The Dynamical Behaviors of the Two-Atom and the Dynamical Casimir Effect in a Non-Stationary Cavity

Abstract

We study the dynamical Casimir effect and the dynamical behaviors of the two-atom in a non-stationary cavity containing two two-level atoms. By solving the problem in a matrix method, we obtain an analytic solution. The results show that the larger of the atom-field coupling coefficients and the coupling coefficient of atoms, the fewer photons generated, but the probability of double excitation of the two-atom increases with the coupling coefficients. The squeezed coefficient enhances the generation rate of the created photons and the possibility of the atoms in the excited states.

Appendix

A note must be added. The parameters M₃₃, M₃₁, M₄₂, M₄₀, M₅₁, M₁₁, M₂₂, M₂₀, M₀₀, L₃₃, L₃₁, L₄₂, L₅₁, L₆₀, L₂₂, L₄₀, L₂₀ and L₁₁ in (19) can be expressed by

$$\begin{array}{@{}rcl@{}} M_{33}&=&|A_{14}|^{2}+|B_{14}|^{2}+|C_{14}|^{2}+|A_{44}|^{2}+|B_{44}|^{2}+|C_{44}|^{2}+B_{44}^{*}C_{44}+B_{44}C_{44}^{*}, \end{array} $$

(22)

$$\begin{array}{@{}rcl@{}} M_{31}&=& A_{14}^{*}B_{14}+ 5C_{14}B_{14}^{*}+A_{24}B_{24}^{*}+A_{34}B_{34}^{*}+A_{44}^{*}D_{44}+ 2A_{44}^{*}B_{44}+A_{44}^{*}C_{44}, \end{array} $$

(23)

$$\begin{array}{@{}rcl@{}} M_{42}&=& A_{14}^{*}B_{14}+C_{14}B_{14}^{*}+A_{44}^{*}B_{44}+A_{44}^{*}C_{44}, \end{array} $$

(24)

$$\begin{array}{@{}rcl@{}} M_{40}&=& 2A_{14}^{*}C_{14}, \end{array} $$

(25)

$$\begin{array}{@{}rcl@{}} M_{51}&=& A_{14}^{*}C_{14}, \end{array} $$

(26)

$$\begin{array}{@{}rcl@{}} M_{11}&=& |B_{14}|^{2}+ 14|C_{14}|^{2}+ 3|A_{24}|^{2}+ 3|A_{34}|^{2}+|D_{44}|^{2}+ 4|B_{44}|^{2}+|C_{44}|^{2}\\ &&+ 2D_{44}^{*}B_{44}+ 2D_{44}B_{44}^{*}+D_{44}^{*}C_{44}+D_{44}C_{44}^{*}+ 2C_{44}B_{44}^{*}+ 2C_{44}^{*}B_{44}, \end{array} $$

(27)

$$\begin{array}{@{}rcl@{}} M_{22}&=& 3|B_{14}|^{2}+ 8|C_{14}|^{2}+|A_{24}|^{2}+|A_{34}|^{2} + |B_{24}|^{2}+|B_{34}|^{2}+ 5|B_{44}|^{2}+ 3|C_{44}|^{2}\\ &&+ D_{44}^{*}B_{44}+D_{44}B_{44}^{*}+D_{44}^{*}C_{44}+D_{44}C_{44}^{*}+ 4C_{44}B_{44}^{*}+ 4C_{44}^{*}B_{44}, \end{array} $$

(28)

$$\begin{array}{@{}rcl@{}} M_{20}&=& 4B_{14}^{*}C_{14}+A_{24}B_{24}^{*}+A_{34}B_{34}^{*}, \end{array} $$

(29)

$$\begin{array}{@{}rcl@{}} M_{00}&=& 4|C_{14}|^{2}+|A_{24}|^{2}+|A_{34}|^{2} . \end{array} $$

(30)

$$\begin{array}{@{}rcl@{}} L_{33} &=& M_{33}\left[\cosh^{6}(-\beta t)+\sinh^{6}(-\beta t)+ 9\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)\right.\\ &&\left.+ 9\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)\right]\\ &&+ (M_{51}+M_{51}^{*})\left[10\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)+ 10\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)\right]\\ &&+ (M_{42}+M_{42}^{*})\left[-4\sinh(-\beta t)\cosh^{5}(-\beta t)-4\sinh^{5}(-\beta t)\cosh(-\beta t)\right.\\ &&\left.- 12\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)\right], \end{array} $$

(31)

$$\begin{array}{@{}rcl@{}} L_{31} &=& M_{33}\left[-3\cosh^{5}(-\beta t)\sinh(-\beta t)-36\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)\right.\\ &&\left.- 21\sinh^{5}(-\beta t)\cosh(-\beta t)\right] \\ &&+ M_{31}\left[\cosh^{4}(-\beta t)+ 3\sinh^{2}(-\beta t)\cosh^{2}(-\beta t)\right]\\ &&+ M_{31}^{*}\left[\sinh^{4}(-\beta t)+ 3\sinh^{2}(-\beta t)\cosh^{2}(-\beta t)\right]\\ &&- 4M_{40}\sinh(-\beta t)\cosh^{3}(-\beta t)-4M_{40}^{*}\sinh^{3}(-\beta t)\cosh(-\beta t)\\ &&- 2M_{22}[\sinh(-\beta t)\cosh^{3}(-\beta t)+\sinh^{3}(-\beta t)\cosh(-\beta t)]\\ &&- M_{51}[50\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)+ 10\sinh(-\beta t)\cosh^{5}(-\beta t)]\\ &&- M_{51}^{*}[30\sinh^{5}(-\beta t)\cosh(-\beta t)+ 30\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)]\\ &&+ M_{42}[24\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)+ 36\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)] \\ &&+ M_{42}^{*}[40\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)\\ &&+ 12\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)+ 8\sinh^{6}(-\beta t)], \end{array} $$

(32)

$$\begin{array}{@{}rcl@{}} L_{42} &=& M_{33}\left[-3\cosh^{5}(-\beta t)\sinh(-\beta t)-9\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)\right.\\ &&\left.- 3\sinh^{5}(-\beta t)\cosh(-\beta t)\right] \\ &&- M_{51}[10\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)+ 5\sinh(-\beta t)\cosh^{5}(-\beta t)]\\ &&- M_{51}^{*}[5\sinh^{5}(-\beta t)\cosh(-\beta t)+ 10\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)]\\ &&+ M_{42}[\cosh^{6}(-\beta t)+ 8\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)+ 6\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)] \\ &&+ M_{42}^{*}[6\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)\\ &&+ 8\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)+\sinh^{6}(-\beta t)], \end{array} $$

(33)

$$\begin{array}{@{}rcl@{}} L_{51} &=& M_{33}\left[3\cosh^{4}(-\beta t)\sinh^{2}(-\beta t)+ 3\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)\right] \\ &&+ M_{51}\left[\cosh^{6}(-\beta t)+ 5\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)\right]\\ &&+ M_{51}^{*}\left[\sinh^{6}(-\beta t)+ 5\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)\right]\\ &&- M_{42}[2\sinh(-\beta t)\cosh^{5}(-\beta t)+ 4\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)]\\ &&- M_{42}^{*}[2\sinh^{5}(-\beta t)\cosh(-\beta t)+ 4\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)], \end{array} $$

(34)

$$\begin{array}{@{}rcl@{}} L_{60} &=& - M_{51}\sinh(-\beta t)\cosh^{5}(-\beta t)-M_{51}^{*}\sinh^{5}(-\beta t)\cosh(-\beta t)\\ &&+ M_{42}\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)+M_{42}^{*}\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)], \end{array} $$

(35)

$$\begin{array}{@{}rcl@{}} L_{22} &=& M_{33}\left[18\cosh^{4}(-\beta t)\sinh^{2}(-\beta t)+ 54\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)+ 9\sinh^{6}(-\beta t)\right] \\ &&- 3(M_{31}+M_{31}^{*})\left[\cosh^{3}(-\beta t)\sinh(-\beta t)+\sinh^{3}(-\beta t)\cosh(-\beta t)\right]\\ &&+ 6(M_{40}+M_{40}^{*})\cosh^{2}(-\beta t)\sinh^{2}(-\beta t)\\ &&+ M_{22}[\cosh^{4}(-\beta t)+\sinh^{4}(-\beta t)+ 4\sinh^{2}(-\beta t)\cosh^{2}(-\beta t)]\\ &&+ (M_{51}+M_{51}^{*})[60\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)+ 30\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)]\\ &&+ (M_{42}+M_{42}^{*})\left[-54\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)-30\sinh^{5}(-\beta t)\cosh(-\beta t)\right.\\ &&\left.- 6\sinh(-\beta t)\cosh^{5}(-\beta t)\right] , \end{array} $$

(36)

$$\begin{array}{@{}rcl@{}} L_{40} &=& M_{33}\left[3\cosh^{4}(-\beta t)\sinh^{2}(-\beta t)+ 12\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)\right] \\ &&- M_{31}\cosh^{3}(-\beta t)\sinh(-\beta t)-M_{31}^{*}\cosh(-\beta t)\sinh^{3}(-\beta t)\\ &&+ M_{40}\cosh^{4}(-\beta t)+M_{40}^{*}\sinh^{4}(-\beta t)+M_{22}\cosh^{2}(-\beta t)\sinh^{2}(-\beta t)\\ &&+ 15M_{51}\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)\\ &&+ M_{51}^{*}[5\sinh^{6}(-\beta t)+ 10\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)]\\ &&- M_{42}[\sinh(-\beta t)\cosh^{5}(-\beta t)+ 14\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)] \\ &&- M_{42}^{*}[9\sinh^{5}(-\beta t)\cosh(-\beta t)+ 6\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)], \end{array} $$

(37)

$$\begin{array}{@{}rcl@{}} L_{20} &=& M_{33}\left[-18\cosh^{3}(-\beta t)\sinh^{3}(-\beta t)-27\sinh^{5}(-\beta t)\cosh(-\beta t)\right] \\ &&+ 6M_{31}\sinh^{2}(-\beta t)\cosh^{2}(-\beta t) + M_{31}^{*}\left[3\sinh^{4}(-\beta t) + 3\sinh^{2}(-\beta t)\cosh^{2}(-\beta t)\right]\\ &&+ M_{20}\cosh^{2}(-\beta t)+M_{20}^{*}\sinh^{2}(-\beta t)\\ &&- 6M_{40}\sinh(-\beta t)\cosh^{3}(-\beta t)-6M_{40}^{*}\sinh^{3}(-\beta t)\cosh(-\beta t)\\ &&- M_{22}[\sinh(-\beta t)\cosh^{3}(-\beta t)+ 5\sinh^{3}(-\beta t)\cosh(-\beta t)]\\ &&- M_{11}\sinh(-\beta t)\cosh(-\beta t)\\ &&- 45M_{51}\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)\\ &&- 15M_{51}^{*}[2\sinh^{5}(-\beta t)\cosh(-\beta t)+\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)]\\ &&+ M_{42}[39\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)+ 6\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)] \\ &&+ M_{42}^{*}\left[30\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)\right.\\ &&\left.+ 3\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)+ 12\sinh^{6}(-\beta t)\right], \end{array} $$

(38)

$$\begin{array}{@{}rcl@{}} L_{11} &=& M_{33}\left[72\cosh^{2}(-\beta t)\sinh^{4}(-\beta t)+ 18\sinh^{6}(-\beta t)+ 9\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)\right] \\ &&- 3(M_{31}+M_{31}^{*})\left[3\cosh(-\beta t)\sinh^{3}(-\beta t)+\sinh(-\beta t)\cosh^{3}(-\beta t)\right]\\ &&- 2(M_{20}+M_{20}^{*})\sinh(-\beta t)\cosh(-\beta t)+ 12(M_{40}+M_{40}^{*})\cosh^{2}(-\beta t)\sinh^{2}(-\beta t)\\ &&+ M_{22}\left[4\sinh^{4}(-\beta t)+ 8\sinh^{2}(-\beta t)\cosh^{2}(-\beta t)\right]\\ &&+ M_{11}\left[\cosh^{2}(-\beta t)+\sinh^{2}(-\beta t)\right]\\ &&+ (M_{51}+M_{51}^{*})[75\sinh^{4}(-\beta t)\cosh^{2}(-\beta t)+ 15\sinh^{2}(-\beta t)\cosh^{4}(-\beta t)]\\ &&- (M_{42}+M_{42}^{*})[42\sinh^{3}(-\beta t)\cosh^{3}(-\beta t)+ 48\sinh^{5}(-\beta t)\cosh(-\beta t)]. \end{array} $$

(39)