^{1,a)}and Ronnie Kosloff

^{1}

### Abstract

Optimal control theory is employed for the task of minimizing the excited-state population of a dye molecule in solution. The spectrum of the excitation pulse is contained completely in the absorption band of the molecule. Only phase control is studied which is equivalent to optimizing the transmission of the pulse through the medium. The molecular model explicitly includes two electronic states and a single vibrational mode. The other degrees of freedom are classified as bath modes. The surrogate Hamiltonian method is employed to incorporate these bath degrees of freedom. Their influence can be classified as electronic dephasing and vibrational relaxation. In accordance with experimental results, minimal excitation is associated with a negatively chirped pulses. Optimal pulses with more complex transient structure are found to be superior to linearly chirped pulses. The difference is enhanced when the fluence is increased. The improvement degrades when dissipative effects become more dominant.

We wish to thank Professor Sanford Ruhman and Omer Nahmias for helpful discussions. This research was supported by the Israel Science Foundation (ISF). The Fritz Haber Center is supported by the Minerva Gesellschaft für die Forschung, GmbH München, Germany.

I. INTRODUCTION

II. THEORY

A. The model

B. Optimization scheme

C. The surrogate Hamiltonian

III. RESULTS AND DISCUSSION

A. Linear chirped pulse

1. Isolated system

2. Dissipative system

B. Nonlinear chirped pulses

1. Isolated system

2. Dissipative system

C. Role of intensity

IV. CONCLUSIONS

### Key Topics

- Chirping
- 65.0
- Excited states
- 39.0
- Dephasing
- 35.0
- Ground states
- 12.0
- Electric dipole moments
- 9.0

## Figures

The pulse intensity spectrum along with absorption (solid line) and fluorescence (dashed line) spectra of LDS750 molecule in acetonitrile. Adapted from Ref. 2.

The pulse intensity spectrum along with absorption (solid line) and fluorescence (dashed line) spectra of LDS750 molecule in acetonitrile. Adapted from Ref. 2.

Excited-state population as a function of the linear chirp for the isolated system . The population is shown at the end of the pulse for the high (solid line) and the low (dashed line) fluences. The duration of the corresponding transform-limited pulse is . Note the different scale for the excited population for two energy regimes.

Excited-state population as a function of the linear chirp for the isolated system . The population is shown at the end of the pulse for the high (solid line) and the low (dashed line) fluences. The duration of the corresponding transform-limited pulse is . Note the different scale for the excited population for two energy regimes.

(Top panel) Time evolution of the excited-state population of the isolated system for the high-energy excitation. The population is calculated for a negatively [, solid line] and a positively [, dashed line] chirped pulses. (Bottom panel) The imaginary part of the transition dipole moment multiplied by the field amplitude.

(Top panel) Time evolution of the excited-state population of the isolated system for the high-energy excitation. The population is calculated for a negatively [, solid line] and a positively [, dashed line] chirped pulses. (Bottom panel) The imaginary part of the transition dipole moment multiplied by the field amplitude.

(Color online) (Top panel) Trajectories of the transition dipole moment renormalized by its maximal amplitude for excitation by the linear negatively chirped pulse . (Bottom panel) Excited-state population at the end of the pulse as a function of the linear chirp parameter . The calculations are performed for the system without dissipation (solid line) and for the system with vibrational relaxation with weak (, dashed line), medium (, dashed-dotted line), and strong (, dotted line) system-bath couplings.

(Color online) (Top panel) Trajectories of the transition dipole moment renormalized by its maximal amplitude for excitation by the linear negatively chirped pulse . (Bottom panel) Excited-state population at the end of the pulse as a function of the linear chirp parameter . The calculations are performed for the system without dissipation (solid line) and for the system with vibrational relaxation with weak (, dashed line), medium (, dashed-dotted line), and strong (, dotted line) system-bath couplings.

(Color online) (Top panel) Trajectories of the transition dipole moment renormalized by its maximal amplitude for excitation by the linear negatively chirped pulse . (Bottom panel) Excited-state population at the end of the pulse as a function of the linear chirp parameter . The calculations are performed for the isolated system (solid line) and for the system with pure electronic dephasing: weak (, dashed line), medium (, dashed-dotted line) and strong (, dotted line) couplings.

(Color online) (Top panel) Trajectories of the transition dipole moment renormalized by its maximal amplitude for excitation by the linear negatively chirped pulse . (Bottom panel) Excited-state population at the end of the pulse as a function of the linear chirp parameter . The calculations are performed for the isolated system (solid line) and for the system with pure electronic dephasing: weak (, dashed line), medium (, dashed-dotted line) and strong (, dotted line) couplings.

Time evolution of the excited-state population of the isolated system for the high-energy excitation. The population is calculated for a linear negatively chirped pulse (solid line) and for the optimal pulse (dashed line) obtained by using a genetic algorithm with the random phases, generated at discrete points of the frequency spectrum. The inset figure shows the temporal profile of the optimized pulse.

Time evolution of the excited-state population of the isolated system for the high-energy excitation. The population is calculated for a linear negatively chirped pulse (solid line) and for the optimal pulse (dashed line) obtained by using a genetic algorithm with the random phases, generated at discrete points of the frequency spectrum. The inset figure shows the temporal profile of the optimized pulse.

(Color) Calculations for nondissipative system. Time-frequency Wigner distribution corresponding to the optimized linear (top panel) and nonlinear (bottom panel) chirped pulses. The right sides show the frequency spectra of the pulses, while their temporal profiles are shown in the upper panels. The phase is expanded in the Taylor series up to the second order (linear chirp) and in the basis of periodic functions (nonlinear chirp).

(Color) Calculations for nondissipative system. Time-frequency Wigner distribution corresponding to the optimized linear (top panel) and nonlinear (bottom panel) chirped pulses. The right sides show the frequency spectra of the pulses, while their temporal profiles are shown in the upper panels. The phase is expanded in the Taylor series up to the second order (linear chirp) and in the basis of periodic functions (nonlinear chirp).

(Top panel) Evolution of the excited-state population of the isolated system for the high-energy excitation. The population is calculated for a linear negatively chirped pulse (solid line) and for the optimal pulse (dashed line) obtained by using a genetic algorithm with the phase expanded in the basis of periodic functions. (Bottom panel) Trajectories of the transition dipole moment renormalized by its maximal amplitude. Solid and dashed lines refer to the linear chirped pulse and its nonlinear analog, respectively.

(Top panel) Evolution of the excited-state population of the isolated system for the high-energy excitation. The population is calculated for a linear negatively chirped pulse (solid line) and for the optimal pulse (dashed line) obtained by using a genetic algorithm with the phase expanded in the basis of periodic functions. (Bottom panel) Trajectories of the transition dipole moment renormalized by its maximal amplitude. Solid and dashed lines refer to the linear chirped pulse and its nonlinear analog, respectively.

(Top panel) Time evolution of the excited-state population of the dissipative system for the high-energy excitation. The population is calculated for a linear negatively chirped pulse (solid line) and for the optimal pulse obtained by using a genetic algorithm (dashed line). (Bottom panel) Trajectories of the transition dipole moment renormalized by its maximal amplitude. Calculations were performed for the system with weak vibrational relaxation and medium pure electronic dephasing .

(Top panel) Time evolution of the excited-state population of the dissipative system for the high-energy excitation. The population is calculated for a linear negatively chirped pulse (solid line) and for the optimal pulse obtained by using a genetic algorithm (dashed line). (Bottom panel) Trajectories of the transition dipole moment renormalized by its maximal amplitude. Calculations were performed for the system with weak vibrational relaxation and medium pure electronic dephasing .

(Color) Calculations for the dissipative system. Time-frequency Wigner distribution corresponding to the optimal linear (top panel) and nonlinear (bottom panel) chirped pulses. Calculations were performed for the system with medium vibrational relaxation and electronic dephasing .

(Color) Calculations for the dissipative system. Time-frequency Wigner distribution corresponding to the optimal linear (top panel) and nonlinear (bottom panel) chirped pulses. Calculations were performed for the system with medium vibrational relaxation and electronic dephasing .

(Color) (Top panel) Time evolution of the excited-state population of the dissipative system for the high-energy excitation. The population is calculated for a linear negatively chirped pulse (solid line) and for the optimal pulse obtained by using a genetic algorithm (dashed line). (Bottom panel) Time-frequency Wigner distribution corresponding to the optimized nonlinear chirped pulse. The calculations were performed for the primary system with the following parameters: the ground-state and the excited-state frequencies and the dimensionless displacement of . The right sides show the frequency spectra of the pulses, while their temporal profiles are shown in the upper panels. The phase is expanded in the basis of periodic functions.

(Color) (Top panel) Time evolution of the excited-state population of the dissipative system for the high-energy excitation. The population is calculated for a linear negatively chirped pulse (solid line) and for the optimal pulse obtained by using a genetic algorithm (dashed line). (Bottom panel) Time-frequency Wigner distribution corresponding to the optimized nonlinear chirped pulse. The calculations were performed for the primary system with the following parameters: the ground-state and the excited-state frequencies and the dimensionless displacement of . The right sides show the frequency spectra of the pulses, while their temporal profiles are shown in the upper panels. The phase is expanded in the basis of periodic functions.

Effect of the intensity for linearly chirped pulses. The value of the linear chirp is plotted as a function of the pulse fluence (the amplitude of the corresponding transform-limited pulse). Calculations were performed for the isolated system (squares) and as well as for a dissipative system (triangles) with f medium vibrational relaxation and medium pure electronic dephasing .

Effect of the intensity for linearly chirped pulses. The value of the linear chirp is plotted as a function of the pulse fluence (the amplitude of the corresponding transform-limited pulse). Calculations were performed for the isolated system (squares) and as well as for a dissipative system (triangles) with f medium vibrational relaxation and medium pure electronic dephasing .

The excited-state population at the end of the pulse as function of the pulse fluence (the amplitude of the corresponding transform-limited pulse). Calculations were performed (top panel) for the system without dissipation and (bottom panel) for the system with medium vibrational relaxation ) and medium pure electronic dephasing . The arrow points to the maximal fluence used in the experiment by Nahmias *et al.* (Ref. 2)

The excited-state population at the end of the pulse as function of the pulse fluence (the amplitude of the corresponding transform-limited pulse). Calculations were performed (top panel) for the system without dissipation and (bottom panel) for the system with medium vibrational relaxation ) and medium pure electronic dephasing . The arrow points to the maximal fluence used in the experiment by Nahmias *et al.* (Ref. 2)

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