Enantioselective Orientation of Chiral Molecules Induced by Terahertz Pulses with Twisted Polarization

Chirality and chiral molecules are key elements in modern chemical and biochemical industries. Individual addressing, and the eventual separation of chiral enantiomers has been and still is an important elusive task in molecular physics and chemistry, and a variety of methods has been introduced over the years to achieve this goal. Here, we theoretically demonstrate that a pair of cross-polarized THz pulses interacting with chiral molecules through their permanent dipole moments induces an enantioselective orientation of these molecules. This orientation persists for a long time, exceeding the duration of the THz pulses by several orders of magnitude, and its dependency on temperature and pulses' parameters is investigated. The persistent orientation may enhance the deflection of the molecules in inhomogeneous electromagnetic fields, potentially leading to viable separation techniques.


I. INTRODUCTION
Molecular chirality was discovered in the 19th century by Louis Pasteur [1] (for a historical excursion, see e.g. [2]), and since then chirality has attracted considerable interest owing to its importance in physics, chemistry, biology, and medicine. Chiral molecules exist in two forms, called left-, and right-handed enantiomers, which are mirror images of each other and cannot be superimposed by translations or rotations [3]. Even though the two enantiomers have many identical chemical and physical properties, e.g. boiling points and moments of inertia, they often differ in their biological activities. This is only one of the reasons why the abilities to differentiate, selectively manipulate, and separate the enantiomers in mixtures containing both of them are of great practical importance. From the point of view of fundamental physics, chiral molecules seem to be promising candidates for experiments aimed at measuring parity violation effects [4].
In this work, we study the orientation of chiral molecules induced by terahertz (THz) pulses with twisted polarization. Since the advent of THz pulse technology, intense THz pulses have been exploited for producing transient field-free orientation of polar molecules of various complexity [41][42][43][44][45][46][47]. Although transient orientation revival spikes may periodically appear on long time scales, the orientation signature generally rides on a zero baseline and its time average is exactly zero. Recently, it was shown that a single linearly polarized THz pulse induces persistent orientation in symmetric-and asymmetric-top molecules, including chiral molecules [48]. In contrast to the transient signal, persistent orientation means that the time-averaged post-pulse orientation degree differs from zero on a long time scale, in the case discussed in this paper -exceeding the duration of the THz pulse by several orders of magnitude.
Here, we theoretically demonstrate that when applied to chiral molecules, a pair of time-delayed cross-polarized THz pulses induces orientation in a direction perpendicular to the plane spanned by the polarizations. The orientation is enantioselective, meaning that the two enantiomers are oriented in opposite directions relative to the plane. We show that the time-averaged projections of the molecular dipole moment onto all three laboratory axes remains nonzero on a nanosecond timescale. The paper is organized as follows. In Sec. II, we briefly summarize our theoretical methods. In Sec. III, we present and analyze the results of classical and fully quantum simulations of THz field-driven molecular rotational dynamics, and Section IV concludes the presentation.

II. METHODS
We carried out classical as well as fully quantum mechanical simulations of the THz field-driven rotational dynamics of chiral molecules. This section outlines the theoretical approaches used in both cases.
Quantum simulation. The Hamiltonian describing the molecular rotation driven by a THz field interacting with the molecular dipole moment is given by [49,50] whereĤ r is the rotational kinetic energy Hamiltonian andĤ int (t) = −μ · E(t) is the molecule-field interaction.
Hereμ is the molecular dipole moment operator and E(t) is the external electric field. In this work, the contributions of higher order interaction terms are small, and are not included. For the quantum mechanical treatment, it is convenient to express the Hamiltonian in the basis of field-free symmetric-top wave functions |JKM [51].
Here J is the value of the total angular momentum (in units of ), while K and M are the values of the projections on the molecule-fixed axis (here the axis with smallest moment of inertia) and the laboratory-fixed Z axis, respectively. The nonzero matrix elements of the asymmetric-top kinetic energy operator are given by [51] JKM where f (J, K) = (J 2 − K 2 )[(J + 1) 2 − K 2 ], A = 2 /2I a , B = 2 /2I b , C = 2 /2I c are the rotational constants (A > B > C), and I a < I b < I c are the molecular moments of inertia. The time-dependent Schrödinger equation i ∂ t |Ψ(t) =Ĥ(t)|Ψ(t) is solved by numerical exponentiation of the Hamiltonian matrix (see Expokit [52]), and a detailed description of our numerical scheme can be found in [48]. In our simulations, the computational basis included all the rotational states with an angular momentum J ≤ 44. For the case of propylene oxide molecules which is used as an example, at a temperature of T = 5 K, initial states with J ≤ 8 are included in the thermal averaging.
Classical simulation. In the classical limit, chiral molecules are modeled as asymmetric tops driven by an external torque. The classical equations of motion for the angular velocities (written in the frame of principal axes of inertia tensor) are the Euler's equations [53] where Ω = (Ω a , Ω b , Ω c ) is the angular velocity vector, I = diag(I a , I b , I c ) is the moment of inertia tensor, and T = (T a , T b , T c ) is the external torque vector. The external torque originates from the electric field, which is defined in the laboratory frame. In order to solve Eq. (4), a time-dependent relation between the molecular frame (the frame of principal axes of inertia) and the laboratory frame is required. Such a relation can be established with the help of a time-dependent unit quaternion, which is used to parametrize the rotation of the rigid body. Quaternions extend complex numbers, and are defined as quadruples of real numbers, q = (q 0 , q 1 , q 2 , q 3 ). The relation between a vector x in the molecular frame and a vector X in the laboratory frame is given by a simple linear transformation x = Q(t)X, where Q(t) is a 3×3 matrix composed of the quaternion's elements [54,55]. The quaternion obeys the following equation of motioṅ where Ω = (0, Ω) is a pure quaternion and the quaternion multiplication rule is implied [54,55]. Equations (4) and (5) are coupled via the torque, T = µ × QE, where µ is the molecular dipole moment vector.
To simulate the behavior of a classical ensemble, we use the Monte Carlo approach. For each individual asymmetric top, we numerically solve the system of Eqs. (4) and (5). We use ensembles consisting of N = 10 7 molecules, which are initially isotropically distributed in space, and the initial angular velocities are given by the Boltzmann distribution where T is the temperature and k B is the Boltzmann constant. The initial uniform random quaternions were generated using the recipe described in [56].

III. RESULTS
We consider propylene oxide (referred to as PPO hereafter) as a typical example of a chiral molecule. Table I summarizes the molecular properties of the right-handed enantiomer, (R)-PPO. Molecular moments of inertia and  the components of molecular dipole moments were computed with the help of the GAUSSIAN software package (CAM-B3LYP/aug-cc-pVTZ method) [57].
The molecules are excited by a pair of delayed crosspolarized THz pulses. The combined electric field of the pulses is modeled using [45] where E 0 is the peak amplitude, f (t) = (1 − 2at 2 )e −at 2 defines the time dependence of each pulse, a determines the temporal width of the pulse, τ is the time delay between the peaks of the two pulses, and e X and e Y are the unit vectors along the laboratory X and Y axes, respectively. The pulses propagate along the laboratory Z axis, while E twists in the XY plane, as shown in Fig. 1. Figure 2 shows the dipole moment projections along the three laboratory axes, µ X , µ Y , and µ Z , as functions of time. The angle brackets · denote the incoherent average of initial thermally populated rotational states, or the ensemble average in the classical case. The parameters of the field are a = 3.06 ps −2 , τ = 0.80 ps, and E 0 = 8.0 MV/cm (corresponding to the peak intensity of 8.5 × 10 10 W/cm 2 ), see Eq. (7) and Fig. 1. Note that THz pulses with peak amplitudes of tens of MV/cm, especially with the use of field enhancement structures [58][59][60], are experimentally available. It is evident from the insets in Fig. 2 that on the short time scale the classical and quantum results are in excellent agreement. Each of the THz pulses is followed by a splash of dipole signal in the direction of the pulse polarization, i.e. initially along the X axis, and then along the Y axis [see the minima in the insets of Figs. 2(a) and 2(b), which are before and after 0 ps, respectively]. This transient orientation induced by single THz pulses is expected and has been observed before [41][42][43][44][45][46][47][48].
However, an unexpected result emerges: a dipole projection along the Z axis (perpendicular to the plane of polarization twisting) appears after the second pulse [see Fig. 2(c)]. This orientation which is unique to chiral molecules is enantioselective, in the sense that the sign of the projection is opposite for the two enantiomers, positive for (S )-PPO and negative for (R)-PPO. The enantioselectivity of the orientation in the Z direction can be formally derived as well [for details, see Appendix A]. Similar enantioselective orientation was observed in chiral molecules optically excited by laser fields with twisted polarization acting on the molecular polarizability [21,22,24].
Furthermore, the classical results clearly show another remarkable feature of the induced orientation. After the field is switched off [t > 2.5 ps, see Fig. 1(b)], the dipole projections along all three laboratory axes persist on the nanosecond timescale. The direct quantum simulation deviates from the classical one on the long time scale, exhibiting quantum beats/revivals [61][62][63]. Nevertheless, as can be seen from Fig. 3, on a timescale of 0.5 ns the time-averaged quantum signals reproduce well the steady state dipole signals obtained by the classical calculation.
Classically, the persistent long-term orientation shown in Figs. 2 and 3 is in fact permanent. In the absence of external torques (external fields), in the laboratory frame the angular momentum vector is conserved, while in the molecular frame the angular momentum follows a fixed trajectory, which can be visualized using the Binet construction [53]. Although the absolute orientation of an asymmetric top in the laboratory frame never recurs, depending on the energy and magnitude of the angular momentum, the projections of the molecular principal axes a or c on the conserved angular momentum vector have a constant sign [53]. As a result, the attained asymptotic values of the orientation factors do not change after the initial dephasing of the ensemble which, according to Fig. 2, takes about 30 ps. On the other hand, quantum mechanically, the notion of welldefined trajectories is invalid, which means that the orientation is simply long-lived and eventually will change its sign. Since the kinetic energy Hamiltonian [see Eq.
(3)] couples rotational states with different K quantum number, the quantum-mechanical asymmetric top does not have permanently oriented eigenstates. Any initially oriented state will eventually oscillate between being oriented and anti-oriented, an effect known as dynamical tunneling [64]. Formally, for a quantum mechanical chiral rotor, one would expect no permanent orientation after the turn-off of all external fields. However, as we show here and as was shown in [23,25], the tunneling timescale may exceed the excitation timescale by orders of magnitude.
Notice, the persistent orientation appearing in the directions of each of the pulses (along the X and Y axes) does not rely on chirality. It was recently shown that single THz pulses applied to symmetric-and asymmetrictop molecules also induce persistent orientation [48]. In contrast, both the appearance and permanency of the orientation along the propagation direction which is perpendicular to both the X and Y axes depend on the chirality of the molecule. Specifically, these effects rely on the lack of molecular symmetry, i.e. all three molecular dipole moment components must differ from zero, µ a , µ b , µ c = 0 [see Appendix B]. For comparison, in the case of optical excitation by the laser fields with twisted polarization, the orientation relies on the existence of the off-diagonal elements of the polarizability tensor, which is a property of chiral molecules as well. In that case, the induced dipole has nonzero projections along all three principal axes, even when the laser field is polarized along only one of the molecular principal axes.
The magnitude of the THz-induced orientation is sensitive to the initial temperature and external field parameters [47,48,65,66]. Through our classical simulations, we carried out an extensive study of the permanent orientation dependence on temperature, T , and the time delay between the two cross-polarized pulses, τ [see Eq. (7)]. The results are summarized in Fig. 4. The permanent values of the dipole projections (denoted by µ X p , µ Y p , and µ Z p ) were obtained by following the field-free dynamics for a sufficiently long time until a steady-state is reached (typically t > 100 ps). Figure 4 shows that for a given rotational temperature, there are one or several disjoint ranges of τ resulting in optimal (largest absolute value) orientation. In general, the optimal time delay between the pulses is shorter for higher temperatures. Note, however, that the temperature dependence of the individual projections µ X p , µ Y p , and µ Z p is non-monotonic. For example, at a fixed time delay τ ≈ 0.5 ps, | µ Z p | increases with temperature up to T ≈ 120 K, after which | µ Z p | begins to decrease.

IV. CONCLUSIONS
We theoretically demonstrated a qualitatively new phenomenon of field-free enantioselective orientation of chiral molecules induced by THz pulses with twisted polarization. The twisted pulse induces orientations along the polarization directions of the two pulses forming the twisted pulse, and this orientation is of the same sign for both enantiomers. In the direction perpendicular to the polarization direction of both pulses, we find that the orientation is of opposite sign for the two enantiomers. The latter effect relies on the molecular chirality, namely on the lack of molecular symmetry, such that the molecular dipole has nonzero projections on all three molecular principal axes. The orientation was shown to persist long after the end of the pulses. We studied the dependence of persistent orientation values on the temperature and the time delay between the two cross-polarized THz pulses. The orientation factors were found to be quite robust against the detrimental effects of temperature provided that the time delay is adjusted appropriately. The orientation dynamics on timescales beyond nanoseconds requires a more detailed analysis, as other effects such as collisions and fine structure effects [67] become important in addition to dynamical tunneling. The relative importance of such effects should be assessed in future works. The orientation persisting on the nanosecond timescale may be measured by means of second (or higher order) harmonic generation [68], and could be used for deflection by inhomogeneous electric fields [69,70] (for an extensive review, see [71]). The enantioselective orientation along the propagation direction may be useful for fast and precise analysis of enentiomeric excess, and may facilitate enantioselective separation using inhomogeneous fields [72]. In the past [20][21][22][23][24][25], related effects induced by optical pulses with twisted polarizations have been reported, and further exploration will examine the combined effect of THz and optical fields together that could maximize the difference in orientations between the two enantiomers. Consider two overlapping THz pulses propagating along the laboratory Z axis, and which are polarized along X and Y axes, respectively. The nonzero matrix elements of the interaction potential can be written as where s = 0, ±1, and p = ±1 (since E where the wave function is given by |Ψ(t) = U (t, 0)|Ψ(0) , withÛ (t, 0) being the evolution operator. Since the discussion here is qualitative, for simplicity we assume that initially the rotor is in the ground rotational state |JKM = |000 , such that |Ψ(t) =Û (t, 0)|000 .
The evolution operator,Û (t, 0) can be expanded in a Dyson serieŝ components of polarization are Note that for the qualitative argument here, the evolution operator in Eq. (A4) is considered aŝ n ′ )|000 = 0. Also, according to Eq. (A1), the allowed transitions are between the states with M and M ± 1, because p = ±1. Hence, it follows from Eq. (A4) that the state changing from 0 to 2M 1 involves an even number of interaction terms (Ĥ int ). Similarly, transitions between states with K quantum number equals 0 and 2K 1 − s involve 2l + s interaction terms with s = 0 [see Eq.
where s = 0, ±1, and L ′ is an integer. As one can see, the components of polarization of the two enantiomers are of opposite signs, and thereby the polarizations satisfy P Appendix B: Orientation of non-chiral molecules Figure 5 shows the orientation signals along the three laboratory axes, µ X (t), µ Y (t), and µ Z (t), for two non-chiral molecules. The first one is the symmetric-top methyl chloride (CH 3 Cl), in which the molecular dipole moment is along the symmetry axis, a axis (µ b = µ c = 0, see Table II). The second molecule, sulfur dioxide (SO 2 ) is an asymmetric-top molecule in which the molecular dipole moment is along the molecular b axis (µ a = µ c = 0, see Table II). The parameters of the THz pulses are similar to Fig. 2. The initial temperature is T = 5 K. The quantum and classical results are in good agreement. As expected, both the symmetric-[Figs. 5(a) and 5(b)] and non-chiral asymmetric-top [Figs. 5(d) and 5(e)] molecules immediately respond to the X-polarized and Y -polarized pulses. However, in contrast to chiral molecules [see Fig. 2], the Z-projection of the dipole moment remains zero [see Figs. 5(c) and 5(f)]. Note that µ Z calculated quantum mechanically is identically zero, while the small amplitude oscillations appearing in the classical results are due to the finite number (N ) of molecules in the ensemble. The lack of orientation in the cases of symmetric-or non-chiral asymmetric-top molecules indicates that all three molecular dipole components (µ a , µ b , µ c ) are indeed required for inducing the perpendicular (in the Z direction) orientation. Other combinations were considered numerically as well (not shown), e.g. asymmetric top molecule having two nonzero molecular dipole components. In all cases, there is no perpendicular orientation, i.e. µ Z = 0.