bundle morphism $f\colon B_{1}\rightarrow B_{2}$ with the properties that:

  1. $(1)$

the range of $\ell_{1}$ consists of vector fields which are projectable via
$f$ and one has

$\displaystyle f_{*}\circ\ell_{1}=\ell_{2},$

  2. $(2)$

$f$ admits enough conservation laws to order $k$. That is, for every $b\in
B_{2}$ there is an open neighbourhood ${\mathcal{O}}$ of $b$ in $B_{2}$ and a
local section $\kappa\colon{\mathcal{O}}\rightarrow B_{2}$ of $f$ for which

$\displaystyle\frac{\rm d}{{\rm
d}t}\bigg{|}_{0}\big{[}\operatorname{Fl}^{\ell_{2}(X)}_{t}\circ\kappa\circ\operatorname{Fl}^{\ell_{1}(X)}_{-t}\big{]}^{k}_{b}=0$

for all $X\in{\mathcal{F}}$ defined in a neighbourhood of $\pi_{B_{2}}(b)$.

Observe, in the notation of Definition 4.18, that if $\sigma$ is any
${\mathcal{F}}$-invariant section of $\pi_{B_{1}}$, then $f\circ\sigma$ is an
${\mathcal{F}}$-invariant section of $\pi_{B_{2}}$: for $X\in{\mathcal{F}}$
defined in a neighbourhood of $x\in\operatorname{dom}(\sigma)$, we have

$\displaystyle\frac{\rm d}{{\rm
d}t}\bigg{|}_{0}\big{[}\operatorname{Fl}^{\ell_{2}(X)}_{t}\circ(f\circ\sigma)\circ\operatorname{Fl}^{X}_{-t}\big{]}^{k}_{x}$
$\displaystyle=\frac{\rm d}{{\rm
d}t}\bigg{|}_{0}\big{[}f\circ\operatorname{Fl}^{\ell_{1}(X)}_{t}\circ\sigma\circ\operatorname{Fl}^{X}_{-t}\big{]}^{k}_{x}$
$\displaystyle={\rm d}[f]^{k}\frac{\rm d}{{\rm
d}t}\bigg{|}_{0}\big{[}\operatorname{Fl}^{\ell_{1}(X)}_{t}\circ\sigma\circ\operatorname{Fl}^{X}_{-t}\big{]}^{k}_{x}=0,$

where $[f]^{k}\colon\Gamma_{k}(\pi_{B_{1}})\rightarrow\Gamma_{k}(\pi_{B_{2}})$
is the smooth map
$\big{(}x,[\sigma]^{k}_{x}\big{)}\mapsto(x,[f\circ\sigma]^{k}_{x})$ induced by
$f$ and ${\rm d}[f]^{k}$ its differential (Definition 2.19). Thus $f$ induces
a morphism
$f^{{\mathcal{F}}}\colon\Gamma_{k}(\pi_{B_{1}})^{{\mathcal{F}}}\rightarrow\Gamma_{k}(\pi_{B_{2}})^{{\mathcal{F}}}$
of diffeological pseudo-bundles, which is surjective since $f$ admits enough
conservation laws to order $k$.

######  Theorem 4.19.

Let $k$ denote any of the symbols $0,1,\dots,\infty,\operatorname{g}$, let
$\pi_{B_{1}}\colon B_{1}\rightarrow M$ and $\pi_{B_{2}}\colon$
$B_{2}\rightarrow M$ be singularly foliated bundles admitting enough
conservation laws to order $k$, and let
$f\colon\pi_{B_{1}}\rightarrow\pi_{B_{2}}$ be a morphism of order $k$ in the
sense of Definition 4.18. Then the identity map
$\operatorname{id}\colon{\mathcal{P}}({\mathcal{F}})\rightarrow{\mathcal{P}}({\mathcal{F}})$
descends, for each $l\leq k$, to a morphism
$\phi^{l}\colon{\mathcal{H}}\big{(}\pi_{B_{1}}^{l,{\mathcal{F}}}\big{)}\rightarrow{\mathcal{H}}\big{(}\pi_{B_{2}}^{l,{\mathcal{F}}}\big{)}$
of diffeological groupoids for which the diagrams

${{\mathcal{H}}\big{(}\pi_{B_{1}}^{l,{\mathcal{F}}}\big{)}}$${{\mathcal{H}}\big{(}\pi_{B_{2}}^{l,{\mathcal{F}}}\big{)}}$${{\mathcal{H}}\big{(}\pi_{B_{1}}^{m,{\mathcal{F}}}\big{)}}$${{\mathcal{H}}\big{(}\pi_{B_{2}}^{m,{\mathcal{F}}}\big{)}}$$\scriptstyle{\phi^{l}}$$\scriptstyle{\Pi_{B_{1}}^{m,l}}$$\scriptstyle{\Pi_{B_{2}}^{m,l}}$$\scriptstyle{\phi^{m}}$
(4.8)

commute for all $m\leq l\leq k$. In particular if $\pi_{B_{1}}$ and
$\pi_{B_{2}}$ admit enough conservation laws to order $\operatorname{g}$ and
$f$ is a morphism of order $\operatorname{g}$, the diagram

${{\mathcal{H}}\big{(}\pi_{B_{1}}^{\operatorname{g},{\mathcal{F}}}\big{)}}$${{\mathcal{H}}\big{(}\pi_{B_{2}}^{\operatorname{g},{\mathcal{F}}}\big{)}}$${{\mathcal{H}}\big{(}\pi_{B_{1}}^{\infty,{\mathcal{F}}}\big{)}}$${{\mathcal{H}}\big{(}\pi_{B_{2}}^{\infty,{\mathcal{F}}}\big{)}}$${\cdots}$${{\mathcal{H}}\big{(}\pi_{B_{1}}^{k+1,{\mathcal{F}}}\big{)}}$${{\mathcal{H}}\big{(}\pi_{B_{1}}^{k,{\mathcal{F}}}\big{)}}$${\cdots}$${{\mathcal{H}}\big{(}\pi_{B_{1}}^{0,{\mathcal{F}}}\big{)}}$${\cdots}$${{\mathcal{H}}\big{(}\pi_{B_{2}}^{k+1,{\mathcal{F}}}\big{)}}$${{\mathcal{H}}\big{(}\pi_{B_{2}}^{k,{\mathcal{F}}}\big{)}}$${\cdots}$${{\mathcal{H}}\big{(}\pi_{B_{2}}^{0,{\mathcal{F}}}\big{)}}$$\scriptstyle{\phi^{\operatorname{g}}}$$\scriptstyle{\phi^{\infty}}$$\scriptstyle{\phi^{k+1}}$$\scriptstyle{\phi^{k}}$$\scriptstyle{\phi^{0}}$

commutes. Here the unlabelled arrows are as in Theorem 4.13. That is, the
hierarchy of holonomy groupoids is functorial.

###### Proof.

Our first task is to show that the map
$\operatorname{id}\colon{\mathcal{P}}({\mathcal{F}})\rightarrow{\mathcal{P}}({\mathcal{F}})$
does indeed descend to a well-defined morphism of each quotient. Suppose then
that $\big{(}\gamma_{1},[X_{1}]^{\operatorname{g}},d_{1}\big{)}$ and
$\big{(}\gamma_{2},[X_{2}]^{\operatorname{g}},d_{2}\big{)}$ in
${\mathcal{P}}({\mathcal{F}})$ satisfy

$\displaystyle
T\big{(}\pi_{B_{1}}^{k,{\mathcal{F}}}\big{)}(\gamma_{1},X_{1},d_{1})\big{(}x,[\sigma]^{k}_{x}\big{)}=T\big{(}\pi_{B_{1}}^{k,{\mathcal{F}}}\big{)}(\gamma_{2},X_{2},d_{2})\big{(}x,[\sigma]^{k}_{x}\big{)}$
(4.9)

for all $\big{(}x,[\sigma]_{x}^{k}\big{)}$ in
$\Gamma_{k}(\pi_{B_{1}})^{{\mathcal{F}}}$. Then since $f$ is a morphism, the
diagram

${{\mathcal{P}}({\mathcal{F}})\times_{s,\pi_{B_{1}}^{k,{\mathcal{F}}}}\Gamma_{k}(\pi_{B_{1}})^{{\mathcal{F}}}}$${{\mathcal{P}}\big{(}\Gamma_{k}(\pi_{B_{1}})^{{\mathcal{F}}}\big{)}}$${{\mathcal{P}}({\mathcal{F}})\times_{s,\pi_{B_{2}}^{k,{\mathcal{F}}}}\Gamma_{k}(\pi_{B_{2}})^{{\mathcal{F}}}}$${{\mathcal{P}}\big{(}\Gamma_{k}(\pi_{B_{2}})^{{\mathcal{F}}}\big{)}}$$\scriptstyle{L_{k}^{1}}$$\scriptstyle{\operatorname{id}\times
f^{{\mathcal{F}}}}$$\scriptstyle{{\mathcal{P}}(f^{{\mathcal{F}}})}$$\scriptstyle{L_{k}^{2}}$

commutes. Therefore, for each $i=1,2$, we have

$\displaystyle
T\big{(}\pi_{B_{2}}^{k,{\mathcal{F}}}\big{)}\big{(}\gamma_{i},[X_{i}]^{\operatorname{g}},d_{i}\big{)}\big{(}f^{{\mathcal{F}}}\big{(}x,[\sigma]^{k}_{x}\big{)}\big{)}$
$\displaystyle=L_{k}^{2}\big{(}\big{(}\gamma_{i},[X_{i}]^{\operatorname{g}},d_{i}\big{)};f^{{\mathcal{F}}}\big{(}x,[\sigma]^{k}_{x}\big{)}\big{)}(d_{i})$
$\displaystyle={\mathcal{P}}(f^{{\mathcal{F}}})\big{(}L_{k}^{1}\big{(}\gamma_{i},[X_{i}]^{\operatorname{g}},d_{i};x,[\sigma]^{k}_{x}\big{)}\big{)}(d_{i})$
$\displaystyle=f^{{\mathcal{F}}}\big{(}T\big{(}\pi_{B_{1}}^{k,{\mathcal{F}}}\big{)}\big{(}\gamma_{i},[X_{i}]^{\operatorname{g}},d_{i}\big{)}\big{(}x,[\sigma]_{x}^{k}\big{)}\big{)}$

for all
$\big{(}x,[\sigma]_{x}^{k}\big{)}\in\Gamma_{k}(\pi_{B_{1}})^{{\mathcal{F}}}$,
so surjectivity of $f^{{\mathcal{F}}}$ together with the hypothesis (4.9)
tells us that
$T\big{(}\pi_{B_{2}}^{k,{\mathcal{F}}}\big{)}\big{(}\gamma_{1},[X_{1}]^{\operatorname{g}},d_{1}\big{)}=T\big{(}\pi_{B_{2}}^{k,{\mathcal{F}}}\big{)}\big{(}\gamma_{2},[X_{2}]^{\operatorname{g}},d_{2}\big{)}$.
Therefore
$\operatorname{id}\colon{\mathcal{P}}({\mathcal{F}})\rightarrow{\mathcal{P}}({\mathcal{F}})$
does indeed descend to a map
$\phi^{k}\colon{\mathcal{H}}\big{(}\pi_{B_{1}}^{k,{\mathcal{F}}}\big{)}\rightarrow{\mathcal{H}}\big{(}\pi_{B_{2}}^{k,{\mathcal{F}}}\big{)}$,
whose smoothness follows from that of $\operatorname{id}$, and which is a
homomorphism by functoriality of $\operatorname{id}$.

We have thus proved that each of the diagrams

${{\mathcal{P}}({\mathcal{F}})}$${{\mathcal{P}}({\mathcal{F}})}$${{\mathcal{H}}\big{(}\pi_{B_{1}}^{k,{\mathcal{F}}}\big{)}}$${{\mathcal{H}}\big{(}\pi_{B_{2}}^{k,{\mathcal{F}}}\big{)}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\Pi_{B_{1}}^{k}}$$\scriptstyle{\Pi_{B_{2}}^{k}}$$\scriptstyle{\phi^{k}}$

commutes, where $\Pi_{B_{i}}^{k}$ is the quotient of
${\mathcal{P}}({\mathcal{F}})$ onto
${\mathcal{H}}\big{(}\pi_{B_{i}}^{k,{\mathcal{F}}}\big{)}$. It follows then
that for $l\leq k$, the diagram

${{\mathcal{P}}({\mathcal{F}})}$${{\mathcal{P}}({\mathcal{F}})}$${{\mathcal{H}}\big{(}\pi_{B_{1}}^{k,{\mathcal{F}}}\big{)}}$${{\mathcal{H}}\big{(}\pi_{B_{2}}^{k,{\mathcal{F}}}\big{)}}$${{\mathcal{H}}\big{(}\pi_{B_{1}}^{l,{\mathcal{F}}}\big{)}}$${{\mathcal{H}}\big{(}\pi_{B_{2}}^{l,{\mathcal{F}}}\big{)}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\Pi_{B_{1}}^{k}}$$\scriptstyle{\Pi_{B_{2}}^{k}}$$\scriptstyle{\phi^{k}}$$\scriptstyle{\Pi_{B_{1}}^{l,k}}$$\scriptstyle{\Pi_{B_{2}}^{l,k}}$$\scriptstyle{\phi^{l}}$

commutes, and then the result follows by Theorem 4.13. ∎

##  5 Outlook

In the author’s estimation, there are three primary questions arising from
this work that have yet to be answered.

Firstly, an assumption that we have had to impose in Definition 4.3 is that
morphisms of sheaves of smooth sections are smooth with respect to the
diffeology of Proposition 3.1. It is far from clear that this assumption is
really necessary. That is, it appears possible that any morphism of sheaves of
smooth sections is automatically smooth with respect to this diffeology.
Indeed, the domain considerations present in Proposition 3.1 are automatically
satisfied by maps arising from morphisms of sheaves, and attempts thus far to
construct a morphism of sheaves which is not smooth with respect to this
diffeology have proved unsuccessful. A proof that any morphism of sheaves of
smooth sections is itself diffeologically smooth would allow us to remove
these seemingly extraneous assumptions.

Secondly, it is clear from Examples 4.4, 4.5 and 4.6 that in many situations,
a singular partial connection on a fibre bundle induces a singular foliation
of its total space by projectable vector fields. This foliation cannot,
however, be expected to meet the requirements of Definition 2.1 (nor the
equivalent definitions using compactly supported vector fields, for instance
the one used in [2]), since projectable vector fields are not closed under
multiplication by arbitrary smooth functions on the total space. This suggests
that Definition 2.1 might to be relaxed to allow for closure of vector fields
under the sheaf of projectable functions. Such a modification will have no
effect on the integration theorem (see [59, Theorem 4.2(e)]). Having relaxed
Definition 2.1, a proof that singularly foliated bundles admit foliations of
their total spaces will require a proof that a presheaf of Lie–Rinehart
algebras (such as the image of a singular partial connection) becomes a sheaf
of Lie–Rinehart algebras under sheafification. This question does not appear
to have been studied in the literature.

Finally, and most importantly, it would be interesting to prove Conjecture
4.17 and to determine whether the techniques of this paper can be generalised
to recover the Androulidakis–Skandalis holonomy groupoid in generality. That
is, for any foliation $(M,{\mathcal{F}})$, one seeks a pseudo-bundle
$\pi_{{\mathcal{B}}}\colon{\mathcal{B}}\rightarrow M$ and a lifting map
${\mathcal{P}}({\mathcal{F}})\times_{s,\pi_{{\mathcal{B}}}}{\mathcal{B}}\rightarrow{\mathcal{P}}({\mathcal{B}})$,
for which quotient of ${\mathcal{P}}({\mathcal{F}})$ by the fibres of the
associated transport functor is isomorphic to the Androulidakis–Skandalis
groupoid. We have two suggestions in this direction.

  1. 1.

It may be possible define a diffeological pseudo-bundle of germs of slices,
whose fibre over $x\in M$ consists of all germs of slices through $x$, using a
diffeology similar to that of the pseudo-bundles of germs given in this paper.
Since flows of elements of ${\mathcal{P}}({\mathcal{F}})$ send slices to
slices, one would obtain the sought-after lifting map using flows. The results
of Garmendia–Villatoro [60] suggest that the resulting holonomy groupoid would
be isomorphic to the Androulidakis–Skandalis groupoid.

  2. 2.

As suggested by one of the referees, one could alternatively attempt to equip
the space $N{\mathcal{F}}:=\sqcup_{x\in M}N_{x}L$ of normal fibres of
$(M,{\mathcal{F}})$ considered in [6] with a diffeology with respect to which
it is a pseudo-bundle over $M$. This pseudo-bundle would admit a (leafwise)
Bott connection [6, p. 374], which could be used to lift elements of
${\mathcal{P}}({\mathcal{F}})$ to $N{\mathcal{F}}$.

### Acknowledgements

This research was supported by the Australian Research Council, through the
Discovery Project grant DP200100729. I extend special thanks to V. Mathai, for
taking me on as a postdoc at the University of Adelaide, and to I.
Androulidakis, for enlightening discussions and correspondence in late 2018
and for encouraging me to think about singular foliations. I thank B.
McMillan, for helpful discussions concerning conservation laws. Finally, I
extend deep thanks to the anonymous referees, whose careful consideration and
critique of the paper have greatly improved its exposition.

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# Strong chiral optical force for small chiral molecules based on electric-
dipole interactions, inspired by the asymmetrical hydrozoan Velella velella

Robert P. Cameron  1,* Duncan McArthur  1 and Alison M. Yao1  1SUPA and
Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK
<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

Drawing inspiration from a remarkable chiral force found in nature, we show
that a static electric field combined with an optical lin$\perp$lin
polarization standing wave can exert a chiral optical force on a small chiral
molecule that is several orders of magnitude stronger than other chiral
optical forces proposed to date, being based on leading electric-dipole
interactions rather than relying on weak magnetic-dipole and electric-
quadrupole interactions. Our chiral optical force applies to most small chiral
molecules, including isotopically chiral molecules, and does not require a
specific energy-level structure. Potential applications range from chiral
molecular matter-wave interferometry for precision metrology and tests of
fundamental physics to the resolution of enantiomers for use in chemistry and
biology.

##  1 Introduction

Interest is growing in the possibility of an optical force that discriminates
between the enantiomers of a small111In this paper, we consider a molecule to
be “small” if it has a mass of $M\sim 10^{2}\,\mathrm{Da}$ or less. chiral
molecule [1, 2, 3]. Such a force could be used in a wealth of potential
applications, including chiral molecular matter-wave interferometry for
precision metrology and tests of fundamental physics [4, 5, 6], the
distillation of chiral molecules in chiral optical lattices as a novel form of
matter [7], and the resolution of enantiomers for use in chemistry and biology
[8, 9, 4, 10, 11, 12, 13]; a task of particular importance [14, 15]. The
chiral optical forces proposed to date, however, are extremely weak, as they
rely on magnetic-dipole and electric-quadrupole interactions. To the best of
our knowledge, no experimental observations of chiral optical forces have been
reported for small chiral molecules, although chiral optical forces have been
demonstrated for large chiral objects, including chiral liquid crystal
microspheres [16, 17, 18, 19, 20, 21, 22, 23] and chiral cantilevers [24].

In this paper, we identify a way forward. Drawing inspiration from a
remarkable chiral force found in nature for the asymmetrical hydrozoan Velella
velella [25, 15], we show that a static electric field combined with an
optical lin$\perp$lin polarization standing wave can exert a chiral optical
force on a small chiral molecule that is several orders of magnitude stronger
than other chiral optical forces proposed to date, being based on electric-
dipole interactions at leading order. Our chiral optical force has the form

$\displaystyle\mathbf{F}$ $\displaystyle\approx
k\mathcal{E}_{z}\mathcal{E}_{y}\mathcal{E}_{x}(\mathtt{A}\alpha_{ZY}\mu_{0X}+\mathtt{B}\alpha_{XZ}\mu_{0Y}+\mathtt{C}\alpha_{YX}\mu_{0Z})\cos(2kZ_{0})\hat{\mathbf{z}},$
(1)

which changes sign if either the electric field or the molecule is inverted,
$\mathcal{E}_{z}\mathcal{E}_{y}\mathcal{E}_{x}$ being a chirally sensitive
product due to the electric field and $\alpha_{ZY}\mu_{0X}$,
$\alpha_{XZ}\mu_{0Y}$, and $\alpha_{YX}\mu_{0Z}$ being chirally sensitive
molecular properties. Molecular anisotropy and rotation are accounted for in
our theory, with orientational effects playing a crucial role. Our chiral
optical force applies to most small chiral molecules, including isotopically
chiral molecules, and does not involve absorption or require a specific
energy-level structure. Potential applications identified elsewhere for other
chiral optical forces [8, 9, 4, 5, 10, 11, 1, 12, 2, 6, 7, 3, 13] should be
enormously more robust and easier to realise using our chiral optical force
instead.

Throughout, we work in vacuum in an inertial frame of reference with time $t$
and position vector
$\mathbf{r}=x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z\hat{\mathbf{z}}$, where $x$,
$y$, and $z$ are laboratory-fixed Cartesian coordinates and
$\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, and $\hat{\mathbf{z}}$ are the
associated unit vectors. SI units are used, $c$ being the speed of light,
$\hbar$ being the reduced Planck constant, $\epsilon_{0}$ being the electric
constant, and $k_{B}$ being Boltzmann’s constant. The Einstein summation
convention is to be understood with respect to lower-case Roman indices
$a,b,\dots\in\\{x,y,z\\}$ and uppercase Roman indices
$A,B,\dots\in\\{X,Y,Z\\}$, where $X$, $Y$, and $Z$ are molecule-fixed
Cartesian coordinates. Complex quantities are decorated with tildes.

##  2 Origin of our chiral optical force

Figure 1: The two distinct mirror-image forms of Velella velella are found on
opposite sides of the Pacific a), having been separated by the wind b). Our
chiral optical force is constructed by analogy c).

This paper was inspired by a remarkable chiral force found in nature. The
asymmetrical hydrozoan Velella velella lives on the surface of the ocean and
has a sail that tilts either antidiagonally or diagonally with respect to a
quasi-rectangular base, giving two distinct mirror-image forms. Specimens
found on the Japanese side of the Pacific are predominantly of the
antidiagonal form whereas those found on the North American side are
predominantly of the diagonal form, as illustrated in Fig. 1 a). It is
believed that both forms arise in roughly equal numbers in the center of the
Pacific and are then separated by the wind [25, 15].

The origin of this chiral force can be understood as follows. A Vellela
velella specimen is orientated by gravity via the normal force. With its sail
blown by the wind, the specimen deflects air molecules from the initial air
channel into a final air channel with an orthogonal component. The
corresponding momentum transfer gives rise to a force with a chiral component.
A subtle but crucial ingredient is that the specimen has preferred
orientations relative to the wind, due to the quasi-rectangular shape of its
base. If all possible orientations were to occur with equal probability
instead, the chiral force would vanish on average. The normal force, initial
air channel and relevant component of the final air channel form either a
left-handed or a right-handed orthogonal triad embodying the chiral
sensitivity of the force, as illustrated in Fig. 1 b).

We construct our chiral optical force by analogy with the above. Instead of a
Velella velella specimen, we consider a chiral molecule. Instead of gravity,
we consider a strong and homogeneous static electric field
$\mathbf{E}_{0}=\mathbf{E}_{0}(\mathbf{r},t)$ given by

$\displaystyle\mathbf{E}_{0}$ $\displaystyle=\mathcal{E}_{z}\hat{\mathbf{z}},$
(2)

where $\mathcal{E}_{z}$ dictates the field’s strength and direction. The
molecule is (partially) orientated by $\mathbf{E}_{0}$ via its permanent
electric-dipole moment. Instead of the wind, we consider an intense and far
off-resonance optical lin$\perp$lin polarization standing wave222Lin$\perp$lin
polarization standing waves are used routinely to laser cool atoms via the
Sisyphus effect [27, 26]. There is little connection with our chiral optical
force. with complex electric field
$\tilde{\mathbf{E}}=\tilde{\mathbf{E}}(\mathbf{r},t)$ given by

$\displaystyle\tilde{\mathbf{E}}$
$\displaystyle=(\mathcal{E}_{y}\hat{\mathbf{y}}\mathrm{e}^{\mathrm{i}kz}+\mathrm{i}\mathcal{E}_{x}\hat{\mathbf{x}}\mathrm{e}^{-\mathrm{i}kz})\mathrm{e}^{-\mathrm{i}\omega
t},$ (3)

where $\mathcal{E}_{y}$ dictates the amplitude and phase of the $y$ polarized
wave, $\mathcal{E}_{x}$ dictates the amplitude and phase of the $x$ polarized
wave, and $\omega$ is the angular frequency of the waves, $k=\omega/c$ being
the associated angular wavenumber. The molecule transfers photons from the $y$
polarized wave to the $x$ polarized wave or vice versa [26]. The corresponding
momentum transfer gives rise to our chiral optical force. To ensure that the
molecule has preferred orientations relative to the $y$ (or $x$) polarized
wave, we take its rotational angular momentum to be quantized along the $y$
(or $x$) axis333In practice, this might be achieved using a static magnetic
field, for example.. If the angular momentum were quantized along the $z$ axis
instead, our chiral optical force would vanish. The static electric field
$\mathcal{E}_{z}\hat{\mathbf{z}}$ and the optical polarization vectors
$\mathcal{E}_{y}\hat{\mathbf{y}}$ and $\mathcal{E}_{x}\hat{\mathbf{x}}$ form
either a left-handed ($\mathcal{E}_{z}\mathcal{E}_{y}\mathcal{E}_{x}<0$) or a
right-handed ($\mathcal{E}_{z}\mathcal{E}_{y}\mathcal{E}_{x}>0$) orthogonal
triad embodying the chiral sensitivity of our force, as illustrated in Fig. 1
c).

##  3 Derivation of our chiral optical force

In this section, we derive Eq. (1) for our chiral optical force, considering
the electromagnetic field defined in Sec. 2 applied to a small, polar,
diamagnetic, chiral molecule in its vibronic ground state with nuclear spins
of $0$ or $1/2$ (see Sec. 3.2).

We perform our derivation in two steps. First, we focus on the molecule’s
vibronic degrees of freedom and derive an expression for the cycle-averaged
optical force with the molecule’s orientation frozen. This is appropriate
assuming the molecule rotates slowly relative to the optical frequency such
that its orientation changes little during an optical period, which will
usually be the case. Second, we focus on the molecule’s rotational degrees of
freedom and average the force with respect to the molecule’s rotational state,
giving our chiral optical force. This two-step approach assumes adiabatic
following and ignores rovibronic coupling, which will usually be of little
consequence in the vibronic ground state. Throughout, we take the molecule’s
center of mass position to be held fixed.

###  3.1 Vibronic degrees of freedom

We take the instantaneous force $\mathbf{f}=\mathbf{f}(t)$ exerted by the
electromagnetic field on the molecule to be given by the Lorentz force law

$\displaystyle\mathbf{f}$
$\displaystyle=\iiint(\rho\mathbf{E}+\mathbf{J}\times\mathbf{B})\,\mathrm{d}^{3}\mathbf{r},$
(4)

where $\rho=\rho(\mathbf{r},t)$ is the molecule’s electric charge density,
$\mathbf{J}=\mathbf{J}(\mathbf{r},t)$ is the molecule’s electric current
density,
$\mathbf{E}=\mathbf{E}(\mathbf{r},t)=\mathbf{E}_{0}+\Re\tilde{\mathbf{E}}$ is
the electric field, $\mathbf{B}=\mathbf{B}(\mathbf{r},t)$ is the magnetic
field, and the region of integration extends over the entire molecule [28,
29].

As the molecule is electrically neutral, we have

$\displaystyle\rho$ $\displaystyle=-\boldsymbol{\nabla}\cdot\mathbf{P}$ (5)
$\displaystyle\mathbf{J}$ $\displaystyle=\frac{\partial\mathbf{P}}{\partial
t}+\boldsymbol{\nabla}\times\mathbf{M},$ (6)

where $\mathbf{P}=\mathbf{P}(\mathbf{r},t)$ is the molecule’s polarization
field and $\mathbf{M}=\mathbf{M}(\mathbf{r},t)$ is the molecule’s
magnetization field [29]. Substituting Eqs. (5) and (6) into Eq. (4) then
making use of the triple product
$(\boldsymbol{\nabla}\times\mathbf{M})\times\mathbf{B}=(\mathbf{B}\cdot\boldsymbol{\nabla})\mathbf{M}-M_{a}\boldsymbol{\nabla}B_{a}$,
integration by parts, and the Faraday-Lenz law
$\boldsymbol{\nabla}\times\mathbf{E}=-\partial\mathbf{B}/\partial t$, we
obtain

$\displaystyle\mathbf{f}$
$\displaystyle=\iiint(P_{a}\boldsymbol{\nabla}E_{a}+M_{a}\boldsymbol{\nabla}B_{a})\,\mathrm{d}^{3}\mathbf{r}+\frac{\mathrm{d}}{\mathrm{d}t}\iiint\mathbf{P}\times\mathbf{B}\,\mathrm{d}^{3}\mathbf{r},$
(7)

which facilitates a multipolar expansion [4].

As the molecule is small, we consider only electric-dipole interactions
explicitly, taking

$\displaystyle\mathbf{P}$
$\displaystyle\approx\boldsymbol{\mu}\,\delta^{3}(\mathbf{r}-\mathbf{R}_{0})$
(8) $\displaystyle\mathbf{M}$ $\displaystyle\approx 0,$ (9)

where $\boldsymbol{\mu}=\boldsymbol{\mu}(t)$ is the molecule’s electric-dipole
moment and
$\mathbf{R}_{0}=X_{0}\hat{\mathbf{x}}+Y_{0}\hat{\mathbf{y}}+Z_{0}\hat{\mathbf{z}}$
is the molecule’s center of mass position, with $X_{0}$, $Y_{0}$ and $Z_{0}$
being the Cartesian coordinates [30]. Substituting Eqs. (8) and (9) into Eq.
(7), we obtain

$\displaystyle\mathbf{f}$
$\displaystyle\approx\mu_{a}\boldsymbol{\nabla}E_{a}(\mathbf{R}_{0})+\frac{\mathrm{d}}{\mathrm{d}t}\left[\boldsymbol{\mu}\times\mathbf{B}(\mathbf{R}_{0})\right],$
(10)

which is the well-established result for the Lorentz force experienced by an
electric dipole [31, 32].

As the static electric field $\mathbf{E}_{0}$ is strong and the standing wave
(with electric field $\Re\tilde{\mathbf{E}}$) is intense, we seek an
expression for our chiral optical force valid to third order in field
components. As Eq. (10) is already linear in field components, it is
sufficient to consider the electric-dipole moment $\boldsymbol{\mu}$ to second
order in field components. Remembering that we are considering only electric-
dipole interactions explicitly, we take

$\displaystyle\boldsymbol{\mu}$
$\displaystyle=\boldsymbol{\mu}_{0}+\Re\tilde{\boldsymbol{\mu}}$ (11)

with

$\displaystyle\tilde{\mu}_{a}$
$\displaystyle\approx\tilde{\alpha}_{ab}(0)E_{0b}+[\tilde{\alpha}_{ab}(\omega)+\tilde{\alpha}_{ab,c}(\omega)E_{0c}]\tilde{E}_{b}$
$\displaystyle+\frac{1}{2}\tilde{\beta}_{abc}(0;0,0)E_{0b}E_{0c}+\frac{1}{2}\tilde{\beta}_{abc}(2\omega;\omega,\omega)\tilde{E}_{b}\tilde{E}_{c}+\frac{1}{2}\tilde{\beta}_{abc}(0;\omega,-\omega)\tilde{E}_{b}\tilde{E}^{\ast}_{c},$
(12)

where $\boldsymbol{\mu}_{0}$ is the molecule’s permanent electric-dipole
moment, $\tilde{\boldsymbol{\mu}}=\tilde{\boldsymbol{\mu}}(t)$ is the
molecule’s complex induced electric-dipole moment,
$\tilde{\alpha}_{ab}=\tilde{\alpha}_{ab}(\Omega)$ is the molecule’s complex
vibronic polarizability,
$\tilde{\alpha}_{ab,c}=\tilde{\alpha}_{ab,c}(\Omega)$444We have simplified the
notation used in [33] by writing
$\tilde{\alpha}_{ab,c}^{(\mu)}\rightarrow\tilde{\alpha}_{ab,c}$. is a
perturbative correction to $\tilde{\alpha}_{ab}$ due to $\mathbf{E}_{0}$, and
$\tilde{\beta}_{abc}=\tilde{\beta}_{abc}(\Omega+\Omega^{\prime};\Omega,\Omega^{\prime})$
is the molecule’s complex vibronic hyperpolarizability, $\Omega$ and
$\Omega^{\prime}$ being angular frequencies [33]. Eq. (12) consists of a
linear static contribution, a linear optical contribution with a linear static
correction, a second-order static contribution, a second-order optical
harmonic contribution, and a second-order optical rectification contribution,
these being all possible first-order and second-order contributions based on
electric-dipole interactions. Substituting Eqs. (11) and (12) into Eq. (10)
then taking an average over one optical period, we find that the cycle-
averaged optical force $\overline{\mathbf{f}}$ is

$\displaystyle\overline{\mathbf{f}}$
$\displaystyle=\frac{\omega}{2\pi}\int_{0}^{2\pi/\omega}\mathbf{f}\,\mathrm{d}t$
$\displaystyle\approx\frac{1}{2}\Re\\{[\tilde{\alpha}_{ab}(\omega)+\tilde{\alpha}_{ab,c}(\omega)E_{0c}]\tilde{E}_{b}(\mathbf{R}_{0})\boldsymbol{\nabla}\tilde{E}^{\ast}_{a}(\mathbf{R}_{0})\\}.$
(13)

Note that $\tilde{\beta}_{abc}$ does not contribute to Eq. (13).

The polarizabilities $\tilde{\alpha}_{ab}$ and $\tilde{\alpha}_{ab,c}$ can be
partitioned as

$\displaystyle\tilde{\alpha}_{ab}$
$\displaystyle=\alpha_{ab}(f)+\mathrm{i}\alpha_{ab}(g)-\mathrm{i}[\alpha^{\prime}_{ab}(f)+\alpha^{\prime}_{ab}(g)]$
(14) $\displaystyle\tilde{\alpha}_{ab,c}$
$\displaystyle=\alpha_{ab,c}(f)+\mathrm{i}\alpha_{ab,c}(g)-\mathrm{i}[\alpha^{\prime}_{ab,c}(f)+\alpha^{\prime}_{ab,c}(g)],$
(15)

where $\alpha_{ab}(f)=\alpha_{ba}(f)$ and $\alpha_{ab,c}(f)=\alpha_{ba,c}(f)$
are the time-even, dispersive contributions; $\alpha_{ab}(g)=\alpha_{ba}(g)$
and $\alpha_{ab,c}(g)=\alpha_{ba,c}(g)$ are the time-even, absorptive
contributions; $\alpha^{\prime}_{ab}(f)=-\alpha^{\prime}_{ba}(f)$ and
$\alpha^{\prime}_{ab,c}(f)=-\alpha^{\prime}_{ba,c}(f)$ are the time-odd,
dispersive contributions; $\alpha^{\prime}_{ab}(g)=-\alpha^{\prime}_{ba}(g)$
and $\alpha^{\prime}_{ab,c}(g)=-\alpha^{\prime}_{ba,c}(g)$ are the time-odd,
dispersive contributions; $f=f_{\Omega}$ indicates a dispersive lineshape, and
$g=g_{\Omega}$ indicates an absorptive lineshape [33]. Substituting Eqs. (2),
(3), (14) and (15) into Eq. (13), we obtain

$\displaystyle\overline{\mathbf{f}}$
$\displaystyle\approx\overline{\mathbf{f}}(f_{\omega})+\overline{\mathbf{f}}(g_{\omega})+\overline{\mathbf{f}}^{\prime}(f_{\omega})+\overline{\mathbf{f}}^{\prime}(g_{\omega})$
(16)

with

$\displaystyle\overline{\mathbf{f}}(f_{\omega})$ $\displaystyle\approx
k[\alpha_{yx}(f_{\omega})+\alpha_{yx,z}(f_{\omega})\mathcal{E}_{z}]\mathcal{E}_{y}\mathcal{E}_{x}\cos(2kZ_{0})\hat{\mathbf{z}},$
(17) $\displaystyle\overline{\mathbf{f}}(g_{\omega})$
$\displaystyle\approx\frac{1}{2}k\\{[\alpha_{yy}(g_{\omega})+\alpha_{yy,z}(g_{\omega})\mathcal{E}_{z}]\mathcal{E}_{y}^{2}-[\alpha_{xx}(g_{\omega})+\alpha_{xx,z}(g_{\omega})\mathcal{E}_{z}]\mathcal{E}_{x}^{2}\\}\hat{\mathbf{z}},$
(18) $\displaystyle\overline{\mathbf{f}}^{\prime}(f_{\omega})$
$\displaystyle\approx-k[\alpha^{\prime}_{yx}(f_{\omega})+\alpha_{yx,z}^{\prime}(f_{\omega})\mathcal{E}_{z}]\mathcal{E}_{y}\mathcal{E}_{x}\sin(2kZ_{0})\hat{\mathbf{z}},$
(19) $\displaystyle\overline{\mathbf{f}}^{\prime}(g_{\omega})$
$\displaystyle\approx 0,$ (20)

where $\overline{\mathbf{f}}(f_{\omega})$ is the time-even, dispersive
contribution; $\overline{\mathbf{f}}(g_{\omega})$ is the time-even, absorptive
contribution; $\overline{\mathbf{f}}^{\prime}(f_{\omega})$ is the time-odd,
dispersive contribution, and $\overline{\mathbf{f}}^{\prime}(g_{\omega})$ is
the time-odd, absorptive contribution.

As the standing wave is far off-resonance, we neglect absorptive
contributions. As the molecule is diamagnetic and thus time-even, we neglect
time-odd contributions. Eqs. (16)-(20) reduce accordingly to

$\displaystyle\overline{\mathbf{f}}$ $\displaystyle\approx
k[\alpha_{yx}(f_{\omega})+\alpha_{yx,z}(f_{\omega})\mathcal{E}_{z}]\mathcal{E}_{y}\mathcal{E}_{x}\cos(2kZ_{0})\hat{\mathbf{z}}.$
(21)

Eq. (21) is our final expression for the cycle-averaged optical force with the
molecule’s orientation frozen.

###  3.2 Rotational degrees of freedom

As the molecule’s nuclear spins are $0$ or $1/2$, we assume that they can be
decoupled from the molecule’s rotational degrees of freedom using a static
magnetic field, for example, and neglect them. Viable isotopes in this regard
include ${}^{1}\mathrm{H}$, ${}^{12}\mathrm{C}$, ${}^{13}\mathrm{C}$,
${}^{15}\mathrm{N}$, ${}^{16}\mathrm{O}$, ${}^{18}\mathrm{O}$, and
${}^{19}\mathrm{F}$. If the molecule had nuclear spins of $1$ or higher, they
would strongly couple to the molecule’s rotational degrees of freedom via
electric-quadrupolar interactions [34, 35, 36], necessitating explicit
consideration.

We take the molecule’s rotational state $|\psi\rangle=|\psi(t)\rangle$ to be
governed by the Schrödinger equation

$\displaystyle\mathrm{i}\hbar\frac{\mathrm{d}|\psi\rangle}{\mathrm{d}t}$
$\displaystyle=(H^{(0)}+V)|\psi\rangle,$

where $H^{(0)}$ is the molecule’s unperturbed rotational Hamiltonian and $V$
is the molecule’s rotational interaction Hamiltonian [37]. $H^{(0)}$ describes
the molecule’s rotational degrees of freedom in the absence of the
electromagnetic field and $V$ accounts for the orientational dependence of the
molecule’s interaction with the electromagnetic field. We assume that
$H^{(0)}\gg|V|$ and treat the interaction perturbatively.

According to basic perturbation theory, the molecule’s rotational energy
eigenstates $|r\rangle$ and associated energy eigenvalues $E_{r}$ are given by

$\displaystyle|r\rangle$ $\displaystyle=|r\rangle^{(0)}+\sum_{q\neq
r}\frac{{}^{(0)}\langle
q|V|r\rangle^{(0)}}{E_{r}^{(0)}-E_{q}^{(0)}}|q\rangle^{(0)}+\dots$ (22)
$\displaystyle E_{r}$ $\displaystyle=E_{r}^{(0)}+{}^{(0)}\langle
r|V|r\rangle^{(0)}+\dots,$ (23)

where the $|r\rangle^{(0)}$ are the molecule’s unperturbed rotational energy
eigenstates, the $E^{(0)}_{r}$ are the associated energy eigenvalues, and we
assume that the rotational interaction Hamiltonian $V$ is diagonal in
degenerate blocks, with no accidental near degeneracies of importance [37].
The molecule’s rotational state $|\psi\rangle$ can be expanded in terms of the
$|r\rangle$ and the $E_{r}$ as

$\displaystyle|\psi\rangle$
$\displaystyle=\sum_{r}\tilde{a}_{r}|\psi_{r}\rangle$ (24)

with

$\displaystyle|\psi_{r}\rangle$
$\displaystyle=\exp(-\mathrm{i}E_{r}t/\hbar)|r\rangle,$ (25)

where the $|\psi_{r}\rangle=|\psi_{r}(t)\rangle$ are the molecule’s stationary
rotational states and the $\tilde{a}_{r}$ are the associated probability
amplitudes [37].

Taking the expectation value of Eq. (21) with respect to Eqs. (24) and (25)
using Eqs. (22) and (23), we find that the cycle-averaged, rotationally
averaged optical force
$\langle\overline{\mathbf{f}}\rangle=\langle\overline{\mathbf{f}}\rangle_{\psi}$
is

$\displaystyle\langle\overline{\mathbf{f}}\rangle$
$\displaystyle=\langle\psi|\overline{\mathbf{f}}|\psi\rangle$
$\displaystyle\approx
k\langle\alpha_{yx}(f_{\omega})+\alpha_{yx,z}(f_{\omega})\mathcal{E}_{z}\rangle\mathcal{E}_{y}\mathcal{E}_{x}\cos(2kZ_{0})\hat{\mathbf{z}}$
(26)

with

$\displaystyle\langle\alpha_{yx}(f_{\omega})+\alpha_{yx,z}(f_{\omega})\mathcal{E}_{z}\rangle$
$\displaystyle\approx\sum_{r}\sum_{s}\tilde{a}^{\ast}_{r}\tilde{a}_{s}\mathrm{e}^{\mathrm{i}(E_{r}^{(0)}-E_{s}^{(0)}+{}^{(0)}\langle
r|V|r\rangle^{(0)}-{}^{(0)}\langle s|V|s\rangle^{(0)}+\dots)t/\hbar}$
$\displaystyle\times\Bigg{[}{}^{(0)}\langle
r|\alpha_{yx}(f_{\omega})+\alpha_{yx,z}(f_{\omega})\mathcal{E}_{z}|s\rangle^{(0)}$
$\displaystyle+\sum_{q\neq r}\frac{{}^{(0)}\langle
r|V|q\rangle^{(0)}{}^{(0)}\langle
q|\alpha_{yx}(f_{\omega})|s\rangle^{(0)}}{E_{r}^{(0)}-E_{q}^{(0)}}$
$\displaystyle+\sum_{q\neq s}\frac{{}^{(0)}\langle
r|\alpha_{yx}(f_{\omega})|q\rangle^{(0)}{}^{(0)}\langle
q|V|s\rangle^{(0)}}{E_{s}^{(0)}-E_{q}^{(0)}}+\dots\Bigg{]}.$ (27)

The first contribution inside the square brackets in Eq. (27) describes the
force that would be obtained if the molecule rotated freely and the remaining
contributions describe perturbative corrections due to the orientational
dependence of the molecule’s interaction with the electromagnetic field. To
evaluate Eqs. (26) and (27), we need an explicit model of the molecule’s
rotational degrees of freedom.

As the molecule is small, chiral and in its vibronic ground state, we treat it
simply as an asymmetric rigid rotor quantized along the $y$
axis555Quantization along the $x$ axis instead sees our chiral optical force
change sign. Quantization along the $z$ axis sees our chiral optical force
vanish., the laboratory-fixed components $\mu_{0a}$,
$\alpha_{ab}(f_{\omega})$, and $\alpha_{ab,c}(f_{\omega})$ being related to
the corresponding molecule-fixed components $\mu_{0A}$,
$\alpha_{AB}(f_{\omega})$, and $\alpha_{AB,C}(f_{\omega})$ via

$\displaystyle\mu_{0a}$ $\displaystyle=\ell_{aA}\mu_{0A},$ (28)
$\displaystyle\alpha_{ab}(f_{\omega})$
$\displaystyle=\ell_{aA}\ell_{bB}\alpha_{AB}(f_{\omega}),$ (29)
$\displaystyle\alpha_{ab,c}(f_{\omega})$
$\displaystyle=\ell_{aA}\ell_{bB}\ell_{cC}\alpha_{AB,C}(f_{\omega}),$ (30)

where the $\ell_{aA}$ are direction cosines, as described in Appendix A.

Recall that we we seek an expression for our chiral optical force valid to
third order in field components. As Eq. (26) is already second order in field
components at leading order, it is sufficient to consider the rotational
interaction Hamiltonian $V$ to linear order in field components. Remembering
once more that we are considering only electric-dipole interactions
explicitly, we take

$\displaystyle V$
$\displaystyle\approx-\boldsymbol{\mu}_{0}\cdot\mathbf{E}_{0},$ (31)

which is the potential energy of the molecule’s permanent electric-dipole
moment $\boldsymbol{\mu}_{0}$ in the static electric field $\mathbf{E}_{0}$.
This form of $V$ is trivially degenerate in diagonal blocks, as the linear
Stark effect vanishes for an asymmetric rigid rotor. We have tacitly assumed
here that
$|-\boldsymbol{\mu}_{0}\cdot\mathbf{E}_{0}|\gg|-[\alpha_{ab}(f_{\omega})+\alpha_{ab,c}(f_{\omega})E_{0c}]\tilde{E}_{a}\tilde{E}_{b}^{\ast}/4|$,
where
$-[\alpha_{ab}(f_{\omega})+\alpha_{ab,c}(f_{\omega})E_{0c}]\tilde{E}_{a}\tilde{E}_{b}^{\ast}/4$
accounts for the orientational dependence of the molecule’s ac Stark
interaction with the standing wave.

Figure 2: For a given configuration of the electric field, our chiral optical
force has opposite signs for opposite enantiomers in the same rotational
state; compare a) with b) and c) with d). For a given enantiomer in a given
rotational state, our chiral optical force has opposite signs depending on
whether the electric field is in left-handed or a right-handed configuration;
compare a) with c) and b) with d).

Let us suppose now that the molecule’s rotational state has the
stationary666When the ac Stark interaction
$-[\alpha_{ab}(f_{\omega})+\alpha_{ab,c}(f_{\omega})E_{0c}]\tilde{E}_{a}\tilde{E}_{b}^{\ast}/4$
is included in the rotational interaction Hamiltonian $V$, we find that
rotational states quantized along the $y$ (or $x$) axis are not necessarily
stationary. This limits the validity of our treatment to suitably short
interaction times. form

$\displaystyle|\psi\rangle$ $\displaystyle=|\psi_{J_{\tau},m}\rangle,$ (32)

where $J$ and $m$ are rotational quantum numbers and $\tau$ is an energy
label, as described in Appendix A. Substituting Eqs. (28)-(32) into Eqs. (26)
and (27) with Eq. (27) considered to linear order in $V$ then making use of
basic symmetry arguments to disregard vanishing contributions, we obtain

$\displaystyle\langle\overline{\mathbf{f}}\rangle$ $\displaystyle\approx
k\mathcal{E}_{z}\mathcal{E}_{y}\mathcal{E}_{x}[\mathtt{a}\alpha_{ZY,X}(f_{\omega})+\mathtt{b}\alpha_{XZ,Y}(f_{\omega})+\mathtt{c}\alpha_{YX,Z}(f_{\omega})$
$\displaystyle+\mathtt{A}\alpha_{ZY}(f_{\omega})\mu_{0X}+\mathtt{B}\alpha_{XZ}(f_{\omega})\mu_{0Y}+\mathtt{C}\alpha_{YX}(f_{\omega})\mu_{0Z}]\cos(2kZ_{0})\hat{\mathbf{z}},$
(33)

where $\mathtt{a}$, $\mathtt{b}$, $\mathtt{c}$, $\mathtt{A}$, $\mathtt{B}$,
and $\mathtt{C}$ are coefficients that depend on the molecule’s rotational
state, as described in Appendix B. The $\mathtt{a}$, $\mathtt{b}$, and
$\mathtt{c}$ contributions are due to perturbation of the molecule’s vibronic
degrees of freedom by the static electric field $\mathbf{E}_{0}$ whereas the
$\mathtt{A}$, $\mathtt{B}$, and $\mathtt{C}$ contributions are due to
perturbation of the molecule’s rotational degrees of freedom. Note that
$\langle\alpha_{yx}(f_{\omega})\rangle$ does not contribute to Eq. (33). The
molecular properties $\alpha_{ZY}(f_{\omega})\mu_{0X}$,
$\alpha_{XZ}(f_{\omega})\mu_{0Y}$, $\alpha_{YX}(f_{\omega})\mu_{0Z}$,
$\alpha_{ZY,X}(f_{\omega})$, $\alpha_{XZ,Y}(f_{\omega})$ and
$\alpha_{YX,Z}(f_{\omega})$ are chirally sensitive, having opposite signs for
opposite enantiomers.

As the vibronic excitations of a molecule are usually much more energetic than
the rotational excitations, we assume that the $\mathtt{a}$, $\mathtt{b}$, and
$\mathtt{c}$ contributions are much smaller than the $\mathtt{A}$,
$\mathtt{B}$, and $\mathtt{C}$ contributions and neglect them. Eq. (33)
reduces accordingly to

$\displaystyle\mathbf{F}$ $\displaystyle\approx
k\mathcal{E}_{z}\mathcal{E}_{y}\mathcal{E}_{x}(\mathtt{A}\alpha_{ZY}\mu_{0X}+\mathtt{B}\alpha_{XZ}\mu_{0Y}+\mathtt{C}\alpha_{YX}\mu_{0Z})\cos(2kZ_{0})\hat{\mathbf{z}},$

where we have simplified our notation by writing
$\langle\overline{\mathbf{f}}\rangle\rightarrow\mathbf{F}=\mathbf{F}_{J_{\tau},m}$
and $\alpha_{AB}(f_{\omega})\rightarrow\alpha_{AB}$. This is our chiral
optical force, as seen in Eq. (1).

Our chiral optical force depends on the chirality of the electric field
$\mathbf{E}$ through the product
$\mathcal{E}_{z}\mathcal{E}_{y}\mathcal{E}_{x}$ and on the chirality of the
molecule through the molecular properties $\alpha_{ZY}\mu_{0X}$,
$\alpha_{XZ}\mu_{0Y}$ and $\alpha_{YX}\mu_{0Z}$, as illustrated in Fig. 2. To
distinguish it from other forces proposed to date, our chiral optical force
might be referred to as a “chiral optical force from
$\mathcal{E}_{z}\mathcal{E}_{y}\mathcal{E}_{x}$” or “COFFEEE”.

##  4 Numerical results

In this section, we present numerical results for our chiral optical force. We
focus on a right-handed electric field configuration
($\mathcal{E}_{z}\mathcal{E}_{y}\mathcal{E}_{x}>0$) and single enantiomers,
remembering that our chiral optical force changes sign if either the electric
field or the molecule is inverted.

For the static electric field, we consider a field strength of
$\mathcal{E}_{z}=1.00\,\mathrm{kV}\,\mathrm{mm}^{-1}$, readily achievable
using a high voltage source. For the standing wave, we consider field
strengths of
$\mathcal{E}_{y}=\mathcal{E}_{x}=1.00\,\mathrm{kV}\,\mathrm{mm}^{-1}$
(intensity $I=\epsilon_{0}c\mathcal{E}_{y}\mathcal{E}_{x}=2.65\times
10^{5}\,\mathrm{W}\,\mathrm{cm}^{-2}$) and a near-infrared wavelength of
$2\pi/k=1.064\,\mu\mathrm{m}$, readily achievable using an Nd:YAG laser. Much
stronger fields are possible, giving rise to even stronger chiral optical
forces than those shown below. To calculate these forces reliably, however, a
non-peturbative treatment of our chiral optical force is required, which is
beyond the scope of this paper.

For the molecule, we calculate the rotational constants $A$, $B$, and $C$ and
the chirally sensitive properties $\alpha_{ZY}\mu_{0X}$,
$\alpha_{XZ}\mu_{0Y}$, and $\alpha_{YX}\mu_{0Z}$ using Gaussian 09, optimizing
the molecule’s geometry at the DFT B3LYP/6-311++G(d,p) level of theory then
employing the DFT B3LYP method with the AUG-cc-pVDZ basis set. Using these
values of $A$, $B$, and $C$, we calculate the molecule’s unperturbed
rotational spectrum as described in Appendix A. Using this spectrum, we
calculate the coefficients $\mathtt{A}$, $\mathtt{B}$, and $\mathtt{C}$ as
described in Appendix B.

Eq. (1) for our chiral optical force can be recast as

$\displaystyle\mathbf{F}$
$\displaystyle\approx-\boldsymbol{\nabla}_{\mathbf{R}_{0}}U$

with

$\displaystyle U$ $\displaystyle=-\frac{1}{2}\Delta U\sin(2kZ_{0})$
$\displaystyle\Delta U$
$\displaystyle=\mathcal{E}_{z}\mathcal{E}_{y}\mathcal{E}_{x}(\mathtt{A}\alpha_{ZY}\mu_{0X}+\mathtt{B}\alpha_{XZ}\mu_{0Y}+\mathtt{C}\alpha_{YX}\mu_{0Z}),$

where $U=U_{J_{\tau},m}(\mathbf{R}_{0})$ is an effective potential energy
surface with depth and sign dictated by $\Delta U=\Delta U_{J_{\tau},m}$. The
‘temperature’ $\Delta U/k_{B}$ serves as a convenient measure of the strength
and sign of our chiral optical force.

###  4.1 Norbornenone

Let us first consider ($1S$,$4S$)-norbornenone with isotopic constitution
${}^{12}\mathrm{C}_{7}{}^{1}\mathrm{H}_{8}{}^{16}\mathrm{O}$. We take the
rotational constants to be $A/2\pi\hbar=3.65\,\mathrm{GHz}$,
$B/2\pi\hbar=2.21\,\mathrm{GHz}$, and $C/2\pi\hbar=1.99\,\mathrm{GHz}$ and the
chirally sensitive molecular properties to be $\alpha_{ZY}\mu_{0X}=2.76\times
10^{-70}\,\mathrm{C}^{3}\,\mathrm{m}^{3}\,\mathrm{J}^{-1}$,
$\alpha_{XZ}\mu_{0Y}=8.72\times
10^{-71}\,\mathrm{C}^{3}\,\mathrm{m}^{3}\,\mathrm{J}^{-1}$, and
$\alpha_{YX}\mu_{0Z}=-4.85\times
10^{-71}\,\mathrm{C}^{3}\,\mathrm{m}^{3}\,\mathrm{J}^{-1}$.

Figure 3: A scatter plot depicting the strength and sign of our chiral optical
force for the rotational states of ($1\textit{S}$,$4\textit{S}$)-norbornenone
in the $J\in\\{0,\dots,10\\}$ manifolds. Each circle corresponds to a
rotational state with $m=0$ and each cross corresponds to a pair of rotational
states with $m=\pm|m|\neq 0$.

Fig. 3 is a scatter plot depicting the strength and sign of our chiral optical
force for rotational states in the $J\in\\{0,\dots,10\\}$ manifolds, expressed
in terms of the ‘temperature’ $\Delta U/k_{B}$. The strongest force is found
for the $|\psi_{1_{0},0}\rangle$ rotational state, giving

$\displaystyle\frac{\Delta U}{k_{B}}$ $\displaystyle=-41.6\,\mu\mathrm{K}.$

Our chiral optical force is negative for some rotational states and positive
for others, however there are $908$ rotational states with $F_{z}<0$ and only
$862$ rotational states with $F_{z}>0$, giving an overall preference for
$F_{z}<0$. We emphasise here than for a given configuration of the electric
field, our chiral optical force has opposite signs for opposite enantiomers in
the same rotational state.

###  4.2 Isotopically chiral difluorocyclohexane

Let us now consider the isotopically chiral form of difluorocyclohexane
constructed by taking the achiral molecule
($1\textit{R}$,$2\textit{S}$)-$1$,$2$-difluorocyclohexane with isotopic
constitution ${}^{12}\mathrm{C}_{6}{}^{1}\mathrm{H}_{10}{}^{19}\mathrm{F}_{2}$
and substituting ${}^{13}\mathrm{C}$ at positions $1$, $5$, and $6$. We take
the rotational constants to be $A/2\pi\hbar=2.76\,\mathrm{GHz}$,
$B/2\pi\hbar=1.88\,\mathrm{GHz}$, and $C/2\pi\hbar=1.39\,\mathrm{GHz}$ and the
chirally sensitive molecular properties to be $\alpha_{ZY}\mu_{0X}=3.75\times
10^{-70}\,\mathrm{C}^{3}\,\mathrm{m}^{3}\,\mathrm{J}^{-1}$,
$\alpha_{XZ}\mu_{0Y}=1.53\times
10^{-71}\,\mathrm{C}^{3}\,\mathrm{m}^{3}\,\mathrm{J}^{-1}$, and
$\alpha_{YX}\mu_{0Z}=2.87\times
10^{-71}\,\mathrm{C}^{3}\,\mathrm{m}^{3}\,\mathrm{J}^{-1}$.

Figure 4: A scatter plot depicting the strength and sign of our chiral optical
force for the rotational states of an isotopically chiral form of
difluorocyclohexane in the $J\in\\{0,\dots,10\\}$ manifolds. Each circle
corresponds to a rotational state with $m=0$ and each cross corresponds to a
pair of rotational states with $m=\pm|m|\neq 0$.

Fig. 4 is a scatter plot depicting the strength and sign of our chiral optical
force for rotational states in the $J\in\\{0,\dots,10\\}$ manifolds. The
strongest force is again found for the $|\psi_{1_{0},0}\rangle$ rotational
state, giving

$\displaystyle\frac{\Delta U}{k_{B}}$ $\displaystyle=-26.4\,\mu\mathrm{K}.$

There are $951$ rotational states with $F_{z}<0$ and only $819$ rotational
states with $F_{z}>0$, giving an overall preference for $F_{z}<0$.

###  4.3 Comparison with other chiral optical forces

It is instructive to compare the results above for our chiral optical force
with the corresponding results for other chiral optical forces proposed to
date.

In [4], for example, we showed that an optical helicity lattice can exert a
chiral optical force on a small chiral molecule of the form

$\displaystyle\mathbf{F}$
$\displaystyle\approx\frac{1}{c}2k\sin\vartheta\mathcal{E}_{\|}\mathcal{E}_{\bot}\cos^{2}\vartheta
G^{\prime}\cos(2k\sin\vartheta Z_{0})\hat{\mathbf{z}},$ (34)

where $2\vartheta$ is the angular separation of the waves comprising the
lattice, $\mathcal{E}_{\|}$ dictates the amplitude and phase of the parallel
polarized wave, $\mathcal{E}_{\bot}$ dictates the amplitude and phase of the
perpendicularly polarized wave, and $G^{\prime}$ is the imaginary part of the
molecule’s vibronic optical activity polarizability pseudoscalar [33]. To
compare the strength of our chiral optical force with the strength of this
helicity-based force, we consider a dimensionless ratio
$\mathcal{R}=\mathcal{R}_{J_{\tau},m}$ given by

$\displaystyle\mathcal{R}$
$\displaystyle=\left|\frac{k\mathcal{E}_{z}\mathcal{E}_{y}\mathcal{E}_{x}(\mathtt{A}\alpha_{ZY}\mu_{0X}+\mathtt{B}\alpha_{XZ}\mu_{0Y}+\mathtt{C}\alpha_{YX}\mu_{0Z})}{2k\sin\vartheta\mathcal{E}_{\|}\mathcal{E}_{\bot}\cos^{2}\vartheta
G^{\prime}/c}\right|,$

where the numerator follows from Eq. (1) and the denominator follows from Eq.
(34).

For the helicity lattice, we consider an angular separation of
$2\vartheta=60.0^{\circ}$ and field strengths of
$\mathcal{E}_{\|}=\mathcal{E}_{\bot}=1.00\,\mathrm{kV}\,\mathrm{mm}^{-1}$,
matching the field strengths
$\mathcal{E}_{y}=\mathcal{E}_{x}=1.00\,\mathrm{kV}\,\mathrm{mm}^{-1}$ of the
standing wave to give a fair comparison.

For ($1S$,$4S$)-norbornenone, we estimate the optical activity polarizability
to be $G^{\prime}=3\times
10^{-36}\,\mathrm{m}\,\mathrm{kg}^{-1}\,\mathrm{s}^{3}\,\mathrm{A}^{2}$ from
vapour-phase measurements [38] together with an appropriate frequency scaling
[5]. Norbornenone is exceptionally well suited to the helicity-based force, as
$G^{\prime}$ is around two orders of magnitude larger than for comparably
sized organic molecules [38]. Even so, we find that our chiral optical force
can be several orders of magnitude larger than the helicity-based force. For
the $|\psi_{1_{0},0}\rangle$ rotational state, we obtain a ratio of

$\displaystyle\mathcal{R}$ $\displaystyle\sim 10^{5}.$

For other molecules with more typical values of $G^{\prime}$, ratios of

$\displaystyle\mathcal{R}$ $\displaystyle\sim 10^{7}$

or higher are possible. That is to say, a disparity in strength of seven
orders of magnitude or more. Even higher ratios can be found at lower
frequencies, as $\mathcal{R}\propto 1/\omega$ far off resonance.

Isotopically chiral molecules are poorly suited to the helicity-based force,
as the optical activity pseudoscalar $G^{\prime}$ is usually very small, its
electronic contributions formally vanishing in the Born-Oppenheimer
approximation [39]. Isotopically chiral molecules are nevertheless amenable to
our chiral optical force, as shown above for an isotopically chiral form of
difluorocyclohexane.

Such comparisons demonstrate that potential applications identified elsewhere
for other chiral optical forces [8, 9, 4, 5, 10, 11, 1, 12, 2, 6, 7, 3, 13]
should be enormously more robust and easier to implement using our chiral
optical force instead, as claimed in Sec. 1.

###  4.4 Relation to other chiroptical phenomena

Most chiroptical phenomena rely on weak magnetic-dipole and electric-
quadrupole interactions [14, 33]. We emphasise here that our chiral optical
force can be orders of magnitude stronger than other chiral optical forces
proposed to date because it is based instead on electric-dipole interactions
at leading order.

Other chiroptical phenomena based on electric-dipole interactions at leading
order include second-order nonlinear sum- and difference-frequency generation
[40, 41], photoelectron circular dichroism [42, 43], second-harmonic
generation circular dichroism from chiral surfaces [44, 45], Rayleigh and
Raman optical activity from chiral surfaces [46], Coulomb-explosion imaging
[47, 48, 49], chiral microwave three-wave mixing [50, 51, 52], photoexcitation
circular dichroism and photoexcitation-induced electron circular dichroism
[53], and anomalous circularly polarized light emission [54]. Our chiral
optical force has elements in common with many of these, in particular its
crucial dependence on orientational effects and its nonlinear character.

##  5 Summary and Outlook

Taking our lead from nature, we have identified a chiral optical force for
small chiral molecules that can be several orders of magnitude stronger than
other chiral optical forces proposed to date. Our chiral optical force should
render potential applications enormously more robust and easier to implement
than previously though possible.

There is much still to be done. For a more accurate description of our chiral
optical force, the molecule’s nuclear spins should be considered explicitly. A
perturbative treatment of our chiral optical force like the one presented in
this paper should be sufficient for chiral molecular matter-wave
interferometry [4, 5, 6] and the distillation of chiral molecules in chiral
optical lattices [7], for example. It remains for us to develop a non-
perturbative treatment for more demanding potential applications, including
the resolution of enantiomers [8, 9, 4, 10, 11, 12, 13].

We note here that many different electromagnetic fields can give rise to
chiral optical forces based on electric-dipole interactions at leading order.
A key ingredient is the use of three mutually orthogonal electric field
components, which can be arranged in either a left-handed or a right-handed
configuration. These chiral optical forces might be referred to as different
“blends” of COFFEEE, the chiral optical force identified in this paper being
the “static/lin$\perp$lin blend”.

We will return to these and related ideas elsewhere.

Funding The Royal Society (URF$\backslash$R1$\backslash$191243).

Acknowledgments Robert P. Cameron is a Royal Society University Research
Fellow and gratefully acknowledges the Royal Society’s support.

Disclosures The authors declare no conflicts of interest.

Data availability No data were generated or analyzed in the presented
research.

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##  Appendix A Asymmetric rigid rotor

The quantum-mechanical description of the asymmetric rigid rotor has been
covered extensively elsewhere [55, 36, 56, 39]. We use the conventions below,
which are slightly unusual as we consider quantization along the $y$ axis
rather than the $z$ axis.

We take the molecule-fixed coordinates $X$, $Y$, and $Z$ to be arranged in a
$\mathrm{III}^{r}$ configuration with direction cosines $\ell_{aA}$ given by

$\displaystyle\ell_{xX}$
$\displaystyle=-\cos\theta\cos\phi\cos\chi+\sin\phi\sin\chi,$
$\displaystyle\ell_{xY}$
$\displaystyle=\cos\theta\cos\phi\sin\chi+\sin\phi\cos\chi,$
$\displaystyle\ell_{xZ}$ $\displaystyle=-\sin\theta\cos\phi,$
$\displaystyle\ell_{yX}$ $\displaystyle=-\sin\theta\cos\chi,$
$\displaystyle\ell_{yY}$ $\displaystyle=\sin\theta\sin\chi,$
$\displaystyle\ell_{yZ}$ $\displaystyle=\cos\theta,$ $\displaystyle\ell_{zX}$
$\displaystyle=\cos\theta\sin\phi\cos\chi+\cos\phi\sin\chi,$
$\displaystyle\ell_{zY}$
$\displaystyle=-\cos\theta\sin\phi\sin\chi+\cos\phi\cos\chi,$
$\displaystyle\ell_{zZ}$ $\displaystyle=\sin\theta\sin\phi,$

where $0\leq\theta\leq\pi$, $0\leq\phi<2\pi$, and $0\leq\chi<2\pi$ are Euler
angles relating $X$, $Y$, and $Z$ to the laboratory-fixed coordinates $x$,
$y$, and $z$. The unperturbed rotational Hamiltonian is thus

$\displaystyle H^{(0)}$
$\displaystyle=\frac{1}{\hbar^{2}}(AJ^{2}_{X}+BJ^{2}_{Y}+CJ^{2}_{Z})$

with

$\displaystyle J_{X}$
$\displaystyle=-\mathrm{i}\hbar\left(\sin\chi\frac{\partial}{\partial\theta}-\csc\theta\cos\chi\frac{\partial}{\partial\phi}+\cot\theta\cos\chi\frac{\partial}{\partial\chi}\right),$
$\displaystyle J_{Y}$
$\displaystyle=-\mathrm{i}\hbar\left(\cos\chi\frac{\partial}{\partial\theta}+\csc\theta\sin\chi\frac{\partial}{\partial\phi}-\cot\theta\sin\chi\frac{\partial}{\partial\chi}\right),$
$\displaystyle J_{Z}$
$\displaystyle=-\mathrm{i}\hbar\frac{\partial}{\partial\chi},$

where $A>B>C$ are the molecule’s equilibrium rotational constants and
$\mathbf{J}$ is the molecule’s rotational angular momentum.

We take the unperturbed rotational energy eigenstates
$|J_{\tau},m\rangle^{(0)}$ and associated energy eigenvalues
$E_{J_{\tau},m}^{(0)}$ to satisfy

$\displaystyle H^{(0)}|J_{\tau},m\rangle^{(0)}$
$\displaystyle=E_{J_{\tau},m}^{(0)}|J_{\tau},m\rangle^{(0)},$
$\displaystyle(J_{X}^{2}+J_{Y}^{2}+J_{Z}^{2})|J_{\tau},m\rangle^{(0)}$
$\displaystyle=\hbar^{2}J(J+1)|J_{\tau},m\rangle^{(0)},$ $\displaystyle
J_{y}|J_{\tau},m\rangle^{(0)}$ $\displaystyle=\hbar m|J_{\tau},m\rangle^{(0)}$

with

$\displaystyle J_{x}$
$\displaystyle=-\mathrm{i}\hbar\left(\sin\phi\frac{\partial}{\partial\theta}+\cot\theta\cos\phi\frac{\partial}{\partial\phi}-\csc\theta\cos\phi\frac{\partial}{\partial\chi}\right),$
$\displaystyle J_{y}$
$\displaystyle=-\mathrm{i}\hbar\frac{\partial}{\partial\phi},$ $\displaystyle
J_{z}$
$\displaystyle=-\mathrm{i}\hbar\left(\cos\phi\frac{\partial}{\partial\theta}-\cot\theta\sin\phi\frac{\partial}{\partial\phi}+\csc\theta\sin\phi\frac{\partial}{\partial\chi}\right),$

where $J\in\\{0,1,\dots\\}$ is the rotational angular momentum quantum number,
$\tau\in\\{-J,\dots,J\\}$ is a label that increases with increasing energy and
$m\in\\{-J,\dots,J\\}$ is the laboratory-fixed rotational angular momentum
projection quantum number.

To help us identify the unperturbed rotational energy eigenstates
$|J_{\tau},m\rangle^{(0)}$ and associated energy eigenvalues
$E_{J_{\tau},m}^{(0)}$ explicitly, we work in a basis of unperturbed symmetric
rigid rotor states $|J,k,m\rangle^{(0)}$ satisfying

$\displaystyle(J_{X}^{2}+J_{Y}^{2}+J_{Z}^{2})|J,k,m\rangle^{(0)}$
$\displaystyle=\hbar^{2}J(J+1)|J,k,m\rangle^{(0)},$ $\displaystyle
J_{Z}|J,k,m\rangle^{(0)}$ $\displaystyle=\hbar k|J,k,m\rangle^{(0)},$
$\displaystyle J_{y}|J,k,m\rangle^{(0)}$ $\displaystyle=\hbar
m|J,k,m\rangle^{(0)},$

where $k\in\\{-J,\dots,J\\}$ is the molecule-fixed rotational angular momentum
projection quantum number, the corresponding wavefunctions
$\langle\theta,\phi,\chi|J,k,m\rangle^{(0)}$ being given by

$\displaystyle\langle\theta,\phi,\chi|J,k,m\rangle^{(0)}$
$\displaystyle=\sqrt{\frac{(J+m)!(J-m)!(J+k)!(J-k)!(2J+1)}{8\pi^{2}}}$
$\displaystyle\times\sum_{\sigma=\mathrm{max}(0,k-m)}^{\mathrm{min}(J-m,J+k)}(-1)^{\sigma}\frac{(\cos\frac{1}{2}\theta)^{2J+k-m-2\sigma}(-\sin\frac{1}{2}\theta)^{m-k+2\sigma}}{\sigma!(J-m-\sigma)!(m-k+\sigma)!(J+k-\sigma)!}\mathrm{e}^{\mathrm{i}m\phi}\mathrm{e}^{\mathrm{i}k\chi}.$

The $|J,k,m\rangle^{(0)}$ render the unperturbed rotational Hamiltonian
$H^{(0)}$ block diagonal in $J$, as

$\displaystyle\langle J^{\prime},k^{\prime},m^{\prime}|H^{0}|J,k,m\rangle$
$\displaystyle=\delta_{J^{\prime}J}\delta_{m^{\prime}m}\Bigg{(}\frac{1}{2}(A+B)J(J+1)\delta_{k^{\prime}k}+\left[C-\frac{1}{2}(A+B)\right]k^{2}\delta_{k^{\prime}k}$
$\displaystyle+\frac{1}{4}(A-B)\\{\sqrt{[J(J+1)-(k-1)(k-2)][J(J+1)-k(k-1)]}\delta_{k^{\prime}k-2}$
$\displaystyle+\sqrt{[J(J+1)-(k+1)(k+2)][J(J+1)-k(k+1)]}\delta_{k^{\prime}k+2}\\}\Bigg{)}.$

The diagonalization of $H^{(0)}$ can completed analytically for
$J\in\\{0,\dots,5\\}$, giving

$\displaystyle|0_{0},0\rangle^{(0)}$ $\displaystyle=|0,0,0\rangle^{(0)},$
$\displaystyle|1_{-1},m\rangle^{(0)}$
$\displaystyle=\frac{1}{\sqrt{2}}(|1,1,m\rangle^{(0)}-|1,-1,m\rangle^{(0)}),$
$\displaystyle|1_{0},m\rangle^{(0)}$
$\displaystyle=\frac{1}{\sqrt{2}}(|1,1,m\rangle^{(0)}+|1,-1,m\rangle^{(0)}),$
$\displaystyle|1_{1},m\rangle^{(0)}$ $\displaystyle=|1,0,m\rangle^{(0)}$

with

$\displaystyle E_{0_{0},m}^{(0)}$ $\displaystyle=0,$ $\displaystyle
E_{1_{-1},m}^{(0)}$ $\displaystyle=B+C,$ $\displaystyle E_{1_{0},m}^{(0)}$
$\displaystyle=A+C,$ $\displaystyle E_{1_{1},m}^{(0)}$ $\displaystyle=A+B,$

for example. The diagonalization of $H^{(0)}$ must be completed numerically
for $J\in\\{6,\dots\\}$.

##  Appendix B Coefficients

The coefficients $\mathtt{a}=\mathtt{a}_{J_{\tau},m}$,
$\mathtt{b}=\mathtt{b}_{J_{\tau},m}$, $\mathtt{c}=\mathtt{c}_{J_{\tau},m}$,
$\mathtt{A}=\mathtt{A}_{J_{\tau},m}$, $\mathtt{B}=\mathtt{B}_{J_{\tau},m}$,
and $\mathtt{C}=\mathtt{C}_{J_{\tau},m}$ are given by

$\displaystyle\mathtt{a}$ $\displaystyle={}^{(0)}\langle
J_{\tau},m|\left(\ell_{yZ}\ell_{xY}\ell_{zX}+\ell_{yY}\ell_{xZ}\ell_{zX}\right)|J_{\tau},m\rangle^{(0)},$
(35) $\displaystyle\mathtt{b}$ $\displaystyle={}^{(0)}\langle
J_{\tau},m|\left(\ell_{yX}\ell_{xZ}\ell_{zY}+\ell_{yZ}\ell_{xX}\ell_{zY}\right)|J_{\tau},m\rangle^{(0)},$
(36) $\displaystyle\mathtt{c}$ $\displaystyle={}^{(0)}\langle
J_{\tau},m|\left(\ell_{yY}\ell_{xX}\ell_{zZ}+\ell_{yX}\ell_{xY}\ell_{zZ}\right)|J_{\tau},m\rangle^{(0)},$
(37) $\displaystyle\mathtt{A}$
$\displaystyle=\sum_{J^{\prime}=0}^{\infty}\sum_{\tau^{\prime}=-J^{\prime}}^{J^{\prime}}\sum_{m^{\prime}=-J^{\prime}}^{J^{\prime}}\frac{2}{E_{J^{\prime}_{\tau^{\prime}},m^{\prime}}^{(0)}-E_{J_{\tau},m}^{(0)}}$
$\displaystyle\times\Re[{}^{(0)}\langle
J_{\tau},m|(\ell_{yZ}\ell_{xY}+\ell_{yY}\ell_{xZ})|J^{\prime}_{\tau^{\prime}},m^{\prime}\rangle^{(0)}{}^{(0)}\langle
J^{\prime}_{\tau^{\prime}},m^{\prime}|\ell_{zX}|J_{\tau},m\rangle^{(0)}],$
(38) $\displaystyle\mathtt{B}$
$\displaystyle=\sum_{J^{\prime}=0}^{\infty}\sum_{\tau^{\prime}=-J^{\prime}}^{J^{\prime}}\sum_{m^{\prime}=-J^{\prime}}^{J^{\prime}}\frac{2}{E_{J^{\prime}_{\tau^{\prime}},m^{\prime}}^{(0)}-E_{J_{\tau},m}^{(0)}}$
$\displaystyle\times\Re[{}^{(0)}\langle
J_{\tau},m|(\ell_{yX}\ell_{xZ}+\ell_{yZ}\ell_{xX})|J^{\prime}_{\tau^{\prime}},m^{\prime}\rangle^{(0)}{}^{(0)}\langle
J^{\prime}_{\tau^{\prime}},m^{\prime}|\ell_{zY}|J_{\tau},m\rangle^{(0)}],$
(39) $\displaystyle\mathtt{C}$
$\displaystyle=\sum_{J^{\prime}=0}^{\infty}\sum_{\tau^{\prime}=-J^{\prime}}^{J^{\prime}}\sum_{m^{\prime}=-J^{\prime}}^{J^{\prime}}\frac{2}{E_{J^{\prime}_{\tau^{\prime}},m^{\prime}}^{(0)}-E_{J_{\tau},m}^{(0)}}$
$\displaystyle\times\Re[{}^{(0)}\langle
J_{\tau},m|(\ell_{yY}\ell_{xX}+\ell_{yX}\ell_{xY})|J^{\prime}_{\tau^{\prime}},m^{\prime}\rangle^{(0)}{}^{(0)}\langle
J^{\prime}_{\tau^{\prime}},m^{\prime}|\ell_{zZ}|J_{\tau},m\rangle^{(0)}],$
(40)

where terms with
$|J^{\prime}_{\tau^{\prime}},m^{\prime}\rangle^{(0)}=|J_{\tau},m\rangle^{(0)}$
are to be excluded from the summations. Eqs. (35)-(40) can be evaluated in
closed form for rotational states in the $J\in\\{0,1,2,3,4\\}$ manifolds, as
the unperturbed rotational energy eigenstates
$|J^{\prime}_{\tau^{\prime}},m^{\prime}\rangle^{(0)}$ and associated energy
eigenvalues $E_{J^{\prime}_{\tau^{\prime}},m^{\prime}}^{(0)}$ of importance
can be found analytically. For $J=0$, we obtain
$\mathtt{a}=\mathtt{b}=\mathtt{c}=\mathtt{A}=\mathtt{B}=\mathtt{C}=0$, which
is unsurprising given that $|0_{0},0\rangle^{(0)}$ has isotropic character
whereas our chiral optical force depends crucially on orientational effects.
For $J=1$, we obtain the results presented in Table 1, where it can be seen
that $\mathtt{a}$, $\mathtt{b}$, $\mathtt{c}$, $\mathtt{A}$, $\mathtt{B}$, and
$\mathtt{C}$ depend on the magnitude but not the sign of $m$ and that they
satisfy relationships like

$\displaystyle\sum_{m=-J}^{J}\mathtt{A}_{J_{\tau},m}$ $\displaystyle=0,$

in accord with the principle of spectroscopic stability [57]. Eqs. (35)-(40)
cannot be evaluated in closed form for rotational states in the
$J\in\\{5,\dots\\}$ manifolds, as some or all of the
$|J^{\prime}_{\tau^{\prime}},m^{\prime}\rangle^{(0)}$ and
$E_{J^{\prime}_{\tau^{\prime}},m^{\prime}}^{(0)}$ of importance need to be
found numerically.

| $|\psi_{1_{-1},m}\rangle$ | $|\psi_{1_{0},m}\rangle$ | $|\psi_{1_{1},m}\rangle$
---|---|---|---
$\mathtt{a}$ | $0$ | $\frac{(3|m|-2)}{10}$ | $\frac{(2-3|m|)}{10}$
$\mathtt{b}$ | $\frac{(2-3|m|)}{10}$ | $0$ | $\frac{(3|m|-2)}{10}$
$\mathtt{c}$ | $\frac{(3|m|-2)}{10}$ | $\frac{(2-3|m|)}{10}$ | $0$
$\mathtt{A}$ | $\frac{(2-3|m|)(B-C)}{5[3(B-C)^{2}-8A(B+C)]}$ | $\frac{(3|m|-2)}{20}\left[\frac{3}{(B-C)}+\frac{1}{(B+3C)}\right]$ | $\frac{(2-3|m|)}{20}\left[\frac{3}{(C-B)}+\frac{1}{(C+3B)}\right]$
$\mathtt{B}$ | $\frac{(2-3|m|)}{20}\left[\frac{3}{(A-C)}+\frac{1}{(A+3C)}\right]$ | $\frac{(2-3|m|)(C-A)}{5[3(C-A)^{2}-8B(A+C)]}$ | $\frac{(3|m|-2)}{20}\left[\frac{3}{(C-A)}+\frac{1}{(C+3A)}\right]$
$\mathtt{C}$ | $\frac{(3|m|-2)}{20}\left[\frac{3}{(A-B)}+\frac{1}{(A+3B)}\right]$ | $\frac{(2-3|m|)}{20}\left[\frac{3}{(B-A)}+\frac{1}{(B+3A)}\right]$ | $\frac{(2-3|m|)(A-B)}{5[3(A-B)^{2}-8C(A+B)]}$
Table 1: The coefficients $\mathtt{a}$, $\mathtt{b}$, $\mathtt{c}$,
$\mathtt{A}$, $\mathtt{B}$, and $\mathtt{C}$ for rotational states in the
$J=1$ manifold.