Adaptive numerical algorithms to simulate the dynamical Casimir effect in a closed cavity with different boundary conditions

We present an alternative numerical approach to compute the number of particles created inside a cavity due to time-dependent boundary conditions. The physical model consists of a rectangular cavity, where a wall always remains still while the other wall of the cavity presents a smooth movement in one direction. The method relies on the setting of the boundary conditions (Dirichlet and Neumann) and the following resolution of the corresponding equations of modes. By a further comparison between the ground state before and after the movement of the cavity wall, we finally compute the number of particles created. To demonstrate the method, we investigate the creation of particle production in vibrating cavities, confirming previously known results in the appropriate limits. Within this approach, the dynamical Casimir effect can be investigated, making it possible to study a variety of scenarios where no analytical results are known. Of special interest is, of course, the realistic case of the electromagnetic field in a three-dimensional cavity, with transverse electric (TE)–mode and transverse magnetic (TM)–mode photon production. Furthermore, with our approach we are able to calculate numerically the particle creation in a tuneable resonant superconducting cavity by the use of the generalized Robin boundary condition. We compare the numerical results with analytical predictions as well as a different numerical approach. Its extension to three dimensions is also straightforward.

Figure 1

(a) Left: Number of particles for the mode field 1, $N_1$, of a massless field inside a cavity with a moving wall under the perturbation $\mathrm{\Omega }=2{\omega }_1$ with DBC. Parameters used: $\epsilon =0.001,\mathrm{\Lambda }={\mathrm{\Lambda }}_m=25$. (b) Right: LogPlot behavior for the number of particles created $N_1$ coefficient for different values of the mass parameter $M$ under the perturbation of $\mathrm{\Omega }=2{\omega }_1$ under DBC, such as $M=1,M=5$, and $M=10$. Parameters used: $\epsilon =0.001,\mathrm{\Lambda }=10,\mathrm{\Lambda }=25$, and ${\mathrm{\Lambda }}_m=25$.

Figure 2

(a) Left: Total number of particles created inside a cavity with DBC and zero mass as function of the total number of field modes $\mathrm{\Lambda }$ considered at a fixed time. Parameters used: $\epsilon =0.01,\mathrm{\Omega }=2{\omega }_1$. (b) Right: Number of particles for the mode field 1, $N_1$, of a massless field inside a cavity with a moving wall under the perturbation $\mathrm{\Omega }=2{\omega }_1$ with DBC (red dashes) and generalized NBC (blue line). Parameters used: $\epsilon =0.01,M=0.5$.

Figure 3

Particle creation when the wall is excited as $\mathrm{\Omega }=2{\omega }_1$ for different fields in the cavity obeying DBC: (a) one-dimensional field $\left(M=0\right)$ and (b) a massive field $M=1$. In all cases, we show different values of $\epsilon$.

Figure 4

Difference of consecutive eigenfrequencies as a function of $V_0=b_{0}cos\left(f_0\right)$, for a fixed value of ${\chi }_0=0.05$ for generalised Robin boundary conditions. The constant line represents the difference of consecutive frequencies for all values of cavity for Dirichlet and generalized Neumann conditions.

Figure 5

(a) Left: Number of particles created $N_1$ under the perturbation of $\mathrm{\Omega }=2{\omega }_1$ for $V_0=20$, and a small perturbation amplitude $\epsilon =0.005$, so as to have $V_0\epsilon \sim 0.1$. For equidistant spectra, the coupling between an infinite number of modes generates a quadratic and linear growth in the number of particles, for short and long timescales respectively. (b) Right: LogPlot behavior of the number of particles created $N_k$ coefficient for a short temporal scale under a perturbation of $\mathrm{\Omega }=2{\omega }_1$ for $V_0=1,{\chi }_0=0.05$, and $\epsilon =0.001$. We show the behavior of the number of particles created for the first eigenstate of eigenfrequency ${\omega }_1=0.8495$ (red and blue dashed lines) and for the second eigenstate of ${\omega }_2=3.2819$ (black dotted line). The difference between the red and blue dashed lines is that we consider different values of ${\mathrm{\Lambda }}_m$. It is shown that $N_k$ grows exponentially for $N_1$ as expected since the spectrum is nonequidistant. Parameters used: $\mathrm{\Lambda }=10$.

Figure 6

(a) Absolute value of $B_1$ as function of time to show the role of the time $t_F$ for which the perturbation is on. It can be seen that the number of particles ${\mathcal{N}}_1={\left|B_1\right|}^2/\left(2{\omega }_1\right)$ is increased as the perturbation time is longer. Parameters used: $\mathrm{\Omega }=2{\omega }_1$, or $V_0=1,\xi =0.05$, and $\epsilon =0.001$. (b) Absolute value of $B_2$ as function of time to show the role of the time $t_F$, which is considerably smaller compared with $B_1$ for the same time.

Figure 7

We show the analytical prediction of Ref. [32] for the number of TE-mode photons created in mode $n$ with a solid red line. With a blue dashed line, we present the curve predicted by the numerical approach of Ruser and with a black dotted line, we show our numerical results. Parameters used: $\epsilon =0.001,M=0.2,L_x=1.0,\mathrm{\Lambda }=10$, and $n=1$.

Figure 8

Time consumed in calculations as a function of the number of modes $\mathrm{\Lambda }$ considered in the electromagnetic field.