# Interplay of Kekulé distortions and laser fields in graphene††thanks: We
dedicate this work to Bertrand in the celebration of his 60th anniversary.

A. Lopez  $|$ ${}^{\textsf{\footnotesize{\mbox{?}}}}$, F. Mireles  $|$
${}^{\textsf{\footnotesize{\mbox{?}}}}$ Corresponding author:
alexlop<EMAIL_ADDRESS>
(Received November 8, 2022, in final form November 28, 2022)

###### Abstract

We study the interplay between a Kekulé patterned distortion in monolayer
graphene and a monochromatic laser irradiation, focusing in the long
wavelength approximation of its Landau level structure. Exploiting the
symmetries of the system, we calculate in the static regime an exact
analytical solution for the energy spectrum and its eigenstates, which in turn
allows us to find close expressions for the polarizations in the valley and
pseudospin degrees of freedom. We find that due to the valley-momentum
coupling, the valley polarization exhibits a distinct magnetic response for
the two $K$-valley components. Owing to that, the introduction of the
radiation field mixes the Landau levels, and it is shown that by tuning the
system to resonance leads to a larger valley-polarization response as compared
to the non-resonant scenario.

Key words: graphene, Landau levels, Kekulé distortion, Floquet theory,
polarization

###### Abstract

Ìè âèâàìî âçàìîçâ’ÿçîê ìæ ñòðóêòóðíèì âèêðèâëåííÿì Êåêóëå ó ìîíîøàðîâîìó
ãðàôåí òà ìîíîõðîìàòèíèì ëàçåðíèì îïðîìíþâàííÿì, ôîêóñóþè óâàãó íà
äîâãîõâèëüîâîìó íàáëèæåíí äëÿ ñòðóêòóðè ðâíâ Ëàíäàó. Âèêîðèñòîâóþè äàí ùîäî
ñèìåòð¿ ñèñòåìè, â ñòàòèíîìó ðåæèì îòðèìàíî òîíèé àíàëòèíèé ðîçâ’ÿçîê äëÿ
åíåðãåòèíîãî ñïåêòðó òà éîãî âëàñíèõ ñòàíâ, ùî, ó ñâîþ åðãó, äîçâîëÿ íàì
çíàéòè íàáëèæåí âèðàçè äëÿ ïîëÿðèçàö¿ ç ñòóïåíÿìè ñâîáîäè ò. çâ. ‘‘äîëèíè’’ òà
ïñåâäîñïíó. Ç’ÿñîâàíî, ùî çàâäÿêè âçàìîä¿ ‘‘äîëèíà-ìïóëüñ’’ ïîëÿðèçàöÿ ïëîñêî¿
çîíè ìà òêî âèðàæåíèé ìàãíòíèé âäãóê äëÿ äâîõ êîìïîíåíòâ $K$-çîíè. Çàâäÿêè
öüîìó ââåäåííÿ ïîëÿ âèïðîìíþâàííÿ çìøó ðâí Ëàíäàó. Ïîêàçàíî, ùî íàëàøòóâàííÿ
ñèñòåìè íà ðåçîíàíñ ïðèçâîäèòü äî ñèëüíøîãî âäãóêó ïîëÿðèçàö¿ ïëîñêî¿ çîíè ó
ïîðâíÿíí ç íåðåçîíàíñíèì ñöåíàðì.

Ключов слова: ãðàôåí, ðâí Ëàíäàó, âèêðèâëåííÿ Êåêóëå, òåîðÿ Ôëîêå, ïîëÿðèçàöÿ

##  1 Introduction

Since the experimental realization of graphene [1, 2, 3] and the related
honeycomb lattice two-dimensional materials, such as silicene [4, 5, 6],
borophene, dichalcogenides, among others, a plethora of proposals to engineer
and harness their electronic and optical properties have emerged. These two-
dimensional materials have also emerged as platforms for the realization of
topological phases of matter within the static scenario [7, 8, 9, 10, 11, 12].
The topological insulating phases occur whenever the system exhibits gapless
states that conduct along the boundaries of the two dimensional material.
These nontrivial topological properties have also been realized in one [13,
14] and three dimensional systems [15]. Within the driven fields regime [16,
17, 18, 19], the emergence of Floquet topological insulators has been put
forward. In this case, the systems in a topologically trivial state in the
static regime are driven to nontrivial topological phases by means of
periodically driven interactions [22, 23, 20, 21, 24, 25, 26]. Thus, in a
Floquet topological insulator, the generation of the boundary gapless edge
states emerges as a consequence of the topological modifications introduced in
the energy band spectrum. Interestingly, these topological phases have also
been experimentally realized in optical lattices [27]. Moreover, the photonic
experimental realization of Floquet topological insulators is experimentally
realized in reference [28] by means of a photonic lattice, exhibiting
topologically protected transport of visible light on the lattice edges.
Additional theoretical work on the optical nonlinear effects in solids [29]
shows that these nonlinear effects can also be properly described within the
framework of Berry curvatures.

Apart from periodically driven interactions, other means to engineer the
topological properties of low dimensional materials have been proposed in the
literature. One interesting realization consists in the generation of a Kekulé
distortion [30, 31, 32] in which the bonding parameters can be modified. This
in turn implies that the size of the unit cell is tripled due to the emergent
dimerization which amounts to the coupling points in the Brillouin zone which
are separated by $\vec{G}=\vec{K}_{+}-\vec{K}_{-}$, with $\vec{K}_{\pm}$ being
the Dirac points in graphene. This distortion in turn, results in the merging
of the two Dirac cones at the center of the Brillouin zone, producing either a
gap (Kek-O) or a superposition of two cones with different Fermi velocities
(Kek-Y) [37, 33, 34, 36, 35], depending upon the bond length between C-C
atoms. The physics of the merging of the Dirac cones has also attracted
attention, as it has been shown that it could lead to semi-Dirac materials in
which, the charge carriers possess an anisotropic energy spectrum as their
long wavelength effective Hamiltonian consists of a combination of linear and
quadratic dispersion relations along two perpendicular directions in the plane
[38, 39, 40, 41]. In addition, the Kekulé distortions can also lead to valley
engineering by strain whose strength could serve as control parameter for
intervalley scattering processes [42]. Upon extending the study to multilayer
systems, there were proposals for bilayer Kekulé distorted graphene in which
the quasiparticle’s masses can be tuned electrostatically. This in turn might
lead to versatile platforms for valleytronics with “multi flavor” models [43].
We refer the interested reader to the review [44] which presents a
comprehensible treatment of the role of strain induced distortions in
graphene. In a recent work, Mojarro et al. [45] analyze the interplay of
Kekulé distortions and electromagnetic radiation perpendicular to the
monolayer graphene sample. One natural extension of the model presented in
reference [45] is to assess what are the consequences of the interplay of
radiation fields and Kekulé distortions in the Landau level structure of
monolayer graphene. Previous works considered the interplay in monolayer
graphene of quantizing magnetic fields and monochromatic radiation, analyzing
the emergent Floquet-Hofstadter spectrum [46, 47] whereas further works
explored the light-induced anomalous Hall effect [48], providing a microscopic
theory to explain the underlying mechanisms arising from the light-matter
interactions. This light induced anomalous Hall effect in graphene was
recently experimentally realized [49], showing that the dependence of the
effect on a gate potential used to tune the Fermi level reveals multiple
features that reflect a Floquet-engineered topological band structure. Further
theoretical work [50] shows that the chirality of light permits switching on
and off the Hall edge conductance, providing additional control of
topologically protected transport.

To explore the issue of the role of radiation fields and Kekulé distortions
and to provide a physical insight into this scenario is the aim of this work.
Therefore, we theoretically analyze the dynamical manipulation of the Landau
level structure of charge carriers on Kekulé distorted graphene when a
periodically driven radiation field is applied perpendicular to the sample. As
Mojarro et al. show, only the so-called KeK-Y bond pattern couples to the
radiation field in the valley degree of freedom. Hence, we focus on this
configuration. To deal with the driven regime, we make use of Floquet’s
theorem to recast the dynamics in an explicitly time-independent fashion,
providing an analytical description of the driven evolution of the relevant
physical quantities. The paper is organized as follows. In section 2 we
present the model for the long wavelength approximation of Landau levels
induced on a Kekulé distorted honeycomb lattice in presence of circularly
polarized electromagnetic radiation, perpendicularly incident upon the sample.
Taking advantage of the rotational invariance of the system, we derive the
exact effective time-independent Floquet Hamiltonian. Next, in section 3 we
present and discuss the main results of the manuscript whereas in section 4 we
present the concluding remarks. Additional calculations are given in the
appendix.

##  2 Model

We study the long wavelength effective Hamiltonian for spinless Dirac fermions
on a distorted honeycomb lattice with a Kekulé bond pattern. The physical
manifestation of the Kekulé distortion implies that the $K$ and $K^{\prime}$
Dirac points or valleys are coupled by the wave vector
$\vec{G}=\vec{K}_{+}-\vec{K}_{-}$ of the Kekulé bond texture and they are
folded onto the center of the superlattice Brillouin zone. For the static
scenario, we consider the Landau levels in monolayer graphene with a Kekulé-Y
bond pattern when the sample is subjected to a perpendicular static quantizing
magnetic field $B$ and to monochromatic radiation circularly polarized
frequency $\omega$. Within these considerations at the $K$ Dirac point, the
Hamiltonian reads $\mathcal{H}(t)=\mathcal{H}_{0}+\mathcal{V}(t)$. The first
term describes a static contribution that in the long wavelength approximation
is given explicitly by [35, 45]

$\mathcal{H}_{0}=v_{\tau}\tau_{0}\otimes\vec{\pi}\cdot\vec{\sigma}+v_{\tau}\vec{\pi}\cdot\vec{\tau}\otimes\sigma_{0},$
(2.1)

where $\vec{\pi}=\vec{p}+e\vec{A}$, with $-e$ being the electron charge,
$(p_{x},p_{y})$ being the momentum measured from the $K$ Dirac point,
$\vec{A}$ is the static vector potential associated to a perpendicular
magnetic field $\nabla\times\vec{A}=\vec{B}$, and $\vec{\sigma}$
($\vec{\tau}$) represents a vector of Pauli matrices in the sublattice
(valley) degree of freedom, whereas $\sigma_{0}$ and $\tau_{0}$ represent the
$2\times 2$ identity matrix, in the corresponding degree of freedom. Finally,
$v_{\sigma}=v_{F}$ for pristine graphene, whereas $v_{\tau}=v_{F}\Delta_{0}$
is the corrected Fermi velocity arising from the valley coupling due to the
Kekulé distortion, within the approximation regime $\Delta_{0}\ll 1$. Before
dealing with the radiation effects, we remark that we can find an exact
analytical solution to the spectrum and eigenstates of $\mathcal{H}_{0}$ by
means of a different approach as the one presented in [35]. As it is discussed
in reference [35], the static magnetic field implies that the valley and
pseudospin contributions do not commute with each other, and yet the spectrum
can be solved exactly. To solve the Schrödinger equation
$\mathcal{H}_{0}|\Psi\rangle=E|\Psi\rangle$, we define the raising and the
lowering operators

$\displaystyle a=\frac{\ell_{B}(\pi_{x}-\mathrm{i}\pi_{y})}{\sqrt{2}\hbar},$
$\displaystyle
a^{\dagger}=\frac{\ell_{B}(\pi_{x}+\mathrm{i}\pi_{y})}{\sqrt{2}\hbar},$ (2.2)

and write down the static Hamiltonian as

$\mathcal{H}_{0}=\hbar\omega_{c}\left(\begin{array}[]{cccc}0&a&\Delta_{0}a&0\\\
a^{\dagger}&0&0&\Delta_{0}a\\\ \Delta_{0}a^{\dagger}&0&0&a\\\
0&\Delta_{0}a^{\dagger}&a^{\dagger}&0\end{array}\right),$ (2.3)

with $\omega_{c}=\sqrt{2}v_{F}/\ell_{B}$ describing the cyclotron frequency.
In addition, $v_{F}\approx 10^{6}$ m/s is the Fermi velocity, whereas
$\ell_{B}^{-2}=eB/\hbar$ is the magnetic length defined in terms of the
strength of the quantizing magnetic field $B$ and $-e$ is the electron charge.
In reference [35], the spectrum of the Hamiltonian in equation (2.3) is solved
using an artificial site energy term of small value that afterwards is made to
vanish, so that the solutions are shown to be independent of the on site
energy parameters. Alternatively, one can instead exploit the symmetries of
the system to solve the Landau level spectrum in a more physically appealing
form. To this end, we notice first that the static Hamiltonian commutes with
the number operator

$\mathcal{N}_{a}=a^{\dagger}a\tau_{0}\otimes\sigma_{0}+\frac{\sigma_{z}+\tau_{z}}{2},$
(2.4)

whereby we can look for simultaneous eigenstates of this number operator and
the static Hamiltonian $\mathcal{H}_{0}$. If we write

$|\Psi_{\sigma\tau}(n)\rangle=\left(\begin{array}[]{c}A^{\tau}_{n}|n-1\rangle\\\
\sigma B^{\tau}_{n}|n\rangle\\\ C^{\tau}_{n}|n\rangle\\\ \sigma
D^{\tau}_{n}|n+1\rangle\end{array}\right),$ (2.5)

with the states $|n\rangle$ being eigenstates of the Hermitian operator
$n=a^{\dagger}a$, i.e., $n=a^{\dagger}a|n\rangle=n|n\rangle$, it is easily
verified that
$\mathcal{N}_{a}|\Psi_{\sigma\tau}(n)\rangle=n|\Psi_{\sigma\tau}(n)\rangle$.
In absence of quantizing magnetic field, the number operator can be identified
with the total angular momentum operator whose conservation is a consequence
of the rotational invariance of the graphene Hamiltonian. In the limit of a
vanishing Kekulé parameter $\Delta_{0}\rightarrow 0$, the expansion
coefficients must satisfy $B^{-}_{n}=A^{-}_{n}$ and $C^{+}_{n}=D^{+}_{n}$,
along with $B^{+}_{n}=A^{+}_{n}=C^{-}_{n}=D^{-}_{n}=0$, as we should recover
the valley degenerate solutions for the Landau levels.

Figure 1: (Colour online) Magnetic field dependence of the first three Landau
levels with the continuous (dashed) lines corresponding to the valley index
$\tau=1$ ($\tau=-1$) as given by equation (2.6). We notice the valley
splitting even for a small Kekulé parameter $\Delta_{0}=0.1$ (see the main
text).

Using this state, and after some algebraic manipulations, the energy spectrum
reads $E_{\sigma\tau}(n)=\sigma E^{\tau}_{n}$ (see appendix for details),
where we have defined:

$E^{\tau}_{n}=\hbar\omega_{c}\sqrt{\left(n+\frac{1}{2}\right)(1+\Delta_{0}^{2})+\frac{1}{2}\tau\sqrt{16n(n+1)\Delta_{0}^{2}+(1+\Delta_{0}^{2})^{2}}}\quad
n=1,2,3,\ldots\,.$ (2.6)

The indexes take on the values $\sigma,\tau=\pm 1$, whereas $n=0,1,2...$
labels the different Landau levels, but one should note that for $n=0$,
equation (2.6) does not apply (this case is treated separately below). This
energy spectrum (figure 1) is equivalent to that obtained in reference [35]
but we avoid the need to introduce a spurious onsite energy term which they
use to obtain their result. From equation (2.6), we observe that in the
vanishing distortion parameter limit $\Delta_{0}\rightarrow 0$, we recover the
degenerate spectrum

$(E^{-}_{n})^{2}=(E^{+}_{n-1})^{2}=n\hbar^{2}\omega_{c}^{2}.$ (2.7)

After some lengthy algebraic operations, we get for the coefficients of the
eigenstate in equation (2.5) the expressions for $\tau=-1$ and $n\neq 0$:

$\displaystyle B^{-}_{n}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{n}^{-}}{\sqrt{n}}\left[\frac{(\epsilon^{-}_{n})^{2}-(n+1)-n\Delta_{0}^{2}}{(\epsilon^{-}_{n})^{2}-(n+1)(1-\Delta_{0}^{2})}\right]A_{n}^{-}\equiv
b^{-}_{n}A^{-}_{n},$ $\displaystyle C^{-}_{n}$ $\displaystyle=$
$\displaystyle\Delta_{0}\left(\frac{\sqrt{n}\epsilon_{n}^{-}+(n+1)b_{n}^{-}}{(\epsilon_{n}^{-})^{2}-(n+1)}\right)A_{n}^{-}\equiv
c^{-}_{n}A^{-}_{n},$ $\displaystyle D^{-}_{n}$ $\displaystyle=$
$\displaystyle\Delta_{0}\sqrt{n+1}\left(\frac{\epsilon_{n}^{-}b_{n}^{-}+\sqrt{n}}{(\epsilon_{n}^{-})^{2}-(n+1)}\right)A_{n}^{-}\equiv
d_{n}^{-}A_{n}^{-},$ (2.8)

whereas those corresponding to $\tau=+1$ and $n\neq 0$, read as

$\displaystyle A^{+}_{n}$ $\displaystyle=$
$\displaystyle\Delta_{0}\sqrt{n}\left(\frac{\epsilon_{n}^{+}c_{n}^{+}+\sqrt{n+1}}{(\epsilon_{n}^{+})^{2}-n}\right)D_{n}^{+}\equiv
a^{+}_{n}D^{+}_{n},$ $\displaystyle B^{+}_{n}$ $\displaystyle=$
$\displaystyle\Delta_{0}\left(\frac{\sqrt{n+1}\epsilon_{n}^{+}+nc_{n}^{+}}{(\epsilon_{n}^{+})^{2}-n}\right)\equiv
b^{+}_{n}D^{+}_{n},$ $\displaystyle C^{+}_{n}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{n}^{+}}{\sqrt{n+1}}\left[\frac{(\epsilon^{+}_{n})^{2}-n-(n+1)\Delta_{0}^{2}}{(\epsilon^{+}_{n})^{2}-n(1-\Delta_{0}^{2})}\right]D_{n}^{+}\equiv
c^{+}_{n}D^{+}_{n},$ (2.9)

and the remaining coefficients $A^{-}_{n}$ and $D^{+}_{n}$ being global
factors, would not be needed for the calculation of physical properties. In
the explicit calculations, we use $a^{-}_{n}=d^{+}_{n}=1$. Here, we have also
introduced the dimensionless quantity
$\epsilon^{\pm}_{n}=E^{\pm}_{n}/\hbar\omega_{c}$. The special case of $n=0$,
corresponding to a valley degenerate zero-energy solution
$E_{\sigma\tau}(0)=0$, with (normalized) eigenstate, reads as:

$|\Psi_{0}\rangle=-\frac{1}{\sqrt{1+\Delta_{0}^{2}}}\left(\begin{array}[]{c}0\\\
-|0\rangle\\\ \Delta_{0}|0\rangle\\\ 0\end{array}\right).$ (2.10)

##  3 Results

We now explore the physical consequences of the interplay of Kekulé and
electromagnetic monochromatic radiation circularly polarized. First, we
analyse the static polarization effects and afterwards we extend the analysis
to the driven scenario.

###  3.1 Static regime

By using the coefficients in equation (2) and (2), we can evaluate the
pseudospin polarization
$\langle\sigma_{z}\rangle^{\pm}_{n}=\langle\Phi^{\pm}_{n}|\tau_{0}\otimes\sigma_{z}|\Phi_{n}^{\pm}\rangle$
and the valley polarizations
$\langle\tau_{z}\rangle^{\pm_{n}}=\langle\Phi{{}^{\pm}_{n}}|\tau_{z}\otimes\sigma_{0}|\Phi{{}^{\pm}_{n}}\rangle$,
for any state $n\neq 0$ with $\pm$ denoting here the Landau level associated
to the electron/hole bands. The pseudospin and valley polarization per Landau
level are given by

$\displaystyle\langle\sigma_{z}\rangle_{n}^{-}$ $\displaystyle=$
$\displaystyle\frac{1-|b^{-}_{n}|^{2}+|c^{-}_{n}|^{2}-|d^{-}_{n}|^{2}}{1+|b^{-}_{n}|^{2}+|c^{-}_{n}|^{2}+|d^{-}_{n}|^{2}},$
$\displaystyle\langle\sigma_{z}\rangle_{n}^{+}$ $\displaystyle=$
$\displaystyle\frac{|a^{+}_{n}|^{2}-|b^{+}_{n}|^{2}+|c^{+}_{n}|^{2}-1}{1+|a^{+}_{n}|^{2}+|b^{+}_{n}|^{2}+|c^{+}_{n}|^{2}},$
$\displaystyle\langle\tau_{z}\rangle_{n}^{-}$ $\displaystyle=$
$\displaystyle\frac{1+|b^{-}_{n}|^{2}-|c^{-}_{n}|^{2}-|d^{-}_{n}|^{2}}{1+|b^{-}_{n}|^{2}+|c^{-}_{n}|^{2}+|d^{-}_{n}|^{2}},$
$\displaystyle\langle\tau_{z}\rangle_{n}^{+}$ $\displaystyle=$
$\displaystyle\frac{|a^{+}_{n}|^{2}+|b^{+}_{n}|^{2}-|c^{+}_{n}|^{2}-1}{1+|a^{+}_{n}|^{2}+|b^{+}_{n}|^{2}+|c^{+}_{n}|^{2}},$
(3.1)

which in the limit of the vanishing Kekulé distortion lead to
$\langle\sigma_{z}\rangle_{n}^{\pm}=0$ and
$\langle\tau_{z}\rangle_{n}^{\pm}=\pm 1$. In the left-hand (right-hand) panel
of figure 2 we show the out of plane static pseudospin (valley) static
polarization for the first three Landau leveles $n=1,2,3$, which show the
valley splitting, even for a small Kekulé parameter $\Delta_{0}=0.1$. We also
find that for $n=0$, these polarizations simply read as:

$\displaystyle\langle\sigma_{z}\rangle_{0}$ $\displaystyle=$
$\displaystyle\frac{-1+\Delta_{0}^{2}}{1+\Delta_{0}^{2}},$
$\displaystyle\langle\tau_{z}\rangle_{0}$ $\displaystyle=$
$\displaystyle\frac{1-\Delta_{0}^{2}}{1+\Delta_{0}^{2}},$ (3.2)

and we verify that $\langle\sigma_{z}\rangle_{0}=-\langle\tau_{z}\rangle_{0}$
is a direct consequence of the condition of null excitations in the $n=0$
Landau level (LL), i.e., $\mathcal{N}_{a}|\Psi_{0}\rangle=0$. The explicit
polarization dependence on $\Delta_{0}$ could be interpreted as a supporting
argument to the initial statement that the bonding pattern couples these two
degrees of freedom.

Figure 2: (Colour online) Left-hand panel: Out of plane static pseudospin
polarization for the first three Landau levels with the thick (thin) lines
corresponding to valley index $\tau=1$ ($\tau=-1$). Right-hand panel: Valley
polarization as given by equations (3.1). We notice the valley sppliting even
for a small Kekulé parameter $\Delta_{0}=0.1$ (see the main text).

###  3.2 Dynamic regime

We now consider the role of the radiation field in modulating the LL structure
of the Kekulé distorted graphene monolayer. To this end, we start from the
ordinary dipolar interaction term $-e\vec{p}\cdot\vec{A}(t)$, which can be
introduced to the tight-binding Hamiltonian via the Peierls substitution.
Under the long wavelength regime of interest we can evaluate, near the Dirac
points, that the effects of the driving field are incorporated through the
minimum coupling prescription, leading to the time-dependent term

$\mathcal{V}(t)=ev_{\sigma}\vec{\sigma}\cdot\vec{A}(t)+ev_{\tau}\vec{\tau}\cdot\vec{A}(t),$
(3.3)

thus, the total Hamiltonian reads

$\mathcal{H}(t)=\mathcal{H}_{0}+\mathcal{V}(t).$ (3.4)

For a monochromatic circularly polarized radiation field impinging
perpendicular to the graphene monolayer, the vector potential can be chosen as
$\vec{A}(t)=A(\cos\omega t,\sin\omega t)$, with $A=\mathcal{E}/\omega$. This
makes the total Hamiltonian periodic in time,
$\mathcal{H}(t+T)=\mathcal{H}(t)$, with $T=2\piup/\omega$ the period of
oscillation of the driving field. The parameters $\mathcal{E}$ and
$\omega=2\piup/T$ correspond to the amplitude and frequency of the radiation
field, respectively. The electric field is given in turn by the standard
relation $\vec{\mathcal{E}}(t)=-\partial_{t}\vec{A}(t)$. Then, the time-
dependent contribution reads

$\mathcal{V}(t)=\xi\left(\begin{array}[]{cccc}0&\mathrm{e}^{-\mathrm{i}\omega
t}&\Delta_{0}\mathrm{e}^{-\mathrm{i}\omega t}&0\\\
\mathrm{e}^{\mathrm{i}\omega t}&0&0&\Delta_{0}\mathrm{e}^{-\mathrm{i}\omega
t}\\\ \Delta_{0}\mathrm{e}^{\mathrm{i}\omega
t}&0&0&\mathrm{e}^{-\mathrm{i}\omega t}\\\
0&\Delta_{0}\mathrm{e}^{\mathrm{i}\omega t}&\mathrm{e}^{\mathrm{i}\omega
t}&0\end{array}\right),$ (3.5)

where we have introduced the effective light-matter coupling strength
$\xi=e\mathcal{E}v_{F}/\omega$. We assume that the beam radiation spot is
large enough compared to the lattice spacing, and we can neglect any spatial
variation.

We can perform a time-dependent unitary transformation
$\mathcal{P}(t)=\mathrm{e}^{-\mathrm{i}\mathcal{N}_{a}\omega t}$, given
explicitly as $\mathrm{e}^{-\mathrm{i}n\omega
t}\text{diag}[\mathrm{e}^{-\mathrm{i}\omega
t},1,1,\mathrm{e}^{\mathrm{i}\omega t}]$, in the form

$\mathcal{H}_{F}=\mathcal{P}^{\dagger}(t)\mathcal{H}(t)\mathcal{P}(t)-\mathrm{i}\hbar\mathcal{P}(t)\partial_{t}\mathcal{P}(t),$
(3.6)

after which we get a time-independent Floquet Hamiltonian [16, 17]

$\mathcal{H}_{F}=\hbar\omega_{c}\tau_{0}\otimes(a^{\dagger}\sigma_{-}+a\sigma_{+})+\hbar\omega_{c}\Delta_{0}(a^{\dagger}\tau_{-}+a\tau_{+})\otimes\sigma_{0}-\mathcal{N}_{a}\hbar\omega+\xi\tau_{0}\otimes\sigma_{x}+\xi\Delta_{0}\tau_{x}\otimes\sigma_{0}.$
(3.7)

Up till now, the results are analytically exact and we could solve numerically
the resulting time-independent Schrödinger equation
$\mathcal{H}_{F}|\Phi_{F}(t)\rangle=\varepsilon|\Phi_{F}(t)\rangle$ to obtain
the quasienergy spectrum $\varepsilon$ and Floquet eigenstates
$|\Phi_{F}(t)\rangle$. Some comments are in order here. Within the approach of
Fourier mode expansion to numerically solve the Floquet Hamiltonian, the
sidebands associated to the periodicity of the quasienergy spectrum yield
$\varepsilon_{N}\,\text{mod}\,{\hbar\omega}$. On the other hand, from the
obtained effective Floquet Hamiltonian given in (3.7) we notice that the role
of the radiation field is actually to couple the different LLs. Indeed, for a
given value of the LL index, the factor $\mathcal{N}_{a}\hbar\omega$ would
represent the discrete energy translations of quasienergies. Moreover, we
clearly see that the last two terms of (3.7) do not commute with the static
Hamiltonian; hence, they would produce the level mixing which no longer
renders the index $n$ a good quantum number. This can be explicitly shown by
expressing the Floquet Hamiltonian in the static eigenbasis (2.5):

$\langle\Phi_{n^{\prime}}^{\sigma^{\prime}\tau^{\prime}}|\mathcal{H}_{F}|\Phi_{n}^{\sigma\tau}\rangle=(E^{\sigma\tau}_{n}-n\hbar\omega)\delta_{nn^{\prime}}\delta_{\sigma\sigma^{\prime}}\delta_{\tau\tau^{\prime}}+V_{nn^{\prime}},$

with the matrix elements of the coupling term

$\displaystyle V_{nn^{\prime}}$ $\displaystyle=$ $\displaystyle\xi\sum_{s=\pm
1}\frac{(b_{n}+c_{n+s}d_{n})\delta_{n^{\prime},n+s}}{\sqrt{(1+|b_{n}|^{2}+|c_{n}|^{2}+|d_{n}|^{2})(1+|b_{n+s}|^{2}+|c_{n+1}|^{2}+|d_{n+s}|^{2})}}$
(3.8) $\displaystyle+$ $\displaystyle\xi\sum_{s=\pm
1}\Delta_{0}\xi\frac{(c_{n}+b_{n+s}d_{n})\delta_{n^{\prime},n+s}}{\sqrt{(1+|b_{n}|^{2}+|c_{n}|^{2}+|d_{n}|^{2})(1+|b_{n+s}|^{2}+|c_{n+s}|^{2}+|d_{n+s}|^{2})}}.$

However, this approach does not provide a clear picture of the most relevant
features of the underlying photoinduced processes. Therefore, in the following
section we introduce physically appealing solutions within appropriate
parameter regimes of experimental relevance.

###  3.3 Results for the driven regime

In the previous section we derived an analytical expression for the Floquet
Hamiltonian. To get a further physical insight, now we assess the role of the
radiation field in different parameter regimes. We explore the valley and
pseudospin polarization dynamics within these different parameter
configurations. In the following discussion we use as the energy scale the
cyclotron frequency $\omega_{c}$. To proceed further, we introduce the shifted
ladder operators $b=a+\xi/\hbar\omega_{c}$, which allow us to rewrite the
exact Floquet Hamiltonian as

$\mathcal{H}_{F}=\hbar\omega_{c}\tau_{0}\otimes(b^{\dagger}\sigma_{-}+b\sigma_{+})+\hbar\omega_{c}\Delta_{0}(b^{\dagger}\tau_{-}+b\tau_{+})\otimes\sigma_{0}-\mathcal{N}_{b}\hbar\omega+\lambda\hbar\omega(b^{\dagger}+b)-\lambda^{2}\hbar\omega,$
(3.9)

with $\lambda=\xi/\hbar\omega_{c}$. In this form, we can assess the role of
the different contributions to the Landau level mixing due to the radiation
field.

We now focus on the light-matter coupling analysis. Within the general
scenario of laser irradiation, at low frequencies compared to the static
bandwidth, the dynamical problem is rather complicated since different Floquet
replica (sidebands) might overlap. Indeed, in the general Floquet formulation
of the problem, it has been shown that the number of sidebands require to
obtain a physically relevant numerical solution of the driven system is to use
a number of modes up to $n_{\text{max}}$ which will depend on whether one is
in the weak $\xi\ll\hbar\omega$ or strong $\xi\gg\hbar\omega$ coupling limit
[53, 54]. Our solution written in terms of shifted operators given in equation
(3.9) shows that the leading order light-matter interaction strength that
couples the different Landau levels is independent of the frequency.
Therefore, the energy corrections (apart from an irrelevant energy shift
$-\lambda^{2}\omega$) would only depend on the amplitude of the radiation
field. This in turn might have the advantage of allowing the study of
experimental configurations in which the stability of the sample is not
frequency dependent in order to avoid damaging the sample [55, 56].

However, the periodicity of the quasienergy spectrum allows for a resonant
behaviour among adjacent Landau levels that turn out to be coupled by the
radiation field. Thus, whenever the effective amplitude of the radiation field
is smaller than the static bandwidth and the system is out of resonance, we
can approximate the Floquet quasienergy spectrum by means of degenerate
perturbation theory. That is, within the weak coupling limit regime of
experimental interest, we can use standard time-independent perturbation
theory to assess the corrections to the energy spectrum $\Delta E(n)$ for
$n\neq 0,1$ which up the aforementioned negative global higher-order energy
shift $-\lambda^{2}\omega$, are given by

$\Delta
E^{\sigma\tau}_{n}=[\sigma(E^{\tau}_{n}+E^{-\tau}_{n-1})-\hbar\omega](f^{\sigma\tau}_{n-1})^{2}+[\sigma(E^{\tau}_{n}+E^{-\tau}_{n+1})-\hbar\omega](f^{\sigma\tau}_{n+1})^{2},$
(3.10)

for which the expressions for the coefficients $f^{\sigma\tau}_{n\pm 1}$ are
somewhat lengthy, so we provide them in the appendix, along with its
derivation.

Figure 3: (Colour online) Magnetic field dependence of the resonant energy
structure due to the photoinduced mixing of the lowest Landau levels $n=0$ and
$n=1$. We have set $e\mathcal{E}\ell_{B_{\text{max}}}=0.1\hbar\omega$, with
$\ell_{B_{\text{max}}}$ being the magnetic length at the largest value of the
static magnetic field. An asymmetric response of the valley components
$\tau=\pm 1$ which could be exploited for photoinduced valley modulations (see
the main text).

Interestingly, only transitions with $\Delta n=\pm 1$ are allowed, such that
higher order transitions are forbidden. Up to the leading order, the corrected
Floquet eigestates read for $n\neq 0,1$

$|\Phi^{\sigma\tau}_{n}\rangle=|\Psi^{\sigma\tau}_{n}\rangle+f^{\sigma\tau}_{n-1}|\Psi^{-\sigma,-\tau}_{n-1}\rangle+f^{\sigma\tau}_{n+1}|\Psi^{-\sigma,-\tau}_{n+1}\rangle.$
(3.11)

The energy corrections given in equation (3.10) show the standard resonant
structure that we now deal with. The energy corrections are determined by
analyzing the subspace of zero energy degeneracy between the static $n=0$ zero
energy level and the $n=1$ Landau level, which occures whenever
$\hbar\omega=\pm E_{1}^{\tau}$. This in turn implies that radiation fields
with right (left) circular polarization state would pick the $+E_{1}^{\tau}$
($-E_{1}^{\tau}$), for any fixed values of the quantizing static magnetic
field $B$ and Kekulé distortion parameter $\Delta_{0}$. Thus, at resonance
these two lowest Landau levels mix to give rise to the leading order
quasienergy solution

$E_{1}^{\tau
s}=s\frac{e\mathcal{E}\ell_{B}}{\sqrt{2}}\frac{b_{1}^{\tau}-\Delta_{0}c_{1}^{\tau}}{\sqrt{1+\Delta_{0}^{2}}\sqrt{|a_{1}^{\tau}|^{2}+|b_{1}^{\tau}|^{2}+|c_{1}^{\tau}|^{2}+|d_{1}^{\tau}|^{2}}},$
(3.12)

with $a_{1}^{-}=d_{1}^{+}=1$. These resonant energies are modulated both by
the static magnetic field and by the amplitude of the radiation field. In
figure 3 we plot $E_{1}^{\tau s}$ as functions of the static magnetic field,
where a clear different resonant response for each valley component is
observed. This in turn can be traced back to the satic energy spectrum shown
in figure 1 where the presence of the Kekulé distortion makes the $\tau=-1$
states get closer to the zero energy lowest Landau level. Thus, we observed a
level “repulsion” in the most distant energy state that couples more strongly
within the resonant regime. In the degenerate subspace spanned by the two
lowest Landau levels, we obtain the Floquet states valid at resonance:

$|\Phi_{1}^{\tau
s}\rangle=\frac{1}{\sqrt{2}}\left(|\Psi_{0}\rangle+s|\Psi^{\tau}_{1}\rangle\right).$
(3.13)

This degenerate state is an equally weighted superposition of the static
orthogonal states corresponding to the two lowest Landau levels, which can be
contrasted to its non-degenerate counterpart

$|\Phi_{1}^{\tau\sigma}\rangle=|\Psi_{1}^{\tau\sigma}\rangle+f^{\tau\sigma}_{1}|\Psi_{0}\rangle,$
(3.14)

where

$f^{\tau\sigma}_{1}=\frac{e\mathcal{E}\ell_{B}}{\sqrt{2(1+\Delta_{0}^{2})(|a_{1}^{\tau}|^{2}+|b_{1}^{\tau}|^{2}+|c_{1}^{\tau}|^{2}+|d_{1}^{\tau}|^{2})}}\left(\frac{b_{1}^{\tau}-\sigma\Delta_{0}c_{1}^{\tau}}{E_{1}^{\tau}-\sigma\hbar\omega}\right).$
(3.15)

Thus, as we would physically expect, within the off resonant scenario, the
relative weight of the Floquet eigenstates mixing the static eigenstates would
be much smaller as compared to the resonant regime.

Moreover, the pseudospin and valley polarizations in the resonant regime are
explicitly found to read in terms of the static polarizations as

$\displaystyle\langle\Phi_{1}^{\tau
s}|\tau_{0}\otimes\sigma_{z}|\Phi_{1}^{\tau s}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\langle\sigma_{0}\rangle+\langle\sigma^{\tau}_{1}\rangle),$
$\displaystyle\langle\Phi_{1}^{\tau
s}|\tau_{z}\otimes\sigma_{0}|\Phi_{1}^{\tau s}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\langle\tau_{0}\rangle+\langle\tau^{\tau}_{1}\rangle),$
(3.16)

whereas the off resonant expressions are

$\displaystyle\langle\Phi_{1}^{\tau\sigma}|\tau_{0}\otimes\sigma_{z}|\Phi_{1}^{\tau\sigma}\rangle$
$\displaystyle=$
$\displaystyle\langle\sigma^{\tau}_{1}\rangle+(f^{\tau\sigma}_{1})^{2}\langle\sigma_{0}\rangle,$
$\displaystyle\langle\Phi_{1}^{\tau\sigma}|\tau_{z}\otimes\sigma_{0}|\Phi_{1}^{\tau\sigma}\rangle$
$\displaystyle=$
$\displaystyle\langle\tau^{\tau}_{1}\rangle+(f^{\tau\sigma}_{1})^{2}\langle\tau_{0}\rangle,$
(3.17)

with $f^{\tau\sigma}_{1}$ being defined in equations (3.14) and (3.15).

##  4 Summary and conclusions

We have addressed the light-matter interaction effects of charge carriers in
monolayer graphene with a Kekulé bond pattern in the so-called KeK-Y
configuration where the radiation field couples to the valley degree of
freedom. By restricting ourselves to the long wavelength approximation, we
first have shown an exact analytical solution to the static Landau level
spectrum for monolayer graphene with a Kekulé bond pattern. For this purpose,
we have used an alternative approach as that given in reference [35].
Following the proposal by Mojarro et al. [45], we have explored the role of
monochromatic radiation incident perpendicularly to the graphene monolayer.
Interestingly, even for a small value of the bonding parameter, within the
weak coupling of the light-matter interaction, a distinctive physical regime
emerges in the resonant driven scenario. We have shown that these resonances
can be addressed by either right-hand or left-hand polarized radiation fields.
In addition, the effective light-matter interaction strength depends only on
the radiation electric field amplitude rather than on its frequency. The
frequency dependence is addressed through the resonant properties of the
driven system. The effective time independent Floquet Hamiltonian shows that
only adjacent Landau levels get coupled. This is explicitly analysed in the
subspace of the lowest Landau level configuration showing the well known level
mixing of degenerate states. Furthermore, the static states get dressed by the
driving interaction and a quasienergy band splitting of the zero energy
solutions can be exploited to enhance the polarization properties of the KeK-Y
graphene system.

## Acknowledgements

A. L. thanks Bertrand for more than 17 years of scientific collaboration and
friendship in which he has always been an inspiration. F. M. acknowledges the
support of DGAPA-UNAM through the project PAPIIT No. IN113920. We dedicate
this work to our colleague and friend Bertrand Berche in the celebration of
his 60th anniversary.

## Appendix

Here, we present the details of calculations of some quantities of interest
presented in the main text.

### Calculation of Landau level spectrum and the expansion coefficients for
the static regime

First we determine the static Landau level spectrum. We start by writing the
KeK-Y Hamiltonian equation (2.3) in the basis given in equation (2.5) which
amounts to writing the Schrödinger equation
$\mathcal{H}_{0}|\Psi_{n}^{\sigma\tau}\rangle=\sigma
E^{\tau}_{n}|\Psi_{n}^{\sigma\tau}\rangle$ explicitly as

$\hbar\omega_{c}\left(\begin{array}[]{cccc}0&\sqrt{n}&\Delta_{0}\sqrt{n}&0\\\
\sqrt{n}&0&0&\Delta_{0}\sqrt{n+1}\\\ \Delta_{0}\sqrt{n}&0&0&\sqrt{n+1}\\\
0&\Delta_{0}\sqrt{n+1}&\sqrt{n+1}&0\\\
\end{array}\right)\left(\begin{array}[]{c}A_{n}^{\tau}\\\ B_{n}^{\tau}\\\
C_{n}^{\tau}\\\
D_{n}^{\tau}\end{array}\right)=E_{n}^{\tau}\left(\begin{array}[]{c}A_{n}^{\tau}\\\
B_{n}^{\tau}\\\ C_{n}^{\tau}\\\ D_{n}^{\tau}\end{array}\right).$ (A.1)

Defining $\epsilon_{n}^{\tau}=E^{\tau}_{n}/\hbar\omega_{c}$ we get the secular
equation:

$\left|\begin{array}[]{cccc}-\epsilon_{n}^{\tau}&\sqrt{n}&\Delta_{0}\sqrt{n}&0\\\
\sqrt{n}&-\epsilon_{n}^{\tau}&0&\Delta_{0}\sqrt{n+1}\\\
\Delta_{0}\sqrt{n}&0&-\epsilon_{n}^{\tau}&\sqrt{n+1}\\\
0&\Delta_{0}\sqrt{n+1}&\sqrt{n+1}&-\epsilon_{n}^{\tau}\\\
\end{array}\right|=0,$ (A.2)

which in turn leads to the equation

$(\epsilon_{n}^{\tau})^{4}-(\epsilon_{n}^{\tau})^{2}[(2n+1)(1+\Delta_{0}^{2})]+n(n+1)(1-\Delta_{0}^{2})^{2}=0.$
(A.3)

Completing the square, we get

$[(\epsilon_{n}^{\tau})^{2}-(n+1/2)(1+\Delta_{0}^{2})]^{2}+n(n+1)(1-\Delta_{0}^{2})^{2}=(n+1/2)^{2}(1+\Delta_{0}^{2})^{2},$
(A.4)

which we rewrite as

$[(\epsilon_{n}^{\tau})^{2}-(n+1/2)(1+\Delta_{0}^{2})]^{2}+n(n+1)(1-2\Delta_{0}^{2}+\Delta_{0}^{4})=n(n+1)(1+2\Delta_{0}^{2}+\Delta_{0}^{4})+\frac{1}{4}(1+\Delta_{0}^{2}),$
(A.5)

that leads to the simpler expression

$[(\epsilon_{n}^{\tau})^{2}-(n+1/2)(1+\Delta_{0}^{2})]^{2}=\frac{1}{4}[16n(n+1)\Delta_{0}^{2}+(1+\Delta_{0}^{2})].$
(A.6)

We get then the result

$\epsilon_{n}^{\tau}=\sqrt{\left(n+\frac{1}{2}\right)(1+\Delta_{0}^{2})+\frac{1}{2}\tau\sqrt{16n(n+1)\Delta_{0}^{2}+(1+\Delta_{0}^{2})^{2}}},\quad
n=1,2,3,\ldots\,.$ (A.7)

To obtain the expansion coefficients for the eigenstate, we write down the
corresponding set of equations

$\displaystyle\sqrt{n}(B_{n}^{\tau}+\Delta_{0}C_{n}^{\tau})=\epsilon_{n}^{\tau}A_{n}^{\tau},$
$\displaystyle\sqrt{n}A_{n}^{\tau}+\Delta_{0}\sqrt{n+1}D_{n}^{\tau}=\epsilon_{n}^{\tau}B_{n}^{\tau},$
$\displaystyle\Delta_{0}\sqrt{n}A_{n}^{\tau}+\sqrt{n+1}D_{n}^{\tau}=\epsilon_{n}^{\tau}C_{n}^{\tau},$
$\displaystyle\sqrt{n+1}(\Delta_{0}B_{n}^{\tau}+C_{n}^{\tau})=\epsilon_{n}^{\tau}D_{n}^{\tau}.$
(A.8)

This set of equations is not linearly independent. We need to select three of
them in order to find solutions for three parameters expressed in terms of the
fourth. In addition, we need to treat separately the solutions for $\tau=\pm
1$ as they have different limiting behavior when $\Delta_{0}\rightarrow 0$. We
show the explicit derivation for the $\tau=-1$ subspace, for which we need to
use the set of coupled equations

$\displaystyle\sqrt{n}(B_{n}^{-}+\Delta_{0}C_{n}^{-})=\epsilon_{n}^{-}A_{n}^{-},$
(A.9)
$\displaystyle\Delta_{0}\sqrt{n}A_{n}^{-}+\sqrt{n+1}D_{n}^{-}=\epsilon_{n}^{-}C_{n}^{-},$
(A.10)
$\displaystyle\sqrt{n+1}(\Delta_{0}B_{n}^{-}+C_{n}^{-})=\epsilon_{n}^{-}D_{n}^{-},$
(A.11)

which in the limit of vanishing Kekulé parameter $\Delta_{0}\rightarrow 0$
reduces to $\epsilon_{n}^{-}=\sqrt{n}$, $A^{-}_{n},B^{-}_{n}\rightarrow 1$ and
$C^{-}_{n},D^{-}_{n}\rightarrow 0$. To solve the system, we apply the standard
algebraic techniques as we show explicitly for $B^{-}_{n}$. Multiplication of
equation (A.5) by $\epsilon_{n}^{-}$ and substitution of equation (A.6), give
us

$\displaystyle\sqrt{n}(B_{n}^{-}+\Delta_{0}C_{n}^{-})=\epsilon_{n}^{-}A_{n}^{-},$
(A.12)
$\displaystyle\epsilon_{n}^{-}\Delta_{0}\sqrt{n}A_{n}^{-}+(n+1)(\Delta_{0}B_{n}^{-}+C_{n}^{-})=(\epsilon_{n}^{-})^{2}C_{n}^{-},$
(A.13)

which upon substitution of $C_{n}^{-}$ on (A.4) can be further reduced to

$B_{n}^{-}+\Delta_{0}\left[\frac{\epsilon_{n}^{-}\Delta_{0}\sqrt{n}A_{n}^{-}+(n+1)\Delta_{0}B_{n}^{-}}{(\epsilon^{-}_{n})^{2}-(n+1)}\right]=\frac{\epsilon_{n}^{-}}{\sqrt{n}}A_{n}^{-},$
(A.14)

which is equivalent to

$B_{n}^{-}\left[1+\frac{(n+1)\Delta^{2}_{0}}{(\epsilon^{-}_{n})^{2}-(n+1)}\right]=\frac{\epsilon_{n}^{-}}{\sqrt{n}}\left[1-\frac{n\Delta^{2}_{0}}{(\epsilon^{-}_{n})^{2}-(n+1)}\right]A_{n}^{-},$
(A.15)

that leads to the result

$B_{n}^{-}=\frac{\epsilon_{n}^{-}}{\sqrt{n}}\left[\frac{(\epsilon^{-}_{n})^{2}-(n+1)-n\Delta_{0}^{2}}{(\epsilon^{-}_{n})^{2}-(n+1)(1-\Delta_{0}^{2})}\right]A_{n}^{-}\equiv
b^{-}_{n}A^{-}_{n}.$ (A.16)

To obtain $C^{-}_{n}$, we use this result. First, we rewrite equation (A.8)

$C_{n}^{-}=\frac{\Delta_{0}(\sqrt{n}\epsilon_{n}^{-}A^{-}_{n}+(n+1)B^{-}_{n}}{(\epsilon^{-}_{n})^{2}-(n+1)},$
(A.17)

factoring out $A^{-}_{n}$, we get

$C_{n}^{-}=\frac{\Delta_{0}(\sqrt{n}\epsilon_{n}^{-}+(n+1)b^{-}_{n}}{(\epsilon^{-}_{n})^{2}-(n+1)}A_{n}^{-}\equiv
c^{-}_{n}A^{-}_{n}.$ (A.18)

To find $D^{-}_{n}$, we rewrite equation (A.5) as

$D_{n}^{-}=\frac{\epsilon_{n}^{-}C^{-}_{n}-\Delta_{0}\sqrt{n}A^{-}_{n}}{\sqrt{n+1}}.$
(A.19)

Using the result for $C^{-}_{n}$, we write this as

$D_{n}^{-}=\frac{\Delta_{0}}{\sqrt{n+1}}\left(\frac{(\epsilon_{n}^{-})^{2}\sqrt{n}+\epsilon_{n}^{-}(n+1)b^{-}_{n}}{(\epsilon^{-}_{n})^{2}-(n+1)}-\sqrt{n}\right)A^{-}_{n},$
(A.20)

which is equivalent to

$D_{n}^{-}=\Delta_{0}\sqrt{n+1}\left(\frac{\epsilon_{n}^{-}b^{-}_{n}+\sqrt{n}}{(\epsilon^{-}_{n})^{2}-(n+1)}\right)A^{-}_{n}\equiv
d^{-}_{n}A^{-}_{n}.$ (A.21)

The other coefficients are calculated in a similar fashion.

### Calculation of the expansion coefficients via perturbation theory

We now determine the expansion coefficients for the Floquet states as obtained
from time- independent perturbation theory. The following results hold for
$n\leqslant 2$ since the $n=1$ Landau level couples with the $n=0$ level,
which as we have found in the main text, is independent of the magnetic field
and its coefficients do not follow the general result given in equation (2.5).
We also give the leading order perturbation result within the off resonance
scenario

$|\Phi^{\tau\sigma}_{n}\rangle=|\Psi^{\tau\sigma}_{n}\rangle+\lambda\sum_{n^{\prime}\neq
n}\frac{\langle\Psi^{-\tau,-\sigma}_{n^{\prime}}|(b+b^{\dagger})|\Psi^{\tau\sigma}_{n}\rangle}{\sigma(E^{\tau}_{n}+E^{-\tau})-(n-n^{\prime})\hbar\omega}|\Psi^{-\tau,-\sigma}_{n^{\prime}}\rangle.$
(A.22)

Using the static eigenstates (2.5), the matrix element

$\langle\Psi^{-\tau,-\sigma}_{n^{\prime}}|b|\Psi^{\tau\sigma}_{n}\rangle,$
(A.23)

is found to be given by

$\langle\Psi^{-\tau,-\sigma}_{n^{\prime}}|b|\Psi^{\tau\sigma}_{n}\rangle=\frac{1}{N^{\tau}N^{-\tau}}(\sqrt{n-1}A^{\tau}_{n}A^{-\tau}_{n^{\prime}}-\sqrt{n}(B^{\tau}_{n}B^{-\tau}_{n^{\prime}}-C^{\tau}_{n}C^{-\tau}_{n^{\prime}}-\sqrt{n+1}D^{\tau}_{n}D^{-\tau}_{n^{\prime}})\delta_{n^{\prime},n-1},$
(A.24)

with
$N^{\tau}=\sqrt{|A^{\tau}_{n}|^{2}+|B^{\tau}_{n}|^{2}+|C^{\tau}_{n}|^{2}+|D^{\tau}_{n}|^{2}}$.

We also obtain

$\langle\Psi^{-\tau,-\sigma}_{n^{\prime}}|b^{\dagger}|\Psi^{\tau\sigma}_{n}\rangle=\frac{1}{N^{\tau}N^{-\tau}}(\sqrt{n}A^{\tau}_{n}A^{-\tau}_{n^{\prime}}-\sqrt{n+1}(B^{\tau}_{n}B^{-\tau}_{n^{\prime}}-C^{\tau}_{n}C^{-\tau}_{n^{\prime}}-\sqrt{n+2}D^{\tau}_{n}D^{-\tau}_{n^{\prime}})\delta_{n^{\prime},n+1},$
(A.25)

upon substitution of these matrix elements in equation (A.26), and performing
the summations by means of the Kronecker symbols, we get

$|\Phi^{\tau\sigma}_{n}\rangle=|\Psi^{\tau\sigma}_{n}\rangle+\sum_{s}f^{\sigma\tau}_{n+s}|\Psi^{-\tau,-\sigma}_{n+s}\rangle.$
(A.26)

where we can infer from the inspection that the coefficients are given by:

$\displaystyle
f^{\sigma\tau}_{n+s}=\left(\frac{e\mathcal{E}\ell_{B}}{\sqrt{2}}\right)$
$\displaystyle\times\frac{\sqrt{n-0.5(1-s)}A^{\tau}_{n}A^{-\tau}_{n+s}-\sqrt{n+0.5(1+s)}(B^{\tau}_{n}B^{-\tau}_{n+s}-C^{\tau}_{n}C^{-\tau}_{n+s})-\sqrt{n+1+0.5(1+s)}D^{\tau}_{n}D^{-\tau}_{n+s}}{\sigma(E^{-\tau}_{n+s}+E^{\tau}_{n})-s\hbar\omega}.$
(A.27)

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