PRL 117, 103001 (2016)
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PHYSICAL REVIEW LETTERS
Rotational Control of Asymmetric Molecules: Dipole- versus
Polarizability-Driven Rotational Dynamics
1
Ran Damari,1,2 Shimshon Kallush,3,4,† and Sharly Fleischer1,2,*
Raymond and Beverly Sackler Faculty of Exact Sciences, School of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel
2
Tel-Aviv University Center for Light-Matter-Interaction, Tel Aviv 6997801, Israel
3
Department of Physics and Optical Engineering, ORT Braude College, P.O. Box 78, 21982 Karmiel, Israel
4
The Fritz Haber Research Center and The Institute of Chemistry, The Hebrew University, Jerusalem 91904, Israel
(Received 5 May 2016; published 1 September 2016)
We experimentally study the optical- and terahertz-induced rotational dynamics of asymmetric molecules
in the gas phase. Terahertz and optical fields are identified as two distinct control handles over asymmetric
molecules, as they couple to the rotational degrees of freedom via the molecular dipole and polarizability
selectively. The distinction between those two rotational handles is highlighted by different types of quantum
revivals observed in long-duration (>100 ps) field-free rotational evolution. The experimental results are in
excellent agreement with random phase wave function (RPWF) simulations [Phys. Rev. A 91, 063420 (2015)]
and provide verification of the RPWF as an efficient method for calculating asymmetric molecular dynamics at
ambient temperatures, where exact calculation methods are practically not feasible. Our observations and
analysis pave the way for orchestrated excitations by both optical and terahertz fields as complementary
rotational handles that enable a plethora of new possibilities in three-dimensional rotational control of
asymmetric molecules.
DOI: 10.1103/PhysRevLett.117.103001
Nonordered media such as gas, liquid, and most solid
phases consist of a large number of molecules oriented in
all possible directions. Such “isotropic” samples lack
association between the molecular and laboratory frames,
and thus their spectroscopic signals are inherently angularly
averaged, impeding the extraction of molecular-framedependent observables. Motivated by lifting this angular
averaging, vast efforts have been put into controlling
molecular angular distributions in gas.
Most of these efforts have focused on molecular alignment
in which the most polarizable molecular axes align with a
chosen lab-frame axis. Recently, the interest is gradually
shifting toward molecular orientation in which the permanent molecular dipoles point preferentially toward a chosen
lab-frame direction (positive or negative lab z axis).
Molecular orientation provides additional desirable features
such as transiently lifted inversion symmetry that enables
optical phenomena governed by even orders of the nonlinear
susceptibility χ ð2nÞ -like generation of even harmonics [1] and
directional molecular ionization or dissociation [2].
Throughout the years, coherent rotational control has yielded
tremendous progress in various tangential fields in addition
to molecular frame spectroscopy [3–5].
Our aims in this work are threefold: to study and
compare between optical- and terahertz-induced rotational
dynamics in asymmetric molecules; to establish singlecycle terahertz (THz) fields as a dipole-selective rotational
handle; and to experimentally verify the random phase
wave function (RPWF) method as an efficient and reliable
method for calculating rotational dynamics of asymmetric
molecules at ambient temperatures.
0031-9007=16=117(10)=103001(6)
Before delving into the dynamics of asymmetric tops, we
note that the vast majority of ultrafast rotational control
studies have focused on linear molecules and symmetric
tops typically modeled as quantum mechanical rigid rotors
with effectively only one moment of inertia. Such molecules show periodic recurrences manifested by anisotropic
distributions with each revival time, given by T rev ¼
1=ð2BcÞ with B the rotational coefficient in [cm−1 ] and
c the speed of light [6,7]. Moreover, most of the efforts
focused on inducing molecular alignment. Significantly
fewer works focused on molecular orientation and even less
on orientation of asymmetric molecules that are at the focus
of this work. In addition to their complicated quantum
rotational dynamics, asymmetric tops introduce highly
demanding computational challenges [8–11].
Laser-induced rotations of asymmetric tops.—The theoretical and experimental foundations of asymmetric-top
rotational dynamics have been laid by Felker [12] more
than two decades ago and have been actively reinforced since
[13–23]. Alignment of asymmetric tops has been achieved by
elliptically polarized pulses [18] and multiple pulses with
controlled polarizations [24]. Existing techniques for the
orientation of asymmetric molecules rely on combinations of
weak electrostatic fields (dc) and ultrashort laser pulses
[25–27]; however, they require low rotational temperatures
achieved via supersonic beam expansion and correspondingly low gas densities that prohibit all-optical detection such
as birefringence measurements and high-harmonic generation. Field-free orientation has been demonstrated in linear
molecules by either mixed-field (ω þ 2ω) excitations [28,29]
or single-cycle THz fields [30–34], both applicable to
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PHYSICAL REVIEW LETTERS
ambient gas temperatures and high densities allowing alloptical probing.
Here we utilize intense THz fields as dipole-selective
rotational handles of asymmetric molecules to induce
orientation and rotational responses that are considerably
different from those induced by optical pulses. We demonstrate THz- and optical-induced dynamics of sulfur
dioxide (SO2 ) molecules, serving as a model for asymmetric tops [10,11,35].
Theoretical.—Asymmetric tops like SO2 are described
by three different moments of inertia as shown in Fig. 1.
The rotational Hamiltonian Ĥ ¼ Ĥ0 þ V̂ðtÞ includes
Ĥ0 ¼ L̂2 a =2I a þ L̂2 b =2I b þ L̂2 c =2I c , the field-free rotational energy, and V̂, the interaction term of the molecule
with the external field. We study the rotational dynamics,
induced by two different excitation fields: an ultrashort
near-IR (optical) pulse (FWHM ∼ 100 fs duration,
ωoptical ¼ 3.75 × 1014 Hz) and a single-cycle THz field
(FWHM ∼ 1 ps, ωTHz ∼ 5 × 1011 Hz).
An ultrashort optical pulse induces rotational dynamics
via the anisotropic polarizability components of the molecule (Fig. 1). The interaction with a linearly polarized
optical pulse is given by V̂ optical ¼ − 14 ε2 ðtÞ½αaa cos2 θa þ
αbb cos2 θb þ αcc cos2 θc where εðtÞ is the pulse field
envelope, αii is the polarizability component along the
molecular frame ii axis, and θi is the angle between the
field polarization and the ith molecular frame axis
[10–12,15]. Thus, the main contribution to the rotational
dynamics comes from the most polarizable axis, the
O-O axis of SO2 (see Fig. 1).
A single-cycle THz field spanning the 3–30 cm−1
range interacts with the molecule via its permanent dipoles
~ rotates the
[31–34]. The interaction term V̂ THz ¼ −μ~ˆ · EðtÞ
dipole axis (b axis of SO2 ; see Fig. 1) to orient in the
direction of the THz field. Since the THz field oscillates at a
much slower rate (∼750-fold slower) than the optical field,
the molecular dipoles are able to follow the field and orient
in its direction via resonant rotational excitation [31,36].
RPWF simulation details.—Simulating rotational
dynamics of asymmetric tops is far more computationally
demanding than linear or symmetric top molecules owing
to the existence of three-coupled rotational degrees of
freedom [8–11]. At ambient temperatures, the large
number of thermally occupied states make exact calculation methods very costly and in many cases practically
FIG. 1. Rotational coefficients and polarizability components
along the three principal axes of the SO2 molecule. The
permanent dipole moment points along the b axis.
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not feasible. Here we use the RPWF method, recently
adopted by us for this purpose [9]. The initial state
of the system is represented by a rotational wave function
P pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k
jθ~k i ¼ JMτ e−EJMτ =kB T =ZeiθJMτ jJMτi with Boltzmann
weighted magnitudes for the wave functions but with
randomly chosen phases θkJMτ . Such an initial state contains
coherences between the rotational states that should be
absent from the thermal ensemble. With an increasing
number of realizations (k), the randomly chosen phases and
corresponding coherences cancel out due to destructive
interferences, and the correct “coherence-free” representation of the initial thermal ensemble is achieved [37–40]. At
later times throughout the quantum evolution, the RPWF
maintains the correct dynamics due to quantum typicality
[41]. We have recently shown [9] that the convergence
efficiency of the RPWF method to exact calculations
increases with the initial temperature of the ensemble
and with the exciting field strength. The calculation of
the optical-induced alignment of SO2 is performed here by
RPWF for the first time and provides yet another test bed
for the integrity of the RPWF by comparison to experimental results (Fig. 2).
In accordance with the spin symmetry selection rules,
only rotational levels that are symmetric with respect to the
b axis were taken into account. Taking the molecular dipole
axis as the z axis of the molecule following the textbook
convention [43], only states with even K participate in the
calculation.
Experimental.—The rotational responses of SO2 to the
optical and the THz fields were measured separately in two
different experimental setups: a time-resolved optical
birefringence setup and time-resolved THz-induced emission measurements briefly described hereafter.
Optical-induced birefringence in SO2 .—The timeresolved birefringence of a 25 torr SO2 gas sample at ambient
temperature is shown in Fig. 2. The experimental setup is
similar to that recently reported in Refs. [10,44] based on the
weak-field polarization detection technique [42,45].
A 100 fs pulse is split to form an optical pump
(∼50 μJ=pulse, 800 nm) and a weak 400 nm probe
(generated via SHG in a BBO crystal), linearly polarized
45° relative to the 800 nm pump. The pump and probe
beams are recombined by a dichroic mirror to propagate
collinearly in the sample cell. The diameter of the pump
beam is reduced by an iris to assure complete overlap with
the probe throughout the sample. Following their interaction with the pump, the molecules assume anisotropic
angular distributions manifested as transient birefringence
of the sample. The polarization changes of the probe due to
its propagation in the birefringent sample are analyzed by a
polarizer and a photodiode [46]. The experimental opticalinduced birefringence setup is described in Supplemental
Material, Sec. 1 [47].
The observed birefringence signal is given by
I Signal ðtÞ∝½αaa ðhcos2 θa iðtÞ −1=3Þþαbb ðhcos2 θb iðtÞ −1=3Þþ
αcc ðhcos2 θc iðtÞ −1=3Þ, with contributions from all three
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PHYSICAL REVIEW LETTERS
FIG. 2. Experimental time-resolved birefringence (solid blue
line) and RPWF-simulated alignment (solid red line) from a 25 torr
SO2 gas sample at ambient temperature, following excitation by a
100 fs optical pulse (800 nm). The J-type revivals (with a period of
26.1 ps) and the C-type revivals (period 28.4 ps) are marked in the
figure. Note the alternating signal phases of the J-type with each
revival and the constant phases of the C-type signals, shown in the
green dashed insets for the first and second J-type and C-type
revivals. The dashed orange inset depicts the weak oscillatory
birefringence signals at the 58–76 ps time frame, with close to
perfect correspondence between the experimental and RPWFsimulated data. From the optical birefringence measurements, the
degree of alignment at the peak of the first J-type revival is
ðcos2 θZa − 1=3Þ ¼ 0.006 [42] (corresponding to 0.25 in the arbitrary units of the birefringence axis). Significantly higher degrees of
alignment are possible with higher intensities of the pump pulse,
controlled pulse duration, and rotationally cold molecules [24].
molecular axes’ polarizabilities, αii (Fig. 1), weighted by the
transient change in the alignment of the axes. The latter are
given by the ensemble-averaged projections of the principal
molecular axes onto the polarization axis of the pump pulse
(laboratory Z axis), hcos2 θi iðtÞ − 1=3 fi ¼ a; b; cg.
The optical-induced birefringence of SO2 in Fig. 2
shows excellent agreement between the experimental (solid
blue line) and the RPWF-simulated data (solid red line). The
dynamics consists of mainly the J-type and C-type revivals
[12], with periods of ½2ðBb þ Bc Þ−1 ¼ 26.1 ps and
½4Bc −1 ¼ 28.4 ps, respectively. The weak birefringence
oscillations observed starting from t ∼ 35 ps are also perfectly captured by the RPWF simulation (shown in the
orange dashed frame in Fig. 2), thereby providing clear
validation of the RPWF method for rotational dynamics.
Minor experimental deviations from the simulated signal are
due to laser fluctuations and photodiode noise (as observed in
the background signal around 15 ps < t < 25 ps).
To examine the actual dynamics associated with these
two types of revivals, we present in Fig. 3 the transient
alignment of the three principal molecular axes onto the
Z-lab axis.
The J-type revivals involve the anisotropic distribution
of all three molecular axes, with the molecular b and c axes
antialigned at the instance of a-axis alignment and vice
versa. This motion is driven via the most polarizable
molecular axis (a axis), forcing its alignment along the
Z-lab axis, and hence entails the antialignment of the b and
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c axes (see J revival cartoon in the Supplemental Material
[47,48]). Correspondingly, the J-type revival period is
given by 1=2½Bb þ Bc , manifesting the rotation around
the b and c molecular axes. Similarly, the C-type revivals
describe rotation around the c axis, with the a axis aligned
at the instance of b-axis antialignment and vice versa (see C
revival cartoon in the Supplemental Material [47,48]);
hence, the C-type revival period is given by 1=½4Bc and
does not depend on the rotational coefficient of the b axis
(Bb ). We note that another periodic feature, the A-type
revivals with a period of 1=½4Ba ¼ 4.9 ps, is captured by
the simulation for axes b and c with opposite contributions
from the two axes. However, due to very similar polarizabilities along the b and c axes, the weighted A-type
signal is absent from the total alignment signal. (For a
simulated alignment of the three molecular axes for the
A-type revivals, see Supplemental Fig. 4 [47].)
THz-induced orientation of SO2 .—We now turn to THzinduced rotational dynamics of the SO2 gas sample. The
experimental setup used for THz-induced orientation measurements is a 4-f reflective setup similar to that used in
Ref. [31] (for details see Supplemental Material, Sec. 2
[47]). An ultrashort laser pulse (100 fs, 800 nm) with
3.5 mJ pulse energy was used to generate a ∼3-uJ singlecycle THz pulse by tilted pulse front optical rectification
[49]. The THz pulse is focused by an off-axis parabolic
reflector into the center of a 10 cm long static gas cell
through a 10 mm wide polytetrafluoroethylene window.
The THz beam is recollimated by an off-axis parabolic
reflector at the output of the cell and focused onto a GaP
crystal (0.5 mm thickness) for its electro-optic sampling
(EOS) by a variably delayed optical readout pulse [50,51].
FIG. 3. RPWF-simulated alignment of the three principal molecular axes along the Z-lab axis hcos2 θi iðtÞ − 1=3 fi ¼ a; b; cg. In the
observed signal, each axis is weighted by its polarizability; hence, the
signal is dominated by the alignment of the (most polarizable) a axis.
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PHYSICAL REVIEW LETTERS
(b)
(c)
FIG. 4. (a) Experimentally (solid blue line) and RPWF-simulated
(solid red line) time-resolved terahertz emission following excitation by a single-cycle THz pulse [shown in inset (b)]. The
decoherence- and decay-free simulated signal was fitted to the
experimental data by phenomenologically including an exponential
decay with τ ¼ 260 ps to account for collisional dephasing and
decay phenomena that are beyond the scope of the current work.
(c) The frequency domain of the experimental signal of Fig. 2(b),
showing a set of frequency peaks separated by 3.4 cm−1 corresponding to the dipole-induced hybrid-type revival given by
1=ð2Ba − Bb − Bc Þ [12]. Estimation of the absolute degree of
orientation at the first orientation peak (t¼9.8 ps) yielded
cosθZb ¼0.01. The estimation was done by comparison between
the FID emission from SO2 and that of OCS [31]. Higher degrees of
orientation can be achieved using more intense THz fields that
overlap with the rotational transition spectrum of the molecule.
The transmitted THz field consists of quasiperiodic THz
free-induction-decay (FID) bursts, emitted upon molecular
orientation governed by the rotational dynamics of the excited
gas. The FID emission profile follows the gradient of the timedependent orientation [31] and is explained here by classical
electrodynamics: Subsequent to their dipole-induced rotational motion, the SO2 molecules exhibit transiently squeezed
angular distributions. At these events, the permanent molecular dipoles are preferentially oriented toward the þZ (or −Z)
lab-frame direction giving rise to the creation and annihilation
of a macroscopic dipole in the sample. These dipole oscillations manifest by electromagnetic field emission with a π
phase relative to the driving field, much like a resonantly
driven antenna. The polarization induced in an antenna is π=2
phase shifted relative to the driving field, and the field emitted
by the antenna is yet another π=2 shifted relative to its
polarization. In analogy, the transient orientation of the SO2
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dipoles is π=2 phase shifted relative to the THz field carrierenvelope phase [31], and the emitted THz burst is another π=2
shifted with respect to the orientation.
Figure 4 shows the time-resolved FID signal emitted
from a 23 torr SO2 sample at room temperature. The
experimental (solid blue line) begins at t ∼ 5 ps after
the THz excitation due to the large amplitude differences
in the obtained FID signal and the exciting THz field (peak
value of 0.87 in the scale of Fig. 4). The RPWF-simulated
signal (solid red curve) is generated by simulating the
orientation of the sample throughout the entire time domain
and introducing a π=2 phase shift as described above.
The ensemble-averaged orientation is the projection of the
dipole vector (positive b-axis direction in SO2 ) onto the
THz polarization axis (lab Z axis), hcos θZb iðtÞ . We note that
the raw EOS signal obtained in the experiment consists of a
large number of satellite signals emanating from various
reflections of both the THz pump and optical probe pulses
in the GaP crystal, the cell windows, and residual water
vapor in the THz beam path outside the cell. In order to
extract the SO2 signal, we measured the response function
of the system by EOS of the evacuated cell and used it to
deconvolve the SO2 signal from the raw EOS data (see
Supplemental Material Sec. 3 [47] for details). While the
extracted signal is in good agreement with the RPWF
simulations, slight discrepancies remain and are attributed
mainly to the not-perfect signal retrieval procedure used
(see Supplemental Material Sec. 3 [47]) and in part to a
water vapor FID signal due to incomplete dry air purging.
Despite the crowded oscillatory character of the SO2
emission in Fig. 4, the fundamental revival features are
clearly observed with a period of ½2Ba − Bb − Bc −1 ∼
9.8 ps corresponding to Felker’s hybrid-type revival [12].
The polarity of the signal alternates with the revival periods
such that the phases at even recurrence signals are π shifted
with respect to the odd ones (marked by “DHyb” for
dipole-induced hybrid type revivals). The existence of the
DHyb revivals as the main feature of Fig. 4(a) is apparent
from its Fourier transform shown in Fig. 4(c) manifesting
several peaks distant by 3.4 cm−1 ¼ 2Ba − Bb − Bc.
During the fundamental emission events (with a 9.8 ps
period), the molecular dipoles preferentially point toward
the positive or negative z-axis direction [see inset sketches
DHyb(1) and DHyb(2) in Fig. 4(a)]. The DHyb revivals
persist for a long time under field-free conditions, enabling
additional laser interactions with the field-free oriented SO2
samples for enhanced rotational control or nonlinear
spectroscopic interrogation of the molecules in their
molecular frame. Note that the periodic orientation revivals
of Fig. 4 (with a 9.8 ps period) are absent from the
alignment results of Fig. 2 and vice versa; the J-type
(26.1 ps) and C-type (28.4 ps) revivals are absent from the
THz-induced dynamics of Fig. 4. Namely, the THz and
optical pulses induce distinct rotational dynamics in asymmetric SO2 molecules and induce selective rotational
motions due to the different nature of their interactions,
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mediated via the permanent dipole and the anisotropic
polarizability, respectively.
Conclusions.—Unlike linear and symmetric top molecules, that have their dipole axis and the most polarizable
axis parallel to each other, in SO2 , as in most asymmetric
molecules, the dipole and the most polarizable axes are not
parallel to one another (in SO2 they are situated perpendicularly). This manifests in distinctively different dynamics
induced by either THz or optical fields, thereby providing
two distinct handles for controlling rotational dynamics in
asymmetric molecules. Well-concerted utilization of these
handles paves a path to realizing three-dimensional control
over molecular angular distributions, achieving much higher
degrees of orientation in rotationally cold molecular ensembles [34], inducing torsion in molecules, and can aid in
breaking down congested rotational dynamics by associating the rotational responses induced by THz and optical
fields separately to their molecular-frame handles.
Coapplication of THz and optical rotational handles can
significantly extend the currently available rotational control
playground to yield novel rotational responses and dynamics unreachable by either of these fields alone.
We thank Ronnie Kosloff (HUJI) for stimulating discussions. This work is supported by the Binational Science
Foundation Grant No. 2012021 (S. K.) and by CMST
COST Action CM1405 MOLIM (S. K.), by the Israeli
Science Foundation Grant No. 1065/14 (S. F.), the Marie
Curie CIG Grant No. 631628 (S. F.), and in part by the
Wolfson family grant (S. F.).
*
Corresponding author.
[email protected]
†
[email protected]
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supplemental/10.1103/PhysRevLett.117.103001for
description of the experimental setups used in the work
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the short cartoons are plotted for the extreme cases of “close
to perfect” alignment of the molecular axes under consideration and do not represent the actual degree of alignment
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