Control of mixed-state quantum systems by a train of short pulses

A density matrix approach is developed for the control of a mixed-state quantum system using a time-dependent external field such as a train of pulses. This leads to the definition of a target density matrix constructed in a reduced Hilbert space as a specific combination of the eigenvectors of a given observable through weighting factors related to the initial statistics of the system. A train of pulses is considered as a possible strategy to reach this target. An illustration is given by considering the laser control of molecular alignment and orientation in thermal equilibrium.

Figure 1

Maximum alignment efficiency as a function of $j_{\mathit{max}}$ (see text) for the molecule LiCl in the case $T=10$ (crosses) and $5\mathrm{K}$ (open circles). Solid and dashed lines, which are just to guide the eye, correspond, respectively, to the optimal maximum and the one for a linear polarization.

Figure 2

Same as Fig. 1, but for orientation.

Figure 3

Maximal alignment efficiencies that can be obtained with a linear polarization (crosses) and associated duration (open circles) as a function of $j_{\mathit{max}}$ (see text) for the molecule LiCl in the case $T=5$ (dashed lines) and $10\mathrm{K}$ (solid lines).

Figure 4

Same as Fig. 3, but for orientation.

Figure 5

Alignment dynamics during and after the train of pulses determined by strategy $S1$, with pulses acting at the global maxima of $\mathrm{Tr}\left[\rho \left(t\right){\cos }^2\theta \right]$, (a) for $\mathrm{Tr}\left[\rho \left(t\right){\cos }^2\theta \right]$ and (b) for $\mathrm{Tr}\left[{\rho }_{\mathit{lin}}^{\left(8\right)}\rho \left(t\right)\right]$. The solid line corresponds to the exactly propagated density operator $\rho \left(t\right)$ and the dashed line to the propagation of $\rho \left(t\right)$ in the subspace ${\mathcal{H}}^{\left(8\right)}$. The train of pulses is displayed on (b); the optimal alignment and the one for a linear polarization in ${\mathcal{H}}^{\left(8\right)}$ are, respectively, indicated by the horizontal solid and dashed lines on (a).

Figure 6

Same as Fig. 5, but for the orientation dynamics.

Figure 7

Same as Fig. 6, but referring to the strategy $S2$, with pulses acting at the maxima of $\mathrm{Tr}\left[{\rho }_{\mathit{lin}}^{\left(8\right)}\rho \left(t\right)\right]$.

Figure 8

Postpulse alignment (a) and orientation (b) dynamics of the molecule LiCl after interaction with a train of short pulses (the time $t=0$ corresponds to the first kick). Solid, dashed, and dash-dotted lines correspond, respectively, to the averages calculated with the exact density operator, the optimal one ${\rho }_{\mathit{opt}}^{\left(8\right)}$, and the one obtained with a linear polarization ${\rho }_{\mathit{lin}}^{\left(8\right)}$. In these two cases, the strategy $S1$ is used.