Analytic pulse design for selective population transfer in many-level quantum systems: Maximizing the amplitude of population oscillations

State-selective preparation and manipulation of discrete-level quantum systems such as atoms, molecules, or quantum dots is the ultimate tool for many diverse fields such as laser control of chemical reactions, atom optics, high-precision metrology, and quantum computing. Rabi oscillations are one of the simplest, yet potentially quite useful mechanisms for achieving such manipulation. Rabi theory establishes that in the two-level systems resonant drive leads to the periodic and complete population oscillations between the two system levels. In this paper an analytic optimization algorithm for producing Rabi-like oscillations in the general discrete many-level quantum system is presented.

Figure 1

Pulse envelope profiles. Time $t$ is on the abscissa and $m\left(t\right)$ is on the ordinate. The dash-dotted line corresponds to a square pulse, the dashed line to a sine pulse, and the dotted line to a sine squared pulse. The pulse is switched on at $t=T_0$ and lasts until $t=T$. In all three cases, the maximum value of the perturbing field intensity field, $F_0$, achieved at time $T-T_0∕2$ is the same. In the case of the three-level system, it is such that ${\sigma }_{\beta p}^2=2.0$ while in case of the many-level system ${\sigma }_{\mathrm{tot}}^2=0.2$.

Figure 2

Optimized frequency plots for the three-level case. Time $t$ is on the abscissa, the perturbation frequency $\omega \left(t\right)$ on the ordinate. The total pulse duration is $7.25\mathrm{ns}$. Two straight solid lines indicate the two resonant frequencies of the system, ${\omega }_{\beta \alpha }$ (upper) and ${\omega }_{\beta p}$ (lower). The remaining three lines are optimized frequencies for the three types of pulse: the dash-dotted for a square pulse, the dashed line for a sine pulse, and the dotted for a sine squared pulse.

Figure 3

Comparison of the resonant and optimized population dynamics. Graphs on the left side present numerical solution to system dynamics for each of pulse types with a resonant perturbation applied, $\omega \left(t\right)={\omega }_{\beta \alpha }$. Graphs on the right-hand side present optimized dynamics. The top row corresponds to a square pulse, the middle to a sine pulse, and the bottom row to a sine square pulse. For the sake of clarity, only $\alpha$ (solid line) and $\beta$ (dashed line) populations are plotted while $p$ population is omitted. Although the optimization clearly does not produce clean two-level dynamics, the increase in the amplitude of population oscillation is nevertheless evident.

Figure 4

Optimized population dynamics of the two targeted levels in the many-level system. Time $t$ is on the abscissa, populations ${\mathrm{\Pi }}_{\alpha ,\beta }\left(t\right)$ are on the ordinate. The top graph corresponds to a square pulse, the middle to a sine pulse and the bottom one to a sine squared pulse. In all cases ${\sigma }_{\mathrm{tot}}^2=0.2$ and the total duration of the pulse is $4.84\mathrm{ns}$. The dotted line on each graph indicates the function $1-m\left(t\right){\sigma }_{\mathrm{tot}}^2$.