Optimal-control approach to field-free three-dimensional alignment of polyatomic molecules

We introduce an optimal control approach that is capable of yielding the best achievable field-free three-dimensional (3-D) alignment in an arbitrary molecule. Our analysis provides the optimal route to the best aligned state along with interesting insights into the 3-D alignment dynamics of polyatomic systems.

Figure 1

Model rotational constants and polarizabilities (in reduced units) as functions of $\kappa$.

Figure 2

Time evolution of $\langle \cos {}^2\theta \rangle {}_T$ (reds, $\langle \cos {}^2\chi \rangle {}_T$ (green lines), and $\langle \cos {}^2\varphi \rangle {}_T$ (blue lines) produced by the initial (dotted lines) and optimized (solid lines) fields for $\kappa =-0.429$. The envelopes of the fields are shown in the inset.

Figure 3

Optimized amplitudes (red circles), centers (green squares), and widths (blue diamonds) of the $x$ (open points) and $y$ pulses (filled points) for a range of $\kappa$ values.

Figure 4

(a) Expectation values of the squared cosines of the Euler angles produced by the optimal fields. (b) Relative aspects of the $x$ and $y$ pulse envelopes of the optimal fields.

Figure 5

Convergence behavior for $\kappa =-0.429$: (a) percentage change between $\langle \cos {}^2\theta \rangle {}_T$ (solid red line), $\langle \cos {}^2\chi \rangle {}_T$ (dashed blue line), and $\langle \cos {}^2\varphi \rangle {}_T$ (dotted black line) at ${\mathrm{t̄}}_f$ produced by the field at a given iteration and those produced by (b) the fully optimized field convergence of ${\varepsilon }_x^0/{\varepsilon }_y^0$ (solid red line), ${\mathrm{t̄}}_x^0-{\mathrm{t̄}}_y^0$ (dashed blue line), and ${\sigma }_x/{\sigma }_y$ (dotted black line) and (c) norm of ${\nabla }_{p}L$ as the optimization progresses.

Figure 6

Snapshots of the time evolution of 3-D alignment produced by the optimized field for $\kappa =-0.429$ at time points 0.062, 0.066, 0.076, 0.094, 0.114, and 0.15 (reduced units).

Figure 7

Alignment produced by (a) the initial field and the fields at iterations (b) 1, (c) 4, and (d) 12 for $\kappa =-0.429$.

Figure 8

Optimized parameters of the elliptically polarized pulse for a range of $\kappa$ values: (a) amplitude and (b) center and width of the envelop; (c) ellipticity angle.

Figure 9

Percentage difference between the alignment observables at ${\mathrm{t̄}}_f$ produced by the optimized two-pulse combination and the single elliptically polarized pulse.

Figure 10

Longtime alignment dynamics produced by the two-pulse combination (solid red line) and the elliptically polarized pulse (dashed blue line) optimized for $\kappa =-0.429$.