Repulsive Casimir-Polder potential by a negative reflecting surface

We present a scheme to generate an all-range long repulsive Casimir-Polder potential between a perfect negative reflecting surface and a ground-state atom. The repulsive potential is stable and does not decay with time. The Casimir-Polder potential is proportional to $z^{-2}$ at short atom-surface distances and to $z^{-4}$ at long atom-surface distances. Because of these advantages, this potential can help in building quantum reflectors, quantum levitating devices, and waveguides for matter waves.

Figure 1

A two-level atom in the ground state is located above the negative reflecting surface, and the atom-surface distance is expressed by $z$.

Figure 2

A scheme for the negative reflection of an electromagnetic wave. For a negative reflecting surface, the incident wave and reflected wave are at the same side of the normal line. When the wave vector of the reflected wave is exactly opposite that of the incident wave, the negative reflection is called perfect.

Figure 3

The monotonic, positive-definite function $I\left(\tilde{z}\right)$ diverges for $\tilde{z}\to 0$ and converges towards zero for $\tilde{z}\to \infty$. These features of function $I\left(\tilde{z}\right)$ make it possible to produce an all-range long repulsive CP force.

Figure 4

If a photon is emitted by the atom, negatively reflected by the surface, and reabsorbed by the atom again, the atom can get recoil momentum from the photon twice and be pushed away from the surface. On average, a repulsion normal to the surface can be produced between the surface and the atom.

Figure 5

Time-ordered graph for interaction of one atom with virtual photons. The thin straight line represents the atom in the ground state, while the thick straight line represents the atom in the excited state. The dashed semicircle represents a photon in mode $\left(\mathit{k},\lambda \right)$. The arrow points out the direction of time evolution. This graph can help in understanding Eq. (43). Once the atom jumps to the excited state and emits a photon, the directions of the polarizations of the excited atom and the photon will be parallel. If the polarization of the photon was changed when it was reflected on the surface, the photon would not interact with the atom again. So only the reflection coefficients $g_{\perp \to \perp }$ and $g_{\parallel \to \parallel }$ appear in Eq. (43).

Figure 6

The functions $V\left(\tilde{z}\right),V_{\text{app}0}\left(\tilde{z}\right)$, and $V_{\text{app}\infty }\left(\tilde{z}\right)$ are plotted. It can be seen that $V\left(\tilde{z}\right)$ converges towards $V_{\text{app}0}\left(\tilde{z}\right)$ for $\tilde{z}\to 0$ and fits $V_{\text{app}\infty }\left(\tilde{z}\right)$ when $\tilde{z}\to \infty$. This result shows the CP potential decays very fast for large atom-surface separation.