Coherent control of atomic excitation using off-resonant strong few-cycle pulses

We study the dynamics of a two-level system driven by an off-resonance few-cycle pulse which has a phase jump $\varphi$ at $t=t_0$, in contrast to many-cycle pulses, under the nonrotating-wave approximation (NRWA). We give a closed form analytical solution for the evolution of the probability amplitude $|C_a(t)|$ for the upper level. Using the appropriate pulse parameters like the phase jump $\varphi$, jump time $t_0$, pulse width $\tau$, frequency $\nu$, and Rabi frequency ${\Omega }_0$ the population transfer after the pulse is gone can be optimized and, for the pulse considered here, an enhancement factor of ${10}^6–{10}^8$ was obtained.

Figure 1

(a) Two-level atomic system, atomic transition frequency $\omega ={\omega }_a-{\omega }_b$, detuning $\Delta =\omega -\nu$, and Rabi frequency $\Omega (t)=\wp \mathcal{E}(t)/\hslash$. (b) Few-cycle sine (dashed line) and cosine (solid line) pulse with Gaussian envelope.

Figure 2

Population left in the upper level $|a\rangle$ after applying an $\Omega (t)={\Omega }_0[\text{sech}(\alpha t)+\text{sech}(\alpha t-3)]$ pulse as a function of the frequency $\nu /\omega$ obtained by numerical solution of Eqs. (1) (dots) and using our approximate analytical result Eq. (6) (solid line). In calculations, we take ${\Omega }_0=0.04\omega$ and $\alpha =0.075\omega$. In (a) $\varphi =0$, whereas in (b) $\varphi =\pi$ and $t_0=0$.

Figure 3

Effect of jump time $t_0$. (a) Probability amplitude $|C_a(\infty )|$ as a function of the phase jump $\varphi$. The phase jump is introduced at the peak of the Gaussian envelope. (b) The symmetric influence on the degree of excitation with respect to the position of $t_0$. The symmetric response is lost for shifted Gaussian input pulses (c) and (d). For numerical calculations, we chose ${\Omega }_0=0.875\omega$, $\nu =0.75\omega$, $\alpha =0.331\omega$, and $\gamma =1.25\omega$.

Figure 4

Effect of $\alpha$. (a) Probability amplitude $|C_a(\infty )|$ varies in the range from ${10}^{-5}~0.7$. (b) Plot of $|C_a(\infty )|$ against normalized pulse width $\gamma \tau$ for fixed $\omega ,\hspace{0.5em}\nu$, and ${\Omega }_0$, and three combinations of the phase jump: $\varphi =0,\hspace{0.5em}\pi /2$, and $\pi$. (c) The temporal evolution for the three combinations used in (b). For numerical simulation we chose ${\Omega }_0=0.875\omega$, $\nu =0.75\omega$, $\gamma =1.25\omega$, and $\alpha =0.331\omega$.

Figure 5

(a) Temporal behavior of $\left|C_a\right|$ for different combinations of $\varphi$. (b) Plot of $|C_a(\infty )|$ against $\nu /\omega$. For numerical simulation we chose ${\Omega }_0=0.875\omega$, $t_0=0,\gamma =1.25\omega$, and $\alpha =0.331\omega$.

Figure 6

(a) Temporal behavior of $\left|C_a\right|$ for difference combination of $\varphi$. (b) Plot of $|C_a(\infty )|$ against ${\Omega }_0$. For numerical simulation in (b), we chose a shifted Gaussian pulse with $t_s=1$, $\nu =0.75\omega$, $t_0=0,\gamma =1.25\omega$, and $\alpha =0.331\omega$. ${\Omega }_0=0.875\omega$ for (a).

Figure 7

Contour plot showing the effect of pulse parameters such as $t_0,\hspace{0.5em}\tau ,\hspace{0.5em}\nu$, and ${\Omega }_0$ on the population left in the excited states $|a\rangle$ in (a), (b), (c), and (d), respectively. The influence of the phase jump time $t_0$ is symmetric with respect to $t_0$, as shown in (a). The parameters used are ${\Omega }_0=0.875\omega$, $\nu =0.75\omega$, $\gamma =1.25\omega ,t_0=0$, and $\alpha =0.331\omega$ as required appropriately. For (c), we used $\alpha =0.110\omega$.