Time dynamics of a multilevel system excited by a train of ultrashort pulses

We present an iterative analytic solution for the time dynamics of the ${}^{87}\text{R}\text{b}$ four-level atoms excited by a train of ultrashort pulses. It enables fast and accurate calculation of the system time evolution circumventing the use of time-consuming numerical procedures. The obtained solution is used to study the complex interaction of atoms with the pulse-train electric field, both in time and frequency domain. We investigate the influence of pulse-train physical parameters, such as the pulse peak power, the repetition frequency and the central laser wavelength on the accumulation of population and coherence in the system characterized by relaxation times larger than the pulse repetition period.

Figure 1

${}^{87}\text{R}\text{b}$ $D_1$ transitions hyperfine structure with frequency splittings indicated in MHz. Ground state $5S_{1/2}$ value is taken from [14] and $5P_{1/2}$ excited-state value is taken from [15].

Figure 2

(Color online) Comparison of the numerical (thin black line) and iterative analytical (points, thick red line) solution of Eqs. (1). Time evolutions of the fractional excited-state populations of ${}^{87}\text{R}\text{b}$ hyperfine levels ${\rho }_{33}\left(5P_{1/2},F=1\right)$ and ${\rho }_{44}\left(5P_{1/2},F=2\right)$ are shown, with the analytical solution evaluated using Eqs. (6, 7, 8, 9, 10, 11) at time intervals equal to the pulse repetition period $T_R$.

Figure 3

(Color online) ${}^{87}\text{R}\text{b}5S_{1/2}$ and $5P_{1/2}$ fractional hyperfine level populations as a function of atomic velocity, induced by the frequency comb excitation. Equilibrium ground-state population distributions (Doppler profiles) are indicated by the dashed lines.

Figure 4

(Color online) ${}^{87}\text{R}\text{b}{\rho }_{33}\left(5P_{1/2},F=1\right)$ population distribution as a function of the number of excitation pulses.

Figure 5

(Color online) Time (a) and frequency (b) domain results for the ${}^{87}\text{R}\text{b}{\rho }_{33}\left(5P_{1/2},F=1\right)$ population as a function of the pulse peak amplitude ${\mathcal{E}}_0$. Results for three different ${\mathcal{E}}_0$ are shown with the values indicated in ${10}^6\text{V}/\text{m}$.

Figure 6

(Color online) Time dynamics of the ${}^{87}\text{R}\text{b}{\rho }_{33}\left(5P_{1/2},F=1\right)$ hyperfine level population as a function of excitation pulse repetition frequency $f_{\text{rep}}$ ($f_{\text{rep}}$ values are indicated in MHz) (a); and the corresponding ${\rho }_{22}\left(5S_{1/2},F=2\right)$ and ${\rho }_{33}\left(5P_{1/2},F=1\right)$ population distributions (b). ${\mathcal{E}}_0$ used in the calculations for different $f_{\text{rep}}$ have been scaled to correspond to the same fs laser time averaged power. Both analytic and numerical solutions for the time dynamics are shown for comparison.

Figure 7

(Color online) Comparison of the results for ${}^{87}\text{R}\text{b}{\rho }_{33}\left(5P_{1/2},F=1\right)$ hyperfine level population time dynamics obtained by numerical calculation (${\omega }_L={\omega }_{23},{\Phi }_R=\pi$, thin black line) and analytic iterative solution (${\omega }_L^{\prime }={\omega }_{23}+\pi /T_R$, points and thick red line) (a); and the effect of the frequency shift of the comb spectrum on the ${\rho }_{33}$ population distribution: black dashed line ${\omega }_L={\omega }_{23},{\Phi }_R=0\left({\omega }_L^{\prime }={\omega }_{23}\right)$, blue solid line ${\omega }_L={\omega }_{23},{\Phi }_R=\pi$ $\left({\omega }_L^{\prime }={\omega }_{23}+\pi /T_R\right)$ (b).