Error as function of the time step for Galerkin (crosses) and collocation (plain lines). Errors compared to a reference solution within the Hagedorn space are shown with dashed lines, and errors compared to a spatially well-resolved reference are shown with solid lines. The TDSE is integrated up to t = 1 with 64 basis functions in (a), and to t = 5 with 8 basis functions in (b).
Projection of to two different Hagedorn spaces, both with K = 32 basis functions. The figures show u(x) with solid lines and its projection with dashed lines. In both cases, ɛ = 1, q = p = 0, P = i/Q. In (a), Q = 1 and in (b), Q = 0.5. The ℓ∞-errors are 0.93 and 1.8 × 10−5, respectively. In (b), the two lines are indistinguishable.
(a) The time evolution of the uncertainty parameter QQ*. The solid line indicates control-based propagation and the dotted semiclassical propagation. The dashed line indicates the control signal. (b) A lin-log plot of the squared modulus of the wave function at time t = 5. The crosses show the location of the collocation points in the control scheme.
The PES of the IBr molecule with sketched wave packets and grids. The equidistant Fourier grid of the state stays fixed, while the Gauss-Hermite grids in the two excited states follow the wave packets.
The real part of ψ2 at time t 0, when the laser pulse peaks. The cross marks the coordinate of strongest coupling with the ground electronic state. The circle marks where the classical coordinate q would have been located if no control were used.
The solution, real parts and moduli, at the three potential energy surfaces at time .
(a) The front of ψ3 in modulus at time . The dashed line indicates , and the cross x δ. As grows, the dashed line will rise above the top of the hump, and x δ will encounter a discontinuity. The response to this is shown in (b). The dashed line shows the control signal for |Q|, and the solid line its actual value.
Level curves for the excited state PES of CO2. The location of the saddle point is indicated by a cross, and the centrepoint of the initial Gaussian wave packet by a ring. The level curves are separated by 0.015 hartree.
Time evolution of the wave function. The modulus of the wave function, |Ψ(s, a)|, is plotted with a separation of 1 between the level curves. The cross indicates the location of the saddle point in the potential. The wave function is plotted at times t = (a) , (b) , (c) , and (d) .
(a) The modulus of Q 2, which determines the spread of the collocation points along the asymmetric stretch coordinate axis, plotted against time. Semiclassical propagation of Q and P is shown with solid lines, and control-based propagation with dashed lines. (b) The time evolution of the maximum pointwise error for the two propagation schemes.
The performance of the controller. |Q 1| (below) and |Q 2| (above) are plotted against time. The square root of the control signal is shown with solid lines, and the actual parameter values with dashed lines.
Level curves for the pointwise error at time for solutions calculated with (a) a control-propagated Hagedorn basis with 322 basis functions, and (b) a Fourier basis with 522 basis functions. The curve separation is 1.0 × 10−5, and the maximum pointwise error is 8.7 × 10−5 and 7.2 × 10−5 for the Hagedorn and Fourier solutions, respectively.
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