Deterministic strong-field quantum control

Strong-field quantum-state control is investigated, taking advantage of the full—amplitude and phase—characterization of the interaction between matter and intense ultrashort pulses via transient-absorption spectroscopy. As an example, we apply the method to a nondegenerate $V$-type three-level system modeling atomic Rb, and use a sequence of intense delayed pulses, whose parameters are tailored to steer the system into a desired quantum state. We show how to experimentally enable this optimization by retrieving all quantum features of the light-matter interaction from observable spectra. This provides a full characterization of the action of strong fields on the atomic system, including the dependence upon possibly unknown pulse properties and atomic structures. Precision and robustness of the scheme are tested, in the presence of surrounding atomic levels influencing the system's dynamics.

Figure 1

(a) Quantum-control scheme based on an optimal sequence of experimentally characterized pulses. (b) Interaction of a quantum system (red, continuous line) with an ultrashort pulse (gray, dashed line), effectively modeled as an instantaneous effect (orange, continuous line).

Figure 2

(a) Transient-absorption-spectroscopy setup used to experimentally reconstruct strong-field interaction operators. (b) Level scheme used to model Rb atoms, aiming at the control of the $V$-type three-level scheme in the box.

Figure 3

(a) Reconstructed SFI operator ${\widehat{U}}^{\mathrm{R}}\left(I\right)$ for $I=3.3\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2$, with bar heights (colors) exhibiting matrix-element amplitudes (phases). (b), (c) Two-pulse scheme implemented for $I_1=3.3\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2, I_2=3.6\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2, \overline{\tau }=198\mathrm{fs}$, and $\overline{\varphi }=1.88\mathrm{rad}$, with (b) a section of the control landscape as a function of $I_1$ and $I_2$, for fixed $\overline{\tau }$ and $\overline{\varphi }$ and (c) the corresponding evolution of the populations of state $|1\rangle$ (blue, continuous), $|2\rangle$ (purple, dashed), and $|3\rangle$ (yellow, dot-dashed), with a reached final state $|{\psi }_{\mathrm{r}}\rangle ={\sum }_{i=1}^3{c}_{\mathrm{r},i}|i\rangle$ featuring $|c_{\mathrm{r},2}{|}^2/{\left|c_{\mathrm{r},3}\right|}^2=2.07$ and $c_{\mathrm{r},1}=0.00$.

Figure 4

[(a),(c)] Reached ground-state population and [(b),(d)] ratio of the populations of the two excited states $|2\rangle$ and $|3\rangle$, averaged over the $N$ best sets of optimization pulse parameters $\{I_1,I_2,\overline{\tau },\overline{\varphi }\}$, as a function of $N$. Mean values are displayed as dashed lines; the surrounding regions (bounded by continuous lines) have an amplitude given by the corresponding standard deviation. Desired final population and ratios are exhibited by black, dotted lines. Reached final populations and ratios are calculated for [(a),(b)] a three-level-only system and [(c),(d)] a five-level system. The best optimization parameters are obtained via minimization of the cost function (25), calculated with SFI operators ${\widehat{U}}^{\mathrm{R}}\left(I\right)$ reconstructed from transient-absorption spectra numerically simulated for [(a),(b)] a three-level and [(c),(d)] a five-level model.

Figure 5

Time evolution of the populations ${\rho }_{11}\left(t\right)$ (blue, continuous), ${\rho }_{22}\left(t\right)$ (purple, dashed), and ${\rho }_{33}\left(t\right)$ (yellow, dotdashed), of a three-level system modeling Rb atoms excited by a two-pulse control sequence, based on the reconstruction of density-matrix strong-field interaction operators. (a) Optimal pulse parameters are determined via minimization of Eq. (43), $I_1=2.9\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2, I_2=3.2\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2, \overline{\tau }=150\mathrm{fs}$, and $\overline{\varphi }=1.13\mathrm{rad}$, with reached final populations ${\rho }_{\mathrm{r},11}=0.00, {\rho }_{\mathrm{r},22}=0.67$, and ${\rho }_{\mathrm{r},33}=0.33$. (b) The same pulse parameters are used as in Fig. 3, $I_1=3.3\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2, I_2=3.6\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2, \overline{\tau }=198\mathrm{fs}$, and $\overline{\varphi }=1.88\mathrm{rad}$, with reached final populations ${\rho }_{\mathrm{r},11}=0.01, {\rho }_{\mathrm{r},22}=0.67$, and ${\rho }_{\mathrm{r},33}=0.32$.