Casimir-Polder potential and transition rate in resonating cylindrical cavities

We consider the Casimir-Polder potential of particles placed inside a metallic cylindrical cavity at finite temperatures, taking account of thermal nonequilibrium effects. In particular, we study how the resonant (thermal nonequilibrium) potential and transition rates can be enhanced by fine tuning the radius of the cavity to match the transition wavelength of the dominant transitions of the particle. Numerical calculations show that the cavity-induced energy-level shift of atoms prepared in low-lying Rydberg states can be enhanced beyond 30 kHz, which is within the range of observability of modern experiments. Because the magnitude of the resonance peaks depends sensitively on the low-frequency dissipation of the cavity metal, experiments in this setup could be a critical test of the disputed thermal correction to the Casimir force between metal plates.

Figure 1

Cross section of the geometry considered: a particle in a vacuum-filled circularly cylindrical cavity.

Figure 2

(a, b) Rotation of the $q$-integration path by an angle $\theta$. (c, d) Resulting paths for $\eta$.

Figure 3

The potential of Rydberg Rb in its $32s_{1/2}$ state at the center ($\rho =0$) of a Au cavity as a function of radius close to the resonance corresponding to the first zero $j_{01}\approx 2.4$ of $J_0$. The cavity resonantly enhances the contribution from the $32s_{1/2}\to 31p_{3/2}$ transition. $R_{01}=799.9\hspace{0.5em}\mu$m as given by Eq. (3.4) is the corresponding perfect conductor resonant radius.

Figure 4

Same setup as Fig. 3, but for the cavity-assisted transition rate from state $32s_{1/2}$ to $32p_{3/2}$.

Figure 5

The dependence of the peak potential at resonance on $\theta =\arg (\varepsilon )$. The solid graph is $U^{\mathrm{res}}$ calculated for a $32s$ Rydberg atom whose $32s_{1/2}\to 31p_{3/2}$ transition is in resonance with the $j_{01}$ zero of $J_0(x)$, varying the phase of $\varepsilon$ throughout the physical range but keeping $\left|\varepsilon \right|$ constant and equal to its value using Eq. (3.5). The dashed graph is the theoretical curve based on the approximate scaling of Eq. (4.1). Both functions have been normalized by their values at $\arg \varepsilon =\pi /2$.

Figure 6

(Left) Casimir-Polder potential at the eight smallest resonant radii acting on isotropic Rb in the state $32s_{1/2}$, whose transition to $31p_{3/2}$ resonates with various modes of a gold cavity described by Eq. (3.5). The radii are chosen according to Eq. (3.17) ($j_{01}$, $j_{02}$, $j_{11}=j_{01}^{\prime }$, $j_{12}=j_{02}^{\prime }$, and $j_{21}$) and Eq. (3.19) ($j_{11}^{\prime }$, $j_{12}^{\prime }$, $j_{21}^{\prime }$). (Right) The potential outside a plane half space.

Figure 7

(Left) Enhanced transition rates at the eight smallest resonant radii acting on isotropic Rb in the state $32s_{1/2}$, whose transition to $31p_{3/2}$ resonates with various modes of in a gold cavity described by Eq. (3.5). The radii are chosen according to Eq. (3.23) ($j_{01}$, $j_{02}$, $j_{11}=j_{01}^{\prime }$, $j_{12}=j_{02}^{\prime }$, and $j_{21}$) and Eq. (3.25) ($j_{11}^{\prime }$, $j_{12}^{\prime }$, $j_{21}^{\prime }$). (Right) The shift outside a plane half space.

Figure 8

Comparison of enhanced and nonenhanced heating rate of ground-state LiH. The heating rate outside a half space [32] shown together with the two highest cylinder resonances [radii from (3.23) and (3.25)] along a central cylinder cross section.