Casimir-Polder energy landscape: Unipolarizable atom and ring

Abstract

The Casimir-Polder interaction energy between a unipolarizable point atom and a unipolarizable dielectric ring has been limited, until now, to the case when the atom is confined on the axis of symmetry of the ring. We find the generalized analytical expression for any position of the atom relative to the ring in terms of complete elliptic integrals. This is aided by the construction of a class of integrals of a Jacobian elliptic function as a linear combination of complete elliptic integrals. Our expression for the interaction energy allows us to investigate the instability of the atom even for the equilibrium points which exists off the axis of symmetry.

Figures (7)

• Figure 1: A unipolarizable point atom with polarizability $\boldsymbol{\alpha}=\alpha_1\hat{\textbf{n}}\hat{\textbf{n}}$ at position ${\bf r}$ above a dielectric ring of radius $a$ with uniform polarizability $\boldsymbol{\sigma}=\sigma_2\hat{\textbf{z}}\hat{\textbf{z}}$. The point atom is a distance $\rho$ away from the axis of the ring. The principal axis $\hat{\bf n}$ of the atomic polarizability $\boldsymbol{\alpha}$ subtends an angle $\theta_1$ with respect to polarizability of the ring and the axis of the symmetry, that is, $\hat{\textbf{n}} \cdot \hat{\textbf{z}}=\cos\theta_1$
• Figure 2: Interaction energy in Eq. (\ref{['res-Ie-m3']}) for $\rho=0$ as a function of $z$. For $\theta_1=0$ the energy has a minimum at $z=0$ and $z=\pm 1.259\,a$, and maximum at $z=\pm 0.956\,a$. For $\theta_1=90^\circ$ the energy has a maximum at $z=0$, and minimum at $z=\pm 0.471\,a$.
• Figure 3: Contours of equal energy in Eq. (\ref{['th1=00ph100-as7']}) plotted as a function of position $x/a$ and $z/a$ of polarizable atom. The legend for the contours is energy measured in units of $E_0$, that is, $E/E_0$.
• Figure 4: Equal energy surfaces for the energy in Eq. (\ref{['th1=00ph100-as7']}) plotted as a function of position, $x/a$, $y/a$, and $z/a$, of the polarizable atom. The inner cyan surface corresponds to $E=-E_0$ and the outer yellow surface if for $E=-0.0065\,E_0$.
• Figure 5: Each of the frames show cross sections of surfaces of equal energy in Eq. (\ref{['res-Ie-m3']}) as contours in the region of space surrounding the unipolarizable dielectric ring of radius $a$ as a function of position $\rho$ and $z$ of the unipolarizable atom. The cross section in each frame is the $x$-$z$ plane. Each of the nine frames correspond to a fixed azimuth angle $\phi_1=0^\circ$ of the polarizability of atom, and the variation in the frames is in the orientation angle $\theta_1$ of the polarizability. The frame on the top left, for $\theta_1=0^\circ$, has three saddle points of equilibrium described by black dots, one at the origin and two (one on each side of ring) on the $z$ axis at $z_0=\pm 1.3\,a$. These three equilibrium points drift away from the $z$ axis as we change the orientation of the polarizability of the atom in the following frames for angles $\theta_1= 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ,$ and then for $\theta_1= 180^\circ, 150^\circ, 135^\circ, 120^\circ, 90^\circ$. The legend for the contours is energy measured in units of $E_0$, that is, $E/E_0$. The contours show an inversion symmetry about the origin between orientations $90^\circ<\theta_1<180^\circ$ and $0^\circ<\theta_1<90^\circ$.