††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Echo-enhanced molecular orientation at high temperatures

Ilia Tutunnikov  AMOS and Department of Chemical and Biological Physics, The
Weizmann Institute of Science, Rehovot 7610001, Israel  Long Xu  AMOS and
Department of Chemical and Biological Physics, The Weizmann Institute of
Science, Rehovot 7610001, Israel  Yehiam Prior<EMAIL_ADDRESS>AMOS and Department of Chemical and Biological Physics, The Weizmann Institute
of Science, Rehovot 7610001, Israel  Ilya Sh. Averbukh
<EMAIL_ADDRESS>AMOS and Department of Chemical and Biological
Physics, The Weizmann Institute of Science, Rehovot 7610001, Israel

###### Abstract

Ultrashort laser pulses are widely used for transient field-free molecular
orientation – a phenomenon important in chemical reaction dynamics, ultrafast
molecular imaging, high harmonics generation, and attosecond science. However,
significant molecular orientation usually requires rotationally cold
molecules, like in rarified molecular beams, because chaotic thermal motion is
detrimental to the orientation process. Here we propose to use the mechanism
of the echo phenomenon previously observed in hadron accelerators, free-
electron lasers, and laser-excited molecules to overcome the destructive
thermal effects and achieve efficient field-free molecular orientation at high
temperatures. In our scheme, a linearly polarized short laser pulse transforms
a broad thermal distribution in the molecular rotational phase space into many
separated narrow filaments due to the nonlinear phase mixing during the post-
pulse free evolution. Molecular subgroups belonging to individual filaments
have much-reduced dispersion of angular velocities. They are rotationally
cold, and a subsequent moderate terahertz (THz) pulse can easily orient them.
The overall enhanced orientation of the molecular gas is achieved with some
delay, in the course of the echo process combining the contributions of
different filaments. Our results demonstrate that the echo-enhanced
orientation is an order of magnitude higher than that of the THz pulse alone.
The mechanism is robust – it applies to different types of molecules, and the
degree of orientation is relatively insensitive to the temperature. The laser
and THz pulses used in the scheme are readily available, allowing quick
experimental demonstration and testing in various applications. Breaking the
phase space to individual filaments to overcome hindering thermal conditions
may find a wide range of applications beyond molecular orientation.

##  I Introduction

Control over the absolute orientation of molecules in space is essential for a
wide variety of physical and chemical processes such as ultra-fast structural
imaging, photon-induced processes, or chemical reactions Stapelfeldt and
Seideman (2003); Ohshima and Hasegawa (2010); Fleischer  _et al._ (2012);
Lemeshko  _et al._ (2013); Koch  _et al._ (2019); Lin  _et al._ (2020).
Diverse optical methods have been developed to align and orient molecules in
the gas phase under field-free conditions. A well-established route for
orienting polar molecules is combining intense non-resonant laser fields with
weak electrostatic fields Friedrich and Herschbach (1999a, b); Sakai  _et al._
(2003); Goban  _et al._ (2008); Ghafur  _et al._ (2009); Holmegaard  _et al._
(2009); Mun  _et al._ (2014); Takei  _et al._ (2016); Omiste and González-
Férez (2016); V. Thesing  _et al._ (2017); another one involves using intense
two-color laser pulses to induce molecular orientation via interaction with
the molecular hyperpolarizability Vrakking and Stolte (1997); Dion  _et al._
(1999); Kanai and Sakai (2001); Takemoto and Yamanouchi (2008); De  _et al._
(2009); Oda  _et al._ (2010); Wu and Zeng (2010); Zhang  _et al._ (2011a);
Frumker  _et al._ (2012); Spanner  _et al._ (2012); Znakovskaya  _et al._
(2014); Lin  _et al._ (2018); Mellado-Alcedo  _et al._ (2020); Wang and
Henriksen (2020a); Xu  _et al._ (2021). For chiral molecules, it is possible
to induce orientation by using laser pulses with twisted polarization
Yachmenev and Yurchenko (2016); Gershnabel and Sh. Averbukh (2018); Tutunnikov
_et al._ (2018); Milner  _et al._ (2019); Tutunnikov  _et al._ (2019, 2020);
Xu (2022). With the advance in terahertz (THz) technology, it has become
possible to orient polar molecules using single- or half-cycle THz pulses
through coupling to the permanent molecular dipole moment Harde  _et al._
(1991); Machholm and Henriksen (2001); Fleischer  _et al._ (2011); Kitano  _et
al._ (2013); Egodapitiya  _et al._ (2014); Damari  _et al._ (2016); Babilotte
_et al._ (2016, 2017); Xu  _et al._ (2020a); Tutunnikov  _et al._ (2021a);
Beer  _et al._ (2022).

The efficiency of the above-mentioned methodologies deteriorates fast with
increasing rotational temperature, and the molecular orientation is typically
just a few percent at room temperature – too low for many practical
applications. High temperature is detrimental to efficient orientation due to
the increased dispersion of the molecular angular velocities with temperature.
In laser-based methods, the degree of orientation may be enhanced by simply
increasing the field intensity. Still, this approach has a practical limit
because high-intensity laser pulses, especially two-color fields, give rise to
significant molecular ionization, an undesired consequence of the strong
field. On the other hand, in the case of THz pulses, the available field
strengths are insufficient for achieving a sizable degree of orientation.
Thus, attaining efficient molecular orientation at room temperatures is still
an open, very challenging problem.

Over the years, several sophisticated proposals for enhancing molecular
alignment and orientation have been put forward. These include applying
optimized sequences of pulses Sh. Averbukh and Arvieu (2001); Leibscher  _et
al._ (2003, 2004); Zhang  _et al._ (2011b, 2016), optimizing pulses’
parameters Nakajima  _et al._ (2012); Mun and Sakai (2018); Mun  _et al._
(2019); Mun and Kim (2020); Md. Maruf Hossain and Sakai (2020); Yu and Wang
(2021); Md. Maruf Hossain  _et al._ (2022); Wang and Henriksen (2020b); Zhou
_et al._ (2021); Xu  _et al._ (2022), and combining one-color laser pulses
with two-color and THz pulses Spanner  _et al._ (2004); Daems  _et al._
(2005); Gershnabel  _et al._ (2006a, b); Kitano  _et al._ (2011); Zhang  _et
al._ (2011c); Tehini  _et al._ (2012); Shu and Henriksen (2013); Liu  _et al._
(2013); Tao  _et al._ (2018). Several schemes were experimentally implemented
Bisgaard  _et al._ (2004); Pinkham  _et al._ (2007); Egodapitiya  _et al._
(2014); Sonoda  _et al._ (2018) and demonstrated improved orientation, mainly
in the low-temperature regime.

Here, we propose an efficient and general approach for achieving significant
field-free molecular orientation by moderate THz pulses even at room
temperature. The method uses a pair of time-delayed pulses arranged in the
echo configuration. The first, non-resonant laser pulse transforms a broad and
smooth thermal probability-density distribution in the molecular rotational
phase-space into multiple filaments (narrow strips). The second (typically
weak) delayed THz excitation is then applied to the “filamented” molecular
ensemble and causes a significant orientation of molecular groups belonging to
different narrow filaments, each evolving independently. The enhanced
ensemble-averaged orientation is achieved after some additional delay when the
contributions of the independent filaments come together by the mechanism of
echo formation. We show that for a proper choice of the excitation parameters,
the amplitude of the orientation echo may be an order of magnitude higher than
the orientation produced by the THz pulse alone. Moreover, the “filamenting”
prepulse suppresses the detrimental effect of the molecular thermal rotational
motion, making the echo amplitude rather insensitive to elevated temperatures.

Echoes are known in various domains of physics. The most notable example is
Hahn’s spin-echo Hahn (1950, 1953) induced by two delayed magnetic field
pulses. They result in a delayed magnetization response appearing after an
additional delay equals the delay between the pulses. Various types of echoes
have been discovered and studied in other fields, including photon echoes
Kurnit  _et al._ (1964); Mukamel (1995), neutron spin-echo Mezei (1972),
cyclotron echoes Hill and Kaplan (1965), plasma-wave echoes Gould  _et al._
(1967), echoes in optically trapped cold atoms Bulatov  _et al._ (1998);
Buchkremer  _et al._ (2000); Herrera  _et al._ (2012), echoes in particle
accelerators Stupakov (1992); Spentzouris  _et al._ (1996); Stupakov (2013);
Sen and Li (2018), and echoes in free-electron lasers Hemsing  _et al._
(2014). In addition, echoes in single quantum systems have been discussed
Tutunnikov  _et al._ (2021b, c) and observed Morigi  _et al._ (2002); Meunier
_et al._ (2005); Qiang  _et al._ (2020).

The enhancement mechanism of the present paper is related to the recently
predicted and observed molecular alignment echoes Karras  _et al._ (2015,
2016); Lin  _et al._ (2016, 2017); Rosenberg  _et al._ (2018); Zhang  _et al._
(2019); Ma  _et al._ (2019); Hartmann  _et al._ (2020); Ma  _et al._ (2020);
Lin  _et al._ (2020); Xu  _et al._ (2020b); Ma  _et al._ (2021). In contrast
to the spin echoes Hahn (1950, 1953), they are based on the phenomenon of
phase space filamentation known in non-linear systems, e.g., in the dynamics
of stellar systems Lynden-Bell (1967) and in accelerator physics Guignard
(1988); Stupakov (1992, 2013). Our paper presents the first application of the
filamentation-based echoes to the problem of molecular orientation.

The paper is organized as follows. We begin with a qualitative physical
description of the scheme using a simplified model of rigid planar rotors.
Then we present the results of three-dimensional classical and fully quantum
simulations of the echo-enhanced orientation of linear and symmetric-top
molecules at room temperature. Finally, we summarize the results and discuss
the prospects of the proposed scheme.

##  II Qualitative discussion

To understand the physical principles behind the enhanced orientation, we
start with a simplified planar rotor model, where a linear polar molecule is
modeled as a rigid rotor restricted to rotate in the $XY$ plane. To begin
with, let us first consider the orientation dynamics induced by a single
orienting kick. The orienting interaction potential is $\propto-\cos(\theta)$,
where $\theta$ is the angle between the molecular axis and the $X$-axis. Such
a potential describes, e.g., the interaction of a single- or half-cycle THz
pulse with the permanent molecular dipole.

FIG1

Figure 1:  Phase space following a single orienting kick at zero initial
rotational temperature.

#### Single orienting kick –

In the impulsive approximation (when the change in $\theta$ during the pulse
is negligible), $\theta$ and the molecule’s angular velocity, $\omega$ after
the orienting kick are given by

$\theta(t)=\theta_{0}+\omega t;\quad\omega=\omega_{0}-P_{or}\sin(\theta_{0}),$
(1)

where $\theta_{0}$ and $\omega_{0}$ are the initial angle and angular
velocity. $P_{or}$ is proportional to the amplitude of the orienting field,
$\varepsilon_{0,or}$ [for details, see Appendix A, Eq. (19)] and has a
physical meaning of the molecular angular velocity change due to the kick.

To study the rotational dynamics of an ideal molecular gas, we consider a
collection of planar rotors that are initially isotropically distributed in
the $XY$ plane. At zero initial temperature, the phase space density before
the kick is presented by a horizontal line $\omega=0$ (see Fig. 1, $t=0_{-}$).
The orienting pulse transforms the initial distribution into a sine-shaped
line ($t=0_{+}$). With time, the curve folds as particles with opposite
angular velocities move in opposite directions along the $\theta$ axis. The
ensemble-averaged degree of orientation, or the orientation factor
$\braket{\cos(\theta)}$ reaches a universal maximum $\approx 58\%$,
independent of $P_{or}$ at $t=1.8/P_{or}$.

At finite, non-zero initial rotational temperature, the initial phase space
density is given by
$p(\theta,\omega,t=0)=(2\pi)^{-3/2}\sigma_{T}^{-1}\exp[-\omega_{0}^{2}/(2\sigma_{T}^{2})]$,
see Fig. 2(a). The parameter $\sigma_{T}=\sqrt{k_{B}T/I}$ defines the thermal
dispersion of angular velocities, where $T$ is the temperature, $k_{B}$ is the
Boltzmann constant and $I$ is the moment of inertia of a single molecule. In
the case of the initial thermal ensemble, the maximum degree of orientation is
determined by the ratio $P_{or}/\sigma_{T}\propto\varepsilon_{0,or}/\sqrt{T}$.
When the orienting kick is relatively strong (or, equivalently, the
temperature is relatively low), the resulting orientation is high, and vice
versa, when the kick strength is insufficient to overcome the thermal
dispersion of angular velocities. The orientation factor after the kick is
given by [for derivation, see Appendix B.1, Eq. (28)]

$\braket{\cos(\theta)}(t)=e^{-\sigma_{T}^{2}t^{2}/2}J_{1}(P_{or}t),$ (2)

where $J_{1}(z)$ is the Bessel function of order one. When time is measured in
units of $1/\sigma_{T}$, the orientation factor depends on the parameter
$P_{or}/\sigma_{T}$. Unfortunately, the currently available THz pulses result
in a weak deformation of the phase space density at room temperature. Shortly
after a typical orienting kick, the phase space density is delocalized and
looks like in Fig. 2(b). When the orienting kick is relatively strong (or,
equivalently, the temperature is relatively low), the phase space distribution
folds like in Fig. 2(c).

FIG2.pdf

Figure 2:  Phase space distribution before and after a single orienting pulse.
(a) At $t=0$, before the kick. (b) After the pulse at $t=\sigma_{T}^{-1}$ for
a relatively weak orienting pulse $P_{or}=0.25\sigma_{T}$. (c) After the pulse
at $t=\sigma_{T}^{-1}$ for a stronger orienting pulse $P_{or}=0.75\sigma_{T}$.

#### Phase space filamentation –

To help a weak orienting pulse overcome the detrimental effect of high
temperatures, we suggest stratifying the rotational phase space density into
numerous separated strips (filaments). This way, the orienting pulse interacts
with many individual narrow filaments rather than a single broadly dispersed
thermal distribution. For stratification, we use a relatively strong prepulse
– a nonresonant linearly polarized short laser pulse that precedes the
orienting one. Intense laser pulses interacting with molecular polarizability
have been widely used for inducing transient field-free molecular alignment
Stapelfeldt and Seideman (2003); Ohshima and Hasegawa (2010); Fleischer  _et
al._ (2012); Lemeshko  _et al._ (2013); Koch  _et al._ (2019). However, here
we are not interested in the alignment effect but focus on the post-alignment
stage characterized by highly stratified rotational phase space density Karras
_et al._ (2015); Lin  _et al._ (2016, 2017) (see below).

A prepulse applied at $t=0$ transforms the rotors variables according to

$\theta(t)=\theta_{0}+\omega
t;\quad\omega=\omega_{0}-P_{pre}\sin(2\theta_{0}),$ (3)

where $P_{pre}$ is proportional to the intensity of the laser prepulse [for
details, see Appendix A, Eq. (23)]. The phase space density after the prepulse
is described by

$p(\theta,\omega,t)=\frac{\sigma_{T}^{-1}}{(2\pi)^{3/2}}\exp\left[-\frac{(\omega+P_{pre}\sin[2\theta-2\omega
t])^{2}}{2\sigma_{T}^{2}}\right]\\!,\\!\\!$ (4)

where we used Eq. (3) to express $\omega_{0}$ in terms of $\omega$ and
$\theta$ after the pulse, and substituted $\omega_{0}$ into the Boltzmann
distribution. The stratifying pulse imprints a sinusoidal shape on the phase-
space density which folds with time. Shortly after the excitation, the phase-
space distribution concentrates near $\theta=0,\pi$, corresponding to
alignment along the $X$ axis [see Fig. 3(a)]. It’s important for our
discussion that with time, the phase space density undergoes filamentation and
turns into a series of almost parallel filaments (strips) separated in angular
velocity by $\pi/t$ [see Eq. (4)]. Note that longer waiting times result in
narrower filaments because the phase space volume is conserved.

FIG3

Figure 3:  Phase space distribution. (a) Shortly after the stratifying kick,
$t=0.4/\sigma_{T}$. (b) Just before the orienting kick, $t=\tau_{-}$, for
$\tau=5/\sigma_{T}$. (c) Just after the orienting kick, $t=\tau_{+}$. (d)
Orientation along the $X$ axis, before the time of the orientation echo at
$t=3\tau$. The regions marked in magenta are shown in the insets. Here
$P_{pre}=2\sigma_{T}$, $P_{or}=0.25\sigma_{T}$.

After a delay $\tau$ (at $t=\tau$), when the filaments are numerous and thin,
[see Fig. 3(b)], we apply the orienting THz kick. Now, each filament resembles
the phase-space density of an isotropic cold molecular ensemble (see Fig. 1,
$t=0_{-}$). Moreover, since the dispersion of angular velocities within each
filament is much smaller than the initial thermal dispersion, the interaction
with the weak orienting pulse results in non-negligible deformation of the
filaments [see Fig. 3(c)].

With time, each filament folds like in Fig. 2(c) developing a bunch. This
folding does not result in substantial orientation just after the orienting
kick because the bunches move relative to each other (they are spaced apart by
$\pi/\tau$ along the $\omega$ axis). Due to the “quasi-quantization” of the
angular velocities, the folded filaments synchronize near $\theta=0,\pi$ after
an additional time delay equal to $\tau$ (at $t\approx 2\tau$, not shown).
This synchronization manifests itself in alignment echo Karras  _et al._
(2015, 2016); Lin  _et al._ (2016, 2017, 2020); Xu  _et al._ (2020b). Similar
synchronization dynamics is behind the echo phenomenon in several other
nonlinear systems Stupakov (1992); Xiang and Stupakov (2009); Stupakov (2013);
Qiang  _et al._ (2020); Tutunnikov  _et al._ (2021b, c).

After an additional delay of $\tau$ (at $t\approx 3\tau$), we witness another
echo, and this is a new observation. The individual bunches re-synchronize in
an asymmetric manner resulting in molecular orientation. First, the bunches
accumulate near $\theta=0$ [just before $t=3\tau$, see Fig. 3(d)], and then
near $\theta=\pi$ [just after $t=3\tau$, not shown]. The closeups in Fig. 3(d)
demonstrate that the bunches in each filament look similar to those seen in
Fig. 2(c). Such re-synchronizations happen again at later times:
$t=5\tau,\,7\tau,\dots$, and manifest themselves in orientation echoes of
higher orders.

FIG4

Figure 4:  Orientation factor following combined excitation by stratifying and
orienting pulses. Here, $P_{pre}=2\sigma_{T}$, $P_{or}=0.1\sigma_{T}$. Note
the enhanced orientation at $t=2\tau$ for both cases of $\tau=10/\sigma_{T}$
and $\tau=20/\sigma_{T}$, with the higher degree of orientation for the longer
delay. With the orienting pulse alone, the maximal orientation factor is
$\approx 0.03$.

Figure 4 shows the time-dependent orientation factor, $\braket{\cos(\theta)}$
for two delays, $\tau=10/\sigma_{T}$ and $\tau=20/\sigma_{T}$. In both cases,
the first orientation echo emerges after a delay equal to $2\tau$. However,
compared to $\tau=10/\sigma_{T}$, the echo amplitude for $\tau=20/\sigma_{T}$
is significantly higher. In line with the qualitative arguments presented
above, longer delays give rise to thinner filaments which, in turn, “helps”
the weak orienting kick.

The orientation factor in the presence of both stratifying and orienting kicks
is given by [for derivation, see Appendix B.2, Eq. (36)]

$\braket{\cos(\theta)}(t)=\Theta(t-\tau)\sum_{\mathrm{odd}\,k}^{\infty}S_{k}(t),$
(5)

where $\Theta(t-\tau)$ is the step-function and

$\\!S_{k}(t)=e^{-\sigma_{T}^{2}(t-k\tau)^{2}/2}J_{k}[P_{or}(t\\!-\\!\tau)]J_{\frac{k-1}{2}}[P_{pre}(k\tau\\!-\\!t)].$
(6)

The orientation signal consists of a series of pulsed responses separated in
time by $2\tau$. The first orientation echo $(k=3)$ is proportional to

$\\!\\!\\!S_{3}(t)\approx
e^{-\sigma_{T}^{2}(t-3\tau)^{2}/2}J_{3}(2P_{or}\tau)J_{1}[P_{pre}(3\tau-t)].$
(7)

Here we used the facts that $J_{3}(z)$ increases with $z$ for $z<4$, and
$t-3\tau$ is relatively small (due to the Gaussian factor). This expression
shows the essential feature of the proposed scheme – for fixed temperature and
stratifying kick (prepulse) strength, $\sigma_{T}$ and $P_{pre}$, the echo
amplitude increases with $P_{or}\tau$. In other words, increasing the delay
$\tau$ is equivalent to increasing the orienting kick strength.

Moreover, the echo amplitude increases with $P_{pre}$ as well. Figure 5 shows
the maximum orientation factor for several delays as a function of $P_{pre}$
(for fixed $P_{or}$). When compared to the maximal orientation achieved by the
orienting kick alone, the addition of the prepulse enhances the orientation
starting from certain $P_{pre}$ values (depending on the delay, $\tau$). Note
that if the prepulse is weak, it has a negative effect, namely the degree of
orientation is even lower than when using only the orienting pulse. The reason
is that a weak prepulse does not generate narrow well-separated filaments
which can be distorted by the weak orienting pulse. Instead, it effectively
slightly increases the dispersion of rotational velocities.

Terms with $k>3$ in Eq. (5) correspond to the orientation echoes of a higher
order that typically (but not always) have lower amplitude compared to the
first echo Xiang and Stupakov (2009); Karras  _et al._ (2015, 2016); Lin  _et
al._ (2016, 2017, 2020).

FIG5

Figure 5:  Maximum orientation factor (absolute value) as a function of
$P_{pre}$ for fixed $P_{or}=0.1/\sigma_{T}$. Note that for the low intensity
prepulses, the degree of orientation is lower than in the absence of a
prepulse altogether. This is a result of very minimal filamentation caused by
the weak prepulses, which, in effect, slightly increases the dispersion of
rotational velocities.

In the classical case, the delay $\tau$ could, in principle, be increased
indefinitely to achieve higher orientation. However, when the delay becomes
comparable to the time of rotational quantum revivals Sh. Averbukh and
Perelman (1989); Robinett (2004), the interplay between echoes and revivals
needs to be taken into account. As was shown previously Herrera  _et al._
(2012); Tutunnikov  _et al._ (2021b), a particular type of echo exists, termed
quantum echo. A series of quantum echoes emerge on a long time scale just
before the quantum revivals. These echoes have a valuable property – in the
limit of weak echo-inducing excitation (the orienting kick in our case), their
amplitude is higher than the classical echoes discussed. We take advantage of
this to achieve even higher molecular orientation with constrained delays.

To illustrate this point, we consider the dynamics of a kicked quantum rigid
2D rotor (for details, see Appendix C). The rotational revival time of the
quantum planar rotor is $T_{rev}=4\pi I/\hbar$, where $I$ is the moment of
inertia. Figure 6 shows the orientation factor in the quantum model in the
presence of the stratifying (prepulse) and orienting kicks. Here, for
demonstration purposes, we choose a short time delay $\tau=0.025T_{rev}$
resulting in a relatively weak classical orientation echo at
$t=3\tau=0.075T_{rev}$. The amplitude of the classical echo is comparable to
the maximal orientation obtained when using the orienting pulse alone,
$\approx 3\%$. In striking contrast, a quantum echo emerges at a delay
$-2\tau$ before the revival, at $t=0.975T_{rev}$ with a much higher amplitude
of $\approx 22\%$. Notice that echoes of the same magnitude (but opposite
sign) emerge just before the half revival.

FIG6

Figure 6:  Orientation factor calculated quantum mechanically at
$T=300\,\mathrm{K}$. Here, the orienting kick is applied after a delay
$\tau=0.025T_{rev}$, which corresponds to $\tau\approx 7.2/\sigma_{T}$. The
excitation strengths, $P_{pre}$ and $P_{or}$ are the same as in Fig. 4. When
only the orienting kick is applied, the maximal orientation is $\approx 0.03$.

Alignment echoes emerging before the quantum revivals, the so-called imaginary
echoes, were studied by us both theoretically and experimentally Lin  _et al._
(2016, 2017).

##  III Three-dimensional simulations

###  III.1 Numerical methods

So far, we considered the orientation enhancement in the simplified model of
2D rigid rotors. In this section, we apply the developed understanding to
“real world” by considering the corresponding three-dimensional classical and
quantum dynamics of linear and prolate symmetric-top molecules driven by the
experimentally available laser and THz fields. The rotational dynamics is
treated within the rigid rotor approximation. We model the time-dependent
electric field, consisting of the delayed non-resonant laser and single-cycle
THz pulses, using

$\mathbf{E}(t)=\varepsilon_{lsr}(t)\cos(\omega
t)\bm{e}_{Z}+\varepsilon_{THz}(t-\tau)\bm{e}_{Z},$ (8)

where $\omega$ is the carrier frequency of the laser pulse,
$\varepsilon_{lsr}(t)=\varepsilon_{0,lsr}\exp[-2\ln 2(t/\sigma_{lsr})^{2}]$,
$\varepsilon_{0,lsr}$ is the peak amplitude, and $\sigma_{lsr}$ is the full
width at half maximum (FWHM) of the laser pulse intensity profile.
$\varepsilon_{THz}(t)=\varepsilon_{0,THz}(1-2\kappa t^{2})\exp(-\kappa t^{2})$
Babilotte  _et al._ (2016, 2017), where $\varepsilon_{0,THz}$ is the peak
amplitude of the THz pulse, $\kappa$ determines the temporal width of the THz
pulse. $\tau$ is the time delay between the laser and THz pulses and
$\mathbf{e}_{Z}$ is a unit vector along the laboratory $Z$ axis. Figure 7
shows the electric field amplitude of the THz pulse as a function of time.

FIG7

Figure 7:  Electric field of the THz pulse, given by
$\varepsilon_{THz}(t)=\varepsilon_{0,THz}(1-2\kappa t^{2})\exp(-\kappa
t^{2})$. Here, $\varepsilon_{0,THz}=1.0\,\mathrm{MV/cm}$,
$\kappa=3.06\,\mathrm{ps}^{-2}$.

#### Classical simulation –

We use the Monte Carlo method to simulate the behavior of a classical
ensemble. Initially, $N=10^{7}$ sample molecules are isotropically distributed
in space. Their angular velocities are given by the Boltzmann distribution
$P(\Omega_{i})\propto\exp[-I_{i}\Omega_{i}^{2}/(2k_{B}T)]$, where $i=x,y,z$
refers to the molecular principal axes of inertia. In the case of a linear
molecule $I_{z}=0$, $I_{x}=I_{y}\equiv I$, while in the case of a prolate
symmetric-top molecule, $I_{z}<I_{x}=I_{y}\equiv I$. For each sample molecule,
the rotational dynamics in the rotating molecular frame is described by
Euler’s equations Goldstein  _et al._ (2002)

$\mathbf{I}\bm{\dot{\Omega}}=(\mathbf{I}\bm{\Omega})\times\bm{\Omega}+\mathbf{T},$
(9)

where $\mathbf{I}=\mathrm{diag}(I_{x},I_{y},I_{z})$ is the moment of inertia
tensor, $\bm{\Omega}=(\Omega_{x},\Omega_{y},\Omega_{z})$ is the angular
velocity, and $\mathbf{T}=(T_{x},T_{y},T_{z})$ is the torque due to the
interactions with the external electric field. The laser field interacts with
the molecular polarizability
$\bm{\alpha}=\mathrm{diag}(\alpha_{xx},\alpha_{yy},\alpha_{zz})$, and the
torque is given by
$\mathbf{T}=[\bm{\alpha}\mathbf{E}_{mol}(t)]\times\mathbf{E}_{mol}(t)$. The
THz field interacts with the molecular permanent dipole moment, $\bm{\mu}$,
and the torque is given by $\mathbf{T}=\bm{\mu}\times\mathbf{E}_{mol}(t)$.
Here, $\mathbf{E}_{mol}(t)$ is the representation of the electric field vector
in the basis of the molecular principal axes. The relation between the
laboratory and the rotating molecular frames is parametrized by a quaternion,
$q=(q_{0},q_{1},q_{2},q_{3})$ Kuipers (1999); Coutsias and Romero (2004). The
quaternion’s equation of motion is $\dot{q}=q\Omega/2$, where the quaternions
multiplication rule is assumed, and
$\Omega=(0,\Omega_{x},\Omega_{y},\Omega_{z})$. The orientation factor at time
$t$ is obtained by averaging over all sample molecules (ensemble average). A
more detailed description of the classical simulations can be found in
Tutunnikov  _et al._ (2019); Xu  _et al._ (2021).

#### Quantum simulation –

The Hamiltonian describing the rotational dynamics in the presence of external
fields [see Eq. (8)] is Krems (2018); Koch  _et al._ (2019)

$\displaystyle H(t)$
$\displaystyle=H_{r}-\mu\varepsilon_{THz}(t)\cos(\theta)-\frac{\Delta\alpha}{4}\varepsilon_{lsr}^{2}(t)\cos^{2}(\theta),$
(10)

where $H_{r}$ is the rotational kinetic energy Hamiltonian, $\mu=|\bm{\upmu}|$
is the permanent molecular dipole moment,
$\Delta\alpha=\alpha_{zz}-\alpha_{xx}=\alpha_{zz}-\alpha_{yy}$ is the
polarizability anisotropy, and $\theta$ is the polar angle between the
molecular symmetry axis and the laboratory $Z$ axis (field polarization axis).
The polarizability interaction term is averaged over the optical cycle.

In the simulations, the wave function $\Psi$ is expressed in the basis of the
eigenfunctions of $H_{r}$. For a linear molecule, these are the spherical
harmonics $Y_{J}^{M}(\theta,\phi)$ ($\phi$ is the azimuthal angle), while for
a symmetric-top molecule, these are Wigner D-functions
$D_{MK}^{J}(\theta,\phi,\chi)$, where $\chi$ is the angle of rotation about
the molecular symmetry axis. The quantum numbers $J$, $M$ and $K$ correspond
to the magnitude of the angular momentum, projection of angular momentum on
the laboratory $Z$ axis, and projection of the angular momentum on the
molecular symmetry axis. The eigenenergies are $E_{J}=BJ(J+1)$ (linear
molecule) and $E_{JK}=CJ(J+1)+(A-C)K^{2}$ (symmetric-top molecule), where
$B=C=\hbar^{2}/(2I)$ and $A=\hbar^{2}/(2I_{z})$. The time-dependent
Schrödinger equation is solved by numerical exponentiation of the Hamiltonian
matrix (see Expokit Sidje (1998)).

In the case of the linear molecule, the orientation factor is given by the
averaged expectation value

$\displaystyle\braket{\cos(\theta)}(t)=\frac{1}{Z}\sum\limits_{JM}\braket{\cos(\theta)}_{JM}(t)e^{-E_{J}/(k_{B}T)},$
(11)

where $Z$ is the partition sum,
$\braket{\cos(\theta)}_{JM}(t)=\braket{\Psi_{JM}(t)}{\cos(\theta)}{\Psi_{JM}(t)}$
for the initial state $\Psi_{JM}(t=0)=Y_{J}^{M}$, and the sum runs over all
the thermally populated eigenstates $Y_{J}^{M}$.

In the case of symmetric-top molecules, due to the increased number of
thermally populated states, we use the random phase wave functions method
(see, e.g. Kallush and Fleischer (2015)) to simulate the behavior of the
thermal ensemble. The orientation factor is given by the average

$\displaystyle\braket{\cos(\theta)}(t)=\frac{1}{N}\sum\limits_{n=1}^{N}\braket{\cos(\theta)}_{n}(t),$
(12)

where $N$ is the number of initial states used, $\braket{\cos(\theta)}_{n}(t)$
is the orientation factor obtained for the initial state $\psi_{n}$

$\displaystyle\psi_{n}=\sum_{JKM}\sqrt{\frac{\epsilon_{K}e^{-E_{JK}/(k_{B}T)}}{Z}}D_{MK}^{J}e^{i\varphi_{n,JKM}},$
(13)

where $\varphi_{n,JKM}\in[0,2\pi)$ is a random number and the sum runs over
all the thermally populated eigenstates $D_{MK}^{J}$. For $\mathrm{CH_{3}I}$
molecule, the statistical weight due to the nuclear spin statistics is given
by (McDowell, 1990)

$\displaystyle\epsilon_{K}=\frac{(2I_{H}+1)^{3}}{3}\left[1+\frac{2\cos(2\pi
K/3)}{(2I_{H}+1)^{2}}\right],$ (14)

with $I_{H}=1/2$.

###  III.2 Results: linear molecule

We consider $\mathrm{OCS}$ as an example linear molecule. The molecular
properties were taken from NIST (DFT, method: CAM-B3LYP/aug-cc-pVTZ) Johnson
(2019): $I=83.021\,\mathrm{amu\,\mathring{A}^{2}}$,
$\mu=0.755\,\mathrm{Debye}$, and
$\Delta\alpha=3.738\,\mathrm{\mathring{A}^{3}}$. The rotational revival time
of the OCS molecule is $T_{rev}=1/(2cB)\approx 82.4\,\mathrm{ps}$, where $c$
is the speed of light.

FIG8_40TW

Figure 8:  Orientation factor for OCS molecules at different temperatures. The
molecules undergo combined excitation by stratifying pulse followed by a THz
pulse with a delay $\tau=7\,\mathrm{ps}$. The temperatures are: (a) 5 K, (b)
50 K, and (c) 300 K, and the maximum orientation factors obtained without the
stratifying laser pulses are 0.0074, 0.0201, and 0.0215, respectively.

Figure 8 shows the time-dependent orientation factor,
$\braket{\cos(\theta)}(t)$ obtained using classical and quantum simulations.
The molecules were excited by the delayed laser and THz pulses [see Eq. (8)]
at different initial rotational temperatures. The delay between the pulses was
set to $\tau=7\,\mathrm{ps}$. The peak intensity of the laser pulse is
$I_{0,lsr}=4\times 10^{13}\,\mathrm{W/cm^{2}}$, FWHM is
$\sigma_{lsr}=100\,\mathrm{fs}$. The THz pulse parameters are
$\varepsilon_{0,THz}=1.0\,\mathrm{MV/cm}$, $\kappa=3.06\,\mathrm{ps}^{-2}$.
Laser and THz pulses with similar parameters are readily available nowadays
Clerici  _et al._ (2013); Oh  _et al._ (2014); Vicario  _et al._ (2014);
Shalaby and Hauri (2015).

Both quantum and classical simulations demonstrate an immediate orientation
response emerging during and shortly after the THz pulse excitation (at
$t\approx 7\,\mathrm{ps}$). The first classical echo appears with a delay
$2\tau$ after the THz pulse, at $t\approx 21\,\mathrm{ps}$. The quantum
mechanical result shows a strong echo signal before the orientation quantum
revival, at $t\approx 75.4\,\mathrm{ps}$. The magnitude of this echo is an
order of magnitude larger than the maximum orientation induced by THz pulse
alone. In agreement with the previous studies Herrera  _et al._ (2012);
Tutunnikov  _et al._ (2021b) and the qualitative analysis based on the
simplified models (see Sec. II), the orientation during the quantum echo is
significantly higher compared to the orientation during the classical echo.
The magnitude of the quantum echo dominates at all temperatures represented in
Fig. 8, including the room temperature, see Fig. 8(c).

Remarkably, the echo amplitude weakly depends on temperature (in the presented
temperature range). This behavior can be understood with the help of the
simplified model introduced in Sec. II. The laser pulse is intense enough to
induce efficient phase-space filamentation at all the represented
temperatures. With time, the number of filaments grows, and they become
thinner. As a result, each molecular subgroup forming a single filament has
much reduced dispersion of angular velocities (caused by chaotic thermal
motion), and it is effectively “cold”. Instead of competing with the highly
dispersed angular velocities of the initial thermal ensemble, the weak THz
pulse competes against the cold narrow filaments. When the change $P_{or}$ of
the rotation velocity due to the orienting pulse exceeds the width of the cold
filaments, the following orientation echo dynamics becomes insensitive to this
width, and, hence, to the initial molecular temperature.

FIG9

Figure 9:  Maximum orientation factor (OCS molecule, $T=300\,\mathrm{K}$,
absolute value) as a function of $\tau$ (the delay between the prepulse and
the orienting THz pulse) in different time windows. (a) During the classical
echo emerging after delay equal $2\tau$ after the THz excitation, $t\approx
3\tau$. (b) During the quantum echo emerging at delay equal $-2\tau$ before
the orientation quantum revival, $t\approx T_{rev}-\tau$. (c) The global
maximum of the orientation factor after the THz pulse. The field parameters
are the same as in Fig. 8. For comparison, the maximum orientation factor
without the stratifying pulse is $\approx 0.0215$.

As noted in the caption of Fig. 8 with the actual THz pulses used here, the
dependence of the maximal orientation obtained using the THz pulse only is
counterintuitive, namely, the maximal orientation factor slightly increases
with temperature instead of decreasing. This is in contrast to the case of
impulsive orienting kicks considered in Sec. II. For relatively long THz
pulses, the frequency contents of the pulse match the transitions between
higher rotational levels and therefore is more effective in orienting higher
temperature ensembles. A similar non-trivial behavior was studied
theoretically in Lapert and Sugny (2012). The dependence of orientation echoes
on the pulse duration and other parameters will be elaborated further in a
forthcoming publication.

As discussed in Sec. II, the echo amplitude increases with the time delay
between the stratifying and orienting pulses (for fixed temperature and field
parameters). Figure 9 shows the maximum orientation during the first classical
echo [at delay $2\tau$ after the THz pulse, see panel (a)] and during the
quantum echo [at delay $-2\tau$ before the orientation revival, see panel (b)]
for relatively short delays $\tau$. For $\tau<15\,\mathrm{ps}$, the magnitude
of both types of orientation echoes increases with $\tau$, which is consistent
with the qualitative discussion in Sec. II. Figure 9(a) shows good
correspondence between the quantum and classical results for $\tau<9$. For
longer delays, the curves diverge. At the local maximum in Fig. 9(b)
($\tau\approx 9\,\mathrm{ps}$), the enhanced (due to the prepulse) degree of
orientation is an order of magnitude higher compared to the case of THz pulse
alone. Figure 9(c) shows the global orientation maximum for various $\tau$.
The dependence on $\tau$ is periodic, and the period equal to half revival,
$T_{\mathrm{rev}}/2\approx 41.2\,\mathrm{ps}$. When $\tau\approx T_{rev}/4$,
the maximum degree of orientation exhibits high-amplitude oscillations. The
reason for this high sensitivity is the coalescence of the echo that appears
_after_ the THz pulse at $t\approx 3\tau=3T_{rev}/4$ and the echo that appears
_before_ the quantum revival at $t\approx T_{rev}-\tau=3T_{rev}/4$. Notice,
the orientation can be further enhanced using a stronger prepulse (see Fig. 5
in Sec. II).

Previously, related schemes for enhancing the THz- and two-color-induced
orientation were applied to cold OCS molecules (at $2\,\mathrm{K}$ in
Egodapitiya  _et al._ (2014), and at $25\,\mathrm{K}$ in Sonoda  _et al._
(2018)). The orientation was substantially enhanced with the help of an
aligning laser pulse. The delay between the aligning and orienting pulses was
set to $T_{rev}/4$. The authors explained the enhancements using quantum
mechanical arguments.

###  III.3 Results: symmetric-top molecule

To demonstrate the robustness of the proposed orientation enhancement
mechanism, we consider an addition example molecule $\mathrm{CH_{3}I}$ that
belongs to the class of symmetric-top molecules. The molecular properties were
taken from NIST (DFT, method: CAM-B3LYP/cc-pVTZ-PP) Johnson (2019):
$I_{x}=I_{y}\equiv I=68.527\,\mathrm{amu\,\mathring{A}^{2}}$,
$I_{z}=3.23\,\mathrm{amu\,\mathring{A}^{2}}$, $\mu=1.697\,\mathrm{Debye}$, and
$\Delta\alpha=2.951\,\mathrm{\mathring{A}^{3}}$.

Figure 10 shows the orientation factor obtained using the quantum simulation
at room temperature. The delay between the pulses was set to
$\tau=9\,\mathrm{ps}$ [see Eq. (8)]. All the parameters are the same as in
Fig. 8, except the lower peak amplitude of the THz pulse,
$\varepsilon_{0,THz}=0.3\,\mathrm{MV/cm}$. On the short time scale, the
quantum and classical results are close. The maximum orientation emerges
before the quantum revival (during the quantum echo). The maximum orientation
obtained using the THz pulse alone at room temperature is ten times lower,
$\approx 1.6\%$.

FIG10

Figure 10:  Orientation factor for $\mathrm{CH_{3}I}$ molecules calculated
quantum mechanically and classically at $T=300\,\mathrm{K}$. The delay between
the stratifying and the THz pulses is $\tau=9\,\mathrm{ps}$. The field
parameters are the same as in Fig. 8, except the THz field intensity
$\varepsilon_{0,THz}=0.3\,\mathrm{MV/cm}$. The maximum orientation factor
obtained using only the THz pulse is $\approx 0.016$.

##  IV Conclusion

In this work, we introduced an echo-based approach for enhancing the field-
free molecular orientation induced by THz pulses at high temperatures. The
enhancement mechanism relies on the general nonlinear phenomenon of phase
space filamentation and requires a single additional non-resonant laser pulse
before the orienting THz excitation. The intense laser pulse effectively
eliminates the detrimental thermal broadening by producing multiple separated
narrow filaments in the molecular rotational phase space. Molecular subgroups
forming individual filaments are effectively cold and easy to orient. The
overall significant orientation of the whole ensemble is achieved in the
course of echoes forming at set time delays. The various echoes emerging from
the combined laser and THz excitation are qualitatively explained using the
transparent and intuitive 2D model. In particular, we show that of many echoes
possible, the so-called quantum echo, gives rise to the especially pronounced
orientation enhancement. In the examples presented in the paper, the maximum
orientation at room temperature reaches $\sim 20\%$. With prepulses of higher
intensity and various optimization procedures, even higher orientation can be
expected. We demonstrate the robustness of the proposed scheme by using fully
quantum simulations of the dynamics of linear (OCS) and symmetric-top
($\rm{CH_{3}I}$) molecules at room temperature. In principle, the temperature
range over which the current methodology is operating is not limited to room
temperature. Higher temperature regimes could be explored, which may help
study materials with low vapor pressure. The high degree of orientation in a
thermal gas paves the way to unprecedented imaging opportunities, chemical
reaction control, and more.

###### Acknowledgements.

L.X. is a recipient of the Sir Charles Clore Postdoctoral Fellowship. This
research was made possible in part by the historic generosity of the Harold
Perlman Family.

##  Appendix A Impulsive approximation

One of the tools for inducing molecular orientation is a half-cycle THz pulse
Egodapitiya  _et al._ (2014). The interaction of such a pulse with the
permanent molecular dipole can be modeled using the potential

$V_{or}(\theta,t)=-\mu\varepsilon_{or}(t)\cos(\theta),$ (15)

where $\mu$ is the molecular dipole moment magnitude, $\varepsilon_{or}(t)$
defines the electric field’s time dependence, and $\theta$ is the polar angle
between the molecular axis and the electric field vector. For the sake of the
model, we assume that $\varepsilon_{or}(t)$ is Gaussian

$\varepsilon_{or}(t)=\varepsilon_{0,or}\exp\left[-t^{2}/\sigma_{or}^{2}\right],$
(16)

where $\varepsilon_{0,or}$ is the peak amplitude of the electric field,
$\sigma_{or}$ is the width of the pulse. The equation of motion for the
molecular angular momentum $L$ is

$\frac{dL}{dt}=-\frac{dV_{or}(\theta)}{d\theta}=-\mu\varepsilon_{or}(t)\sin(\theta).$
(17)

In the impulsive approximation, when the change in $\theta$ is negligible
during the pulse, the change in $L$ due to the orienting interaction is

$\Delta L_{or}=-\sqrt{\pi}\mu\sigma_{or}\varepsilon_{0,or}\sin(\theta_{0}),$
(18)

where $\theta_{0}$ is the angle at the moment of the pulse, and $P_{or}$ is
defined as

$P_{or}=\frac{\mu\sqrt{\pi}}{I}\sigma_{or}\varepsilon_{0,or},$ (19)

where $I$ is the molecule’s moment of inertia.

A standard tool for inducing molecular alignment is non-resonant femtosecond
laser pulses. We use such a pulse as a prepulse to stratify the phase space,
see Sec. II. The interaction of a molecule with the prepulse is modeled as

$V_{pre}(\theta,t)=-\frac{\Delta\alpha}{4}\varepsilon_{pre}^{2}(t)\cos^{2}(\theta),$
(20)

where $\Delta\alpha$ is the polarizability anisotropy of the molecule. The
potential in Eq. (20) is averaged over the laser’s optical cycle. The function
$\varepsilon_{pre}(t)$ is the slowly varying Gaussian envelope of the laser
pulse

$\varepsilon_{pre}(t)=\varepsilon_{0,pre}\exp\left[-2\ln(2)t^{2}/\sigma_{pre}^{2}\right],$
(21)

where $\varepsilon_{0,pre}$ is the peak amplitude of the laser’s electric
field, $\sigma_{pre}$ is the full width at half maximum of the laser pulse
intensity. In the impulsive approximation

$\Delta
L_{pre}=-\sqrt{\frac{\pi}{\ln(16)}}\frac{\Delta\alpha}{4}\sigma_{pre}\varepsilon_{0,pre}\sin(2\theta_{0}),$
(22)

and $P_{pre}$ is defined as

$P_{pre}=\sqrt{\frac{\pi}{\ln(16)}}\frac{\Delta\alpha}{4I}\sigma_{pre}\varepsilon_{0,pre}.$
(23)

##  Appendix B Orientation factor

Here, we derive the formulas for the orientation factor

$\braket{\cos(\theta)}(t)=\braket{\mathrm{Re}(e^{i\theta})}(t)=\mathrm{Re}\left[\braket{e^{i\theta}}(t)\right],$
(24)

where the angle brackets denote ensemble average, for two cases: (i)
excitation by an orienting pulse alone, and (ii) excitation by a prepulse
followed by an orienting kick.

###  B.1 Single orienting pulse

The time-dependent orientation factor after the orienting excitation is given
by the real part of

$\displaystyle\braket{e^{i\theta}}(t)$
$\displaystyle=(2\pi)^{-3/2}\sigma_{T}^{-1}$
$\displaystyle\times\int_{-\infty}^{\infty}\int_{0}^{2\pi}\exp\\{i[\theta_{0}+\omega_{0}t-P_{or}\sin(\theta_{0})t]\\}$
$\displaystyle\times\exp[-\omega_{0}^{2}/(2\sigma_{T}^{2})]\,d\theta_{0}d\omega_{0}$
$\displaystyle=(2\pi)^{-3/2}\sigma_{T}^{-1}\int_{-\infty}^{\infty}e^{i\omega_{0}t}\exp[-\omega_{0}^{2}/(2\sigma_{T}^{2})]\,d\omega_{0}$
$\displaystyle\times\int_{0}^{2\pi}e^{i\theta_{0}}\exp\left[-iP_{or}\sin(\theta_{0})t\right]\,d\theta_{0},$
(25)

where $\sigma_{T}=\sqrt{k_{B}T/I}$, and
$\theta(t)=\theta_{0}+\omega_{0}t-P_{or}\sin(\theta_{0})t$ [see Eq. (1)].
Using the Jacobi–Anger expansion

$e^{iz\sin(\theta)}=\sum_{k=-\infty}^{\infty}e^{ik\theta}J_{k}(z),$ (26)

where $J_{k}(z)$ is the Bessel function of integer order $k$, we can represent
the second integral in Eq. (25) as

$\displaystyle\int_{0}^{2\pi}e^{i\theta_{0}}\exp\left[-iP_{or}\sin(\theta_{0})t\right]\,d\theta_{0}$
$\displaystyle=\sum_{k=-\infty}^{\infty}\int_{0}^{2\pi}e^{i\theta_{0}}e^{-ik\theta_{0}}J_{k}(P_{or}t)\,d\theta_{0}$
$\displaystyle=2\pi J_{1}(P_{or}t),$ (27)

where all the terms in the sum, except for $k=1$, vanish. Finally,

$\braket{\cos(\theta)}(t)=e^{-\sigma_{T}^{2}t^{2}/2}J_{1}(P_{or}t).$ (28)

###  B.2 Stratifying prepulse followed by orienting kick

Just after the stratifying kick (prepulse), the rotor’s variables are

$\displaystyle\theta_{1}(t)$ $\displaystyle=\theta_{0}+\omega_{1}t,$ (29)
$\displaystyle\omega_{1}$ $\displaystyle=\omega_{0}-P_{pre}\sin(2\theta_{0}),$

After a delay equal to $\tau$, at $t=\tau$, we apply the orienting kick after
which the rotor’s variables are

$\displaystyle\theta_{2}(t)$
$\displaystyle=\theta_{1}(\tau)+(t-\tau)\omega_{2},$ (30)
$\displaystyle\omega_{2}$
$\displaystyle=\omega_{1}-P_{or}\sin[\theta_{1}(\tau)].$

The orientation factor is given by the real part of

$\displaystyle\braket{e^{i\theta}}(t)$
$\displaystyle=\Theta(t-\tau)\frac{(2\pi)^{-3/2}}{\sigma_{T}}\int_{-\infty}^{\infty}\int_{0}^{2\pi}\exp\left[i\theta_{2}(t)\right]$
$\displaystyle\times\exp[-\omega_{0}^{2}/(2\sigma_{T}^{2})]\,d\theta_{0}d\omega_{0},$
(31)

where $\Theta(t-\tau)$ is the step function. We use Eqs. (29) and (30) to
express $\theta_{2}$ in terms of $\theta_{0}$ and $\omega_{0}$. Next, we use
the Jacobi–Anger expansion

$\displaystyle\exp$
$\displaystyle\left\\{iP_{or}(t-\tau)\sin[-\theta_{0}-\omega_{0}\tau+P_{pre}\sin(2\theta_{0})\tau]\right\\}$
$\displaystyle=\sum_{k=-\infty}^{\infty}\left(\exp\left\\{-ik[\theta_{0}+\omega_{0}\tau-
P_{pre}\sin(2\theta_{0})\tau]\right\\}\right.$ $\displaystyle\left.\times
J_{k}[P_{or}(t-\tau)]\right)$ (32)

Combining the terms proportional to $\omega_{0}$ in the exponent, we get
$\exp[i\omega_{0}t-ik\omega_{0}\tau]$. Integral $d\omega_{0}$ then yields

$\displaystyle\int_{-\infty}^{\infty}\exp[i\omega_{0}(t-k\tau)]e^{-\omega_{0}^{2}/(2\sigma_{T}^{2})}\,d\omega_{0}$
$\displaystyle=\sqrt{2\pi}\sigma_{T}e^{-\sigma_{T}^{2}(t-k\tau)^{2}/2}.$ (33)

The rest of the terms in the exponent read

$\displaystyle\exp[i\theta_{0}-iP_{pre}\sin(2\theta_{0})t-ik\theta_{0}+ikP_{pre}\sin(2\theta_{0})\tau]$
$\displaystyle=$
$\displaystyle\exp[i\theta_{0}(1-k)+iP_{pre}\sin(2\theta_{0})(k\tau-t)]$
$\displaystyle=$
$\displaystyle\sum_{m=-\infty}^{\infty}\exp[i\theta_{0}(1-k+2m)]J_{m}[P_{pre}(k\tau-t)].$
(34)

Integral $d\theta_{0}$ then yields

$\displaystyle\int_{0}^{2\pi}\sum_{m=-\infty}^{\infty}\exp[i\theta_{0}(1-k+2m)]J_{m}[P_{pre}(k\tau-t)]\,d\theta_{0}$
$\displaystyle=2\pi\sum_{m=-\infty}^{\infty}\delta(1-k+2m)J_{m}[P_{pre}(k\tau-t)]$
$\displaystyle=\begin{cases}2\pi
J_{\frac{k-1}{2}}[P_{pre}(k\tau-t)],&\mathrm{odd}\;k,\\\
0,&\mathrm{\mathrm{even\;}}k.\end{cases}$ (35)

Overall, the orientation signal reads

$\displaystyle\braket{\cos(\theta)}(t)=\Theta(t-\tau)\sum_{\mathrm{odd}\,k}^{\infty}S_{k}(t),$
(36)

where

$\displaystyle S_{k}(t)$
$\displaystyle=e^{-\sigma_{T}^{2}(t-k\tau)^{2}/2}J_{k}[P_{or}(t-\tau)]$
$\displaystyle\times J_{\frac{k-1}{2}}[P_{pre}(k\tau-t)].$ (37)

##  Appendix C Quantum rigid rotor

The energy eigenstates of the quantum rotor restricted to the $XY$ plane are
$\psi_{n}=\exp(i\phi n)/\sqrt{2\pi}$, where $n=0,\pm 1,\pm 2,\dots$. The
energies (in atomic units, $\hbar=1$) are $E_{n}=n^{2}/(2I)$, where $I$ is the
moment of inertia. Like in the classical case, we treat the effect of the
stratifying (prepulse) and orienting kicks in the impulsive approximation. The
prepulse transforms an initial wave function $\psi_{-}$ into
$\psi_{+}=\exp[iIP_{pre}\cos^{2}(\theta)]\psi_{-}$, where $P_{pre}$ is given
by Eq. (23). Similarly, for the orienting kick
$\psi_{+}=\exp[iIP_{or}\cos(\theta)]\psi_{-}$, where $P_{or}$ is given by
$\eqref{eq:App-A-P-THz}$.

Each of the basis wave functions, $\psi_{n}$ is “kicked” and propagated in
time. The final observable, the ensemble averaged orientation factor,
$\braket{\cos(\theta)}$ is obtained by thermal averaging over the various
expectation values
$\braket{\Psi_{n}(\theta,t)}{\cos(\theta)}{\Psi_{n}(\theta,t)}$, where
$\Psi_{n}(\theta,t)$ is the wave function at time $t$ corresponding to the
initial state $\psi_{n}$.

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