Numerical approach to simulating interference phenomena in a cavity with two oscillating mirrors

We study photon creation in a cavity with two perfectly conducting moving mirrors. We derive the dynamic equations of the modes and study different situations concerning various movements of the walls, such as translational or breathing modes. We can even apply our approach to one- or three-dimensional cavities and reobtain well-known results of cavities with one moving mirror. We compare the numerical results with analytical predictions and discuss the effects of the intermode coupling in detail as well as the nonperturbative regime. We also study the time evolution of the energy density and provide analytic justifications for the different results found numerically.

Figure 1

(a) Behavior of the $|B_1{|}^2$ coefficient as a function of the dimensionless time for different values of $M$ under the perturbation of $\mathrm{\Omega }=2{\omega }_1$. The gray solid (upper) curve represents the analytical solution for the number of particles created when $M=10$, while the blue solid (lower) curve is the numerical solution obtained for the right-moving mirror. (b) Behavior of the $|B_1{|}^2$ coefficient as a function of the dimensionless time for smaller values of $M$, i.e., $M=0.01, M=0.05$, and $M=1$. Parameters used are $\epsilon =0.001, \mathrm{\Lambda }=10$.

Figure 2

In both plots, we can see the behavior of the $|B_1{|}^2$ coefficient of the mode ${\omega }_1$ of the field as a function of the dimensionless time. For times smaller than $1/\epsilon$, the behavior can be fitted by a quadratic curve, while for times $\mathrm{\Omega }t\sim 1/\epsilon$ it can be fitted with a linear-in-time curve. For very long times, we find an exponential behavior, as shown in Ref. [31]. (a) We use ${\epsilon }_R=0.01$ and. (b) We set ${\epsilon }_R=0.001$. Parameters used are $M=0, \mathrm{\Lambda }=10$.

Figure 3

Energy density for a moving wall located at $x=R\left(t\right)$ as a function of the dimensionless time. The density of energy grows quadratically with time in the one-dimensional case of one moving mirror. Parameters are $\epsilon =0.001, \mathrm{\Lambda }=10$. In our units, energy is measured in units of $1/L_0$.

Figure 4

In this plot, we have used $\epsilon =0.01, \mathrm{\Lambda }=10$ for the blue solid curves, $\mathrm{\Lambda }=50$ for the red dashed curves, and $\mathrm{\Lambda }=100$ for the black dotted ones. In the upper values of the ${\left|B\right|}^2$ axis we plot $|B_1{|}^2$ for the $n=1$ mode of the field, while in the lower part of the axis we show $|B_2{|}^2$ of the field as a function of the dimensionless time. We need to use a large number of modes $\mathrm{\Lambda }$ in order to get total convergence into the final values. This plots show the dependence on $\mathrm{\Lambda }$ and a tendency to exponential growth of created particles.

Figure 5

(a) ${\left|B\right|}^2$ coefficient as a function of the dimensionless time for different field modes: $|B_3{|}^2$ (red dotted line), $|B_5{|}^2$ (blue dashed line), and $|B_7{|}^2$ (blue solid line) under ${\mathrm{\Omega }}_R={\mathrm{\Omega }}_L={\omega }_5$ and ${\varphi }_R=\pi$. (b) ${\left|B\right|}^2$ coefficient for different field modes: $|B_3{|}^2, |B_4{|}^2$, and $|B_5{|}^2$ for ${\mathrm{\Omega }}_R={\mathrm{\Omega }}_L={\omega }_4$ and ${\varphi }_R=0$. Parameters used are ${\varphi }_L=0, \epsilon =0.01, \mathrm{\Lambda }=10$.

Figure 6

We present the behavior of an odd field mode $|B_3{|}^2$ as a function of the dimensionless time for different values of ${\varphi }_R$ under a perturbation $\mathrm{\Omega }={\omega }_5$. In the case of ${\varphi }_L={\varphi }_R=0$ (dotted gray line), we see there is another slope (solid red line) which corresponds to a dephasing of ${\varphi }_L={\varphi }_R=\pi /4$. Both cases are examples of a translational oscillation mode, a case which is considered separately. With the blue dashed line we present ${\varphi }_L=0$ and ${\varphi }_R=\pi /4$, and we show ${\varphi }_L=0$ and ${\varphi }_R=0.35\pi$ with the dot-dashed black line. Both examples show exponential growth with time of the number of particles created with excitation $\mathrm{\Omega }=w_5$ for mode $n=3$. Parameters used are $\epsilon =0.01, \mathrm{\Lambda }=10$.

Figure 7

(a) Solid lines are for $|B_8{|}^2$ and $|B_5{|}^2$; the others lines correspond to $|B_3{|}^2, |B_2{|}^2$, and $|B_1{|}^2$. Parameters used are $\epsilon =0.01, \mathrm{\Lambda }=10, M=10$. (b) All modes are parametrically excited for a bigger value of the mass. Parameters used are $\epsilon =0.001, \mathrm{\Lambda }=10, M=50$. Excitation frequency $\mathrm{\Omega }={\omega }_2+{\omega }_3$.

Figure 8

(a) For the breathing modes, we have the ${\left|B\right|}^2$ coefficient versus time for different field modes $n=2$ ($|B_2{|}^2$), $n=4$ ($|B_4{|}^2$), and $n=6$ ($|B_6{|}^2$) under ${\mathrm{\Omega }}_R={\mathrm{\Omega }}_L={\omega }_4$ and ${\varphi }_L=\pi$. Field modes $n=1$ ($|B_1{|}^2$) and $n=3$ ($|B_3{|}^2$) seem to get excited at longer times. (b) Energy density as a function of the dimensionless time. Parameters used are $\epsilon =0.01, \mathrm{\Lambda }=10$. In our units, energy is measured in units of $1/L_0$.

Figure 9

(a) $B_1$ coefficient for field mode 1 under ${\mathrm{\Omega }}_R={\mathrm{\Omega }}_L={\omega }_5$ and ${\varphi }_L=0$ for different values of the mass: $M=0.01$ (red dotted line), $M=1$ (blue dashed line), and $M=5$ (black solid curve). (b) Energy as a function of time. Parameters used are $\epsilon =0.01, \mathrm{\Lambda }=10$. In our units, energy is measured in units of $1/L_0$.