Effect of curvature and confinement on the Casimir-Polder interaction

Modifications of Casimir-Polder interactions due to confinement inside a cylindrical cavity and due to curvature in- and outside the cavity are studied. We consider a perfectly conducting cylindrical shell with a single particle (atom or macroscopic sphere) located next to its interior or exterior surface or two atoms placed inside the shell. By employing the scattering approach, we obtain the particle-cavity interaction and the modification of the two-particle interaction due to the cavity. We consider both retardation and thermal effects. While for the atoms a dipole description is sufficient, for the macroscopic sphere we sum (numerically) over many multipole fluctuations to compute the interaction at short separations. In the latter limit we make comparisons to the proximity approximation and a gradient expansion and find agreement. Our results indicate a confinement-induced suppression of the force between atoms. General criteria for suppression and enhancement of Casimir interactions due to confinement are discussed.

Figure 1

Amplitudes of the interior Casimir-Polder interaction between an atom and a cylindrical shell as a function of $\delta =d/R_c$. Shown are the limiting analytical results for short distances [dotted curves; see Eqs. (17) and (18)] and for the case where the atom is close to the axis of the cylinder [dashed curves; see Eqs. (19) and (20)]. Thick curves correspond to the full numerical result for $f_0^E\left(\delta \right)$ [Eq. (15)] and $f_0^M\left(\delta \right)$ [Eq. (16)].

Figure 2

The same as Fig. 1, for the classical limit. Shown are the analytical results for short surface distances $\ell$ [dotted curves; see Eqs. (25) and (26)] and for the case where the atom is close to the axis of the cylinder [dashed curves; see Eqs. (27) and (28)]. Thick curves represent the full numerical result for $f_{\mathrm{cl}}^E\left(\delta \right)$ [Eq. (23)] and $f_{\mathrm{cl}}^M\left(\delta \right)$ [Eq. (24)].

Figure 3

Amplitudes of the exterior Casimir-Polder interaction between an atom and a cylindrical shell as a function of $1/\delta =R_c/d$. Shown are the limiting analytical results for short distances [dotted curves; see Eqs. (38) and (39)] and the case where the atom is far away from the cylinder [dashed curves; see Eqs. (40) and (41)]. Thick curves correspond to the full numerical result for $g_0^E\left(\delta \right)$ [Eq. (36)] and $g_0^M\left(\delta \right)$ [Eq. (37)].

Figure 4

The same as Fig. 3, for the classical limit. Shown are the analytical results for short surface distances $\tilde{\ell }$ [dotted curves; see Eqs. (46) and (47)] and the case where the atom is far away from the cylinder [dashed curves; see Eqs. (48) and (49)]. Thick curves represent the full numerical result for $g_{\mathrm{cl}}^E\left(\delta \right)$ [Eq. (44)] and $g_{\mathrm{cl}}^M\left(\delta \right)$ [Eq. (45)].

Figure 5

Numerical result for the zero-temperature Casimir energy (normalized to the PFA) for a sphere inside a cylinder for different ratios of the radii, ranging from $r_s=R_s/R_c=0.1$ [bottom-most (blue) curve] to $r_s=0.9$ [topmost (green) curve] in steps of $0.1$, as a function of $\ell /R_c<1-r_s$. Curves terminate at the positions where the sphere is located at the axis of the cylinder $\left(\ell /R_c=1-r_s\right)$. The straight lines originating from unity represent the first correction to the PFA; see Eq. (62).

Figure 6

The same as Fig. 5, but for the classical high-temperature limit.

Figure 7

Log-log plot of the numerical results for the functions $e_0\left(\eta \right)$ as a function of $\eta =h/R_c$ for the three combinations of polarizations (solid curves). Dashed curves represent the long-distance limits $\left(\eta \gg 1\right)$ given by Eqs. (77), (78), and (79), and dotted curves correspond to the short-distance limits $\left(\eta \ll 1\right)$ given by Eqs. (75) and (76).

Figure 8

Log-log plot of the numerical results for the functions $e_{cl}\left(\eta \right)$ as a function of $\eta =h/R_c$ for the two combinations of polarizations (solid curves). Dashed curves represent the long-distance limits $\left(\eta \gg 1\right)$ given by Eqs. (81) and (82), and dotted curves correspond to the short-distance limits $\left(\eta \ll 1\right)$ given by Eq. (80).