Transition from weak- to strong-field coherent control

We trace out the transition from weak- to strong-field coherent control, focusing on multiphoton population transfer in atomic sodium. We show how dynamic Stark shifts and ground-state depletion mark the departure from perturbative to strong-field excitation, and how a time-domain picture yields insight into the dynamics when a static resonance, frequency-domain approach breaks down. Experimental measurements are interpreted with the help of calculations that numerically integrate the Schrödinger equation.

Figure 1

(Color online) (a) Experimental setup. (b) Na energy levels.

Figure 2

$3p$-$4s$ signal as function of energy for a spatially filtered beam and a Gaussian beam. The presence of Rabi oscillations is detected only when using the filtering system. Note that the signals for both the unfiltered and filtered laser beams are each normalized to themselves. Inset: filtered (solid line) and not filtered (dashed line) spatial profiles measured at the interaction region.

Figure 3

(a) Ratio of $4s$ population for a GA maximized pulse to an unshaped one vs intensity. (b) Same as (a) but for the $7p$ state. $I_0$ is the peak intensity required for a $\pi$ pulse on the $3s\text{−}4s$ transition. Center wavelength is ${\lambda }_0=780\mathrm{nm}$. TL denotes transform limited (unshaped).

Figure 4

(a) Experimental measurement of the $4s$ population as function of pulse energy (equivalent to peak intensity) and $\pi$ flip position. (b) Lineouts of (a) for three different pulse energies. (c) Simulation of the $4s$ population as function of peak intensity and $\pi$ flip position. (d) Lineouts of (c) for different intensities. The solid, dashed, and dotted lines are for $I=0.01I_0$, $0.5I_0$, and $I_0$, respectively. The dark regions correspond to higher population transfer.

Figure 5

(a) Calculation of the population transfer using the left-hand side of Eq. (16) as function of peak intensity (for an unshaped pulse) and $\pi$ flip position. (b), (c), (d) Evolution of $\alpha \left(t\right)$ (left axis, solid line) and $I\left(t\right)$ (right axis, dashed line) in time for different $\pi$ phase flip positions (see text). All three are for the same pulse energy, corresponding to the minimum energy required for a $\pi$ pulse, as defined by Eq. (16) with $\alpha \left(t\right)=0$. The straight dotted lines mark the change of $\alpha \left(t\right)$ between the two maxima of $I\left(t\right)$. In (a) the dark regions correspond to higher population transfer.

Figure 6

(a) Measurement of the $7p$ population for three different energies as function of the $\pi$ flip position. (b) Simulation of the $7p$ population for equivalent peak intensities as function of the $\pi$ flip position. The enhancement obtained at weak fields at ${\lambda }_0=781.2\mathrm{nm}$ washes out for high fields. In all graphs, the solid line is for low intensities $I\approx 0.1I_0$, dashed line is for intermediate energies $I\approx 0.4I_0$, and dotted line is for $I\approx I_0$.