# Formation of dark excitons in monolayer transition metal dichalcogenides
by a vortex beam: optical selection rules

Omadillo Abdurazakov<EMAIL_ADDRESS>Department of Physics, University
of Texas at El Paso, El Paso, Texas 79968, USA  Chunqiang Li  Department of
Physics, University of Texas at El Paso, El Paso, Texas 79968, USA  Yun-Pil
Shim<EMAIL_ADDRESS>Department of Physics, University of Texas at El Paso, El
Paso, Texas 79968, USA

###### Abstract

Monolayer transition metal dichalcogenides host tightly-bound excitons, which
dominate their optoelectronic response even at room temperatures. Light beams
are often used to study these materials with the polarization - often termed
as the spin angular momentum of the light - providing the mechanism for
exciting excitonic states. Light beams, however, can also carry an orbital
angular momentum by creating helical structures of their phase front. In this
work, we consider a Laguerre-Gaussian beam possessing an orbital angular
momentum in addition to the spin angular momentum to create excitons in
monolayer transition metal dichalcogenides. We derive optical selection rules
that govern the allowed transitions to various exciton series using symmetry
arguments. Our symmetry considerations show that we can create dark excitons
using these high-order optical beams opening up new avenues for creating long-
lived dark excitons with the potential of exploiting them in quantum
information processing and storage.

††preprint: APS/123-QED

##  I  Introduction

When a photon of a certain frequency illuminates a semiconducting crystal, in
response, an exciton can be formed. During the process, an electron excited to
the conduction band binds with its hole left in the valence band through the
Coulomb interaction. Because ordinary light does not couple to the electron
spin, it is conserved in the process. For the direct band-gap semiconductors,
this kind of excitation is called a bright exciton and they are short-lived
because an electron can quickly recombine with the hole emitting a photon. It
is also possible for an electron to flip its spin via some nontrivial
processes during the exciton formation. In contrast to the bright excitons,
this kind of excitation is long-lived because the promoted electron can not
radiatively recombine without flipping its spin. Therefore, it is called a
dark exciton and makes a promising candidate for a solid-state qubit due to
its long recombination lifetime and coherence[1, 2] and has lately become an
active field of research in its own right [3, 4].

The family of quasi-two-dimensional (quasi-2D) semiconducting crystals called
monolayer transition metal dichalcogenides (ML TMDs) can host excitons of
strikingly large binding energies compared to those of traditional
semiconducting crystals such as GaAs[5]. This is partly due to the weak
dielectric screening and geometric confinement in quasi-2D structures.
Although an indirect band-gap semiconductor in bulk, when exfoliated down to a
monolayer[6], a TMD crystal transitions into a direct band-gap semiconductor
[7, 8]. Therefore, the optical response of these materials to light is
dominated by bright exciton formation and consequent photoluminescence signal
at sub band-gap frequencies. Due to the presence of heavy transition metal
atoms in ML TMDs, there is a strong (moderate) spin-orbit interaction in the
valence (conduction) band. Combined with the lack of spatial inversion
symmetry inside the monolayer crystal, the conduction and valance bands at the
$\pm K$ momentum points (or valleys) are split into two bands each that can
individually host only one type of electron spin species. These bands at the
two valleys are related through the time-reversal symmetry. Consequently, the
electron spin and valley degrees of freedom are locked and each valley can be
selectively addressed with circularly polarized light [9, 10, 11, 12].

Because of its valley physics and tightly bound excitons that are manifest
even at room temperatures, a ML TMD crystal is an ideal platform for hosting
stable and long-lived dark excitons. However, creating and controlling dark
excitons in these materials is challenging as ordinary light beams do not
couple to the electron spin. Nevertheless, these dark excitons can be induced
to decay via radiative means by applying very strong in-plane magnetic
fields[13] or coupling to the surface plasmon polaritons[14]. Although these
methods offer some pathways to access the spin-forbidden dark excitons and to
control their lifetimes, they are limited to extremely high magnetic fields
and extremely low temperatures. Therefore, purely optical means to access and
control the dark exciton states would lead to a new practical platform for
many quantum applications including quantum information technology.

The polarization of light, which is used to control individual valleys in ML
TMDs, corresponds to the spin angular momentum (SAM) of light and can be
transferred to matter as a mechanical torque[15]. Light has another
fundamental degree of freedom called orbital angular momentum (OAM). The laser
beams possessing OAM can be created with holographic phase-plates and were
already realized decades ago[16]. They are often called vortex beams and have
been shown to induce optical transitions that are not allowed otherwise and to
strongly modify optical selection rules[17, 18]. They have been used to
transfer orbital angular momentum to macroscopic particles[19], individual
atoms[20] and electrons[18], and to create Rydberg excitons in atoms [21], and
even momentum-forbidden dark excitons in ML TMDs[22].

In this work, we theoretically investigate the possible optical transitions
with a vortex beam in ML TMD systems. Specifically, we consider a Laguerre-
Gaussian laser beam, a type of vortex beam, to create spin-forbidden dark
excitons as well as bright excitons in ML TMD crystals. We derive optical
selection rules for such transitions based on the symmetry of the ML TMD
crystal structure, the spatial structure of the beams, and the exciton
envelope functions. We show that, in the quadrupole coupling regime, B-type
dark excitons with various envelope functions can be created selectively in
each valley. As vortex beams with high OAM can be readily created in optics
labs nowadays[23], this all-optical means to create dark excitons in ML TMDs
should be feasible.

The outline of the paper is as follows. First, we describe our method to
identify the allowed optical transition to create excitons in these materials
in Sec.II where we consider the symmetry of the electronic bands, the spatial
profile of vortex beams, and exciton envelope functions in terms of the
irreducible representations of the pertinent point group the monolayer crystal
belongs to. In Sec.III, we derive and discuss optical selection rules for the
bright and dark excitons. And in Sec.IV, we conclude and discuss the
implication of our results to future studies.

##  II  Methods

Figure 1:  Monolayer transition metal dichalcogenide (ML TMD) crystal and band
structure and a vortex beam. (a) The side view and (b) top view of ML TMD
crystal spanned with a unit cell (demarcated by a blue-dashed line) consisting
of one transition metal and one chalcogen atom with the lattice constant of
$a$.(c) The principal axis with threefold rotation symmetry, a mirror plane
perpendicular to this axis, and (d) three twofold rotation axes lying on the
mirror plane are shown. (e) The first Brillouin zone of ML TMD and a few high
symmetry momentum points in it are shown. (f) An effective two-band model with
the spin-orbit coupling splitting around $K$ and $-K$ momentum points and the
corresponding irreducible representations of the double-group relevant at the
$-K/K$ points. The conduction band splitting is exaggerated for clarity. The
possible types of exciton transitions are symbolically shown by the arrows.
Here, the solid (dashed) lines designate a A(B) type exciton. The bright
(dark) excitons are shown by thick yellow (thin black) lines. Note that the
arrows are shifted laterally for clarity, but the transitions predominantly
occur at the valley center. (g) The helical wavefront and phases of optical
vortex beams are plotted for a few values of the orbital angular momentum of
light. (h) Schematic of the vortex beam created exciton in a ML TMD. The
vortex beam propagates along the $z$-axis, the direction that is perpendicular
to the monolayer sample.

In this work, we are interested in understanding the exciton formation in a
monolayer TMD crystal under the excitation of a vortex beam. By using symmetry
arguments regarding the crystal band structure and light beam, we identify
allowed and forbidden optical transitions from the ground state of the crystal
to various exciton states when the light beam possesses both spin and orbital
angular momenta. Such an analysis grants us the optical selection rules that
govern the formation of bright and dark excitons in the $\pm K$ valleys of a
monolayer TMD crystal. As the band gap energy in this class of materials
predominantly lies in the optical spectrum, we only consider the direct
optical transition between the spin-split valence and conduction bands at the
$\pm K$ points. The symmetry of the electron wavevector around these points
can be derived by constructing a tight-binding band and studying its behavior
under the pertinent symmetry operations that belong to the relevant symmetry
group[24].

In the single electron band picture, an exciton with the $1s$ envelope
function is formed through the binding of an electron excited into the
conduction band minimum and the hole left on the valence band maximum[25]. The
exciton symmetry is defined as the direct product of the irreducible
representations of the exciton envelope function, the conduction band, and the
valence band, namely
$\Gamma_{X}=\Gamma_{\text{env}}\otimes\Gamma_{c}\otimes\Gamma_{v}^{*}$. For
the optical transition from the crystal ground state to an exciton state to
occur, $\Gamma_{\text{X}}$ must contain the irreducible representation of the
driving field operator
$\Gamma_{\text{beam}}=\Gamma_{\text{order}}\otimes\Gamma_{\text{OAM}}\otimes\Gamma_{\text{SAM}}$(see
Appendix A). Here, $\Gamma_{\text{order}}$ is the order of the light-crystal
coupling such as dipole or quadrupole. For the dipole coupling, the basis
function representing its symmetry is proportional to the identity, which is
represented by $\Gamma_{1}$ in the point group of $C_{3h}$, whereas for the
quadrupole coupling the basis function is proportional to the longitudinal
coordinate $z$, which is represented by $\Gamma_{4}$.

To find the symmetry of the electronic bands at the $\pm K$ points, we can
construct a tight-binding band and assign an irreducible representation to
each band. The symmetry of the exciton envelope function is found from
solutions of the 2D Hydrogen model, and the symmetry of the driving field
operator is found by representing the polarization and the spatial structure
of the Laguerre-Gaussian beam basis functions of the threefold rotation
operator $C_{3}$.

In the bulk form, TMDs are indirect-gap semiconductors formed by weakly
interacting layers through van-der Waals forces. A monolayer of TMD, however,
is a direct-gap semiconductor with a hexagonal crystal structure. The side and
top view of a ML TMD crystal are depicted in Figs.1(a) and (b), respectively.
The crystal unit cell contains one chalcogen atom and one transition metal
atom. The crystal structure is invariant under certain transformation
operations. They include a threefold rotation around the principal axis
perpendicular to the monolayer ($C_{3}$), a reflection around the mirror plane
($\sigma$) perpendicular to the principal axis, three two-fold rotations
around the axes lying on the mirror plane ($C_{2}$), and the improper rotation
around the principal axis followed by a mirror reflection ($S_{3}$). The set
of all these operations belongs to the group of $D_{3h}$.

In the reciprocal space, the first Brillouin zone of the ML TMD crystal, which
is also of a hexagonal structure, has a few high symmetry momentum points such
as $\Gamma$, $K$, and $-K$ as shown in Fig.1(e). Although the electron wave
function at $\Gamma$ points inherits the point group of the crystal lattice,
the symmetry at $\pm K$ points is lowered to that of $C_{3h}$. A tight-biding
wave function at these particular momentum points can be constructed from the
dominant $d$ atomic orbitals of the transition metal atoms. According to the
first-principle calculations [26], the $d_{z^{2}}$ atomic orbitals dominate
the conduction band, whereas the $d_{x^{2}-y^{2}}$ and $d_{xy}$ atomic
orbitals dominate the valence band at $\pm K$ points. One can construct a
tight-binding band by symmetrizing these orbital with respect to the threefold
rotation operator $C_{3}^{+}$ relevant to the monolayer crystal symmetry.
Here, the plus sign indicates that the three-fold rotation operator acts in
the counter clockwise direction around the principal axis. Considering the
spin-orbit splitting of the valence and conduction bands at these particular
momentum points, we can study the behavior of the constructed bands under the
symmetry operation of the relevant double-point group using the Tab.A.I and
assign corresponding irreducible representation(irrep). The schematic diagrams
for the bands and their irreps at $\pm K$ points are shown in Fig.1(f).

The Laguerre-Gaussian beams are high-order solutions to the paraxial wave
equation in cylindrical coordinates[27]. These high-order Gaussian modes
exhibit more complex spatial and phase structures compared to the regular
Gaussian beams. More importantly, due to the helical structure of their phase
front, they carry orbital angular momentum along the propagation direction in
addition to their spin angular momentum or polarization.

The spatial profile of the beam propagating along the $z$-axis in the
cylindrical coordinates is given by
$\bm{A}(\rho,\phi,z)=\bm{\epsilon}A_{p}^{l}(\rho,\phi,z)e^{ikz}$ where

$\displaystyle A_{p}^{l}(\rho,\phi,z)$ $\displaystyle=$ $\displaystyle
C_{pl}\frac{w_{0}}{w(z)}\left(\frac{\sqrt{2}\rho}{w(z)}\right)^{|l|}L_{p}^{|l|}\left(\frac{2\rho^{2}}{w^{2}(z)}\right)$
(1) $\displaystyle\times$ $\displaystyle
e^{-\rho^{2}/w^{2}(z)}e^{i\Psi}e^{il\phi}$

and the phase factor contains the following terms

$\displaystyle\Psi=k\rho^{2}/2R(z)+(2p+|l|+1)\arctan{(z/z_{R})}$ (2)

where $C_{pl}=\sqrt{2p!/\pi(|l|+p)!}$ is the normalization constant,
$R(z)=z+z_{R}^{2}/z$ is the beam front curvature, and
$w(z)=w_{0}\sqrt{1+z^{2}/z_{R}^{2}}$ is the spot size which increases as a
function of the distance $z$ from the beam waist $w_{0}$. The beam acquires a
nontrivial spatial structure through the associated Laguerre function
$L_{p}^{|l|}(\rho,z)$. The intensity profile of the beam is of a ring-like
structure. Its radial index $p$, which can be zero or any positive integer,
indicates the number of rings when the beam is projected on a screen. Its
azimuthal index $l$, which can be any integer number, indicates the amount of
orbital angular momentum the beam carries. If this index is zero, a Gaussian
beam with a plane wave phase is recovered. For $p=0$ the beam intensity has a
single ring structure whose radius scales as $\propto\sqrt{l}$ at the beam
waist. Here, $z_{R}=\pi w_{0}^{2}/\lambda$ is the Raleigh range, within which
the beam spot size stays nearly constant. The first term in the phase factor
gives a spherical curvature to the wave, whereas the latter term is called a
Gouy phase, which develops as the beam propagates and mostly varies inside the
Raleigh range. And $\lambda$ is the wavelength of the light beam. The most
significant phase term contains $l\phi+kz$. As depicted in Fig.1(g), the
points of the constant phase make a helical surface along the $z$-axis. For
$|l|>1$ there will be as many helices intertwined together. The center of the
beam where the phase is undefined and the light intensity goes to zero is
called a vortex. The lower panels in Fig.1(g) showcase how the phase changes
as a function of the azimuthal angle $\phi$ for corresponding values of $l$.

The light polarization vector $\bm{\epsilon}$ of the vortex beam depicted in
Fig.1(h) lies parallel to the two-dimensional sample and perpendicular to the
beam propagation direction. Here, the beam can be circularly polarized either
in the left-handed direction ($\sigma^{-}$) or the right-handed direction
($\sigma^{+}$). It is also possible employ a beam polarized perpendicular to
the sample plane and it is termed as $\pi$-polarized beam. There are at least
two ways of achieving it. First, if a Gaussian beam is directed along the two-
dimensional sample plane, then its polarization is perpendicular to the
sample[28]. Therefore, one could modify the beam-sample setup by changing the
angle between them so that there is a field component that is perpendicular to
the sample with a finite amount of $\pi$-polarization. Another way to create a
$\pi$-polarized beam is to tightly focus the vortex beam so that it is no
longer in the paraxial regime and has a longitudinal component. This non-
paraxial vortex beam can posses polarization that is along the beam axis that
comes from the finite longitudinal field component, albeit with much weaker
amplitude. This component scales as the first order in the parameter
$1/(kw_{0})$ which is $\sim 10^{4}$ times smaller than the transverse field
component for a typical vortex beam spot $w_{0}$ within the optical
diffraction limit[29, 30]. Therefore, in this work, we also consider the
possibility of having a $\pi$-polarized beam for the sake of completeness in
deriving the optical selection rules.

Exciton env. | Basis function | Irrep
---|---|---
$1s$ | $1$ | $\Gamma_{1}$
$2p_{+}$ | $(x+iy)$ | $\Gamma_{2}$
$2p_{-}$ | $(x-iy)$ | $\Gamma_{3}$
$3d_{+}$ | $(x+iy)^{2}$ | $\Gamma_{3}$
$3d_{-}$ | $(x-iy)^{2}$ | $\Gamma_{2}$
Table 1: Basis functions and the irreps of various exction envelope functions.

Because of the quasi-2D structure of ML TMDs, the electric field lines between
electrons and holes are screened well in the plane of the crystal and poorly
screened outside. This renders the dielectric constant of the material
dependent on the relative distance of the exciton envelope function.
Therefore, to solve the exciton problem in ML TMD quantitatively one needs to
resort to an elaborate numerical framework such as the Bethe-Salpeter
equations[31]. However, we adopt a framework that treats the exciton problem
as a 2D hydrogen model with an effective dielectric screening because we are
primarily interested in the symmetry properties of the exciton envelope
function rather than its exact quantitative form. It was shown that one can
embed the inhomogeneous screening effects into an effective dielectric
constant that is averaged over the radius of an exciton and obtain a 2D analog
of the Hydrogen like solutions for exciton envelope functions with quite
accurate binding energies [32]. Then, the exciton envelope solutions will be
of the form $\Phi_{nm}(\bm{r})=R_{n|m|}e^{im\phi}$ in real space[33, 34]. The
radial function $R$ depends on the principal quantum number
$n=\\{1,2,3,\ldots\\}$ and the absolute value of the magnetic quantum number
$m=\\{0,\pm 1,\ldots,\pm(n-1)\\}$. It also depends on the relative coordinate
of the electron and hole, the exciton radius which contains the effective
dielectric constant. Because of the spherical symmetry of the radial function
both in real and reciprocal space, the phase factor entirely defines the
symmetry of the exciton envelope function. Because
$e^{im\phi}\propto(x+(-1)^{m}iy)^{m}$, one can assign an irreducible
representation to each exciton state depending on the sign or the value of $m$
(see Tab. 1). When $m=0$, the irrep of the envelope function is $\Gamma_{1}$,
while for $m=\pm 1$, it is $\Gamma_{2}$ and $\Gamma_{3}$, respectively. For
$m>1$, it will be $n$ direct products of $\Gamma_{2}$ or $\Gamma_{3}$.

##  III  Results

The experimental absorption measurements[8] and first-principles
calculations[35] show that TMDs clearly exhibit A (B) excitons that are formed
due to the direct optical transitions between the conduction band and spin-
orbit split higher (lower) valance band. Here, we present the optical
selection rules for bright A(B) and dark A(B) excitons created by vortex beams
in a tabular form below. In addition to the usual dipole coupling, we also
consider quadrupole coupling. The quadrupole transitions become important when
the electric field has a spatial gradient. Due to their complex spatial
structure, high-order laser beams such as the Laguerre-Gaussian beams,
especially those with high values of OAM, have a strong special field
gradient. Although the oscillator strength for the quadrupole transition is
much weaker than that of the dipole transition for ordinary light beams, the
vortex beams can have sizable quadrupole effects that are even comparable to
those of the dipole coupling at very high values of OAM because of their
transverse field gradient around the center of the vortex[36, 37]. Therefore,
this second-order transition can no longer be ignored and needs a special
consideration, especially for the vortex beams with large OAM. With the
availability of the tools to create vortex beams with extremely high OAM[23],
significant quadrupole exciton transition rates are plausible. To evaluate the
optical transition amplitudes and dark exciton recombination lifetimes
quantitatively, one needs to resort to elaborate numerical calculations such
as solving the Bethe-Salpeter equations combined with the ab-initio band
structure calculations, which will be the topics of future studies. As the
vortex beams carry non-zero OAM, we also derive optical selection rules for
the exciton envelope functions with higher magnetic quantum numbers
complementary to the $1s$ excitons.

###  III.1 Bright Excitons

$l(\text{OAM})$ | $-5$ | $-4$ | $-3$ | $-2$ | $-1$ | $0$ | $+1$ | $+2$ | $+3$ | $+4$ | $+5$
---|---|---|---|---|---|---|---|---|---|---|---
-K | A | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$
| B | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$
K | A | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$
| B | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$
Table 2: Bright A and B excitons in the dipole coupling regime: Optical selection rules for bright excitons created by laser beams carrying orbital angular momentum (OAM) and spin angular momentum (SAM). $l(\text{OAM})$ | $-5$ | $-4$ | $-3$ | $-2$ | $-1$ | $0$ | $+1$ | $+2$ | $+3$ | $+4$ | $+5$
---|---|---|---|---|---|---|---|---|---|---|---
-K | A | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$
| B | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$
K | A | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$
| B | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$
Table 3: Bright A and B excitons in the quadrupole coupling regime: Optical
selection rules for bright excitons created by laser beams carrying orbital
angular momentum (OAM) and spin angular momentum (SAM).

In the single-electron picture, the bright excitons are formed when an
electron and a hole of the same spins pair via the Coulomb attraction. The
optical selection rules for $1s$ bright excitons in the dipole coupling regime
are presented in Tab.2. The OAM of the vortex beam is in the range of $|l|\leq
5$. When the beam has no OAM ($l=0$), the vortex beam reduces to the
fundamental Gaussian beam with a circular polarization, and the well-known
valley polarization optical selection rules are recovered[9]; the $\sigma^{-}$
polarized light can only create an exciton at the $K$ valley and the
$\sigma^{+}$ polarized light can only create an exciton at the $-K$ valley.
The two valleys and polarization directions are related through the time-
reversal symmetry. In this case, the same selection rules apply to both A and
B excitons. When the OAM of light $l$ is nonzero, the selection rules are
modified as seen in the Tab.2. The allowed and disallowed transitions with
$\sigma^{+}(\sigma^{-})$ polarization alternates for a given valley and has a
periodicity of three units of the OAM of light, which is inherited from the
threefold rotation symmetry of the monolayer crystal. These results for bright
excitons agree well with the recent study[38] where the authors derived
similar optical selection rules for the bright $1s$ excitons. However, we are
mainly focused on dark excitons formation in these 2D crystals. We also show
the selection rules for bright excitons in the quadrupole coupling regime in
Tab.3. We can see that we can only create excitons with particular values of
$l$ using the beams that are $\pi$-polarized, i.e., directed along the light
propagation axis. However, in the paraxial wave regime, the Laguere-Gauss beam
is only polarized in the transverse direction. Therefore, no optical
transitions are allowed in this regime for circularly polarized light.
Nonetheless, one can employ tightly focused vortex beams that can have a
polarization along the beam propagation axis[30].

###  III.2 Dark Excitons

The spin-forbidden dark excitons or simply dark excitons in this work are
formed when an electron and a hole with opposite spins pair via the Coulomb
attraction. A light beam can not create these states because the electric
field of light does not couple to the electron spin. However, the electronic
bands are not purely made of one type of spin species, but mixture of both
spin up and spin down species. This spin mixing is due to the spin-orbit
coupling and the interaction with remote bands[28]. The conduction and valence
bands, therefore, contain a dominant spin component and a smaller component
for the opposite spin. The transition between these bands is decided by the
symmetry properties of the bands and the light.

The optical selection rules for $1s$ dark excitons using vortex beams with
$|l|\leq 5$ are presented in Tab.4. Here, the optical selection rules are
entirely modified compared to the bright exciton cases. We can see that no
circularly polarized light can excite dark excitons. Only when the beam is
polarized in the direction perpendicular to the monolayer plane, one can
create dark excitons. For $l=0$ only dark A-exciton can be created with the
$\pi$ polarized beam that is directed parallel to the sample, which was
demonstrated in photoluminescence experiments on tungsten-based TMDs[28].
However, the optical transition corresponding to the dark B-exciton is
categorically forbidden for any polarization direction at $l=0$. For $|l|>0$
dark B excitons can be excited with $\pi$ polarized light at particular values
of $l$.

As mentioned earlier, the quadrupole transitions can be appreciable when a
vortex beam is tightly focused and has a high value of OAM resulting in a
strong spatial field gradient[36]. Observing that the selection rules have a
fixed periodicity with respect to the values of OAM, one can easily find the
allowed optical transitions at very high values of the beam OAM where the
quadrupole transitions may become considerable. In Tab.5, we present the
selection rules resulting from the quadrupole transitions for the A- and
B-type dark excitons. Both types of dark excitons can be readily created with
circularly polarized light at particular values of OAM. At $l=0$ or when the
laser beam has a simple Gaussian profile, the A-type dark exciton transitions
are strictly forbidden. This is the case for all values of OAM with
$l\pmod{3}=0$. With other finite OAM ($l\pmod{3}\neq 0$), A-type dark excitons
can be created with circularly polarized lights, but the selection rule is the
same for both $+K$ and $-K$ valleys. On the other hand, the B-type dark
exciton transitions are valley-dependent. For example, it is allowed with a
$\sigma^{+}$($\sigma^{-}$) polarized beam at the $K$($-K$) valley at $l=0$.
Therefore, we can selectively create dark B excitons at a particular valley
with circularly polarized light. This shows that the selection rules are
manifestly altered in the presence of a nonzero OAM of light and starkly
differ for the A and B dark excitons due to the symmetry considerations
stemming from not only the band but also the spatial profile of the vortex
beam, which is determined by the value of its OAM.

$l(\text{OAM})$ | $-5$ | $-4$ | $-3$ | $-2$ | $-1$ | $0$ | $+1$ | $+2$ | $+3$ | $+4$ | $+5$
---|---|---|---|---|---|---|---|---|---|---|---
-K | A | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$
| B | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$
K | A | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$
| B | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$ | $\oslash$ | $\pi$ | $\oslash$
Table 4: Dark A and B excitons in the dipole coupling regime: Optical selection rules for dark excitons created by laser beams carrying orbital angular momentum (OAM) and spin angular momentum (SAM). $l(\text{OAM})$ | $-5$ | $-4$ | $-3$ | $-2$ | $-1$ | $0$ | $+1$ | $+2$ | $+3$ | $+4$ | $+5$
---|---|---|---|---|---|---|---|---|---|---|---
-K | A | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$
| B | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$
K | A | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$
| B | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$
Table 5: Dark A and B excitons in the quadrupole coupling regime: Optical
selection rules for dark excitons created by laser beams carrying orbital
angular momentum (OAM) and spin angular momentum (SAM).

###  III.3 Exciton Envelope Functions

The experimental and theoretical studies have shown that the excitonic
properties of 2D TMDs differ starkly from those in the bulk. This is mainly
due to non-local Coulomb screening in and out of the plane of the monolayer
TMD. For example, the optical reflectance measurements in a typical ML TMD,
$\text{WSe}_{2}$, show that the exciton series in these materials clearly
deviate from the 2D Hydrogen series[39]. The two-photon excitation
spectroscopy combined with the many-body perturbation theory within the GW
self-energy approximation and the Bethe-Salpeter Equations also attest to the
same[40]. However, these calculations also showed that there is a one-to-one
correspondence between the exciton wave function symmetry in ML TMDs and 2D
Hydrogen model. By treating the exciton problem in ML TMDs as a 2D Hydrogen
model for the sake of exciton wave function symmetry, we explore the higher-
lying exciton states such as $2p_{+}$ and $2p_{-}$. The symmetries of the
exciton states such as $3d_{\pm}$ and $4f_{\pm}$ can be reduced to the
symmetries of $2p_{\pm}$.

In Tab.6, we tabulated the optical selection rules for the bright A and B
excitons of the $1s$ and $2p_{\pm}$ series in the dipole coupling regime. Both
A- and B- type bright excitons follow the same selection rules. The selection
rules for the $2p_{\pm}$ excitons differ from those of the $1s$ excitons,
nevertheless, the order of the light polarization with respect to the OAM
stays the same – the pattern has the periodicity of three units of OAM.
Because dark exciton states can not be realized with circularly polarized
light in the dipole coupling regime(see Tab.4), we only display the selection
rules for dark excitons in the quadrupole regime in Tab.7 and Tab.8 for the
$1s$ and $2p_{\pm}$ series. It is possible to create dark excitons with vortex
beams in the quadrupole coupling regime for various exciton states. Using an
appropriate vortex beam, we can selectively excite $1s$ or $2p_{\pm}$ dark A
(B) excitons. In Fig.2 and Fig.3, we pictorially show the allowed A and B dark
exciton transitions from the crystal ground state to the $1s$, $2p_{+}$, and
$2p_{-}$ exciton states for the vortex beam for a particular value of the
light OAM ($l=1$). According to the diagrams, the selection rules are the same
in both valleys for the dark A excitons and differ for the dark B excitons.

$l(\text{OAM})$ | $-5$ | $-4$ | $-3$ | $-2$ | $-1$ | $0$ | $+1$ | $+2$ | $+3$ | $+4$ | $+5$
---|---|---|---|---|---|---|---|---|---|---|---
1s | -K | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$
| K | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$
$2p_{+}$ | -K | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$
| K | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$
$2p_{-}$ | -K | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$
| K | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$
Table 6: Bright A and B excitons in the dipole coupling regime: Optical selection rules for the bright A and B excitons created by laser beams carrying orbital anglar momentum (OAM) and spin angular momentum (SAM) for $1s$ and $2p_{\pm}$ exciton envelope functions. Figure 2:  Dark A excitons in the quadrupole coupling regime: Diagram describes the allowed transitions from the crystal ground state to the $1s$ and $2p_{\pm}$ exciton states for the dark B excitons when the vortex beam has $l=1$ orbital angular momentum. The symmetries of all higher lying exciton states can be reduces to the symmetries of these three states. $l(\text{OAM})$ | $-5$ | $-4$ | $-3$ | $-2$ | $-1$ | $0$ | $+1$ | $+2$ | $+3$ | $+4$ | $+5$
---|---|---|---|---|---|---|---|---|---|---|---
1s | -K | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$
| K | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$
$2p_{+}$ | -K | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$
| K | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$
$2p_{-}$ | -K | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$
| K | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$
Table 7: Dark A excitons in the quadrupole coupling regime: Optical selection rules for the dark A excitons created by laser beams carrying orbital angular momentum (OAM) and spin angular momentum (SAM) for $1s$ and $2p_{\pm}$ exciton envelope functions. Figure 3:  Dark B excitons in the quadrupole coupling regime: Diagram describes the allowed transitions from the crystal ground state to the $1s$ and $2p_{\pm}$ exciton states for the dark B excitons when the vortex beam has $l=1$ orbital angular momentum. The symmetries of all higher lying exciton states can be reduces to the symmetries of these three states. $l(\text{OAM})$ | $-5$ | $-4$ | $-3$ | $-2$ | $-1$ | $0$ | $+1$ | $+2$ | $+3$ | $+4$ | $+5$
---|---|---|---|---|---|---|---|---|---|---|---
1s | -K | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$
| K | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$
$2p_{+}$ | -K | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$
| K | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$
$2p_{-}$ | -K | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$
| K | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$ | $\oslash$ | $\sigma^{-}$ | $\sigma^{+}$
Table 8: Dark B excitons in the quadrupole coupling regime: Optical selection
rules for the dark B-excitons created by laser beams carrying orbital angular
momentum (OAM) and spin angular momentum (SAM) for $1s$ and $2p_{\pm}$ exciton
envelope functions.

##  IV  Conclusion

Using symmetry arguments, we have derived optical selection rules for the
spin-forbidden dark excitons and bright excitons in a monolayer TMD crystal
excited by a vortex beam. In the dipole coupling regime, creating A and B
bright excitons follows the same selection rules in individual valleys. In
addition to $1s$ excitons, we can also selectively create $2p_{\pm}$ excitons.
In this work, we showed the necessary conditions for creating various dark
excitons in terms of the optical selection rules. Specifically, the vortex
beams can be used to create A and B dark excitons in the quadrupole coupling
regime and we provided the tables for optical selection rules for such
transitions. This demonstrates the usefulness of the vortex beams with a
nonzero orbital angular momentum in generating otherwise forbidden dark
excitons, which are long-lived and have long coherence times. Optically
created quantum states such as these dark exciton states can be a useful tool
in many quantum device applications. For example, this could be an efficient
method of transduction between semiconductor spin qubits with great tunability
for information processing and photonic qubits with a long coherence time to
use for communications.

###### Acknowledgements.

C.L. acknowledges financial support from the National Science Foundation via
grant NSF DMR-1827745.

$C_{3h}$ | $E$ | $C_{3}^{+}$ | $C_{3}^{-}$ | $\sigma_{h}$ | $S_{3}^{+}$ | $S_{3}^{-}$ | $\bar{E}$ | $\bar{C}_{3}^{+}$ | $\bar{C}_{3}^{-}$ | $\bar{\sigma}_{h}$ | $\bar{S}_{3}^{+}$ | $\bar{S}_{3}^{-}$ | bases
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\Gamma_{1}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $R$
$\Gamma_{2}$ | $1$ | $\omega$ | $\omega^{*}$ | $1$ | $\omega$ | $\omega^{*}$ | $1$ | $\omega$ | $\omega^{*}$ | $1$ | $\omega$ | $\omega^{*}$ | $x+iy$
$\Gamma_{3}$ | $1$ | $\omega^{*}$ | $\omega$ | $1$ | $\omega^{*}$ | $\omega$ | $1$ | $\omega^{*}$ | $\omega$ | $1$ | $\omega^{*}$ | $\omega$ | $x-iy$
$\Gamma_{4}$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $z$
$\Gamma_{5}$ | $1$ | $\omega$ | $\omega^{*}$ | $-1$ | $-\omega$ | $-\omega^{*}$ | $1$ | $\omega$ | $\omega^{*}$ | $-1$ | $-\omega$ | $-\omega^{*}$ |
$\Gamma_{6}$ | $1$ | $\omega^{*}$ | $\omega$ | $-1$ | $-\omega^{*}$ | $-\omega$ | $1$ | $\omega^{*}$ | $\omega$ | $-1$ | $-\omega^{*}$ | $-\omega$ |
$\Gamma_{7}$ | $1$ | $-\omega$ | $-\omega^{*}$ | $i$ | $-i\omega$ | $i\omega^{*}$ | $-1$ | $\omega$ | $\omega^{*}$ | $-i$ | $i\omega$ | $-i\omega^{*}$ | $\chi(1/2,+1/2)$
$\Gamma_{8}$ | $1$ | $-\omega^{*}$ | $-\omega$ | $-i$ | $i\omega^{*}$ | $-i\omega$ | $-1$ | $\omega^{*}$ | $\omega$ | $i$ | $-i\omega^{*}$ | $i\omega$ | $\chi(1/2,-1/2)$
$\Gamma_{9}$ | $1$ | $-\omega$ | $-\omega^{*}$ | $-i$ | $i\omega$ | $-i\omega^{*}$ | $-1$ | $\omega$ | $\omega^{*}$ | $i$ | $-i\omega$ | $i\omega$ |
$\Gamma_{10}$ | $1$ | $-\omega^{*}$ | $-\omega$ | $i$ | $-i\omega^{*}$ | $i\omega$ | $-1$ | $\omega^{*}$ | $\omega$ | $-i$ | $i\omega^{*}$ | $-i\omega$ |
$\Gamma_{11}$ | $1$ | $-1$ | $-1$ | $i$ | $-i$ | $i$ | $-1$ | $1$ | $1$ | $-i$ | $i$ | $-i$ |
$\Gamma_{12}$ | $1$ | $-1$ | $-1$ | $-i$ | $i$ | $-i$ | $-1$ | $1$ | $1$ | $i$ | $-i$ | $i$ |
Table A.I: Character table for the $C_{3h}$ double-group[41]. Here, $\omega=\exp(i2\pi/3)$ and $\omega^{*}=\exp(-i2\pi/3)$. | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{4}$ | $\Gamma_{5}$ | $\Gamma_{6}$ | $\Gamma_{7}$ | $\Gamma_{8}$ | $\Gamma_{9}$ | $\Gamma_{10}$ | $\Gamma_{11}$ | $\Gamma_{12}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\Gamma_{1}$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{4}$ | $\Gamma_{5}$ | $\Gamma_{6}$ | $\Gamma_{7}$ | $\Gamma_{8}$ | $\Gamma_{9}$ | $\Gamma_{10}$ | $\Gamma_{11}$ | $\Gamma_{12}$
$\Gamma_{2}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{1}$ | $\Gamma_{5}$ | $\Gamma_{6}$ | $\Gamma_{4}$ | $\Gamma_{10}$ | $\Gamma_{12}$ | $\Gamma_{8}$ | $\Gamma_{11}$ | $\Gamma_{7}$ | $\Gamma_{9}$
$\Gamma_{3}$ | $\Gamma_{3}$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{6}$ | $\Gamma_{4}$ | $\Gamma_{5}$ | $\Gamma_{11}$ | $\Gamma_{9}$ | $\Gamma_{12}$ | $\Gamma_{7}$ | $\Gamma_{10}$ | $\Gamma_{8}$
$\Gamma_{4}$ | $\Gamma_{4}$ | $\Gamma_{5}$ | $\Gamma_{6}$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{9}$ | $\Gamma_{10}$ | $\Gamma_{7}$ | $\Gamma_{8}$ | $\Gamma_{12}$ | $\Gamma_{11}$
$\Gamma_{5}$ | $\Gamma_{5}$ | $\Gamma_{6}$ | $\Gamma_{4}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{1}$ | $\Gamma_{8}$ | $\Gamma_{11}$ | $\Gamma_{10}$ | $\Gamma_{12}$ | $\Gamma_{9}$ | $\Gamma_{7}$
$\Gamma_{6}$ | $\Gamma_{6}$ | $\Gamma_{4}$ | $\Gamma_{5}$ | $\Gamma_{3}$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{12}$ | $\Gamma_{7}$ | $\Gamma_{11}$ | $\Gamma_{9}$ | $\Gamma_{8}$ | $\Gamma_{10}$
$\Gamma_{7}$ | $\Gamma_{7}$ | $\Gamma_{10}$ | $\Gamma_{11}$ | $\Gamma_{9}$ | $\Gamma_{8}$ | $\Gamma_{12}$ | $\Gamma_{6}$ | $\Gamma_{1}$ | $\Gamma_{3}$ | $\Gamma_{4}$ | $\Gamma_{5}$ | $\Gamma_{2}$
$\Gamma_{8}$ | $\Gamma_{8}$ | $\Gamma_{12}$ | $\Gamma_{9}$ | $\Gamma_{10}$ | $\Gamma_{11}$ | $\Gamma_{7}$ | $\Gamma_{1}$ | $\Gamma_{5}$ | $\Gamma_{4}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{6}$
$\Gamma_{9}$ | $\Gamma_{9}$ | $\Gamma_{8}$ | $\Gamma_{12}$ | $\Gamma_{7}$ | $\Gamma_{10}$ | $\Gamma_{11}$ | $\Gamma_{3}$ | $\Gamma_{4}$ | $\Gamma_{6}$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{5}$
$\Gamma_{10}$ | $\Gamma_{10}$ | $\Gamma_{11}$ | $\Gamma_{7}$ | $\Gamma_{8}$ | $\Gamma_{12}$ | $\Gamma_{9}$ | $\Gamma_{4}$ | $\Gamma_{2}$ | $\Gamma_{1}$ | $\Gamma_{5}$ | $\Gamma_{6}$ | $\Gamma_{3}$
$\Gamma_{11}$ | $\Gamma_{11}$ | $\Gamma_{7}$ | $\Gamma_{10}$ | $\Gamma_{12}$ | $\Gamma_{9}$ | $\Gamma_{8}$ | $\Gamma_{5}$ | $\Gamma_{3}$ | $\Gamma_{2}$ | $\Gamma_{6}$ | $\Gamma_{4}$ | $\Gamma_{1}$
$\Gamma_{12}$ | $\Gamma_{12}$ | $\Gamma_{9}$ | $\Gamma_{8}$ | $\Gamma_{11}$ | $\Gamma_{7}$ | $\Gamma_{10}$ | $\Gamma_{2}$ | $\Gamma_{6}$ | $\Gamma_{5}$ | $\Gamma_{3}$ | $\Gamma_{1}$ | $\Gamma_{4}$
Table A.II: Multiplication table for the $C_{3h}$ double-group[41].

##  Appendix A Symmetry analysis

###  A.1 Band symmetry

We label the electronic bands of our effective two-band (spin-split) model
with their respective irreducible representations in the double-point group
$C_{3h}$ relevant at the $\pm K$ momentum points. To assign a specific irrep
to the valence (conduction) band we construct a tight-binding wave function by
symmetry-adapted atomic orbitals with respect to the threefold rotation
symmetry of the crystal. Applying the threefold rotation operator $C_{3}$ on
this wave function gives an eigenvalue that can be used to read off their
respective irreps from the character table Tab.A.I.

According to the DFT band structure calculations[26], the valence (conduction)
band at the $\pm K$ momentum points in the Brillouin zone of the ML TMD
crystal are predominantly made of the $d_{x^{2}-y^{2}}$ and $d_{xy}$
($d_{z^{2}}$) atomic orbitals of the transition metal atoms. Using these
atomic orbitals as the basis functions[9] that are symmetrized with respect to
the threefold rotation operator $C_{3}^{+}$ relevant to the monolayer crystal
symmetry, we can construct a tight-binding wave function[42]. For the valence
(conduction) bands denoted by $\alpha$ index at the $K$ momentum point of the
reduced Brillouin zone, the wave function that satisfies the Block theorem is
then given by

$\displaystyle\psi_{\alpha}^{K}(\bm{r})=\frac{1}{\sqrt{N}}\sum_{n}e^{i\bm{K}\cdot(\bm{R}_{n}+\bm{\zeta})}\phi_{\alpha}(\bm{r}-\bm{R}_{n}-\bm{\zeta}).$
(3)

Here, the summation is run over the lattice vectors $\bm{R}_{n}$, and
$\bm{\zeta}=(\frac{1}{2},\frac{1}{2\sqrt{3}})a$ is the position of the
transition metal atom relative to the hexagon center $h$ inside the unit
cell(see Fig.1(b)). The momentum point $K=\frac{2\pi}{a}(\frac{2}{3},0)$ or
the $K$ valley is take relative to the Brillouin zone center $\Gamma=(0,0)$
(see Fig.1(e)). The valence band constructed from the atomic orbitals at this
valley is $\phi_{v}(\bm{r})\propto\frac{1}{\sqrt{2}}(d_{x^{2}-y^{2}}+id_{xy})$
and the same for the conduction band is $\phi_{c}(\bm{r})\propto d_{z^{2}}$.

Thus, the eigenvalue of the operator $C_{3}^{+}$ has two components: one comes
from the rotation of the atomic orbital around its own center by $2\pi/3$
degrees counter-clockwise, and the other comes from the change of the Block
phase after the threefold rotation. We note that the rotation operator affects
the phase change differently depending on the rotation center. We took the
hexagon center $h$ to be the rotation center. One could also equally take the
transition metal atom to be the rotation center. However, such a choice would
alter the band irreps, but the selection rules would not change. By applying
the $C_{3}^{+}$ on the valence band at the $K$ valley
$C_{3}^{+}\psi_{v}^{K}(\bm{r})=\eta_{c}\psi_{v}^{K}(\bm{r})$ we obtain its
eigenvalue $\eta_{c}=\mu_{v}\nu_{v}$, which has two components coming from the
change in the Block phase $\mu_{v}$ and the rotation of the atomic orbital
$\nu_{v}$. In our case, we obtain $\mu_{v}=e^{-i\frac{2\pi}{3}}$. The atomic
orbital for the valence band can be shown to be proportional to a spherical
harmonic function as $\phi_{v}(\bm{r})\propto Y_{2}^{2}(\bm{r})$ and
$\nu_{v}=e^{+i\frac{2\pi}{3}}$. So the application of the $C_{3}^{+}$ operator
on $Y_{2}^{2}(\bm{r})$ yields $\nu_{v}=e^{-i\frac{2\pi}{3}2}\equiv
e^{+i\frac{2\pi}{3}}$. Consequently, the total eigenvalue $\eta_{v}=1$. From
the character table Tab.A.I we see that this eigenvalue of the $C_{3}^{+}$
corresponds to the $\Gamma_{1}$ irrep for the valency band at the $K$ valley
without considering the spin-orbit interaction.

Similarly, we can follow the same procedure for the conduction band. The
eigenvalue component coming from the Block phase change is the same in this
case as well $\mu_{c}=e^{-i\frac{2\pi}{3}}$. The atomic orbital for the
conduction band is also proportional to a spherical harmonic function
$\phi_{c}(\bm{r})\propto Y_{2}^{0}(\bm{r})$. Therefore, the eigenvalue
$\nu_{c}=e^{-i\frac{2\pi}{3}0}\equiv 1$. Consequently, the total eigenvalue
$\eta_{c}=e^{-i\frac{2\pi}{3}}$. From the character table Tab.A.I we see that
this eigenvalue, on the other hand, corresponds to the $\Gamma_{3}$ irrep for
the valency band at the $K$ valley without considering the spin-orbit
interaction.

Taking the spin-orbit splitting of the valence band into account[43] gives the
following irreps for the spin-split valence bands:
$\Gamma_{1}\otimes\Gamma_{7}=\Gamma_{7}$ and
$\Gamma_{1}\otimes\Gamma_{8}=\Gamma_{8}$ (see Tab.A.II). In this case, the
spin-split band with a positive spin-orbit interaction energy lies higher in
the band diagram [see Fig.1(f)]. Similarly, we can also label the spin-split
conduction bands with their respective irreps:
$\Gamma_{3}\otimes\Gamma_{7}=\Gamma_{11}$,
$\Gamma_{3}\otimes\Gamma_{8}=\Gamma_{9}$. The order of the spin-split
conduction bands can be inferred from the first-principles band structure
calculations because the origin of the spin-orbit splitting in the conduction
band is non-trivial and due to the competition between the minority orbitals
coming from the chalcogen atom and the interaction with other nearby
bands[26]. The irreps of the bands at the $-K$ valley are related to those of
the bands at the $K$ valley by the time-reversal symmetry.

###  A.2 Vortex beam profile

During the analysis of the spatial profile of the beam, we make use of the
cylindrical and spherical coordinates interchangeably. The vortex beam
propagates along the $z$-axis, which is perpendicular to the ML TMD crystal
lying on the $xy$ plane [see Fig.1(h)]. By placing a ML TMD crystal around the
beam waist ($z\approx 0$), we can obtain a much simplified spatial profile for
the fundamental radial mode ($p=0$) of the Laguerre-Gauss beam where the
complex Gouy phase vanishes. We limit our attention to the fundamental radial
mode ($p=0$) as $p$ does not affect the selection rules qualitatively. In the
vicinity of the beam vortex ($\rho<<w_{0}$):

$\displaystyle
A^{l}(\rho,\phi,z)=\sqrt{\frac{2^{|l|+1}}{\pi|l|!}}\frac{\rho^{|l|}}{w_{0}^{|l|+1}}e^{il\phi}.$
(4)

In the spherical coordinates, the vortex beam assumes the following form

$\displaystyle A^{l}(r,\theta,\phi)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2^{|l|+1}}{\pi|l|!}}\frac{r^{|}l|}{w_{0}^{|l|+1}}(\sin\theta)^{|l|}e^{il\phi}$
(5) $\displaystyle=$
$\displaystyle(-1)^{l}\sqrt{\frac{2^{3|l|+2}|l|!}{(2|l|+1)!}}\frac{r^{|l|}}{w_{0}^{|l|}}Y_{|l|}^{l},$

where we have used the identity
$\left(\sin\theta\right)^{|l|}e^{il\phi}=(-1)^{l}2^{|l|}|l|!\sqrt{4\pi/(2|l|+1)}Y_{|l|}^{l}$.
Here, $Y_{|l|}^{l}$ is a spherical harmonic function of degree (order) $|l|$
($l$). As the spherical harmonic function $Y_{|l|}^{\pm l}\propto(x\pm
iy)^{|l|}$, we can easily assign an irrep for specific values of the beam OAM
$l$. For example, for $l=1$($-1$) it is $\Gamma_{2}$ ($\Gamma_{3}$) and for
$l=0$ it is $\Gamma_{1}$. For high values of $l$ one needs to compute a direct
product of $l$ number of $\Gamma_{2}$ or $\Gamma_{3}$ using the irrep
multiplication table for the double-group of $C_{3h}$.

###  A.3 Transition matrix element

Fermi’s Golden rule states that a direct optical transition rate from the
crystal ground state to the excited state is given by

$\displaystyle
R=\frac{2\pi}{\hbar}\sum_{X}|\braket{\Psi_{X}}{\hat{H}_{\text{int}}}{0}|^{2}\delta(E_{X}-E_{0}-\hbar\omega),$
(6)

where $\hat{H}_{\text{int}}$ is the light-crystal interaction Hamiltonian and
$\omega$ is the light frequency. The sum is taken over all final (exciton)
states. The exciton wave function is a triple product of the electron-hole
envelope function, the electron and hole Bloch wavefunctions at the conduction
and valence bands, respectively[44]

$\displaystyle\Psi_{X}(\bm{r}_{e},\bm{r}_{h})=\sum_{\bm{k}}F_{nm}(\bm{k})\psi_{c\bm{k}}(\bm{r}_{e})\psi_{v\bm{k}}^{*}(\bm{r}_{h})$
(7)

where $F_{nm}(\bm{k})=R_{n|m|}(k)e^{im\phi}$ is the Fourier transform of the
2D exciton envelope function $\Phi_{nm}(\bm{r})$. The symmetry of the envelope
function is entirely defined by the factor
$\exp(im\phi)\propto(x+(-1)^{m}iy)^{m}$. Depending on the value of $m$, it
constitutes a basis for the irreps $\Gamma_{2}$ or $\Gamma_{3}$. Subsequently,
the product of those function form a basis for the exciton representation

$\displaystyle\Gamma_{X}=\Gamma_{\text{env}}\otimes\Gamma_{c}\otimes\Gamma_{v}^{*}.$
(8)

In other words, this triple direct product represents the symmetry of an
exciton state.

In the Coulomb gauge, one can show that the interaction Hamiltonian

$\displaystyle\hat{H}_{int}$ $\displaystyle\propto$ $\displaystyle
e^{ikz}\bm{r}\cdot\bm{\epsilon}A^{l}$ (9) $\displaystyle\approx$
$\displaystyle(-1)^{l}\sqrt{\frac{2^{3|l|+2}|l|!}{(2|l|+1)!}}\frac{r^{|l|+1}}{w_{0}^{|l|}}(1+ikz)Y_{1}^{s}Y_{|l|}^{l}.$
(10)

Here, the spherical harmonic function $Y_{|l|}^{l}$ ($Y_{1}^{s}$) expresses
the orbital (spin) angular momentum of light. The right (left) circularly-
polarized light is designated with $s=+1(-1)$ values. When the light beam is
polarized along the direction perpendicular to the crystal plane, $s=0$. As
these functions form bases for the irreps in the double-point group $C_{3h}$
relevant at the $\pm K$ valleys, we can assign corresponding irreps depending
on the values of $l$ and $s$. The symmetry properties of the spin degree of
freedom are represented by $\Gamma_{2}$ ($\Gamma_{3}$) for $s=+1$ ($-1$) and
$\Gamma_{4}$ for $s=0$ since $Y_{1}^{\pm 1}\propto(x\pm iy)$ and
$Y_{1}^{0}\propto z$, respectively. Because the basis function
$Y_{|l|}^{l}\propto(x+(-1)^{l}iy)^{l}$, the irreps $\Gamma_{2}/\Gamma_{3}$ or
their $l$th power (i.e., their $l$ direct products) can represent the symmetry
properties of this basis function. The order of the light-crystal coupling has
the symmetry of $\Gamma_{1}$ for the dipole ($e^{ikz}\approx 1$) and
$\Gamma_{4}$ for the quadrupole ($e^{ikz}\approx 1+ikz$) transition. Thus, the
symmetry of the vortex beam is equal to

$\displaystyle\Gamma_{\text{beam}}=\Gamma_{\text{order}}\otimes\Gamma_{\text{OAM}}\otimes\Gamma_{\text{SAM}}.$
(11)

For an optical transition from the crystal ground state to an excitonic state
to occur, the transition matrix element
$\braket{\Psi_{X}}{\hat{H}_{\text{int}}}{0}$ must be nonzero. For that to
happen the triple direct product of the exciton state, vortex beam, and the
crystal ground state must span the totally symmetric irrep of the group
$C_{3h}$, which is $\Gamma_{1}$. The symmetry of the crystal ground state is
also represented by $\Gamma_{1}$. Thus, the allowed exciton transitions
satisfy the following direct product equation

$\displaystyle\Gamma_{X}^{*}\otimes\Gamma_{\text{beam}}\otimes\Gamma_{1}=\Gamma_{1}.$
(12)

This statement is also known as the matrix element theorem[25, 45], which
states that the symmetry of the final (exciton) state must contain symmetry of
the driving field operator for an excitation from the ground state to occur.

## References

  * Poem  _et al._ [2010] E. Poem, Y. Kodriano, C. Tradonsky, N. H. Lindner, B. D. Gerardot, P. M. Petroff, and D. Gershoni, Accessing the dark exciton with light, Nature Physics 6, 993 (2010).
  * Schwartz  _et al._ [2015] I. Schwartz, E. R. Schmidgall, L. Gantz, D. Cogan, E. Bordo, Y. Don, M. Zielinski, and D. Gershoni, Deterministic writing and control of the dark exciton spin using single short optical pulses, Phys. Rev. X 5, 011009 (2015).
  * Madéo  _et al._ [2020] J. Madéo, M. K. L. Man, C. Sahoo, M. Campbell, V. Pareek, E. L. Wong, A. Al-Mahboob, N. S. Chan, A. Karmakar, B. M. K. Mariserla, X. Li, T. F. Heinz, T. Cao, and K. M. Dani, Directly visualizing the momentum-forbidden dark excitons and their dynamics in atomically thin semiconductors, Science 370, 1199 (2020).
  * Rustagi and Kemper [2018] A. Rustagi and A. F. Kemper, Photoemission signature of excitons, Phys. Rev. B 97, 235310 (2018).
  * Wang  _et al._ [2018] G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie, T. Amand, and B. Urbaszek, Colloquium: Excitons in atomically thin transition metal dichalcogenides, Rev. Mod. Phys. 90, 021001 (2018).
  * Novoselov  _et al._ [2005] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, Two-dimensional atomic crystals, Proceedings of the National Academy of Sciences 102, 10451 (2005).
  * Mak  _et al._ [2010] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Atomically thin ${\mathrm{mos}}_{2}$: A new direct-gap semiconductor, Phys. Rev. Lett. 105, 136805 (2010).
  * Splendiani  _et al._ [2010] A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli, and F. Wang, Emerging photoluminescence in monolayer ${\mathrm{mos}}_{2}$, Nano Letters 10, 1271 (2010).
  * Xiao  _et al._ [2012] D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Coupled spin and valley physics in monolayers of ${\mathrm{mos}}_{2}$ and other group-vi dichalcogenides, Phys. Rev. Lett. 108, 196802 (2012).
  * Cao  _et al._ [2012] T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu, P. Tan, E. Wang, B. Liu, and J. Feng, Valley-selective circular dichroism of monolayer molybdenum disulphide, Nature Communications 3, 10.1038/ncomms1882 (2012).
  * Mak  _et al._ [2012] K. F. Mak, K. He, J. Shan, and T. F. Heinz, Control of valley polarization in monolayer MoS2 by optical helicity, Nature Nanotechnology 7, 494 (2012).
  * Zeng  _et al._ [2012] H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Valley polarization in MoS2 monolayers by optical pumping, Nature Nanotechnology 7, 490 (2012).
  * Zhang  _et al._ [2017] X.-X. Zhang, T. Cao, Z. Lu, Y.-C. Lin, F. Zhang, Y. Wang, Z. Li, J. C. Hone, J. A. Robinson, D. Smirnov, S. G. Louie, and T. F. Heinz, Magnetic brightening and control of dark excitons in monolayer WSe2, Nature Nanotechnology 12, 883 (2017).
  * Zhou  _et al._ [2017] Y. Zhou, G. Scuri, D. S. Wild, A. A. High, A. Dibos, L. A. Jauregui, C. Shu, K. D. Greve, K. Pistunova, A. Y. Joe, T. Taniguchi, K. Watanabe, P. Kim, M. D. Lukin, and H. Park, Probing dark excitons in atomically thin semiconductors via near-field coupling to surface plasmon polaritons, Nature Nanotechnology 12, 856 (2017).
  * Beth [1936] R. A. Beth, Mechanical detection and measurement of the angular momentum of light, Phys. Rev. 50, 115 (1936).
  * Allen  _et al._ [1992] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes, Phys. Rev. A 45, 8185 (1992).
  * Konzelmann  _et al._ [2019] A. M. Konzelmann, S. O. Krüger, and H. Giessen, Interaction of orbital angular momentum light with rydberg excitons: Modifying dipole selection rules, Phys. Rev. B 100, 115308 (2019).
  * Schmiegelow  _et al._ [2016] C. T. Schmiegelow, J. Schulz, H. Kaufmann, T. Ruster, U. G. Poschinger, and F. Schmidt-Kaler, Transfer of optical orbital angular momentum to a bound electron, Nature Communications 7, 10.1038/ncomms12998 (2016).
  * Friese  _et al._ [1998] M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Optical alignment and spinning of laser-trapped microscopic particles, Nature 394, 348 (1998).
  * Picón  _et al._ [2010] A. Picón, A. Benseny, J. Mompart, J. R. V. de Aldana, L. Plaja, G. F. Calvo, and L. Roso, Transferring orbital and spin angular momenta of light to atoms, New Journal of Physics 12, 083053 (2010).
  * Rodrigues  _et al._ [2016] J. D. Rodrigues, L. G. Marcassa, and J. T. Mendonça, Excitation of high orbital angular momentum rydberg states with laguerre–gauss beams, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 074007 (2016).
  * Simbulan  _et al._ [2021] K. B. Simbulan, T.-D. Huang, G.-H. Peng, F. Li, O. J. Gomez Sanchez, J.-D. Lin, C.-I. Lu, C.-S. Yang, J. Qi, S.-J. Cheng, T.-H. Lu, and Y.-W. Lan, Selective photoexcitation of finite-momentum excitons in monolayer mos2 by twisted light, ACS Nano 15, 3481 (2021), pMID: 33566571, https://doi.org/10.1021/acsnano.0c10823 .
  * Fickler  _et al._ [2016] R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, Quantum entanglement of angular momentum states with quantum numbers up to 10,010, Proceedings of the National Academy of Sciences 113, 13642 (2016).
  * Liu  _et al._ [2014] G.-B. Liu, H. Pang, Y. Yao, and W. Yao, Intervalley coupling by quantum dot confinement potentials in monolayer transition metal dichalcogenides, New Journal of Physics 16, 105011 (2014).
  * Yu and Cardona [2010] P. Y. Yu and M. Cardona,  _Fundamentals of Semiconductors_ (Springer Berlin Heidelberg, 2010).
  * Kormányos  _et al._ [2015] A. Kormányos, G. Burkard, M. Gmitra, J. Fabian, V. Zólyomi, N. D. Drummond, and V. Fal’ko, $k\cdot p$ theory for two-dimensional transition metal dichalcogenide semiconductors, 2D Materials 2, 022001 (2015).
  * Kogelnik and Li [1966] H. Kogelnik and T. Li, Laser beams and resonators, Proc. IEEE Inst. Electr. Electron. Eng. 54, 1312 (1966).
  * Wang  _et al._ [2017] G. Wang, C. Robert, M. M. Glazov, F. Cadiz, E. Courtade, T. Amand, D. Lagarde, T. Taniguchi, K. Watanabe, B. Urbaszek, and X. Marie, In-plane propagation of light in transition metal dichalcogenide monolayers: Optical selection rules, Phys. Rev. Lett. 119, 047401 (2017).
  * Lax  _et al._ [1975] M. Lax, W. H. Louisell, and W. B. McKnight, From maxwell to paraxial wave optics, Phys. Rev. A 11, 1365 (1975).
  * Quinteiro  _et al._ [2017] G. F. Quinteiro, F. Schmidt-Kaler, and C. T. Schmiegelow, Twisted-light–ion interaction: The role of longitudinal fields, Phys. Rev. Lett. 119, 253203 (2017).
  * Echeverry  _et al._ [2016] J. P. Echeverry, B. Urbaszek, T. Amand, X. Marie, and I. C. Gerber, Splitting between bright and dark excitons in transition metal dichalcogenide monolayers, Phys. Rev. B 93, 121107 (2016).
  * Olsen  _et al._ [2016] T. Olsen, S. Latini, F. Rasmussen, and K. S. Thygesen, Simple screened hydrogen model of excitons in two-dimensional materials, Phys. Rev. Lett. 116, 056401 (2016).
  * Shinada and Sugano [1966] M. Shinada and S. Sugano, Interband optical transitions in extremely anisotropic semiconductors. i. bound and unbound exciton absorption, Journal of the Physical Society of Japan 21, 1936 (1966), https://doi.org/10.1143/JPSJ.21.1936 .
  * Yang  _et al._ [1991] X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong, and W. Y. Ching, Analytic solution of a two-dimensional hydrogen atom. i. nonrelativistic theory, Phys. Rev. A 43, 1186 (1991).
  * Qiu  _et al._ [2013] D. Y. Qiu, F. H. da Jornada, and S. G. Louie, Optical spectrum of ${\mathrm{mos}}_{2}$: Many-body effects and diversity of exciton states, Phys. Rev. Lett. 111, 216805 (2013).
  * Lembessis and Babiker [2013] V. E. Lembessis and M. Babiker, Enhanced quadrupole effects for atoms in optical vortices, Phys. Rev. Lett. 110, 083002 (2013).
  * Schmiegelow and Schmidt-Kaler [2012] C. Schmiegelow and F. Schmidt-Kaler, Light with orbital angular momentum interacting with trapped ions, The European Physical Journal D 66, 10.1140/epjd/e2012-20730-4 (2012).
  * Ishii  _et al._ [2019] S. Ishii, N. Yokoshi, and H. Ishihara, Optical selection rule of monolayer transition metal dichalcogenide by an optical vortex, Journal of Physics: Conference Series 1220, 012056 (2019).
  * Chernikov  _et al._ [2014] A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi, Y. Li, O. B. Aslan, D. R. Reichman, M. S. Hybertsen, and T. F. Heinz, Exciton binding energy and nonhydrogenic rydberg series in monolayer ${\mathrm{ws}}_{2}$, Phys. Rev. Lett. 113, 076802 (2014).
  * Ye  _et al._ [2014] Z. Ye, T. Cao, K. O’Brien, H. Zhu, X. Yin, Y. Wang, S. G. Louie, and X. Zhang, Probing excitonic dark states in single-layer tungsten disulphide, Nature 513, 214 (2014).
  * Koster  _et al._ [1963] G. Koster, J. Dimmock, R. Wheeler, and H. Statz,  _Properties of the Thirty-two Point Groups_, M.I.T. Press research monographs (M.I.T. Press, 1963).
  * Ashcroft and Mermin [1976] N. W. Ashcroft and N. Mermin,  _Solid State Physics_ (Brooks/Cole, Florence, KY, 1976).
  * Dresselhaus  _et al._ [2007] M. S. Dresselhaus, G. Dresselhaus, and A. Jorio,  _Group theory_ , 2008th ed. (Springer, Berlin, Germany, 2007).
  * Elliott [1957] R. J. Elliott, Intensity of optical absorption by excitons, Phys. Rev. 108, 1384 (1957).
  * Hopfdeld [1960] J. Hopfdeld, Fine structure in the optical absorption edge of anisotropic crystals, Journal of Physics and Chemistry of Solids 15, 97 (1960).


††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Experimental verification of Arcsine laws in mesoscopic non-equilibrium and
active systems

Raunak Dey  Avijit Kundu  Biswajit Das  Ayan Banerjee<EMAIL_ADDRESS>Department of Physical Sciences, IISER Kolkata

###### Abstract

A large number of processes in the mesoscopic world occur out of equilibrium,
where the time course of the system evolution becomes immensely important –
they being driven principally by dissipative effects. Non-equilibrium steady
states (NESS) represent a crucial category in such systems – which are widely
observed in biological domains – especially in chemical kinetics in cellular
processes [1], and molecular motors [2]. In this study, we employ a model NESS
stochastic system which comprises of an colloidal microparticle, optically
trapped in a viscous fluid and externally driven by a temporally correlated
colored noise, and show that the work done on the system and the work
dissipated by it – both follow the three Lévy arcsine laws. These statistics
remain unchanged even in the presence of a perturbation generated by a
microbubble at close proximity to the trapped particle. We confirm our
experimental findings with theoretical simulations of the systems. Our work
provides an interesting insight into the NESS statistics of the meso-regime,
where stochastic fluctuations play a pivotal role.

††preprint: APS/123-QED

In 1940, Paul Lévy introduced three laws that described the long-term
statistics of a one-dimensional Wiener process, often used to describe the
trajectory of Brownian motion [3]. For the simplest case of Brownian motion,
the particle’s position coordinate starts from zero. It has an equally
probable incremental step in either direction, which does not depend on its
past trajectory – a key signature of a Markovian process. Also, the increments
have a zero-centered Gaussian nature where the variance of the distribution
increases linearly with time. For such a process, three variables can be
defined: a) the proportion of time the fluctuating variable stays above zero:
$T_{+}=\frac{1}{t}\int_{0}^{t}\theta(W(t^{\prime})-\langle
W(t^{\prime})\rangle)dt^{\prime}$, where $\theta(x)$ is the Heaviside function
which takes values ’1’ and ’0’ when its argument is positive and negative,
respectively, b) the ratio of the last time when the variable had changed its
sign to the total time: $T_{last}=sup\\{t\in[0,1]:W(t)=0\\}$, c) the ratio of
time when the variable has attained its global maximum to the total time:
$T_{max}$. $W(T_{max})=sup\\{W(t),t\in[0,1]\\}$. These variables are scaled
from $0$ to $1$, and they can be shown to have a normalized probability
density function,
$\text{PDF}(T_{+}/T_{last}/T_{max})=\frac{1}{\pi\sqrt{T(1-T)}}$, as long as
the root variable follows a Wiener process. It follows that the cumulative
distribution function (CDF) of these three variables follow the arcsine law
given by:

$\text{CDF}(T)=\int_{0}^{T}\text{PDF}(T^{\prime})dT^{\prime}=\dfrac{2}{\pi}\text{arcsin}(\sqrt{T})$
(1)

for $T=T_{+}/T_{last}/T_{max}\in[0,1]$.

A large number of processes in the natural sciences follow the properties of
the Wiener process, and consequently, the arcsine laws. Notably, operations in
the microscopic regime have low degrees of freedom, where fluctuations can
dominate the system’s thermodynamic properties. These fluctuations are modeled
successfully using the arcsine distributions. In recent times, the electron
current in cold quantum dots [4], the net number of steps of a molecular motor
[5], position fluctuation of a particle in a periodic potential [6], and the
work done on the system in micrometer-sized stochastic Carnot engines [7],
have been shown to follow the arcsine distribution. Apart from non-equilibrium
systems, entirely disparate systems including the quantum state of a dressed
photon in a fiber probe [8], the fluctuation of stock prices [9], waiting time
distributions of human dynamics [10], leads in competitive sports like
football or basketball [11] as well as various random walk models [12], are
examples that follow arcsine distributions. These laws can even be modified to
include the behavior of fractional Brownian motion [13], non-Markovian
processes [14], and processes that display anomalous diffusion [15].

At mesoscopic length scales, the thermal fluctuations in the system play an
important role since the energy of the system is in the order of $k_{B}T$.
Very often, these systems which include Brownian motors [16] and engines [17],
operate far from equilibrium by deriving energy from the external
fluctuations, whereas their dissipation is dictated by the rate of entropy
generated [18]. An interesting class of processes is those that are non-
equilibrium, but in steady state – since, with the sophisticated
micromanipulation techniques enabled by optical trapping, these can be studied
in great detail even experimentally. However, apart from an experimental study
of the work done in a Brownian Carnot engine [7], there exists a lacuna
towards determining and validating arcsine properties of stochastic variables
in mesoscopic systems, both periodic and non-periodic, experimentally. Indeed,
microscopic engines developed around Carnot and Stirling cycles [17], as well
as Brownian ratchets and motors [16], or even the simple manoeuvre of dragging
a particle to and fro at a constant velocity [19] – all represent
experimentally demonstrable periodic systems. In the last example, the free
energy may not change over many periods – since, though it contains both
steady state and transient effects – the particle may always relax back to
equilibrium. However, for non-periodic systems that remain in a steady state
out of equilibrium, the physics of operation is entirely different, with
$(\Delta F\neq 0)$.

On another note, interest regarding systems driven with colored noise [20, 19]
has been growing in the scientific community, primarily due to the fact that
these serve as a tunable model in understanding real-world biological systems
at the microscopic level – especially where phenomena such as non-Gaussian
displacement statistics [21], active dissipative fluctuations [22], and flows
[23] may become prominent. There is an obvious requirement to study the long-
term statistics of stochastic micro-currents that drive these systems in a
steady state, since any deviations from the arcsine laws will capture “aging”
[15] and “non-markovian” [13] behavior that determine the efficiency of these
micro-engines. For example, if a fluctuating current does not satisfy the
Markov properties of a typical diffusing Wiener process, it will deviate from
the arcsine behavior. Thus, it is possible to infer about underlying processes
from the long-term statistics of these fluctuating currents, without scanning
the system holistically.

Figure 1: Work done on the system and work dissipated by the system
represented schematically, as obtained from experimental data. (a) The work
dissipated by the probe over time plotted in units of $k_{B}T$ after
subtracting the mean value of dissipating current. Values of the Lévy
variables $T_{+}$,$T_{last}$,$T_{max}$ are displayed for this window in red,
green, blue color coding, respectively. (b) The fraction of time the
stochastic current $W_{d}$ (in red) lies above the average $\langle
W_{d}\rangle$ over many cycles. (c) The cumulative work done $(W)$ and work
dissipated $(W_{d})$ by the engine in units of $k_{B}T$ are plotted over time,
which are positive and negative, respectively. (d) The probability
distributions of the work done and work dissipated per unit sampling time
$\widetilde{W}$ and $\widetilde{W_{d}}$ plotted on different axial scales. (e)
Schematics of the process, where the trap center is modulated by an Ornstein
Uhlenbeck process of a constant amplitude to proceed from an equilibrium to a
non-equilibrium steady state (ESS to NESS).

In this paper, we experimentally demonstrate that both the work done by the
colored noise and work dissipated by the micro-system in the process, follow
Lévy arcsine statistics in a model NESS system. In our study, we trap a
micron-sized polystyrene probe particle in water in a harmonic potential
generated by optical tweezers, and drive it externally with a colored noise
generated by the Ornstein-Uhlenbeck (OU) process. The work done in such a
process changes the effective free energy of the system irreversibly. To
maintain the temperature of the system, which is out of equilibrium with its
surroundings, work needs to be done continuously by an externally added noise.
A small component of that work is dissipated as heat, mainly due to viscous
effects of the fluid where the probe is embedded. However, the presence of a
microbubble in the fluid medium causes the generation of flow fields, so that
the effective velocity of a trapped mesoscopic probe in the close vicinity of
the bubble gets modified, along with the spatial diffusion coefficient
profile. We observe that even in this case, the Lévy arcsine statistics are
not modified. We finally conclude that the arcsine statistics of the
fluctuations of an optically confined Brownian probe in a Newtonian fluid
remain unaffected with change in amplitude and time constant of an externally
added noise – as long as the observation period is much longer than the time
constant of the added noise – and in the presence of flow fields provided the
flow velocity the probe experiences is constant. (See Supplementary
Information).

Figure 2:  Statistics of work dissipated as obtained from simulation. The
results for the three arcsine laws for $T_{+}$ (in red), $T_{last}$ (in green)
and $T_{max}$ (in blue) from a simulation of the time series, where the CDF
and PDF are plotted in (a) and (b).

We can describe the dynamics of our system with the overdamped Langevin
equation as:

$\displaystyle\dot{x}(t)$
$\displaystyle=-\dfrac{x(t)-\lambda(t)}{\tau_{d}}+\sqrt{2D}\zeta(t)$ (2)
$\displaystyle\dot{\lambda}(t)$
$\displaystyle=-\dfrac{\lambda(t)}{\tau_{0}}+\sqrt{2A}\xi(t)$ (3)

In Equation(2), $D$ is the diffusion constant and $\tau_{d}=\gamma/k$ is the
relaxation time of the harmonic trap, where $k$ is the trap stiffness and
$\gamma$ is the drag coefficient related by the Stokes-Einstein relation as
$D\gamma=k_{B}T$. In Equation(3), $\lambda(t)$ is the Ornstein-Uhlenbeck noise
that drives the center of the harmonic trap, with a time constant $\tau_{0}$,
and amplitude proportional to $A$. $\zeta(t)$ and $\xi(t)$ are zero-centered
unit variance delta correlated white noise with no mutual correlations,
$\langle\xi(t)\zeta(t^{\prime})\rangle=0$. The OU noise has an exponentially
decaying correlation
$\langle\lambda(t)\lambda(t^{\prime})\rangle=A\tau_{0}\exp(-|t-t^{\prime}|/\tau_{o})$
and a power spectrum that falls off as $f^{-\alpha}$, $\alpha\approx 2$. In
presence of the microbubble the system can be modelled with a suitable
addition of the flow field $(u_{d})$, $\dot{x}\rightarrow\dot{x}-u_{d}$ and by
values of $\tau_{d}$ and $D$ that get scaled as a function of the spatial
distance from the microbubble. (see Ref. [23]).

Figure 3:  Action of a microbubble in close vicinity of the optically trapped
probe. (a) Schematic of the experimental probe particle-micobubble system. The
microbubble is grown on a pre-existing absorbing pattern/trail in the sample
chamber (see Methods). The different dimensions of the bubble and . (b)
Histogram of the position fluctuation – clearly it develops a skew in the
presence of the flow field generated by the microbubble.

The incremental work in each small step can be calculated as
$\widetilde{W}=dW=(\partial U(x(t),x_{0}(t))/\partial x_{0}(t))*dx_{0}(t)$,
where $x_{0}(t)$ is the parameter varying in the non equilibrium process – the
center of the trap in our case. On the other hand the work done in small
dissipative steps $(\widetilde{W_{d}})$ is due to the viscous drag on the
probe embedded in the bath. We use a tightly focused laser beam to form a
harmonic trapping potential that varies with time, given by
$U(x(t),\lambda(t))=k[x(t)-\lambda(t)]/2$, and trap a spherical probe particle
(polystyrene) of diameter $3\ \mu m$, where the trap stiffness is $k=19.7\pm
0.1$ pN/$\mu$m. We modulate the trap-center with an Ornstein Uhlenbeck noise
given by $\lambda(t)$ which is generated by Eq 3, with its time constant for
exponentially decaying correlation $\tau_{0}=2.5$ ms, and an amplitude equal
to $A=0.3\times(0.6\times 10^{-6})^{2}\ m^{2}/s$. The effective operating
temperature of the system is approximately $588$ K. For the next stage of the
experiment, we form a microbubble of diameter $21\pm 0.6\ \mu m$ (see Methods)
using another laser [24] such that its surface remains $d=10\ \mu m$ away from
the mean position of the center of the probe particle. We sample the
displacement of the colloidal particle at a rate of 10 kHz for 100 seconds. We
repeat these experiments for varying A, $\tau_{0}$, d, and discuss the results
in the Supplementary Information. The cumulative work done on the system and
work dissipated by the system is given by,

$\displaystyle W$ $\displaystyle=\int-k[x(t)-\lambda(t)]*d\lambda(t)$ (4)
$\displaystyle W_{d}$ $\displaystyle=\int\gamma\lambda(t)*dx(t)$ (5)

where the period of integration should be chosen such that it is greater than
$\tau_{0}$. The symbol ${}^{\prime}*^{\prime}$ represents the Stratonovich
product for a stochastic variable. [Note that for two stochastic variables
$\alpha_{t}$ and $\beta_{t}$, the Stratonovich product is given by
$\int_{0}^{\tau}f(\alpha_{t})*d\beta_{t}=\sum_{t}\frac{f(\alpha_{t})+f(\alpha_{t+\Delta
t})}{2}(\beta_{t+\Delta t}-\beta_{t})$, where t runs from $0$ to $\tau-\Delta
t$.]

Figure 4:  Arcsine laws tabulated for all the cases. The work dissipated by
the stochastic engine as well as the work done in our experiments are used to
plot the three arcsine laws for $T_{+}$, $T_{last}$ and $T_{max}$ in red,
green and blue colours, respectively.

Results: The position fluctuations for the probe lies in the order of tens of
nanometers. The external source of colored noise, whose diffusion constant
$(A=0.3\times(0.6\times 10^{-6})^{2}\ m^{2}/s)$ is of the order of the
diffusion constant of water, increases the variance of the probe fluctuations
and drives it out of equilibrium to an elevated temperature of 588 K. However
the temperature of the water (environment), surrounding our driven micro-
colloidal system does not increase more than 1K, as energy dissipated by the
viscous forces as well as heat absorbed from the laser [25] is negligible
compared to the work done on the system. We show the results from a section of
the experimental data as an example in Fig. 1. Note that we consider the work
done on the system as positive, while the work dissipated by the system is
negative. Fig. 1(a) shows the fluctuating current for dissipated work after
the average fluctuation for all the segments is subtracted from the time
series. Further, Fig. 1(b), shows that the total time the time series of
$W_{d}$ lies above the average current $\langle W_{d}\rangle$, is 0.02 s.
Similarly, we calculate from this section of data, that the last instant of
time when the stochastic current was positive is 0.81 s, and the time instant
when it reached the global maximum within this section of the data is 0.04 s.
Although the work done and dissipated are stochastic in short times, over a
long time, they accumulate cumulatively. We plot the histogram of the work
done (dissipated) in small steps $\widetilde{W}$ ($\widetilde{W_{d}}$) in Fig.
1(d). The work on the micro-system is in the order of $k_{B}T$ – out of which
approximately 0.1% is dissipated into the fluid due to friction. We represent
the process schematically in Fig. 1(e). The stochastic currents of work done
(dissipated) on (by) the system follow the properties of a Wiener process –
they start at zero, their small increments are independent of the past
(Markov) and are Gaussian with its variance increasing linearly. [26] The time
series of the position variable has embedded correlation because of the
correlated noise, and due to the effect of the trap – so that those variables
do not follow the Wiener properties. Next, we simulate the stochastic
trajectories of a probe perturbed by an OU process and verify all the results
obtained in the experiment. The cumulative density function work dissipated
(done) by (on) such a system follows an arcsine distribution (for all the
three variables), which we show in Fig. 2(a). Similarly, in Fig. 2(b), we
demonstrate an example that the PDF of the stochastic current follows
$\frac{1}{\pi\sqrt{T(1-T)}}$.

When we perform the experiment in the presence of a microbubble as we display
in Fig. 3(a), an active flow field is generated, which constricts and skews
the motion of the probe in one direction – which we demonstrate in Fig. 3(b),
where the Gaussian nature of the position probability distribution function is
clearly observed to be skewed in the presence of the flow generated by the
bubble. Note that the work done depends on the probe’s velocity through the
medium, which now has an additive term due to the flow induced by the
microbubble. However, since the probe’s fluctuation is minimal compared to its
distance from the surface of the microbubble, we can assume the strength of
the flow field to unchanged, so that the arcsine nature of the statistics also
remains undisturbed.

While obtaining the statistics, we divide the data into segments of 1 s
($10^{4}$ points) each, which is much greater than the time constant of the
underlying process, $\tau_{d}(=2.5\ ms)$. Note that this is important towards
obtaining robust statistics. When more data points are sampled, a better fit
is obtained for the arcsine laws. In Fig. 4 we tabulate all the experimental
plots and show that the CDF of $T_{+}$, $T_{last}$, and $T_{max}$ obtained
from both the work done and work dissipated by the system follow Lévy arcsine
statistics.

Conclusion: We unambiguously demonstrate that the stochastic currents
associated with a Markov process in a non-equilibrium steady-state system in
the presence of viscous forces obey Lévy arcsine statistics, leading to
experimentally measurable variables which also demonstrate the same
statistics. Thus, we observe that – even in the presence of colored noise,
temporally correlated perturbations, or active flow-fields – the statistical
characteristics of the work exchange between the system and its surroundings
are not modified – indeed they all follow arcsine laws. Our method can be
extended to other systems in non-equilibrium, or even to study transient
phenomena in physics. Most importantly, any deviations from arcsin laws can
indicate non-Markovian behavior or anomalous diffusion in the underlying
processes that generate the stochastic current, thus providing useful insight
into the system dynamics before probing it in detail.

### Methods

Particle trapping and manipulation: We perform all experiments on a double-
distilled aqueous dispersion of spherical polystyrene particles (Sigma-Aldrich
LB30) having a diameter 3 $\mu m$. We build a custom sample chamber of
thickness 100 $\mu m$ with double-sided sticky tape between two coverslips and
mount the sample chamber on a motorized stage. A Gaussian beam (1064 nm)
tightly focused with high numerical aperture oil immersion objective (100x,
NA=1.3) of a standard oil immersion objective an inverted microscope (Olympus
IX71) traps the particle at a height 15 $\mu m$ from the lower surface of the
sample chamber, to mitigate surface forces. The laser passes through an
acousto-optic modulator (BRIMROSE-AOM), and the first-order beam traps and
modulates the particle perpendicular to the beam’s direction with input noise
generated through the Ornstein-Uhlenbeck process. We focus a second co-
propagating beam of wavelength 785 nm and measure the backscattered intensity
by position-sensitive photo-detectors (PDA100A2) to sample the one-directional
trajectory of the probe at a spatio-temporal resolution of 1 nm - 10 kHz.
Additionally, we use the autocorrelation of the time series of a trapped
particle and the noise to calibrate the fluctuation of the probe from volts
(measured by the photodiode) to nm.

Generation of microbubble: For the experiments involving the microbubble, we
employ a coverslip patterned by a polyoxometalate material [24], absorbing at
1064 nm as the bottom surface of the sample chamber while keeping everything
else unchanged. We use a second laser operating at a 1064 nm wavelength to
generate a microbubble of diameter 21 $\mu m$ – the remains constant for
unchanging laser power [24] throughout the time-scale of the experiment. We
trap the particle at the same height as the radius of the bubble and calibrate
the length scale of our setup with the company manufactured integrated
software, which we also verify from the size of the probe.

## References

  * Grima [2010] R. Grima, An effective rate equation approach to reaction kinetics in small volumes: Theory and application to biochemical reactions in nonequilibrium steady-state conditions, The Journal of chemical physics 133, 07B604 (2010).
  * Foglino  _et al._ [2019] M. Foglino, E. Locatelli, C. Brackley, D. Michieletto, C. Likos, and D. Marenduzzo, Non-equilibrium effects of molecular motors on polymers, Soft matter 15, 5995 (2019).
  * Lévy [1940] P. Lévy, On some homogeneous stochastic processes, Compositio mathematica 7, 283 (1940).
  * Sánchez  _et al._ [2013] R. Sánchez, B. Sothmann, A. N. Jordan, and M. Büttiker, Correlations of heat and charge currents in quantum-dot thermoelectric engines, New Journal of Physics 15, 125001 (2013).
  * Schmiedl and Seifert [2008] T. Schmiedl and U. Seifert, Efficiency of molecular motors at maximum power, EPL (Europhysics Letters) 83, 30005 (2008).
  * Pigolotti  _et al._ [2017] S. Pigolotti, I. Neri, É. Roldán, and F. Jülicher, Generic properties of stochastic entropy production, Physical review letters 119, 140604 (2017).
  * Barato  _et al._ [2018] A. C. Barato, É. Roldán, I. A. Martínez, and S. Pigolotti, Arcsine laws in stochastic thermodynamics, Physical review letters 121, 090601 (2018).
  * Saigo [2018] H. Saigo, Quantum probability for dressed photons: The arcsine law in nanophotonics, in  _Progress in Nanophotonics 5_ (Springer, 2018) pp. 79–106.
  * Dale and Workman [1980] C. Dale and R. Workman, The arc sine law and the treasury bill futures market, Financial analysts journal 36, 71 (1980).
  * Baek  _et al._ [2008] S. K. Baek, T. Y. Kim, and B. J. Kim, Testing a priority-based queue model with linux command histories, Physica A: Statistical Mechanics and its Applications 387, 3660 (2008).
  * Clauset  _et al._ [2015] A. Clauset, M. Kogan, and S. Redner, Safe leads and lead changes in competitive team sports, Physical Review E 91, 062815 (2015).
  * Bel and Barkai [2006] G. Bel and E. Barkai, Random walk to a nonergodic equilibrium concept, Physical Review E 73, 016125 (2006).
  * Sadhu  _et al._ [2018] T. Sadhu, M. Delorme, and K. J. Wiese, Generalized arcsine laws for fractional brownian motion, Physical review letters 120, 040603 (2018).
  * Singh and Kundu [2019] P. Singh and A. Kundu, Generalised ‘arcsine’laws for run-and-tumble particle in one dimension, Journal of Statistical Mechanics: Theory and Experiment 2019, 083205 (2019).
  * Akimoto  _et al._ [2020] T. Akimoto, T. Sera, K. Yamato, and K. Yano, Aging arcsine law in brownian motion and its generalization, Physical Review E 102, 032103 (2020).
  * Reimann [2002] P. Reimann, Brownian motors: noisy transport far from equilibrium, Physics reports 361, 57 (2002).
  * Blickle and Bechinger [2012] V. Blickle and C. Bechinger, Realization of a micrometre-sized stochastic heat engine, Nature Physics 8, 143 (2012).
  * Toyabe  _et al._ [2010] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Experimental demonstration of information-to-energy conversion and validation of the generalized jarzynski equality, Nature physics 6, 988 (2010).
  * Mestres  _et al._ [2014] P. Mestres, I. A. Martinez, A. Ortiz-Ambriz, R. A. Rica, and E. Roldan, Realization of nonequilibrium thermodynamic processes using external colored noise, Physical Review E 90, 032116 (2014).
  * Saha and Marathe [2019] A. Saha and R. Marathe, Stochastic work extraction in a colloidal heat engine in the presence of colored noise, Journal of Statistical Mechanics: Theory and Experiment 2019, 094012 (2019).
  * Krishnamurthy  _et al._ [2016] S. Krishnamurthy, S. Ghosh, D. Chatterji, R. Ganapathy, and A. Sood, A micrometre-sized heat engine operating between bacterial reservoirs, Nature Physics 12, 1134 (2016).
  * Albay  _et al._ [2021] J. A. Albay, Z.-Y. Zhou, C.-H. Chang, and Y. Jun, Shift a laser beam back and forth to exchange heat and work in thermodynamics, Scientific reports 11, 4394 (2021).
  * Manikandan  _et al._ [2021] S. K. Manikandan, S. Ghosh, A. Kundu, B. Das, V. Agrawal, D. Mitra, A. Banerjee, and S. Krishnamurthy, Quantitative analysis of non-equilibrium systems from short-time experimental data (2021), arXiv:2102.11374 [cond-mat.soft] .
  * Ghosh  _et al._ [2020] S. Ghosh, A. D. Ranjan, S. Das, R. Sen, B. Roy, S. Roy, and A. Banerjee, Directed self-assembly driven mesoscale lithography using laser-induced and manipulated microbubbles: Complex architectures and diverse applications, Nano letters 21, 10 (2020).
  * Peterman  _et al._ [2003] E. J. Peterman, F. Gittes, and C. F. Schmidt, Laser-induced heating in optical traps, Biophysical journal 84, 1308 (2003).
  * Durrett [2019] R. Durrett,  _Probability: theory and examples_ , Vol. 49 (Cambridge university press, 2019).
  * Martinez  _et al._ [2013] I. A. Martinez, E. Roldán, J. M. Parrondo, and D. Petrov, Effective heating to several thousand kelvins of an optically trapped sphere in a liquid, Physical Review E 87, 032159 (2013).

##  I Supplementary Information

###  I.1 Calculating effective temperature

When white noise is added to a system, the effective temperature can be easily
calculated in by adding the variance of the noise to the data; however this
assumption breaks down when the added noise has a finite temporal correlation,
given by, $\lambda(t)\lambda(t^{\prime})=\sigma^{2}T(t-t^{\prime})$. The
effective temperature is related by
$\dfrac{2k_{B}T}{\gamma}+\dfrac{\sigma^{2}}{\gamma^{2}}\leq\dfrac{2k_{B}T_{\text{eff}}}{\gamma}$
– however, the equality holds only for the case of delta correlated noise. We
can calculate the effective temperatures using the relation
$\text{Var}(x^{\prime})=\frac{k_{B}T^{\prime}}{k}$ such that we get
$\dfrac{2\text{Var}(x)}{\tau_{d}}+\dfrac{\text{Var}(\lambda)}{\tau_{0}}=\dfrac{2k_{B}T_{\text{add}}}{\tau_{d}}$,
where $T_{\text{add}}$ is the additive temperature. The effective temperature
$(T_{\text{eff}})$ is calculated directly from the variance of the position
fluctuation. In Fig. 1 we show that the effective temperature as well as the
additive temperatures increase with increasing amplitude of the added
Ornstein-Uhlenbeck noise. In all the cases we note that
$T_{\text{eff}}>T_{\text{add}}>T$; $T_{\text{eff}}$ will be equal to
$T_{\text{add}}$ iff, the added noise is temporally delta correlated [27].

Figure 1: The additive temperatures $(T_{\text{add}})$ and the effective
temperatures $(T_{\text{eff}})$ for the non equilibrium steady state (NESS)
system plotted along with the standard errors for varying amplitude of the
input OU noise. Figure 2: Logarithmic plot of the scaled error of the arcsine
fit, which decreases on increasing the number of points considered to obtain
the statistics.

###  I.2 Accuracy

To obtain the arcsine statistics, we must integrate much more data points than
those in a single decay cycle of the temporally correlated modulation.
Naturally, using a larger number of data points improve the accuracy of the
statistics. We define error by the absolute deviation of the obtained
statistic from the arcsine distribution, divided by the number of points used.
Fig. 2 shows in the logarithmic plot that the error decreases in a power law
with the increasing number of data points, where the exponent $\approx-0.9$

Figure 3: Characteristics of the added Ornstein-Unhelbeck noise. (a) The
autocorrelation of the added noise which we fit with the theoretical
exponential decay function with amplitude $A\tau_{0}$ and decay constant
$\tau_{0}$. (b) The logarithmic plot of the power spectral density of the
noise in the Fourier domain. The amplitude falls off as $1/f^{\alpha}$, where
$\alpha$ is an exponent denoting the characteristics of the noise. Figure 4:
The Mean squared displacement (MSD) plotted for short time scales. (a) and (b)
show the MSD of the $W_{d}$ and $W$ variables, obtained from the micro-engine
in absence of the bubble. (c) and (d) show the MSD of $W_{d}$ and $W$ in
presence of the bubble’s flow field. The MSD increases linearly with time as
expected for a Wiener variable. Figure 5: Change in the amplitude of the input
OU noise (a) and (b) show all the three Arcsine laws for $T_{+}$, $T_{last}$
and $T_{max}$ in red, green and blue respectively for the case when input
amplitude is $A_{1}=0.1\times(0.6\times 10^{-6})^{2}\ m^{2}/s$. In (c) the
distribution of $x$ and $\lambda$ is plotted. We repeat the same for
(d),(e),(f) for the case, $A_{2}=0.2\times(0.6\times 10^{-6})^{2}\ m^{2}/s$
Figure 6: Change in the distance of the probe from the microbubble: (a) and
(b) the three Arcsine laws for $T_{+}$, $T_{last}$ and $T_{max}$ in red, green
and blue where distance from the microbubble is $5\mu m$ and $15\mu m$,
respectively. (c) The variation in the dissipated work for the two distances
$5\mu m$ and $15\mu m$ are in red and blue respectively.

###  I.3 Ornstein Uhlenbeck noise.

Fig. 3 shows that the OU process generates a colored noise that has an
exponentially decaying correlation and a power spectrum that fits a power law.
We can use the autocorrelation property of the added noise to calibrate its
amplitude. In Fig. 3(a) we plot the autocorrelation for $\lambda(t)$ where
$\tau_{0}=25\ ms$. In Fig. 3(b), we plot the power spectrum of the noise in a
logarithmic plot. By fitting the data points with $1/f^{\alpha}$, we get
$\alpha\approx 1.8$, which confirms that it is colored noise.

###  I.4 Mean Square displacement

The main criterion for any variable to follow a Wiener process (and therefore
the Lévy arcsine statistics) is that its variance should increase linearly
with time – which implies that the MSD of the variable should increase
linearly with increasing time lags. To demonstrate this, we plot the MSD of
the variable in Fig.. 4, which is required to characterize the work done by
the external modulation as well as the work dissipated for both the cases
where the microbubble is present and absent. By fitting the MSD with a power-
law, we observe the exponent is approximately equal to 1 for all the cases,
which is a necessary condition to be followed.

###  I.5 Variation of amplitude of the added noise

All our results are demonstrated for the case where the amplitude of the added
noise is given by $A=0.3\times(0.6\times 10^{-6})^{2}\ m^{2}/s$. Changing the
amplitude of the input noise changes the heat dissipated by the system and the
magnitude of the work done associated with the process. With increasing
amplitude, the production rate of entropy also increases linearly, so does the
system’s effective temperature. However, the arcsine nature of the concerned
variables, $W$ and $W_{d}$ remains intact. In Fig. 5(a) & (b) and (d) & (e),
we plot the arsine statistics for two different amplitudes
$A_{1}=0.1\times(0.6\times 10^{-6})^{2}\ m^{2}/s$ and
$A_{2}=0.2\times(0.6\times 10^{-6})^{2}\ m^{2}/s$. In Fig. 5(c) & (f), we show
that the effective spread of $x-\lambda$ is in tens of nanometers, which
increases with increasing amplitude of the added noise. It has been
demonstrated in Ref [23] that increase in the entropy production rate or the
work dissipated is not infinite but has an upper-bound.

###  I.6 Change of the distance from microbubble

The ambient flow field changes with changing distance from the microbubble –
as a result, the effective velocity of our trapped colloidal particle also
gets modified. The flow field skews the symmetry of the position distribution
of the trapped particle. In Ref [23], we observe how the flow-field and the
current increase when we trap the particle at the close proximity of the
bubble. However, in Fig. 6 we show that the arcsine laws for $W_{d}$ do not
get modified when the distance changes because the properties of the Wiener
process are still applicable. In our case, we have reported calculations for
the center of the probe $10\mu m$ away from the surface in the main
manuscript. In Fig. 6(a) and (b), we show results for distances of $5\mu m$
and $15\mu m$, respectively. In the presence of the microbubble, the motion of
the probe gets spatially constrained. Further away from the bubble, the
dissipated work (see Fig. 6(c)), as well as entropy production, increases to
an upper bound, which is predicted by thermodynamic uncertainty relations.