Numerical investigation of photon creation in a three-dimensional resonantly vibrating cavity: Transverse electric modes

The creation of TE-mode photons in a three-dimensional perfectly conducting cavity with one resonantly vibrating wall is studied numerically. We show that the creation of TE-mode photons in a rectangular cavity is related to the production of massive scalar particles on a time-dependent interval. The equations of motion are solved numerically which allows us to take into account the intermode coupling. We compare the numerical results with analytical predictions and discuss the effects of the intermode coupling in detail. The numerical simulations reveal that photon creation in a three-dimensional resonantly vibrating cavity can be maximized by arranging the size of the cavity such that certain conditions are realized. In particular, the creation of TE-mode photons in the lowest-frequency mode $(1,1,1)$ is most efficient in a noncubic cavity where the size of the nondynamical dimensions is roughly 11 times larger than the size of the dynamical dimension. We discuss this effect and its relation to the intermode coupling in detail.

Figure 1

(Color online) Number of particles created in the resonant mode $n=1$ for mass parameters $M=0.2$, 0.7, 2, and 3.5 in comparison with the analytical prediction (33).

Figure 2

(Color online) Particle spectra for different mass parameters $M=3.5$, 2, 0.7, and 0.2 at time $t=6700$ corresponding to Fig. 1. The spectra are shown for $k_{\max }=10$ (dots) and $k_{\max }=20$ (squares) to demonstrate numerical stability.

Figure 3

(Color online) (a) Number of particles created in the modes $k=1$, $2$, and $3$ for the mass parameter $M=0.7$ corresponding to the spectrum shown in Fig. 2. Part (b) shows $N_3\left(t\right)$ and part (c) $N_2\left(t\right)$ in each case for the two resolutions $\mathrm{\Delta }t=0.01$ (circles) and $\mathrm{\Delta }t=0.005$ (solid lines). The particle numbers calculated for times at which the mirror has returned to its initial position are accented by “$+$” and the background motion is shown for comparison as well.

Figure 4

(Color online) (a) Number of particles created in the modes $k=1$, 2, 3, 4, and $5$ for the mass parameter $M=0.2$ corresponding to the spectrum shown in Fig. 2. Part (b) shows $N_5\left(t\right)$ and part (c) $N_2\left(t\right)$ in each case for the two resolutions $\mathrm{\Delta }t=0.01$ (circles) and $\mathrm{\Delta }t=0.005$ (solid lines).

Figure 5

(Color online) Number of particles created in the resonance mode $n=1$ at time $t=2000$ as a function of the mass parameter $M$. The solid line shows the analytical prediction, Eq. (33). Arrows pointing towards particular mass values of $M$ mark masses for which Eq. (33) is not valid because of exact intermode coupling. The coupled modes are given in brackets $\left[\right(1,k\left)\right]$. No numerical results are shown in the plot for those cases. Most of the numerical results are shown for different values of the cutoff $k_{\max }$ to underline stability.

Figure 6

(Color online) Particle spectra for mass parameter $M=0.4$ at times $t=1500$, 3000, 4500, and $6700$. Each spectrum is shown for values $k_{\max }=10$ (dots) and $k_{\max }=20$ (squares) to indicate numerical stability.

Figure 7

(Color online) (a) Number of particles created in the modes $k=1$, 2, 3, and $4$ for the mass parameter $M=0.4$ corresponding to the spectra shown in Fig. 6. Part (b) shows $N_3\left(t\right)$ and part (c) $N_2\left(t\right)$ in each case for the two resolutions $\mathrm{\Delta }t=0.01$ (circles) and $\mathrm{\Delta }t=0.005$ (solid lines).

Figure 8

(Color online) Total particle number $N$ (circles) and number of particles created in the mode $k=1$ $N_1$ (squares) for mass parameters $M=0.2$, 0.15, 0.1, and $0.05$ together with the analytical predictions for the massless case, Eq. (6.5) (dashed line) and Eq. (6.10) (solid line), of [15] to demonstrate the convergence of the solutions towards the $M=0$ case (see also Fig. 4 of [53]). The cutoff parameter $k_{\max }=30$ was used in the simulations.

Figure 9

(Color online) Particle spectra for $\omega =2{\Omega }_1^0$ and mass parameter $M=\sqrt{2}\pi$ yielding exact coupling between the modes $n=1$ and $k=5$. Dots correspond to $k_{\max }=10$ and squares to $k_{\max }=20$.

Figure 10

(Color online) Number of particles created in the modes $n=1$ and $k=5$ for $\omega =2{\Omega }_1^0$ and $M=\sqrt{2}\pi$ corresponding to the particle spectra depicted in Fig. 9. The numerical results are compared to the analytical predictions, Eq. (54) (solid line) and Eq. (55) (dashed line) of [33]. The numerical results shown correspond to the cutoff parameter $k_{\max }=20$ which guarantees stability.

Figure 11

(Color online) Particle spectra for $\omega =2{\Omega }_1^0$ and mass parameter $M=\sqrt{7∕8}\pi$ yielding exact coupling between the modes $n=1$ and $k=4$. Dots correspond to $k_{\max }=10$ and squares to $k_{\max }=20$.

Figure 12

(Color online) Number of particles created in the modes $n=1$ and $k=4$ for $\omega =2{\Omega }_1^0$ and mass parameter $M=\sqrt{7∕8}\pi$ corresponding to the particle spectra depicted in Fig. 11.

Figure 13

(Color online) Particle spectra for $\omega =2{\Omega }_1^0$ and mass parameter $M=\sqrt{5}\pi$. Dots correspond to $k_{\max }=40$ and squares to $k_{\max }=50$.

Figure 14

(Color online) Number of particles created in the modes $n=1$, $k=7$, and $l=17$ for $\omega =2{\Omega }_1^0$ and mass parameter $M=\sqrt{5}\pi$ corresponding to the particle spectra depicted in Fig. 13.

Figure 15

(Color online) Number of particles created in the resonance mode $n=2$ as function of the mass parameter $M$. The solid line corresponds to the analytical prediction (33). Note that the first three values in the spectrum are not numerically stable due to an insufficient $k_{\max }$.

Figure 16

(Color online) Particle spectra for $\omega =2{\Omega }_2^0$ and mass parameters $M=\sqrt{13∕8}\pi$ and $M=\sqrt{7∕2}\pi$.

Figure 17

(Color online) Number of particles created in the modes $k=2$ and $7$ for $M=\sqrt{13∕8}\pi$ and $k=2$ and $8$ for $M=\sqrt{7∕2}\pi$, respectively, corresponding to the particle spectra shown in Fig. 16.

Figure 18

(Color online) The function $d_k\left(t\right)$ [Eq. (A11)] for $\omega =2{\Omega }_1^0$ and $M=\sqrt{2}\pi$ [panels (a) and (b)] and $M=0.4$ [panels (c) and (d)]. In any case $d_k\left(t\right)$ is shown for $k=1,\dots ,5$ (upper bands) and the last five values $k=k_{\max }-4,\dots ,k_{\max }$ (lower bands). With “err” we denote the preset values for the relative and absolute error used in the numerical simulations performed with, in these cases, the Runge-Kutta Prince-Dormand method.