# Quantum beats of coherent 1s 2s excitons in two dimensional transition metal
dichalcogenides

Nizar Chaouachi1, Sihem Jaziri1,2
1Faculté des Sciences de Tunis, Laboratoire de Physique de la Matière
Condensée, Département de Physique, Université Tunis el Manar, Campus
Universitaire 2092 Tunis, Tunisia
2 Faculté des Sciences de Bizerte, Laboratoire de Physique des Matériaux
Structure et Propriétés , Université de Carthage,7021 Jarzouna, Tunisia

(August 28, 2024)

###### Abstract

Abstract

Motivated by recent experimental measurement of the intrinsic excitonic wave-
function in 2D Transition-metal dichalcogenides (TMDs) by angle-resolved
photoemission spectroscopy (ARPES), we developed a theoretical study to
resolve some characteristics of these excitons and some of the many open
issues in these systems. The system is assumed to be embedded in an
environment with average dielectric constant $\kappa$, below which
electrostatic interactions in the corresponding TMD layer are screened. We
adopt the long range approximation, which gives the electron-hole interaction
in the Rytova-Keldysh form. Latter allows understanding the role of screening
in TMDs structures. The bound state 1s, 2s… energy eigenvalues for the two-
dimensional are reformulated in momentum space leads to an integral form of
the Wannier equation. The eigenfunctions are then expanded in terms of
spherical harmonics. To evaluate the dynamic of the angle-resolved
photoemission spectrum arising from the dissociation of excitons given their
steady- states 1s, 2s.. expressions, we follow the semi perturbative
theoretical description developed by previous calculations. We discuss the
dielectric environment effect on the dispersive features of the spectrum for
different 1s, 2s,… exciton distributions. Quantum beat signatures in
photoemission intensity demonstrate coherent coupling between 1s and 2s
excitons. the beating contribution due to excitonic coherence is also
discussed.The periodic oscillations arising from coherent superposition
states, quantum beats, enable exploration of novel coherent phenomena.

MoS2, exciton binding energy, tr-ARPES, TMDs, coherence

###### pacs:

Valid PACS appear here

##  I Introduction

The exciton interaction with the electromagnetic field may lead to significant
phenomenas. Moreover the bosonic character of excitons allow us to reveal
Bose-Einstein-Condensate-like phenomena can be described as coherence of
excitons [1][2]. Using femto-second lasers, it is possible to create two
coherent excitons with two different energy levels simultaneously[3]. The
oscillatory behavior of the light intensity emitted by the two-levels excited
system is defined as quantum beats[4]. However only limited observations were
reported For the quantum beats in excitonic complexes in Transition Metal
Dichalcogenides (TMDs) [5][6][7][8][9][10].
Two-dimensional TMDs are materials composed from one layer of transition metal
(M,W$\cdots$) between two layers of chalcogenides (S,Se,Te)
[11][12][13][14][15][16][17]. TMDs family have exhibited many intrinsic
electronic and optical properties like semiconductors with direct band gap in
visible spectrum, metals, superconductors, semi-metals Weyl semi-metals, Mott-
insultors and important spin orbit coupling [16][17][18][19][20][21][22].
All these properties have enlarged the application field of these materials
from ultra-thin transistor, photovoltaic panels, energy stocking to new
domains like spintronics, Valleytronics,
plamonics[16][23][24][25][26][27][28]. The optical response in TMDs semi-
conductors are dominated by excitonic effects. Originating from strong reduced
dielectric screening, two-dimensional transition metal dichalcogenides exhibit
strongly bound excitons with an energy overcomes the room temperature. Thus it
is possible to manipulate these quasi-particles [29][30][31][32][33][34].
Recent studies have shown Rydberg states in monolayer $\mathrm{MoS_{2}}$
[35][36] which make it an ideal platform for studying coherent phenomena.
Motivated by recent experimental measurement of the intrinsic excitonic wave-
function as well as by time and angle-resolved photoemission spectroscopy (tr-
ARPES) [37][38], we develop a theoretical study to resolve some
characteristics of these excitons and some of the many open issues in these
systems.
Time resolved ARPES have provided a plethora of information about non-
equilibrium states because of its controllable environment as well as the high
resolution on energetic, angular and time scales using ultra-short lasers
pulses [39][40]. This technique have demonstrated a success in investigating
strongly correlated systems [41], exotic phases such as Weyl semi-metals [42],
superconductors [43], charge density wave [44] and topological insulators
[45].
Among its advantages is that it maps a large energy spectrum as well as the
momentum. Hence it is possible to observe various excitons including A, B
excitons, dark, spin and momentum forbidden excitons along with excitonic
insulators [45][46][47][48][49]. In this article we first describe excitons in
2D materials and the environment effect on the density of probability and the
energy using Wannier equation. The second section is dedicated to the
theoretical formalism of the time resolved photoemission spectroscopy resulted
from excitons in 2D semi-conductors with direct gap based on Freerick et al.
[50] which provides a description of time resolved photoemission spectroscopy
within non-equilibrium states, and A.F Kemper et al. [51][52] for theoretical
description of photoemission of exciton in bulk materials with indirect gap.
The results of photoemission spectra and the effect of environment for
excitons as well as the quantum beats are discussed and compared to other
results in the third part.

##  II Exciton in Transition Metal Dichakcogenides

We consider a model two-band 2D TMD semiconductor within the effective mass
approximation. Note that we take into account the conduction band minima and
the valence band maxima at the K or K’ valley, which are crucial for the
formation of momentum exciton states.

A monolayer is always embedded between two media. In this case, it is more
convenient to describe the screened interaction by Rytova-Keldysh potential
$V_{RK}(r)=\frac{-e^{2}}{4\pi\varepsilon_{0}\kappa
r_{s}}\\{H_{0}(\frac{r}{r_{s}})-Y_{0}(\frac{r}{r_{s}})\\}$, where $H_{0}(x)$
and $Y_{0}(x)$ are the zero order Sturve and Newmann functions,
r=$\|\vec{r}_{e}-\vec{r}_{h}\|$ is the in plane relative distance between the
electron and hole, $r_{s}=\frac{\varepsilon_{r}d}{2\kappa}$ is the effective
screening length, $\kappa=\frac{\varepsilon_{t}+\varepsilon_{b}}{2}$ is the
average dielectric constant of the two media surrounding the monolayer where
$\varepsilon_{b}$ is the dielectric constant of the bottom medium and
$\varepsilon_{t}$ is the dielectric constant of the top medium, d is the
monolayer width and $\varepsilon_{r}$ is its dielectric constant
[53][54][55][56]

Figure 1: Schematic illustration of field lines between electron and hole in
the monolayer embedded between two media

To obtain access to the wave-functions and eigenvalues of excitons, we solve
the eigenvalue problem of the following Hamiltonian [57][33] :

$H_{X_{Q,k}}=\frac{\hbar^{2}Q^{2}}{2M}+\frac{\hbar^{2}k^{2}}{2\mu}+V_{RK}(r)+E_{g}$
(1)

Here $\vec{Q}=\vec{k}_{e}-\vec{k}_{h}$ is the center of mass momentum,
$\vec{k}=\frac{m_{h}\vec{k}_{e}+m_{e}\vec{k}_{h}}{m_{h}+m_{e}}$ is the
relative momentum , $M=m_{e}+m_{h}$ is the total mass ,
$\mu=\frac{m_{e}m_{h}}{m_{h}+m_{e}}$ is the reduced mass of the exciton and
$E_{g}$ the band gap energy of the material.
$\frac{\hbar^{2}k^{2}}{2\mu}+V_{RK}(r)$ defines the in-plane relative motion
hamiltonian $H_{rel}$. Hence we write the momentum space Wannier equation of
relative motion in the unit of Bohr radius
$a_{X}=\frac{\varepsilon_{r}\hbar^{2}}{\mu e^{2}}$ :

$\displaystyle
q^{2}\Phi_{nm}(\vec{q})-\frac{1}{\pi}\int\frac{\Phi_{nm}(\vec{q}^{\prime})d\vec{q}}{|\vec{q}-\vec{q}^{\prime}|}$
$\displaystyle+\frac{1}{\pi}\int\frac{\Phi_{nm}(\vec{q}^{\prime})d\vec{q}}{|\vec{q}-\vec{q}^{\prime}|+\vec{q}_{s}}=E^{*}_{nm}\Phi_{nm}(\vec{q})$
(2)

$\vec{q}=\vec{k}a_{X}$ is the dimensionless relative wave vector and
$q_{s}=\frac{a_{X}}{r_{s}}$. Here the first two terms representing $H_{hyd}$
which is the effective 2D hydrogenic-like Hamiltonian but we replace
$\varepsilon_{r}$ by $2\kappa$. Therefore, the modified eigenvalues and the
effective Bohr radius are
$E^{hyd}_{nm}=-\frac{\varepsilon_{r}^{2}}{\kappa^{2}}\frac{R_{X}}{(n-1/2)^{2}}$
knowing that $R_{X}=\frac{\mu e^{4}}{2\varepsilon_{r}^{2}\hbar^{2}}$ is the
effective Rydberg and
$a^{*}_{X}(n)=\frac{\kappa}{\varepsilon_{r}}a_{X}(n-1/2)$ where n=1,2,3 …is
the principal quantum number, we notice that the new eigenvalues depend
explicitly on the two dielectric constants $\varepsilon_{t}$ and
$\varepsilon_{b}$ .The corresponding eigenstate are described in cylindrical
coordinates [58] :

$\displaystyle\varphi_{nm}(q,\phi)=(-\mathit{i})^{m}\sqrt{2\pi}(\frac{\frac{2}{n-1/2}}{q^{2}+(\frac{1}{n-1/2})^{2}})^{\frac{3}{2}}$
$\displaystyle\times
P^{|m|}_{n-1}(\frac{q^{2}-(\frac{1}{n-1/2})^{2}}{q^{2}+(\frac{1}{n-1/2})^{2}})e^{-\mathit{i}m\phi}$
(3)

Marking that $P^{|m|}_{\textit{n}-1}(x)$ is the associated Legendre
Polynomial, $\phi$ is the polar angle m= -(n-1)…n-1 is the magnetic number but
for this work we will consider only the states $\lambda=ns$ i.e m=0.
To find the energies of relative exciton motion by the screened potential, we
numerically solve the eigenvalue problem equation (2) by direct
diagonalization over the basis of eigenstates of the 2D hydrogen-like exciton
expressed as [59] :

$\Phi_{\lambda}(q)=\sum_{\lambda^{\prime}}c_{\lambda^{\prime}}\varphi_{\lambda^{\prime}}(q)$
(4)

The coefficients $c_{\lambda}^{\prime}$ are real numbers for ns states
obtained by the diagonalization of the matrix. Accordingly the exciton state
with a generic finite center of mass momentum $\vec{Q}$ is [57][60] :

$\displaystyle\ket{X_{\lambda,K}(\vec{Q})}=$
$\displaystyle\frac{1}{\sqrt{A}}\sum_{\vec{k}}\Phi^{*}_{\lambda}(\vec{k})a^{{\dagger}}_{c}(\vec{K}+\vec{k}+\frac{m_{e}}{M}\vec{Q})$
(5) $\displaystyle\times a_{v}(\vec{K}+\vec{k}-\frac{m_{h}}{M}\vec{Q})\ket{0}$

Where A is the ML area ,
$a^{{\dagger}}_{c}(\vec{K}+\vec{k}+\frac{m_{e}}{M}\vec{Q})/a_{v}(\vec{K}+\vec{k}-\frac{m_{h}}{M}\vec{Q})$
create an electron/hole in the conduction/valence band within the valley K and
$\ket{0}$ is the ground state with completely filled valence band. This state
have the following energy :

$E_{X_{\lambda,\vec{Q}}}=\frac{\hbar^{2}Q^{2}}{2M}+E_{\lambda}+E_{g}$ (6)

noting that $E_{\lambda}$ is the relative motion energy for the $\lambda$
state derived from the solutions of equation (2) and
$\frac{\hbar^{2}Q^{2}}{2M}$ is the exciton center of mass kinetic energy and
$E_{G}$ is the gap energy of the MoS2 ML.

##  III Theoretical formalism of two photons Time and Angle Resolved
PhotoEmission Spectroscopy

To appraise excitonic spectrum we use many body hamiltonians. Foremost we
suppose that all charge carriers have same spin direction [61]. Also the
exciton density is assumed to be low in order to neglect exciton-exciton
interactions [62]. The process is described by two photons where the first
photon correspond to the pump photon with Visible near Infra-red energy.
Accordingly this photon leads to creation of excitons in the K valley
($\frac{4\pi}{3a}\vec{k}_{x}$) where a is the lattice parameter) which will
couple with the electromagnetic field expressed by the following hamiltonian :

$H^{(1)}(t)=H_{X}+E_{0}+H_{vc}(t)$ (7)

$H_{X}=\sum_{\lambda}E_{X_{\lambda}}X^{{\dagger}}_{\lambda,\vec{K}}X_{\lambda,\vec{K}}$
describes the exciton energy levels obtained from equation 6 corresponding to
the state $\lambda$ with zero center of mass
$\ket{X_{\lambda,\vec{K}}}=\frac{1}{\sqrt{A}}\sum_{\vec{k}}\phi^{*}_{\lambda}(\vec{k}+\vec{K})a^{{\dagger}}_{c}(\vec{K}+\vec{k})a_{v}(\vec{K}+\vec{k})\ket{0}$.
$E_{0}$ is the energy of the ground state $\ket{0}$ which portrays the unbound
electrons in the valence band. The excitons with $\vec{Q}=\vec{0}$ can couple
directly with photons and lead to optical absorption. Moreover, the
$H_{vc}(t)$ describes the exciton-photon coupling where we place ourselves in
Coulomb gauge i.e $\vec{\nabla}\cdot\vec{A}=0$, which implies that the
potential vector $\vec{A}$ is only restricted only to his in-plane component.
We consider the dipole approximation since our pump photon is in visible
domain i.e the wave length is larger than the distance between the charge
carriers $r$ ($\lambda_{vis}\gg r$). Thus $H_{vc}(t)$ can be expressed as :

$H_{vc}=-\sum_{\lambda}\vec{\epsilon}_{vc}(t)\vec{p}_{\lambda}(t)$ (8)

Here
$\vec{\epsilon}_{vc}(t)=\vec{\epsilon}_{0}e^{\frac{-(t-t_{c})^{2}}{2\sigma_{c}^{2}}}\cos(\Omega(t-t_{c}))$
is the classical electromagnetic pump pulse centred in $t_{c}$ with Gaussian
profile, $\Omega$ is the frequency in the visible light, $\sigma_{c}$ is the
temporal pulse width. Moreover
$\vec{p}_{\lambda}(t)=\vec{d}_{\lambda}(X_{\lambda}^{{\dagger}}e^{\mathit{i}\frac{E_{X_{\lambda}}}{\hbar}t}+X_{\lambda}e^{-\mathit{i}\frac{E_{X_{\lambda}}}{\hbar}t})$
is the electric dipole operator [63] and $d_{\lambda}$ is the transition
matrix element.
After certain time delay from creating the non-equilibrium state, the probe
photon with energy higher than the material’s work function [39][64] will be
absorbed by the bound electron. the many body Hamiltonian of the photoemitted
electron is expressed as :

$H^{(2)}(t)=H_{f}+H_{cf}(t)$ (9)

where
$H_{f}=\sum_{\vec{k}_{e}}(\frac{\hbar^{2}k_{e}^{2}}{2m_{e}}+W)a_{f}^{{\dagger}}(\vec{k}_{e})a_{f}(\vec{k}_{e})$
is the electron kinetic energy after being ejected from the solid with
momentum $\vec{k}_{e}$, $a_{f}^{{\dagger}}(\vec{k}_{e})/a_{f}(\vec{k}_{e})$ is
the creation/annihilation operator of free electron. $W$ denotes the work
function of the monolayer. Additionally,
$H^{2}_{int}=\sum_{\vec{k}_{e},\vec{k^{\prime}}}\hbar\Omega_{R}^{cf}(t)a_{f}^{{\dagger}}(\vec{k}_{e})a_{c}(\vec{k^{\prime}}+\vec{K})+h.c$
will annihilate an electron from non equilibrium state and create it in the
vacuum.
$\Omega_{R}^{cf}(t)=\frac{d^{cf}\epsilon_{1}}{\hbar}e^{\mathit{i}\omega_{ph}t}e^{-\frac{(t-t_{p})^{2}}{2\sigma_{p}^{2}}}$
is time dependent Rabi frequency with low magnitude of the electromagnetic
field $\epsilon_{1}$ and $d^{cf}$ is the transition matrix element,
$\hbar\omega_{ph}$ is the photon energy in the Xrays-Ultra-violet domain [40],
$\sigma_{p}$ is the probe pulse temporal width, $t_{p}$ is the moment when the
pump pulse is maximal and $d^{cf}$ is the dipole transition element which also
assumed to be momentum independent.

Following the evolution of the system described in the Appendix A and B
bellow, the probability can be expressed in terms of $\hbar\omega$ and
$\vec{k}$ :

$\displaystyle
P_{\vec{k},\hbar\omega}(t_{d})=\sum_{\lambda,\lambda^{\prime}}P^{\lambda,\lambda^{\prime}}_{\vec{k},\hbar\omega}$
(10) $\displaystyle=$
$\displaystyle\sum_{\lambda,\lambda^{\prime}}2\pi\sigma_{p}^{2}\frac{(d^{cf}\epsilon_{1})^{2}}{\hbar^{2}}|\alpha_{\lambda^{\prime}}^{*}(0)||\alpha_{\lambda}(0)|\Phi_{\lambda^{\prime}}(k+K)\Phi_{\lambda}(k+K)$
$\displaystyle\times
e^{-\frac{\sigma_{p}^{2}}{4\hbar^{2}}(E_{X_{\lambda}}-E_{X_{\lambda^{\prime}}})}e^{-\frac{\sigma_{p}^{2}}{\hbar^{2}}(\hbar\omega+\frac{\hbar^{2}(k+K)^{2}}{2m_{h}}-\frac{(E_{X_{\lambda}}+E_{X_{\lambda^{\prime}}})}{2})^{2}}$
$\displaystyle\times
e^{\frac{\mathit{i}}{\hbar}((E_{X_{\lambda}}-E_{X_{\lambda^{\prime}}})(t_{c}-t_{p})}$

The probability is expressed as product of two wave functions, oscillating
term, Gaussian term that represents the energy conservation rule and Gaussian
term with the energy difference. Henceforth it can be represented in a table :

$P_{\lambda,\lambda}({\vec{k},\hbar\omega})$ | … | $P_{\lambda,\lambda^{\prime}}({\vec{k},\hbar\omega})$
---|---|---
$\vdots$ | $\ddots$ | $\vdots$
$P_{\lambda^{\prime},\lambda}({\vec{k},\hbar\omega})$ | … | $P_{\lambda^{\prime},\lambda^{\prime}}({\vec{k},\hbar\omega})$

The diagonal terms i.e $\lambda=\lambda^{\prime}$ are the probabilities of
exciton in the state $\lambda$. We notice that the oscillating term and the
Gaussian term with energy difference are equal to 1. moreover the Gaussian
term that reveal the energy conservation rule implies that the spectrum is
intense around the $\lambda$ exciton energy. Furthermore the non-diagonal
terms constitute the coherent states obtained via the interference of the
excitons polarization. We notice that these terms are assified as an
oscillation with frequency described by the energy levels difference.
Additionnaly the Gaussian term with energy difference presuppose that the
signal intensity derived from coherent states is weaker when the two energy
levels are far from each other. And the energy conservation term implies that
the signal is crucial around the mean value of the two energies.

##  IV Photoemission spectrum of exciton in monolayer $\mathrm{MoS_{2}}$

Our candidate to study theoretical excitonic signals is the TMD monolayer of
MoS2. The electron-hole interaction will be described by Rytova-Keldysh
potential for different $\kappa$ and compared to the unscreened Coulomb
potential. Recently Chunhao Guo et al. [65] have shown that the gap energy is
modified by the dielectric environment. In this work we assume that the gap
energy of the monolayer will remain unchanged under the effect of the media.
The parameters used to determine exciton eigenvalues and eigenfunctions as
well as the substrate dielectric constants (SiO2, hBN) are listed in the table
bellow.

$\mu(m_{e})$ | $\varepsilon_{r}$ | d(Å) | W(eV) | a(Å) |  $E_{g}$(eV) | $\varepsilon_{hBN}$ | $\varepsilon_{SiO_{2}}$
---|---|---|---|---|---|---|---
0.28 [66] | 4.26 [66] | 6.15 [14] | 6.12 [67] | 3.183 [67] | 2.16 [35] | 2.1 [34] | 4.5 [34]

Table 1 : 2H-MoS2 parameters, SiO2 and $hBN$ dielectric constants

The convergence of the numerical diagonalisation is obtained for first 10
states but we only restrict ourselves to 1s and 2s states. For zero magnetic
field Molas et al.[68] have shown that the only the first 5 $\ket{ns}$ states
can be observed. Thus we only consider contribution of these states in
equation (4) to the eigenstates 1s and 2s.
The figure (2) shows the effect of the dielectric environment on energy levels
for excitons in a monolayer of $\mathrm{MoS_{2}}$ :

Figure 2: Exciton binding energy levels for MoS2 placed on SiO2
($\kappa=1.55$), MoS2 placed on hBN ($\kappa=2.75$), hBN encapsulation
($\kappa=4.5$) and for unscreened Coulomb interaction

Substrate | $-E_{1s}(eV)$ | $-E_{2s}(eV)$
---|---|---
Unscreened Coulomb | 0.851 | 0.094
SiO2/MoS2/vac | 0.671 | 0.308
hBN/MoS2/vac | 0.348 | 0.1107
hBN/MoS2/hBN | 0.208 | 0.0519

Table 2 : 1s and 2s exciton binding energy for different dielectric
environment.

The figure (2) shows that the 1s binding energy for different media decreases
from 671 meV for $\kappa=1.55$ to 208 meV for $\kappa=4.5$. The dielectric
environment modify the binding energy of 1s exciton more than 2s exciton. This
was explained by Cudazzo et al which implies that when the distance between
electron and hole increase i.e larger quantum number, Keldysh potential tend
to become unscreened Coulomb potential [69][70]. This was also shown by Molas
et al. [68].
We found the binding energies difference $E_{2s}-E_{1s}$ is smaller for
important screening effect $(\kappa=4.5$) than for low screening effect
($\kappa=1.55$). It goes from $\frac{32R_{X}}{9}(753meV)$ in Coulomb potential
to 363 (156) meV for MoS2 on SiO2 ( hBN encapsulation ) respectively.

(a) (b) (c) (d)

Figure 3: Photoemission spectrum of 1s exciton for unscreened Coulomb
potential (3(a)) 1s excitons in MoS2 placed on SiO2 (3(b)) ,hBN (3(c)),
between two hBN (3(d))

The results are similar to experimental work of M.Puppin et al.[71] Mikel K.L
et al.[37] and Julien Madéo et al.[47], which illustrate that the excitons are
located between the valence and conduction band within the high symmetry
points. The spectrum is intense around the exciton energy and the K valley.
However we notice that when $\kappa$ increases, the excitonic signal
approaches to the conduction band.
For larger $\kappa$ (larger effective Bohr radius) the exciton spectrum
becomes narrower along momentum axis which implies that for monolayer embedded
between 2 hBN layers only holes around K point can contribute in exciton
creation. Unlike when the monolayer placed on SiO2 or hBN where even holes
with momentum $\vec{k}_{h}$ far from K point can contribute to exciton
formation which is demonstrated by the valence band-like curve in the figures
(3(b))(3(c)). We conclude that for SiO2 and hBN substrates, the Arpes signal
reflects clearly the dispersion relation of the valence band.
Moreover, the exciton squared wave-function can be deduced via the integration
of the probability on the energy axis. In the figure (4) we represent the
density of probability of excitons for 1s and 2s states and for different
dielectric environments.

(a)

(b)

Figure 4:  Density of probability for the 1s state (4(a)) and 2s state (4(b))
for MoS2 placed on SiO2 ($\kappa=1.55$), MoS2 placed on hBN ($\kappa=2.75$),
hBN encapsulation ($\kappa=4.5$) and for unscreened Coulomb interaction

Accordingly, figures (4(a)) and (4(b)) represent the densities of the
probability in the relative momentum space. We remark for the same dielectric
environment that the density of probability of 1s state is more spread in the
momentum space than 2s state because the effective Bohr radius is related to
$\kappa$. The larger $\kappa$ gets ( larger radius) the more narrow the
density becomes. Meanwhile the magnitude density of probability for 2s state
is higher. The results are relevant since the larger the radius gets the
broader the wave-function squared becomes in the real space [59].
To observe the photoemission spectrum resulted from 1s exciton, we set
$\hbar\Omega$ to be equal to the 1s exciton energy. the probe energy in vacuum
Ultra-violet domain $\hbar\omega_{ph}=12eV$ and the pump/probe temporal width
$\sigma_{c}=\sigma_{p}=10fs$. Keeping in mind that the smaller the temporal
probe width the larger the spectrum along the energy axis. Hence the figure
(3) illustrates the photoemission spectra derived from equation (10) of the 1s
exciton for different dielectric environments. The momentum integrated
$P_{1s}(\vec{k},\hbar\omega)$ is numerically calculated. It provides
information about the position and the intensity of the signal on the energy
axis.

Figure 5: Momentum integrated spectra of excitons in MoS2 for different
dielectric environments, the width of the curve is controlled by the temporal
probe width $\sigma_{p}$

the peaks are located respectively in 1.489eV for SiO2/MoS2/vac, 1.852eV for
hBN/MoS2/vac and 1.952eV for hBN/MoS2/hBN. Furthermore the intensity is more
important when the monolayer is encapsulated between two hBN layers because of
its high transparency that preserves excitonic signals, also high dielectric
constant and large band gap which prevent exciton quenching [37].
A resonant excitation is convenient to observe 1s excitonic signals. However
since the density of probability of 2s state is larger than 1s state as
discussed in section 2, it is trivial that the signal of 2s exciton will be
more intense. Thus to observe the coherent terms, it is important to choose
the pump energy $\hbar\Omega$ in way that we obtain equal contribution from
both exitons. Therefore we numerically calculate the contrast between the two
excitons signals :

$C=\frac{|P_{1s}^{max}-P_{2s}^{max}|}{P_{1s}^{max}+P_{2s}^{max}}$ (11)

Where $P_{1s}^{max}(P_{2s}^{max})$ correspond to the maximum value of the
excitonic 1s (2s) signals. $\hbar\Omega$ is set in a way that we obtain a
minimum contrast $P_{1s}^{max}\simeq P_{2s}^{max}$. Thus we set
$\hbar\Omega=1.66eV$ for MoS2 placed on SiO2, $\hbar\Omega=1.91eV$ for MoS2
placed on hBN and $\hbar\Omega=2.04eV$ for MoS2 encapsulated between two hBN
layers.
The dependency on the time delay in non-diagonal terms will lead to a quantum
beats. We first investigate excitonic signals for MoS2 placed on hBN.

(a) (b) (c)

Figure 6: Photoemission spectrum of coherent excitons in MoS2 placed on hBN.
Noting that the 1s and 2s excitonic spectra are time invariant. However the
interference spectrum is oscillating in time

The figure (6) the MoS2 is placed on hBN. We capture the photoemission spectra
for different time delays $t_{b}=t_{c}-t_{p}$. We have
$t_{b}=\frac{\pi\hbar}{2|E_{X_{1s}}-E_{X_{2s}}|}$ (6(a)),
$t_{b}=\frac{\pi\hbar}{|E_{X_{1s}}-E_{X_{2s}}|}$(6(b)) and
$t_{b}=\frac{2\pi\hbar}{|E_{X_{1s}}-E_{X_{2s}}|}$ (6(a)). The 2s spectra is
narrower than 1s signal along the momentum axis and close to the conduction
band along the energy axis. Since
$P_{\lambda}(\vec{k},\hbar\omega)\propto|\Phi_{\lambda}(\vec{k})|^{2}$, the
figure (4) showed that 2s exciton denisty is less spread in k space than 1s
exciton density. This suggests also that only holes with momentum near the
valley K can contribute to 2s exciton formation.

The beats signals are formed in the region between the two excitons spectrum.
As time progresses, the polarization interference displays periodic
fluctuations. We notice that the interference represents a maxima for $\Delta
E\Delta t=\pi\hbar$ ($t_{c}-t_{p}=8.8fs$), minima for $\Delta E\Delta
t=2\pi\hbar$ ($t_{c}-t_{p}=17.6fs$) and for $\Delta E\Delta
t=\frac{\pi}{2}\hbar$ ($t_{c}-t_{p}=4.4fs$), $P_{1s,2s}$ and $P_{2s,1s}$
cancel each other.
In order to observe the beat evolution through time, we numerically calculate
the momentum integrated probability for excitons in MoS2 placed on hBN for
different value of probe temporal width $\sigma_{p}$ and we set
$\sigma_{c}=10fs$.

(a) (b) (c)

Figure 7: momentum integrated probability of coherent excitons in MoS2 placed
on hBN for $\sigma_{p}=15fs$ (7(a)), $\sigma_{p}=7fs$ (7(b)) and
$\sigma_{p}=4fs$ (9(a))

The probe temporal width $\sigma_{p}$ is an important parameter to observe the
beats. Therefore for MoS2 on hBN substrate we notice when $\sigma_{p}$ is
large, the beat signals are suppressed. On the other side when the pulse
temporal width is low we obtain an overlap between the two signals arising
from 1s and 2s excitons. Therefore we set the temporal width $\sigma_{p}=7fs$
for MoS2 placed on SiO2 and hBN, $\sigma_{p}=10fs$ for monolayer encapsulated
between two hBN layers and $\sigma_{p}=4fs$ for unscreened Coulomb potential :

(a) (b) (c) (d)

Figure 8: Photoemission spectrum of coherent excitons for unscreened Coulomb
potential (8(a)), for MoS2 placed on SiO2 (8(b)), placed on hBN (8(c)), MoS2
encapsulated between two hBN layers (8(d))

The figure (8) shows that the two excitons are formed simultaneously. The
coherent exciton signals appear in the region between two energy levels.
Therefore the two excitons are behaving like a single quasi particle. Moreover
for hBN/MoS2/hBN (Figure (8(d))) the two excitons signals are closer than for
SiO2/MoS2/vac (Figure (8(b))) and hBN/MoS2/vac (Figure (8(c))) which enhance
the quantum interference hBN encapsulation. As result, it yields to obtain
different oscillation frequency for every substrate. To gain a deeper
understanding about these beats and the periodic oscillation we plot the
momentum integrated probability.

(a) (b) (c) (d)

Figure 9: Momentum integrate of Photoemission spectrum of coherent excitons in
MoS2 placed of SiO2 (9(a)), placed on hBN (9(b)), MoS2 encapsulated between
two hBN layers (9(c)) and unscreened Coulomb potential (9(d))

We notice in the figure (9) that the beat effect is important for hBN
substrate and hBN encapsulation. However we notice fo SiO2 substrate the beat
is slightly observed because of the important energy difference. The
oscillation period $T=\frac{2\pi\hbar}{|E_{X_{1s}}-E_{X_{2s}}|}$ is influenced
by the environment. We notice that when $\kappa$ increases, the period of
oscillation becomes larger which make encapsulating MoS2 between two hBN layer
appropriate to observe this phenomena since $T=26.6fs$. However for for MoS2
placed on hBN we have $T=17.6fs$ which yields to multiple oscillation before
electron recombination. Moreover the oscillation period of 2s-3s excitons is
larger since the two levels are closer. For MoS2 placed on hBN we illustrate
the momentum integrated Photoemission spectra in the figure bellow :

Figure 10: The momentum integrated spectra of coherent 2s-3s exciton in MoS2
placed on hBN. we assume that the 3s state is not affected by the dielectric
environment and the interaction between electron and hole is unscreened
Coulombian. to obtain this coherent state we set $\sigma_{c}=\sigma_{p}=15fs$
and $\hbar\Omega=2.08eV$

In the Figure (10) we notice that the period of oscillation is around
$T=51fs$. Thus we conclude that it is possible to control this oscillation by
choosing the two energy levels.

##  V Conclusion

Ultimately, Unlike the bulk semi-conductors, the dielectric environment have
an impact on the binding energy and the probability of presence of 2D
excitons. As a result it yields to changing the photoemission signals of
excitons for different substrates. Momentum integrated spectra have
demonstrated that the intensity is more important for 2D TMD materials placed
on substrate with large dielectric constant. Furthermore, using the adequate
parameters such as the pump energy will lead to the observation of coherent
excitons. The interference of the excitons polarization display oscillations
where the frequency is determined by the energy difference. This interference
created quantum beats between 1s and 2s excitons. The dielectric environment
plays a crucial role in the beats period. for high dielectric constants
($\kappa=4.5)$ we observe a decrease of the oscillation frequency.

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##  Appendix A  the time dependent wave function after the pump

In the moment $t=t_{0}$ the system is in the ground state $\ket{0}$.
Afterwards the pump acts at the moment $t_{c}-\sigma_{c}$ and ends at
$t_{c}+\sigma_{c}$. Hence the dynamic of the state is described using
Heisenberg description, the state $\ket{\Psi(t)}$ :

$\displaystyle\ket{\Psi(t)}=U_{X}(t,t_{0})\ket{0}$
$\displaystyle=e^{-\frac{\mathit{i}}{\hbar}\int_{t_{0}}^{t}(H_{X}+E_{0}+H_{vc}(\tau))d\tau}\ket{0}$
$\displaystyle=\prod_{\lambda}e^{-\frac{\mathit{i}}{\hbar}\int_{t_{0}}^{t}(H_{X_{\lambda}}+E_{0})d\tau}D_{\lambda}(\alpha_{\lambda})\ket{0}$
(12)

$D_{\lambda}(\alpha_{\lambda})=e^{\alpha_{\lambda}X^{{\dagger}}_{\lambda}-\alpha^{*}_{\lambda}X_{\lambda}}$
is analogue to the the displacement operator established by R.J Glaubert to
describe coherent systems [72][73]. Furthermore
$\alpha_{\lambda}=\frac{\mathit{i}}{\hbar}\int_{t_{c}-\sigma_{c}}^{t_{c}+\sigma_{c}}\epsilon_{vc}(\tau)d_{\lambda}e^{\mathit{i}\frac{E_{X_{\lambda}}}{\hbar}\tau}d\tau$
is the complex number but since the pump pulse is zero far from
[$t_{c}-\sigma_{c},t_{c}+\sigma_{c}$] we can extend the integral to
]$-\infty,+\infty$[. We pose $\tau^{\prime}=\tau-t_{c}$ in
$\vec{\epsilon}_{vc}(\tau)$ expression and we neglect therm with
$e^{-\frac{\sigma_{c}^{2}}{2\hbar^{2}}(E_{X_{\lambda}}+\hbar\Omega)^{2}}$
because it.s very small compared to the term with
$e^{\frac{-\sigma_{c}^{2}}{2\hbar^{2}}(E_{X_{\lambda}}-\hbar\Omega)^{2}}$ :

$\alpha_{\lambda}(0)\simeq\mathit{i}\sqrt{\pi}\frac{\sigma_{c}}{\hbar}\epsilon_{0}d_{\lambda}e^{-\frac{\sigma_{c}^{2}}{2\hbar^{2}}(E_{X_{\lambda}}-\hbar\Omega)^{2}}e^{\frac{\mathit{i}}{\hbar}E_{X_{\lambda}}t_{c}}$
(13)

The complex number $\alpha_{\lambda}(0)$ depends on the transition matrix
element, the electromagnetic field, a phase factor
$e^{\frac{\mathit{i}}{\hbar}E_{X_{\lambda}}t_{c}}$ as well as the probe energy
and the $\lambda$ exciton energy difference. Thus $|\alpha_{\lambda}(0)|$ tend
to 0 if we choose $\hbar\Omega$ far from the exciton energy $E_{X_{\lambda}}$.
The action of the displacement operator on the ground state $\ket{0}$ is
creating a coherent state $\ket{\alpha_{\lambda}}$ with eigenvalues
$\alpha_{\lambda}(t)=\alpha_{\lambda}(0)e^{-\mathit{i}\frac{E_{X_{\lambda}}}{\hbar}t}$.
This allows us to write the time dependent state of the system as [74] :

$\ket{\Psi(t)}=\prod_{\lambda}e^{-\frac{\mathit{i}}{\hbar}(E_{0}+H_{X_{\lambda}})(t-t_{0})}e^{-\frac{|\alpha_{\lambda}(0)|^{2}}{2}}e^{\alpha_{\lambda}(t)X^{{\dagger}}_{\lambda}}\ket{0}$
(14)

##  Appendix B The time dependent wave-function after the probe

the probe pulse will take a place at $t_{p}-\sigma_{p}$ and ends at
$t_{p}+\sigma_{p}$. Thus the evolution of the system after the probe is given
by :

$\displaystyle\ket{\Psi^{F}(t)}=U_{p}(t,t_{0})\ket{\Psi(t)}$
$\displaystyle=e^{-\frac{\mathit{i}}{\hbar}\int_{t_{0}}^{t}H_{f}d\tau}e^{-\frac{\mathit{i}}{\hbar}\int_{t_{0}}^{t}H_{cf}(\tau)d\tau}\ket{\Psi(t)}$

Supposing that the probe pulse magnitude is low. Hence we treat $H_{cf}(\tau)$
as perturbation which will allow us to linearise $U_{p}(t,t_{0})$ to obtain
[75] :

$\displaystyle\ket{\Psi^{F}(t)}\simeq
e^{-\frac{\mathit{i}}{\hbar}H^{f}(t-t_{0})}$
$\displaystyle\times(1-\frac{\mathit{i}}{\hbar}\int_{t_{0}}^{t}U_{X}(t,\tau)H_{cf}(\tau)U_{X}(\tau,t_{0})d\tau)\ket{0}$

We note that $a_{f}(\vec{k}_{e})\ket{\Psi(t)}=0$ and the momentum conservation
rule implies that $\vec{k}_{e}-\vec{k^{\prime}}-\vec{K}=\vec{0}$ i.e the
electron undergoes direct transition from conduction band to vacuum. Thus the
first order term have the following expression. :

$\displaystyle
U_{p}^{(1)}(t,t_{0})=-\frac{\mathit{i}}{\hbar}\int_{t_{0}}^{t}\tilde{U}(t,\tau_{1})\sum_{\vec{k^{\prime}}}\hbar\Omega_{R}^{cf}(\tau_{1})$
$\displaystyle\times
a_{f}^{{\dagger}}(\vec{k^{\prime}}+\vec{K})a_{c}(\vec{k^{\prime}}+\vec{K})\tilde{U}(\tau_{1},t_{0})d\tau_{1}$
(17)

With
$\tilde{U}(t,\tau_{1})=e^{-\frac{\mathit{i}}{\hbar}H^{f}(t-\tau_{1}})U_{X}(t,\tau_{1})$.
First step is to evaluate $a_{c}(\vec{k^{\prime}})\tilde{U}(\tau_{1},t_{0})$
using the following commutation relation [A,f(B)]=[A,B]$\frac{df(B)}{dB}$. the
momentum conservation rule presuppose that
$\vec{k^{\prime}}+\vec{K}=\vec{K}+\vec{k}$. As consequence we obtain :

$\displaystyle a_{c}(\vec{k^{\prime}})\tilde{U}(\tau_{1},t_{0})$
$\displaystyle=$
$\displaystyle(\sum_{\lambda}\alpha_{\lambda}(\tau_{1})\Phi_{\lambda}(\vec{K}+\vec{k}))a_{v}(\vec{K}+\vec{k})$
(18) $\displaystyle\times
e^{-\frac{\mathit{i}}{\hbar}(\frac{\hbar^{2}(\vec{K}+\vec{k})^{2}}{2m_{e}}+W+H_{X}+E_{0})(\tau_{1}-t_{0})}$
$\displaystyle\times\prod_{\lambda}e^{-\frac{|\alpha_{\lambda}|^{2}}{2}}e^{\alpha_{\lambda}(t)X^{{\dagger}}_{\lambda}}$

The probability to detect a photoemitted electron in the state $\ket{f}$
resulted from excitonic states is :

$\displaystyle
P(t_{d})=\left|\left|\bra{f}\ket{\Psi^{F}(t)}\right|\right|^{2}$
$\displaystyle=\frac{(d^{cf}\epsilon_{1})^{2}}{\hbar^{2}}\int_{t_{0}}^{t}d\tau_{2}\int_{t_{0}}^{t}d\tau_{1}e^{-\frac{(\tau_{2}-t_{p})^{2}}{2\sigma_{p}^{2}}}e^{-\frac{(\tau_{1}-t_{p})^{2}}{2\sigma_{p}^{2}}}$
$\displaystyle\times e^{\frac{\mathit{i}}{\hbar}(\hbar\omega-
E_{v}(\vec{K}+\vec{k})-\sum_{\lambda}E_{X_{\lambda}})(\tau_{2}-\tau_{1})}$
$\displaystyle\times(\sum_{\lambda^{\prime}}\alpha^{*}_{\lambda^{\prime}}(\tau_{2})\Phi^{*}_{\lambda^{\prime}}(\vec{K}+\vec{k}))(\sum_{\lambda}\alpha_{\lambda}(\tau_{1})\Phi_{\lambda}(\vec{K}+\vec{k}))$

with $\hbar\omega=\frac{\hbar^{2}k^{2}}{2m_{e}}+W-\hbar\omega_{ph}$ is the
energy difference of the ejected electron. Bearing in mind that when the pump
take place and the electron is photoemitted leaving a whole in the solid band,
the energy of the system becomes $E_{0}-E_{v}(\vec{K}+\vec{k})$ where
$E_{v}(\vec{K}+\vec{k})=-\frac{\hbar^{2}k^{2}}{2m_{h}}$ is the energy of the
electron in the valence band. We expand the integrals of $\tau_{1}$ and
$\tau_{2}$ from [$t_{0},t]$ to ]$-\infty,+\infty$[.