# Rotation of polarization of light propagating through
a gas of molecular super-rotors

Ilia Tutunnikov  AMOS and Department of Chemical and Biological Physics, The
Weizmann Institute of Science, Rehovot 7610001, Israel  Uri Steinitz  AMOS and
Department of Chemical and Biological Physics, The Weizmann Institute of
Science, Rehovot 7610001, Israel  Soreq Nuclear Research Centre, Yavne
8180000, Israel  Erez Gershnabel  AMOS and Department of Chemical and
Biological Physics, The Weizmann Institute of Science, Rehovot 7610001, Israel
Jean-Michel Hartmann  Laboratoire de Météorologie Dynamique/IPSL, CNRS, École
Polytechnique, Institut Polytechnique de Paris, Sorbonne Université, École
Normale Supérieure, PSL Research University, F-91120 Palaiseau, France
Alexander A. Milner  Department of Physics and Astronomy, The University of
British Columbia, Vancouver V6T-1Z1, Canada  Valery Milner
<EMAIL_ADDRESS>Department of Physics and Astronomy, The University of
British Columbia, Vancouver V6T-1Z1, Canada  Ilya Sh. Averbukh
<EMAIL_ADDRESS>AMOS and Department of Chemical and Biological
Physics, The Weizmann Institute of Science, Rehovot 7610001, Israel

###### Abstract

We present a detailed theoretical and experimental study of the rotation of
the plane of polarization of light traveling through a gas of fast-spinning
molecules. This effect is similar to the polarization drag phenomenon
predicted by Fermi a century ago and it is a mechanical analog of the Faraday
effect. In our experiments, molecules were spun up by an optical centrifuge
and brought to the super-rotor state that retains its rotation for a
relatively long time. Polarizability properties of fast-rotating molecules
were analyzed considering the rotational Doppler effect and Coriolis forces.
We used molecular dynamics simulations to account for intermolecular
collisions. We found, both experimentally and theoretically, a nontrivial
nonmonotonic time dependence of the polarization rotation angle. This time
dependence reflects transfer of the angular momentum from rotating molecules
to the macroscopic gas flow, which may lead to the birth of gas vortices.
Moreover, we show that the long-term behavior of the polarization rotation is
sensitive to the details of the intermolecular potential. Thus, the
polarization drag effect appears as a novel diagnostic tool for the
characterization of intermolecular interaction potentials and studies of
collisional processes in gases.

##  I Introduction

Light traveling through matter is affected by the motion of the medium.
Already in 1818, A.-J. Fresnel considered this effect for a hypothetical
“aether drag” (Fresnel, 1818). The Fizeau experiment (Fizeau, 1851) seemingly
demonstrated it; Fizeau showed that the phase and velocity of light change
(being dragged) while propagating through a moving dielectric medium (flowing
water) depending on the flow direction. Later, J. J. Thomson argued that not
only the phase (or velocity) but also the light’s polarization should be
affected when propagating through a rotating aether (Thomson, 1885), thus
creating a transverse drag that leads to the rotation of the plane of
polarization. Eventually, the concept of “luminiferous aether” was refuted by
the Michelson-Morley experiment (Michelson and Morley, 1887). E. Fermi
considered the combination of Fizeau’s experiment and Thomson’s ideas, and
suggested that light traveling through a rotating dielectric experiences a
“polarization drag” (Fermi, 1923). It took half a century to test the theory
experimentally (Jones, 1976), and support it with further theoretical advances
(Player and Jones, 1976; Evans, 1992; Nienhuis  _et al._ , 1992). In the
pioneering experiment (Jones, 1976), R. V. Jones used a rapidly rotating glass
cylinder that changed the laser’s polarization direction by a few
microradians. About a decade ago, the following experiment (Franke-Arnold  _et
al._ , 2011) enhanced the polarization drag by about four orders of magnitude
by using near-resonant slow light in a rotating ruby cylinder.

Detecting polarization rotation in gases is a much more difficult task,
primarily due to the medium’s low density. J.-B. Biot measured the optical
rotation caused by gases of chiral molecules (Biot, 1819) under hazardous
conditions, resulting in an accidental explosion of a 30 meter-long pipe
filled with turpentine vapor.

Another major hurdle is the rapid rotation of a gaseous medium required to
induce a measurable polarization drag. However, recently we proposed a
workaround (Steinitz and Sh. Averbukh, 2020), suggesting that instead of
mechanically rotating a bulky dielectric object as a whole, one could excite a
fast unidirectional rotation of individual microscopic particles. Current
laser techniques (for recent reviews, see, e.g., (Ohshima and Hasegawa, 2010;
Fleischer  _et al._ , 2012; Lemeshko  _et al._ , 2013; Koch  _et al._ , 2019))
including cross-polarized pulse pairs (Fleischer  _et al._ , 2009; Kitano  _et
al._ , 2009; Zhdanovich  _et al._ , 2011), chiral pulse trains (Zhdanovich
_et al._ , 2011), polarization-shaped pulses (Karras  _et al._ , 2015), and,
especially, optical centrifuges (Karczmarek  _et al._ , 1999; Villeneuve  _et
al._ , 2000; Yuan  _et al._ , 2011; Korobenko  _et al._ , 2014; Owens  _et
al._ , 2018; MacPhail-Bartley  _et al._ , 2020), can bring molecules in the
gas phase to very fast spinning. When the molecules are excited to extremely
high rotational states, with their angular momentum reaching hundreds of
$\hbar$, they become super-rotors, which are rather resistant to collisions.
Gases of unidirectionally rotating super-rotors have been studied both
theoretically and experimentally. They have many unique optical and kinetic
properties; most relevant to this study are the rotational Doppler shift
(Korech  _et al._ , 2013; Korobenko  _et al._ , 2014; Steinitz  _et al._ ,
2014) and inhibited rotational relaxation rate (Korobenko  _et al._ , 2014;
Khodorkovsky  _et al._ , 2015; Steinitz  _et al._ , 2016; Korobenko, 2018;
Toro  _et al._ , 2013; Milner  _et al._ , 2014; Murray  _et al._ , 2015,
2016).

Most recently, following the theoretical proposal (Steinitz and Sh. Averbukh,
2020), polarization drag (also known as the mechanical Faraday effect) in a
molecular gas was experimentally demonstrated for the first time (Milner  _et
al._ , 2021). The experiments used the optical centrifuge pulses applied to
various gases at ambient conditions. The molecules were spun up to high
angular velocities, compensating for the gas’s low density (compared to
rotating solids). The results demonstrated unprecedentedly high specific
optical rotatory power (Rumble, 2021) (i.e., polarization rotation angle per
unit propagation length per unit density), three orders of magnitude higher
than in the above record slow-light experiment (Franke-Arnold  _et al._ ,
2011), and correspondingly almost nine orders of magnitude higher than in
Jones’ work (Jones, 1976).

Here, we present a detailed theoretical analysis of the polarization rotation
of light propagating through a gas of fast-spinning molecules, and report new
experimental results of time-resolved measurements of the polarization drag.
Our study’s central subject is the dynamics of the polarization rotation angle
following the molecular excitation by the optical-centrifuge pulse. We argue
that the measured time-dependence of the polarization drag reflects the
exchange of angular momentum (due to intermolecular collisions) between the
microscopic rotational degrees of freedom and the macroscopic gas flow. This
time dependence provides indirect access to the details of generation of
laser-induced gas vortices (Steinitz  _et al._ , 2012, 2016).

The paper is organized as follows. In the next Sec. II, we derive an explicit
expression for the polarization rotation angle using a simplified two
dimensional model, in which the molecules are confined to a plane
perpendicular to the light propagation direction. Section III contains
practical formulas for the angle of polarization rotation in the fully three
dimensional case. Secion IV focuses on the relaxation dynamics of the
polarization rotation process. Section V describes the experimental setup,
presents the experimental results and compares them with the theoretical
predictions. Section VI summarizes the paper.

##  II Qualitative description

Consider an electromagnetic wave propagating through a non-magnetic,
homogeneous medium, whose macroscopic properties are time-independent. The
wave equation for the electric field vector, $\mathbf{E}$ is:

$\nabla^{2}\mathbf{E}-\frac{1}{c^{2}}\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\varepsilon}}_{r}\frac{\partial^{2}\mathbf{E}}{\partial
t^{2}}=0,$ (1)

where tensor
$\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\varepsilon}}_{r}$
is the relative permittivity, $c=1/\sqrt{\mu_{0}\varepsilon_{0}}$ is the speed
of light in vacuum, $\mu_{0}$ is the magnetic permeability of vacuum, and
$\varepsilon_{0}$ is the permittivity of vacuum. In the case of molecular
gases,
$\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\varepsilon}}_{r}$
is related to the molecular polarizability tensor
$\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}$ by

$\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\varepsilon}}_{r}=1+\frac{N}{\varepsilon_{0}}\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}},$
(2)

where $N$ is the number density, and
$\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}$
is the average (over all molecular orientations) polarizability. Substituting
Eq. (2) into Eq. (1) yields

$\nabla^{2}\mathbf{E}-\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{E}}{\partial
t^{2}}=\frac{N\braket{\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\boldsymbol{\alpha}}}}{\varepsilon_{0}c^{2}}\frac{\partial^{2}\mathbf{E}}{\partial
t^{2}},$ (3)

Letting $\mathbf{E}=(E_{X},E_{Y})^{T}\exp[i(\omega t-kz)]$ (where superscript
$T$ denotes transpose), allows us to rewrite Eq. (3) as

$\begin{split}\left(k^{2}-\frac{\omega^{2}}{c^{2}}\right)E_{X}&=\frac{N\omega^{2}}{\varepsilon_{0}c^{2}}[\braket{\alpha_{XX}^{(\omega)}}E_{X}+\braket{\alpha_{XY}^{(\omega)}}E_{Y}],\\\
\left(k^{2}-\frac{\omega^{2}}{c^{2}}\right)E_{0Y}&=\frac{N\omega^{2}}{\varepsilon_{0}c^{2}}[\braket{\alpha_{YX}^{(\omega)}}E_{X}+\braket{\alpha_{YY}^{(\omega)}}E_{Y}],\end{split}$
(4)

where $k$ is the wave number, the superscript $(\omega)$ means that we
consider the medium response at the same frequency as the driving field. To
simplify the solution of the system of equations in Eq. (4), we assume
$\braket{\alpha_{XX}^{(\omega)}}=\braket{\alpha_{YY}^{(\omega)}}$ and
$\braket{\alpha_{YX}^{(\omega)}}=-\braket{\alpha_{XY}^{(\omega)}}$. These
assumptions will be justified in Sec. III, when considering the rotational
excitation of molecules by the optical centrifuge.

Changing variables to $E^{\pm}=E_{X}\pm iE_{Y}$, we obtain the relative
permittivities for the two circular polatizations

$\begin{split}\varepsilon_{r}^{\pm}=&1+\frac{N}{\varepsilon_{0}}[\braket{\alpha_{XX}^{(\omega)}}\mp
i\braket{\alpha_{XY}^{(\omega)}}].\\\ \end{split}$ (5)

Finally, the angle of polarization rotation is given by (Fowles, 1989)

$\displaystyle\Delta\Phi$ $\displaystyle=\frac{\omega
L}{2c}\left(\sqrt{\varepsilon_{r}^{+}}-\sqrt{\varepsilon_{r}^{-}}\right)\approx-i\frac{\omega
L}{2c}\frac{N}{\varepsilon_{0}}\braket{\alpha_{XY}^{(\omega)}},$ (6)

where $L$ is the propagation length and the relative permittivities are very
close to unity, as is typical in gases.

To obtain an explicit expression for $\braket{\alpha_{XY}^{(\omega)}}$ in the
case of linear molecules, we begin with a two-dimensional model in which the
molecules rotate in the $XY$ laboratory plane. We analyze the fully three
dimensional case, including intemolecular interactions, in the following
sections. Figure 1(a) shows a diatomic molecule rotating in the $XY$ plane
with an angular velocity $\Omega\mathbf{\hat{Z}}$, where $\hat{\mathbf{Z}}$ is
the unit vector along the $Z$ axis. At the input plane, the electric field
$\mathbf{E}$ of the probe light is modeled by
$\mathbf{\mathbf{E}}=E_{0}\exp(i\omega t)\hat{\mathbf{X}}$, where
$\hat{\mathbf{X}}$ is the unit vector along the $X$ axis. Prior to probing,
the molecules are assumed to be excited by another (pump) laser pulse. Here,
we neglect the effect of inhomogeneity stemming from a non-uniform laser
profile and the pulsed nature of the laser. For the qualitative modeling, we
assume that the molecule consists of nuclei at a fixed relative distance and a
single polarization charge $q$ having a mass $m$. The nuclei attract the
charge with a restoring electric force which has components along the
molecular bond axis $x$ and along the perpendicular axis $y$

$\begin{split}F_{x,\mathrm{r}}&=-k_{\parallel}x,\qquad
F_{y,\mathrm{r}}=-k_{\perp}y\end{split}.$ (7)

This simple molecular model is capable of capturing the essential physics, and
agrees well with previous studies on the polarizability of rotating objects
(B. Baranova  _et al._ , 1979; Pan  _et al._ , 2019). For example, it
demonstrates sensitivity to the anisotropy of the molecule.

FIG. 1: Simplified 2D model. The molecule is restricted to rotate in the $XY$
plane.

In the rotating frame, there are inertial forces acting on the mass

$F_{x,\mathrm{i}}=2m\Omega\dot{y}+m\Omega^{2}x,\;F_{y,\mathrm{i}}=-2m\Omega\dot{x}+m\Omega^{2}y,$
(8)

including the Coriolis force (first terms) and centrifugal one (second terms).
In the molecular frame [see Fig. 1(b)], the electric field components read

$\negthickspace E_{x}=E_{0}e^{i\omega t}\cos(\gamma+\Omega
t),\;E_{y}=-E_{0}e^{i\omega t}\sin(\gamma+\Omega t).$ (9)

Here, $\gamma+\Omega t$ is the instantaneous angle between the $x$ axis and
the electric field defined by

$\cos(\gamma+\Omega t)=\hat{\mathbf{x}}\cdot\hat{\mathbf{E}}\;.$ (10)

Combining the restoring forces [see Eq. (7)], inertial forces [see Eq. (8)],
and the force due to the electric field, yields the following coupled
equations of motion

$\begin{split}\negthickspace
m\ddot{x}=&(m\Omega^{2}\\!-\\!k_{\parallel})x\\!+\\!2m\Omega\dot{y}\\!+\\!qE_{0}e^{i\omega
t}\cos(\gamma\\!+\\!\Omega t),\\\ \negthickspace
m\ddot{y}=&(m\Omega^{2}\\!-\\!k_{\perp})y\\!-\\!2m\Omega\dot{x}\\!-\\!qE_{0}e^{i\omega
t}\sin(\gamma\\!+\\!\Omega t).\end{split}$ (11)

We solve this set of equations to get the driven steady-state solutions $x(t)$
and $y(t)$. By multiplying them by $q$, we obtain the induced dipole. The
dipole components in the laboratory frame are found by projection on the $X$
and $Y$ axes. Shortly after the end of the exciting pulses, the molecules
become dispersed in angle due to different angular velocities and therefore we
average the dipole moment over the angle $\gamma$. The averaging nullifies the
induced dipole components oscillating at the shifted frequencies $\omega\pm
2\Omega$, which is consistent with the derivation of Eq. (6). It is worth
mentioning that in our experiments, special efforts were taken to eliminate
the effect of residual linear birefringence (e.g., due to the quantum
revivals) on the measured value of the polarization drag angle (see Sec. V for
details).

Expanding the dipole components (averaged over $\gamma$) in a power series up
to the first order in $\Omega$ and substituting the polarizabilities of a non-
rotating molecule

$\begin{split}\alpha_{\parallel}(\omega)&=\frac{q^{2}}{k_{\parallel}-m\omega^{2}},\quad\alpha_{\perp}(\omega)=\frac{q^{2}}{k_{\perp}-m\omega^{2}},\end{split}$
(12)

we get

$\begin{split}d_{X}&=E_{0}e^{i\omega
t}\frac{\alpha_{\parallel}+\alpha_{\perp}}{2},\\\
d_{Y}&=\frac{iE_{0}e^{i\omega
t}m\omega}{q^{2}}(\alpha_{\parallel}-\alpha_{\perp})^{2}\Omega.\end{split}$
(13)

The off-diagonal component of the polarizability tensor is given by

$\braket{\alpha_{XY}}=\frac{\braket{d_{Y}}}{E_{0}e^{i\omega
t}}=\frac{im\omega}{q^{2}}(\alpha_{\parallel}-\alpha_{\perp})^{2}\braket{\Omega},$
(14)

where the angle brackets denote averaging over the angular velocity, $\Omega$.
We can express the angle of polarization rotation using Eq. (6) in terms of
polarizabilities in Eq (12) and their derivatives (with respect to the driving
frequency $\omega$)

$\displaystyle\Delta\Phi$
$\displaystyle=\frac{1}{2}(\alpha_{\parallel}^{\prime}-\alpha_{\perp}^{\prime})\frac{\alpha_{\parallel}-\alpha_{\perp}}{\alpha_{\parallel}+\alpha_{\perp}}\frac{\omega
NL}{2c\varepsilon_{0}}\braket{\Omega}.$ (15)

When $\alpha_{\parallel}=\alpha_{\perp}$, i.e. when the tensor of
polarizability is isotropic, $\Delta\Phi=0$. This is consistent with the
results of (B. Baranova  _et al._ , 1979). When $\alpha_{\perp}=0$, i.e. the
charge is restricted to move along the molecular axis,

$\Delta\Phi=\frac{\alpha_{\parallel}^{\prime}}{2}\frac{\omega
N}{2\varepsilon_{0}}\frac{L}{c}\braket{\Omega},$ (16)

which is the same as in (Steinitz and Sh. Averbukh, 2020). Equation (15) shows
that the polarization rotation angle is proportional to the average molecular
angular velocity, which is along the $Z$ axis in the considered 2D case. Under
certain conditions, the same holds for the three dimensional case too. Notice
that we considered only the induced dipole oscillating at the frequency
$\omega$ (the frequency of the input field), which is justified when the
molecules are isotropically distributed in the $XY$ plane.

We can rewrite Eq. (15) in an approximate form without using the
polarizability components’ derivatives $\alpha_{\parallel}^{\prime}$ and
$\alpha_{\perp}^{\prime}$. We know from Eq. (12) that
$\alpha_{\parallel}^{\prime}{\alpha_{\parallel}}^{-2}=\alpha_{\perp}^{\prime}\alpha_{\perp}^{-2}$,
and therefore $\alpha_{\parallel}^{\prime}=f^{2}\alpha_{\perp}^{\prime}$,
where $f=\alpha_{\parallel}/\alpha_{\perp}$. Then, the derivative of the
average 3-dimensional polarizability is
$\bar{\alpha}^{\prime}=(2\alpha_{\perp}^{\prime}+\alpha_{\parallel}^{\prime})/3=\alpha_{\perp}^{\prime}(f^{2}+2)/3$.
Substitution of
$\alpha_{\parallel}^{\prime}-\alpha_{\perp}^{\prime}=3\bar{\alpha}^{\prime}(f^{2}-1)/(f^{2}+2)$
in Eq. (15) leads to:

$\Delta\Phi\approx\frac{3}{2}\frac{(f-1)^{2}}{f^{2}+2}\frac{\omega
N\bar{\alpha}^{\prime}}{2\varepsilon_{0}}\frac{L}{c}\braket{\Omega}.$ (17)

The dispersion relation expressed by the difference of the group and phase
refractive indices is $n_{g}-n_{\phi}=\omega
N\bar{\alpha}^{\prime}/(2\varepsilon_{0})$, and therefore

$\displaystyle\Delta\Phi\approx\left[\frac{3}{2}\frac{(f-1)^{2}}{f^{2}+2}\right](n_{g}-n_{\phi})\frac{L}{c}\braket{\Omega}.$
(18)

Using Eq. (18), we can estimate the specific rotatory power per unit average
molecular angular velocity $[\alpha]/\braket{\Omega}=\Delta\Phi/(\rho
L\braket{\Omega})$ (where $\rho$ is the mass density), for a variety of
molecules. For example, for $\mathrm{O}_{2}$, $\mathrm{N}_{2}$,
$\mathrm{CO}_{2}$ we get [in units of
$\mathrm{deg\,(g/mL)^{-1}\,dm^{-1}\,(rad/ps)^{-1}}$] $77$, $25$, $104$,
respectively. To determine $f=\alpha_{\parallel}/\alpha_{\perp}$, the
molecular data [($\alpha_{\parallel}-\alpha_{\perp}$) and
$(\alpha_{\parallel}+2\alpha_{\perp})/3$] was taken from (Bridge and
Buckingham, 1966). The dispersion relations were taken from (Zhang  _et al._ ,
2008; Křen, 2011) ($\mathrm{O}_{2}$), (Börzsönyi  _et al._ , 2008)
($\mathrm{N}_{2}$) and (Bideau-Mehu  _et al._ , 1973) ($\mathrm{CO}_{2}$).
Notice that the measurements in (Bridge and Buckingham, 1966) were done at a
wavelength of $6328\,\textup{\AA}$, while in our experiments the wavelength of
the probe light is $4000\,\textup{\AA}$. Nevertheless, since none of the
example molecules has a resonance in the visible region of the spectrum, the
obtained values of $f$ are good approximations. At the average molecular
angular velocity $\braket{\Omega}\sim 10\,\mathrm{rad/ps}$, achieved using
typical optical centrifuge pulse (see Fig. 3), the specific rotatory power
$[\alpha]$ may approach $1000\,\mathrm{deg\,(g/mL)^{-1}\,dm^{-1}}$, about
three orders of magnitude higher than in (Franke-Arnold  _et al._ , 2011).

##  III Three dimensional case

We generalise the analysis and find the ensemble averaged polarizabilities
when the molecules rotate in three dimensions [the derivation is detailed in
Appendix A, see Eqs. (43)-(45)]:

$\displaystyle\braket{\alpha_{XX}}$
$\displaystyle=\frac{1}{2}[\alpha_{\parallel}(\omega)+\alpha_{\perp}(\omega)]$
$\displaystyle-\frac{1}{2}[\alpha_{\parallel}(\omega)-\alpha_{\perp}(\omega)]\Braket{\frac{\Omega_{X}^{2}}{\Omega^{2}}},$
(19)

$\displaystyle\braket{\alpha_{YY}}$
$\displaystyle=\frac{1}{2}[\alpha_{\parallel}(\omega)+\alpha_{\perp}(\omega)],$
$\displaystyle-\frac{1}{2}[\alpha_{\parallel}(\omega)-\alpha_{\perp}(\omega)]\Braket{\frac{\Omega_{Y}^{2}}{\Omega^{2}}},$
(20) $\displaystyle\braket{\alpha_{XY}}$
$\displaystyle=-\frac{1}{2}(\alpha_{\parallel}-\alpha_{\perp})\Braket{\frac{\Omega_{X}\Omega_{Y}}{\Omega^{2}}}$
$\displaystyle+\frac{i}{2}(\alpha_{\parallel}^{\prime}-\alpha_{\perp}^{\prime})\frac{\alpha_{\parallel}-\alpha_{\perp}}{\alpha_{\parallel}+\alpha_{\perp}}\braket{\Omega_{Z}},$
(21)

where $\Omega_{X},\Omega_{Y},\Omega_{Z}$ are the components of the molecular
angular velocity, $\boldsymbol{\Omega}$, and $\Omega$ is its magnitude. Also,
$\braket{\alpha_{XY}}=\braket{\alpha_{XY}^{\dagger}}$. Equations (19)-(21) are
valid when the angular momentum of each molecule is conserved during the
measurement. In the pump-probe experiments in (Milner  _et al._ , 2021) and
the present study, this criterion is satisfied, because the probe duration is
much shorter than the characteristic times of collisional relaxation (see
Sections IV and V).

FIG. 2: Envelope of the optical centrifuge electric field. Here, the duration
of the optical centrifuge pulse is $\tau=75\,\mathrm{ps}$. The peak amplitude
corresponds to intensity of $I_{0\mathrm{c}}=3.5\,\mathrm{TW/cm^{2}}$.

To prepare an optically active molecular sample, i.e. a sample inducing a
measurable polarization rotation, we create molecular unidirectional rotation
in a gas of oxygen $(\mathrm{O}_{2})$ molecules. One of the most efficient
tools for this purpose is the optical centrifuge for molecules (Karczmarek
_et al._ , 1999; Villeneuve  _et al._ , 2000; Yuan  _et al._ , 2011; Korobenko
_et al._ , 2014; MacPhail-Bartley  _et al._ , 2020) – a laser pulse, whose
linear polarization undergoes an accelerated rotation around its propagation
direction. The electric field of optical centrifuge is modeled using

$\mathbf{E}_{\mathrm{c}}=E_{0\mathrm{c}}f(t;\tau)\cos(\omega_{\mathrm{c}}t)[\cos(\beta
t^{2})\hat{\mathbf{X}}+\sin(\beta t^{2})\hat{\mathbf{Y}}],$ (22)

where $E_{0\mathrm{c}}$ is the electric field amplitude, $\beta$ is the
angular acceleration of the centrifuge (here,
$\beta=0.3\,\mathrm{rad/ps^{2}}$). The function $f(t;\tau)$ defines the pulse
envelope’s time dependence (see Fig. 2), where $\tau$ is the duration of the
pulse (here, $\tau=75\,\mathrm{ps}$). The electric field of the optical
centrifuge pulse polarizes the molecules. The induced dipole interacts with
the same field resulting in a torque. This torque drives the molecules to
follow the vector $\mathbf{E}_{\mathrm{c}}$ that rotates in the $XY$ plane.
Depending on their orientation and angular momentum at the beginning of the
laser pulse, a fraction of the molecules is “captured” by the optical
centrifuge and those are accelerated to higher angular velocities. The axes of
these molecules tend to be perpendicular to the $Z$ direction, so they behave
more as in the two-dimensional case.

To simulate the rotational-translational dynamics of the molecules at ambient
conditions we use classical molecular-dynamics simulations (CMDS), which are
carried out as described in (Hartmann and Boulet, 2012; Steinitz  _et al._ ,
2016) where more information can be found. Briefly, one first initializes the
center of mass positions (randomly, but not too close to each other), and the
velocities as well as the molecular axis orientation and angular momentum
vector so that they verify the Boltzmann statistics for a gas at thermodynamic
equilibrium. These four quantities define the state of each molecule, and are
then propagated in time as follows. First, at each time step, we calculate the
force and torque applied to each particle by its neighbors. This is done using
the known positions and orientations of all molecules and an input
intermolecular potential (from (Bouanich, 1992), for
$\mathrm{O_{2}}$–$\mathrm{O_{2}}$). During the centrifugation process, for
each particle we compute the torque exerted by the laser’s electric field
according to the (known) instantaneous orientation of the molecular axis and
the polarizability anisotropy $(\alpha_{\parallel}-\alpha_{\perp})$. We add it
to the torque exerted by the neighboring molecules. After computing the forces
and torques, the molecules positions, orientations, linear and angular
velocities are evolved classically through the time step. The CMDS
simulations, which use periodic boundary conditions, nearest neighbors spheres
and the Verlet algorithm (Allen and Tildesley, 1987), provide a full
description of the translational and rotational states of each molecule at all
times. From these, distributions and mean values of various dynamical
quantities, including the averages appearing in Eqs. (19)-(21), can be
derived.

Our CMDS simulations show that at the end of the optical centrifuge
excitation,
$\braket{\Omega_{X}^{2}/\Omega^{2}}\approx\braket{\Omega_{Y}^{2}/\Omega^{2}}$
and $\braket{\Omega_{XY}^{2}/\Omega^{2}}\ll 1$. Accordingly,
$\braket{\alpha_{XX}}\approx\braket{\alpha_{YY}}$, and
$\braket{\alpha_{XY}}\approx-\braket{\alpha_{XY}}$. This verifies the
assumptions we made in the derivation of the formula for the polarization
rotation angle in Eq. (6). Therefore, we may substitute the formula in Eq. (6)
with $\braket{\alpha_{XY}}$ from Eq. (21) to obtain

$\Delta\Phi\approx\frac{1}{2}(\alpha_{\parallel}^{\prime}-\alpha_{\perp}^{\prime})\frac{\alpha_{\parallel}-\alpha_{\perp}}{\alpha_{\parallel}+\alpha_{\perp}}\frac{\omega
NL}{2c\varepsilon_{0}}\braket{\Omega_{Z}},$ (23)

which is similar to the expression obtained in the 2D case [see Sec. II, Eq.
(15)]. Alternatively, the polarization rotation angle may be written as

$\Delta\Phi\approx\left[\frac{3}{2}\frac{(f-1)^{2}}{f^{2}+2}\right](n_{g}-n_{\phi})\frac{L}{c}\braket{\Omega_{Z}},$
(24)

similarly to Eq. (18).

##  IV Relaxation dynamics

Figure 3 shows the computed average projection of the molecular angular
velocity on the $Z$ axis, $\braket{\Omega_{Z}}$ (right ordinate) following the
optical centrifuge pule excitation for various peak intensities. The angle of
polarization rotation, $\Delta\Phi$ (left ordinate) was obtained using Eq.
(23) with frequency-dependent polarizabilities taken from (Hettema  _et al._ ,
1994) (for details, see Appendix B). After the pulse, the total angular
momentum (as a vector) of all the molecules is conserved. It is the sum of
rotational angular momenta of each molecule about its center of mass and
angular momenta of the centers of mass about the origin (center of the optical
centrifuge focal plane). The time-dependence of $\braket{\Omega_{Z}}$ (or,
equivalently, of the average $Z$ component of the angular momentum,
$\braket{L_{Z}}$) seen in Fig. 3 is the indirect evidence of the angular
momentum transfer process. Due to intermolecular collisions, the rotational
angular momentum is transferred into the center of mass angular momentum.
Under certain conditions, this process leads to the birth of a macroscopic
vortex, which was theoretically analyzed in (Steinitz  _et al._ , 2012, 2016).

FIG. 3: Average $Z$ projection of the molecular angular velocity (right
ordinate) and angle of polarization rotation (left ordinate) after excitation
by the optical centrifuge pulse calculated for a gas of $\mathrm{O}_{2}$
molecules. Pulse duration is $\tau=75\,\mathrm{ps}$, centrifuge angular
acceleration is $\beta=0.3\,\mathrm{rad/ps^{2}}$. Initial temperature is
$T=300\,\mathrm{K}$, pressure is $0.5\,\mathrm{bar}$. Propagation length is
$L=1\,\mathrm{mm}$.  FIG. 4: Distributions of $Z$ projection of the molecular
angular velocity at the end of the optical centrifuge pulse. Pulse parameters
are the same as in Fig. 3. The expected terminal angular velocity (denoted by
vertical dashed line) is $[\Omega_{Z}]_{\mathrm{exp}}\approx
2\beta\tau=45\,\mathrm{rad/ps}$. Initial thermal distribution is denoted in
red.

Notice that for $t<1\,\mathrm{ns}$ there is a visible nonmonotonic behavior (a
“bump”) in the graphs of $\braket{\Omega_{Z}}$ and $\Delta\Phi$ (see the inset
in Fig. 3). To explain why they both first increase before thermalization and
decay to zero, let us consider the results shown in Fig. 4. The latter
displays the distributions of $\Omega_{Z}$ at the end of the centrifuge pulse
for various peak intensities (calculated under collision-free conditions). We
can identify the following main features, confirmed by multiple experimental
observations (e.g. (Korobenko  _et al._ , 2014; MacPhail-Bartley  _et al._ ,
2020)). (i) The distribution is bimodal with a clear separation between the
molecules that have been caught by the centrifuge (right hand side peak) and
those which have not (left hand side peak). (ii) The distribution of the
accelerated molecules peaks, as expected, near the terminal value of the
centrifuge angular speed ($2\beta\tau$). (iii) As the energy and peak
intensity of the pulse increase, so does the relative number of centrifuged
molecules (at the fixed initial temperature used here, $T=300\,\mathrm{K}$).
This comes from the fact that the centrifuge becomes more and more efficient
in capturing (and keeping) molecules thanks to the laser-induced torque
increasing with the amplitude of the laser electric field. This explains the
increase of $\Omega_{Z}$ (and thus $\Delta\Phi$) with the pulse intensity
shown in Fig. 3. For a detailed theoretical analysis of centrifuge-driven
molecular dynamics, see (Steinitz  _et al._ , 2016; Armon and Friedland, 2016,
2017). (iv) Last but not least, comparing the distribution of the non-
centrifuged molecules with the initial Boltzmann distribution (shown in red),
we find that the centrifuge preferentially catches those molecules that were
initially rotating with the polarization vector (i.e., have a positive
$\Omega_{Z}$). This implies that, when the centrifuge is turned off,
$\braket{\Omega_{Z}}$ for the non-centrifuged molecules (integration over the
left hand side peak) is negative, while $\braket{\Omega_{Z}}$ for the
centrifuged molecules (integration over the right hand side peak) is positive.
This last feature, together with the fact (discussed below) that the
collisional thermalization (i.e. isotropization) of the rotational angular
momenta is faster for slowly spinning molecules than for very fast rotors,
explains the presence of a bump in the drag angle at early times shown in the
insert of Fig. 3. Figure 5 separately shows $\braket{\Omega_{Z}}$ for
centrifuged and non-centrifuged molecules. The molecules were divided into the
two groups based on their angular velocity at the end of the laser pulse,
$\Omega_{Z}>20\,\mathrm{rad/ps}$ (for centrifuged) and
$\Omega_{Z}<20\,\mathrm{rad/ps}$ (for non-centrifuged), in agreement with Fig.
4. After the excitation, $\braket{\Omega_{Z}}$ for the non-centrifuged is
negative, while the value for the accelerated molecules the value is positive.
Figure 5 also shows that the decays rates are very different for the two
groups of molecules, significantly smaller for the centrifuged molecules than
for the non-centrifuged ones. This results in a “bump” in the plot of total
$\braket{\Omega_{Z}}$ obtained by summing the two contributions. The quicker
thermalization of the slowly spining molecules compared to the fast ones, is
due to the fact that intermolecular interactions are more efficient in
changing the angular momentum of a slow rotor than when the molecule is
spinning very fast (a “super-rotor” effect discussed in (Korobenko  _et al._ ,
2014; Khodorkovsky  _et al._ , 2015; Steinitz  _et al._ , 2016)). Indeed, in
this latter case, only very strong (and thus rare) intermolecular collisions
involving a high relative speed and a small impact parameter are efficient.
For completeness, we note that similar arguments were invoked in the study of
the polarization of the electronic spin of centrifuged $\mathrm{O}_{2}$
molecules (Floß  _et al._ , 2018) in order to explain the nonmonotonic time
dependence of the collisional dissipation of macroscopic magnetization.

FIG. 5: Average $Z$ projection of the molecular angular velocity (right
ordinate) and angle of polarization rotation (left ordinate) for centrifuged
and non-centrifuged molecules (see the text). Here,
$I_{0\mathrm{c}}=3.5\,\mathrm{TW/cm^{2}}$. The rest of the parameters are the
same as in Fig. 3.  FIG. 6: Scheme of the experimental setup. Top: femtosecond
pulses with the central wavelength of $792\,\mathrm{nm}$ (upper, red) and
$398\,\mathrm{nm}$ (lower, blue) are used for creating the centrifuge and the
probe pulses, respectively. The pulses are shaped, delayed with respect to one
another, combined in a collinear geometry and focused in a gas cell. Bottom:
after passing through the gas sample, probe pulses are filtered out from the
centrifuge light and sent to the time-gated polarization analyzer, implemented
with a boxcar integrator. $PC$: Pockels cell, $\lambda/2$: zero-order half-
wave plate, $P$: polarizer, $M$: metallic mirror, $DM$: dichroic dielectric
mirror, $L$: lens, $BC$: Berek compensator (Newport 5540), $WP$: Wollaston
prism (Thorlabs WP10-A), $BD$: balanced detector (Thorlabs PDB220A2).
Alternatively, the probe pulses may be sent to a Raman spectrometer to
characterize the rotation of the centrifuged molecules.

##  V Experimental demonstration

We have developed a sensitive experimental setup for the polarization drag
measurements. It is schematically shown in Fig. 6 and has been described in
detail in a recent publication (Milner  _et al._ , 2021). It consists of an
optical centrifuge, an oxygen gas cell, and the detection optics. To achieve
the required sensitivity of $\sim 1\,\upmu\mathrm{rad}$, the conventional
centrifuge was modified in the following way.

A stationary quarter-wave plate at the output of the centrifuge pulse shaper
was replaced by a Pockels cell ($PC$), which enabled us to modulate the
direction of the centrifuge rotation between CW and CCW at the frequency of
$37\,\mathrm{Hz}$. A lock-in amplifier was employed to record the signal at
the modulation frequency. All mirrors delivering the centrifuge pulses to the
gas sample ($M,\;DM$) were oriented as close to the normal incidence as
possible (unlike the traditional $45^{\circ}$ geometry). This minimized the
distortions of the centrifuge polarization, which introduced a preferential
axis of the field polarization and produced an undesired linear anisotropy in
the sample. The latter resulted in the unpredictable rotation of the probe
polarization, which contaminated (and often dominated) the polarization drag
signal. To further suppress any residual linear anisotropy (e.g. due to the
sudden rising edge of the centrifuge), we passed the centrifuge beam through a
half-wave plate, mechanically rotated at a few revolutions per second. In
addition to flipping the centrifuge direction, which was taken into account in
our analysis, the rotating wave plate randomized the direction of the rising
edge, averaging its effect on the polarization rotation to zero. The
successful cancelation of these linear artifacts was ensured by the
appropriately long time constant ($\sim 1\,\mathrm{s}$) of the lock-in
amplifier.

To eliminate additional systematic errors, the method of controlling the
rotational frequency of the centrifuged molecules has also been modified. In
all of our previous studies, this frequency of the super-rotors was controlled
by truncating the centrifuge pulse in time, which resulted in an early
termination of the accelerated molecular rotation and correspondingly to a
lower rotational frequency. This method involved significant changes of the
total pulse energy by up to $50\%$. Given the high centrifuge intensities,
large variations in the pulse energy can lead to changes in the local heating
of optical elements, potentially affecting their birefringence and modifying
the polarization of the probe pulses passing through them. To suppress any
effects of the pulse energy on the sensitive polarimetry setup, we developed a
new technique of centrifuge “piercing”, described in detail in (Amani  _et
al._ , 2021). As schematically illustrated in Fig. 6, a short $2\,\mathrm{ps}$
notch was introduced in the field envelope of an optical centrifuge by means
of a spectral filter in the centrifuge pulse shaper. The notch interrupts the
accelerated rotation of molecules at any desired rotational frequency.
Controlling that frequency was executed by moving the notch position in time,
which was accompanied by less than 2% variation in pulse energy.

The centrifuge pulses were focused in a cell filled with oxygen gas at room
temperature and at pressure of $0.5\,\mathrm{bar}$. The focusing lens $L$ with
a focal length of $10\,\mathrm{cm}$ provided a length of the centrifuged
region of about $1\,\mathrm{mm}$ and a peak intensity of up to
$5\,\mathrm{TW/cm^{2}}$. Probing the gas of oxygen super-rotors was done with
short probe pulses (pulse lengths of $3\,\mathrm{ps}$) delayed with respect to
the centrifuge. The probe pulses were derived from the same laser system,
spectrally narrowed down to the bandwidth of $0.1\,\mathrm{nm}$, and frequency
doubled to $400\,\mathrm{nm}$ for the ease of separating them from the
excitation light. Care was taken to make the probe focal spot smaller than the
corresponding size of the centrifuge beam to minimize the effects of pointing
instability.

The detection scheme consisted of two alternate channels: the Raman channel
and the polarization drag channel. In the former, the probe pulses (filtered
out from the centrifuge light) were circularly polarized and sent to a
spectrometer. Coherent scattering from the centrifuged oxygen molecules
resulted in Raman spectra with well-resolved peaks, corresponding to
individual rotational quantum states (Korobenko  _et al._ , 2014). The
magnitude of the Raman shift was translated to the rotational frequency, which
was set to the desired final value by moving the position of the frequency
piercing in the centrifuge shaper (Amani  _et al._ , 2021).

In the second detection channel, a very sensitive method for measuring the
small degree of polarization rotation was implemented by means of an optical
configuration depicted inside the dashed gray rectangle at the bottom of Fig.
6. A Berek compensator ($BC$) and a half-wave plate were used to align the (in
this case, linear) probe polarization at $45^{\circ}$ with respect to the axes
of a Wollaston prism. This equalized the intensity of light in both arms of a
differential balanced detector resulting in a zero signal. As soon as probe
polarization undergoes rotation in the sample medium, the balance shifts
towards one of the photo-diodes yielding a non-zero signal, whose sign
indicates the direction of rotation. The signal from the amplified balanced
detector was gated around the arrival time of the probe pulse by means of a
boxcar integrator, and then sent to the lock-in amplifier.

To calibrate the absolute value of the polarization rotation angle, we
inserted the same Pockels cell in the probe beam and used it to modulate the
ellipticity of the probe polarization by a known amount at the same frequency
and under the same experimental conditions. To convert the modulated
ellipticity into the oscillations of the polarization angle, an additional
quarter-wave plate was added after the Pockels cell (with the optical axes of
the two elements oriented at $45^{\circ}$ with respect to each other).
Finally, to eliminate possible systematic errors, each measurement was
repeated with two orthogonal states of linear probe polarization. Since the
two probe polarizations should result in signals with opposite signs, the
final drag angle was calculated as half the difference between the two
signals.

FIG. 7: Experimentally measured decay of the polarization drag signal in the
gas of oxygen super-rotors under the pressure of $0.5\,\mathrm{bar}$. The
centrifuge was adjusted for the terminal rotation frequency of
$7.2\,\mathrm{THz}$. The inset in the upper right corner shows the separately
measured high resolution signal between $200\,\mathrm{ps}$ and
$600\,\mathrm{ps}$, where it was normalized to the average drag value in that
time window. Vertical error bars represent the standard error of the mean over
$\sim 10^{5}$ laser pulses.

Our experimental results are shown in Fig. 7. Here, the gas of oxygen
molecules was kept under the pressure of $0.5\,\mathrm{bar}$. By piercing the
centrifuge around $75\,\mathrm{ps}$ from its front edge, the rotational
frequency of the centrifuged molecules was set at $7.2\,\mathrm{THz}$ (as
found from the corresponding Raman spectra). The observed time dependence of
the polarization rotation angle reproduces quite well the results of our
theoretical analysis shown in Fig. 3. A transient increase of the drag signal
by a few percent during the first three hundred picoseconds after the
centrifuge turn off is clearly seen in the experimental data (inset in Fig.
7). As discussed in Section IV, this bump at the beginning of the curve is a
result of a quickly decaying contribution of the slow molecular rotors,
spinning in the _opposite direction_ with respect to the centrifuge rotation.
Note that the shape of the experimentally recorded time dependence is slightly
different from the theoretical predictions (both in the exact magnitude of the
early bump, and the long-term decay profile) due to the nonuniform profile of
the laser beam, which is hard to quantify and therefore to take into account
in the numerical simulations by calculating several peak intensities (see Fig.
3). Moreover, the predicted relaxation dynamics are very sensitive to both the
intensity of the pulse (see Fig. 3) and the repulsive part of the
intermolecular potential (see Appendix C), two poorly known input parameters
of the simulations.

##  VI Conclusions

We presented a detailed theoretical and experimental study of the recently
predicted (Steinitz and Sh. Averbukh, 2020) and experimentally demonstrated
(Milner  _et al._ , 2021) mechanical Faraday effect in a gas of fast-spinning
molecules. This phenomenon, related to Fermi’s prominent polarization drag
effect (Fermi, 1923), was examined here in an ensemble of molecules spun up by
an optical centrifuge and brought to the super-rotor state that retains its
rotation for a relatively long time. To get a deeper qualitative insight into
the physics of the polarization drag, we considered a simple model of
molecular polarizability that accounts for both the rotational Doppler effect
and Coriolis force. This treatment was combined with molecular-dynamics
simulations to account for the collisional effects in the gas. Notably, our
approach not only caught the main qualitative features of the time-dependent
polarization drag signals measured in our experiments, but also accurately
described them quantitatively. In particular, our study explained a non-
monotonic time-dependence of the polarization drag angle observed at the early
stages of the field-free evolution, just after the end of the optical
centrifuge pulses, and provided a detailed information about the long-time
decay of this angle. As was discussed before (Steinitz  _et al._ , 2012,
2016), collisional relaxation of the unidirectional molecular rotation results
in the appearance of vortex flows in gases, and the observed time-dependence
of the polarization drag may be considered as an evident manifestation of this
phenomenon.

Apart from the experimental demonstration of a fundamental physical effect,
the presented results have several practical aspects. Polarization drag
measurements may be regarded as a new diagnostic method for characterizing
molecular rotation and its relaxation, an addition to the continuously
expanding toolbox (e.g., (Lin  _et al._ , 2015; Mizuse  _et al._ , 2015; He
_et al._ , 2019; Karamatskos  _et al._ , 2019; Bert  _et al._ , 2020) to name
just a few). In particular, the observed non-exponential decay of the drag
angle reflects the inhibited relaxation rate of the fast spinning molecular
super-rotors and depicts their gradual thermalization over a long time-scale.
We also demonstrate how this new ability to measure rotational decay may
provide valuable information on intermolecular interaction potential.

###### Acknowledgements.

This work was partially supported by the Israel Science Foundation (Grant No.
746/15) and Canada Foundation for Innovation. I. A. acknowledges support as
the Patricia Elman Bildner Professorial Chair and thanks the UBC Department of
Physics and Astronomy for hospitality extended to him during his sabbatical
stay.

##  Appendix A Polarizabilities - 3D case

In this Appendix, we derive the polarizability of an ensemble of polarization
charges of mass $m$ and having charge $q$. Each charge experiences restoring
forces applied by the molecular nuclei

$\begin{split}\mathbf{F}_{x,\mathrm{r}}&=-k_{1}x\hat{\mathbf{x}},\;\mathbf{F}_{y,\mathrm{r}}=-k_{2}y\hat{\mathbf{y}},\;\mathbf{F}_{z,\mathrm{r}}=-k_{3}z\hat{\mathbf{z}}.\end{split}$
(25)

where $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$ and $\hat{\mathbf{z}}$ are unit
vectors along the three axes of a body-fixed frame associated with each
molecule. The $z$ axis coincides with the direction of the molecular angular
momentum vector $\mathbf{L}$. The molecular bond coincides with the $x$ axis.
Linear molecules rotate uniformly with angular velocity $\Omega$ in the plane
perpendicular to vector $\mathbf{L}$. The orientation of the $z$ axis in the
laboratory fixed $XYZ$ frame is arbitrary.

The input electric field is modeled using $\mathbf{E}=2E_{0}\cos(\omega
t)\hat{\mathbf{X}}$, or in complex notation

$\mathbf{E}=E_{0}e^{i\omega t}\hat{\mathbf{X}}.$ (26)

In the rotating frame, there are inertial forces acting on the mass

$\mathbf{F}_{\mathrm{i}}=-2m\boldsymbol{\Omega}\times\mathbf{v}-m\boldsymbol{\Omega}\times(\boldsymbol{\Omega}\times\mathbf{r}),$
(27)

including the Coriolis (first term) and centrifugal (second term) force. Here,
$\boldsymbol{\Omega}=\Omega\hat{\mathbf{z}}$,
$\mathbf{r}=x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z\hat{\mathbf{z}}$,
$\mathbf{v}=\dot{\mathbf{r}}=\dot{x}\hat{\mathbf{x}}+\dot{y}\hat{\mathbf{y}}+\dot{z}\hat{\mathbf{z}}$.
Substitution yields

$\mathbf{F}_{\mathrm{i}}=2m\Omega(\dot{y}\hat{\mathbf{x}}-\dot{x}\hat{\mathbf{y}})+m\Omega^{2}(x\hat{\mathbf{x}}+y\hat{\mathbf{y}}).$
(28)

For our purposes, it is convenient to represent $\mathbf{E}$ in a basis $B$,
which includes vector $\hat{\mathbf{z}}$, $\hat{\mathbf{E}}_{xy}$ (normalized
projection of $\mathbf{E}$ on the $xy$ plane), and the third vector formed by
a cross product of the preceding two vectors [see Fig. 8(a)], i.e.

$B=\\{\mathbf{B}_{1},\mathbf{B}_{2},\mathbf{B}_{3}\\}=\\{\hat{\mathbf{E}}_{xy},\hat{\mathbf{z}}\times\hat{\mathbf{E}}_{xy},\hat{\mathbf{z}}\\}.$
(29)

FIG. 8: (a) Basis vectors, see Eq. (29). Notice that $\hat{\mathbf{z}}$ has an
arbitrary orientation in space. (b) The $x$ and $y$ axes rotate in the plane
spanned by $\hat{\mathbf{E}}_{xy}$ and
$\hat{\mathbf{z}}\times\hat{\mathbf{E}}_{xy}$.

In terms of basis $B$, the electric field reads

$\displaystyle\mathbf{E}$
$\displaystyle=(\hat{\mathbf{E}}_{xy}\cdot\mathbf{E})\hat{\mathbf{E}}_{xy}+[\mathbf{E}\cdot(\hat{\mathbf{z}}\times\hat{\mathbf{E}}_{xy})](\hat{\mathbf{z}}\times\hat{\mathbf{E}}_{xy})$
$\displaystyle+(\hat{\mathbf{z}}\cdot\mathbf{E})\hat{\mathbf{z}},$ (30)

where

$\mathbf{E}_{xy}=\mathbf{E}-(\mathbf{E}\cdot\hat{\mathbf{z}})\hat{\mathbf{z}},\quad\hat{\mathbf{E}}_{xy}=\frac{\mathbf{E}_{xy}}{|\mathbf{E}_{xy}|}$
(31)

and $|\cdot|$ denotes the magnitude of a vector. We want to find the electric
field projections along the rotating $x$ and $y$ axes. As shown by Fig. 8(b),
the $x$ axis rotates in the plane with angular velocity $\Omega$, therefore
the relative angle between the $x$ axis and $\mathbf{E}_{xy}$ increases with
time, such that the electric field projections are

$\begin{split}E_{x}=&E_{0}|[\hat{\mathbf{X}}-(\hat{\mathbf{X}}\cdot\hat{\mathbf{z}})\hat{\mathbf{z}}]|e^{i\omega
t}\cos(\gamma+\Omega t),\\\
E_{y}=&-E_{0}|[\hat{\mathbf{X}}-(\hat{\mathbf{X}}\cdot\hat{\mathbf{z}})\hat{\mathbf{z}}]|e^{i\omega
t}\sin(\gamma+\Omega t),\end{split}$ (32)

where $\gamma+\Omega t$ is the instantaneous angle between $\hat{\mathbf{x}}$
and $\hat{\mathbf{E}}_{xy}$, and the angle $\gamma$ is defined by

$\cos(\gamma+\Omega t)=\hat{\mathbf{x}}\cdot\hat{\mathbf{E}}_{xy}$ (33)

To simplify the notation, we define two constants $c_{1}$ and $c_{2}$

$\begin{split}c_{1}&\equiv|[\hat{\mathbf{X}}-(\hat{\mathbf{X}}\cdot\hat{\mathbf{z}})\hat{\mathbf{z}}]|\\\
c_{2}&\equiv\hat{\mathbf{X}}\cdot\hat{\mathbf{z}}\end{split}$ (34)

Combining the restoring forces [see Eq. (25)], fictitious forces [see Eq.
(28)], and the force due to the electric field, yields the following coupled
equations of motion

$\begin{split}\negthickspace\negthickspace
m\ddot{x}=&(m\Omega^{2}\\!-\\!k_{1})x\\!+\\!2m\Omega\dot{y}\\!+\\!qE_{0}c_{1}e^{i\omega
t}\cos(\gamma\\!+\\!\Omega t),\\\ \negthickspace\negthickspace
m\ddot{y}=&(m\Omega^{2}\\!-\\!k_{2})y\\!-\\!2m\Omega\dot{x}\\!-\\!qE_{0}c_{1}e^{i\omega
t}\sin(\gamma\\!+\\!\Omega t),\\\ \negthickspace\negthickspace
m\ddot{z}=&-k_{3}z+qc_{2}e^{i\omega t}.\end{split}$ (35)

We solve this set of equations and then

  1. 1.

We multiply $x(t)$, $y(t)$ and $z(t)$ by $q$ to get the induced dipole moment
components in the molecular frame

  2. 2.

Express the dipole in terms of the $B$ basis

$q\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\mathbf{R}}_{z}(\gamma+\Omega
t)\left[\begin{array}[]{c}x(t)\\\ y(t)\end{array}\right],$

where
$\overset{\text{\text{\tiny$\bm{\leftrightarrow}$}}}{\mathbf{R}}_{z}(\gamma+\Omega
t)$ is the canonical rotation matrix about the $z$ axis.

  3. 3.

Average over the angle $\gamma$ [see Eq. (33)], assuming that it is uniformly
distributed in the interval $[0,2\pi]$.

  4. 4.

Expand the resulting expression in a power series up to the first order in
$\Omega$.

  5. 5.

Substitute
$k_{1}=m\omega_{\parallel}^{2},\,k_{2}=m\omega_{\perp}^{2},\,k_{3}=m\omega_{\perp}^{2}$.

After carrying out the listed manipulations, the induced dipole components
[see Eq. (29) and Fig. 8] read

$\begin{split}d_{B_{1}}&=-\frac{c_{1}q^{2}E_{0}e^{i\omega
t}(2\omega^{2}-\omega_{\parallel}^{2}-\omega_{\perp}^{2})}{2m(\omega_{\parallel}^{2}-\omega^{2})(\omega_{\perp}^{2}-\omega^{2})},\\\
d_{B_{2}}&=\frac{ic_{1}E_{0}q^{2}e^{i\omega
t}\omega(\omega_{\parallel}^{2}-\omega_{\perp}^{2}){}^{2}\Omega}{m(\omega_{\parallel}^{2}-\omega^{2}){}^{2}(\omega_{\perp}^{2}-\omega^{2}){}^{2}},\\\
d_{B_{3}}&=\frac{c_{2}q^{2}E_{0}e^{i\omega
t}}{m(\omega_{\perp}^{2}-\omega^{2})}.\end{split}$ (36)

Overall, the induced dipole vector reads

$\mathbf{d}=d_{B_{1}}\hat{\mathbf{E}}_{xy}+d_{B_{2}}(\hat{\mathbf{z}}\times\hat{\mathbf{E}}_{xy})+d_{B_{3}}\hat{\mathbf{z}}.$
(37)

The polarizability contribution along the $X$ axis is given by

$\displaystyle\alpha_{XX}=\frac{\hat{\mathbf{X}}\cdot\mathbf{d}}{E_{0}e^{i\omega
t}}$
$\displaystyle=-\frac{q^{2}(2\omega^{2}-\omega_{\parallel}^{2}-\omega_{\perp}^{2})}{2m(\omega_{\parallel}^{2}-\omega^{2})(\omega_{\perp}^{2}-\omega^{2})}[1-(\hat{\mathbf{X}}\cdot\hat{\mathbf{z}})^{2}]$
$\displaystyle+\frac{q^{2}}{m(\omega_{\perp}^{2}-\omega^{2})}(\hat{\mathbf{X}}\cdot\hat{\mathbf{z}})^{2},$
(38)

where we used

$\begin{split}\hat{\mathbf{X}}\cdot\hat{\mathbf{E}}_{xy}=&\frac{\hat{\mathbf{X}}\cdot[\hat{\mathbf{X}}-(\hat{\mathbf{X}}\cdot\hat{\mathbf{z}})\hat{\mathbf{z}}]}{|[\hat{\mathbf{X}}-(\hat{\mathbf{X}}\cdot\hat{\mathbf{z}})\hat{\mathbf{z}}]|}=\frac{1-(\hat{\mathbf{X}}\cdot\hat{\mathbf{z}})^{2}}{c_{1}},\\\
\hat{\mathbf{X}}\cdot(\hat{\mathbf{z}}\times\hat{\mathbf{E}}_{xy})&=\frac{\hat{\mathbf{X}}\cdot(\hat{\mathbf{z}}\times\hat{\mathbf{X}})}{c_{1}}=0.\end{split}$
(39)

The polarizability contribution along the $Y$ axis reads

$\displaystyle\alpha_{XY}=\frac{\hat{\mathbf{Y}}\cdot\mathbf{d}}{E_{0}e^{i\omega
t}}$
$\displaystyle=\frac{q^{2}(2\omega^{2}-\omega_{\parallel}^{2}-\omega_{\perp}^{2})}{2m(\omega_{\parallel}^{2}-\omega^{2})(\omega_{\perp}^{2}-\omega^{2})}(\hat{\mathbf{X}}\cdot\hat{\mathbf{z}})(\hat{\mathbf{Y}}\cdot\hat{\mathbf{z}})$
$\displaystyle+\frac{iq^{2}\omega(\omega_{\parallel}^{2}-\omega_{\perp}^{2}){}^{2}\Omega}{m(\omega_{\parallel}^{2}-\omega^{2}){}^{2}(\omega_{\perp}^{2}-\omega^{2}){}^{2}}(\hat{\mathbf{Z}}\cdot\hat{\mathbf{z}})$
$\displaystyle+\frac{q^{2}}{m(\omega_{\perp}^{2}-\omega^{2})}(\hat{\mathbf{X}}\cdot\hat{\mathbf{z}})(\hat{\mathbf{Y}}\cdot\hat{\mathbf{z}}),$
(40)

where we used

$\begin{split}\hat{\mathbf{Y}}\cdot\hat{\mathbf{E}}_{xy}=\frac{\hat{\mathbf{Y}}\cdot[\hat{\mathbf{X}}-(\hat{\mathbf{X}}\cdot\hat{\mathbf{z}})\hat{\mathbf{z}}]}{|[\hat{\mathbf{X}}-(\hat{\mathbf{X}}\cdot\hat{\mathbf{z}})\hat{\mathbf{z}}]|}&=-\frac{(\hat{\mathbf{X}}\cdot\hat{\mathbf{z}})(\hat{\mathbf{Y}}\cdot\hat{\mathbf{z}})}{c_{1}},\\\
\negthickspace\hat{\mathbf{Y}}\cdot(\hat{\mathbf{z}}\times\hat{\mathbf{E}}_{xy})=\frac{\hat{\mathbf{Y}}\cdot(\hat{\mathbf{z}}\times\hat{\mathbf{X}})}{c_{1}}&=\frac{(\hat{\mathbf{Z}}\cdot\hat{\mathbf{z}})}{c_{1}}.\end{split}$
(41)

We express $\alpha_{XX}$ and $\alpha_{XY}$ in terms of the angular velocity
vector ($\boldsymbol{\Omega}=\Omega\hat{\mathbf{z}}$) projections
$(\Omega_{X},\Omega_{Y},\Omega_{Z})$ and the polarizabilities
$\alpha_{\parallel}(\omega)$ and $\alpha_{\perp}(\omega)$ of a non-rotating
molecule

$\alpha_{\parallel}(\omega)=\frac{q^{2}}{m(\omega_{\parallel}^{2}-\omega^{2})},\quad\alpha_{\perp}(\omega)=\frac{q^{2}}{m(\omega_{\perp}^{2}-\omega^{2})}.$
(42)

Then, we carry out averaging over the angular velocities

$\displaystyle\braket{\alpha_{XX}}$
$\displaystyle=\frac{1}{2}[\alpha_{\parallel}(\omega)+\alpha_{\perp}(\omega)]$
$\displaystyle-\frac{1}{2}[\alpha_{\parallel}(\omega)-\alpha_{\perp}(\omega)]\Braket{\frac{\Omega_{X}^{2}}{\Omega^{2}}},$
(43) $\displaystyle\braket{\alpha_{XY}}$
$\displaystyle=-\frac{1}{2}(\alpha_{\parallel}-\alpha_{\perp})\Braket{\frac{\Omega_{X}\Omega_{Y}}{\Omega^{2}}}$
$\displaystyle+\frac{i}{2}(\alpha_{\parallel}^{\prime}-\alpha_{\perp}^{\prime})\frac{\alpha_{\parallel}-\alpha_{\perp}}{\alpha_{\parallel}+\alpha_{\perp}}\braket{\Omega_{Z}}.$
(44)

Similarly, it can be shown that

$\displaystyle\braket{\alpha_{YY}}$
$\displaystyle=\frac{1}{2}[\alpha_{\parallel}(\omega)+\alpha_{\perp}(\omega)],$
$\displaystyle-\frac{1}{2}[\alpha_{\parallel}(\omega)-\alpha_{\perp}(\omega)]\Braket{\frac{\Omega_{Y}^{2}}{\Omega^{2}}},$
(45)

and $\braket{\alpha_{XY}}=\braket{\alpha_{XY}^{\dagger}}$. Notice that in the
case of isotropic molecular ensemble
($\braket{\Omega_{X}^{2}/\Omega^{2}}=\braket{\Omega_{Y}^{2}/\Omega^{2}}=1/3$
and $\braket{\Omega_{X}\Omega_{Y}/\Omega^{2}}=0$), we get the well known
result,
$\braket{\alpha_{XX}}=\braket{\alpha_{YY}}=(\alpha_{\parallel}+2\alpha_{\perp})/3$,
while the off diagonal elements vanish.

Assuming $\braket{\Omega_{X}\Omega_{Y}/\Omega^{2}}\approx 0$, Eq. (44) reduces
to

$\braket{\alpha_{XY}}=\frac{i}{2}(\alpha_{\parallel}^{\prime}-\alpha_{\perp}^{\prime})\frac{\alpha_{\parallel}-\alpha_{\perp}}{\alpha_{\parallel}+\alpha_{\perp}}\braket{\Omega_{Z}}.$
(46)

Finally, substitution into Eq. (6) yields

$\Delta\Phi=\frac{1}{2}(\alpha_{\parallel}^{\prime}-\alpha_{\perp}^{\prime})\frac{\alpha_{\parallel}-\alpha_{\perp}}{\alpha_{\parallel}+\alpha_{\perp}}\frac{\omega
NL}{2c\varepsilon_{0}}\braket{\Omega_{Z}}.$ (47)

When $\alpha_{\parallel}=\alpha_{\perp}$, i.e. when the tensor of
polarizability is isotropic, $\Delta\Phi=0$. This is consistent with the
results of (B. Baranova  _et al._ , 1979). When $\alpha_{\perp}=0$, i.e. the
charge is restricted to move along the molecular axis,

$\Delta\Phi=\frac{\alpha_{\parallel}^{\prime}}{2}\frac{\omega
NL}{2c\varepsilon_{0}}\braket{\Omega_{Z}},$ (48)

which is the same as in (Steinitz and Sh. Averbukh, 2020).

##  Appendix B Frequency dependent polarizabilities of oxygen molecule

The evaluation of the angle of polarization rotation [see Eq. (47)] requires
$\alpha_{\parallel}(\omega)$ and $\alpha_{\perp}(\omega)$, where $\omega$ is
the frequency of the driving (probe) field, and their derivatives. According
to (Hettema  _et al._ , 1994) (see Eq. 4, Table IV), the frequency-dependent
polarizabilities (in atomic units) of molecular oxygen ($\mathrm{O}_{2}$) are
given by

$\displaystyle\alpha_{\parallel}(\omega)$
$\displaystyle=\sum_{k=0}^{3}S_{\parallel}^{(-2k-2)}\omega{}^{2k},$ (49)
$\displaystyle\alpha_{\perp}(\omega)$
$\displaystyle=\sum_{k=0}^{3}S_{\perp}^{(-2k-2)}\omega{}^{2k}.$ (50)

Note that the superscripts $(-2k-2)$ are indices, while $\omega{}^{2k}$
actually means “omega raised to the power $2k$”. The numerical values of the
coefficients (in atomic units)
$S_{\parallel,\perp}^{(-2)},\,S_{\parallel,\perp}^{(-4)},\,S_{\parallel,\perp}^{(-6)},\>S_{\parallel,\perp}^{(-8)}$
are

$\displaystyle S_{\parallel}$
$\displaystyle=[14.993,\,67.040,\,520.995,\,4680.275],$ (51) $\displaystyle
S_{\perp}$ $\displaystyle=[7.834,\>13.988,\,39.826,\,151.640],$ (52)

for $k=0,\dots,3$ (see Table IV in (Hettema  _et al._ , 1994)). Higher order
coefficients, $S_{\parallel,\perp}^{(-10)},S_{\parallel,\perp}^{(-12)}\dots$,
are neglected. For $\omega=0$, the polarizabilities are
$\alpha_{\parallel}(0)=S_{\parallel}^{(-2)}\approx 15\,\mathrm{a.u.}$ and
$\alpha_{\perp}(0)=S_{\perp}^{(-2)}\approx 7.8\,\mathrm{a.u.}$, which is close
to the values for $\mathrm{O}_{2}$ molecule provided in NIST database (NIS, )
(calculated polarizabilities). In our experiments, we used a $\sim
400\,\mathrm{nm}$ wavelength probe pulse, which corresponds to
$\omega=0.11\,\mathrm{a.u.}$ The approximate values of polarizabilities and
their derivatives at $\omega=0.11\,\mathrm{a.u.}$ are
$\alpha{}_{\parallel}=15.80\,\mathrm{a.u.}$,
$\alpha{}_{\parallel}=8.00\,\mathrm{a.u.}$,
$\alpha{}_{\parallel}^{\prime}=18.04\,\mathrm{a.u.}$,
$\alpha{}_{\perp}^{\prime}=3.30\,\mathrm{a.u.}$

##  Appendix C Dependence on the intermolecular interaction potential

To test the sensitivity of the calculated time dependence of the drag angle on
the intermolecular forces, we carried calculations using three different
$\mathrm{O}_{2}$–$\mathrm{O}_{2}$ atom-atom anisotropic potentials, modeled
using

$\displaystyle\negthickspace\negthickspace V_{n-6}$
$\displaystyle=\sum_{i=1,2}\sum_{j=1,2}4\varepsilon_{n-6}\left[\left(\frac{\sigma_{n-6}}{R_{ij}}\right)^{n}-\left(\frac{\sigma_{n-6}}{R_{ij}}\right)^{6}\right]$
$\displaystyle+\frac{q_{i}q_{j}}{R_{ij}},$ (53)

where $i$ and $j$ run over the atoms of the first and second molecule,
respectively. Here $n=12$ (the usual 12-6 Lennard-Jones potential), $10$, or
$14$. $R_{ij}$ is the distance between atom $i$ of the first molecule and atom
$j$ of the second molecule. The “reference” potential used, which corresponds
to the choice $n=12$, was proposed in (Bouanich, 1992) where the
$\mathrm{O}_{2}$ electric quadrupole (i.e. the charges $q_{i}$) was fixed and
the $\varepsilon_{12-6}$ and $\sigma_{12-6}$ parameters for O–O interactions
were obtained from fits of measured second-virial coefficients at temperatures
between $200\,\mathrm{K}$ and $400\,\mathrm{K}$. This potential was later on
used in requantized Molecular Dynamics Simulations of the shape of pure
$\mathrm{O}_{2}$ absorption lines, leading to excellent agreements with
measured data (Hartmann  _et al._ , 2013). Starting from this $V_{12-6}$
potential, we generated $V_{10-6}$ and $V_{14-6}$ potentials, with
$\sigma_{12-6}=\sigma_{10-6}=\sigma_{14-6}$ and associated values of
$\varepsilon_{10-6}$ and $\varepsilon_{14-6}$ for which the O–O interaction
has the same minimum value $-\varepsilon_{12-6}$ (note that since the same
$\sigma$ is used, the three atom-atom potentials are also identical and equal
to zero for the distance $R_{ij}=\sigma_{12-6}$).

The three potentials are plotted in Fig. 9, which shows that in the range
where the potential is comparable to the kinetic temperature, the values are
very close. This implies that using these three potentials would lead, for
quantities essentially sensitive to the potential well and lower part of the
repulsive front (such as the virial coefficients and line shapes near room
temperature mentioned above, and the isotropization of the angular momentum of
noncentrifuged molecules as discussed below), to very similar results.

FIG. 9: Interatomic interaction potential between two oxygen atoms, see Eq.
(53).

In contrast, as expected, the inset in Fig. 9 shows that in the repulsive
region with large interaction energies, differences between the three
potentials are apparent. The 10-6 (resp. 14-6) potential being significantly
weaker and less steep (resp. larger and steeper) than the 12-6 potential.
Intermolecular forces being given by the gradient of the potential, it is
obvious that they significantly increase (for collisions involving short
distances) when going from the 10-6 to the 14-6 potential, with the 12-6
potential being in between these two. One can thus expect that the three
potentials should lead to significantly different decay rates (including that
of the angular momentum) of molecules which have been centrifuged up to very
high angular velocities (magnitude and orientation) since changing their
angular velocities requires extremely efficient collisions which only occur
when the strongly repulsive part of the potential is involved (i.e. short
intermolecular distances).

Obviously, in this case, the dissipation time should decrease when going from
the 10-6 to the 14-6 potential. The above qualitative arguments are confirmed
by Fig. 10. The latter displays, for $\mathrm{O}_{2}$ gas at
$0.5\,\mathrm{bar}$, the average $Z$ components of the angular velocity after
excitation by a $75\,\mathrm{ps}$ long optical centrifuge with peak intensity
of $3.5\,\mathrm{TW/cm^{2}}$. Similar to Fig. 5, the contributions of
centrifuged and non-centrifuged molecules have been separated. As expected,
the various potentials lead to very close results in the latter case, while
significant differences are obtained in the former. These results suggest that
studying the thermalization of optically-centrifuged molecules appears as an
interesting tool to test (and improve) the strongly repulsive region of
intermolecular potentials.

FIG. 10: Average $Z$ projection of the molecular angular velocity (left
ordinate) and angle of polarization rotation (right ordinate) for a
$\tau=75\,\mathrm{ps}$ long centrifuge having peak intensity of
$I_{0\mathrm{c}}=3.5\,\mathrm{TW/cm^{2}}$ applied to pure $\mathrm{O}_{2}$ gas
at temperature $T=300\,\mathrm{K}$ and pressure $0.5\,\mathrm{bar}$. The
propagation length is $L=1\,\mathrm{mm}$. (a) Centrifuged molecules (b) Non-
centrifuged molecules.

## References

  * Fresnel (1818) A. Fresnel, Letter from Augustin Fresnel to François Arago, on the influence of the movement of the Earth on some phenomena of optics, Ann. Chim. Phys. 9, 57 (1818).
  * Fizeau (1851) A.-H. Fizeau, The Hypotheses Relating to the Luminous Aether, C. R. Acad. Sci. 33, 349 (1851).
  * Thomson (1885) J. J. Thomson, Note on the Rotation of the Plane of Polarization of Light by a Moving Medium, Math. Proc. Camb. Philos. Soc. 5, 250 (1885).
  * Michelson and Morley (1887) A. A. Michelson and E. W. Morley, On the relative motion of the Earth and the luminiferous ether, Am. J. Sci. 34, 333 (1887).
  * Fermi (1923) E. Fermi, On the rotation of the plane of polarization in a rotating medium, Rend. Mat. Acc. Lincei (1923).
  * Jones (1976) R. V. Jones, Rotary “aether drag”, Proc. R. Soc. London, Ser. A 349, 423 (1976).
  * Player and Jones (1976) M. A. Player and R. V. Jones, On the dragging of the plane of polarization of light propagating in a rotating medium, Proc. R. Soc. London, Ser. A 349, 441 (1976).
  * Evans (1992) M. W. Evans, Electrodynamics of a Rotating Body: Relativistic Theory of Circular and Axial Birefringence, Int. J. Mod. Phys. B 6, 3043 (1992).
  * Nienhuis  _et al._ (1992) G. Nienhuis, J. P. Woerdman, and I. Kuščer, Magnetic and mechanical Faraday effects, Phys. Rev. A 46, 7079 (1992).
  * Franke-Arnold  _et al._ (2011) S. Franke-Arnold, G. Gibson, R. W. Boyd, and M. J. Padgett, Rotary Photon Drag Enhanced by a Slow-Light Medium, Science 333, 65 (2011).
  * Biot (1819) J.-B. Biot, Sur les rotations que certaines substances impriment aux axes de polarisation des rayons lumineux, Mémoires de l’Académie royale des sciences II, 41 (1819).
  * Steinitz and Sh. Averbukh (2020) U. Steinitz and I. Sh. Averbukh, Giant polarization drag in a gas of molecular super-rotors, Phys. Rev. A 101, 021404(R) (2020).
  * Ohshima and Hasegawa (2010) Y. Ohshima and H. Hasegawa, Coherent rotational excitation by intense nonresonant laser fields, Int. Rev. Phys. Chem 29, 619 (2010).
  * Fleischer  _et al._ (2012) S. Fleischer, Y. Khodorkovsky, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, Molecular Alignment Induced by Ultrashort Laser Pulses and Its Impact on Molecular Motion, Isr. J. Chem 52, 414 (2012).
  * Lemeshko  _et al._ (2013) M. Lemeshko, R. V. Krems, J. M. Doyle, and S. Kais, Manipulation of molecules with electromagnetic fields, Mol. Phys. 111, 1648 (2013).
  * Koch  _et al._ (2019) C. P. Koch, M. Lemeshko, and D. Sugny, Quantum control of molecular rotation, Rev. Mod. Phys. 91, 035005 (2019).
  * Fleischer  _et al._ (2009) S. Fleischer, Y. Khodorkovsky, Y. Prior, and I. Sh. Averbukh, Controlling the sense of molecular rotation, New J. of Phys. 11, 105039 (2009).
  * Kitano  _et al._ (2009) K. Kitano, H. Hasegawa, and Y. Ohshima, Ultrafast Angular Momentum Orientation by Linearly Polarized Laser Fields, Phys. Rev. Lett. 103, 223002 (2009).
  * Zhdanovich  _et al._ (2011) S. Zhdanovich, A. A. Milner, C. Bloomquist, J. Floß, I. Sh. Averbukh, J. W. Hepburn, and V. Milner, Control of molecular rotation with a chiral train of ultrashort pulses, Phys. Rev. Lett. 107, 243004 (2011).
  * Karras  _et al._ (2015) G. Karras, M. Ndong, E. Hertz, D. Sugny, F. Billard, B. Lavorel, and O. Faucher, Polarization shaping for unidirectional rotational motion of molecules, Phys. Rev. Lett. 114, 103001 (2015).
  * Karczmarek  _et al._ (1999) J. Karczmarek, J. Wright, P. Corkum, and M. Ivanov, Optical centrifuge for molecules, Phys. Rev. Lett. 82, 3420 (1999).
  * Villeneuve  _et al._ (2000) D. M. Villeneuve, S. A. Aseyev, P. Dietrich, M. Spanner, M. Y. Ivanov, and P. B. Corkum, Forced molecular rotation in an optical centrifuge, Phys. Rev. Lett. 85, 542 (2000).
  * Yuan  _et al._ (2011) L. Yuan, S. W. Teitelbaum, A. Robinson, and A. S. Mullin, Dynamics of molecules in extreme rotational states, Proc. Natl. Acad. Sci. U.S.A 108, 6872 (2011).
  * Korobenko  _et al._ (2014) A. Korobenko, A. A. Milner, and V. Milner, Direct observation, study, and control of molecular superrotors, Phys. Rev. Lett. 112, 113004 (2014).
  * Owens  _et al._ (2018) A. Owens, A. Yachmenev, S. N. Yurchenko, and J. Küpper, Climbing the rotational ladder to chirality, Phys. Rev. Lett. 121, 193201 (2018).
  * MacPhail-Bartley  _et al._ (2020) I. MacPhail-Bartley, W. W. Wasserman, A. A. Milner, and V. Milner, Laser control of molecular rotation: Expanding the utility of an optical centrifuge, Rev. Sci. Instrum 91, 045122 (2020).
  * Korech  _et al._ (2013) O. Korech, U. Steinitz, R. J. Gordon, I. Sh. Averbukh, and Y. Prior, Observing molecular spinning via the rotational Doppler effect, Nature Photon. 7, 711 (2013).
  * Steinitz  _et al._ (2014) U. Steinitz, Y. Prior, and I. Sh. Averbukh, Optics of a Gas of Coherently Spinning Molecules, Phys. Rev. Lett. 112, 013004 (2014).
  * Khodorkovsky  _et al._ (2015) Y. Khodorkovsky, U. Steinitz, J.-M. Hartmann, and I. Sh. Averbukh, Collisional dynamics in a gas of molecular super-rotors, Nat. Commun. 6, 7791 (2015).
  * Steinitz  _et al._ (2016) U. Steinitz, Y. Khodorkovsky, J.-M. Hartmann, and I. Sh. Averbukh, Dynamics and Hydrodynamics of Molecular Superrotors, ChemPhysChem 17, 3795 (2016).
  * Korobenko (2018) A. Korobenko, Control of molecular rotation with an optical centrifuge, J. Phys. B 51, 203001 (2018).
  * Toro  _et al._ (2013) C. Toro, Q. Liu, G. O. Echebiri, and A. S. Mullin, Inhibited rotational quenching in oriented ultra-high rotational states of CO2, Mol. Phys. 111, 1892 (2013).
  * Milner  _et al._ (2014) A. A. Milner, A. Korobenko, J. W. Hepburn, and V. Milner, Effects of ultrafast molecular rotation on collisional decoherence, Phys. Rev. Lett. 113, 043005 (2014).
  * Murray  _et al._ (2015) M. J. Murray, H. M. Ogden, C. Toro, Q. Liu, D. A. Burns, M. H. Alexander, and A. S. Mullin, State-Specific Collision Dynamics of Molecular Super Rotors with Oriented Angular Momentum, J. Phys. Chem. A 119, 12471 (2015).
  * Murray  _et al._ (2016) M. J. Murray, H. M. Ogden, C. Toro, Q. Liu, and A. S. Mullin, Impulsive Collision Dynamics of CO Super Rotors from an Optical Centrifuge, Chem. Phys. Chem 17, 3692 (2016).
  * Milner  _et al._ (2021) A. A. Milner, U. Steinitz, I. Sh. Averbukh, and V. Milner, Observation of Mechanical Faraday Effect in Gaseous Media, Phys. Rev. Lett. 127, 073901 (2021).
  * Rumble (2021) J. R. Rumble, ed.,  _CRC Handbook of Chemistry and Physics_, 102nd ed. (2021).
  * Steinitz  _et al._ (2012) U. Steinitz, Y. Prior, and I. Sh. Averbukh, Laser-induced gas vortices, Phys. Rev. Lett. 109, 033001 (2012).
  * Fowles (1989) G. R. Fowles,  _Introduction to Modern Optics_ (Dover, New York, 1989).
  * B. Baranova  _et al._ (1979) N. B. Baranova, B. Ya. Zeldovich, and R. V. Jones, Coriolis contribution to the rotatory ether drag, Proc. R. Soc. Lond. A 368, 591 (1979).
  * Pan  _et al._ (2019) D. Pan, H. Xu, and F. J. García de Abajo, Circular dichroism in rotating particles, Phys. Rev. Lett. 123, 066803 (2019).
  * Bridge and Buckingham (1966) N.-J. Bridge and A. D. Buckingham, The polarization of laser light scattered by gases, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 295, 334 (1966).
  * Zhang  _et al._ (2008) J. Zhang, Z. H. Lu, and L. J. Wang, Precision refractive index measurements of air, $\mathrm{N}_{2}$, $\mathrm{O}_{2}$, $\mathrm{Ar}$, and $\mathrm{CO}_{2}$ with a frequency comb, Appl. Opt. 47, 3143 (2008).
  * Křen (2011) P. Křen, Comment on “Precision refractive index measurements of air, N2, O2, Ar, and CO2 with a frequency comb”, Appl. Opt. 50, 6484 (2011).
  * Börzsönyi  _et al._ (2008) A. Börzsönyi, Z. Heiner, M. P. Kalashnikov, A. P. Kovács, and K. Osvay, Dispersion measurement of inert gases and gas mixtures at $800\,\mathrm{nm}$, Appl. Opt. 47, 4856 (2008).
  * Bideau-Mehu  _et al._ (1973) A. Bideau-Mehu, Y. Guern, R. Abjean, and A. Johannin-Gilles, Interferometric determination of the refractive index of carbon dioxide in the ultraviolet region, Opt. Commun. 9, 432 (1973).
  * Hartmann and Boulet (2012) J.-M. Hartmann and C. Boulet, Quantum and classical approaches for rotational relaxation and nonresonant laser alignment of linear molecules: A comparison for CO2 gas in the nonadiabatic regime, J. Chem. Phys 136, 184302 (2012).
  * Bouanich (1992) J.-P. Bouanich, Site-site Lennard-Jones potential parameters for N2, O2, H2, CO and CO2, J. Quant. Spectrosc. Radiat. Transfer 47, 243 (1992).
  * Allen and Tildesley (1987) M. P. Allen and D. J. Tildesley,  _Computer Simulation of Liquids_ (Clarendon Press, Oxford, 1987).
  * Hettema  _et al._ (1994) H. Hettema, P. E. S. Wormer, P. Jørgensen, H. J. A. Jensen, and T. Helgaker, Frequency-dependent polarizabilities of $\mathrm{O_{2}}$ and van der Waals coefficients of dimers containing $\mathrm{O_{2}}$, J. Chem. Phys 100, 1297 (1994).
  * Armon and Friedland (2016) T. Armon and L. Friedland, Capture into resonance and phase-space dynamics in an optical centrifuge, Phys. Rev. A 93, 043406 (2016).
  * Armon and Friedland (2017) T. Armon and L. Friedland, Quantum versus classical dynamics in the optical centrifuge, Phys. Rev. A 96, 033411 (2017).
  * Floß  _et al._ (2018) J. Floß, C. Boulet, J.-M. Hartmann, A. A. Milner, and V. Milner, Electronic spin polarization with an optical centrifuge, Phys. Rev. A 98, 043401 (2018).
  * Amani  _et al._ (2021) P. Amani, A. A. Milner, and V. Milner, Selective rotational control in mixtures of molecular super-rotors, J. Chem. Phys. 155, 124201 (2021).
  * Lin  _et al._ (2015) K. Lin, Q. Song, X. Gong, Q. Ji, H. Pan, J. Ding, H. Zeng, and J. Wu, Visualizing molecular unidirectional rotation, Phys. Rev. A 92, 013410 (2015).
  * Mizuse  _et al._ (2015) K. Mizuse, K. Kitano, H. Hasegawa, and Y. Ohshima, Quantum unidirectional rotation directly imaged with molecules, Sci. Adv. 1, e1400185 (2015).
  * He  _et al._ (2019) Y. He, L. He, P. Lan, B. Wang, L. Li, X. Zhu, W. Cao, and P. Lu, Direct imaging of molecular rotation with high-order-harmonic generation, Phys. Rev. A 99, 053419 (2019).
  * Karamatskos  _et al._ (2019) E. T. Karamatskos, S. Raabe, T. Mullins, A. Trabattoni, P. Stammer, G. Goldsztejn, R. R. Johansen, K. Długołecki, H. Stapelfeldt, M. J. J. Vrakking, S. Trippel, A. Rouzée, and J. Küpper, Molecular movie of ultrafast coherent rotational dynamics of OCS, Nat. Commun. 10, 3364 (2019).
  * Bert  _et al._ (2020) J. Bert, E. Prost, I. Tutunnikov, P. Béjot, E. Hertz, F. Billard, B. Lavorel, U. Steinitz, I. Sh. Averbukh, and O. Faucher, Optical imaging of coherent molecular rotors, Laser Photonics Rev. 14, 1900344 (2020).
  * (60) NIST Computational Chemistry Comparison and Benchmark Database,
NIST Standard Reference Database Number 101
Release 21, August 2020, Editor: Russell D. Johnson III.

  * Hartmann  _et al._ (2013) J.-M. Hartmann, V. Sironneau, C. Boulet, T. Svensson, J. T. Hodges, and C. T. Xu, Collisional broadening and spectral shapes of absorption lines of free and nanopore-confined O2 gas, Phys. Rev. A 87, 032510 (2013).