Shaping of the time evolution of field-free molecular orientation by THz laser pulses

We present a theoretical study of the shaping of the time evolution of field-free orientation of linear molecules. We show the extent to which the degree of orientation can be steered along a desired periodic time-dependent signal. The objective of this study is not to optimize molecular orientation but to propose a general procedure to precisely control rotational dynamics through the first moment of the molecular axis distribution. Rectangular and triangular signals are taken as illustrative examples. At zero temperature we compute the quantum states leading to such field-free dynamics. A tera-Hertz laser pulse is designed to reach these states by using optimal control techniques. The investigation is extended to the case of nonzero temperature. Due to the complexity of the dynamics, the control protocol is derived with a Monte Carlo simulated annealing algorithm. A figure of merit based on the Fourier coefficients of the degree of orientation is used. We study the robustness of the control process against temperature effects and amplitude variations of the electric field.

Figure 1

Time evolution of the orientation of the OCS molecule at zero temperature for $j_{max}=20$. (a) and (c) The ideal rectangular and triangular signals, while (b) and (d) display the degree of orientation when the initial state is $|{\psi }_T\rangle$. The parameters $r$ and $A_0$ are, respectively, set to $r=1/\sqrt{2}$ and 1. The Gibbs-Wilbraham phenomenon has been corrected.

Figure 2

Time evolution of the sawtooth shaped orientation of the OCS molecule for different values of $j_{\max }$. The black and red (dark gray) solid lines represent the response without and with the correction to the Gibbs-Wilbraham phenomenon. The parameters $A_0$ and $r$ are set to 1 and $1/\sqrt{2}$.

Figure 3

(Top) Time evolution of the electric field and its normalized Fourier transform. (Bottom) Evolution of the figure of merit ${\mathcal{F}}_0$ as a function of the number of iterations and the corresponding degree of orientation $\langle cos\theta \rangle$ for the OCS molecule. Numerical parameters are set to $j_{\max }=9, A_0=1$, and $r=1/\sqrt{2}$. The electric field $E\left(t\right)$ is expressed in a.u., the other quantities are dimensionless.

Figure 4

Optimization results in the case of a sawtooth signal for the CO molecule at $T=10$ K. (a) The evolution of the figure of merit as a function of the number of iterations. (b) The corresponding optimal electrical field. The time evolution of $\langle cos\theta \rangle \left(t\right)$ is depicted in (c). (d) A comparison of the modulus of the Fourier coefficients of $\langle cos\theta \rangle \left(t\right)$ and of an ideal sawtooth. The electric field $E\left(t\right)$ and the Fourier coefficients are expressed in a.u., the other quantities are dimensionless.

Figure 5

Same as Fig. 4 but for $T=30$ K. In (b), the dashed red line is the fitted electric field (see the text for details). The electric field $E\left(t\right)$ and the Fourier coefficients are expressed in a.u., the other quantities are dimensionless.

Figure 6

Robustness of the time evolution of $\langle cos\theta \rangle$ for the CO molecule against temperature effects (a) and amplitude variations of the control field (b). The results in (a) were obtained with the fitted electric field $E\left(t\right)$ given by the expression (20) at three different temperatures: 15 K (in dashed red), 30 K (in solid black line), and 50 K (blue dots). The numerical parameters used for the field are displayed in Table 2 column (a), the maximum amplitude is set to $E_m=1.25\times {10}^{-4}$ a.u. The results in (b) have been obtained at 30 K with the same electric field but with a maximum amplitude reduced by factors 2 and 4 (in solid black and red lines, respectively) and increased by 25% (blue dots).

Figure 7

Optimization results in the case of a rectangular signal for the CO molecule at $T=10$ K. (a) The evolution of the figure of merit as a function of the number of iterations. (b) and (c) The optimal electrical field and the corresponding time evolution of the degree of orientation. A comparison between the modulus of the Fourier coefficients of $\langle cos\theta \rangle \left(t\right)$ and of an ideal rectangular signal is presented in (d). The electric field $E\left(t\right)$ and the Fourier coefficients are expressed in a.u., the other quantities are dimensionless.

Figure 8

Same as Fig. 7 but for $T=30$ K. In (b), the dashed red line is the fitted electric field (see the text for details). The electric field $E\left(t\right)$ and the Fourier coefficients are expressed in a.u., the other quantities are dimensionless.

Figure 9

Same as Fig. 6 but for the fitted electric field with the parameters given in column (b) of Table 2.

Figure 10

Time evolution of the degree of orientation (left) and of its time derivative (right) at $T=0$ K. The first rotational period corresponds to the application of the electric field, the three others to field-free dynamics. The time derivative function is expressed in arbitrary units.