Nonequilibrium Casimir-Polder plasmonic interactions

We investigate how the combination of nonequilibrium effects and material properties impacts on the Casimir-Polder interaction between an atom and a surface. By addressing systems with temperature inhomogeneities and laser interactions, we show that nonmonotonous energetic landscapes can be produced where barriers and minima appear. Our treatment provides a self-consistent quantum theoretical framework for investigating the properties of a class of nonequilibrium atom-surface interactions.

Figure 1

The different contributions to the interaction energy $U$ at $T_{\mathrm{f}}=T_{\mathrm{a}}=0$ for an isotropic atom above a gold half-space (Drude model with ${\mathrm{\Omega }}_{\mathrm{p}}=9\mathrm{eV}$ and $\mathrm{\Gamma }=35\mathrm{meV}$) as a function of the atom-wall distance $L$. The (black) solid lines correspond to the total interaction energy $U$, the (red) dotted lines to $U_{\text{f}}$, and the (green) dashed lines to $U_{\text{a}}$. Distances are in units of the plasma wavelength ${\lambda }_{\mathrm{p}}=2\pi c/{\mathrm{\Omega }}_{\mathrm{p}}$, and energies are normalized to the total interaction energy $U_0$ at $L/{\lambda }_{\mathrm{p}}=1$. The top panel corresponds to a case where the atomic resonant frequency is below the asymptotic plasmonic frequency ${\mathrm{\Omega }}_{\mathrm{sp}}$ (${\omega }_{\mathrm{a}}=0.85{\mathrm{\Omega }}_{\mathrm{sp}}$), while the bottom panel corresponds to the opposite situation (${\omega }_{\mathrm{a}}=1.13{\mathrm{\Omega }}_{\mathrm{sp}}$).

Figure 2

Out-of-equilibrium interaction energy $U_{\mathrm{oe}}$ for a rubidium atom above a gold half-space as a function of the distance for different thermal populations of the plasmonic branch. The (black) solid line shows the thermal equilibrium situation ($T_{\mathrm{sp}}=T=300\mathrm{K}$). Two imbalanced configurations with $T=300\mathrm{K}$ are also shown, corresponding to $T_{\mathrm{sp}}=1100\mathrm{K}$ (red dotted) and $T_{\mathrm{sp}}=2000\mathrm{K}$ (green dashed). Inset: Dispersion relation ${\omega }_{\mathrm{sp}}\left(k\right)$ for the surface plasmon. Plasmonic modes with frequencies below (above) the atomic transition frequency ${\omega }_{\text{a}}$ give an attractive (repulsive) contribution to $U_{\mathrm{oe}}$.

Figure 3

Kretschmann configuration of the system under investigation.

Figure 4

Total out-of-equilibrium interaction energy $U_{\mathrm{oe}}$ for a rubidium atom in front of a plasmon-excited glass-gold structure (configuration of Fig. 3) at $T=300$ K. The contour plot shows the dependence on the atom-surface distance $L$ and on the incidence angle of the excitation laser ${\theta }_{\mathrm{i}}$. From the latter we subtract the total internal reflection angle ${\theta }_{\mathrm{T}}$ for the bare glass-vacuum interface. The frequency of the laser is ${\omega }_l=24.6\times {10}^{14}$ rad/s, which implies ${\theta }_{\mathrm{T}}\simeq 34.{23}^{\circ }$. The laser is blue-shifted with respect to the main transitions of rubidium, whose polarizability is modeled according to Eq. (42). The power and waist of the beam within the glass are, respectively, $P_l=200$ mW and $w_l=180\mu \mathrm{m}$. The thick dashed curve marks the position of the potential-barrier maximum height as a function of ${\theta }_{\mathrm{i}}$. The other curves are contour lines.

Figure 5

Total out-of-equilibrium atom-surface interaction energy $U_{\mathrm{oe}}$ as a function of $L$ for the same configuration of Fig. 4. Results for three different values of the incidence angle ${\theta }_{\mathrm{i}}$ of the exciting laser beam are shown, corresponding to vertical slices in Fig. 4.

Figure 6

Maximum height of the atom-surface potential barrier $U_{\max }$ as a function of the exciting-laser incidence angle for different values of the laser frequency (see legend). Inset: Imaginary part of the polarizability ${\alpha }_{\mathrm{Rb}}$ [Eq. (42)] over its zero-frequency value $\alpha \left(0\right)$ as a function of the frequency $\omega$. The four vertical lines mark the laser frequencies used in the main plot.

Figure 7

Total out-of-equilibrium interaction energy $U_{\mathrm{oe}}$ for a rubidium atom at a distance $L$ from a structure like that of Fig. 3. The present plot refers to a gold surface excited by two laser beams, of frequencies ${\omega }_l^{\mathrm{b}}=24.6\times {10}^{14}$ rad/s and ${\omega }_l^{\mathrm{r}}=21.0\times {10}^{14}$ rad/s. These frequencies are, respectively, blue- and red-shifted with respect to the main transitions of rubidium, whose polarizability is modeled according to Eq. (42). The total internal reflection angles at the bare glass-vacuum interface result as ${\theta }_{\mathrm{T}}^{\mathrm{b}}=34.{23}^{\circ }$ for ${\omega }_l^{\mathrm{b}}$ and ${\theta }_{\mathrm{T}}^{\mathrm{r}}=34.{63}^{\circ }$ for ${\omega }_l^{\mathrm{r}}$. We assumed $P_l=1.2$ W and $w_l=180\mu \mathrm{m}$ as power and beam waist for both beams within the glass. The plotted curves correspond to different incidence angles of the blue-shifted beam ${\theta }_{\mathrm{i}}^{\mathrm{b}}$ (cf. legend), while we fixed ${\theta }_{\mathrm{T}}^{\mathrm{r}}-{\theta }_{\mathrm{i}}^{\mathrm{r}}\simeq 0.{502}^{\circ }$, corresponding to the surface-plasmon resonance condition for the red-shifted laser.

Figure 8

(a) Depth of the potential well generated by the total atom-surface interaction in the two-laser configuration of Fig. 7. The well depth is shown as a function of the incidence angles ${\theta }_{\mathrm{i}}^{\mathrm{r}}$ and ${\theta }_{\mathrm{i}}^{\mathrm{b}}$ of the two beams. The thick (black) contour indicates the zero-depth boundary, beyond which the well disappears. The two dots marks the incidence angles to which panels (b) and (c) refer. Panels (b) and (c) show the depth of the out-of-equilibrium potential well as a function of the red- and blue-shifted laser powers $P_{\text{r}}$ and $P_{\text{b}}$. Panel (b) corresponds to both lasers satisfying their surface-plasmon resonance condition, i.e., ${\theta }_{\mathrm{T}}^{\mathrm{r}}-{\theta }_{\mathrm{i}}^{\mathrm{r}}\simeq 0.{502}^{\circ }$ and ${\theta }_{\mathrm{T}}^{\mathrm{b}}-{\theta }_{\mathrm{i}}^{\mathrm{b}}\simeq 0.{691}^{\circ }$. For panel (c) we just changed ${\theta }_{\mathrm{T}}^{\mathrm{b}}-{\theta }_{\mathrm{i}}^{\mathrm{b}}\simeq 0.{757}^{\circ }$, correspoding to the maximum depth observed in the left panel. In both (b) and (c), the thick dot-dashed line marks $P_{\text{r}}=P_{\text{b}}$. All other parameters are the same as in Fig. 7.

Figure 9

Out-of-equilibrium atom-surface potential $U_{\mathrm{oe}}$ for a rubidium atom at distance $L$ from a structure like that of Fig. 3 in the presence of two laser beams of frequency ${\omega }_l=24.6\times {10}^{14}$ rad/s, power $P_l=200$ mW, and waist $w_l=180\mu \mathrm{m}$. We consider the beams as counterpropagating along the $x$ axis, which implies that $U$ depends on $x$ but not on $y$. All the physical parameters of the system are the same as Fig. 4.