^{1,a)}, Mirta Rodríguez

^{1}, Julio Santos

^{1}, Attila Karpati

^{2}and Viktor Szalay

^{2}

### Abstract

It has been suggested that appropriate periodic sequences of laser pulses can maintain molecular alignment for arbitrarily long times [J. Ortigoso, Phys. Rev. Lett.93, 073001 (2004)]. These aligned states are found among the cyclic eigenstates of truncated matrix representations of the one-period time propagator . However, long time localization of periodic driven systems depends on the nature of the spectrum of their exact propagator; if it is continuous, eigenstates of finite-basis propagators cease to be cyclic, in the long time limit, under the exact time evolution. We show that, for very weak laser intensities, the evolution operator of the system has a point spectrum for most laser frequencies, but for the laser powers needed to create aligned wave packets it is unknown if has a point spectrum or a singular continuous spectrum. For this regime, we obtain error bounds on the exact time evolution of rotational wave packets that allow us to determine that truncated aligned cyclic states do not lose their alignment for millions of rotational periods when they evolve under the action of the exact time propagator.

Financial support from the Spanish Government, under Project Nos. FIS2004-02558 and FIS2007-61686 is acknowledged. The authors also thank the Hungarian Spanish Intergovernmental Science and Technology Cooperation Program for support through Project Nos. ESP-17/2006 and HH2006-0023. M.R. is grateful to the Ministerio de Educación y Ciencia of Spain for a Ramón y Cajal grant.

I. INTRODUCTION

II. CYCLIC STATES FOR TIME-DEPENDENT SYSTEMS

III. ROTATIONAL CYCLIC STATES AND ALIGNED WAVE PACKETS

IV. EXISTENCE OF ROTATIONAL CYCLIC STATES WHEN

A. Some basic definitions regarding the spectra of self-adjoint operators

B. Perturbation of operators with dense spectrum

C. Consequences of the existence of a singular continuous component in the spectrum of the Floquet Hamiltonian

D. Rigorous results on the nature of the spectrum of the Floquet Hamiltonian for a molecule in a laser pulse train

V. ERROR BOUNDS FOR FINITE-BASIS CALCULATIONS OF FLOQUET STATES

VI. RESULTS

VII. SUMMARY AND CONCLUSIONS

### Key Topics

- Wave functions
- 30.0
- Eigenvalues
- 23.0
- Hilbert space
- 16.0
- Molecular spectra
- 10.0
- Angular momentum
- 7.0

## Figures

Upper panel shows the time dependence of molecular alignment, given by , during several laser pulses for two different initial states. The time envelope of the laser is the black line. The calculation corresponds to the parameters in Eq. (7), in units, and . The middle panel shows the composition of the wave function that gives maximum alignment (red), in the rotational basis set, at two times, before the launch of the laser field (left) and when the maximum of the last laser pulse takes place. The lower panel shows the same for the wave function for which the alignment is smaller (blue). A global phase factor has been eliminated for the wave functions at the time when the intensity reaches the maximum value.

Upper panel shows the time dependence of molecular alignment, given by , during several laser pulses for two different initial states. The time envelope of the laser is the black line. The calculation corresponds to the parameters in Eq. (7), in units, and . The middle panel shows the composition of the wave function that gives maximum alignment (red), in the rotational basis set, at two times, before the launch of the laser field (left) and when the maximum of the last laser pulse takes place. The lower panel shows the same for the wave function for which the alignment is smaller (blue). A global phase factor has been eliminated for the wave functions at the time when the intensity reaches the maximum value.

Density plots of two eigenstates of , Eq. (7), in the basis set, for , , , , and . The two cyclic states shown in Fig. 1 were calculated from these states. The eigenstate in (b) is the misaligned state and it occupies a larger number of basis functions than the aligned state in (a). In order to make visible the long tails the plots represent , Eq. (11). Coefficients for which have been eliminated.

Density plots of two eigenstates of , Eq. (7), in the basis set, for , , , , and . The two cyclic states shown in Fig. 1 were calculated from these states. The eigenstate in (b) is the misaligned state and it occupies a larger number of basis functions than the aligned state in (a). In order to make visible the long tails the plots represent , Eq. (11). Coefficients for which have been eliminated.

In the upper row are shown the eigenvalues of in the first Brillouin zone (see text) as a function of , calculated with , . Column (a) is for a frequency resonant with the transition . Column (b) is for a nonresonant frequency, such that , where and . Column (c) corresponds to , . The middle row represents the same eigenvalues but the label of the axis is an index that identifies eigenvalues by their quasienergy ordering. The lower row gives the level spacing distributions.

In the upper row are shown the eigenvalues of in the first Brillouin zone (see text) as a function of , calculated with , . Column (a) is for a frequency resonant with the transition . Column (b) is for a nonresonant frequency, such that , where and . Column (c) corresponds to , . The middle row represents the same eigenvalues but the label of the axis is an index that identifies eigenvalues by their quasienergy ordering. The lower row gives the level spacing distributions.

Same as Fig. 3 but instead of 16 000.

Same as Fig. 3 but instead of 16 000.

Lower panel gives the logarithm of the error bounds for the three lowest quasienergy Floquet eigenstates of , Eq. (7), for , (both in units), and , as a function of the value to which the matrix representation of is truncated. The upper panel shows the corresponding quasienergies. The lines connect the different points according to quasienergy ordering and they are plotted only to guide the eye but have no other meaning. The same color is used in both panels to identify a given quasienergy and its error bound.

Lower panel gives the logarithm of the error bounds for the three lowest quasienergy Floquet eigenstates of , Eq. (7), for , (both in units), and , as a function of the value to which the matrix representation of is truncated. The upper panel shows the corresponding quasienergies. The lines connect the different points according to quasienergy ordering and they are plotted only to guide the eye but have no other meaning. The same color is used in both panels to identify a given quasienergy and its error bound.

The three cyclic vectors that were used to plot Fig. 5 for and . The left column gives the composition of the cyclic states at , and right column at . A global phase factor has been eliminated for the wave functions at . Colors are used to identify the cyclic states with their quasienergies and error bounds shown in Fig. 5.

The three cyclic vectors that were used to plot Fig. 5 for and . The left column gives the composition of the cyclic states at , and right column at . A global phase factor has been eliminated for the wave functions at . Colors are used to identify the cyclic states with their quasienergies and error bounds shown in Fig. 5.

Lower panel gives the logarithm of the error bounds for three Floquet eigenstates of , Eq. (7), for , (both in units), , and , as a function of the value to which the matrix representation of is truncated. The upper panel shows the corresponding quasienergies. The lines connect the different points according to quasienergy ordering and they are plotted only to guide the eye but have no other meaning. The same color is used in both panels to identify a given quasienergy and its error bound.

Lower panel gives the logarithm of the error bounds for three Floquet eigenstates of , Eq. (7), for , (both in units), , and , as a function of the value to which the matrix representation of is truncated. The upper panel shows the corresponding quasienergies. The lines connect the different points according to quasienergy ordering and they are plotted only to guide the eye but have no other meaning. The same color is used in both panels to identify a given quasienergy and its error bound.

Lower panel gives the logarithm of the error bounds for five Floquet eigenstates of , Eq. (7), for , (both in units), , and , as a function of the value to which the matrix representation of is truncated. The upper panel shows the corresponding quasienergies. The lines connect the different points according to quasienergy ordering and they are plotted only to guide the eye but have no other meaning. The same color is used in both panels to identify a given quasienergy and its error bound.

Lower panel gives the logarithm of the error bounds for five Floquet eigenstates of , Eq. (7), for , (both in units), , and , as a function of the value to which the matrix representation of is truncated. The upper panel shows the corresponding quasienergies. The lines connect the different points according to quasienergy ordering and they are plotted only to guide the eye but have no other meaning. The same color is used in both panels to identify a given quasienergy and its error bound.

Cyclic vectors, at , for the same parameter values as in Fig. 8. Here and columns (a) , (b) , and (c) . Colors match those used in Fig. 8.

Cyclic vectors, at , for the same parameter values as in Fig. 8. Here and columns (a) , (b) , and (c) . Colors match those used in Fig. 8.

Cyclic vectors, at , for the same parameters as in Fig. 8. Here and columns (a) , (b) , and (c) . Colors match those used in Fig. 8.

Cyclic vectors, at , for the same parameters as in Fig. 8. Here and columns (a) , (b) , and (c) . Colors match those used in Fig. 8.

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