Analytic results for the population dynamics of a driven dipolar molecular system

We theoretically investigate the dipolar molecular system driven by monochromatic periodic, linear, parabolic, and ${\mathrm{sech}}^2$ forms external fields, respectively. The two-level Hamiltonian model with nonzero diagonal dipole matrix elements is adopted to describe the population dynamics of the driven dipolar molecule, and the corresponding exact solutions are presented in terms of the confluent Heun equations without the generalized rotating-wave approximation. The analytic solutions derived here are valid in the whole parameter space.

Figure 1

The instantaneous eigenvalues with respect to time for different cases. The solid and dashed lines correspond to the cases for the dipolar molecule ($\mu =1.0$) and the normal atomic media ($\mu =0.0$), respectively. In our numerical calculation, other parameters are chosen as ${\omega }_0=\omega =1.0$, ${\mathrm{\Omega }}_0=1.4$ and $\mathrm{\Omega }\left(t\right)={\mathrm{\Omega }}_{0}cos\left(\omega t\right)$.

Figure 2

The evolution of the population $|C_i{|}^2$ for the dipolar molecule driven by a periodic field: (a) exact solutions; (b) approximate solutions. The solid and dash lines correspond to $|C_1{|}^2$ and $|C_2{|}^2$, respectively. In our numerical calculation, the parameters are chosen as the same to ones in Fig. 1, except for $\mathrm{\Omega }\left(t\right)={\mathrm{\Omega }}_{0}cos\left[\omega (t-t_0)\right]$ with $t_0=10$. At the initial time ($t=0$), only the ground state is populated $[c_1=1$ in Eq. (1)].

Figure 3

The evolution of the population $|C_i{|}^2$ for the dipolar molecule driven by a linear field. The solid and dash lines correspond to $|C_1{|}^2$ and $|C_2{|}^2$, respectively. $t_c$ denotes the critical time defined in the former. Here the parameters are chosen as the same to ones in Fig. 1, except for $\mathrm{\Omega }\left(t\right)={\mathrm{\Omega }}_0(t-t_0)$ with $t_0=10$.

Figure 4

The evolution of the population $|C_i{|}^2$ for the dipolar molecule driven by a quadratic nonlinear field. The solid and dashed lines correspond to $|C_1{|}^2$ and $|C_2{|}^2$, respectively. Here the parameters are chosen as the same to ones in Fig. 1, except for $\mathrm{\Omega }\left(t\right)={\mathrm{\Omega }}_0{(t-t_0)}^2$ with $t_0=10$.