Dynamical Casimir effect for a massless scalar field between two concentric spherical shells

In this work we consider the dynamical Casimir effect for a massless scalar field—under Dirichlet boundary conditions—between two concentric spherical shells. We obtain a general expression for the average number of particle creation, for an arbitrary law of radial motion of the spherical shells, using two distinct methods: by computing the density operator of the system and by calculating the Bogoliubov coefficients. We apply our general expression to breathing modes: when only one of the shells oscillates and when both shells oscillate in or out of phase. Since our results were obtained in the framework of the perturbation theory, under resonant breathing modes they are restricted to a short-time approximation. We also analyze the number of particle production and compare it with the results for the case of plane geometry.

Figure 1

(Color online) Map of the solutions of the transcendental equation [11]. The colors correspond to different values of the number $l$: the black lines are for $l=0$, the red ones for $l=1$, and for $l=2$ the blue lines. The solid, dashed, and dotted lines correspond to the cases $s=1$, $s=2$, and $s=3$, respectively.

Figure 2

(Color online) Plot of $(r_o-r_i)\mid c_{l1s^{\prime }}^{\alpha }\mid$ against the ratio $r_o∕r_i$. The solid and dashed lines correspond to $s^{\prime }=1$ and $s^{\prime }=2$, respectively. The black line correspond to $l=0$ and $\alpha =i,o$. The blue and red lines are for $\alpha =i$ and $\alpha =o$, respectively, both with $l=1$.

Figure 3

Plot of $N_{lms}\left(t\right)∕\left(ϵ\varpi t\right)$ against $\varpi ∕{\omega }_{01}\left(0\right)$, in the resonance condition, for the cases (a), (b), (c), and (d) for few values of $l$ and $s$. We have set $r_o=2r_i$.