^{1}, S. Ramakrishna

^{1}and Tamar Seideman

^{1,a)}

### Abstract

We apply optimal control theory to explore and manipulate rotational wavepacket dynamics subject to a dissipative environment. In addition to investigating the extent to which nonadiabatic alignment can make a useful tool in the presence of decoherence and population relaxation, we use coherent rotational superpositions as a simple model to explore several general questions in the control of systems interacting with a bath. These include the extent to which a pure state can be created out of a statistical ensemble, the degree to which control theory can develop superposition states that resist dissipation, and the nature of environments that prohibits control. Our results illustrate the information content of control studies regarding the dissipative properties of the bath and point to the strategies that optimize different targets in wavepacket alignment in nonideal environments. As an interesting aside, the method is used to illustrate the limit where the coherence-based approach to molecular alignment converges to traditional incoherent approaches.

We are grateful to the Department of Energy, Grant No. DAAD19-03-R0017, for support of our research on the alignment problem.

I. INTRODUCTION

II. OPTIMAL CONTROL THEORY FOR DISSIPATIVE MOLECULAR ALIGNMENT

A. The system Hamiltonian and density operators

B. Control fields for maximizing alignment that is distributed in time

III. RESULTS AND DISCUSSION

A. Maximizing molecular alignment during a specified time interval

1. Nondissipative dynamics

2. Effects of population relaxation

3. Effects of temperature

4. Effects of pure decoherence

B. Construction of prespecified rotational superposition states

1. Control in a nondissipative environment

2. Effects of population relaxation

3. Effects of increased temperature

IV. CONCLUSIONS

### Key Topics

- Angular momentum
- 12.0
- Electric fields
- 10.0
- Relaxation times
- 9.0
- Optical coherence tomography
- 8.0
- Coherent control
- 6.0

## Figures

Alignment over a pre-specified time interval. Averaged alignment, , for different values of the Gaussian width parameter that determines the required duration of the alignment: (a) (solid curve); (dashed curve); (b) (solid curve); (dashed curve). (c) -state populations for (solid curve), (dashed curve), and (dotted curve), showing the nearly complete transfer of the population to the state. In all cases, the pressure and the temperature are 0 and , as indicated by the vertical dashed line. We use reduced time units, , where is the rotational constant.

Alignment over a pre-specified time interval. Averaged alignment, , for different values of the Gaussian width parameter that determines the required duration of the alignment: (a) (solid curve); (dashed curve); (b) (solid curve); (dashed curve). (c) -state populations for (solid curve), (dashed curve), and (dotted curve), showing the nearly complete transfer of the population to the state. In all cases, the pressure and the temperature are 0 and , as indicated by the vertical dashed line. We use reduced time units, , where is the rotational constant.

Short-time Fourier transform of optimal fields at 0 pressure and with (a) a delta function window, (b) , and (c) . The arrow indicates the target time at which the algorithm seeks to maximize the alignment, , and the color bar provides the intensity in relative units. Time is given, as in Fig. 1, in units of , energy is given in units of , and , where is the Boltzmann constant and the temperature. Note that the generalized boundary condition allows the field to extend past .

Short-time Fourier transform of optimal fields at 0 pressure and with (a) a delta function window, (b) , and (c) . The arrow indicates the target time at which the algorithm seeks to maximize the alignment, , and the color bar provides the intensity in relative units. Time is given, as in Fig. 1, in units of , energy is given in units of , and , where is the Boltzmann constant and the temperature. Note that the generalized boundary condition allows the field to extend past .

(a) The expectation value of at , for (⋯); (---); (solid curve). (b) The population measure [Eq. (12)] for the parameters of panel (a). (c) The corresponding coherence measure . Here as indicated by the vertical dashed line.

(a) The expectation value of at , for (⋯); (---); (solid curve). (b) The population measure [Eq. (12)] for the parameters of panel (a). (c) The corresponding coherence measure . Here as indicated by the vertical dashed line.

Average alignment maximized at (vertical dashed line) for (⋯), (---); (solid curve). The pressure increases from left to right as 0 [panels (a), (d), and (g)]; , [panels (b), (e), and (h)]; (panels c, f, i). The temperature increases from top to bottom as [(a)–(c)]; [(d)–(f)]; [(g)–(i)]. For CO in Ar, this corresponds to 10, 20, and , respectively.

Average alignment maximized at (vertical dashed line) for (⋯), (---); (solid curve). The pressure increases from left to right as 0 [panels (a), (d), and (g)]; , [panels (b), (e), and (h)]; (panels c, f, i). The temperature increases from top to bottom as [(a)–(c)]; [(d)–(f)]; [(g)–(i)]. For CO in Ar, this corresponds to 10, 20, and , respectively.

Population of a two-level superposition (here , ) with a predetermined composition at and . [(a)–(c)] , [(d)–(f)] . [(a) and (d)] Time evolution of the population in rotational levels : (diamonds), (++++), (solid curve), (–⋅–⋅), and (---). [(b) and (e)] The optimal fields for the transitions in panels (a) and (d), respectively. The major subpulse features of the field are approximately separated by , the natural timescale of the system. [(c)and (f)] The Fourier transform of the fields in panels (b) and (e), respectively. The dashed lines indicate the energies of transition from state to state in the model system (CO in Ar). The optimal pulse for the case is essentially the pulse for the study, shifted to cover the last portion of the pulse, leaving the initial near zero. Correspondingly, the frequency spectrum of the pulse is identical to that of the pulse.

Population of a two-level superposition (here , ) with a predetermined composition at and . [(a)–(c)] , [(d)–(f)] . [(a) and (d)] Time evolution of the population in rotational levels : (diamonds), (++++), (solid curve), (–⋅–⋅), and (---). [(b) and (e)] The optimal fields for the transitions in panels (a) and (d), respectively. The major subpulse features of the field are approximately separated by , the natural timescale of the system. [(c)and (f)] The Fourier transform of the fields in panels (b) and (e), respectively. The dashed lines indicate the energies of transition from state to state in the model system (CO in Ar). The optimal pulse for the case is essentially the pulse for the study, shifted to cover the last portion of the pulse, leaving the initial near zero. Correspondingly, the frequency spectrum of the pulse is identical to that of the pulse.

Evolution of rotational populations for generically constructed optimal superpositions [panels (a), (d), and (h)] and corresponding superpositions created by the optimal control algorithm [panels (b), (e), and (i)] at , and . The optimal fields corresponding to panels (b), (e), and (i) are shown in panels (c), (f), and (j), respectively. is denoted by a solid curve, by (⋯), by (––⋅––⋅), by (- - -), by (–⋅–⋅), by (–⋅⋅–), by large (–⋅–⋅), and by small (---). For clarity, only states relevant to the studied dynamics are shown. Panels (a)–(c) consider a target superposition of the and with 60% of the population in the former and 40% in the latter. (The closest kinematically allowed populations, see Ref. 56, are and .) In this case the superposition lifetime is about . Panels (d)–(f) consider a target superposition of the and , with which the superposition lifetime is . Panels (h)–(j) seek to produce a , superposition with a lifetime of . For low energy superposition, the effective pulse is of duration of the relaxation time and acts just before . As , increase, multiple subpulses of increasingly high frequency develop and earlier portions of the pulse are utilized.

Evolution of rotational populations for generically constructed optimal superpositions [panels (a), (d), and (h)] and corresponding superpositions created by the optimal control algorithm [panels (b), (e), and (i)] at , and . The optimal fields corresponding to panels (b), (e), and (i) are shown in panels (c), (f), and (j), respectively. is denoted by a solid curve, by (⋯), by (––⋅––⋅), by (- - -), by (–⋅–⋅), by (–⋅⋅–), by large (–⋅–⋅), and by small (---). For clarity, only states relevant to the studied dynamics are shown. Panels (a)–(c) consider a target superposition of the and with 60% of the population in the former and 40% in the latter. (The closest kinematically allowed populations, see Ref. 56, are and .) In this case the superposition lifetime is about . Panels (d)–(f) consider a target superposition of the and , with which the superposition lifetime is . Panels (h)–(j) seek to produce a , superposition with a lifetime of . For low energy superposition, the effective pulse is of duration of the relaxation time and acts just before . As , increase, multiple subpulses of increasingly high frequency develop and earlier portions of the pulse are utilized.

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