Effect of rotational-state-dependent molecular alignment on the optical dipole force

The properties of molecule-optical elements such as lenses or prisms based on the interaction of molecules with optical fields depend in a crucial way on the molecular quantum state and its alignment created by the optical field. However, in previous experimental studies, the effects of state-dependent alignment have never been included in estimates of the optical dipole force acting on the molecules while previous theoretical investigations took the state-dependent molecular alignment into account only implicitly. Herein, we consider the effects of molecular alignment explicitly and, to this end, introduce an effective polarizability which takes proper account of molecular alignment and is directly related to the alignment-dependent optical dipole force. We illustrate the significance of including molecular alignment in the optical dipole force by a trajectory study that compares previously used approximations with the present approach. The trajectory simulations were carried out for an ensemble of linear molecules subject to either propagating or standing-wave optical fields for a range of temperatures and laser intensities. The results demonstrate that the alignment-dependent effective polarizability can serve to provide correct estimates of the optical dipole force, on which a state-selection method applicable to nonpolar molecules could be based. We note that an analogous analysis of the forces acting on polar molecules subject to an inhomogeneous static electric field reveals a similarly strong dependence on molecular orientation.

carried out for an ensemble of linear molecules subject to either propagating or standingwave optical fields for a range of temperatures and laser intensities. The results demonstrate that the alignment-dependent effective polarizability can serve to provide correct estimates of the optical dipole force, on which a state-selection method applicable to nonpolar molecules could be based. We note that an analogous analysis of the forces acting on polar molecules subject to an inhomogeneous static electric field reveals a similarly strong dependence on molecular orientation. *Corresponding author zhao@unist.ac.kr

Introduction
Current laser technology has made it possible to generate spatially and temporally well defined optical fields -whether propagating or standing-wave -that can be used to manipulate molecular motion and create molecule-optical elements, such as lenses [1][2][3] and prisms [4,5], as well as to decelerate molecules [6][7][8][9][10][11]. Manipulating the translation of other than spherical molecules by optical fields entails manipulating their rotation first. The field hybridizes the rotational states of the molecules and thereby creates directional states in which their induced dipole moments are aligned with respect to the polarization vector of the field. Only in such directional states are the molecular body-fixed dipole (or higher) moments accessible in the laboratory frame and can be acted upon by space-fixed fields in order to achieve efficient manipulation of their translation. Whether the directional states will be created and hence strong optical dipole forces exerted by the optical field on the molecules is contingent upon the symmetry of the molecules' polarizability tensor: only for an anisotropic polarizability whose principal components are not all equal to one another will the hybridization of rotational states by the interaction with an optical field take place [12][13][14][15][16].
For instance, the polarizability tensor of every linear molecule is anisotropic, with the principal polarizability component along the molecular axis exceeding that perpendicular to it; this makes all linear molecules amenable to facile manipulation by an optical field.
In most of the previous experimental studies, a rotation-averaged molecular polarizability was used to analyze the experimental data [1,2,4,[6][7][8]10], in which neither the effects of the rotational state nor of the molecular alignment on the optical dipole force were considered. Although it was demonstrated during the last decade that the optical dipole force is modified by the field-induced molecular alignment, the state dependence of the molecular polarizability was still ignored [9,11]. Not until very recently was the rotational-statedependent molecular polarizability used to interpret the transverse dispersion of CS 2 molecules subject to pulsed optical standing waves, although the molecular alignment effect was neglected [5].
In contrast to the above experimental studies, their theoretical counterparts made use of the state dependence of molecular alignment, since the degree of molecular alignment intrinsically depends on the rotational state of the molecule [12][13][14]16]. Translational motion of molecules subject to propagating laser fields was traced with quantum-mechanical [17], hybrid quantum-classical [17][18][19] and classical [20] trajectory methods, in which the statedependent molecular alignment was included in the Hamiltonian of the system under study.
However, the trajectories were calculated without considering the relation between the molecular alignment and the optical dipole force. Therefore, in these theoretical approaches, the effect of the alignment on the optical dipole force was not explicitly included.
Furthermore, these studies focused on the low-intensity [20,21] or the high-intensity limits [17][18][19][20], wherein the molecules are hardly aligned or the degree of their alignment approaches its maximum, respectively. However, in particular when molecules travel through an optical standing wave, some of them probe the wave's whole intensity range --from zero to the peak value. Therefore, the intermediate intensity range, in which the molecules are aligned partially, is key for understanding the relation between the molecular alignment and the optical dipole force. The intensity range varies for different initial rotational states of the molecules and hence the alignment for which they intrinsically allow. For instance, at a certain laser intensity, the high-field limit is reached for molecules in the rotational ground state, whereas molecules occupying rotationally excited states may be hardly affected by the same laser field.
The rotational-state-dependent molecular alignment is a precondition for exerting the rotational-state-dependent optical dipole force, which, in turn, is the basis for selecting nonpolar molecules that occupy a specific quantum state. There has been continuous interest in the separation and discrimination of molecules by using nonresonant laser beams [4,5,9,17,[21][22][23][24][25], which includes isotope and spin isomer separation. New techniques based on the nonresonant optical dipole force would complement the methods for state selection of polar molecules via inhomogeneous static electric fields [26][27][28][29][30][31]. Therefore, the development of new tools for separating nonpolar molecules requires the proper evaluation and optimization of the optical rotational-state-dependent dipole force.
Based on our analysis presented herein, we introduce a new kind of effective polarizability that explicitly connects the state-dependent molecular alignment with the optical dipole force. The state-dependent variation of such an effective polarizability is directly reflected in the optical dipole force. The calculated optical dipole force is then utilized to investigate the deflection and dispersion of CS 2 (X 1 ) molecules subject to either a propagating laser field or an optical standing wave as an example. The intensity of interest in this study includes the intermediate range for each rotational state as well as the low-and the high-intensity ranges. We demonstrate that the new effective polarizability that includes the effect of the rotational-state-dependent alignment can be used as a general guide in assessing the optical dipole force. Therefore, our study provides an approach toward the optimization of the state-dependent optical dipole force which is a prerequisite for the quantum state selection of nonpolar molecules. We also examine the counterpart of the new effective polarizability that arises in the context of the dipole force exerted on polar molecules by an inhomogeneous static electric field. 1(a). The second scheme pertains to the experiment by Sun et al. [5], who studied the transverse dispersion of molecules brought about by pulsed optical standing waves, as illustrated in Fig. 1(b). In Fig. 1, "IR1" and "IR2" denote pulsed, linearly polarized infrared (IR) laser beams. In the second scheme, the requisite pulsed standing wave is generated by two counter-propagating beams IR1 and IR2, cf. Fig. 1(b). In what follows, we consider a supersonic beam of CS 2 molecules whose rotational temperature T is assumed to be 1 or 35 K.
The former temperature can be achieved by state-of-the-art molecular beam sources [32] and the latter was used in previous experimental studies [5,9,11]. The molecular beam crosses the IR beam or the optical standing wave at right angles. We choose the infrared laser beam (IR1), the molecular beam, and the laser polarization directions to be along the x, z, and y axis, respectively. The coordinate origin is at the focal point of IR1 and IR2 and the time origin (t=0) is given by the maximum intensity of the pulsed laser beam. The respective intensities of the propagating and the standing waves can be written as follows: Here, I 0 ,  0 , , and  are the peak intensity, waist radius (e −2 radius), pulse duration (full width at half maximum, FWHM), and wavelength of IR1 and IR2, respectively. We choose  0 = 23.5 m,  = 10 ns, and  = 1064 nm for our study. Since the rotational period B/ħ = 49 ps for CS 2 (whose rotational constant B is 0.109 cm −1 ) is much smaller than the pulse duration , the rotational motion of the molecule in the laser field is adiabatic [33].
Below, we trace the trajectories of 10 6 molecules starting at their initial positions (x 0 , y 0 , z 0 ) with initial velocities (v 0x , v 0y , v 0z ) at t = −30 ns, whose passage through one of the two laser fields results in a transverse velocity distribution g(v x , v y ) at t = 30 ns.

Theory
The interaction potential between a 1  molecule (such as the linear ground-state CS 2 ) and a laser field of intensity I is where  || and ⊥ are the polarizability components parallel and perpendicular to the molecular axis,  the polar angle between the molecular axis and the laser polarization axis (i.e., the y axis), and Z 0 the vacuum impedance.
In the absence of the field (I=0), U = 0, in which case the molecule undergoes free field that affects molecular translation exceeds the rotational period by several orders of magnitude [17], the translational motion is governed by the following approximate potential: Here, which is the polarizability component along the space-fixed laser polarization axis [18], and was termed the effective polarizability in previous experimental works [9,11]. The interaction potential, Eq. (3), and the space-fixed polarizability component, Eq. (4), depend on the degree of molecular alignment <cos 2 > J,M and so does the resulting dipole force The two terms in the curly brackets of Eq. is counterintuitive, since the rotational states of |M| º 0, whose rotational plane includes the laser polarization axis, are expected to be aligned by the lower laser intensity. We tentatively attribute this behavior to the inclination in I (min) , which is closely correlated to I (+) .
This behavior comes about as follows: As the laser intensity I increases, the for (J -|M|) even and The approach used to develop the F-effective polarizability can also be applied to the interaction between a dipole moment and an inhomogeneous static electric field E(r), whose potential is given by: Here,  eff is the space-fixed electric dipole moment. The interaction results in a dipole force given by: The explicit formula for the dipole force has been approximated by  eff "E(r) [28,30,31] by neglecting the second term (d eff /dE)E, which is a counterpart of  J,M (I) in the optical dipole force. However, the magnitudes of the first and second terms are the same when  eff is a linear function of E. The absolute ground states of para and ortho water [35] can be approximated using this condition. Furthermore, the second term which is, by the Hellmann-Feyman theorem [15,36], proportional to the orientation cosine, can be an order of magnitude respectively. In order to illustrate the spatial variation of the two potentials, we consider the potential U J,M (z) along y =  0 /2 at t = 0 for the former, Eq. (12a), and the potential U J,M (x) along the x axis, at t = 0 for the latter, Eq. (12b), i.e.: respectively. Concentrating on the change of the transverse velocity of the molecules, we examine the following optical dipole forces: The effect of rotational-state-dependent alignment on the interaction potential and the optical dipole force is studied in two ways: by increasing J with M = 0 and by increasing |M| for a given J. In Figs. 4-6, we depict the negative interaction potential and the corresponding dipole force using black and red solid curves, respectively. For comparison, we also plot the two functions calculated without considering the molecular alignment, namely, using observable such as a molecular velocity [1,2,5,9].

C. Velocity distributions
In this section, we make use of the optical dipole force that depends on the rotationalstate-dependent molecular alignment to calculate the transverse velocity distribution g(v x , v y ) of molecules that have passed through the propagating wave or the optical standing wave, cf.
Since, for the first scheme in Fig. 1, we neglect the x-dependence of the propagating laser intensity in Eq. (1a), we use the velocity profile h(v y ) along the v y axis as a proxy for g(v x , v y ).
In the second scheme -i.e. that is the molecular dispersion caused by the optical standing wave with a relatively low I 0 -we consider molecules passing near y = 0, which allows us to use the velocity profile h(v x ) along the v x axis [5].
The Monte Carlo sampling method is used to select the initial velocity (v 0x , v 0y , v 0z ), the initial position (x 0 , y 0 , z 0 ), and the initial rotational state |J,M> of each individual molecule.
We follow the approximation used in the previous work by Sun et al. [5] for Here, t detection and t simul are the detection and the total simulation time, respectively. From this initial point, the individual molecule arrives at the detection plane z = v mp t detection at t = t detection . x 0 and y 0 are chosen randomly from a 600 m × 3 m rectangle, whose y-center is set to  0 /2 and 0 for the first and the second schemes, respectively. The initial rotational state follows the Boltzmann distribution e -Bj(j+1)/kT /q r , where k is Boltzmann's constant and q r the rotational partition function. Furthermore, the weighted center of the red solid line calculated for T =1 K is shifted further toward slower velocities than the one calculated for T = 35 K.
In order to explain these aspects, we plot the v y (z) value of various initial rotational states with v 0y , v 0z , and y 0 set to 0, v mp , and  0 /2, respectively. Figures 8(a)-(c) show v y (z) for J ≤ 4, J = 10, and J = 20, respectively. The red and black solid curves in Fig. 8(b) illustrate  Fig. 7(b). Furthermore, the degree of the state-dependent alignment is larger for low-J states, which results in a larger velocity change as shown in Fig. 8(a).
Therefore, the red solid curve for T = 1 K is shifted further than the one pertaining to T = 35 K, at which temperature the less-aligned molecules in high-J states dominate the h(v y ) distribution.
Using the same format as Fig. 8, Fig. 9 shows Our work therefore provides a way for estimating and tailoring the state-dependent optical dipole force more accurately, thus paving the way for developing separation techniques that can be applied to any polarizable molecules, including nonpolar ones: mixtures of nonpolar conformers, isotopes of homonuclear diatomic molecules, or of their spin isomers.