Center for Theoretical Physics of Complex Systems, IBS School University of
Science and Technology Doctor of Philosophy August 2023

# Light-matter interaction in 2D materials in weak and strong-coupling regimes

Dogyun Ko

###### Abstract

This thesis studies light-matter interactions in strong and weak coupling
regimes. In the first part, we study the formation and propagation of exciton-
polariton condensates in different microcavities in the strong coupling
regime. Exciton-polaritons are composite quasiparticles created as a result of
the strong coupling between microcavity photons and quantum well excitons.

In the first part of the thesis, we take a system of exciton-polaritons in a
Kagome lattice and show that an initially localized condensate propagates in a
specific direction in space in the presence of anisotropy in the lattice, and
the initially localized condensate experiences revivals. We also study the
formation of exciton-polariton condensates in two different lowest energy
states at an exciton-polariton microcavity and the transition from the higher
energy state to the ground state under pulsed and continuous wave excitation
conditions by using various pump profiles.

In the second part of the thesis, we study the valley selection rules for the
optical transitions from impurity states to the conduction band in two-
dimensional Dirac materials, taking a monolayer of MoS2 as an example, we
focus on the weak light-matter coupling regime. We find the spectrum of the
light absorption coefficients and calculate the photon-drag electric current
density due to the impurity-band transitions.

###### Acknowledgements.

I’m deeply grateful to my supervisor Prof. Ivan Savenko for giving me the
excellent opportunity to perform doctoral research in his group and for his
support and expert advice. I would also like to thank my colleague Dr. Meng
Sun for the helpful discussions for the entire projects we worked on together.
I also thank Prof. Vadim Kovalev, Prof. Yuri Rubo, Prof. Alexei Andreanov, Dr.
Anton Parafilo, and Dr. Ihor Vakulchyk for valuable discussions. I am also
grateful to my collaborators at KAIST, in particular, Dr. Daegwang Choi, Prof.
Hyoungsoon Choi, and Prof. Yong-Hoon Cho for their fruitful cooperation with
our team.

###### Contents

  1. 1 Introduction
    1. 1.1 Excitons in semiconductors
      1. 1.1.1 Excitons in bulk semiconductors
      2. 1.1.2 Excitons in quantum wells
      3. 1.1.3 Excitons in 2D transition metal dichalcogenides
        1. 1.1.3.1 Transition metal dichalcogenides
        2. 1.1.3.2 Screening effect
        3. 1.1.3.3 Valley excitons
    2. 1.2 Microcavity photons
      1. 1.2.1 Semiconductor microcavities
      2. 1.2.2 Dispersion of cavity photons
    3. 1.3 Exciton-polaritons
      1. 1.3.1 Dispersion of polariton states
      2. 1.3.2 Polariton condensation
        1. 1.3.2.1 The basics of conventional Bose-Einstein condensation
        2. 1.3.2.2 Bose-Einstein quasi condensation of exciton-polaritons
        3. 1.3.2.3 Excitaion of polariton condensates by resonant and nonresonant pumping
  2. 2 Partial revivals of localized condensates in distorted lattices
    1. 2.1 Introduction
    2. 2.2 Tight-binding model on kagome lattice
    3. 2.3 The formation and propagation of polaritons in the lattice
    4. 2.4 Distorted lattice
    5. 2.5 Summary of the chapter
  3. 3 Observation of a quantized vortex vanishment in exciton-polariton system
    1. 3.1 Introduction
    2. 3.2 Formation of vortex states
    3. 3.3 Pulsed-excitation regime
    4. 3.4 Continuous-wave excitation regime
    5. 3.5 Summary
  4. 4 Valleytronics of impurity states in two-dimensional Dirac materials
    1. 4.1 Introduction
    2. 4.2 Hamiltonian and eigenstate
    3. 4.3 Electric current density
    4. 4.4 Symmetry analysis of the electric current density
    5. 4.5 Light absorption coefficient
    6. 4.6 Summary
  5. 5 Summary
  6. 6 Appendix
    1. 6.1 Dynamics of polariton compact localized states in kagome x-squeezed strip
    2. 6.2 Detailed calculation of $j_{0x}$ and $j_{1x}$
      1. 6.2.1 Derivation of $j_{0x}$
      2. 6.2.2 Derivation of $j_{1x}$

###### List of Figures

  1. 1.1 Band structure of GaAs
  2. 1.2 Structure of GaAs quantum well with AlGaAs barriers
  3. 1.3 Band structure of MoS2
  4. 1.4 Screening effects in monolayer TMDs
  5. 1.5 Band structures of a monolayer MoS2
  6. 1.6 Polariton dispersions and Hopfield coefficients with various values of detuning
  7. 1.7 Resonant and nonresonant excitation
  8. 2.1 Tight-binding description for kagome chains
  9. 2.2 System schematic
  10. 2.3 Evolution of a polariton condensates in a kagome strip
  11. 3.1 System schematic
  12. 3.2 Pump and condensates profiles.
  13. 3.3 Energy of vortex and ground states with various beam diameters
  14. 3.4 Evolution of polariton vortex with different pump diameter
  15. 3.5 Number of particles in condensate and reservoirs
  16. 3.6 Polariton condensate under continuous wave exciation
  17. 4.1 System schematic
  18. 4.2 Current density $j_{0x}$
  19. 4.3 Current density $j_{1x}$
  20. 4.4 Absorbance spectrum
  21. 6.1 Kagome $x$-squeezed strip

##  Chapter 1 Introduction

###  1.1 Excitons in semiconductors

####  1.1.1 Excitons in bulk semiconductors

Light absorption in semiconductors results in the creation of a free electron
in a conduction band and a hole in a valence band. A quasiparticle called
excitons can be formed by the Coloumb interaction between a conduction-band
electron and a valence-band hole, resulting in a bound state similar to a
hydrogen atom. The energy spectrum of a hydrogen atom is a sum of the kinetic
energy and binding energy of a hydrogen atom [1],

$E_{H}=\frac{\hbar^{2}k^{2}}{2(m_{0}+m_{p})}-\frac{\hbar^{2}}{2m_{0}a^{2}_{0}n^{2}},$
(1.1)

where $n$ is the principal quantum number, $m_{0}$ is the mass of a free
electron and $m_{p}$ is the mass of a proton.
$a_{0}=4\pi\hbar^{2}\epsilon_{0}/(e^{2}m_{0}$) is the Bohr radius for a
hydrogen atom. $\epsilon_{0}$ is the vacuum permittivity. The energy levels
for the hydrogen atom can be used to define the binding energy for an exciton
by replacing $a_{0}$ into
$a_{ex}=4\pi\hbar^{2}\epsilon_{0}\epsilon_{r}/e^{2}m_{ex}$ , which is the Bohr
radius for an exciton with the relative permittivity $\epsilon_{r}$ and the
effective mass $m_{ex}=\frac{m_{e}m_{h}}{m_{e}+m_{h}}$, and $m_{e}(m_{h})$ is
the mass of an electron(a hole) in a conduction(valence) band and
$\epsilon_{0}$ is the dielectric constant of the semiconductor. Such a
replacement gives the energy of the $n^{th}$ exciton as

$E_{ex}=\frac{\hbar^{2}k^{2}}{2(m_{e}+m_{h})}-\frac{\hbar^{2}}{2m_{ex}a_{ex}^{2}n^{2}},$
(1.2)

The substitution of the Coloumb attraction implies that the binding energy of
the exciton is weaker than the binding energy of a hydrogen atom because the
effective mass is significantly reduced and the permittivity for the
semiconductor is larger than the permittivity in Vacuum. For example, in a
GaAs semiconductor, the relative permittivity is the value of
$\epsilon_{r}=13.1$ and the effective mass is $m_{ex}=0.0591$, resulting in
the binding energy
$\frac{m_{ex}}{m_{0}}\frac{13.6\text{eV}}{\epsilon_{r}^{2}}=4$ meV which is
several order of magnitude smaller than the binding energy of the hydrogen
atom $13.6$ eV. Due to the weaker binding energy of the excitons than the
hydrogen atom, the Bohr radius of the exciton becomes much larger than the
hydrogen Bohr radius. Based on the value of the Bohr radius, excitons can be
categorized into two types, Frenkel excitons, and Wannier excitons. For the
Frenkel excitons, the Bohr radius is less than the distance between the
lattice sites, so that the electron and the hole are located in the same
lattice point. Due to the large Bohr radius for the Wannier excitons, the
electron, and hole are separated by multiple unit cells.

####  1.1.2 Excitons in quantum wells

A quantum well(QW) is a thin layer of semiconductor, where the movement of
particles is restricted in the direction perpendicular to the surface of the
layer, whereas the particles freely move in the other directions. A QW usually
represents a layer of a small band gap semiconductor embedded in between two
other layers made of materials with higher band gaps, thus acting as barriers.
These structures are usually fabricated with metal-organic chemical vapor
deposition [2] or molecular beam epitaxy [3]. A typical example of a quantum
well is the gallium arsenide(GaAs) quantum well sandwiched between the
aluminum gallium arsenide(AlGaAs). GaAs is a direct band gap semiconductor in
which the minimum of the conduction band lies at the same point in the k-space
as the maximum of the valence band. The electronic structure of GaAs at $300$K
is pictured in Fig.(1.1). The conduction band is separated from the valence
band by $E_{g}=1.519$eV at the temperature T$=0$K and $1.424$eV at T$=300$K
for GaAs. The valence band is split into the heavy hole band, the light hole
band, and the split-off hole band, originating from the mixture of p-symmetry
atomic orbitals. Fig.(1.2) shows the band structure for GaAs quantum well with
AlGaAs barriers at $0$K. The thickness of the quantum well determines the
confinement of the electron and hole wave functions between the barriers. Due
to the confinement effect in GaAs quantum well, the bands for the light and
heavy holes are split at $k=0$ as compared to the bulk GaAs semiconductor.
Quantum confinement also affects the binding energy of excitons. The exciton
binding energy in the quantum well case is increased by a factor of four
compared to the case of bulk excitons [4].

Figure 1.1:  Band structure of GaAs. The size of the direct band gap of GaAs
at room temperature is 1.42eV. The valence band is split into subbands by
light holes, heavy holes, and a split-off band.  Figure 1.2:  Energy band
diagram of GaAs quantum well embedded between AlGaAs. The band gaps for GaAs
and AlGaAs at $0$K are 1.52eV and 1.95eV, respectively. The light hole(LH),
heavy(HH), and the ground state of the conduction band(CB) are represented by
their wave functions.

####  1.1.3 Excitons in 2D transition metal dichalcogenides

#####  1.1.3.1 Transition metal dichalcogenides

Transition metal dichalcogenides (TMDs) are a family of 2D materials with the
form MX2 where M corresponds to the transition metal elements (e.g. Mo, W) and
X denotes the chalcogen elements (e.g. S, Se, Te). For the monolayer of TMD,
such as MoS2, the metal element molybdenum is located at the center of the
trigonal prismatic geometry coordinated by six chalcogen atoms of sulfurs with
strong covalent bonds. Multiple layers can be attached by weak van der Waals
interaction. The structure of MoS2 is depicted in Fig.(1.3). The band
structure of MoS2 monolayer displays a direct band gap located at the K points
in reciprocal space, whereas bulk MoS2 exhibits the maximum of the valence
band located at the $\Gamma$ point and the minimum of the conduction band is
located at a point between $\Gamma$, and $K$ points in the Brillouin zone,
which indicates that bulk MoS2 has an indirect band gap [5].

Figure 1.3: (a). The bulk unit cell of MoS2. (b). Schematic of MoS2 monolayer
structure. Electronic band structures of the (c) bulk, (d) quadrilayer, (e)
bilayer, and (f) monolayer MoS2. The indirect band gap increases with the
decrease in the number of layers, but the energy for direct excitonic
transition rarely changes with the thickness of layers. This picture is taken
from [5]

.

#####  1.1.3.2 Screening effect

In bulk semiconductors, excitons can be described by the Bohr model of the
hydrogen atom as negatively and positively charged particles bound to each
other. However, to describe the energy of excitons in monolayer TMDs, the
model for the hydrogen atom is not suitable anymore due to the change of the
dielectric environment, resulting in the dielectric screening effects, which
is depicted in Fig.1.4(a). An alternative model accounting for the
inhomogeneous dielectric environment is introduced to demonstrate the exciton
energies in the thin layers. Another model can be built from the equation
derived by Keldysh [6] to describe the screened Coulomb potential between two
point charges in a thin two-dimensional dielectric layer,

$V(r)=-\frac{\pi
e^{2}}{2r_{0}}\left[H_{0}(\frac{r}{r_{0}})-Y_{0}(\frac{r}{r_{0}})\right],$
(1.3)

where $e$ is the elementary charge,
$r_{0}=d\epsilon/(\epsilon_{a}+\epsilon_{b})$ is the effective screening
length, $d$ is the thickness of the thin layer, $\epsilon$ is the dielectric
constant for the thin layer, $\epsilon_{a}$ and $\epsilon_{b}$ are the
dielectric constants for the surrounding medium and $H_{0}$ and $Y_{0}$ are
the Struve and Neumann functions, respectively. Fig.1.4.(b) shows the
experimental results for the exciton energies in WS2, which agree greatly with
theoretical calculations based on the Keldysh model. The ground state of the
excitons shows the largest deviation compared to the hydrogen model for the
Wannier excitons, and for the 2S state, it still shows a significant
disagreement. In contrast, the higher excited states for the excitons fit the
tendency of the 2D hydrogen model. This can be understood by the change in the
spatial separation between an electron and a hole consisting of an exciton in
a monolayer. As the quantum number of the exciton state increases, the exciton
binding energy decreases, which leads to an increase in the distance between
the electron and hole. This results in a larger fraction of the electric field
leaking out of the layer, where the screening effect is strongly suppressed.
Therefore, the hydrogen atom model can describe the $n=3-5$ excitons where the
screening effect is highly suppressed, whereas the $n=1,2$ excitons exhibit
non-hydrogenic behavior.

Figure 1.4:  (a). Schematic representation of the exciton dielectric screening
in the bulk and monolayer TMDs. (b). The energy of the states of the excitons
in WS2 monolayer as a function of the quantum number n. These pictures are
adopted from [7].

#####  1.1.3.3 Valley excitons

The spin-orbit coupling in MoS2 originating from d orbitals split the valence
and conduction bands on subbands. The valence band splits extensively, whereas
the conduction band remains almost spin degenerate. The splitting gives rise
to two types of excitonic transition: _A_ exciton(ground state exciton) and
_B_ exciton(higher spin-orbit split state). The energy of _A_ exciton is 100
to 200 meV lower than _B_ exciton energy, corresponding to the splitting of
the valence band due to strong orbital coupling in MoS2.

Another peculiar property of monolayer TMDs is the appearance of the valley
degrees of freedom [8]. The additional degree of freedom results from the
broken inversion symmetry, leading to the nonequivalence of the two valleys at
K and K’ points of the two-dimensional hexagonal Brillouin zone as depicted in
Fig.1.5.(a). The broken inversion symmetry and the strong spin-orbital
coupling lead to the valley-dependent optical selection rules, as shown in
Fig.1.5.(b). A right-handed polarized light can selectively excite the
excitons in K valley. Conversely, a left-handed polarized light can excite the
excitons in K’ valley by satisfying the conservation of the angular momentum.

Figure 1.5:  (a). Schematic band structure of MoS2 monolayer at K and -K
points (b). Valley optical selection rules. The interband transitions at K(-K)
points are exclusively coupled with photons with optical helicity
$\sigma^{+}(\sigma^{-})$. Solid(Dahsed) lines denotes These pictures are
adopted from [9].

###  1.2 Microcavity photons

####  1.2.1 Semiconductor microcavities

In a semiconductor microcavity, the confinement of photons is attained by an
optical resonator constructed by two opposing mirrors. The microcavity
consists of two distributed Bragg reflectors (DBRs) separated by the cavity
layers. DBRs represent a one-dimensional structure of an alternating sequence
of layers of semiconductors or dielectrics with different refractive indices
$n_{i}$. The thickness $d_{i}$ of the layers is chosen by the relation
$n_{i}d_{i}=\lambda/4$, such that they fulfill the Bragg condition resulting
in the constructive interference of the waves reflected from the multiple
interfaces. In addition, the distance between the two Bragg mirrors should be
proportional to an integer number of half-wavelength of light in the medium.
By fulfilling the conditions, photons can be trapped in the cavity between the
mirrors.

####  1.2.2 Dispersion of cavity photons

The dispersion of photons in the planar microcavity is different as compared
to the photon dispersion in free space due to confinement. In the case that
photons are trapped in the z-direction, the dispersion relation can be written
as

$E_{ca}=\frac{\hbar c}{n_{ca}}|\mathbf{k_{ca}}|=\frac{\hbar
c}{n_{ca}}\sqrt{k_{z}^{2}+k_{\parallel}^{2}},$ (1.4)

where $\hbar$ is the reduced Planck constant, $c$ is the speed of light in the
medium with reflective index $n_{ca}$, $k_{z}$ is the wave vector of light
perpendicular to the cavity, and $k_{\parallel}^{2}=k_{x}^{2}+k_{y}^{2}$ is
the in-plane components of the wave vector $\mathbf{k}_{ca}$. The component of
the wave vector $k_{z}$, which is perpendicular to the cavity, is
significantly larger than the in-plane wave vector $k_{\parallel}$, and the
energy dispersion of cavity photon can be approximated with the Talyor
expansion as

$E_{ca}\approx\frac{\hbar
c}{n_{ca}}k_{\perp}(1+\frac{k_{\parallel}^{2}}{2k_{\perp}^{2}})=E_{ca}(k_{\parallel}=0)+\frac{\hbar^{2}k_{\parallel}^{2}}{2m_{ca}},$
(1.5)

where $E_{ca}(k_{\perp}=0)=\hbar k_{\perp}c/n_{ca}$ and $m_{ca}=\hbar
k_{\perp}n_{ca}/c$ is the effective mass of cavity photons .

###  1.3 Exciton-polaritons

Exciton-polaritons (polaritons) are quasiparticles created due to the strong
coupling between cavity photons and quantum well excitons. Since polaritons
are composite particles, polaritons inherit properties from their excitonic
and photonic components, such as low effective mass and strong nonlinearity.
As interacting bosons with low effective mass, polaritons are promising for
investigating quantum collective phenomena, such as non-equilibrium Bose-
Einstein condensation [10], and the formation of quantum vortices at room
temperatures [11].

####  1.3.1 Dispersion of polariton states

The formation of polaritons can be described by the model of two coupled
oscillators. The Hamiltonian for the coupling reads

$H=\sum_{k_{\parallel}}[E_{exc}(k_{\parallel})\rho^{\dagger}_{k_{\parallel}}\rho_{k_{\parallel}}+E_{cav}(k_{\parallel})\phi^{\dagger}_{k_{\parallel}}\phi_{k_{\parallel}}+\hbar\Omega(\rho^{\dagger}_{k_{\parallel}}\phi_{k_{\parallel}}+\phi^{\dagger}_{k_{\parallel}}\rho_{k_{\parallel}})],$
(1.6)

where $E_{exc}$ is the energy of an exciton in a quantum well, and $E_{cav}$
is the energy of a single cavity photon. The operators
$\rho^{\dagger}_{k_{\parallel}}$ and $\phi^{\dagger}_{k_{\parallel}}$ are
creation operators for excitons and cavity photons with the in-plane wave
vector $k_{\parallel}$, and $\rho_{k_{\parallel}}$ and $\phi_{k_{\parallel}}$
are the annihilation operators of the exciton and cavity photon, and
$\hbar\Omega$ describes the interaction strength. The Hamiltonian can be also
written in the matrix form,

$H=\begin{pmatrix}E_{cav}&\hbar\Omega\\\ \hbar\Omega&E_{exc}\end{pmatrix}$
(1.7)

Two eigenvalues can be found by diagonalization of this matrix, yielding

$E_{UP}(k_{\parallel})=\frac{1}{2}(E_{cav}(k_{\parallel})+E_{exc}(k_{\parallel}))+\frac{1}{2}\sqrt{4(\hbar\Omega)^{2}+(E_{cav}(k_{\parallel})-E_{exc}(k_{\parallel}))^{2}},$
(1.8)
$E_{LP}(k_{\parallel})=\frac{1}{2}(E_{cav}(k_{\parallel})+E_{exc}(k_{\parallel}))-\frac{1}{2}\sqrt{4(\hbar\Omega)^{2}+(E_{cav}(k_{\parallel})-E_{exc}(k_{\parallel}))^{2}}.$
(1.9)

Here, $E_{UP}$ is the upper polariton state energy, which has higher energy
than the lower polariton state energy $E_{LP}$. Both upper and lower polariton
branches can be characterized by their photonic and excitonic fractions,
$\absolutevalue{C(k_{\parallel})}^{2}$ and
$\absolutevalue{X(k_{\parallel})}^{2}$, which are known as the Hopfield
coefficients [12] and
$\absolutevalue{C(k_{\parallel})}^{2}+\absolutevalue{X(k_{\parallel})}^{2}=1$.
These coefficients obey

$\absolutevalue{C(k_{\parallel})}^{2}=\frac{1}{2}(1-\frac{\Delta(k_{\parallel})}{\sqrt{\Delta^{2}(k_{\parallel})+(2\hbar\Omega)^{2}}})$
(1.10)
$\absolutevalue{X(k_{\parallel})}^{2}=\frac{1}{2}(1+\frac{\Delta(k_{\parallel})}{\sqrt{\Delta^{2}(k_{\parallel})+(2\hbar\Omega)^{2}}}),$
(1.11)

where $\Delta(k_{\parallel})$ is the energy difference between cavity photons
and excitons. Fig.(1.6) shows the polariton dispersions and the Hopfield
coefficients with the different values of detuning.

Due to their photonic properties, polaritons can rapidly propagate. Moreover,
they can be optically excited and have short lifetime of the order of 10ps
compared to the lifetime of excitons, which is on the order of 100ps up to a
few ns. Photons can exchange their energy with optical phonons through the
conservation of momentum and energy. In contrast, photons can not interact
with acoustic phonons because the conservation laws are not satisfied.
However, due to the excitonic component, polaritons can exchange their energy
with acoustic phonons, resulting in the relaxation process in the lower
polariton branch. Thanks to the polariton-polariton interaction, polaritons
can accumulate at $k=0$ forming Bose-Einstein condensation.

Figure 1.6:  (a-c). Energies of the upper polariton, lower polariton, bare
cavity photon, and bare exciton as a function of in-plane wave vector
$k_{\parallel}$ with different values of detuning -10 meV, 0 meV, and +10 meV,
respectively. (d-e). The photonic and excitonic fractions of the Hopfield
coefficients correspond to the dispersions.

####  1.3.2 Polariton condensation

#####  1.3.2.1 The basics of conventional Bose-Einstein condensation

Bose-Einstein condensate(BEC) is a phenomenon characterized by the macroscopic
occupation of the ground state in thermal equilibrium. Einstein extended the
theory developed by Bose who studied the statistics of photons [13]. Thus
Einstein describes a non-interacting gas of massive atoms in an equilibrium
state and trapped in a box. He concluded that the lowest energy state is
occupied by a massive number of particles below some critical temperatures
[14]. However, the interaction between particles and peculiar excitation
spectra are essential to understand BEC. Bogoliubov developed a theoretical
formulation for a weakly interacting Bose gas and predicted the spectrum
behaves as phonon in low momentum region. [15, 16] The first experimental
observation of BEC was performed in the gas of rubidium atoms cooled down to
$170$nK by the group of Carl Wieman and Eric Cornell [17], which was followed
by the observation of the BEC in other atomic gases: Lithium [18] and Sodium
[19].

#####  1.3.2.2 Bose-Einstein quasi condensation of exciton-polaritons

Imamoğlu first suggested the theoretical description of the quasi-condensation
of polaritons while studying the coherence of polariton Bose-Einstein
condensation [20]. The first attempts to observe the polariton condensation
were demonstrated with a GaAs microcavity containing a low number of quantum
wells. The dependence of lower polariton emissions on the pump intensity and
time evolution of lower polariton emission at $k_{\parallel}=0$ [21]. The
condensation of exciton-polaritons was clearly observed for the first time by
Kasprzak et al. [10]. In the experiment, they used a CdTe/cdMgTe microcavity
containing sixteen quantum wells and the Rabi splitting was 26 meV. A
polariton condensate was also observed in a planar GaAs microcavity in 2009
[22]. However, it is essential to point out that, unlike the conventional BEC,
the polariton BEC is not an equilibrium state. It can even consist of several
condensates emitting at different energies [23].

#####  1.3.2.3 Excitaion of polariton condensates by resonant and nonresonant
pumping

The first step to achieve the condensation of polaritons is to increase the
polariton population, which can be done by external pumping. There exist two
principal nonresonant and resonant pumping. In the nonresonant laser
excitation scheme depicted in Fig.1.7(a), free charge carriers are excited at
the energy above the upper polariton branch, which then relax to the lower
polariton branch via acoustic and optical phonon interaction [12]. At high in-
plane wave vectors, the polaritons for the lower polariton branch are mostly
exciton-like, resulting in relatively long lifetimes, a large density of
states, and heavy effective masses. However, when the particles arrive at the
dispersion inflection point where the actual strong coupling part of the lower
polariton branch begins, polaritons become more photonic-like. Consequently,
the lifetime of polaritons decreases by about two orders of magnitude, and the
effective mass of polaritons is highly reduced by several orders of magnitude,
resulting in a decrease in the density of states. From the inflection point,
to reach the ground state at $k_{\parallel}=0$, polaritons should lose their
energy about 5 to 10 meV. However, the relaxation to the ground state can not
be attained by the interaction with the acoustic phonons since the required
energy is larger than the energy of the acoustic phonons, which is usually
less than 1meV. The acoustic phonon-mediated relaxation process takes longer
than the polariton lifetime at the strong coupling regime, resulting in the
particles accumulating at the inflection point. This phenomenon is referred to
as the bottleneck effect, which was reported by Tassone [24]. This obstacle
can be suppressed by increasing the number of polariton pumps, resulting in
more efficient energy relaxation to the ground state via polariton-polariton
scattering above a certain excitation density threshold.

The formation and time evolution of polariton condensates by the nonresonant
excitation can be analyzed in detail based on a driven-dissipative Gross-
Piaevskii equation coupled with reserovirs [25],

$\mathrm{i}\hbar\frac{\partial\psi}{\partial
t}=-\frac{\hbar^{2}\nabla^{2}}{2m_{p}}\psi-\frac{\mathrm{i}\hbar}{2\tau_{p}}\psi+(g+\frac{i\hbar
R}{2})n_{A}\psi+gn_{I}\psi+\alpha|\psi|^{2}\psi,$ (1.12) $\frac{\partial
n_{I}}{\partial t}=P-\frac{n_{I}}{\gamma_{I}}-\frac{n_{I}}{\tau_{R}},$ (1.13)
$\frac{\partial n_{A}}{\partial
t}=-\frac{n_{A}}{\gamma_{A}}+\frac{n_{A}}{\tau_{R}}-R|\psi|^{2}n_{A}$ (1.14)

Eq.(1.12) describes the polariton condensate, $\psi$ is the coherent polariton
field in the lower polariton branch. $m_{p}$ is the mass of polaritons,
$\tau_{p}$ is the lifetime of polaritons, and $g$ is the energy exchange term
between the reservoir particles and the polariton condensate. $R$ describes
the rate of exciton scattering into the polariton condensate, and $\alpha$ is
nonlinear repulsive polariton interactions.

Eq.(1.13) describes the rate equation for the inactive reservoirs. Indeed, the
external pump creates an inactive exciton. By the nonradiative recombination,
the inactive excitons disappear with the decay rate $1/\gamma_{I}$ or they can
be replenished into the active reservoir with $1/\tau_{R}$. The rate equation
for the active reservoir is given by Eq.(1.14). The particles in the active
reservoir disappear radiatively at the rate $1/\gamma_{A}$, or they are
scattered down to the polariton condensate at the rate $R$.

Another method to create a polariton condensate is using a resonant excitation
into the lower polariton band [26]. The resonant pump results in optical
parametric oscillation, which can help polaritons overcome the bottleneck
effect. Optical parametric oscillators are resonators based on an optical gain
through a nonlinear medium. The nonlinear resonator generates a second-order
nonlinear process that a coherent pump passing by the medium, produces the
coherent signal and idler outputs [27]. Polaritons can experience a stimulated
scattering process when injected by a coherent pump on the lower polariton
branch with the angles near the inflection points depicted in Fig. 1.7. (b).
By exciting the polaritons with a resonant pump above the threshold at the
momentum $k_{p}$ of the inflection point, two polaritons can scatter into two
different states, which are a zero momentum and a higher momentum $2k_{p}$ in
such a way that the polariton can achieve the energy and momentum
conservation. Due to the bosonic statistic of the polaritons, a stimulated
scattering process takes place, and the interaction between polaritons allows
the particles to be populated into the ground state at $k=0$.

Figure 1.7:  (a). Schematic representation of the formation of the polariton
condensate by nonresonant excitations. UP(LP) stands for the upper(lower)
polariton branch, and $\eta_{I}(\eta_{A})$ represents the inactive(active)
reservoirs. (b). Schematic of the resonant excitation.

##  Chapter 2 Partial revivals of localized condensates in distorted lattices

###  2.1 Introduction

Through classical trajectories, coherent states [28] of simple quantum systems
show nondestructive dynamics, but the wave packets are generally dissipative
in time. This spreading reverses if the energies of the system are rational
numbers, leading to complete or fractional revivals of the quantum states as,
e.g., in the case of the Brown states [29] of the hydrogen atom [30].
Particles loaded in a flat band (FB) of a periodic potential [31]. exhibits
unique quantum transport properties. The particles form the compact localized
state (CLS) [32] by occupying only a few neighboring sites of the lattice in
an ideal case described by the tight-binding model. The CLS is an eigenstate
of the single-particle Hamiltonian, and thus it does not propagate nor does it
spread in time. However, in reality, perfect localization is hard to achieve
due to the approximate flatness of the band, interparticle interactions, and
finite lifetime of particles. These relentless obstacles lead us to the
natural questions: Are FBs actually feasible? How do reasonably small
deformations of a FB affect the dynamics of initially perfect CLSs? Moreover,
can one at all speak about any (quasi-)CLSs in realistic circumstances?

We address these issues by studying the dynamics of exciton-polaritons (later
polaritons) loaded in a nearly perfect FB [33]. Under external coherent or
incoherent excitation, polaritons might condense [10, 34], demonstrating
macroscopic occupation of a particular state—phenomenon also referred to as
polariton lasing. The state in question can be localized or trapped, and there
exist multiple experimental scenarios of polariton trapping, either in a
single state or in a periodic network. In particular, polaritons can be
exposed to spatially-periodic acoustic waves [35] or created in etched
microcavities. Polariton condensation in artificial periodic potentials has
recently become a remarkably active field of research. Polaritons develop new
transport properties if loaded in honeycomb [36], kagome [37, 38], or 1D and
2D Lieb [39, 40, 41] lattices, occasionally forming topologically protected
[42, 43, 44, 45, 46, 47] and single-particle quasi-flat bands (which we will
also refer to as FBs in what follows).

Several questions remain open and disputable in this field. In particular,
polariton condensates in FBs possess a relatively short coherence length. It
is unclear if this is the consequence of a disorder or if condensate
fragmentation is a generic property of out-of-equilibrium systems loaded in a
FB. However, a periodic long-range order can appear spontaneously in
resonantly-driven cavities [48]. Moreover, condensed out-of-equilibrium
particles should not necessarily occupy the lowest energy state (typically the
$\Gamma$ point). There emerge $\pi$-condensates (at the edge of a band) in 1D
potentials [49] and $d$-condensates in 2D square lattices [50]. The choice of
the condensate phase is controlled by the polariton-polariton interaction,
which can also lead to a space-time intermittency regime in microcavities with
periodic potentials (lattices) [51].

###  2.2 Tight-binding model on kagome lattice

We start by using a regular tight-binding model on a kagome chain. It has a FB
in the case of uniform hoppings. By adjusting the hopping coefficients in
horizontal bonds [see Fig. 2.1(a)], we lift the FB degeneracy and study the
unitary evolution of the system initialized in one of the CLSs, which occupies
sites of a single hexagon [Fig. 2.1(b)]. Namely, the hoppings on the
horizontal bonds, indicated by the red color in Fig. 2.1(a), have half the
value of the non-horizontal bonds. This mimics the lattice distortion of the
continuous model discussed below. The period of the revivals and the spreading
velocity is mostly affected by altering the ratio of the hopping coefficients.

Figure 2.1: Tight-binding description. (a) A kagome chain, with red
(horizontal) bonds having half the hopping integrals of the other (black)
bonds. State of the system: (b) $t=0$ (arb. units), the initial state is the
exact CLS; (c) $t=4.7$; (d). $t=12.4$. (e-f) Density plot of CLS revivals
evolution in time with a kagome chain and kagome lattice, respectively: every
point of the y-axis is a hexagon. The color denotes the value of the overlap
with a CLS. The vertical line in (e) indicates the time shown in (c), and the
black dots represent the two copies of the original CLS.

Due to the distortion, the CLS ceases to be an exact eigenstate, and it
spreads over the chain with time. We observe a signature of revivals of the
CLS at short times [Fig. 2.1(c)] and nonperfect twins of the original CLS that
appear at shifted positions. We note that this is a partial revival since the
whole state of the quantum system does not fully coincide with its original
state, and it is characterized by the presence of a few CLS condensates
further away from the creation center. Eventually, the CLS is completely
destroyed [Fig. 2.1(d)]. This destruction occurs faster for stronger
distortions.

Figures 2.1(e) and 2.1(f) illustrate the entire process of the CLS partial
revivals and their eventual destruction in a kagome chain (e) when we leave a
single layer of hexagons composed of triangles and full 2D kagome lattice (f).
We plot the overlap of every hexagon in the chain with a CLS and the time
evolution of the overlaps. One observes relatively clear signatures of
revivals at short times, but they are gradually suppressed at longer times.
Similar but weaker signatures also occur in a 2D kagome lattice, where a
stripe of sites with different couplings is introduced. The revivals of the
initial CLS in the tight-binding model are weaker than the ones seen in the
continuous model, which we discuss next.

###  2.3 The formation and propagation of polaritons in the lattice

We study the formation and propagation of polaritons coupled with an
incoherent excitonic reservoir, using the evolution equations

$\displaystyle\mathrm{i}\hbar\frac{\partial\psi}{\partial
t}=\mathcal{H}\psi+\mathcal{P}-\frac{\mathrm{i}\hbar}{2\tau_{p}}\psi+\frac{\mathrm{i}\hbar
Gn_{R}}{2}\psi+\alpha|\psi|^{2}\psi+{\color[rgb]{0,0,0}gn_{R}\psi},$ (2.1)
$\displaystyle\frac{\partial n_{R}}{\partial
t}=P_{in}-\frac{n_{R}}{\tau_{R}}-G|\psi|^{2}n_{R},$ (2.2)
$\displaystyle\mathcal{H}=-\frac{\hbar^{2}}{2m_{p}}\nabla^{2}+V$ (2.3)

where $\psi$ is a polariton macroscopic wave function, $\mathcal{H}$ is the
Hamiltonian, responsible for the propagation of the particles with the
effective mass $m_{p}$ in a kagome lattice potential $V$, shown in Fig.
2.2(a); $\mathcal{P}$ is the coherent pumping term; $\tau_{p}$ and $\alpha$
are the polariton lifetime and polariton-polariton interaction constant,
respectively; g is polariton-reservoir interaction constant; $n_{R}$ is the
reservoir particle density, which is incoherently pumped by the continuous
wave (CW) and homogeneous in space term $P_{in}$ and has a finite lifetime
$\tau_{R}$; the reservoir and polaritons are coupled by a phenomenological
constant $G$.

Taking $V=0$ inside the lattice sites (pillars) and $V=30$ meV outside the
pillars, we first solve the eigenvalue problem $\mathcal{H}\psi=E\psi$ in the
framework of the continuous model, and we find the band structure, shown in
Fig. 2.2(b). Unlike the tight-binding model, we note that the third band here
is only approximately flat. To create a CLS of the nearly flat band, we employ
a coherent Laguerre-Gaussian (LG) pump at the center of the lattice, as shown
in Fig. 2.2(c), which is given by

$\mathcal{P}=P_{0}\left(\frac{r}{R}\right)^{l}L^{0}_{l}\left(\frac{r^{2}}{R^{2}}\right)exp\left[-\left(\frac{r^{2}}{R^{2}}\right)+\mathrm{i}(l\phi-\omega_{0}t)\right],$
(2.4)

where $P_{0}$ is the pump amplitude, $r$ is the radial distance from the
center of the plaquette, $\phi$ is the azimuth angle, $R=0.7\,\mu\mathrm{m}$
is the radius of the pumping ring, $l=3$ determines the phase difference
between neighboring lattice sites (adjacent pillars), which in our case is
$\pi$ [see Fig. 2.2(d)]. Furthermore, $\omega_{0}$ is the pump frequency,
which we put equal to the frequency of the third band of the kagome lattice at
the $\Gamma$ point. For short pulses, the spreading in energy is greater than
the width of the flat band, so that the exact coincidence of frequencies is
not necessary for the CLS excitation.

Figure 2.2: System schematic. (a) A compact localized state can be created at
the center of the 2D Kagome lattice, occupying six pillars. (b) The band
structure shows the two lowest dispersive bands and the highest non-dispersive
ones. (c) The intensity of Laguerre-Gaussian (LG) pumping that aligns well
with the hexagonal structure of the kagome lattice. (d). Phase of the LG beam.

We note that it is convenient to excite CLSs in the kagome lattice with the
Laguerre-Gaussian pump, as compared to the Lieb lattice [52], due to the
absence of parasite pumping of the sites outside the CLS, i.e., the sites,
where the population should be zero as a result of destructive interference.

We apply this coherent pump for a short time, thus making a short pulse of
about $0.5\,\mathrm{ps}$ duration, to generate the CLS and let it evolve with
the support of the continuous background incoherent pumping $P_{in}$.  In the
absence of $P_{in},$ the population of the system quickly decreases. The
subthreshold pumping $P_{in}<P_{th}=(G\tau_{R}\tau_{P})^{-1}$ helps to prolong
the existence of the condensate by compensating the losses due to the finite
lifetime. Nevertheless, this background incoherent pump does not hinder the
revival phenomenon, as shown below.

It is essential that the main excitation comes from the resonant Laguerre-
Gaussian pulse. We assume this pulse alone to be strong enough to create the
substantial macroscopic occupation of CLS, such that its evolution can be
described by mean-field equation even without additional non-resonant pumping.
The non-resonant pumping improves the applicability of the Gross-Pitaevskii
equation by additional feeding of this macroscopic state. It is assumed that
the number of particles is still macroscopic after revivals, so the
semiclassical description remains valid. On the other hand, it is necessary to
keep the background non-resonant pumping below the threshold. Otherwise, the
effect would be masked by the spontaneous condensate formation.

###  2.4 Distorted lattice

Similarly to the tight-binding case, we introduce anisotropy in the system, as
it is shown in Fig. 2.3, and solve Eqs. (2.1) and (2.2) numerically. As a
result, we observe (i) quasi-1D propagation of the CLSs in specific
directions, which can be seen as partial jumps, and (ii) revivals of the CLSs
at their original position and different positions in space.

Figure 2.3:  (a) The kagome $y$-squeezed strip. (b) Red pillars are squeezed
vertically. Snapshots of CLS at 1 ps (c), 14 ps (d), 18 ps (e), and 32 ps (f).
The vertical lines indicate times, shown in (d-f). (g) Temporal evolution of
the number of particles in CLS at the center of the strip for different
polariton-reservoir interaction constants: a manifestation of the partial
revivals. (Inset) Comparison of the time evolution of the CLS weight in the
tight-binding (red) and continuous (blue) model.

We study the kagome strips squeezed in $y$-direction, as shown in Figs.
2.3(a,b). $x$ squeezed kagome strips is disscused in appendix 6.1. We note
that experimentally, two methods exist to squeeze the lattice. One way is to
etch the lattice with a defective line, where the distances are slightly
different. Another way is to apply unidirectional mechanical stress to the
lattice, which will cause effective deformation of the lattice.

In this work, we simulate the former method. The lattice constant is
3$\mu\mathrm{m}$. The radius of the pillars is $0.65\mu\mathrm{m}$ and the
pillars are shifted as $0.1\mu\mathrm{m}$. The revival of the CLS on a kagome
$y$-squeezed strip is shown in Fig. 2.3(c-f). The LG beam shines in the center
of the lattice for $0.5$ ps. Figure 2.3(c) shows the particle density when the
coherent pump is switched off. The condensate now propagates both to the left
and the right, and the CLS at the center of the strip completely disappears at
about $14\,\mathrm{ps}$. At the same time there appear new CLSs next to the
center. See Fig. 2.3(d). Further on, the condensate continues to propagate and
creates new CLSs in neighboring sites, but the CLS at the center is partially
restored at $32\,\mathrm{ps}$, see Fig. 2.3(e). Figure 2.3(f) shows that the
CLS at the center of the strip disappears again, and the condensate reaches
the boundary of the system. Figure 2.3(g) shows the influence of the
interaction constant $g$ on revivals. By increasing $g$, the number of
particles in the strip decreases. This figure also proves that the revival
phenomenon still occurs with higher polariton-reservoir interaction.

We considered a clean system and disregarded any structural disorder. Typical
potential disorder in polariton lattices of 0.2 meV is much smaller than the
height of 30 meV of the potential barriers in the kagome lattice. Thus we do
not expect strong suppression of the revivals. We also note that the effect
has an essentially local character: the particles created by the initial pulse
spread and come back involving a few lattice sites only. Therefore,
experimentally one can try to find a place in the lattice with the
particularly weak disorder to observe the revivals.

###  2.5 Summary of the chapter

We reported on the peculiar unexpected dynamics of bosonic condensates loaded
in a distorted kagome lattice, where the transport of particles manifests
itself in jumps and partial quantum revivals of compact localized states. In
the tight-binding model, the dynamics of the initial CLS take place by its
reappearance in the neighboring sites, but the revival of the original CLSs is
not observed. In contrast, in the continuous model, the deformation of the
nearly flat band by squeezing the lattice in either a horizontal or vertical
direction leads to visible revivals of the polariton wave packet.

The physical mechanism of the revival phenomenon is constructive interference
of waves reflected from the lattice while propagating uni-dimensionally. In
the usual one-dimensional systems, the back-scattering leads to the Anderson
localization, resulting in disordered localized states of particles. In
contrast, the localized states in FB lattices are prepared in specific CLS
form, and the back-scattering produces revivals of these CLSs without change
in their shape and size. A background pump can also support this effect,
opening new possibilities for developing all-optical logical elements based on
polariton condensates in nearly flat bands.

We expect that our work will stimulate the research towards building new
quantum devices, such as kagome optomechanical lattice [53]. Moreover,
exciton-polariton delocalization and revivals can be used for information
transfer and storage [54].

##  Chapter 3 Observation of a quantized vortex vanishment in exciton-
polariton system

###  3.1 Introduction

A superfluid behaves differently as compared to conventional fluids due to its
macroscopic quantum coherence. To describe the superfluid state, the single-
valuedness of the wave function is introduced and it gives rise to the
quantized circulation in units of $\hbar/m$, where $\hbar$ is the reduced
Planck constant, and $m$ is the mass of the particle. An integer multiple of
$2\pi$ phase winding around the vortex core is carried in the superfluid with
the quantization of circulation and the density of the superfluid is depleted.

Starting from the first prediction of quantized circulation in superfluid
helium by Onsager [55], the creation of quantized vortices has been
extensively studied in conventional single-component bosonic superfluids such
as liquid helium [56, 57] and cold atomic gases [58, 59]. Due to the
macroscopic phase winding, a quantized vortex is considered to be a
topologically stable object. Therefore, these vortices can be potentially
utilized as quantized information bits [60]. The macroscopic phase winding
associated with a quantized vortex makes it a topologically stable object;
hence, these vortices can be potentially used as quantized information bits
[60]. Their topological stability has extensively drawn attention regarding
particle injection, and decay dynamics [61, 62, 63].

Exciton-polaritons (later polaritons) provide an excellent platform for
investigating the stability of a topological vortex in a non-equilibrium
superfluid [11, 64]. A polariton is a quasiparticle created as a result of the
strong coupling between a cavity photon and a quantum-well exciton in a
semiconductor microcavity. Polaritons inherit most of the properties from
their constituents. Due to the photonic component, polaritons have a short
lifetime which results in the non-equilibrium physics [65, 66], and the
excitonic constituent gives rise to the interaction between polaritons.
Moreover, the bosonic nature of polaritons allows the formation of macroscopic
quantum phenomena such as polariton condensation [10], quantized circulation
[67], and superfluidity [68]. In particular, numerous theoretical [69, 70, 71,
72] and experimental methods have been widely developed to generate quantized
vortices in polariton systems, including the circular grating structures [73],
resonant excitation by Laguerre–Gaussian (LG) beams [74], and optical
parametric oscillations [75].

In addition, polariton superfluids have been also investigated to achieve
quantized vortex states even under nonresonantly pumped LG beam excitation
[76]. The combination of the photonic constituent of polariton and the non-
equilibrium nature of the polariton condensates creates a unique opportunity
for studying the time-resolved dynamics of these vortices. In fact, the
creation of multiply charged vortices [77, 78] and decay of vortices have been
investigated, such as vortex–anti-vortex pair annihilation [79, 80], vortex
annihilation through spiraling out of condensates [81], and the destruction of
the condensate itself [75]. However, the dynamics associated with vanishing a
single polariton vortex within the polariton condensate lifetime have not been
reported.

###  3.2 Formation of vortex states

In this part, we show the vanishment of a single quantized vortex in a
polariton superfluid. To create a vortex, we used a nonresonant LG beam, which
imprints the orbital angular momentum (OAM) of the pump beam to the polariton
condensate [76]. The energy-integrated photoluminescence (PL) image shows that
the vortex contains an incomplete depletion of the superfluid density at the
center. Based on the careful spectral analysis of the vortex, we discovered
that the polaritons form two different modes of condensation and the
incomplete depletion of the superfluid density at the vortex core is an
artifact of the time-averaged measurement of the multimode condensate, as
evidenced by energy-resolved interferometry experiments. We found that only
the higher energy state of the multimode condensate carries angular momentum,
referred to as a quantum vortex, by separately extracting the phase profile of
the wave functions in the two different states. Furthermore, time-resolved
spectroscopy exhibits that the excited (vortex) state relaxes to the
stationary ground state in a pulsed LG beam excitation, which serves as a
direct experimental signature of a quantized vortex vanishment from the
system. Such a transition is demonstrated by modifying the diameter of the LG
pumping beam.

We can attain a steady state where the excited- and ground states coexist
under continuous-wave (CW) pumping. The experimental observations are
supported by a theoretical model based on the driven-dissipative Gross-
Pitaevskii equation coupled with incoherent and coherent reservoirs; hence, we
suggest a mechanism for OAM transfer from the pump beam to the polariton
condensate that is based on the microscopic coherent properties of the latter.
Furthermore, we describe the transition of the system from the first excited
state characterized by the vortex and a concrete OAM to the intermediate
regime, when two lowest-energy states coexist, and finally, to the ground-
state condensate with zero OAM after vanishment of the vortex.

Figure 3.1:  (a) Schematic of the system optically pumped with a nonresonant
LG beam with OAM $\absolutevalue{l}=1$. Above the pumping threshold,
polaritons are condensed, forming two energy states with different density
distributions. (b) Energy-integrated PL image of the condensates above the
pumping threshold. (c) Phase map of (b) shows $2\pi$ phase winding. (d) Real-
space spectrum along line $x=0$ in (b). (e) The calculated real-space spectrum
for the condensate.

In the experiment, a microcavity that contains GaAs quantum wells(QW) is used
and for the nonresonant excitation, the LG beam is illuminated to the
microcavity, as depicted in Fig. 3.1(a). In the microcavity, an optical cavity
is constructed between a pair of distributed Bragg reflectors with a set of
GaAs/AlAs QW stacks. To reach as low as $5K$, the microcavity sample was put
inside a cryostat(with an optical window). A Ti: sapphire laser which contains
an energy of $1.72eV(720nm)$ was used as a pulsed and a CW excitation to pump
the electron-hole plasma into the QWs nonresonantly. An LG beam was generated
as the Gaussian laser beam passed a $2\pi$ winding phase mask, which resulted
in an OAM of the $l=+1$ beam Fig. 3.2(a). Polaritons formed condensates with a
quantized vortex above the pumping threshold. The energy-integrated PL image
is shown in Fig. 3.1(b) and the phase map is depicted in Fig. 3.1(c) under
pulsed excitation. In the phase map, the measurement of $2\pi$ phase winding
implies that the superfluid carries a circulation of $\hbar/m$. Fig. 3.1(d)
shows the real-space spectrum along $x=0$. From the spectrum, the condensate
is not in a uniform static state but two distinctly quantized states. We will
refer to the higher- and lower-energy states as the excited and ground states
for simplicity. For the theoretical description of the experiment, we used a
driven-dissipative Gross-Pitaevskii equation coupled with incoherent and
coherent reservoirs, where the latter carries the same OAM as the LG beam. We
assume that the OAM of the pumping source is almost erased in the scattering
processes, and only a small amount remains. The OAM is revived in the
polariton condensate owing to the interaction with a weak coherent reservoir
and the coherent properties of the condensate. Fig. 3.1(e) exhibits a
simulation result of multistate condensate similar to that in the experiment.

Figure 3.2:  Profiles of the pump and multimode condensates (a) LG pump
intensity distribution; energy resolved PL images of (b) excited state at
$1592.37meV$, and (c) ground state at $1591.69meV$ under pulsed excitation;
(d) phase map of LG beam having OAM $l=+1$; phase maps of (e) excited state
and (f) ground state. A $2\pi$ phase winding in (e) indicates a quantized
vortex in an excited state. Panels (g), (h), and (i) show calculated real-
space images of LG beam excitation and polariton condensates when the system
is stationary (either ground or excited state). White circles in panels (h)
and (i) illustrate the nominal diameter (6.0 $\mu m$) of the LG beam
corresponding to the maximum intensity spots of the simulated beam (g).

Energy-resolved spatial interferometry based on the tomographic measurement
protocol [82, 83, 84, 85, 86] is employed to study the phase information and
density profile in each state. The experimental procedure was similar to the
modified Mach-Zehnder interferometry, which is particularly utilized to
visualize quantized vortices. Fig. 3.1(d) shows the excited and ground state
which contains the energy $1592.37meV$ and $1591.69meV$ and they are
represented with solid and dotted lines, respectively. To represent the
spatial densities of polaritons in the excited- and ground energy levels, we
show energy-resolved PL intensity under pulsed excitation which is depicted in
Fig. 3.2(b) and Fig. 3.2(c). The excited state shows the density minimum in
the center and the ground state reveals a density distribution in Gaussian
form. Fig. 3.2(e) and Fig. 3.2(f) show the phase maps attained from the
interference patterns of these two states. The excited state exhibits a $2\pi$
phase winding with respect to the density minimum point, indicating a
quantized vortex. The phase winding of the vortex Fig. 3.2(e) was identical to
the OAM of the LG pump Fig. 3.2(d). However, the phase map for the ground
state indicated no such phase winding, and the phase was relatively flat aside
from the radial gradient Fig. 3.2(f). In other words, the ground state did not
carry any vortices. The simulated LG beam Fig. 3.2(g) resulted in polariton
condensate formation at the two energy levels Fig. 3.1(e). The real-space
emission profiles in the steady state for each of these energy levels are
shown in Fig. 3.2(h) and Fig. 3.2(i), exhibiting consistency with the
experiment. (To simulate the formation of the ground- and excited states, we
used the CW excitation instead of the exact femtosecond-order pulsed
excitation).

###  3.3 Pulsed-excitation regime

Figure 3.3:  (a) Real-space spectra of PL profiles in the pulsed-excitation
regime for different pumping beam diameters of $4.3,5.2,5.5$, and $6.4\mu m$.
Beam diameter is defined as the distance between two maxima of laser profile
Fig. 3.2(a). (b) The energy of ground- (blue circle) and first excited (red
diamond) states as a function of the beam diameter. The numerical simulation
of ground- (open circle) and excited states (open diamond) are plotted.

The time-integrated real-space spectra measurement reveals that the relative
intensity of the PL emission from the two states depends on the diameter of
the LG beam, as shown in Fig. 3.3(a). Below the diameter of the beam $4.5\mu
m$, only the ground state is populated. However, the population of the excited
state starts to increase as the beam diameter increases, and, at a
sufficiently large diameter, the ground state becomes almost invisible.

This can be explained by a confinement effect: if the beam size is
sufficiently small, the excited state is pushed out of the potential trap.
This can be understood as a confinement effect: the excited state is pushed
out of the potential trap if the beam spot is sufficiently small. The
confinement effect is reduced when the diameter of the beam increases,
resulting in a redshift Fig. 3.3(b).

The condensate can be formed in the excited state when the energy of the first
excited state is sufficiently low enough to fit inside the potential trap.
This behavior was confirmed via numerical calculations. The numerical
calculations and experimental results are depicted in Fig. 3.3(b). The figure
exhibits the energy of the excited state and ground state with different pump
diameters. Due to the reduced blueshift with smaller confinement, the slope of
the data downwards with an increase in the diameter of the beam. To determine
the exact nature of the multistate condensate under the pulsed excitation, the
time-integrated measurement is not suitable since it does not allow us to
figure out where polaritons occupy the two distinct energy states sequentially
or simultaneously. We performed time-resolved real-space spectroscopy which
measures the relative intensity of the two different states in particular time
intervals to distinguish between the two possible scenarios, depicted in Figs.
3.4(a) - Figs. 3.4(c). The results show that the excited state which carries
the vortex firstly forms and then sequentially the ground state appears.
Eventually, the system experienced a transition from the vortex carrying
excited state to the ground state. Note that in a smaller LG beam diameter
Fig. 3.4(a), polaritons occupy the “Gaussian” ground state dominantly
(appearing as single lobe along the $y$ axis at $x=0$) in the measured time
evolution. However, in the case of intermediate LG beam diameter Fig. 3.4(b),
polaritons occupy the “annular” excited state (appearing as double lobes along
the $y$ axis at $x=0$) initially, and they experience a transition to the
ground state as time proceeds. In the case of a larger LG beam diameter Fig.
3.4(c), polaritons show only the excited state over the whole time duration.
Simulations also confirmed these measurements. Our simulations indicate that
the transition phenomena were accompanied by the competition between two
energy states and the lowering of the potential barrier [87] as time proceeds
under the pulsed excitation. In fact, the transition to the lower-energy state
occurred due to scattering processes [88]. Polaritons try to occupy the vortex
core, which is the center of the potential trap when the population of the
ground state increases. Therefore, the excited state competes with the growing
ground state to occupy the vortex core and the ground state to push the
excited state out of the interior of the potential barrier. In pulsed
excitation, the gradual depletion of the exciton reservoir over time
diminishes the potential trap as compared to CW excitation. Then, the excited
state is pushed out faster, and the system remains in the ground state. The
entire process can take longer than the condensate lifetime, and polaritons
decay before the transition occurs for sufficiently large beam diameters,
shown in Figs. 3.4(c). For smaller beam diameters Fig. 3.4(a), before
polaritons completely decay, the excited state can be pushed out more easily;
hence, an earlier transition to the ground state can be observed. Although the
total OAM of the polaritons will not reach complete unity in practice because
of inherent fluctuations in the sample and the nonideality of the LG beam, the
simulation explains well what we see in the experiments Fig. 3.4(d): a smaller
beam size resulted in the sooner loss of the total OAM of the system. We
investigated the dependence of the OAM on the size of the LG beam and its
intensity in the regime of the pulsed exaction.

To numerically investigate the evolution of the exciton-polariton condensate,
we used the driven-dissipative Gross–Pitaevskii equation as follows:

$i\hbar\partial_{t}\psi=-\frac{\hbar^{2}\psi^{2}}{2m}\psi-\frac{i\hbar}{2\tau}\psi+\frac{i\hbar
R\eta_{A}}{2}\psi+\frac{i\hbar
g}{2}\phi+\alpha\absolutevalue{\psi}^{2}\psi+\beta(\eta_{A}+2\eta_{I})\psi+\beta\absolutevalue{\phi}^{2}\psi.$
(3.1)

Here, $\psi(r,t)$ is the complex-valued macroscopic wave function of the
lower-branch polaritons; the effective mass of the polaritons was $m=0.7\times
10^{-4}m_{e}$, where $m_{e}$ represents the free electron mass, the particle
lifetime $\tau=5ps$, and the repulsive polariton–polariton interaction was
$\alpha=10^{-3}meV\cdot\mu m^{2}$. To include the energy relaxation and
scattering dynamics of particles created by the off-resonant optical
excitation, we considered two incoherent reservoirs: an active reservoir
$\eta_{A}$ and an inactive reservoir $\eta_{I}$. The polariton condensate was
replenished by the active reservoir controlled by a phenomenal parameter
$R=0.02ps^{-1}\cdot\mu m^{2}$. The intensity of the repulsive interaction
between the condensate and reservoir particles was $\beta=10^{-2}meV\cdot\mu
m^{2}$. Furthermore, we assumed a small fraction of coherent excitation $phi$
to be inherited by the condensate (with $g=1ps^{-1}$). This allowed one to
achieve identical vorticity in the excitation laser and polariton condensate.
The coherent polaritons interacted with the condensate, as enabled by the term
$\beta\absolutevalue{\phi}^{2}\psi$ (which is small and negligible). The
following equation governs the dynamics of the active reservoir:

$\partial_{t}\eta_{A}=\frac{\eta_{I}}{\lambda}-R\absolutevalue{\psi}^{2}\eta_{A}-\frac{\eta_{A}}{\gamma}.$
(3.2)

Here, the lifetime of the reservoir particles is $\gamma=300ps$. The active
reservoir is refilled by the inactive reservoir at a rate of
$\lambda^{-1}=120ps^{-1}$. The inactive reservoir is pumped directly using an
external incoherent pumping source $\absolutevalue{P_{i}}$. We assumed that
the lifetime of the inactive reservoir particles is the same as that of the
active reservoir. Hence, the following equation is obtained:

$\partial_{t}\eta_{I}=\absolutevalue{P_{i}}-\frac{\eta_{I}}{\gamma}-\frac{\eta_{I}}{\lambda}.$
(3.3)

The weak coherent excitation field obeys the evolution equation as follows:

$i\hbar\partial_{t}\phi=iP_{c}exp[-i\omega(t)t]-\frac{i\hbar
R_{in}}{2}\absolutevalue{\phi}^{2}\phi.$ (3.4)

Here, $P_{c}$ is the coherent pumping source, which breaks the central
symmetry and hence transfers the winding number information from the laser
pump to the condensate. The frequency of this coherent pump $\omega(t)$ was
selected randomly at each time step according to the uniform distribution in
range [ min($E_{pl}/\hbar$), max($E_{pl}/\hbar$) ], where $E_{pl}$ is the
spectrum of the lower polariton branch. To saturate the gain of this coherent
reservoir, we introduced a nonlinear loss term controlled by the parameter
$R_{in}=1ps^{-1}\mu m^{2}$ (last term in the equation above). The LG pumping
term used in the simulation (both coherent pump $P_{c}$ and incoherent pump
$P_{i}$ ) reads

$P(r)=P_{0}\sqrt{2}\frac{r}{R}exp[-\frac{r^{2}}{R^{2}}+i\tan^{-1}{(\frac{y}{x})}],$
(3.5)

where $R$ is the pumping radius, and $P_{0}$ is the pumping intensity. Because
the external pumping energy is significantly far above the lower polariton
branch, the coherent pumping we used was several orders of magnitude smaller
than the incoherent pump. Therefore, the number of particles in the coherent
reservoir was much smaller compared with the occupation of the incoherent
reservoirs in Fig. 3.5(a). Hence, we claim that the macroscopic occupation of
the system was due to incoherent pumping. Using the equations above, we
investigated the condensate density’s dependence on the pumping’s size in the
pulsed exaction regime An LG pump with duration $t_{p}=100ps$ was used.

$P(r)=P_{0}\sqrt{2}\frac{r}{R}exp[-\frac{r^{2}}{R^{2}}+i\tan^{-1}(\frac{y}{x})]\theta(t)\theta(t_{p}-t)$
(3.6)

The parameters are the same as those used in CW excitation conditions. Fig.
3.5(b) shows the number of condensed particles as a function of time for
different pump diameters (as before, the coherent pump intensity was
approximately two orders of magnitude smaller than the incoherent pump
intensity). The condensate will form faster if the pumping intensity is fixed
and the pump diameter increases from $4.3$ to $6.4\mu m$.

Figure 3.4:  Time-resolved spectroscopy of polariton vortex in pulsed LG pump
regime for different beam radii: (a) $5.2$, (b) $6.3$, and (c) $7.1\mu m$.
Data were measured along a straight line through the center of the beam (at
$x=0\mu m$). (d) Simulation of OAM evolution for different beam diameters with
the same initial condition. The pumping term was turned off after $100ps$ to
simulate the pulsed excitation case.  Figure 3.5:  (a). Number of condensate
and reservoir particles in the system as functions of time for a single
simulation. Blue line: number of particles in polariton condensate. Red line:
number of particles in the inactive reservoir. Yellow line: number of
particles in active reservoir. Purple line: number of particles in the
coherent reservoir. Initial states were as follows: zero initial condition for
the coherent and active reservoir, steady state for the inactive reservoir,
and random initial condition for polariton condensate. (b). Number of
particles in condensate for different pump diameters. A random initial
condition was applied for polariton condensate; The initial conditions are the
same as (a).  Figure 3.6:  The steady state of partially rotating polariton
superfluid under CW LG beam excitation. (a) A real-space spectrum of polariton
condensates. (b) Energy-integrated PL image of the system. (c) Calculated
polariton distribution [simulation of (b)]. Energy-resolved PL images of
excited- and ground states are shown in panels (d) and (e), respectively;
insets show corresponding phase maps.

The most recent experiment [89] describes the decay of vortices in polariton
superfluids under net angular-momentum conserving conditions. The result
exhibits the simultaneous appearance of the vortices with OAM $l=+1$ and
$l=-1$ and the system decay into a purely zero-OAM ground state under the
excitation of $l=0$ ring-shaped pumping. In contrast to their work, our
observation shows that a single quantized vortex with a net OAM is
spontaneously relaxed in an angular-momentum-nonconserving manner under an
intentional excitation of the $\absolutevalue{l}=1$ LG beam. The reason for
the transition in our observation is that a single vortex is generated by
nonresonant pump excitation. However, a resonantly created polariton vortex
cannot experience such a transition since the resonant pump cannot effectively
generate a spatial potential from the excitonic reservoir. Unlike the vortices
in an equilibrium superfluid (which do not evaporate over time), the vortices
in polariton condensates might decay if the vortex escapes from the system.

We observed the vanishment of the vortex based on time-resolved measurements
of the energy-state transition, which cannot be treated as a direct
observation of an escaping vortex. We considered this scenario as a distinct
possibility, as revealed by our simulations, which show the disappearance of
the singularity from the center of the trap. Regardless of the actual
mechanism through which the vortex escapes from the system, the experiments,
conducted by our collaborators, present a knob with which one can control the
timing of the switch between the polariton condensates with and without a
quantized vortex.

###  3.4 Continuous-wave excitation regime

For the comparison with pulsed excitation, we also investigated the creation
of a polariton vortex under a CW excitation with $l=-1$ OAM beams in the same
experimental setup; subsequently, we observed two lowest-energy states Fig
3.6(a). The energy-integrated PL image of the system, depicted in Fig 3.6(b),
was consistent with the theoretical simulations in Fig 3.6(c). Fig 3.6(d) and
Fig 3.6(e) show the energy-resolved PL images of the excited- and ground
states in the CW excitation regime, respectively. The insets describe the
corresponding phase profiles for each state. The main distinction from the
pulsed excitation regime is that the CW pump beam consistently provides the
potential barrier, which blocks the excited state from being escaped from the
trap. Without diminishing the potential trap during time evolution, the
excited state cannot be pushed out of the barrier only by the competition
between the two distinct energy states. Therefore, to maintain a steady state,
there exists a balance among the decay, pump, and interstate transition
processes. We emphasize that this steady state represents an intriguing regime
in which a vortex-carrying and ground state coexist.

###  3.5 Summary

In conclusion, we explicitly demonstrated time-resolved spectroscopy shows a
single quantum vortex vanishing in an exciton-polariton superfluid pumped by a
pulsed nonresonant LG beam. By using various diameters of the LG pump, we
investigated the competition between the vortex-carrying excited state and
ground state, which determines the population of the two distinct states of
condensates and affects the transition between the states under the pulsed
excitation regime.

The experimental observations are greatly consistent with the numerical
calculations using the driven-dissipative Gross-Pitaevskii equation coupled
with incoherent and coherent reservoirs. The coherent reservoir gives a rise
to the breaking of the rotational symmetry and the revival of the OAM of the
nonresonant excitation with the subsequent formation of a quantum vortex with
a predefined winding number.

We proposed experimental and theoretical approaches to manipulate multistate
polariton condensates through potential landscape engineering in semiconductor
microcavities; these tools have broad implications for investigating light-
matter interaction and stability of topological objects in semiconducting
devices and developing polariton devices for quantum communications in the
near future.

##  Chapter 4 Valleytronics of impurity states in two-dimensional Dirac
materials

###  4.1 Introduction

The critical idea of valleytronics is in using the valley index as an
additional active degree of freedom of charge carriers [90, 9] in gapped
graphene [90], monolayers of transition metal dichalcogenides (TMDs) [9],
among other two-dimensional Dirac materials. One of the representatives of
TMDs is MoS2: a material with a structure composed of molybdenum atoms
sandwiched between pairs of sulfur atoms. In contrast to graphene, it is
characterized by the inversion symmetry breaking, and it possesses a large
band gap with a width in the optical range, absent in monolayer graphene [91].
It represents a direct band gap material with the minima of the conduction
band and maxima of the valence band located at points $K$ and $K^{\prime}$ in
reciprocal space. Moreover, electrons in MoS2 are subject to strong spin-
orbital interaction, which also makes it different from graphene, where the
spin-orbital interaction is relatively weak. This latter property, due to the
electrons occupying d-orbitals in MoS2, results in an extra band splitting
[92].

It has been shown that the interband (between the conduction and valence
bands) transitions in Dirac materials are valley-sensitive: at a given
circular polarization of the external electromagnetic perturbation, the
interband transitions occur predominantly in one valley since the electrons in
each valley couple with a specific polarization of light [93]. Switching to
the opposite circular polarization changes the valley where the interband
transitions take place [94]. These optical selection rules are fulfilled for
interband optical transitions, where the electron momentum is a good quantum
number.

However, each material is to some extent disordered: it contains impurities,
some of which are unintentional and emerge due to the imperfections of the
growth technique, whereas some of the impurities are embedded intentionally to
enhance the electronic (or other) properties of the sample. As a result,
additional donor and acceptor energy levels emerge in the vicinity of the
conduction and valence bands, respectively. Then, if the sample is exposed to
external light with the frequency smaller than the band gap of the material,
the optical properties become determined mainly by the electron transitions
from donor impurities to the conduction band and from the valence band to the
acceptor states [95]. In this case, the electron states on impurities are
characterized by some quantum numbers instead of the translational momentum
due to localization. The theoretical description of optical transitions from
these states to the bands and the analysis of the corresponding optical
selection rules, which consider the valley quantum number, represent an
important problem of valley optoelectronics, which has been studied
theoretically. In particular, work [96] presents a numerical study of a
particular type of disorder: vacancy defects.

Figure 4.1: System schematic. (a) A monolayer of MoS2 (with impurities)
exposed to a circularly polarized electromagnetic (EM) field of light (green
cylinder). (b) The band structure of MoS2. The band gap is $\Delta$, and the
system is exposed to an EM field with frequency $\omega$.

Usually, the energy gap between the impurity states and the conduction band
corresponds to the terahertz (THz) frequency range (10-20 meV). It can be used
to design (pulsed) terahertz radiation detectors. A polarized optical signal
is transformed into an electric current (to be measured) in such a detector.
The analysis of the optical selection rules here is thus of utmost importance
for applications. We want also to mention another potential application of the
theory of impurity-band transitions. It has been recently proposed that TMD
monolayers containing atomic impurities can be used as single-photon emitters
[97]. Utilizing artificially-created atomic vacancies, one can achieve the
single-photon regime of operation. This is one of the fast-developing topics
of research recently.

For fundamental purposes and to design the devices, it is useful to study the
general properties of defects of any type, and an analytical analysis would be
beneficial here. However, one of the problems to face is that the simple
model, which assumes the impurity potential energy to be the Dirac delta-
function [98], is not applicable in the case of a Dirac Hamiltonian since the
electron wave function becomes singular exactly at the center of coordinates
[99]. We build a theory of impurity-band transitions in 2D Dirac materials,
utilizing and modifying the model of zero-radius impurity potential, which is
frequently used for the description of shallow impurities in semiconductors
and semiconductor nanostructures [99, 100]. We investigate the optical
properties of disordered TMDs, examining the light absorption and photoinduced
transport effects, accounting for the spin-orbital coupling of electrons. We
study the behavior of drag electric current density and the absorption
coefficient for different vital parameters of the sample and different
polarizations of the incident light. It should be mentioned that the
generation of the electric current in 2D Dirac materials due to the interband
optical transitions (the photon and phonon drag effects) has been extensively
studied [101, 102, 103, 104, 105], but the impurity-band transitions have not
been addressed.

###  4.2 Hamiltonian and eigenstate

Light absorption is governed by microscopic transitions of electrons from the
bound impurity states to the conduction band. The Hamiltonian of the electron
bound at the attractive potential $u(\textbf{r})$ reads

$H=\left(\frac{\Delta}{2}\sigma_{z}+\textbf{v}\cdot\textbf{p}\right)\otimes\mathbbm{1}-\frac{\lambda\eta}{2}(\sigma_{z}-\mathbbm{1})\otimes\hat{s}_{z}+u(\textbf{r}),$
(4.1)

where $\Delta$ is a band gap, $\textbf{v}=v(\eta\sigma_{x},\sigma_{y})$ is the
velocity operator, p is the electron momentum, and $\sigma_{\alpha}$ with
$\alpha=x,y,z$ the Pauli matrices of pseudospin. The index $\eta=\pm 1$
indicates the valley; $\lambda$ is intrinsic spin-orbital coupling;
$\hat{s}_{z}$ is the matrix of the electron spin. The first term in
Hamiltonian (4.1) describes a two-band model of gapped graphene (or a band
structure of a TMD material). We consider a shallow impurity potential,
$u(\textbf{r})$; thus, we assume that the ionization potential of the donor is
much smaller than $\Delta$.

To find the eigenfunctions and eigenenergies, we write the Schrödinger
equation in the momentum representation,

$\displaystyle\begin{pmatrix}\frac{\Delta}{2}-E&vp_{-}\\\
vp_{+}&-\frac{\Delta}{2}+s\lambda\eta-E\end{pmatrix}\chi_{s,m}(\textbf{p})$
(4.2)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\int\frac{d\textbf{p}^{\prime}}{(2\pi\hbar)^{2}}u({\textbf{p}-\textbf{p}^{\prime}})\chi_{s,m}(\textbf{p}^{\prime})=0,$

where $s=\pm 1$, $p_{\pm}=\eta p_{x}\pm ip_{y}=pe^{\pm i\eta\varphi}$ with
$\varphi$ the angle of the vector p with respect to $x$-axis, $m$ is the
eigenvalue of the z-projection of the electron angular momentum (the quantum
number which characterizes electron localized on impurity),

$\displaystyle
u(\textbf{p}-\textbf{p}^{\prime})=2\pi\int^{\infty}_{0}rdru(r)J_{0}(|\textbf{p}-\textbf{p}^{\prime}|r)$
(4.3)
$\displaystyle~{}~{}~{}~{}~{}=2\pi\sum_{k}\int^{\infty}_{0}rdru(r)J_{k}(pr)J_{k}(p^{\prime}r)\cos
k(\varphi-\varphi^{\prime}),$

where $J_{k}(x)$ are the $k$-order Bessel functions. We search for the spinor
eigenfunctions in the form,

$\chi_{s,m}(\textbf{p})=\left(\begin{array}[]{c}a_{s,m}(p)e^{im\varphi}\\\
b_{s,m}(p)e^{i(m+\eta)\varphi}\end{array}\right)$ (4.4)

since this form reflects the axial symmetry, and the eigenstates are
characterized by the angular momentum projection with quantum number $m$.
Substituting Eq. (4.4) in Eq. (4.2) and performing the integration over
$\varphi^{\prime}$, we find the system of equations for the coefficients
$a_{s,m}$ and $b_{s,m}$,

$\displaystyle 0=\begin{pmatrix}\frac{\Delta}{2}-E&vp\\\
vp&-\frac{\Delta}{2}+s\lambda\eta-E\end{pmatrix}\begin{pmatrix}a_{s,m}(p)\\\
b_{s,m}(p)\end{pmatrix}$ (4.5)
$\displaystyle+\int\limits^{\infty}_{0}rdru(r)\int\limits^{\infty}_{0}\frac{p^{\prime}dp^{\prime}}{\hbar^{2}}\begin{pmatrix}J_{m}(pr)J_{m}(p^{\prime}r)a_{s,m}(p^{\prime})\\\
J_{(m+\eta)}(pr)J_{(m+\eta)}(p^{\prime}r)b_{s,m}(p^{\prime})\end{pmatrix}.$

To draw principal conclusions, we can now simplify these equations. Only the
low-lying impurity states are occupied for a shallow impurity
($\epsilon^{i}_{s,m,\eta}\ll\Delta$) and low enough temperatures. Then, we can
consider the transitions from impurity states corresponding to $m=0$ and
$m=\pm 1$ levels only. Assuming that the potential of each impurity $u(r)$ is
sharply peaked in the vicinity of its center $r=0$ and rapidly decreases with
$r$, we can take the Bessel functions under the integral $r=0$. For the $m=0$
state, $J_{0}(0)=1$ and $J_{\eta=\pm 1}(0)=0,$ and we find the simplified form
of Eq. (4.5),

$\begin{split}\begin{pmatrix}\epsilon^{i}_{s,0,\eta}&vp\\\
vp&-\Delta+s\lambda\eta\end{pmatrix}\begin{pmatrix}a_{s,0}\\\
b_{s,0}\end{pmatrix}+\begin{pmatrix}Au_{0}\\\ 0\end{pmatrix}=0,\end{split}$
(4.6)

where

$A=\int^{\infty}_{0}\frac{p^{\prime}dp^{\prime}}{\hbar^{2}}a_{s,0}(p^{\prime});~{}~{}~{}u_{0}=\int^{\infty}_{0}u(r)rdr.$
(4.7)

The solution of Eq .(4.6) reads

$\begin{pmatrix}a_{s,0}\\\ b_{s,0}\end{pmatrix}=-\hbar
v\sqrt{\frac{2\pi\epsilon^{i}_{s,0,\eta}}{\Delta}}\begin{pmatrix}\frac{\Delta-s\lambda\eta}{(vp)^{2}+\epsilon^{i}_{s,0,\eta}(\Delta-s\lambda\eta)}\\\
\frac{vp}{(vp)^{2}+\epsilon^{i}_{s,0,\eta}(\Delta-s\lambda\eta)}\end{pmatrix}.$
(4.8)

For the $m=1,\eta=-1$ state, $J_{0}(0)=1$, $J_{1}(0)=0$, and $J_{2}(0)=0$, Eq.
(4.5) can be simplified,

$\begin{split}\begin{pmatrix}\epsilon^{i}_{s,1,-1}&vp\\\
vp&-\Delta-s\lambda\end{pmatrix}\begin{pmatrix}a_{s,1}\\\
b_{s,1}\end{pmatrix}+\begin{pmatrix}0\\\ Bu_{0}\end{pmatrix}=0,\end{split}$
(4.9)

where now

$B=\int^{\infty}_{0}\frac{p^{\prime}dp^{\prime}}{\hbar^{2}}b_{s,1}(p^{\prime}).$
(4.10)

The solution of Eq. (4.9) reads

$\begin{pmatrix}a_{s,1}\\\ b_{s,1}\end{pmatrix}=-\hbar
v\sqrt{\frac{2\pi(\Delta^{2}-\lambda^{2})}{\Delta\epsilon^{i}_{s,1,-1}}}\begin{pmatrix}\frac{vp}{(vp)^{2}+\epsilon^{i}_{s,1,-1}(\Delta+s\lambda)}\\\
\frac{-\epsilon^{i}_{s,1,-1}}{(vp)^{2}+\epsilon^{i}_{s,1,-1}(\Delta+s\lambda)}\end{pmatrix}.$
(4.11)

The energy $\epsilon^{i}_{s,1,-1}$ of this state can also be found using the
definition Eq. (4.10). We see that within the framework of the shallow-
impurity model, the state $m=1$ forms ‘under’ the $\eta=-1$ valley (and vice
versa). In other words, the following rule holds $m+\eta=0$ for $m=\pm 1$
states.

The electron states in the wave function describes the conduction band,

$\psi_{s,\eta}(\textbf{p})=\left(\begin{array}[]{c}\cos\left(\frac{\theta_{s,\eta}}{2}\right)\\\
\sin\left(\frac{\theta_{s,\eta}}{2}\right)e^{i\eta\varphi_{\textbf{p}}}\end{array}\right),$
(4.12)

where we use the notations
$\cos\theta_{s,\eta}=(\Delta-s\lambda\eta)/2E_{s,\eta}(\textbf{p})$ and
$\sin\theta_{s,\eta}=\eta vp/E_{s,\eta}(\textbf{p})$ with the conduction band
electron energy $E_{c}(\textbf{p})=s\eta\lambda/2+E_{s,\eta}(\textbf{p})$,
$E_{s,\eta}(\textbf{p})=\sqrt{(vp)^{2}+\left[(\Delta-s\lambda\eta)/2\right]^{2}}$.
Since the transitions from the impurity state with a given valley number
$\eta$ to the conduction band of the other valley $\eta^{\prime}\neq\eta$ are
strongly suppressed due to the significant distance between the valleys in the
reciprocal space [106], the main contribution to the light absorption comes
from the impurity-band transitions with the same valley number
$\eta^{\prime}=\eta$. The circularly polarized EM field case is the most
interesting from the point of view of applications. The Hamiltonian describing
the interaction of electrons with the external EM perturbation reads
$\hat{V}(\textbf{}{r},t)=-e\textbf{v}\cdot\mathbf{A}(\textbf{r},t)$, where
$\mathbf{A}(\mathbf{r},t)=\mathbf{A}_{0}exp(i\mathbf{kr}-i\omega
t)+\mathbf{A}^{*}_{0}exp(-i\mathbf{kr}+i\omega t)$ is the vector potential of
EM field. Here, $\textbf{A}_{0}=A_{0}\hat{x}+A_{0}i\sigma\hat{y}$ with
$\sigma$ the light polarization, $\hat{x}$ and $\hat{y}$ the unity vectors in
the corresponding directions in direct space, $\mathbf{k}$ and $\omega$ being
the photon wave vector and frequency, respectively.

###  4.3 Electric current density

Figure 4.2: Spectrum of electric current density due to the transitions from
$m=0$, $\eta=-1$ impurity states to spin-up conduction band states for the
polarization of light $\sigma=-1$ (red) and $\sigma=1$ (blue);
$\gamma=\epsilon^{i}_{1,0,-1}/\hbar$, where $\epsilon^{i}_{1,0,-1}=10$ meV is
the energy of impurity counted from the bottom of the conduction band. Black
curves show the positive and negative contributions to the current in
$\sigma=-1$ case. We used the density of impurities $n_{i}\approx 5\times
10^{12}$ cm-2; electron relaxation time $\tau=2\times 10^{-13}$ s, velocity
$v=at/\hbar$, the lattice constant of MoS2 $a=3.193$ Å, effective hopping
integral $t=1.10$ eV [107], amplitude of light $A_{0}=3.8\times 10^{12}$
eV$\cdot$ s/C$\cdot m$, the band gap $\Delta=1.16$ eV, and the spin-orbit
coupling strength $\lambda=75$ meV.

The general expression for the (partial) component of photon-drag electric
current density, corresponding to electron transitions from the impurity state
with a quantum number $m$ to the conduction band, reads $(\alpha=x,y)$

$j_{m\alpha}=\frac{2\pi
en_{i}\tau}{\hbar}\int\frac{v_{\alpha}(\textbf{p})d\textbf{p}}{(2\pi\hbar)^{2}}|M_{m}(\textbf{p},\textbf{k})|^{2}\delta(E_{c}(\textbf{p})-E_{i}-\hbar\omega),$
(4.13)

where $n_{i}$ is the impurity concentration, $e$ is the elementary charge,
$\tau$ is the electron relaxation time in the conduction band,
$M_{m}(\mathbf{p},\mathbf{k})=\langle\psi_{s,\eta}(\mathbf{p})|\hat{V}|\chi_{s,m}(\mathbf{p}-\mathbf{k})\rangle$
is the matrix element of impurity-band transitions, and the electron velocity
components read
$v_{x}(\textbf{p})=\eta\sin\theta_{\textbf{p}}\cos\varphi_{\textbf{p}}$ and
$v_{y}(\textbf{p})=-i\eta\sin\theta_{\textbf{p}}\sin\varphi_{\textbf{p}}$;
$E_{i}=\Delta/2-\epsilon^{i}_{s,m,\eta}$ is the impurity energy level. The
detailed calculations for electric current density are discussed in appendix
6.2

Figure 4.3: Spectrum of electric current density due to the transitions from
$m=1$, $\eta=-1$ (solid) and $m=-1$, $\eta=1$ (dashed) impurity states to
spin-up conduction band states for the polarizations of light $\sigma=-1$
(red) and $\sigma=1$ (blue); $\gamma=\epsilon^{i}_{1,1,-1}/\hbar$. The other
parameters are taken the same as in Fig. 4.2.

Considering the $m=0$ impurity state, we find the corresponding matrix
element,

$\displaystyle|M_{0}(\textbf{p,k})|^{2}$ $\displaystyle=$
$\displaystyle(evA_{0})^{2}(\hbar
v)^{2}\frac{2\pi\epsilon^{i}_{s,0,\eta}}{\Delta}$
$\displaystyle\times\left\\{\frac{(\eta+\sigma)^{2}v^{2}(\textbf{p}-\textbf{k})^{2}\cos^{2}\left(\frac{\theta_{s,\eta}}{2}\right)}{\Big{[}v^{2}(\textbf{p}-\textbf{k})^{2}+\epsilon^{i}_{s,0,\eta}(\Delta-s\lambda\eta)\Big{]}^{2}}\right.\left.+\frac{(\eta-\sigma)^{2}(\Delta-s\lambda\eta)^{2}\sin^{2}\left(\frac{\theta_{s,\eta}}{2}\right)}{\Big{[}v^{2}(\textbf{p}-\textbf{k})^{2}+\epsilon^{i}_{s,0,\eta}(\Delta-s\lambda\eta)\Big{]}^{2}}\right\\}.$

Without the loss of generality, let us choose $\mathbf{k}$ to be directed
along the x-axis ($\mathbf{k}=k\hat{x}$). Then, in Eq. (4.13) only the term
containing $\cos\varphi_{\textbf{p}}$ survives and $j_{y}=0$, which reflects
the fact that the photon-drag current should be directed along the photon wave
vector. Substituting Eq. (4.3) in Eq. (4.13), we find

$\begin{split}&j_{0x}=\beta_{0}^{\prime}\Theta[\delta\hbar\omega_{s,0,\eta}]\frac{k\pi}{v}\frac{(\Delta-s\lambda\eta)+\delta\hbar\omega_{s,0,\eta}}{\Big{[}(\delta\hbar\omega_{s,0,\eta})^{2}+(\Delta-s\lambda\eta)\hbar\omega\Big{]}^{2}}\\\
&\times\frac{\delta\hbar\omega_{s,0,\eta}}{(\Delta-s\lambda\eta)+\delta\hbar\omega_{s,0,\eta}}\Bigg{\\{}\frac{4(\Delta-s\lambda\eta)^{2}}{(\delta\hbar\omega_{s,0,\eta})^{2}+(\Delta-s\lambda\eta)\hbar\omega}\\\
&\times\delta\hbar\omega_{s,0,\eta}(\eta-\sigma)^{2}+\Big{[}(\Delta-s\lambda\eta)+\delta\hbar\omega_{s,0,\eta}\Big{]}(\eta+\sigma)^{2}\\\
&\times\left[\frac{4\Big{(}(\Delta-s\lambda\eta)+\delta\hbar\omega_{s,0,\eta}\Big{)}\delta\hbar\omega_{s,0,\eta}}{(\delta\hbar\omega_{s,0,\eta})^{2}+(\Delta-s\lambda\eta)\hbar\omega}-2\right]\Bigg{\\}},\end{split}$
(4.15)

where $\beta_{0}^{\prime}=en_{i}\tau
v^{2}\epsilon_{i}(evA_{0})^{2}/\hbar\Delta$ and
$\delta\hbar\omega_{s,m,\eta}=\hbar\omega-\epsilon^{i}_{s,m,\eta}$. Figure 4.2
shows the spectrum of electric current density for different polarizations of
light and $m=0$, $\eta=-1$. Interesting to note that in the case of the
polarization of light $\sigma=-1$, the electric current flows in the opposite
direction in some regions of frequencies, and then it changes its direction. A
similar behavior (of the inversion of the direction of the electric current
density) was demonstrated in work [101]. Mathematically, it happens due to an
interplay of different terms in Eq. (LABEL:EqCurrent0), shown as dashed
curves. Such behavior is not observed for $\sigma=1$. For the case $m=1$ (and,
correspondingly, $\eta=-1$), we find

$\displaystyle|M_{1}(\textbf{p},\textbf{k})|^{2}$ $\displaystyle=$
$\displaystyle(evA_{0})^{2}(\hbar
v)^{2}\frac{2\pi(\Delta^{2}-\lambda^{2})}{\Delta\epsilon^{i}_{s,1,-1}}$
$\displaystyle\times\left\\{\frac{(\sigma-1)^{2}(\epsilon^{i}_{s,1,-1})^{2}\cos^{2}\left(\frac{\theta_{s,-1}}{2}\right)}{\Big{[}v^{2}(\textbf{p}-\textbf{k})^{2}+\epsilon^{i}_{s,1,-1}(\Delta+s\lambda)\Big{]}^{2}}\right.+\left.\frac{(\sigma+1)^{2}v^{2}(\textbf{p}-\textbf{k})^{2}\sin^{2}\left(\frac{\theta_{s,-1}}{2}\right)}{\Big{[}v^{2}(\textbf{p}-\textbf{k})^{2}+\epsilon^{i}_{s,1,-1}(\Delta+s\lambda)\Big{]}^{2}}\right\\}.$

Again, only the x-component of the current is finite,

$\begin{split}j_{1x}=\beta_{1}^{\prime}\Theta[\delta\hbar\omega_{s,1,-1}]\frac{k\pi}{v}\frac{(\Delta+s\lambda)+\delta\hbar\omega_{s,1,-1}}{\Big{(}(\delta\hbar\omega_{s,1,-1})^{2}+(\Delta+s\lambda)\hbar\omega\Big{)}^{2}}\\\
\times\frac{\delta\hbar\omega_{s,1,-1}}{(\Delta+s\lambda)+2\delta\hbar\omega_{s,1,-1}}\Bigg{\\{}\frac{(\Delta+s\lambda)+\delta\hbar\omega_{s,1,-1}}{(\delta\hbar\omega_{s,1,-1})^{2}+(\Delta+s\lambda)\hbar\omega}\\\
\times
4(\epsilon{{}^{i}_{s,1,-1}})^{2}(\sigma-1)^{2}+\delta\hbar\omega_{s,1,-1}(\sigma+1)^{2}\\\
\times\left(\frac{4\Big{(}(\Delta+s\lambda)+\delta\hbar\omega_{s,1,-1}\Big{)}\delta\hbar\omega_{s,1,-1}}{(\delta\hbar\omega_{s,1,-1})^{2}+(\Delta+s\lambda)\hbar\omega}-2\right)\Bigg{\\}},\end{split}$
(4.17)

where
$\beta_{1}^{\prime}=\beta_{0}^{\prime}[\Delta^{2}-\lambda^{2}]/\epsilon_{i}^{2}$.

Figure 4.3 shows the spectrum of electric current density for $m=-1$ and
$m=1$. We choose $\eta=-1$ for $m=1$ and $\eta=1$ for $m=-1$ . Also, the
$\sigma=1$ polarization gives the region of positive and negative electric
currents for $m=1$ (blue solid curve). For $m=-1$, the $\sigma=-1$ polarized
light gives such electric current (red dashed curve). For a given $\sigma$, we
have optical transitions in both the K (for $m=0$ or 1) and K′ (for $m=1$ or
0) valleys. They can sum up or partially compensate each other.

###  4.4 Symmetry analysis of the electric current density

Let us now analyze the formulas for the electric current density [Eqs.
(LABEL:EqCurrent0) and (4.17)] from the symmetry point of view and compare
them with the case of a graphene monolayer [108]. Single-layer graphene
(without a substrate) possesses the $D_{6h}$ point group, while single-layer
MoS2 has $D_{3h}$ point group. However, for both the groups, the fourth-rank
(generalized conductivity) tensor $\Phi_{\alpha\beta\gamma\mu}$ is the same
[109].

The general expression for the electric current density reads [108]

$j_{x}=T_{1}k_{x}\frac{|E_{x}|^{2}+|E_{y}|^{2}}{2}+T_{2}k_{x}\frac{|E_{x}|^{2}-|E_{y}|^{2}}{2},$
(4.18)

where $E_{x}$ and $E_{y}$ are the components of the electric field; $T_{1}$
and $T_{2}$ are constants describing linear photon drag effect. Since the
electric field is circularly polarized in our case, $\mathbf{E}=E_{0}(1,i)$,
only the first term in Eq. (4.18) remains, and the other one vanishes since
$|E_{x}|=|E_{y}|$. We see that Eqs. (LABEL:EqCurrent0) and (4.17) obey the
symmetry properties of the system.

###  4.5 Light absorption coefficient

Furthermore, let us study the light absorption coefficient for the $m-$th
impurity state. It is defined as the ratio of the energy flux of absorbed
photons and the average energy flux of incident photons [110],
$\alpha_{m}(\hbar\omega)=\hbar\omega W_{m}/P$, where $P$ are the average of
the Poynting flux for the light intensity [111],
$P=n_{r}c\epsilon_{0}\omega^{2}A_{0}^{2}/2$, where $n_{r}$ is the refractive
index of MoS2 and $\epsilon_{0}$ is the vacuum permittivity. The probability
of light absorption in a given valley $\eta$ and from a particular impurity
state $m$ is given by the Fermi golden rule,

$W_{m}(\omega)=\frac{2\pi
n_{i}}{\hbar}\int\frac{d\textbf{p}}{(2\pi\hbar)^{2}}|M_{m}(\textbf{p},0)|^{2}\delta(E_{c}(\textbf{p})-E_{i}-\hbar\omega).$
(4.19)

For the transition from $m=0$ impurity state, we find

$\begin{split}&\alpha_{0}=\frac{2\pi
n_{i}e^{2}v^{2}\epsilon^{i}_{s,0,\eta}}{n_{r}c\omega\Delta\epsilon_{0}}\Theta[\delta\hbar\omega_{s,0,\eta}]\frac{\delta\hbar\omega_{s,0,\eta}}{(\Delta-s\lambda\eta)+2\delta\hbar\omega_{s,0,\eta}}~{}~{}~{}\\\
&\times\frac{\sqrt{4\Big{(}\Delta-s\lambda\eta+\delta\hbar\omega_{s,0,\eta}\Big{)}\delta\hbar\omega_{s,0,\eta}+(\Delta-s\lambda\eta)^{2}}}{\Big{(}(\delta\hbar\omega_{s,0,\eta})^{2}+(\Delta-s\lambda\eta-\epsilon^{i}_{s,0,\eta})\hbar\omega\Big{)}^{2}}\\\
&\times\left\\{(\eta+\sigma)^{2}\Big{(}\Delta-s\lambda\eta+\delta\hbar\omega_{s,0,\eta}\Big{)}^{2}\right.\left.+(\eta-\sigma)^{2}(\Delta-s\lambda\eta)^{2}\right\\},\end{split}$

and for $m=1,\eta=-1$ state,

$\begin{split}\alpha_{1}&=\frac{2\pi
n_{i}e^{2}v^{2}}{n_{r}cw\Delta\epsilon_{0}}\Theta[\delta\hbar\omega_{s,1,-1}]\frac{\Delta^{2}-\lambda^{2}}{\epsilon^{i}_{s,1,-1}}\frac{(\Delta+s\lambda)+\delta\hbar\omega_{s,1,-1}}{(\Delta+s\lambda)+2\delta\hbar\omega_{s,1,-1}}\\\
\times&\frac{\sqrt{4\Big{(}\Delta+s\lambda+\delta\hbar\omega_{s,1,-1}\Big{)}\delta\hbar\omega_{s,1,-1}+(\Delta+s\lambda)^{2}}}{\Big{(}(\delta\hbar\omega_{s,1,-1})^{2}+(\Delta+s\lambda-\epsilon^{i}_{s,1,-1})\hbar\omega\Big{)}^{2}}\\\
\times&\left\\{(\sigma-1)^{2}(\epsilon^{i}_{s,1,-1})^{2}+(\sigma+1)^{2}(\delta\hbar\omega_{s,1,-1})^{2}\right\\}.\end{split}$
(4.20)

Figure 4.4: Spectrum of absorbance for $m=-1$ (red), $m=0$ (green), and $m=1$
(blue); $\sigma=1$ (dashed) and $\sigma=-1$ (solid).

Figure 4.4 shows the spectra of absorbance. For the transitions from the state
$m=1$, the magnitude of absorbance is higher for the $\sigma=-1$ light, but
the valley dependence disappears by increasing the photon energy. For the
transitions from the state $m=0$, both the polarized lights give a comparable
contribution.

It is enlightening to compare the matrix element corresponding to impurity-
band transitions with the matrix element for the interband transitions,
$|M_{cv}(\mathbf{}{p})|^{2}$ [93, 112]. The valley selectivity for interband
transitions is, to a large extent, satisfied only at small values of momentum
$p$, giving $|M_{cv}(0)|^{2}\propto(\eta+\sigma)^{2}$. In our case, the
transitions from $m=0$ impurity states are strongly suppressed due to
$|M_{0}(0,0)|^{2}\rightarrow 0$, whereas for $m=\pm 1$ we find $|M_{m=\pm
1}(0,0)|^{2}\propto\epsilon_{i}^{2}(\sigma+\eta)^{2}$ under the condition
$m+\eta=0$. It means that the valley selectivity takes place for orbital
impurity states $m=\pm 1$ (and thus, we have $\exp(im\varphi)\neq 1$),
reflecting the chirality of the band electron wavefunction. These general
conclusions are supported by numerical analysis. For instance, the absorption
coefficient in Fig. 4.4 is large in the vicinity of the threshold for
$m=1,\eta=-1$ state at $\sigma=-1$ polarization.

###  4.6 Summary

We have studied the selection rules for the light-induced transitions from
impurity states to the conduction band in two-dimensional gapped Dirac
materials. We calculated and investigated the absorption coefficient and the
photon-drag-induced electric current. For clarity, we used the shallow
impurity potential model. Nevertheless, this model correctly reflects the
selection rules of any impurity possessing azimuthal symmetry. Thus, our
conclusions on the optical selection rules are sufficiently general.

##  Chapter 5 Summary

Chapter 1 is devoted to a general introduction to exciton-polaritons. We
discussed the formation of excitons in bulk semiconductors, quantum well, and
transition metal dichalcogenide materials. We derived the dispersion of
exciton-polaritons and the formation of polariton condensates with a brief
description of the history in experiments and numerical modeling to simulate
the appearance and evolution of polariton condensates.

In chapter 2, we studied the partial quantum revivals of compact localized
states loaded in a distorted kagome lattice. In the tight-binding model, the
reappearance of the initial compact-localized states took place in the
neighboring sites of the lattice, but the revival of the original compact-
localized states is not observed. In contrast, in the continuous model,
apparent revivals of the polariton compact-localized states were observed by
squeezing the lattice in either a vertical or horizontal direction which
deforms the nearly-flat band.

In chapter 3, we demonstrated a quantum polariton vortex vanishment excited by
a ring-shaped Laguerre-Gaussian pump. The competition between ground and
vortex states was observed in pulsed excitation by changing the diameter of
the Laguerre-Gaussian pump. It affected the transition between the two states
and determines the multistate condensate population. For the simulation, we
utilized the driven-dissipative Gross-Pitaevskii equation coupled with
incoherent and coherent reservoirs. An external pumping source directly
provided particles in the inactive reservoir. Particles in the active
reservoir were refilled from the inactive reservoir and then the polariton
condensate was replenished by the active reservoir We showed that the
numerical simulation was consistent with the experimental results.

In chapter 4, we studied the valley selection rules for the light-induced
transitions from impurity states to the conduction band in two-dimensional
gapped Dirac materials. We calculated the absorption coefficient and the
photon-drag-induced electric current for the various transitions with
different light polarizations. The electric current induced by the transition
from impurities showed the flow of opposite direction in some frequency
regions with specific polarization of light. For the transitions from the
impurity state $m=1$, the absorption coefficient had a higher magnitude for
the $\sigma=-1$ light, but the valley dependence vanished by increasing the
frequency of the light. However, with the impurity state $m=0$, both the
polarized lights showed comparable behaviors.

##  Chapter 6 Appendix

###  6.1 Dynamics of polariton compact localized states in kagome x-squeezed
strip

We discuss the kagome strips squeezed in $x$-direction, as is shown in Fig.
6.1(a) and (b). Blue pillars are at the original position of the kagome
lattice, whereas red pillars are shifted closer to each other horizontally to
make the adjacent pillars touch each other. We illuminate the LG beam at the
center of the strip [Fig. 6.1(c)], using the following parameters:
$\tau_{p}=40\,\mathrm{ps}$, $G=0.005\,\mathrm{ps}^{-1}\cdot\mu\mathrm{m}^{2}$,
$\alpha=0.1\,\mu\mathrm{eV}\cdot\mu\mathrm{m}^{2}$ and
$\tau_{R}=100\,\mathrm{ps}$. We note that with the Laguerre-Gaussian pump, it
is very convenient to excite CLSs in kagome lattice compared with, e.g., Lieb
lattice [52] due to the absence of parasite pumping of the sites where we
expect to have destructive interference.

The condensate propagates in both up and down directions. The initial CLS
disappears completely at about $80\,\mathrm{ps}$, while two CLS condensates
are formed next to the original one [see Fig. 6.1(d)]. Later at around
$120\,\mathrm{ps}$, the condensates partially move back and form the CLS at
its original position, which constitutes the revival phenomenon, shown in Fig.
6.1(e). We note that this is a partial revival since the whole state of the
quantum system does not fully coincide with its original state, and it is
characterized by the presence of a few CLS condensates further away from the
creation center.

Figure 6.1(f) shows the dynamics of the number of polaritons in the original
CLS [at the center of the strip, as in 6.1(c)] as a ratio of the total number
of particles in the whole strip for different incoherent pumping intensities,
$P_{in}$. By increasing $P_{in}$, the number of particles in the strip also
increases. This figure also provides additional confirmation of the revival
effect.

Figure 6.1(g) shows the ratio of the total number of polaritons in the strip
(consisting of nine plaquettes) and the total number of particles in the
system. If $P_{in}$ is less than the pumping threshold
$P_{th}=(G\tau_{R}\tau_{P})^{-1}$, the total number of particles decays. When
$P_{in}=P_{th}$, the losses due to the finite lifetime become saturated due to
the compensation by the incoherent pumping reservoir. Nevertheless, this
background incoherent pump does not hinder the phenomenon we are interested
in: the revival of the CLS persists, as it is shown in Fig. 6.1(c-e) with the
condition $P_{in}=P_{th}$.

Figure 6.1:  (a) A kagome $x$-squeezed strip. (b) Red pillars are squeezed
horizontally. (c) Snapshot of CLSs at 10 ps. (d) At 80 ps, the CLS at the
center vanishes, and neighboring CLS’s emerge. (e) At 120 ps, one can see the
partial revival of the CLS at the center. (f) Time evolution of the ratio of
the CLS polaritons at the central plaquette of the lattice and the total
number of polaritons for different incoherent pumping intensities. (g) Time
evolution of the ratio of the whole CLS particles and the total number of
particles with different incoherent pumping intensities.

###  6.2 Detailed calculation of $j_{0x}$ and $j_{1x}$

####  6.2.1 Derivation of $j_{0x}$

To show the detailed derivation of $j_{0x}$, we start from the matrix element
$M_{0}(\textbf{p},\textbf{k})$ in Eq. 4.3.

$\displaystyle M_{0}(\textbf{p},\textbf{k})$
$\displaystyle=\bra{\psi_{s,\eta}(\textbf{p})}\hat{V}\ket{\chi_{s,0}(\textbf{p}-\textbf{k})}$
$\displaystyle=e(A_{0}\hat{\textbf{x}}+A_{0}i\sigma\hat{\textbf{y}})\hbar
v\sqrt{\frac{2\pi\epsilon^{i}_{s,0,\eta}}{\Delta}}\begin{pmatrix}\cos(\frac{\theta_{s,\eta}}{2})&\sin(\frac{\theta_{s,\eta}}{2})e^{i\eta\varphi_{\textbf{p}}}\end{pmatrix}$
$\displaystyle\times(v\begin{pmatrix}0&\eta\\\
\eta&0\end{pmatrix}\hat{\textbf{x}}+v\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}\hat{\textbf{y}})\begin{pmatrix}\frac{\Delta-s\lambda\eta}{(vp)^{2}+\epsilon^{i}_{s,0,\eta}(\Delta-s\lambda\eta)}\\\
\frac{vp}{(vp)^{2}+\epsilon^{i}_{s,0,\eta}(\Delta-s\lambda\eta)}\end{pmatrix}$
$\displaystyle=(evA_{0})(\hbar
v)\sqrt{\frac{2\pi\epsilon^{i}_{s,0,\eta}}{\Delta}}\Bigg{\\{}(\eta+\sigma)\frac{v(\textbf{p}-\textbf{k})}{v^{2}(\textbf{p}-\textbf{k})^{2}+\epsilon^{i}_{s,0,\eta}(\Delta-s\lambda\eta)}\cos(\frac{\theta_{s,\eta}}{2})$
$\displaystyle+(\eta-\sigma)\frac{\Delta-s\lambda\eta}{v^{2}(\textbf{p}-\textbf{k})^{2}+\epsilon_{i}(\Delta-s\lambda\eta)}\sin(\frac{\theta_{s,\eta}}{2})e^{i\eta\varphi_{\textbf{p}}}\Bigg{\\}}.$
(6.1)

Based on Eq. 4.13, the current density $j_{0x}$ can be written as

$j_{0x}=\frac{2\pi
en_{i}\tau}{\hbar}\int\frac{v_{x}(\textbf{p})d\textbf{p}}{(2\pi\hbar)^{2}}|M_{0}(\textbf{p},\textbf{k})|^{2}\delta(E_{c}(\textbf{p})-E_{i}-\hbar\omega).$
(6.2)

By plugging Eq. 6.1 into Eq. 6.2,

$\displaystyle j_{0x}$ $\displaystyle=\frac{2\pi
en_{i}\tau}{\hbar}\frac{1}{(2\pi\hbar)^{2}}\int\frac{vp}{E_{c}(\textbf{p})}\cos\varphi_{\textbf{p}}d\textbf{p}(evA_{0})^{2}(\hbar
v)^{2}\frac{2\pi\epsilon_{i}}{\Delta}$
$\displaystyle\times\Bigg{\\{}\frac{(\eta+\sigma)^{2}v^{2}(\textbf{p}-\textbf{k})^{2}\cos^{2}(\frac{\theta_{s,\eta}}{2})}{\Big{[}v^{2}(\textbf{p}-\textbf{k})^{2}+\epsilon_{i}(\Delta-s\lambda\eta)\Big{]}^{2}}+\frac{(\eta-\sigma)^{2}(\Delta-\lambda\eta)^{2}\sin^{2}(\frac{\theta_{s,\eta}}{2})}{\Big{[}\Big{(}v(\textbf{p}-\textbf{k})\Big{)}^{2}+\epsilon_{i}(\Delta-s\lambda\eta)\Big{]}^{2}}\Bigg{\\}}\delta(E_{c}(\textbf{p})-E_{i}-\hbar
w).$ (6.3)

The current density can be split into two contributions
$j_{0x}=j^{1}_{0x}+j^{2}_{0x}$

$\displaystyle j^{1}_{0x}=$
$\displaystyle\beta^{\prime}_{0}\int\frac{vp}{E_{\uparrow}}\cos\varphi
d\textbf{p}\frac{(\eta+\sigma)^{2}\Big{(}v(\textbf{p}-\textbf{k})\Big{)}^{2}}{\Big{[}\Big{(}v(\textbf{p}-\textbf{k})\Big{)}^{2}+\epsilon_{i}(\Delta-\lambda\eta)\Big{]}^{2}}\cos^{2}(\frac{\theta_{s,\eta}}{2})\delta(E_{c}(\textbf{p})-E_{i}-\hbar
w),$ (6.4) $\displaystyle j^{2}_{0x}=$
$\displaystyle\beta^{\prime}_{0}\int\frac{vp}{E_{\uparrow}}\cos\varphi
d\textbf{p}\frac{(\eta-\sigma)^{2}(\Delta-\lambda\eta)^{2}}{\Big{[}\Big{(}v(\textbf{p}-\textbf{k})\Big{)}^{2}+\epsilon_{i}(\Delta-\lambda\eta)\Big{]}^{2}}\sin^{2}(\frac{\theta_{s,\eta}}{2})\delta(E_{c}(\textbf{p})-E_{i}-\hbar
w)$ (6.5)

The definition of $\cos{\theta_{s,\eta}}$ and $\sin{\theta_{s,\eta}}$ reads

$\cos^{2}{\frac{\theta_{s,\eta}}{2}}=\frac{1+\cos{\theta_{s,\eta}}}{2}=\frac{2E_{s,\eta}(\textbf{p})+(\Delta-s\lambda\eta)}{4E_{s,\eta}(\textbf{p})},$
(6.6)
$\sin^{2}{\frac{\theta_{s,\eta}}{2}}=\frac{1-\cos{\theta_{s,\eta}}}{2}=\frac{2E_{s,\eta}(\textbf{p})-(\Delta-s\lambda\eta)}{4E_{s,\eta}(\textbf{p})}.$
(6.7)

By assumption $p\gg\textbf{k}$, the terms in $j_{1}$ and $j_{2}$ can be
simplified as

$\displaystyle\frac{1}{\Big{[}\Big{(}v(\textbf{p}-\textbf{k})\Big{)}^{2}+\epsilon_{i}(\Delta-\lambda\eta)\Big{]}^{2}}$
$\displaystyle\approx\frac{1}{\Big{[}(vp)^{2}+\epsilon_{i}(\Delta-\lambda\eta)\Big{]}^{2}}\Bigg{\\{}1+\frac{4pv^{2}\cos\varphi
k}{(vp)^{2}+\epsilon_{i}(\Delta-\lambda\eta)}\Bigg{\\}},$ (6.8)
$\displaystyle\frac{(\textbf{p}-\textbf{k})^{2}}{\Big{[}\Big{(}v(\textbf{p}-\textbf{k})\Big{)}^{2}+\epsilon_{i}(\Delta-\lambda\eta)\Big{]}^{2}}$
$\displaystyle\approx\frac{1}{\Big{[}(vp)^{2}+\epsilon_{i}(\Delta-\lambda\eta)\Big{]}^{2}}\Bigg{\\{}p^{2}-2pk\cos\varphi$
$\displaystyle+\frac{4p^{3}v^{2}\cos\varphi
k}{(vp)^{2}+\epsilon_{i}(\Delta-\lambda\eta)}\Bigg{\\}}.$ (6.9)

By substituting Eq. (6.6) - Eq. (6.9) into Eq. (6.4) and Eq. (6.5), we obtain

$\displaystyle j^{1}_{0x}=$
$\displaystyle\beta^{\prime}_{0}\Theta[\delta\hbar\omega_{s,0,\eta}]\frac{k\pi}{v}\frac{(\eta+\sigma)^{2}\Big{[}(\Delta-s\lambda\eta)+\delta\hbar\omega_{s,0,\eta}\Big{]}^{2}}{\Big{[}(\delta\hbar\omega_{s,0,\eta})^{2}+(\Delta-s\lambda\eta)\hbar\omega\Big{]}^{2}}\frac{\delta\hbar\omega_{s,0,\eta}}{(\Delta-s\lambda\eta)+\delta\hbar\omega_{s,0,\eta}}$
(6.10)
$\displaystyle\times\left(\frac{4\Big{(}(\Delta-s\lambda\eta)+\delta\hbar\omega_{s,0,\eta}\Big{)}\delta\hbar\omega_{s,0,\eta}}{(\delta\hbar\omega_{s,0,\eta})^{2}+(\Delta-s\lambda\eta)\hbar\omega}-2\right),$
(6.11) $\displaystyle j^{2}_{0x}=$
$\displaystyle\beta^{\prime}_{0}\Theta[\hbar\omega-\epsilon_{i}]\frac{k\pi}{v}\frac{4(\eta-\sigma)^{2}(\Delta-s\lambda\eta)^{2}}{(\Delta-s\lambda\eta)+\delta\hbar\omega_{s,0,\eta}}\frac{\Big{[}(\Delta-s\lambda\eta)+\delta\hbar\omega_{s,0,\eta}\Big{]}(\delta\hbar\omega_{s,0,\eta})^{2}}{\Big{[}(\delta\hbar\omega_{s,0,\eta})^{2}+(\Delta-s\lambda\eta)\hbar\omega\Big{]}^{3}}.$
(6.12)

, where $\beta_{0}^{\prime}=en_{i}\tau
v^{2}\epsilon_{i}(evA_{0})^{2}/\hbar\Delta$ and
$\delta\hbar\omega_{s,m,\eta}=\hbar\omega-\epsilon^{i}_{s,m,\eta}$. The
summation of $j_{1}$ and $j_{2}$ gives Eq. (4.3)

####  6.2.2 Derivation of $j_{1x}$

With the same method to obtain $M_{0}(\textbf{p},\textbf{k})$,
$M_{1}(\textbf{p},\textbf{k})$ reads

$\displaystyle M_{1}(\textbf{p},\textbf{k})$
$\displaystyle=\bra{\psi_{s,\eta}(\textbf{p})}\hat{V}\ket{\chi_{s,1}(\textbf{p}-\textbf{k})}.$
$\displaystyle=(evA_{0})(\hbar
v)\sqrt{\frac{2\pi(\Delta^{2}-\lambda^{2})}{\Delta\epsilon^{i}_{s,1,-1}}}\Bigg{\\{}\frac{(\sigma-1)\cos(\frac{\theta_{s,-1}}{2})}{v^{2}(\textbf{p}-\textbf{k})^{2}+\epsilon^{i}_{s,1,-1}(\Delta+s\lambda)}$
$\displaystyle-\frac{(\sigma+1)v(\textbf{p}-\textbf{k})\sin(\frac{\theta_{s,-1}}{2})}{v^{2}(\textbf{p}-\textbf{k})^{2}+\epsilon^{i}_{s,1,-1}(\Delta+s\lambda)}e^{-i\varphi_{\textbf{p}}}\Bigg{\\}}.$
(6.13)

Based on Eq. (4.13), the current density $j_{1x}$ can be written as

$j_{1x}=\frac{2\pi
en_{i}\tau}{\hbar}\int\frac{v_{x}(\textbf{p})d\textbf{p}}{(2\pi\hbar)^{2}}|M_{1}(\textbf{p},\textbf{k})|^{2}\delta(E_{c}(\textbf{p})-E_{i}-\hbar\omega).$
(6.14)

By substituting Eq. (6.2.2) into Eq. (6.14).

$\begin{split}j_{1x}=&\frac{2\pi
en_{i}\tau}{\hbar}\frac{1}{(2\pi\hbar)^{2}}\int\frac{vp}{E_{c}(\textbf{p})}\cos\varphi_{\textbf{p}}d\textbf{p}(evA_{0})^{2}(\hbar
v)^{2}\frac{2\pi(\Delta^{2}-(\lambda)^{2})}{\Delta\epsilon_{i}}\\\
&\times\Bigg{\\{}\frac{(\sigma-1)^{2}(\epsilon^{i}_{s,1,-1})^{2}}{\Big{[}\Big{(}v(\textbf{p}-\textbf{k})\Big{)}^{2}+\epsilon^{i}_{s,1,-1}(\Delta+s\lambda)\Big{]}^{2}}\cos^{2}(\frac{\theta_{s,-1}}{2})\\\
&+\frac{(\sigma+1)^{2}v^{2}(\textbf{p}-\textbf{k})^{2}}{\Big{[}\Big{(}v(\textbf{p}-\textbf{k})\Big{)}^{2}+\epsilon^{i}_{s,1,-1}(\Delta+s\lambda)\Big{]}^{2}}\sin^{2}(\frac{\theta_{s,-1}}{2})\Bigg{\\}}\delta(E_{c}(\textbf{p})-E_{i}-\hbar
w).\end{split}$ (6.15)

Let
$\beta_{1}^{\prime}=\beta_{0}^{\prime}[\Delta^{2}-\lambda^{2}]/\epsilon_{i}^{2}$
and split $j_{1x}$ into $j^{1}_{1x}$ and $j^{2}_{1x}$.

$\displaystyle j^{1}_{1x}=$
$\displaystyle\beta^{\prime}_{1}\int\frac{vp}{E_{c}(\textbf{p})}\cos\varphi_{\textbf{p}}d\textbf{p}\frac{(\eta+\sigma)^{2}\epsilon_{i}^{2}\cos^{2}(\frac{\theta_{s,-1}}{2})}{\Big{[}\Big{(}v(\textbf{p}-\textbf{k})\Big{)}^{2}+\epsilon_{i}(\Delta-\lambda\eta)\Big{]}^{2}}\delta(E_{c}(\textbf{p})-E_{i}-\hbar
w),$ (6.16) $\displaystyle j^{2}_{1x}=$
$\displaystyle\beta^{\prime}_{1}\int\frac{vp}{E_{c}(\textbf{p})}\cos\varphi_{\textbf{p}}d\textbf{p}\frac{(\eta-\sigma)^{2}v^{2}(\textbf{p}-\textbf{k})^{2}\sin^{2}(\frac{\theta_{s,-1}}{2})}{\Big{[}\Big{(}v(\textbf{p}-\textbf{k})\Big{)}^{2}+\epsilon_{i}(\Delta-\lambda\eta)\Big{]}^{2}}\delta(E_{c}(\textbf{p})-E_{i}-\hbar
w).$ (6.17)

By substituting Eq. (6.6) - Eq. (6.9) into Eq. (6.16) and Eq. (6.17),

$\displaystyle j^{1}_{1x}=$
$\displaystyle\beta^{\prime}_{1}\Theta[\delta\hbar\omega_{s,0,\eta}]\frac{k\pi}{v}\frac{(\sigma+1)^{2}\Big{[}(\Delta+s\lambda)+\delta\hbar\omega_{s,1,-1}\Big{]}}{\Big{[}(\delta\hbar\omega_{s,1,-1})^{2}+(\Delta+s\lambda)\hbar\omega\Big{]}^{2}}\frac{(\delta\hbar\omega_{s,1,-1})^{2}}{(\Delta+s\lambda)+2\delta\hbar\omega_{s,1,-1}}$
$\displaystyle\times\left(\frac{4\Big{(}(\Delta+s\lambda)+\delta\hbar\omega_{s,1,-1}\Big{)}\delta\hbar\omega_{s,1,-1}}{(\delta\hbar\omega_{s,1,-1})^{2}+(\Delta+s\lambda)\hbar\omega}-2\right).$
(6.18) $\displaystyle j^{2}_{1x}=$
$\displaystyle\beta^{\prime}_{1}\Theta[\hbar\omega-\epsilon_{i}]\frac{k\pi}{v}\frac{4(\sigma-1)^{2}(\epsilon^{i}_{s,1,-1})^{2}}{(\Delta+s\lambda)+2\delta\hbar\omega_{s,1,-1}}\frac{\Big{[}(\Delta+s\lambda)+\delta\hbar\omega_{s,1,-1}\Big{]}(\delta\hbar\omega_{s,1,-1})}{\Big{[}(\delta\hbar\omega_{s,1,-1})^{2}+(\Delta+s\lambda)\hbar\omega\Big{]}^{3}}.$
(6.19)

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