the scheme to the wall with $\alpha =6$ and use it to compute the renormalized energy density and pressure inside the cavity for various values of the conformal parameter. The consistency of the results is verified by comparison to their numerical counterparts and verification of the trace anomaly and the conservation law. Finally, we use the approximation scheme to reproduce the energy density inside the quadratic wall, which was previously calculated exactly but with some uncertainty.

In this study, first, the phenomenon of multistability in the Lü system is found, which shows the coexistence of two different point attractors and one chaotic attractor. These coexisting attractors are dependent on initial conditions of the system while the parameters of the system are fixed. Then, the Lü system is transformed to a Kolmogorov-type system, which includes the conservative torque consisting of the inertial torque and the internal torque, the dissipative torque, and the external torque. Moreover, by analyzing the combination of different types of torques and investigating the cycling of energy based on the Casimir function and Hamiltonian function, the interaction between the external torque and other torques is found to be the main reason for the Lü system to generate chaos. Finally, by investigating the Casimir function, it is found that the boundary of the Lü system is only related to system parameters.

The response of a superconductor to a gravitational wave is shown to obey a London-like constituent equation. The Cooper pairs are described by the Ginzburg–Landau free energy density embedded in curved spacetime. The lattice ions are modeled by quantum harmonic oscillators characterized by quasi-energy eigenvalues. This formulation is shown to predict a dynamical Casimir effect since the zero-point energy of the ionic lattice phonons is modulated by the gravitational wave. It is also shown that the response to a gravitational wave is far less for the Cooper pair density than for the ionic lattice. This predicts a “charge separation effect” which can be used to detect the passage of a gravitational wave.

The influence of Casimir and van der Waals forces on the instability of vibratory micro and nano-bridge gyroscopes with proof mass attached to its midpoint is studied. The gyroscope subjected to the base rotation, Casimir and van der Waals attractions is actuated and detected by electrostatic methods. The system has two coupled bending motions actuated by the electrostatic and Coriolis forces. First a system of nonlinear equations for the flexural-flexural deflection of beam gyroscopes is derived using the extended Hamilton's principle. In modeling, the nonlinearities due to mid-plane stretching, electrostatic forces, including fringing field, Casimir and van der Waals attractions, are considered. The method of homotopy perturbation is used to solve the equations of equilibrium, with the solution validated by numerical methods. In addition, the effect of nondimensional parameters on the instability and deflection of the gyroscope is investigated. The data presented can be used in the design of vibratory micro/nano gyroscopes.

We discuss some geometrical and dynamical properties of Rabinovich systems.

We present calculations of the zero-temperature Casimir interaction between two freestanding graphene sheets as well as between a graphene sheet and a substrate. Results are given for undoped graphene and for a set of doping levels covering the range of experimentally accessible values. We describe different approaches that can be used to derive the interaction. We point out both the predicted power law for the interaction and the actual distance dependence.