Vacuum fluctuations and radiation reaction contributions to the resonance dipole-dipole interaction between two atoms near a reflecting boundary

We investigate the resonance dipole-dipole interaction energy between two identical atoms, one in the ground state and the other in the excited state, interacting with the electromagnetic field in the presence of a perfectly reflecting plane boundary. The atoms are prepared in a correlated (symmetric or antisymmetric) Bell-type state. Following a procedure due to Dalibard et al. [J. Dalibard et al., J. Phys. (Paris) 43, 1617 (1982); J. Phys. (Paris) 45, 637 (1984)], we separate the contributions of vacuum fluctuations and radiation reaction (source) field to the resonance interaction energy between the two atoms and show that only the source field contributes to the interatomic interaction, while vacuum field fluctuations do not. By considering specific geometric configurations of the two-atom system with respect to the mirror and specific choices of dipole orientations, we show that the presence of the mirror significantly affects the resonance interaction energy and that different features appear with respect to the case of atoms in free space, for example, a change in the spatial dependence of the interaction. Our findings also suggest that the presence of a boundary can be exploited to tailor and control the resonance interaction between two atoms, as well as the related energy transfer process. The possibility of observing these phenomena is also discussed.

Figure 1

Pictorial representation of the physical system of two atoms aligned along the $z$ axis, perpendicular to the plate.

Figure 2

Plot of $\delta E/\delta E_0$ as a function of the interatomic distance $L$ in the near zone for atoms aligned perpendicular to the mirror. The dipole moments are both assumed to be oriented in the $z$ direction (green line) or in the $x$ direction (blue line). The interaction energy is enhanced (green line) or suppressed (blue line) compared to that in the vacuum, depending on the orientation of the two dipole moments. The parameters are chosen such that ${\omega }_0=4.17\text{eV}$, $z_A=2\times {10}^{-2}{\text{eV}}^{-1}$, and ${\mu }_x^A={\mu }_z^A={\mu }_x^B={\mu }_z^B=1.024\times {10}^{-3}{\text{eV}}^{-1}$.

Figure 3

Plots of the resonance interaction energy as a function of the interatomic distance $L$ for atoms aligned perpendicular to the mirror and for different atom-plate distances $z_A$ in (a) the near-zone limit $L\ll {\omega }_0^{-1}$ and (b) and (c) the far-zone limit $L\gg {\omega }_0^{-1}$. The two dipole moments are both assumed to be oriented in the $x$ direction, with ${\mu }_x^A={\mu }_x^B=1.024\times {10}^{-3}{\text{eV}}^{-1}$ and ${\omega }_0=4.17\text{eV}$. (a) The resonance interaction is suppressed with respect to that in the vacuum (black solid line) and increases by increasing the atom-plate distance $z_A$. The parameters have been chosen such that $z_A=2.0\times {10}^{-2}{\text{eV}}^{-1}$ (blue dashed line), $z_A=2.5\times {10}^{-2}{\text{eV}}^{-1}$ (green dotted line), and $z_A=3.5\times {10}^{-2}{\text{eV}}^{-1}$ (red long-dashed line). (b) and (c) The resonance interaction has oscillations in space and is enhanced or inhibited compared to the case of atoms in vacuo, depending on the distances $L$ and $z_A$ of atom $A$ from the plate. The numerical values of $z_A$ range from $z_A\sim 2.5\times {10}^{-2}{\text{eV}}^{-1}\ll {\omega }_0^{-1}$ to $z_A\sim 5.5{\text{eV}}^{-1}\gg {\omega }_0^{-1}$.

Figure 4

Pictorial representation of the physical system of two atoms aligned in the $y$ direction, parallel to the reflecting plate.

Figure 5

Plots of the resonance interaction energy as a function of the interatomic distance $D$ for atoms aligned parallel to the plate in (a) the near-zone limit ($D\ll {\omega }_0^{-1}$) and for atoms very close to the plate and (b) and (c) the far-zone limit $D\gg {\omega }_0^{-1}$, at different atom-plate distances. The dipole moments are both assumed to be oriented in the $x$ direction. (a) The interaction is inhibited by the presence of the mirror and approaches that of atoms in the vacuum (black solid line) when the distance $z$ of atoms $A$ and $B$ from the mirror is increased. The parameters are chosen such that $z=2.0\times {10}^{-2}{\text{eV}}^{-1}$ (blue dashed line), $z=2.5\times {10}^{-2}{\text{eV}}^{-1}$ (green dot-dashed line), $z=3.5\times {10}^{-2}{\text{eV}}^{-1}$ (red dotted line), ${\omega }_0=4.17\text{eV}$, and ${\mu }_x^A={\mu }_x^B=1.024\times {10}^{-3}{\text{eV}}^{-1}$. The black solid line represents the resonance interaction between two atoms in the vacuum. (b) and (c) The resonance interaction shows oscillations in space and can be suppressed or enhanced with respect to that between atoms in the unbounded space by varying the distances involved. The parameters are chosen such that ${\omega }_0=4.17\text{eV}$, ${\mu }_x^A={\mu }_x^B=1.024\times {10}^{-3}{\text{eV}}^{-1}$, and $2.5\times {10}^{-2}\le D\le 6.5{\text{eV}}^{-2}$.

Figure 6

Plots of the resonance interaction energy as a function of the interatomic distance $D$ for atoms with dipole moments along $y$ and $z$ in (a) the near-zone limit $D\ll {\omega }_0^{-1}$ and (b) the far-zone limit $D\gg {\omega }_0^{-1}$, with ${\mu }_A$ assumed to be in the $y$ direction and ${\mu }_B$ in the $z$ direction. The parameters are chosen such that ${\omega }_0=4.17\text{eV}$ and ${\mu }_y^A={\mu }_z^B=1.024\times {10}^{-3}{\text{eV}}^{-1}$. (a) The interaction decreases by increasing the distance $z$ of atoms $A$ and $B$ from the mirror. (b) The interaction energy oscillates with distance and it is of the same order of magnitude as that for atoms in the vacuum.

Figure 7

Comparison between the resonance energies as a function of the interatomic distance $D$ for different orientations of the dipole moments. The atoms are assumed to be very close to the mirror. The plot shows that the interaction decays more slowly with distance compared to the case of parallel dipole moments. The values of the parameters used are $z=2.5\times {10}^{-2}{\text{eV}}^{-1}$, ${\omega }_0=4.17\text{eV}$, and ${\mu }_x^A={\mu }_x^B=1.024\times {10}^{-3}{\text{eV}}^{-1}$.