## 1. Introduction

Since its recent experimental realization [1], graphene has attracted a lot of research interest because of its remarkable transport properties. At low energy, single-layer graphene (SLG) has a *linear dispersion spectrum*, and charge carriers can be described as massless, chiral, Dirac fermions. Moreover, SLG is a zero-gap semiconductor, has a high mobility of charge carriers and features an unconventional half-integer quantum Hall effect which can be measured at room temperature [2]; therefore, it has been the focus of many experimental [3] and theoretical analyses [4, 5], and it is expected to have several potential applications in carbon-based spintronics. Thus, the study of spin-related transport phenomena is one of the most active research fields in graphene. One important breakthrough was the demonstration of a quantum spin Hall effect in SLG [6]. This in turn relies on intrinsic spin-orbit coupling (SOC), which although weak compared to other energy scales in the problem [7] opens a gap in the energy spectrum, making SLG into a topological insulator. Another possibility for SOC is due to interactions with a substrate, the presence of electric fields or curvature of the SLG. Therefore, it is called extrinsic or Rashba spin-orbit coupling (RSOC). The RSOC is believed to be responsible for spin polarization [8], and spin relaxation phenomena in graphene [9]. In addition, in the presence of an external magnetic field

Indeed, one interesting aspect of the new graphene-like materials is that they are predicted to support the so-called edge states which are related to the realization of new topological aspects of matter. As it has been acknowledged in the recent literature, since the discovery of the quantum Hall effect [10, 11], the quest for new topological states of matter has attracted a great deal of attention (for a recent review, see Ref. [12]). For instance, some proposals to topologically generate and characterize non-trivial phases in graphene have paved the road for research on how to implement models of non-trivial topological states in different physical systems ranging from semiconducting quantum wells [13], superconductors [14] and neutral [15] and cold atoms [16], just to mention a few.

In Ref. [13], a model for quantum spin Hall (QSH) state in telluride-based quantum wells was analysed. Yet, it was shown afterwards that this model really corresponds to a topological insulator state [17]. In general, a topological insulator is a quantum state of matter where a given material supports edge states that counter-propagate on its boundary. The existence of these edge states allows for non-dissipative transport, which is a most-wanted property in technological implementations, for instance, in quantum information and nanotechnological devices. In principle, the topological-insulating phase is protected by time-reversal symmetry. However, a time-reversal-symmetry-broken QSH state proposal was recently put forward by means of ferromagnetic leads attached to the sample [18].

Upon introduction of time-periodic dynamical modulation, for instance, in semiconducting quantum wells, with a zincblende structure, it has been recently shown that AC driving can induce a topological phase transition leading to the so-called Floquet topological insulator (FTI) phases [19]. This means that a non-trivial topological phase can be induced in a system that in equilibrium behaves trivially [20–22], that is, it does not possess non-dissipative or gapless edge states.

In this chapter, we show some of our results on the photoinduced effects on the Dirac fermions of graphene-like systems presenting two specific scenarios. In Section 2, we describe the Landau levels (LLs) in graphene when a laser is incident perpendicularly to the sample [23]. We show that an exact effective Floquet Hamiltonian can be found from which the dynamics is afterwards described [24–26]. Here, we show that the quasi-energy spectrum presents a level-dependent photoinduced gap and we also show that coherent Landau level states can be synchronized in order to produce Rabi oscillations and quantum revivals [27]. In Section 3, we describe how the energy spectrum and pseudospin polarization in monolayer silicene [28–33] can be manipulated beyond the so-called off-resonant regime [34, 35] when strong radiation effects are taken into account [36–38]. We find the explicit and realistic parameter regime for the realization of a single-valley polarized state in silicene. In Section 4, an outlook is given whereas in Section 5 we present some technical details of the calculations employed because although we know there are several works describing the dynamical aspects of periodically driven systems, we wanted to be self-contained. Another reason for delving into such details is that we think this explicit technical aspect could be useful for both students and researchers interested in the field.

## 2. Technical aspects of periodically driven systems

### 2.1. Floquet Fourier-mode approach

Before delving into the models of interest, let us present a summary of an important tool in the description of time-dependent Hamiltonian dynamics when the interaction term is periodic in time. These kinds of interactions are ubiquitous in physical systems ranging from cold atoms, cavity QED, superconducting interferometric devices, lasers, and so on. Let us then consider a generic time-dependent periodic Hamiltonian

The solution of the dynamics for the evolution operator

is formally given by

with

with

where

In the following, we would like to explicitly describe the Floquet-Fourier mode strategy. For this purpose, we assume that we can solve the dynamics of the free part _{,} where

In order to analyse the evolution equation,

we take advantage of the periodicity of the Hamiltonian so we can resort to Floquet’s theorem [24, 25]. For this purpose, we define an auxiliary Hermitian Hamiltonian

along with the so-called Floquet states

such that

which are periodic functions of time,

We can verify that the states

are also eigenstates of the Hamiltonian *first Brillouin zone*

Using the periodic temporal basis

we write the Fourier-mode expansion

Now, we use the expansion

Multiplication by

and we have used the simplifying notation

Then, the quasi-energies

where

For example, for a two-level problem we could take the interaction in the form _{,} we find that

### 2.2. Off-resonant approximation

In some physical scenarios, the frequency of the radiation field is way much larger than any other energy scale in the problem. Within this regime, one can derive an approximate Floquet Hamiltonian that captures the essence of the photoninduced bandgaps. In order to describe this so-called off-resonant regime, let us begin by explicitly showing the effective Floquet Hamiltonian: For this purpose, let us start from the general periodic Floquet Hamiltonian given in Eq. (1)

where

(18) |

where the interaction submatrices are defined as

For a monochromatic harmonic perturbation, this reduces to a block-tridiagonal matrix

and to simplify the notation we have set _{,} we will have

with each

From the first and last equations, we get

such that we get an effective equation for

which explicitly reads

For

so we get the effective approximate Floquet Hamiltonian, valid for large frequencies

With a similar procedure, one can show that for

These results essentially imply that a photoinduced energy bandgap can be induced by means of the dressed Floquet states that emerge from the off-resonant condition. Needless to say that sometimes the photoinduced energy bandgaps can be tiny; yet, in some circumstances one only needs to make sure that there is a gap, however, small and the topological properties of the driven system can be qualitatively different as those of the undriven (gapless) system.

## 3. Photoinduced effects of Landau levels

In this section, we summarize our results reported in Ref. [23] where we theoretically analyse the dynamical manipulation of the LL structure of charge carriers on suspended monolayer graphene when a periodically driving radiation field is applied perpendicular to the sample. For this purpose, we focus on the low-energy properties of non-interacting spinless charge carriers in a suspended monolayer graphene subject to a perpendicular, uniform and constant magnetic field

where *v*_{F} ∼ 10^{6}*m/s* is the Fermi velocity in graphene. In addition, the canonical momenta _{,} the Hamiltonian Eq. (29) at each K (K’) Dirac point, which corresponds to

where the annihilation and creation operators are defined by standard relations as

The eigenenergies of the Hamiltonian (29) are then

with

for

Due to time-reversal symmetry, we have

where

which explicitly reads

with the effective coupling constant

periodic in time

Thus, let us focus on the K point physics and afterwards, we can make the necessary substitutions. In order to simplify the notation, we set

the time-dependent interaction potential can be rewritten as

Now, we invoke Floquet’s theorem which states that the time evolution operator of the system induced by a periodic Hamiltonian can be written in the form [24]

with

Accordingly, for our problem we can find approximate solutions to the dynamics by modifying slightly the analytical strategy presented in Ref. [39]. Then, one finds that the excitation number operator,

which commutes with the Hamiltonian

such that the time-dependent Schrödinger equation

can be transformed with a time-independent operator

where

where the effective coupling to the radiation field is

The associated mean energies are in turn found as

which are invariant under

As can be seen in **Figure 1**, these mean energies are plotted as function of the quantizing magnetic field

### 3.1. Discussion

So far, we have shown that upon introduction of a perpendicularly radiation field in the Terahertz frequency domain, the Landau-level-quantized scenario can be turned into a level-dependent-gapped system. As shown in reference, this energy bandgap effects can be traced via the oscillations of the pseudospin polarization as well as the temporal evolution of the autocorrelation function for an initially prepared coherent superposition of the static Landau-level configuration. Thus, due to the natural connection between coherent-state superpositions and applications in quantum optics, one could expect that the experimental consequences of such radiation effects on the Landau-level structure of graphene would have some applications within experimentally accessible parameter regimes in the quantum optics realm.

## 4. Irradiated silicene

As expected, the family of new two-dimensional materials with graphene-like properties has grown in the recent years. However, we show in this section that, although similar in lattice structure, both materials can be experimentally found to have different physical properties. In this section, we focus our attention on silicene which consists of a two-dimensional honeycomb lattice structure of silicon atoms analogous to that of graphene. Some works have reported the synthesis of silicene [30–32]. Silicene has a corrugated or buckled lattice structure that makes the silicon atoms in one sublattice to be perpendicularly displaced with respect to the other sublattice. For this reason, when a perpendicular electric field

In particular, since its intrinsic spin-orbit coupling is much larger than that of pristine graphene, an interesting interplay among intrinsic spin-orbit and electric field effects was predicted to appear because the bandgap can be electrically controlled. Moreover, the addition of an exchange potential term (which physically could represent the proximity effect due to coupling of ferromagnetic leads) allows for topological quantum phase transitions in the static regime [33]. Furthermore, in the presence of circularly polarized electromagnetic radiation, the realization of the so-called single Dirac cone phase in silicene has been recently proposed. At this topological phase, it is found that well-defined spin-polarized states are supported at every Dirac point. Moreover, within this configuration different spin components propagate in opposite directions giving rise to a pure spin current [34]. Yet, these photoinduced topological phase changes [20, 21] reported by Ezawa [34] were derived under the off-resonant assumption, that is, dynamical processes such that the frequency (coupling strength) of the radiation field is much larger (smaller) than any other energy scale in the problem. Under these assumptions, it is possible to derive an effective time-independent Floquet Hamiltonian [24, 25] with a tiny photoinduced bandgap correction that stems from virtual photon absorption and emission processes. Since the sign of the bandgap term (i.e., the effective bandgap) determines important topological properties of the material, it is vital both for potential practical implementations, for instance, in technological realizations of silicene-based devices, and from a fundamental point of view, to effectively achieve manipulation of this quantity.

In this section, we show via an exactly solvable model, where in order to detect physically relevant photoinduced effects in the energy band structure of silicene under strong circularly polarized electromagnetic radiation in the terahertz (frequency) domain one needs to go beyond the aforementioned off-resonant approximation. Indeed, we find that a zero momentum, the obtained exactly solvable time-dependent Hamiltonian, suggests the range of control parameters that physically might lead to experimentally feasible realization of new topological phases in silicene.

### 4.1. Model

Let us consider the Dirac cone approximation to describe the dynamics of non-interacting charge carriers in silicene subject to a perpendicular, uniform and constant electric field

where

describe the spin-orbit coupling associated to the next nearest neighbour hopping and nearest neighbour tight-binding formulation, respectively.

The term

Within the approximation *intense* radiation field represented by the time-dependent vector potential

with

Using the standard minimal coupling prescription given as

In the following, we explore the emerging photoinduced dynamical features at zero momentum since this scenario allows for an exact analytical solution to the dynamical evolution equations. Given the fact that we have an exact analytical solution, we can explore the low-, intermediate-, and strong-coupling regimes of the charge carriers in silicene under the radiation field. We then discuss this exact solution and argue the need to explore either the intermediate or strong light-matter-coupling regimes in order to obtain experimentally observable modifications in the physical properties of the system within the irradiation configuration.

### 4.2. Physics at *k* = 0

In this subsection, we explicitly analyse another exactly solvable model for a graphene-like system. Let us then focus on the zero-momentum scenario for which the extrinsic spin-orbit term *z*-component of spin

Using the unitary transformation

we get the effective time-independent Floquet Hamiltonian

Thus, the zero-momentum quasi-energy spectrum is given as

where

On the other hand, the zero-momentum exact Floquet eigenstates are

with

Some comments are in order at this point. The exact quasi-energy spectrum resembles the solution for the Rabi problem and one could expect that Rabi oscillations should appear in the dynamical evolution of this zero-momentum solution. Moreover, since we are interested in analysing the behaviour of a topological quantity not in terms of the momentum variable but assuming as a toy model that one explores the

After some algebra, we get

where

Thus, within this limit, the average out-of-plane pseudospin polarization

Thus, the main advantage of finding semi-analytical solutions to the dynamical evolution is that closed expressions for the physical quantities of interest can be found in such a way that one can provide further insight into the nature of the physical mechanisms involved and how their interplay leads to a given behaviour of the polarization, charge current and so on.

Let us now assume that the system is initially prepared in the arbitrary state

with

In addition, the one-period mean-value pseudospin polarization

with

where

In particular, for initial states that have zero polarization (

Setting the value **Figure 2** the mean pseudospin polarization for the different spin and valley

From this figure, we find that within the low coupling regime (

## 5. Conclusions

In this chapter, we have described some photoinduced consequences of using periodically driven interactions in graphene-like systems. In essence, these examples provide new insights on the non-trivial behaviour of systems taken out of equilibrium and given the fact that Floquet’s theorem allows for a dynamical analysis in terms of an equivalent static description of the physics via the Floquet states, one can infer new physical aspects that can emerge on these systems when subject to such periodic interactions. We consider that these approaches do provide an arena for analysing some interesting phenomena beyond the static limit. The extension of the results presents more involved scenarios, where numerical tools can profit from the simple models described in this chapter and we hope the interested reader could profit from the material presented. One could further explore the dynamical features at finite momentum but refer the reader to Ref. [35] where it is proven that the intermediate light-matter-coupling regime is the physically correct scenario for describing the emergent topological features such as the realization of the single-valley polarized state in silicene. Needless to say that other quantities of interest such as charge and spin currents can be treated within the approximate scenarios discussed in the chapter, they go beyond the main interest of the chapter and are thus not further discussed here.