Enhanced Photo-excitation and Angular-Momentum Imprint of Gray Excitons in WSe2 Monolayers by Spin–Orbit-Coupled Vector Vortex Beams

A light beam can be spatially structured in the complex amplitude to possess orbital angular momentum (OAM), which introduces an extra degree of freedom alongside the intrinsic spin angular momentum (SAM) associated with circular polarization. Furthermore, superimposing two such twisted light (TL) beams with distinct SAM and OAM produces a vector vortex beam (VVB) in nonseparable states where not only complex amplitude but also polarization is spatially structured and entangled with each other. In addition to the nonseparability, the SAM and OAM in a VVB are intrinsically coupled by the optical spin–orbit interaction and constitute the profound spin–orbit physics in photonics. In this work, we present a comprehensive theoretical investigation, implemented on the first-principles base, of the intriguing light–matter interaction between VVBs and WSe2 monolayers (WSe2-MLs), one of the best-known and promising two-dimensional (2D) materials in optoelectronics dictated by excitons, encompassing bright exciton (BX) as well as various dark excitons (DXs). One of the key findings of our study is that a substantial enhancement of the photoexcitation of gray excitons (GXs), a type of spin-forbidden DX, in a WSe2-ML can be achieved through the utilization of a 3D-structured TL with the optical spin–orbit interaction. Moreover, we show that a spin–orbit-coupled VVB surprisingly allows for the imprinting of the carried optical information onto GXs in 2D materials, which is robust against the decoherence mechanisms in the materials. This suggests a promising method for deciphering the transferred angular momentum from structured light to excitons.


I. INTRODUCTION
A spatially structured light beam with a cylindrically twisted phase front introduces quantized orbital angular momenta (OAM), L z = , which serves as a novel degree of freedom for light alongside spin angular momentum (SAM), S z = ± , associated with polarization of light.[1][2][3][4] Such a cylindrically structured beam, also referred to as twisted light (TL) or optical vortex (OV), characterized by an unbounded quantum number has been demonstrated advantageous in a variety of advanced photonic and quantum applications, ranging from optical tweezers, [5,6] optical trapping, [7,8] highresolution optical microscope, [9][10][11] optical communication, [12,13] to high dimensional quantum information.[14][15][16][17] Besides, the co-existence of SAM and OAM in a structured light beam gives rise to intriguing optical spin-orbit-coupled phenomena, [18][19][20] including photonic spin Hall effect, [21][22][23][24] spin-based plasmonics, [25] photonic wheel, [26,27] optical transverse spin, [28] and longitudinal field of light.[29][30][31][32][33] Furthermore, a structured light beam can be tailored by the controlled superposition of TLs with distinct SAM and OAM, forming a vector vortex beam (VVB) in nonseparable states, where not only the complex amplitude but also the polarization of light are spatially struc-tured and entangled with each other.[34][35][36][37][38] The exceptional characteristics of VVBs as light sources have been demonstrated to enable advanced photonics applications, [39] particle acceleration, [40,41] vector beam multiplexing communication, [42,43] high dimensional quantum entanglement, [44,45] and vector vortex quantum steering.[46] The non-separability of the SAM and OAM, further coupled by the optical spin-orbit interaction (SOI), in a VVB embodies the profound spin-orbit physics of optics and naturally affect its interaction with matters, which, however, remain largely unexplored so far.Following the rapid advancement in the TL-based optics, [47] it is timely crucial to investigate the physics of the interaction between structured lights and the emergent nano-materials suited for the prospective TL-based optoelectronics.
Atomically thin transition-metal dichalcogenide monolayer (TMD-ML) is one of the most promising optoelectronic 2D materials with superior light-matter interactions that are dictated by excitons.[48][49][50][51][52] In TMD-MLs, excitons are strongly bound by the enhanced Coulomb interaction, leading to the atypical band dispersion and exciton fine structures associated with the diverse degrees of freedom inherent in excitons, including spin and valley properties as well as the center-of-mass motion of exciton.[53][54][55][56][57] The remarkable exciton fine structure of a TMD-ML enables the unambiguous spectral resolution of diverse exciton complexes, such as the BX and various DX states, [58] each possessing distinct degrees of freedom.In darkish W-based TMD-MLs, e.g.WSe 2 , [59] the intravalley repulsive exchange energy combined with the conduction band splitting shifts the dipole-allowed bright exciton states upwards by tens of meV and leave the spinforbidden dark exciton doublet as the excitonic ground states.Furthermore, the lowest doublet of dark excitons undergoes valley-mixing, resulting from weak intervalley exchange interaction, and exhibits a slight energy splitting, yielding a completely dark exciton and a slightly optically active state known as a gray exciton (GX).[60][61][62] Notably, GXs have recently garnered significant attention due to their possession of the both advantages from bright excitons (BXs) as well as dark excitons (DXs), i.e. long lifetime and brightness.[56,63] These characteristics are highly desirable for future dark-exciton-based quantum technologies and devices.[64,65] Nevertheless, optically accessing the GX states remains a nontrivial task and usually needs the additional aid of external fields or post-processed structures of samples, such as in-plane magnetic fields, [59,62,66] plasmonic fields, [67] or photonic crystals in close proximity.[56,68] The fascinating attributes of twisted light have recently stimulated a few pioneering investigations concerning their interactions with bright excitons in 2D systems.[69][70][71][72][73][74][75][76][77][78] However, the exploration of the interplay between twisted light and GXs remains an area that is still largely unexplored.
In this study, we present a comprehensive theoretical investigation based on first principles, focusing on the interaction between spin-orbit-coupled VVBs and exciton states in a WSe 2 monolayer, including both BX and GX.We reveal that structured lights can serve as an exceptional light source enabling optically enhance the photo-excitation of GXs in a WSe 2 -ML through the coupling of the longitudinal field component associated with the SOI.Furthermore, we show that a spin-orbit-coupled VVB enables the imprinting of optical information onto the optical transitions of GXs in the 2D materials.
In Section II, we begin by reviewing the electromagnetic theory of structured light and introducing the formalism for twisted lights in the Laguerre-Gaussian (LG) modes, and present the generalized theory for the lightmatter interaction between generic structured light and excitons in 2D materials.
In Section III, we present the calculated results and engage in a thorough physical discussion.We calculate the momentum-dependent optical matrix elements of the twisted-light-excited exciton states in a WSe 2 -ML.Specifically, we focus on the photo-excitation of GXs in a WSe 2 -ML by spin-orbit-coupled VVBs that are formed by the controlled superposition of two twisted lights with distinct angular momenta.Finally, in Section IV, we conclude our work.

II. THEORY
A. Theory of Laguerre-Gaussian

Vector potentials in the real space
To describe a twisted light with SAM (σ ) and OAM ( ), we begin with the ansatz of the vector potential in the Lorentz gauge, A σ pq0,L (r) = εσ u p (r)e iq0z , that satisfies the paraxial Helmholtz equation, [79] where r = (x, y, z) = (ρ, z) is the 3D coordinate position, εσ = 1 √ 2 (x + iσŷ) is the transverse polarization labelled by the optical helicity σ = ±1, (p) is the index of the azimuthal (radial) mode of light, and q 0 is the wave number of light propagating along the z-direction.[80] Next, the solved vector potential of a TL in the LG mode from the vectorial Helmholtz equation in the paraxial approximation is transformed to that in the Coulomb gauge, which is normally adopted by the standard theory of light-matter interaction, [81,82] via the transformation equation, which is established by equalizing the electric field expressed in terms of the vector potential in the Coulomb gauge and that in the Lorentz gauge as shown by Refs.[83,84].For brevity, hereafter we shall remove the superscript C, q 0 , and p and preserve only the indices of SAM (σ) and OAM ( ) for the vector potential of a twisted LG beam in the fundamental radial mode (p = 0), which will be under the main discussion of this work.In the Rayleigh range where the amplitude of light remains nearly constant along the z coordinate, the vector potential of a circularly polarized LG TL in the Coulomb gauge is solved as A σ, (r) = e iq0z A σ, (ρ) = e iq0z [ε σ A (ρ) + ẑA σ, z (ρ)], being a 3D-structured light with the both transverse and longitudinal components, [84] which are, respectively, given by where ρ = x 2 + y 2 , φ = tan −1 (y/x) is the azimuthal angle, A 0 is the amplitude of light, = 0, ±1, ±2, ±3, ... (p = 0, 1, 2, ...) is the index of azimuthal (radial) mode, is the associated Laguerre polynomial, and w 0 is the beam waist of light beam.Throughout this work, we consider the beam waist, w 0 = 1.5µm, and the wavelength, λ 0 = 2π/q 0 = FIG.1.
(a) Schematics of a vector vortex beam (VVB) formed by the superposition of two twisted lights with distinct angular momenta as a light source for the photo-generation of excitons in a WSe2-ML.BS is the abbreviation of beam splitter.In a VVB, not only the complex amplitude but also the polarization of light are structured spatially.The left square inset shows the spatially varied polarization of a VVB considered in Fig. 4 532nm, of the twisted light with the wave number q 0 that is resonant to the exciton transition, E X B0 =1.7eV .[85,86] In Eq.( 3), one notes that the strength of the longitudinal field in a TL increases with reducing w 0 and critically depends on the signs of σ and .The product of σ appearing Eq.( 3) manifests the effect of optical SOI in the longitudinal field component.
Remarkably, the longitudinal field, A σ, z (ρ), in Eq.( 3) is imposed by the phase term of total angular momentum (TAM), e i(σ+ )φ , while the transverse field, A (ρ), in Eq.( 2) is structured with the OAM only.As the electric field of a light beam is E = iωA A in the Coulomb gauge, those 3D-structured TLs with longitudinal field components naturally enable the photo-excitation of the exciton states with out-of-plane dipole moments, such as the GX state of a TMD-ML, as shown by Fig. 1(e).

Vector potentials in the momentum space
For the integration with the light-matter interaction based on the exciton band structures of 2D materials, it is necessary to transform the vector potentials into the angular spectrum representation through a 2D Fourier transform.Following Eqs.( 2) and ( 3), the Fourier transforms of the complex transverse component of ) is derived as and the longitudinal one as as detailed in Section SII of Supplemental Material, where q = q/|q| with q = (q , q 0 ), q = (q x , q y ) and φ q = tan −1 (q y /q x ).The complex-valued radial function is F| [77] where represents the Bessel function of the first kind of order | |. [87] In turn, the vector potential as a function of coordinate position in the real space can be expressed as A σ, (ρ) = q A σ, (q )e iq •ρ , via the inverse Fourier transform.[73,77] The appearance of A (q ) in Eq.( 5) accounts for that the longitudinal field in a TL fully inherits the OAM-encoded transverse spatial structures described by Eq.( 4).Notably, the term (ε σ • q) = (ε σ • q )/q appearing in Eq.( 5) manifests itself as the optical SOI that couples the optical spin (ε σ ) and the in-plane momentum component (q ) carried by the longitudinal field.Alternatively, (ε σ • q) = sin θq √ 2 e iσφq with sin θ q ≡ q q 2 +q 2 0 can be expressed in the spherical coordinates, showing that the optical spin σ is fully transferred to the longitudinal field and the strength of optical SOI increases with increasing q .Combining A (q ) and (ε σ • q), the longitudinal field expressed by Eq.( 5) is shown imprinted by (σ + ) ≡ J, which is the total angular momentum of TL in the paraxial regime.[18] Fig. 2(b) and (c) [(d) and (e)] show the squared magnitude, the real part, and the imaginary part of the complex vector potential A (q ) [A σ, z (q )], as functions of q for the polarized TLs in the LG modes with p = 0 and the optical angular momenta, (σ, ) = (1, 1) and (σ, ) = (−1, −1), respectively.Basically, the squared magnitudes of the vector potentials of the TLs carrying finite OAM (| | > 0) in the fundamental radial mode (p = 0) present ring-shaped distributions over the q plane, whose ring sizes increase with increasing .[77] This indicates that the TLs with greater comprise the more components of large q and, according to the momentum-conservation law, likely couple the more exciton states with large in-plane momentum, Q.Moreover, the effects of optical SOI become more important in the TLs with greater .One also notes that the ring size of the q -dependent magnitudes of the longitudinal component A σ, z (q ) 2 is unequal but slightly larger than that of the transverse one, A (q ) 2 .With no effects of SOI, the transverse component of vector potential is decoupled from SAM (see Eq.( 4)) and remains the same for σ = +1 and σ = −1.Indeed, the patterns of Re A =1 q and Im A =1 q of Fig. 2(a.2)-(a.3)are shown dumbbelllike to reflect the OAM = 1 carried by the TL.As pointed out previously, the longitudinal field in a TL inherits the total angular momentum, J = σ + , of the light.Thus, as seen in Fig. 2(d.2) and (d.3) [(e.2) and (e.3)], the in-plane patterns of the real and imaginary parts of A σ=±1, =±1 z (q ) are double-dumbbell-like to reflect the TAM, J = σ + = ±2.

B. Exciton fine structures of TMD monolayers: DFT-based studies
For the studies of exciton, we employ the theoretical methodology developed by Ref. [57,88] to solve the Bethe-Salpeter equation (BSE) established in firstprinciples for the exciton fine structure spectra of encapsulated 2D materials.First, we calculate the quasi-particle band structures, nk , and the Bloch wave functions, ψ nk (r), of WSe 2 -MLs by using the first principles Quantum Espresso package [89,90] in the density-functional theory (DFT) with the consideration of SOI. Figure S1 in Supplemental Material shows the calculated quasi-particle band structure of WSe 2 -ML (See Section SI of Supplemental Material for details).In terms of the calculated Bloch states, the exciton states of a 2D material is expressed as where Ω is the area of the 2D material, ĉ † ck ( ĥ † v−k ) is defined as the particle operator creating the electron (hole) of wavevector k (−k) in conduction band c (valence band v), |GS denotes the ground state of the material, Λ SQ (vck) is the amplitude of the electron-hole configuration ĉ † ck+Q ĥ † v−k |GS and corresponds to the solution of the Bethe-Salpeter equation (BSE) for the exciton in momentum space, S is the band index of the exciton state, Q is the center-of-mass momentum of exciton and Ω denotes the area of the 2D material.By using the Wannier90 package, [91] we transform the calculated Bloch states into a complete set of maximally localized Wannier function (MLWF) basis, in which the Kohn-Sham Hamiltonian in DFT is reformulated as a tight-binding matrix with small dimension.In the Wannier tight binding scheme, we establish and are able to efficiently solve the BSE with the Coulomb kernel consisting of the screened e-h direct interaction and unscreened exchange interaction to calculate the momentum-space wave function, Λ SQ (vck), and the energy, E X SQ , of the exciton state, |S, Q (See Section SI of Supplemental Material for more details ).With the enhanced e-h Coulomb interaction in a 2D material, the low-lying exciton fine structure spectrum of a TMD-ML is featured with significant fine structure splitting, spectrally resolving the BX and various DX states.For a WSe 2 -ML, the DX states as the exciton ground states are spectrally significantly lower than the bright ones by ∼ 48.8meV, as shown in Fig. 1(b) and (c).[92,93] Carefully examining the lowest DX states, one notes a small splitting between the DX doublet resulting.Combined with the spin-orbit interaction of quasi-particle, the inter-valley exchange interaction splits the lowest DX doublet and turns one of them, referred to as gray exciton (GX), to be slightly bright.[61] With finite Q, the inter-valley e-h exchange interaction splits the valley exciton BX bands into a quasi-linear upper, |B+, Q , and parabolic lower band |B−, Q .[94,95] At the light cone edge where |Q| = q 0 ≡ Q c , the valley splitting between the upper and lower BX bands is merely 1-2 meV, much smaller than the energy separation of BX and DX/GX states.The transition dipole moment of an exciton state is evaluated by ), where d vk,ck ≡ e ψ vk | r |ψ ck = e im0( ck − vk ) ψ vk |p|ψ ck is the dipole moment of single-electron transition evaluated by using the theoretical method described in Section SI of Supplemental Material, [88,[95][96][97] with p the operator of linear momentum and m 0 (|e|) the mass (the magnitude of the charge) of free electron.coupled by the spin-orbital interaction (SOI).Because of the SOI, the polarization field is not purely transverse but possesses also the longitudinal field component.In the Coulomb gauge, the transverse (gray arrows) and longitudinal (red arrows) fields are parallel to the transverse and longitudinal components of the vector potential, A (q ) and A σ, z (q ), respectively.The gray circular arrow represents the projection of circular polarization onto the x-y plane.(b.1)-(b.3):The distributions of the squared magnitude, real part, and imaginary part of the transverse component, A =1 (q ), of the vector potential for the TL with (σ, ) = (1, 1) over the q -plane.The dumbbell-like pattern of Re(A =1 (q )) and Im(A =1 (q )) reflects the optical OAM, = 1, carried by the TL.The length of the white scale bar is, q = 0.1q0, for reference.z (q ) and A σ=−1, =−1 z (q ), of the vector potentials of the same TLs.Differing from the transverse components, the distribution patterns of Re(A ±1,±1 z (q )) and Im(A ±1,±1 z (q )) over the the q -plane are double-dumbbell-like, resulting from the TAM, J = σ + = ±2 carried by the longitudinal components.
(out-of-plane) oriented.Neglecting the very slight variation of dipole moments with respect to Q, the transition dipoles of the upper and lower BX, and the GX states are described by D is the magnitude of dipole moment of BX (GX).In addition to the strong exciton-photon interaction, the fine structure spectrum of WSe 2 -ML consisting of various exciton states with distinctly oriented dipoles serves as an excellent test bed to explore the distinct field components in the 3D-structured lights.In turn, twisted lights carrying controlled SAM and OAM enable us selectively access and distinguish a variety of exciton states of 2D materials.

C. Exciton-light interaction
In the time-dependent perturbation theory, the Hamiltonian of light-matter interaction with respect to a light described by the vector potential A(r) is given by H LM I ≈ |e| 2m0 A(r) • p in the weak field and rotating wave approximations.[98] Accordingly, the optical matrix element of an exciton state, |S, Q , is derived as M σ, SQ = 1 [77] which measures the amplitude of the optical transi-tion of the exciton state, |S, Q , induced by the incident TL carrying the angular momenta σ and , In terms of the optical matrix element, the Fermi's golden rule formulates the rate of incoherently photo-exciting the finitemomentum exciton state, |S, Q , by using a TL with (σ, ), as , where ρ( ω) is the density of states of light in the range of angular frequency between ω and ω + dω.In the electric dipole approximation, one derives where Q) is the Fourier transform of the vector potential of structured light with the transverse and longitudinal components as given by Eqs.( 4) and (5), and E g = c1K − v1K is the energy gap of the material, where c 1 (v 1 ) is the lowest conduction (topmost valence) band.The optical matrix elements of Eq.( 6) for BX and GX states under the excitation of a TL in the LG mode with (σ, ) are derived in the cylindrical coordinate and explicitly shown as below, where the exponential term e i(σ+ )φ Q accounts for the TAM transfer from a TL to a GX and the term sin arises from the SOI, which makes a normally incident TL forbidden to excite a GX with Q = 0 but enhances the photo-generate of GX states with large Q as increasing .Examining the Q-dependence of the optical matrix element of an exciton allows us to infer its angle-dependent optical properties [63,77,99] thereby inferring the optically transferred TAM in the excited GX state.Since the valley splitting between the lower and upper BX bands is merely of ∼ 1 meV and normally spectrally unresolvable, as seen in Fig. 1(c), [100] the total transition rate of the BX doublet, |B±, Q under the photo-excitation of a TL can be counted by Γ σ, B,Q By contrast, the transition rate of a GX state that is spectrally well apart from the BX states can be evaluated by the optical matrix element of the specific state alone, Γ σ, GQ ∝ | M σ, GQ | 2 .

III. RESULTS AND DISCUSSION
A. Photo-excitation of exciton by a single twisted light Figure 3(a) and (b) shows the contour plots of the optical transition rates, Γ σ, SQ , as functions of Q for the finitemomentum BX and GX states of a WSe 2 -ML incident by polarized TLs with (σ, ) = (1, 1), (1,5) and (1,15).Overall, the Γ σ, SQ for the non-zero = 1, 5, 15 exhibit similar ring-shaped patterns over the Q-plane, with the ring sizes increasing with increasing .This indicates that a TL with greater enables the photo-generation of the exciton states (both BX and GX ones) with larger Q, whose superposition forms a spatially more localized wave packet as previously pointed out by Ref. [77].Analytically, one can show that the a TL with mostly likely excite the finite momentum BX state with Q = q = 2( + 1)/w 0 , where the square of the magnitude of A (Q) is maxima so that | Q=q = 0. Figure 3(c) shows the total transition rates of Γ σ, S ∝ Q Γ σ, SQ , which take into account the all finite-momentum states of BX and GX excited by the TLs with = 0, 1, ...15.Notably, the rate of photo-exciting the GX superposition states, Γ σ, G , using a TL with is shown linearly increasing with increasing , while the rate of photo-exciting the BX ones, Γ σ, B , remain nearly unchanged against .Increasing the OAM of the incident TL from = 1 to = 15, Γ σ, G is enhanced by over one order of magnitude.Theenhanced photo-generation of GX is associated with the term of SOI, (ε σ • q) = q 2(q 2 +q 2 0 ) e iσφq ≈ 1 √ 2 q q0 e iσφq ∝ q , in the longitudinal field of TL as expressed by Eq.( 5).Recall that q ∝ √ + 1.Thus, with increasing of a TL, the in-plane component of momentum, q , carried by the TL increases, and so do the strength of the optical SOI and the magnitude of the longitudinal field, A σ, z (Q), of Eq.( 5).
Despite the phase term of TAM, e i(σ+ )φ Q , encoded in the complex optical matrix elements of BX and GX states as shown in Eqs.( 7) and ( 8), the phase information is not preserved in the squared magnitude of the optical matrix elements.These squared magnitudes, which measure the optical transition rates of the exciton states under incoherence conditions, cannot show the transferred angular momenta to the exciton states.However, we will demonstrate that the combination of TLs with different optical angular momenta, forming so-called vector vortex beams (VVBs), serves as an exceptional light source, which enables the revelation of the transferred optical angular momenta from TLs to the GXs even when incoherence conditions are present.

B. Photo-excition of exciton by using a VVB
Generally, the superposition of two TLs, denoted by |σ and |σ , respectively, can be expressed by in terms of the azimuthal angles α and the polar angle β in the representation of higher order Poincaré sphere.[35,36,101,102] As presented in Fig. 4. In Fig. 4, the north and south poles represent the TL basis in the single LG modes, which are |σ and |σ , respectively.The superposition state, |σ , σ ; α, β , with β = 0, π is represented by points located on the sphere surface in between the poles.In fact, the superposition state of structured light, |σ , σ ; α, β , with distinct SAM (σ = −σ) and OAM ( = ) forms a VVB, [34,37,103] which is structured in both polarization and amplitudes over the 3D space, [104] and is prospective in the frontier photonic applications, [105] e.g.laser material processes, [106,107], optical encoding/decoding in communication, [108] and microscopy.[109] The vector potential of such a VVB is given by A σ, ,−σ, (q ; α, β) = cos (β/2) A σ, (q ) + e iα sin (β/2) A −σ, (q ) and leads to the corresponding complex optical matrix element for an exciton in the state Following Eq.( 4), one can show that the squared magnitude of the transverse component of the vector potential in angular spectrum representation is A σ, ,−σ, (q ; α, β) 2 is independent of the azimuthal angle, φ q .Hence, A 1 1 ,−1 1 (q ; α, β) 2 exhibits the isotropic contours over the q -plane, as shown in GQ for the TL-excited finite-momentum GX states.All of the contour plots follow the same colormap on the leftmost side.For reference, the length of the horizontal bar in white color represents the magnitude of 0.1Qc.(c) The total transition rate of all TL-excited finite-momentum BX (green) and GX states (blue) as a function of of TL.Note that the transition rate of a GX linearly increases with increasing , while that of a BX remains nearly unchanged against .
Fig. 4(a), and does not preserve the optical information of carried by the TL basis that is encoded in the phase term, e i φq , of Eq.( 4).In Fig. 4, the dark circular panels present the spatially varying polarizations of the VVBs over the q -plane.[35,105,110] By contrast, the squared magnitude of the longitudinal component of the vector potential of the same VVB is derived as |A σ, ,−σ, z ] and shown φ q -dependent, as long as β = 0, π and ∆J ≡ J − J = (σ + ) − (σ + ) = 0.
Further, from Eq.( 7) one can derive the total transition rate of the spectrally unresolvable BX doublet with Q under the excitation of a VVB, Γ σ, ,−σ, B,Q As expected, the transition rate of BX doublet Γ σ, ,−σ, BQ excited by a VVB is shown φ Q -irrelevant and exhibit an isotropic distribution over the Q plane, as shown by Fig. 5(a) for the VVB with (σ, ) = (1, 1) and (σ , ) = (−1, −1).For a GX, the transition rate, Γ σ, ,σ , where The first two terms in Eq.( 10) can be viewed as the sum of the squared magnitude of the optical matrix element of GX under the excitation of the two noninterfered TL-basis of the VVB, which depends only on the magnitude of Q and remains invariant with varying φ Q .The last cross-term arises from the coherent interference between the two TL basis and explicitly shows the φ Q -dependence, which is importantly associated with the difference of TAM between the TL basis, ∆J.As (α − ∆| |) π/2 is simply a constant phase offset, the cross-term ∝ cos ∆J φ Q + α − ∆| | π 2 , is varied sinusoidally with the winding number, i.e. n = |∆J|, by rotating φ Q . 2 , for the VVB states at the poles and equator of higher-order Poincaré sphere over the q -plane.For reference, the length of the horizontal bar in white color represents the magnitude of q = 0.1Qc.The dark circular panels present the spatially varying polarizations of the VVBs over the q -plane.The pure states of TL basis at the poles (β = 0, π) possess circular polarization.By contrast, the maximal superposition states of VVB at the equatorial points (β = π/2), are linearly polarized along the direction depending on q .(b) Density plots of the squared magnitudes of the longitudinal components of the vector potentials, A 1,1,−1,−1 z (q ; α, β) 2 , of the same VVBs as presented in (a).While the pattern of A 1,1,−1,−1 (q ; α, β) 2 always remain isotropic as varying the geometric angles of the superposition states of VVB, those of A 1,1,−1,−1 z (q ; α, β = π/2) 2 of the equatorial superposition states exhibit anisotropic patterns, possessing the rotational symmetry associated with the finite = 0 carried by the TL basis of the VVB.
Therefore, by utilizing non-separable VVBs as light sources, one can decode the angular momentum difference (∆J) within the VVB by analyzing the angledependent optical spectrum that is correlated with the Q-dependence of Γ σ, ,σ G,Q , [63]   , for the GX states.
As expected from the preceding analysis, the donutlike distribution of the Γ +1,1,−1,−1 B,Q (α, β) over the Qspace for the BX doublet under the excitation of the superposition TLs remains invariant against the varied α and β (see SIII).By contrast, the distribution of the Γ +1,1,−1,−1 G,Q (α, β) over the Q-plane for the GX states varies with changing the geometric angles, α and β.In particular, at the equator (β = π/2) where the VVB is the maximal superposition of TLs, the φ Q -varying (α, π/2) for the GX states excited by the VVBs in the higher order modes with = 2, 3, 4 are presented in Fig. S2 of Supplemental Material, confirming the formalism for extracting the transferred angular momentum from the n-fold rotational symmetry of the Q-dependent pattern of the magnitudes of the optical matrix elements of the GXs by VVBs.Note that the angular-momenta are encoded in the n-fold petal-like pattern of the magnitude of transition rate of GX and should be robust against decoherence in materials.

IV. CONCLUSION
In conclusion, we present a comprehensive investigation based on first principles, focusing on the light-matter interaction between structured lights carrying optical angular momenta and tightly bound excitons in 2D materials.We show that the photo-excitation of a specific type of spin-forbidden dark excitons, i.e. gray exciton, is greatly enhanced by the incident twisted lights that carry orbital angular momentum and possess the longitudinal field component associated with the interaction between spin and orbital angular momenta.Moreover, we investigate the superposition of two twisted lights with distinct SAM and OAM, resulting in the formation of a vector vortex beam (VVB) that is spatially engineered in both complex amplitude and polarization as well.Our research demonstrates that a spin-orbit-coupled VVB in a non-separable form surprisingly allows for the imprinting of the carried optical information onto gray excitons in 2D materials, which is robust against the decoherence mechanisms in materials.These studies unveil the indispensable role of gray excitons in twisted-light-based optoelectronics and suggest the utilization of VVB for transferring optical information onto 2D materials.
FIG. 1.(a) Schematics of a vector vortex beam (VVB) formed by the superposition of two twisted lights with distinct angular momenta as a light source for the photo-generation of excitons in a WSe2-ML.BS is the abbreviation of beam splitter.In a VVB, not only the complex amplitude but also the polarization of light are structured spatially.The left square inset shows the spatially varied polarization of a VVB considered in Fig.4.The right rectangular inset presents the vector field of a circularly polarized twisted light propagating along the z-axis located at some in-plane position.(b) The exciton band structure of a WSe2-ML sandwiched by semi-infinite hBN layers calculated by solving the BSE in the Wannier tight-binding scheme established on the first-principles base.(c) The exciton fine structure of the low-lying exciton states, comprising the valley-split bright exciton bands (green circles) and the lowest gray (blue circles) and dark exciton ones (un-filled circles).(d) [(e)] shows the transverse component, D X, SQ [longitudinal component, D X,z SQ ], of the BSE-calculated transition dipole moments of the bright (green lines) and gray exciton (blue lines) states at the edge of the light cone, Qc ≈ q0 ≡ 2π/λ0.D X B0 = |D X B0 | represents the magnitude of the dipole momentum of the bright exciton state at Q = 0.
Figure 1(d) shows the in-plane and out-of-plane projections of the transition dipole of the exciton states in the fine structure of WSe 2 -ML.The transition dipole moments of the BX (GX) states are shown mainly in-plane

FIG. 2 .
FIG. 2. (a)The polarization field (pink arrows) of a TL with the SAM σ and OAM coupled by the spin-orbital interaction (SOI).Because of the SOI, the polarization field is not purely transverse but possesses also the longitudinal field component.In the Coulomb gauge, the transverse (gray arrows) and longitudinal (red arrows) fields are parallel to the transverse and longitudinal components of the vector potential, A (q ) and A σ, z (q ), respectively.The gray circular arrow represents the projection of circular polarization onto the x-y plane.(b.1)-(b.3):The distributions of the squared magnitude, real part, and imaginary part of the transverse component, A =1 (q ), of the vector potential for the TL with (σ, ) = (1, 1) over the q -plane.The dumbbell-like pattern of Re(A =1 (q )) and Im(A =1 (q )) reflects the optical OAM, = 1, carried by the TL.The length of the white scale bar is, q = 0.1q0, for reference.(c.1)-(c.3):Same as (b.1)-(b.3)but for the TL with (σ, ) = (−1, −1).Note that the transverse components of the vector potentials for the TLs with the opposite angular momenta remain the same in the squared magnitudes, as shown by (b.1) and (c.1).(d.1)-(e.3):Same as (b.1)-(c.3)but for the longitudinal components, A σ=1, =1 FIG. 2. (a)The polarization field (pink arrows) of a TL with the SAM σ and OAM coupled by the spin-orbital interaction (SOI).Because of the SOI, the polarization field is not purely transverse but possesses also the longitudinal field component.In the Coulomb gauge, the transverse (gray arrows) and longitudinal (red arrows) fields are parallel to the transverse and longitudinal components of the vector potential, A (q ) and A σ, z (q ), respectively.The gray circular arrow represents the projection of circular polarization onto the x-y plane.(b.1)-(b.3):The distributions of the squared magnitude, real part, and imaginary part of the transverse component, A =1 (q ), of the vector potential for the TL with (σ, ) = (1, 1) over the q -plane.The dumbbell-like pattern of Re(A =1 (q )) and Im(A =1 (q )) reflects the optical OAM, = 1, carried by the TL.The length of the white scale bar is, q = 0.1q0, for reference.(c.1)-(c.3):Same as (b.1)-(b.3)but for the TL with (σ, ) = (−1, −1).Note that the transverse components of the vector potentials for the TLs with the opposite angular momenta remain the same in the squared magnitudes, as shown by (b.1) and (c.1).(d.1)-(e.3):Same as (b.1)-(c.3)but for the longitudinal components, A σ=1, =1

FIG. 3 .
FIG. 3. (a1)-(a3): Density plots of the optical transition rates, Γ σ BQ as functions of Q for the finite momentum BX states of a WSe2-ML under the excitation of polarized TLs with the SAM and OAM, (σ, ) = (1, 1), (1, 5) and (1, 15), respectively.(b1)-(b3): Density plots of Γ σGQ for the TL-excited finite-momentum GX states.All of the contour plots follow the same colormap on the leftmost side.For reference, the length of the horizontal bar in white color represents the magnitude of 0.1Qc.(c) The total transition rate of all TL-excited finite-momentum BX (green) and GX states (blue) as a function of of TL.Note that the transition rate of a GX linearly increases with increasing , while that of a BX remains nearly unchanged against .