Controlling the sense of molecular rotation

We introduce a new scheme for controlling the sense of molecular rotation. By varying the polarization and the delay between two ultrashort laser pulses, we induce unidirectional molecular rotation, thereby forcing the molecules to rotate clockwise/counterclockwise under field-free conditions. We show that unidirectionally rotating molecules are confined to the plane defined by the two polarization vectors of the pulses, which leads to a permanent anisotropy in the molecular angular distribution. The latter may be useful for controlling collisional cross-sections and optical and kinetic processes in molecular gases. We discuss the application of this control scheme to individual components within a molecular mixture in a selective manner.

The essence of coherent control is to drive a molecular system towards specific behavioural goal. The goals are usually set as the enhanced population of a specific vibrational or electronic state and the tools are ultrashort laser pulses which are modulated either in the time or frequency domains. Here we present a double pulse scheme for controlling the sense (clockwise / counter clockwise) of the molecular rotation.
Laser induced molecular rotation and alignment has received significant attention in recent years. In the last decade, interest in the field has increased, mainly due to the improving capabilities to control the laser pulse characteristics (such as time duration and temporal shape), which in turn leads to potential applications offered by controlling the angular distribution of molecules. Since the typical rotational motion is 'slow' (~10 ps) with respect to the typical short pulse (~50 fs), effective rotational control and manipulation are in reach. In the liquid phase, molecular alignment following excitation by a strong laser pulse was observed in the seventies 1 , and proposed as a tool for optical gating. In the early experiments, picosecond laser pulses were used for the excitation, and deviation of the refractive index from that of an isotropic gas was utilized as a measure of the alignment 2,3 . More recently, this research area has been revisited both theoretically and experimentally (for a recent review, see Ref. 4 ). Temporal rotational dynamics of pulse-excited molecules was studied 5,6,7 , and multiple pulse sequences giving rise to the enhanced alignment were suggested, 8,9 and realized 10,11,12,13 . Further manipulations such as optical molecular centrifuge and alignment-dependent strong field ionization of molecules were demonstrated 13,14,15 . Selective rotational excitation in bimolecular mixtures was suggested and demonstrated in the mixtures of molecular isotopes 16 and spin isomers 17 . Transient molecular alignment has been shown to compress ultrashort light pulses 18,19 and it is successfully used in controlling high harmonic generation 20,21,22 .
Other experiments were reported in which transient grating techniques were employed for detailed studies of molecular alignment and deformation 23,24 . In the past few years, molecular alignment became a common tool in attosecond studies, in particular, in experiments for probing molecular bond structures 25,26 .
In practically all the previous works in the field of laser molecular alignment (with the exception of Ref. 14 ), the rotational motion was enhanced, but the net total angular momentum delivered to the molecules remained zero, and for a good reason. For single pulse schemes, as well as for techniques using multiple pulses polarized in the same direction, no preferred sense of rotation exists due to the axial symmetry of excitation.
In order to inject angular momentum to the medium and to force the molecules to rotate with a preferred sense of rotation, one has to break the axial symmetry. This has been previously demonstrated by Karczmarek et. al 14 who used two oppositely chirped, circularly polarized pulses overlapping in time and space, thereby creating a linearly polarized pulse, rotating unidirectionally and accelerating in a plane.
In solid state, controlled unidirectional rotation of induced polarization by impulsive excitation of two-fold degenerate lattice vibrations was demonstrated in α -quartz 27 .
In this paper, we propose a double pulse scheme for breaking the axial symmetry and for inducing unidirectional molecular rotation under field-free conditions. Pictorially, our double pulse control scheme is sketched in Figure 1. An ultrashort laser pulse (red arrow), linearly polarized along the z axis, is applied to the molecular ensemble and induces coherent molecular rotation. The molecules rotate under field-free conditions until they reach an aligned state, in which they are temporarily confined in a narrow cone around the polarization direction of the first pulse. At this moment, a second pulse, linearly polarized at 45 degrees to the first one, is applied, inducing unidirectional (clockwise, in our case) molecular rotation. In the present paper, we discuss the dependence of the induced angular momentum on the pulse intensities, and the delay between pulses. Moreover, we focus on the molecular angular distribution when the molecules are subject to unidirectional rotation, and show that it is confined in the plane defined by the two polarization vectors of the pulses. This anisotropic angular distribution is characterized by the observable 2 cos ϕ which is referred to as the azimuthal factor. This highly anisotropic angular distribution induced by the breaking of the axial symmetry may offer an efficient way to control the kinetic and optical properties of the gas medium.
We consider the problem by modelling the molecules as driven rigid rotors interacting with a linearly polarized laser field. Within this model the Hamiltonian of the linear molecules is given by where Ĵ is the angular momentum operator , θ is the angle between the polarization vector of the field (defining the z axis) and the molecular axis, and I is the moment of inertia of the molecule. The latter is related to the molecular rotational constant /(4 ) I cB π = , where is Planck constant and c is the speed of light. The interaction term is given by where ( ) t ε is the envelope amplitude of the laser field, and α , α ⊥ are the parallel and perpendicular components of the polarizability tensor, respectively.
We simulated the proposed scheme quantum-mechanically by two independent methods. The first of them is mainly analytical: it uses spectral decomposition of the time-dependent rotational wave function, and relies heavily on angular momentum algebra for calculating the observable quantities (see Appendix A for the mathematical details,). Such an approach has been widely used in the past by many groups, including ourselves 8,9 , to analyze multipulse alignment,. The second method uses direct numerical simulation of the driven ensemble of quantum rotors by means of Finite-Difference Time-Domain (FDTD) approach (for details, see Appendix B).
In both cases, the laser pulses were approximated as δ functions, and their integrated 'strength' was characterized by a dimensionless pulse strength parameter Physically, the parameter P represents a typical increase of the molecular angular momentum (in the units of ) due to the interaction with the pulse.
All simulations were performed at finite temperature, and the results were averaged over a thermal molecular ensemble. Moreover, recently the same problem was studied by us classically with the help of the Monte Carlo method, the details are given elsewhere 28   θ°= ± to the field.
As the next step we explore the dependence of the induced angular momentum on the strength 1,2 P of the two laser pulses. Figure 3 shows 2 cos θ of the aligned state just before 1 2 rev T as a function of the first pulse strength 1 P . At higher pulse power, the alignment factor saturates at ~ 0.9. Figure 4 shows the angular momentum induced by the double-pulse excitation. We scan the power of the second pulse ( 2 P ) for each of the 1 P values shown in Figure 3.     Nitrogen molecules, the difference between these two time delays is about 200 fs.
In a mixture of two species such as molecular isotopes 16 one can find time delays when one species is aligned while the other is anti-aligned at the same time.
Application of the second pulse at this time moment will result in the opposite senses of rotation induced for the two species, which can be potentially used for their physical separation.
In Figure 7 we show a calculation similar to the one in Figure 6    At higher temperatures and higher excitation powers, the fractional revivals are better observed. In figure 9 we plot 2 cos ϕ as a function of time for 1 2 10 for Nitrogen molecules.   decomposition is a non-trivial task that can be done by several methods (see, e.g. 8,29 ).
In our case, we introduce the artificial "time"-parameter τ , and consider the τ - We convert the problem to the solution of a set of coupled differential equations for the coefficients , ( ) l m C τ . At 0 τ = only the coefficient corresponding to the initial eigenstate is non-zero (and equal to 1). The after-pulse wavefunction ( ) ψ + is given by The needed set of differential equations is produced by differentiating (6) with respect to τ , and projecting both sides of the resulting equation to ' ' m l Y : We write 2 cos θ in terms of the spherical harmonic functions: For the integration of the product of three spherical harmonic functions, we use the Wigner 3j symbol: The equations (7) become: From the equations above, one can see that the interaction term ( After some algebra we express the angular-dependent functions in (10) as: To consider the action of the second pulse in the impulsive approximation (similar to (6), (7)) we have to solve the following set of coupled differential equations: In contrast to the case of the first pulse, different m -states are now coupled to states with 1 and 2 m m ± ± , thus leading to the generation of more complex angular wave packets. By solving the above equations on the interval 0 1 τ ≤ ≤ , one is able to decompose the resulting wavepackets in the basis of rotor eigenstates, and to define their time-dependent dynamics.
As mentioned in the main text, the induced unidirectional rotation is accompanied by the confinement of the molecular angular distribution to the plane defined by polarization vectors of the two pulses. We introduce an observable Here we used the fact that 2 i e ϕ + couples only m and 2 m − eigenstates.
For the overlap integral of associated Legendre polynomials in (13)