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

In a recent paper, we formulated a theory of nonrelativistic 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 nonretarded 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 nonretarded 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.

Figure 1

The Feynman propagator for the displacement field, given in Eq. (48), describes the propagation of photons near a slab of absorptive material with anisotropic and frequency-dependent permittivity ${\epsilon }_{\sigma }\left(\omega \right)$. It is calculated exactly starting from the quantum model for the light-matter interactions described in Secs. 1 and 2.

Figure 2

The dimensionless $F_{\parallel }$ of Eq. (60) that enters the shift (59) for an electromagnetic response of the form (68), as a function of the dimensionless parameter $Z{\omega }_{mg}$, i.e., the distance from the surface measured in atomic transition wavelengths. The solid straight line shows the retarded limit for the perfect reflector, which is the leading-order term in the asymptotic expansion of $F_{\parallel }$, given by Eqs. (69) and (70). In the presence of damping (red, dot-dashed line), i.e., for nonzero ${\gamma }_{\parallel }$, even for relatively large values of $\mathcal{Z}{\omega }_{mg}$, the corrections to the leading-order behavior are still appreciable, especially for poor conductors. Without damping (blue, dashed line), they are much smaller.

Figure 3

For small $\mathcal{Z}{\omega }_{mg}$, we choose to plot $F^{\rho }/\left(\mathcal{Z}{\omega }_{mg}\right)$, which according to Eq. (75) approaches a constant as $\mathcal{Z}{\omega }_{mg}\to 0$, so that the energy shift in Eq. (59) behaves as ${\mathcal{Z}}^{-3}$ in the nonretarded limit. The solid lines represent the results of the exact numerical integration of Eq. (73), whereas the dashed lines come from the corresponding approximate formulas in Eqs. (74) and (75).

Figure 4

For large $\mathcal{Z}{\omega }_{mg}$, the coefficients $F^{\rho }$ according to Eq. (75) approach constants, which implies that the energy shift in Eq. (59) behaves as ${\mathcal{Z}}^{-4}$ in the retarded regime. The solid lines represent results of the exact numerical integration of Eq. (73), whereas the dashed lines have been calculated from the corresponding approximate formulas in Eqs. (76) and (77). Note that the corrections to the leading-order terms may be significant for small values of ${\omega }_P/{\omega }_{mg}$.

Figure 5

Exact values of $F^{\parallel }$ of Eq. (82) with $\epsilon \left(\omega \right)$ as in Eq. (78), with and without the losses in the material. In the case of no damping (blue, dashed line), $F^{\parallel }$ approaches a constant value of $1/2$ for increasing $\mathcal{Z}{\omega }_{mg}$, which is how a perfect reflector would behave in the retarded regime. When the material is absorbing (red, dot-dashed line), $F^{\parallel }$ still approaches a constant value (i.e., the energy shift still vanishes as $1/{\mathcal{Z}}^4$), but the proportionality factor depends on the dimensionless combination $L{\sigma }_{\parallel }\left(0\right)$; cf. Eq. (86).

Figure 6

The function $F^{\perp }$ of Eq. (83) for an isotropic conductor (blue, dashed line) and for an $\epsilon \left(\omega \right)$ as in Eq. (78), where the conductivity of the slab in the direction normal to the surface is suppressed and the slab's optical response in this direction is that of a nondispersive dielectric (red, dot-dashed line). As $\mathcal{Z}{\omega }_{mg}$ increases, the difference between the energy shifts in the two cases vanishes, which confirms our conclusion that the anisotropy of the conductor does not matter in the retarded regime.

Figure 7

The function $F^{\perp }/\mathcal{Z}{\omega }_{mg}$ of Eq. (83) for an isotropic conductor (blue, dashed line) and for an anisotropic conductor with an $\epsilon \left(\omega \right)$ as in Eq. (78) (red, dot-dashed line). For small distances $\mathcal{Z}{\omega }_{mg}$, the anisotropy of the material has a considerable impact on the Casimir-Polder interaction. In this example, the Casimir-Polder force at $\mathcal{Z}{\omega }_{mg}\approx 1$ is reduced by about $25\%$ due to the suppression of the conductivity in the direction normal to the slab's surface.