Analytical solutions for a two-level system driven by a class of chirped pulses

We present analytical solutions for the problem of a two-level atom driven by a class of chirped pulses. The solutions are given in terms of Heun functions. By use of the appropriate chirping parameters, an enhancement of four orders of magnitude in the population transfer is obtained.

Figure 1

(a) Two-level atomic system, atomic transition frequency $\omega ={\omega }_a-{\omega }_b$, detuning $\Delta =\omega -\nu$, and Rabi frequency $\Omega (t)=\wp \mathcal{E}(t)/2\hslash$. (b) Classical electromagnetic field $E(t)=\text{exp}(-{\alpha }^2{t}^2)\text{cos}(\nu t+t^2)$.

Figure 2

(I) Heun equation case: Chirping function $\varphi (\tau )$ given by Eq. (15) for $\xi =10$ and (a) $\delta >1$ and (b) $\delta \approx$ 1. (c) Electric field $E(\tau )$ for varying $\xi$ and $\delta =1.01$. (d) Probability amplitudes for the upper level $|a\rangle$ for the corresponding fields in (c). $|C_a(\tau )|$ is given by Eq. (18).

Figure 3

(II) Confluent Heun equation case: (a) Profile of the electric field $E(\tau )$ for varying $\zeta$. (b) Probability amplitudes for the upper level $|a\rangle$ for the corresponding fields in (a). The pulse envelope $\Omega (\tau )$ and are given by Eq. (27).

Figure 4

Effect of chirping on the population left in the excited states $|a\rangle$. (a) Heun equation and (b) confluent Heun equation. The inset shows the dip (minima) in the population left for the Heun case. For the calculations, $\beta =2.5,\gamma =0.25,$ and $\delta =1.01.$ (c) Chirping function for the confluent Heun case, $\mathrm{\varphi ̇}(\tau ).$

Figure 5

Contour plot showing the effect of chirping and the detuning on the population left in the excited states $|a\rangle$. (a) and (b) correspond to the Heun and (c) and (d) to the confluent Heun case. For calculations, (a) $\gamma =0.50,\hspace{0.5em}\delta =1.01,$ (b) $\gamma =1,\hspace{0.5em}\delta =1.01,$ (c) $\gamma =0.5$, and (d) $\gamma =1.$