Robust two-level system control by a detuned and chirped laser pulse

We propose and demonstrate a robust control scheme by an ultrafast nonadiabatic chirped laser pulse, designed for targeting coherent superpositions of two-level systems. Robustness against power fluctuation is proved by our numerical study and a proof-of-principle experiment performed with femtosecond laser interaction on cold atoms. They exhibit for the final driven dynamics a cusp on the Bloch sphere, corresponding to a zero curvature of fidelity. This solution is particularly simple and thus applicable to a wide range of potential applications.

Figure 1

Robustness map $g(\mathrm{\Theta },c_2^{\prime })$ for ${\mathrm{\Delta }}^{\prime }=0.637$. The negative curvature $g$ is plotted as a function of the pulse area $\mathrm{\Theta }$ and the chirp rate $c_2^{\prime }$. The most robust point is located at B, while $g_{\mathbf{A}}=0.11,g_{\mathbf{C}}=0.05$, and $g_{\mathrm{Rabi}}=0.62$. (b) Fidelity curves $\mathcal{F}\left(\gamma \right)$: The fidelity curve is calculated at B as a function of the fluctuation $\gamma$ (see text for definition) and compared with cases A and C and also with the Rabi-type evolution for the same excited-state probability $P_e=0.5$.

Figure 2

$\mathrm{\Theta }$-parametrized trajectories on Bloch sphere. The final state after a chirped- and detuned-pulse interaction is plotted with a Bloch vector. The trajectories through A, B, and C in Fig. 1 are plotted.

Figure 3

Probability $P_e(\mathrm{\Theta },c_2)$ of the excited state ($5P_{1/2}$ of rubidium) after the detuned and chirped Gaussian pulse excitation as a function of the pulse area $\mathrm{\Theta }$ and the chirp rate $c_2$, while the detuning and laser bandwidth are fixed at $\delta =3.5$ nm and $\mathrm{\Delta }{\omega }_{\mathrm{FWHM}}=3.1\times {10}^{13}$ rad/s, respectively. (a) Experimentally obtained probability map $P_e^{\mathrm{expt}}(\mathrm{\Theta },c_2)$, to which the measured atom population was converted using Rabi-oscillation calibration measurements (see text for detail). (b) Theoretical result $P_e^{\mathrm{TDSE}}(\mathrm{\Theta },c_2)$, obtained using the TDSE calculation for an atom cloud of a Gaussian profile with a diameter 47% of the Gaussian laser beam diameter. (c) The behavior of the population $P_e\left(\mathrm{\Theta }\right)$ for selected chirp rates $c_2=4000$, $8000$, and $16000{\mathrm{fs}}^2$ shows good agreement with the TDSE calculation (solid lines). The star and arrow symbols represent the robust condition.

Figure 4

Robust control conditions in the parameter space (${\mathrm{\Delta }}^{\prime },c_2^{\prime },\mathrm{\Theta }$): Red circles represent the parameters for detuned and chirped pulses that induce robust control to various target probabilities; the solid and dashed lines are the numerical fit and its projections to each plane, respectively. The inset shows the $\mathrm{\Theta }$ trajectories on the Bloch sphere, corresponding to the robust control red circles, each featuring a cusp.