# Perspective on Some Recent and Future Developments in Casimir Interactions

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Nontrivial Topology in Materials

_{x}Sb

_{1-x}, layered Hg(Cd)Te structures, half-Heuslers, and others [15]. TIs differ from ordinary materials as their bulk is typically gapped or insulating, while the surface states are gapless and they are a consequence of the unique band structure of the material. In fact, because of the insensitivity of the topology to perturbations, the surface states are considered to be rather stable.

## 3. Nonlinear Optical Response

## 4. Dynamical Casimir Effect

^{2}, and $\lambda \sim 3$ cm we obtain $\frac{N}{\tau}\sim {10}^{-5}$ photons/s, or one photon pair every two days. However, the effect can be significantly amplified inside cavities with oscillating boundaries, provided the frequency of oscillations is close to the doubled eigenfrequency $2{\omega}_{0}$ of some field mode when the conditions of parametric resonance are satisfied. The main result, obtained within quite different schemes [77,78,79,80], is a simple formula for the number of quanta, created from the initial vacuum state during time $t$,

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Attraction and repulsion at $T\to \infty $ and ${\theta}_{1}=-{\theta}_{2}=\theta $. The forbidden region is given in black. (Figure is taken from reference [18]).

**Figure 2.**(

**a**) Fabry–Perot cavity of two graphene-like materials under an external electric field ${E}_{z}$ and laser light $\mathsf{\Lambda}$; (

**b**) Casimir force phase diagram of two identical dissipationless graphene-like materials with $\frac{d{\lambda}_{SO}}{\hslash c}=1$. (Figure taken from reference [42]).

**Figure 3.**Casimir force between Weyl semimetals filled with chiral medium normalized to the one for perfect metals ${F}_{0}=-\frac{\hslash c{\pi}^{2}}{240{a}^{4}}$ as a function of separation. Here $\delta {k}_{z}=\mathcal{V}B$ ($\mathcal{V}$ —Verdet constant, $B$ —applied magnetic field); $\delta {k}_{z}=0$ corresponds to a vacuum filled gap, while the nonvanishing $\delta {k}_{z}=\pm 2\times {10}^{5}{\mathrm{m}}^{-1}$. In addition, $\frac{{\sigma}_{xy}}{c}=265\times {10}^{6}{\text{}\mathrm{m}}^{-1}$. The curves marked with $S$ correspond to ${\sigma}_{xy}^{1}={\sigma}_{xy}^{2}$, while the rest correspond to ${\sigma}_{xy}^{1}=-{\sigma}_{xy}^{2}$. The line marked as “large” correspond to $\frac{{\sigma}_{xy}^{1}}{c}=2650\times {10}^{6}{\mathrm{m}}^{-1}$ (Figure is taken from reference [60]).

**Figure 4.**Casimir pressure between a perfect mirror and a nonlinear material (see inset sketch), as a function of distance at $T=0$ K. For simplicity, the force is given for a real, frequency-independent linear permittivity as labeled, and a symmetric nonlinear response, ${\chi}_{iikk}^{\left(3\right)}={\chi}_{ikki}^{\left(3\right)}={\chi}_{ikki}^{\left(3\right)}={\chi}_{ikik}^{\left(3\right)}={\chi}_{iiii}^{\left(3\right)}/3=2\times {10}^{-16}\frac{{m}^{2}}{{V}^{2}}$, with a value measured for glass fused with silver nanoparticles [72]. Figure adapted from Reference [70].

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Woods, L.M.; Krüger, M.; Dodonov, V.V.
Perspective on Some Recent and Future Developments in Casimir Interactions. *Appl. Sci.* **2021**, *11*, 293.
https://doi.org/10.3390/app11010293

**AMA Style**

Woods LM, Krüger M, Dodonov VV.
Perspective on Some Recent and Future Developments in Casimir Interactions. *Applied Sciences*. 2021; 11(1):293.
https://doi.org/10.3390/app11010293

**Chicago/Turabian Style**

Woods, Lilia M., Matthias Krüger, and Victor V. Dodonov.
2021. "Perspective on Some Recent and Future Developments in Casimir Interactions" *Applied Sciences* 11, no. 1: 293.
https://doi.org/10.3390/app11010293