^{1,a)}, Hans-Dieter Meyer

^{2}and Volkhard May

^{3}

### Abstract

The multiconfiguration time-dependent Hartree (MCTDH) method is combined with the optimal control theory(OCT) to study femtosecond laser pulse control of multidimensional vibrational dynamics. Simulations are presented for the widely discussed three-electronic-level vibronic coupling model of pyrazine either in a three or four vibrational coordinate version. Thus, for the first time OCT is applied to a four-coordinate system. Different control tasks are investigated and also some general aspects of the OCT-MCTDH method combination are analyzed.

Financial support by the *Deutsche Forschungsgemeinschaft* through *Sonderforschungsbereich 450* is gratefully acknowledged.

I. INTRODUCTION

II. THE MODEL

III. OPTIMAL CONTROL SCHEME COMBINED WITH THE MCTDH METHOD

IV. NUMERICAL RESULTS

A. Reference calculations: Neglect of the State

B. The coupling between the and the state: Acceleration versus suppression of internal conversion

C. Acceleration of the internal conversion: Choice of different vibrational target states at the electronic state PES

D. Application of the four-mode pyrazine model

V. CONCLUSIONS

### Key Topics

- Wave functions
- 20.0
- Probability theory
- 11.0
- Excited states
- 10.0
- Optical coherence tomography
- 10.0
- Ground states
- 8.0

## Figures

Laser pulse control of pyrazine described by a three-mode two-electronic-state model. Solid line: -state population; dashed line: -state population, lower part of the figure: temporal behavior of the optimal pulse field strength. The target state of the control task is given by the complete state to be reached at (the unit 0.1 at the axis corresponds to an electric field strength of about .

Laser pulse control of pyrazine described by a three-mode two-electronic-state model. Solid line: -state population; dashed line: -state population, lower part of the figure: temporal behavior of the optimal pulse field strength. The target state of the control task is given by the complete state to be reached at (the unit 0.1 at the axis corresponds to an electric field strength of about .

Laser pulse control of pyrazine described by a three-mode two-electronic-state model. Solid line: -state population, dashed line: -state population, lower part of the figure: temporal behavior of the optimal pulse field strength. The target state of the control task should be reached at and is given by a vibrational state in the state following from the vibrational ground-state function displaced into the -state equilibrium position (the unit 0.1 at the axis corresponds to an electric field strength of about .

Laser pulse control of pyrazine described by a three-mode two-electronic-state model. Solid line: -state population, dashed line: -state population, lower part of the figure: temporal behavior of the optimal pulse field strength. The target state of the control task should be reached at and is given by a vibrational state in the state following from the vibrational ground-state function displaced into the -state equilibrium position (the unit 0.1 at the axis corresponds to an electric field strength of about .

Laser pulse control of pyrazine described by a three-mode two-electronic-state model. Solid line: -state population, dashed line: -state population, lower part of the figure: temporal behavior of the optimal pulse field strength. The target state of the control task should be reached at and is given by a vibrational state in the state following from the vibrational ground-state function displaced into (the unit 0.1 at the axis corresponds to an electric field strength of .

Laser pulse control of pyrazine described by a three-mode two-electronic-state model. Solid line: -state population, dashed line: -state population, lower part of the figure: temporal behavior of the optimal pulse field strength. The target state of the control task should be reached at and is given by a vibrational state in the state following from the vibrational ground-state function displaced into (the unit 0.1 at the axis corresponds to an electric field strength of .

Temporal evolution of the reduced -state coordinate probability distributions referring to the control task of Fig. 3. Upper panel: mode 1 and lower panel: mode .

Temporal evolution of the reduced -state coordinate probability distributions referring to the control task of Fig. 3. Upper panel: mode 1 and lower panel: mode .

Control yield for a laser pulse driven vibrational wave packet motion into vibrational states displaced with respect to either the or the coordinate. The upper panel shows the yield vs the target wave packet position along the coordinate and the lower panel the yield vs the coordinate. In any case the wave packet position with respect to the two other coordinates is given by the -state equilibrium position. The solid line gives the overall control yield and the dashed line the renormalized yield (see also text).

Control yield for a laser pulse driven vibrational wave packet motion into vibrational states displaced with respect to either the or the coordinate. The upper panel shows the yield vs the target wave packet position along the coordinate and the lower panel the yield vs the coordinate. In any case the wave packet position with respect to the two other coordinates is given by the -state equilibrium position. The solid line gives the overall control yield and the dashed line the renormalized yield (see also text).

Laser pulse controlled maximization of the overall -state population (upper panel, ) and of the -state population (lower panel, ). Solid line: -state population, dashed line: -state population, dashed-dotted line: -state population, thin solid line: temporal behavior of the respective optimal pulse (the unit 0.1 at the axis corresponds to an electric field strength of about ).

Laser pulse controlled maximization of the overall -state population (upper panel, ) and of the -state population (lower panel, ). Solid line: -state population, dashed line: -state population, dashed-dotted line: -state population, thin solid line: temporal behavior of the respective optimal pulse (the unit 0.1 at the axis corresponds to an electric field strength of about ).

Temporal evolution of the reduced -state coordinate probability distributions referring to the control task of the upper panel of Fig. 6. From the top to the bottom: mode 1, mode , and mode .

Temporal evolution of the reduced -state coordinate probability distributions referring to the control task of the upper panel of Fig. 6. From the top to the bottom: mode 1, mode , and mode .

Laser pulse controlled wave packet formation in the state with the target wave packet located at . Solid line: -state population, dashed line: -state population, dashed-dotted line: -state population, thin solid line: temporal behavior of the respective optimal pulse (the unit 0.1 at the axis corresponds to an electric field strength of ).

Laser pulse controlled wave packet formation in the state with the target wave packet located at . Solid line: -state population, dashed line: -state population, dashed-dotted line: -state population, thin solid line: temporal behavior of the respective optimal pulse (the unit 0.1 at the axis corresponds to an electric field strength of ).

Laser pulse controlled wave packet formation in the state with the target wave packet located at . Solid line: -state population, dashed line: -state population, dashed-dotted line: -state population, thin solid line: temporal behavior of the respective optimal pulse (the unit 0.1 at the axis corresponds to an electric field strength of about ).

Laser pulse controlled wave packet formation in the state with the target wave packet located at . Solid line: -state population, dashed line: -state population, dashed-dotted line: -state population, thin solid line: temporal behavior of the respective optimal pulse (the unit 0.1 at the axis corresponds to an electric field strength of about ).

Laser pulse controlled wave packet formation in the state with the target wave packet located at . Solid line: -state population, dashed line: -state population, dashed-dotted line: -state population, thin solid line: temporal behavior of the respective optimal pulse [note the absence of any time dependence of ; the unit 0.1 at the axis corresponds to an electric field strength of about ].

Laser pulse controlled wave packet formation in the state with the target wave packet located at . Solid line: -state population, dashed line: -state population, dashed-dotted line: -state population, thin solid line: temporal behavior of the respective optimal pulse [note the absence of any time dependence of ; the unit 0.1 at the axis corresponds to an electric field strength of about ].

Temporal evolution of the reduced -state coordinate probability distributions referring to the control task of Fig. 10. From the top to the bottom: mode 1, mode , and mode .

Temporal evolution of the reduced -state coordinate probability distributions referring to the control task of Fig. 10. From the top to the bottom: mode 1, mode , and mode .

Laser pulse controlled maximization of the overall -state population (upper panel, ) and the -state population (lower panel, ) using the four-coordinate model. Solid line: -state population, dashed line: -state population, dashed dotted line: -state population, thin solid line: temporal behavior of the respective optimal pulse (the unit 0.1 at the axis corresponds to an electric field strength of ).

Laser pulse controlled maximization of the overall -state population (upper panel, ) and the -state population (lower panel, ) using the four-coordinate model. Solid line: -state population, dashed line: -state population, dashed dotted line: -state population, thin solid line: temporal behavior of the respective optimal pulse (the unit 0.1 at the axis corresponds to an electric field strength of ).

Temporal evolution of the reduced -state coordinate probability distributions using the four-mode model. The control task refers to that of the upper panel of Fig. 12. From the top to the bottom: mode 1, mode , mode , and mode .

Temporal evolution of the reduced -state coordinate probability distributions using the four-mode model. The control task refers to that of the upper panel of Fig. 12. From the top to the bottom: mode 1, mode , mode , and mode .

Temporal evolution of the total population of the (dashed line) and states (dashed-dotted line) after an impulsive excitation of the state. Upper panel: three-mode model, lower panel: four-mode model.

Temporal evolution of the total population of the (dashed line) and states (dashed-dotted line) after an impulsive excitation of the state. Upper panel: three-mode model, lower panel: four-mode model.

Temporal evolution of the reduced -state coordinate probability distributions after an impulsive excitation of the state and in using the three-mode model. From the top to the bottom: mode 1, mode , and mode .

Temporal evolution of the reduced -state coordinate probability distributions after an impulsive excitation of the state and in using the three-mode model. From the top to the bottom: mode 1, mode , and mode .

Temporal evolution of the reduced -state coordinate probability distributions after an impulsive excitation of the state and in using the four-mode model. From the top to the bottom: mode 1, mode , mode , and mode .

Temporal evolution of the reduced -state coordinate probability distributions after an impulsive excitation of the state and in using the four-mode model. From the top to the bottom: mode 1, mode , mode , and mode .

Laser pulse driven maximization of the -state population of pyrazine described in the three-mode model. The target state population is drawn vs the number of iterations of the OCT equations and for different sets of computations listed in Table III. Solid line: set dashed-line: set dashed-dotted line: set .

Laser pulse driven maximization of the -state population of pyrazine described in the three-mode model. The target state population is drawn vs the number of iterations of the OCT equations and for different sets of computations listed in Table III. Solid line: set dashed-line: set dashed-dotted line: set .

## Tables

Parameters of the three-electronic-state vibronic coupling model for pyrazine [see Eq. (2)] based either on three (Refs. 15 and 17) or four vibrational modes (Refs. 19 and 20). The oscillation periods of the vibrational modes and their equilibrium positions in the two excited states are also given (, , and in eV, in fs). The vertical electronic excitation energies amount to and .

Parameters of the three-electronic-state vibronic coupling model for pyrazine [see Eq. (2)] based either on three (Refs. 15 and 17) or four vibrational modes (Refs. 19 and 20). The oscillation periods of the vibrational modes and their equilibrium positions in the two excited states are also given (, , and in eV, in fs). The vertical electronic excitation energies amount to and .

Control yield and renormalized yield obtained within the control tasks described in Sec. IV C. The results are ordered with respect to the target state specified by the electronic level and the localization of the coupling mode vibrational wave packet. A second set of results, where the position of the target state on the coupling mode has been changed, are also tabularized and are listed in parentheses. The wave packets in the tuning modes are localized at the equilibrium position in their respective electronic states. The final time and the parameter are also given (for details see text).

Control yield and renormalized yield obtained within the control tasks described in Sec. IV C. The results are ordered with respect to the target state specified by the electronic level and the localization of the coupling mode vibrational wave packet. A second set of results, where the position of the target state on the coupling mode has been changed, are also tabularized and are listed in parentheses. The wave packets in the tuning modes are localized at the equilibrium position in their respective electronic states. The final time and the parameter are also given (for details see text).

Performance of OCT-MCTDH approach (the control task refers to the maximization of the -state population at and described in the three-mode pyrazine model). Listed are different sets of calculations (see text). Every one is ordered with respect to the vibrational modes and characterized by a certain number of single particle functions as well as particular natural populations (NP). Both numbers are ordered with respect to their electronic state contributions (, , and states—from the left to the right). The basic time step (in fs) of the propagation as well as the performance (for the machine see text) given by the overall computational time (in h) are also presented.

Performance of OCT-MCTDH approach (the control task refers to the maximization of the -state population at and described in the three-mode pyrazine model). Listed are different sets of calculations (see text). Every one is ordered with respect to the vibrational modes and characterized by a certain number of single particle functions as well as particular natural populations (NP). Both numbers are ordered with respect to their electronic state contributions (, , and states—from the left to the right). The basic time step (in fs) of the propagation as well as the performance (for the machine see text) given by the overall computational time (in h) are also presented.

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