Introduction

Controlling solids with light constitutes a modern approach to induce novel functionalities. A specific framework within this broader effort is Floquet engineering. Floquet engineering refers to inducing dynamics that are captured by an effective Floquet Hamiltonian in a system by periodic driving. For a non- or weakly interacting system this approach describes effective single-particle states that form a natural basis for the driven system. These states are the Floquet-Bloch bands of the electrons, in analogy to the Bloch bands of the equilibrium system. These Floquet-Bloch bands can have qualitatively distinct features from the Bloch bands of the non-driven system1,2,3,4. A striking example are Floquet topological insulators5,6,7, for which applications in spintronics8 have been discussed. A general overview on spintronics can be found in9. A specific realization of Floquet topological insulators is monolayer graphene illuminated with circularly polarized light, for which the band structure approaches the Haldane model in the high-frequency limit10,11. However, while the ground state of the equilibrium Haldane model forms indeed a topological insulator, which manifests in a quantized Hall conductance, the Hall conductance of optically driven graphene is not topologically quantized, but of geometric-dissipative origin12,13. This observation is part of the larger challenge of an unambiguous detection of the Floquet-Bloch bands in a solid. We note that the geometric properties of bands in periodically driven lattices have been demonstrated in ultracold atom experiments14,15,16 as well as helical wave guides17,18 and classical settings19. Signatures of Floquet-Bloch bands have been seen in angle-resolved photoelectron spectroscopy20 and approaches for observing related pseudospin textures have been proposed21,22. In this context, the effects of Auger recombination23 and scattering decoherence24 on the electron dynamics in graphene have been discussed. The transport properties of similar Floquet systems have been discussed25 and the high-frequency probing limit has been explored26. However, a smoking-gun in the transport measurements of solids is lacking.

In this paper, we propose to detect light-induced Floquet band gaps in graphene via the optical longitudinal transport. We determine the optical conductivity as a function of the probing frequency and the driving field strength, which displays resonant features. We present an interpretation of these features in terms of the Floquet-Bloch band dispersion and the effective occupation of these states. These occupations are determined by the dissipation and the driving field, which balance out to form the steady state. We include the dissipation processes in our master equation approach that we use to describe the system. With this, we attribute the observable resonant features in the optical conductivity to two transition processes. One occurs between bands inside the same Floquet zone and the other between adjacent bands of neighboring Floquet zones. These processes compete in their impact on the optical conductivity, which can result in vanishing and even negative optical conductivity for specific frequencies and driving field strengths. In general, we show that the conductivity depends on the relative occupation of the Floquet bands. We also point out that the relative occupation is in qualitative agreement with a comoving band velocity, to be defined below. In particular, we show that there are regimes of driving field strengths that show an effective inversion of Floquet band populations. These are in the regimes in which negative optical conductivity is achieved. Therefore, as a second point besides the demonstration of Floquet-Bloch bands in solids, our proposal shows non-trivial control of the transport properties of solids, induced by light.

Results and discussion

Model Hamiltonian

We consider a circularly polarized laser with frequency ωd = 2π × 48 THz ≈ 200 meV and variable field strength Ed, which illuminates a graphene layer from perpendicular direction. The electromagnetic forces drive the electrons into a steady state. We propose to measure the longitudinal AC conductivity of this steady state in the optical frequency domain. The conductivity displays frequency regimes in which its magnitude is increased compared to the non-driven graphene layer, and regimes in which it is decreased. These frequency regimes derive from resonances between the Floquet states, which in turn depend on the driving field strength. As a result, these frequency regimes can be tuned to overlap, resulting in a partial cancellation. In particular, the band gap Δ0 at the Dirac point can be overshadowed, in general, by other features. However, we point out a regime in which it can be identified unambiguously.

The Hamiltonian of light-driven graphene, close to the Dirac point is given by

$$H=\mathop{\sum}\limits_{k}{c}_{{{{{{{{\bf{k}}}}}}}}}^{{{{\dagger}}} }{{{{{{{\bf{h}}}}}}}}({{{{{{{\bf{k}}}}}}}}){c}_{{{{{{{{\bf{k}}}}}}}}},$$
(1)

where \({c}_{{{{{{{{\bf{k}}}}}}}}}={({c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{A}}}}}}}}},{c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{B}}}}}}}}})}^{{{{{{{{\rm{T}}}}}}}}}\) and ck,i are the fermionic annihilation operators of an electron with momentum k and sublattice index i = A, B. Invoking the edge-bulk correspondance, the transport properties of the periodic bulk captured by Eq. (1) directly translate to localized edge modes of finite systems. The Hamiltonian of a single momentum k is

$${{{{{{{\bf{h}}}}}}}}({{{{{{{\bf{k}}}}}}}})=\hslash {v}_{{{{{{{{\rm{F}}}}}}}}}({q}_{{{{{{\mathrm{x}}}}}}}{\sigma }_{{{{{{\mathrm{x}}}}}}}+{q}_{{{{{{\mathrm{y}}}}}}}{\sigma }_{{{{{{\mathrm{y}}}}}}}),$$
(2)

with

$${q}_{{{{{{\mathrm{x}}}}}}}={k}_{{{{{{\mathrm{x}}}}}}}+\frac{{E}_{{{{{{{{\rm{d}}}}}}}}}}{{\omega }_{{{{{{{{\rm{d}}}}}}}}}}\sin ({\omega }_{{{{{{{{\rm{d}}}}}}}}}t)-\frac{{E}_{{{{{{{{\rm{L}}}}}}}}}}{{\omega }_{{{{{{{{\rm{L}}}}}}}}}}\cos ({\omega }_{{{{{{{{\rm{L}}}}}}}}}t),$$
(3)
$${q}_{{{{{{\mathrm{y}}}}}}}={k}_{{{{{{\mathrm{y}}}}}}}+\frac{{E}_{{{{{{{{\rm{d}}}}}}}}}}{{\omega }_{{{{{{{{\rm{d}}}}}}}}}}\cos ({\omega }_{{{{{{{{\rm{d}}}}}}}}}t),$$
(4)

where vF ≈ 106 m s−1 is the Fermi velocity. ki are the momentum components and σi are the Pauli matrices. Ed and ωd are field strength and frequency of the driving laser. EL and ωL are the same quantities for the longitudinal probing field.

We simulate the dynamics via a master equation approach, expanding on previous work by some of the authors12. The density matrix of the system factorizes in momentum space, as ρ = ∏kρk. Each ρk matrix operates on a four dimensional Hilbert space, given by the states \(\left|0\right\rangle\), \({c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{A}}}}}}}}}^{{{{\dagger}}} }\left|0\right\rangle\), \({c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{B}}}}}}}}}^{{{{\dagger}}} }\left|0\right\rangle\), \({c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{B}}}}}}}}}^{{{{\dagger}}} }{c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{A}}}}}}}}}^{{{{\dagger}}} }\left|0\right\rangle\). We include doubly and unoccupied states to determine two-time correlation functions, and thereby frequency-resolved quantities.

In addition to the unitary time evolution induced by the Hamiltonian in Eq. (2), we include dissipation via Lindblad operators defined in the instantaneous eigenbasis of the driven system, to describe the dissipative environment due to degrees of freedom not included in the Hamiltonian. We include a dephasing term γz, a decay term γ− and a term with decay rate γbg that models particle exchange of the graphene layer to a supporting substrate backgate. This model provides a realistic discription of the non-equilibrium electron dynamics12.

We choose the coefficients γz = 1 THz, γ− = 2.25 THz and γbg = 2.5 THz. This sets the scale for the broadening of the effective bands in the single-particle correlation function as well as the optical conductivity. These values are a factor of 10 smaller than those estimated12 for the experimental setup of McIver et al.13. Our predictions apply to high-mobility samples, e.g., BN-encapsulated graphene. For larger values, such as those that are realized in the work of McIver et al.13, resolving the gap features that we describe in the following, would require larger driving frequencies and stronger driving. Throughout this work we use the temperature T = 80 K, which is the same as the setup of McIver et al.13. We fix the value of the chemical potential at μ = 0.

Electron distribution

As a first observable we display the momentum-resolved and energy-resolved electron distribution inspired by Freericks et al.27

$$n({{{{{{{\bf{k}}}}}}}},\omega )=\int\nolimits_{{\tau }_{1}}^{{\tau }_{2}}\int\nolimits_{{\tau }_{1}}^{{\tau }_{2}}{{{{{{{\mathcal{G}}}}}}}}({{{{{{{\bf{k}}}}}}}},{t}_{2},{t}_{1})\frac{{e}^{i\omega ({t}_{2}-{t}_{1})}}{{({\tau }_{2}-{\tau }_{1})}^{2}}{{{{{{{\rm{d}}}}}}}}{t}_{2}{{{{{{{\rm{d}}}}}}}}{t}_{1},$$
(5)

which is manifestly real-valued with

$${{{{{{{\mathcal{G}}}}}}}}({{{{{{{\bf{k}}}}}}}},{t}_{2},{t}_{1})=\langle {c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{A}}}}}}}}}^{{{{\dagger}}} }({t}_{2}){c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{A}}}}}}}}}({t}_{1})\rangle +\langle {c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{B}}}}}}}}}^{{{{\dagger}}} }({t}_{2}){c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{B}}}}}}}}}({t}_{1})\rangle$$
(6)

for which its complex conjugate corresponds to the exchange of t1 and t2. We use the time interval [τ1, τ2] as the probing interval. We choose τ1 such that the system has reached its steady state. τ2 − τ1 is a sufficiently long probing time of the order of hundreds of driving periods 2π/ωd that is also commensurate with the probing period 2π/ωL. We note that this quantity provides a prediction for trARPES measurements27. In Fig. 1 we show n(k = ∣k∣, ω) for the driving field strength Ed = 26 Me V m−1. We note that a similar result was presented in previous work by some of the authors12. The electron distribution of the steady state is consistent with the effective band structure predicted by Floquet theory and identifies the non-equilibrium electron occupation of these Floquet-Bloch bands.

Fig. 1: Electron distribution of light-driven graphene, revealing its Floquet states.
figure 1

The electron distribution n(k, ω) of graphene driven with circularly polarized light at the driving frequency ωd = 2π × 48 THz ≈ 200 meV and field strength Ed = 26 MV m−1. The distribution n(k, ω) depends only on the momentum k = ∣k∣. This quantity displays the steady state occupation of the Floquet-Bloch band structure. The one-photon resonance gap Δ1 at k = ωd/(2vF), the two-photon gap Δ0 at the Dirac point, and the two-photon gap at k = ωd/vF are highlighted for clarity. vF is the Fermi velocity. Additionally, the complementary gaps ωd − Δ1 and ωd − Δ2 are indicated. The dotted lines indicate the Floquet energies of the first Floquet zone.

Full size image

We label the band gaps as Δm, based on their location mωd/(2vF) in momentum space for small driving field strength Ed → 0, as shown in Fig. 1. Due to the periodicity in frequency space of the Floquet spectrum, there is a complementary gap ωd − Δm for any given band gap Δm, with m > 0. These complementary gaps are also visible in the optical conductivity of the system. They reduce the conductivity at the corresponding frequency, rather than enhance it. The gap Δ0 at the Dirac point does not exhibit this behavior, as discussed later.

Optical conductivity

The second observable that we present is the longitudinal optical conductivity. We propose to measure this quantity experimentally, to compare to the predictions made here. In Fig. 2 we show the real part of the total optical conductivity of the system as a function of the driving field strength Ed. This is obtained from our master equation approach as

$${\sigma }_{{{{{{{{\rm{r}}}}}}}}}({\omega }_{{{{{{{{\rm{L}}}}}}}}})={{{{{{{\rm{Re}}}}}}}}\left[\frac{{j}_{{{{{{\mathrm{x}}}}}}}({\omega }_{{{{{{{{\rm{L}}}}}}}}})}{{E}_{{{{{{\mathrm{x}}}}}}}({\omega }_{{{{{{{{\rm{L}}}}}}}}})}\right],$$
(7)

with the longitudinal current and electric field

$${j}_{{{{{{\mathrm{x}}}}}}}({\omega }_{{{{{{{{\rm{L}}}}}}}}})={n}_{{{{{{{{\rm{s}}}}}}}}}{n}_{{{{{{{{\rm{v}}}}}}}}}e{v}_{{{{{{{{\rm{F}}}}}}}}}\mathop{\sum}\limits_{{{{{{{{\bf{k}}}}}}}}}\int\nolimits_{\tau }^{\tau +\frac{2\pi }{{\omega }_{{{{{{{{\rm{L}}}}}}}}}}}{{{{{{{\rm{Tr}}}}}}}}({\rho }_{{{{{{{{\bf{k}}}}}}}}}(t){\sigma }_{{{{{{\mathrm{x}}}}}}}){e}^{i{\omega }_{{{{{{{{\rm{L}}}}}}}}}t}{{{{{{{\rm{d}}}}}}}}t,$$
(8)
$${E}_{{{{{{\mathrm{x}}}}}}}({\omega }_{{{{{{{{\rm{L}}}}}}}}})=\int\nolimits_{\tau }^{\tau +\frac{2\pi }{{\omega }_{{{{{{{{\rm{L}}}}}}}}}}}({E}_{{{{{{{{\rm{d}}}}}}}}}\cos ({\omega }_{{{{{{{{\rm{d}}}}}}}}}t)+{E}_{{{{{{{{\rm{L}}}}}}}}}\sin ({\omega }_{{{{{{{{\rm{L}}}}}}}}}t)){e}^{i{\omega }_{{{{{{{{\rm{L}}}}}}}}}t}{{{{{{{\rm{d}}}}}}}}t,$$
(9)

where τ is a point in time where the system has reached its steady state. ns = nv = 2 are the spin- and valley-degeneracies. e is the electron charge. σr(ωL) is obtained for the probing field EL = 10 V m−1. We have verified that the conductivity obtained in this manner is the linear response and that the sum over k includes sufficiently many points surrounding the Dirac point.

Fig. 2: Longitudinal optical conductivity of light-driven graphene.
figure 2

The real part of the optical conductivity of graphene driven at the frequency ωd = 2π × 48 THz ≈ 200 meV as a function of the driving field strength Ed (a) and a cut at half the driving frequency ωL = ωd/2 = 2π × 24 THz (b). The dashed lines show the various band gaps Δm as given by Floquet theory. The gap Δ0 becomes clearly visible above values of ωL ≈ 2π × 14 THz ≈ 60 meV and Ed ≈ 28 MV m−1. We also see the complementary resonant features at ωd − Δm, with m > 0.

Full size image

As we demonstrate in Fig. 2a, σr(ωL) displays resonant features that match the band gaps of the Floquet spectrum. The energy gap Δ0 increases with increasing field strength Ed, in a monotonuous fashion. The energy gaps Δm, with m > 0, first increase with Ed, then reach a maximum at \({E}_{{{{{{{{\rm{d}}}}}}}}}={E}_{{{{{{{{\rm{d}}}}}}}},\max }^{(m)}\), then decrease, and ultimately reach 0 at \({E}_{{{{{{{{\rm{d}}}}}}}}}={E}_{{{{{{{{\rm{d}}}}}}}},{{{{{{{\rm{band}}}}}}}}}^{(m)}\). At this driving strength the gap is located at k = 0 and merges with Δ0.

The magnitude of σr(ωL) at the resonance Δ1, i.e., the magnitude of σr(Δ1), displays a maximum for \({E}_{{{{{{{{\rm{d}}}}}}}}} < {E}_{{{{{{{{\rm{d}}}}}}}},\max }^{(1)}\), relative to its background, and a minimum for \({E}_{{{{{{{{\rm{d}}}}}}}}} > {E}_{{{{{{{{\rm{d}}}}}}}},\max }^{(1)}\). The magnitude of σr(ωL) at ωd − Δ1, displays the complementary behavior. σr(ωd − Δ1) has a minimum for \({E}_{{{{{{{{\rm{d}}}}}}}}} < {E}_{{{{{{{{\rm{d}}}}}}}},\max }^{(1)}\), and a maximum for \({E}_{{{{{{{{\rm{d}}}}}}}}} > {E}_{{{{{{{{\rm{d}}}}}}}},\max }^{(1)}\). Note that this does not happen for Δ0 due to the lack of a complementary gap ωd − Δ0 as can be seen in Figs. 1 and 3.

Fig. 3: Electron distribution at the Dirac point.
figure 3

The electron distribution n(k = 0, ω) at the Dirac point as a function of the driving field strength Ed. The driving frequency is ωd = 2π × 48 THz ≈ 200 meV. The scaling behavior of the gap at the Dirac point is \({{{\Delta }}}_{0}=2\sqrt{{v}_{{{{{{{{\rm{F}}}}}}}}}^{2}{E}_{{{{{{{{\rm{d}}}}}}}}}^{2}/{\omega }_{{{{{{{{\rm{d}}}}}}}}}^{2}+{\omega }_{{{{{{{{\rm{d}}}}}}}}}^{2}/4}-{\omega }_{{{{{{{{\rm{d}}}}}}}}}\), where vF is the Fermi velocity. The vertical dotted line indicates \({E}_{{{{{{{{\rm{d}}}}}}}}}={E}_{{{{{{{{\rm{d}}}}}}}},{{{{{{{\rm{band}}}}}}}}}^{(1)}.\) The horizontal dotted lines indicate Floquet zone boundaries. The dashed lines show the Floquet energies at the Dirac point (See Supplementary Note 1) that are formally constrained to be inside the first Floquet zone. The occupations stay confined within the Floquet bands adiabatically connected to the bare graphene and one replica outwards. There are no complementary gaps at k = 0.

Full size image

We note that in the limit of Ed → 0, the optical conductivity σr(ωL) approaches the value \(\frac{1}{4}\frac{{e}^{2}}{\hslash }\) for non-zero frequencies. An example for this is visible in Fig. 2b, for ωL = 2π × 24 THz. Additionally, we obtain a peak at ωL = 0, which is the Drude peak broadened by the dissipative terms. We show the real part of the longitudinal conductivity σr(ωL) for Ed = 0 in Supplementary Note 2.

We obtain analytical expressions for Δ0 and \({E}_{{{{{{{{\rm{d}}}}}}}},{{{{{{{\rm{band}}}}}}}}}^{(m)}\) by considering the Hamiltonian in Eq. (2) at the Dirac point and without probing, i.e., k = 0 and EL = 0. This has the time-dependent Rabi solutions

$$\left|+\right\rangle \sim {e}^{i({\omega }_{{{{{{{{\rm{d}}}}}}}}}t/2+\pi /4)}\left(\begin{array}{c}\cos ({{\Omega }}t)-\frac{i{\omega }_{{{{{{{{\rm{d}}}}}}}}}\sin ({{\Omega }}t)}{2{{\Omega }}}\\ {e}^{-i{\omega }_{{{{{{{{\rm{d}}}}}}}}}t}\frac{{E}_{{{{{{{{\rm{d}}}}}}}}}\sin ({{\Omega }}t)}{{{\Omega }}\omega }\end{array}\right),$$
(10)
$$\left|-\right\rangle \sim {e}^{-i({\omega }_{{{{{{{{\rm{d}}}}}}}}}t/2+\pi /4)}\left(\begin{array}{c}-{e}^{i{\omega }_{{{{{{{{\rm{d}}}}}}}}}t}\frac{{E}_{{{{{{{{\rm{d}}}}}}}}}\sin ({{\Omega }}t)}{{{\Omega }}{\omega }_{{{{{{{{\rm{d}}}}}}}}}}\\ \cos ({{\Omega }}t)+\frac{i{\omega }_{{{{{{{{\rm{d}}}}}}}}}\sin ({{\Omega }}t)}{2{{\Omega }}}\end{array}\right),$$
(11)

where

$${{\Omega }}=\sqrt{\frac{{v}_{{{{{{{{\rm{F}}}}}}}}}^{2}{E}_{{{{{{{{\rm{d}}}}}}}}}^{2}}{{\omega }_{{{{{{{{\rm{d}}}}}}}}}^{2}}+\frac{{\omega }_{{{{{{{{\rm{d}}}}}}}}}^{2}}{4}}.$$
(12)

The gap at the Dirac point is given by Δ0 = 2Ω − ωd. This expression is also the Aharanov-Anandan phase of this system2. In the weak driving limit this gap follows the expected perturbative behavior11\({{{\Delta }}}_{0}\approx {v}_{{{{{{{{\rm{F}}}}}}}}}^{2}{E}_{{{{{{{{\rm{d}}}}}}}}}^{2}/{\omega }_{{{{{{{{\rm{d}}}}}}}}}^{3}\) whereas in the strong driving limit it develops a linear dependence on Ed as Δ0 ≈ vFEd/ωd. We use the full expression for Δ0 to find the driving strengths \({E}_{{{{{{{{\rm{d}}}}}}}},{{{{{{{\rm{band}}}}}}}}}^{(m)}\), since they occur whenever the gap Δ0 spans a multiple of ωd. By setting 2Ω − ωd = mωd, \(m\in {\mathbb{N}}\), we find

$${E}_{{{{{{{{\rm{d}}}}}}}},{{{{{{{\rm{band}}}}}}}}}^{(m)}\,=\,{v}_{{{{{{{{\rm{F}}}}}}}}}^{-1}\sqrt{\frac{m}{2}+\frac{{m}^{2}}{4}}{\omega }_{{{{{{{{\rm{d}}}}}}}}}^{2}.$$
(13)

We display the Dirac gap in Fig. 3, and compare it to the electron distribution at k = 0, of the steady state. We observe that the two maxima of the electron distribution that emerge from ω = 0 follow the prediction of ±Δ0, even as Δ0 grows larger than the Floquet zone boundary at ωd. Therefore, Δ0 is a more natural energy scale to predict the resonances at k = 0 for large driving intensities, than the direct band gap that is strictly smaller than ωd. For increasing field strength Ed, the occupation of the upper two bands decreases. The occupation of complementary gaps is zero throughout Fig. 3.

As visible in Fig. 2a, the conductivity vanishes around the probing frequency ωL ≈ 2π × 12 THz ≈ 50 meV and the driving field strength of Ed ≈ 39 MV m−1. Here the first gap Δ1 decreases with increasing Ed and creates a negative contribution that suppresses σr(ωL) to zero. For higher order gaps, e.g., Δ2 and Δ3 in Fig. 2a, the same phenomenon even leads to a sign change in the conductivity. Whenever a gap is in the regime of decreasing with increasing Ed, and no other resonance contributes positively and too strongly to the conductivity, the negative contributions can cancel the background and result in net negative optical conductivity. Such a total negative conductivity in the system amounts to an out-of-phase response to a probing field with a probing frequency in the regime in which σr(ωL) is negative. In principle this can be utilized to obtain electrical gain out of the system, where the required energy is effectively taken from the driving.

Momentum-resolved conductivity

In order to gain further insight into the origin of the features in the total optical conductivity shown in Fig. 2, we explore the momentum resolved contributions to the conductivity. In Fig. 4 we resolve the contributions to the conductivity along the kx and ky directions relative to the Dirac point in momentum space, defined as

$${\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}({{{{{{{\bf{k}}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}})=\frac{{n}_{{{{{{{{\rm{s}}}}}}}}}{n}_{{{{{{{{\rm{v}}}}}}}}}e{v}_{{{{{{{{\rm{F}}}}}}}}}| {{{{{{{\bf{k}}}}}}}}| }{{E}_{x}({\omega }_{{{{{{{{\rm{L}}}}}}}}})}\int\nolimits_{\tau }^{\tau +\frac{2\pi }{{\omega }_{{{{{{{{\rm{L}}}}}}}}}}}{{{{{{{\rm{Tr}}}}}}}}({\rho }_{{{{{{{{\bf{k}}}}}}}}}(t){\sigma }_{x}){e}^{i{\omega }_{{{{{{{{\rm{L}}}}}}}}}t}{{{{{{{\rm{d}}}}}}}}t.$$
(14)

Here, we include the linear scaling with the absolute momenta ∣k∣ in polar coordinates. Direct interband transitions between neighboring Floquet bands give rise to resonant features in \({\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}({{{{{{{\bf{k}}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}})\) that match the Floquet band energy differences Δϵ(k) and ωd − Δϵ(k) (See Supplementary Note 1). These resonant features contribute to the conductivity with alternating signs. The sign changes occur close to the band gap locations, but slightly shifted towards (away from) the Dirac point in case the gap size increases (decreases) with respect to the field strength Ed. For gaps that do not change with respect to Ed, i.e., gaps at their maximum, this shift vanishes. Therefore, the accumulated contributions across gaps net either positive or negative conductivity depending on the change in gap size with respect to field strength Ed. This is consistent with the enhancements and reductions in σr(ωL) at the gaps Δm, with m > 0, and their complementary gaps ωd − Δm, seen in Fig. 2a.

Fig. 4: Momentum-resolved optical conductivity of light-driven graphene.
figure 4

The momentum-resolved contributions to the optical conductivity of driven graphene along the kx (a) and ky (b) momentum directions. The driving frequency is ωd = 2π × 48 THz ≈ 200 meV and the field strength is Ed = 34 MV m−1. For these parameters the gap at the Dirac point roughly matches half the driving frequency such that Δ0 ≈ ωd/2. The dashed lines indicate the Floquet band energy differences Δϵ(k) and ωd − Δϵ(k) (See Supplementary Note 1).

Full size image

For probing frequencies ωL that are not resonant with a given band gap, \({\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}({{{{{{{\bf{k}}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}})\) does not vanish in general. This results in a background conductivity that can obscure the gap Δ0 at the Dirac point in particular, as is the case for \({E}_{{{{{{{{\rm{d}}}}}}}}} < {E}_{{{{{{{{\rm{d}}}}}}}},\max }^{(1)}\) in Fig. 2a. Since the band gaps Δm, with m > 0, and their complementary gaps ωd − Δm have a maximum at the field strength \({E}_{{{{{{{{\rm{d}}}}}}}}}={E}_{{{{{{{{\rm{d}}}}}}}},\max }^{(m)}\), there is always a range that no gap Δm, with m > 0, reaches that is centered around ωL = ωd/2. In this range, it is the gap Δ0 that is visible predominantly. The overall behavior of the gaps is self-similar with respect to the driving frequency ωd. Therefore in this system, there always exists a reliable range of probing frequencies where the gap Δ0 can be observed.

The Floquet interband transitions resonant with Δϵ(k) occur inside a given Floquet zone. The ones resonant with ωd − Δϵ(k) occur across Floquet zone boundaries. Hence, we refer to them as intra-Floquet \({\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{intra}}}}}}}}}({{{{{{{\bf{k}}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}})\) and inter-Floquet \({\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{inter}}}}}}}}}({{{{{{{\bf{k}}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}})\) contributions to the conductivity, respectively. To distinguish the two we write

$${\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}({{{{{{{\bf{k}}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}})={\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{intra}}}}}}}}}({{{{{{{\bf{k}}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}})+{\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{inter}}}}}}}}}({{{{{{{\bf{k}}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}})+{\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{bg}}}}}}}}}({{{{{{{\bf{k}}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}}),$$
(15)

where \({\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{bg}}}}}}}}}({{{{{{{\bf{k}}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}})\) is a remaining background contribution accounting for the ωL → 0 behavior in \({\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}({{{{{{{\bf{k}}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}})\). Figure 4b shows that \({\sigma }_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{bg}}}}}}}}}({k}_{{{{{{\mathrm{y}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}})\approx 0\). We fit a function of two Lorentzians located at Δϵ(k) and ωd − Δϵ(k) with the same fixed width Γ = 1 THz to the numerical results of \({\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}({k}_{{{{{{\mathrm{y}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}})\). Specifically, we use

$${\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{fit}}}}}}}}}({k}_{{{{{{\mathrm{y}}}}}}},{\omega }_{{{{{{{{\rm{L}}}}}}}}})= \, \frac{{{\Gamma }}}{\pi }\frac{{\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{intra}}}}}}}}}({k}_{{{{{{\mathrm{y}}}}}}})}{{{{\Gamma }}}^{2}+{({\omega }_{{{{{{{{\rm{L}}}}}}}}}-{{\Delta }}\epsilon )}^{2}}\\ \, +\ \frac{{{\Gamma }}}{\pi }\frac{{\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{inter}}}}}}}}}({k}_{{{{{{\mathrm{y}}}}}}})}{{{{\Gamma }}}^{2}+{({\omega }_{{{{{{{{\rm{L}}}}}}}}}-{\omega }_{{{{{{{{\rm{d}}}}}}}}}+{{\Delta }}\epsilon )}^{2}}$$
(16)

as a fitting function.

The conductivity features derive from the transitions between the Floquet bands, and are therefore related to the occupation of these bands. We define the relative occupation

$${{\Delta }}n(k)=\mathop{\sum}\limits_{{{{{{\mathrm{m}}}}}}\in {\mathbb{Z}}}{n}_{{{{{{\mathrm{m}}}}}}}^{-}(k)-{n}_{{{{{{\mathrm{m}}}}}}}^{+}(k),$$
(17)

where \({n}_{{{{{{\mathrm{m}}}}}}}^{\pm }(k)\) is the occupation at momentum k of the mth upper (lower) Floquet band given by integrating n(k, ω) from \((m-\frac{1}{4}\pm \frac{1}{4}){\omega }_{{{{{{{{\rm{d}}}}}}}}}\) to \((m+\frac{1}{4}\pm \frac{1}{4}){\omega }_{{{{{{{{\rm{d}}}}}}}}}\).

Figure 5 shows the momentum-resolved intra-Floquet conductivity \({\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{intra}}}}}}}}}({k}_{{{{{{\mathrm{y}}}}}}})\) which is determined via fitting as described above, as well as the effective relative occupation Δn(k) of the Floquet bands as functions of the field strength Ed. They are in good qualitative agreement with each other. Both quantities display tongues with alternating signs and zero-crossings seperating them that agree very well between Δn(k) and \({\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{intra}}}}}}}}}({k}_{{{{{{\mathrm{y}}}}}}})\). The zero-crossings touch the vF k-axis at mωd/2 and the Ed-axis at \({E}_{{{{{{{{\rm{d}}}}}}}},{{{{{{{\rm{band}}}}}}}}}^{(m)}\), \(m\in {\mathbb{N}}\). The solid white lines show the location of the Floquet band gaps, i.e., where the radial band velocity vanishes, i.e., ∂k ϵ = 0. They roughly follow the zero-crossings of Δn(k) and \({\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{intra}}}}}}}}}({k}_{{{{{{\mathrm{y}}}}}}})\) while showing small, but clear deviations. We observe that for \({E}_{{{{{{{{\rm{d}}}}}}}},\max }^{(m)} < {E}_{{{{{{{{\rm{d}}}}}}}}} < {E}_{{{{{{{{\rm{d}}}}}}}},{{{{{{{\rm{band}}}}}}}}}^{(m)}\) the steady state displays an inversion of the Floquet bands, which creates a negative contribution to the optical conductivity. The dotted white lines indicate where the comoving radial band velocity ∂Πϵ with Π = vF(k + Ed/ωd) vanishes, i.e., ∂Πϵ = 0. These lines show improved agreement with the zero-crossings of Δn(k) and \({\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{intra}}}}}}}}}({k}_{{{{{{\mathrm{y}}}}}}})\). Further, there is an overall resemblance between ∂Πϵ(k), Δn(k), and \({\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{intra}}}}}}}}}({k}_{{{{{{\mathrm{y}}}}}}})\) (See Supplementary Note 1).

Fig. 5: Effective Floquet state occupation and intra-Floquet conductivity.
figure 5

The effective occupation Δn(k) (a) and the fitted intra-Floquet conductivity \({\tilde{\sigma }}_{{{{{{{{\rm{r}}}}}}}}}^{{{{{{{{\rm{intra}}}}}}}}}({k}_{{{{{{\mathrm{y}}}}}}})\) (b) as functions of the field strength Ed. The solid white lines are given by the locations in momentum space of the band gaps Δm, with m > 0. The dotted white lines are given by the zero-crossings of a type of comoving band velocity ∂ΠΔϵ = 0, where Π = vF(k + Ed/ωd) (See Supplementary Note 1). The dashed line is given by \(k=\frac{{E}_{{{{{{{{\rm{d}}}}}}}}}}{{\omega }_{{{{{{{{\rm{d}}}}}}}}}}\).

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We summarize that the momentum-resolved optical conductivity shows two types of interband processes across the effective Floquet band structure. These resonant processes correspond to the energy differences Δϵ(k) and ωd − Δϵ(k) between Floquet bands and contribute both positively and negatively. We relate these conductivities to the effective relative occupation of the Floquet bands Δn(k). We find that effective inversions of the Floquet bands correspond to reductions in the conductivity which can lead to a sign change in the total optical conductivity. These band inversions at the Floquet gaps and their reductions of the optical conductivity systematically occur in regimes of decreasing gap sizes with respect to the driving field strength.

Conclusion

In conclusion, we have proposed the longitudinal optical conductivity of illuminated graphene as a realistic observable to detect Floquet band gaps. We have shown that this quantity displays the Floquet gaps as functions of the driving intensity and the probing frequency. In particular, we have pointed out a regime in which the band gap at the Dirac point can be detected. All band gaps except for the band gap at the Dirac point, first increase with the driving intensity, approach a maximal value, and then decrease. For the increasing regime, the optical conductivity displays a positive contribution. For the decreasing regime, it displays a negative contribution that can amount to a total negative conductivity at the given frequency. We point out that this negative contribution derives from an inversion of the occupation of the Floquet bands. Therefore, the proposed experiment not only provides an unambiguous detection of Floquet bands, but also demonstrates dynamical control of transport in solids with light.

Methods

Driven graphene dynamics

We express the driven graphene Hamiltonian in a four-level description, spanned by the states \(\left|0\right\rangle\), \({c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{A}}}}}}}}}^{{{{\dagger}}} }\left|0\right\rangle\), \({c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{B}}}}}}}}}^{{{{\dagger}}} }\left|0\right\rangle\) and \({c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{B}}}}}}}}}^{{{{\dagger}}} }{c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{A}}}}}}}}}^{{{{\dagger}}} }\left|0\right\rangle\). The \({c}_{{{{{{{{\bf{k}}}}}}}},{{{{{{{\rm{A}}}}}}}}/{{{{{{{\rm{B}}}}}}}}}^{({{{\dagger}}} )}\) are the annihilation (creation) operators at the momentum k in the sublattice A/B. The Hamiltonian H is defined in Eq. (1). We factorize the density matrix in momentum space as ρ = Πkρk and simulate the dissipative dynamics using the Lindblad-von Neumann master equation

$$\dot{\rho }=i[\rho ,H]+\mathop{\sum}\limits_{{{{{{\mathrm{j}}}}}}}{c}_{{{{{{\mathrm{j}}}}}}}({L}_{{{{{{\mathrm{j}}}}}}}\rho {L}_{{{{{{\mathrm{j}}}}}}}^{{{{\dagger}}} }-\frac{1}{2}\{{L}_{{{{{{\mathrm{j}}}}}}}^{{{{\dagger}}} }{L}_{{{{{{\mathrm{j}}}}}}},\rho \}),$$

where the sum over j goes over the momentum-dependent Lindblad operators

$${L}_{z}= \, V({c}_{{{{{{\bf{k}}}}}},{{{{{\rm{A}}}}}}}^{{{\dagger}} }{c}_{{{{{{\bf{k}}}}}},{{{{{\rm{A}}}}}}}-{c}_{{{{{{\bf{k}}}}}},{{{{{\rm{B}}}}}}}^{{{\dagger}} }{c}_{{{{{{\bf{k}}}}}},{{{{{\rm{B}}}}}}}){V}^{{{\dagger}} }\\ {L}_{+} = \,V({c}_{{{{{{\bf{k}}}}}},{{{{{\rm{B}}}}}}}^{{{\dagger}} }{c}_{{{{{{\bf{k}}}}}},{{{{{\rm{A}}}}}}}){V}^{{{\dagger}} }\\ {L}_{-} = \, V({c}_{{{{{{\bf{k}}}}}},{{{{{\rm{A}}}}}}}^{{{\dagger}} }{c}_{{{{{{\bf{k}}}}}},{{{{{\rm{B}}}}}}}){V}^{{{\dagger}} }\\ {L}_{{{{\mathrm{l}}}}} = \, V\left(\begin{array}{rclc}0&{\delta }_{{{{\mathrm{l}}}},-1}&{\delta }_{{{{\mathrm{l}}}},-2}&0\\ {\delta }_{{{{\mathrm{l}}}},1}&0&0&{\delta }_{{{{\mathrm{l}}}},-3}\\ {\delta }_{{{{\mathrm{l}}}},2}&0&0&{\delta }_{{{{\mathrm{l}}}},-4}\\ 0&{\delta }_{{{{\mathrm{l}}}},3}&{\delta }_{{{{\mathrm{l}}}},4}&0\end{array}\right){V}^{{{\dagger}} }$$

with l = ±1, ±2, ±3, ±4. δl,i is the Kronecker-Delta and V is the transformation into the instantaneous eigenbasis of h(k) defined in Eq. (2). The dissipation coefficients cj fulfill the conditions

$${c}_{z}= \, {\gamma }_{z}\\ {c}_{+}+{c}_{-} = \, {\gamma }_{-}\qquad {c}_{+}={c}_{-}\exp \{-2\epsilon \beta \}\\ {c}_{1}+{c}_{-1} = \, {\gamma }_{{{{{{\rm{bg}}}}}}}\qquad {c}_{-1}={c}_{1}\exp \{+\epsilon \beta \}\\ {c}_{2}+{c}_{-2} = \, {\gamma }_{{{{{{\rm{bg}}}}}}}\qquad {c}_{-2}={c}_{2}\exp \{-\epsilon \beta \}\\ {c}_{3}+{c}_{-3} = \, {\gamma }_{{{{{{\rm{bg}}}}}}}\qquad {c}_{-3}={c}_{3}\exp \{-\epsilon \beta \}\\ {c}_{4}+{c}_{-4} = \, {\gamma }_{{{{{{\rm{bg}}}}}}}\qquad {c}_{-4}={c}_{4}\exp \{+\epsilon \beta \}$$

with \(\beta ={({k}_{{{{{{{{\rm{B}}}}}}}}}T)}^{-1}\). ±ϵ are the instantaneous eigenenergies of h(k). This approach is also detailed in previous work12.

In order to calculate the electron distribution, we first calculate the two-point correlation functions \(\langle {c}_{{{{{{{{\bf{k}}}}}}}},i}^{{{{\dagger}}} }({t}_{2}){c}_{{{{{{{{\bf{k}}}}}}}},i}({t}_{1})\rangle\). We do this by acting with ck,i on the density matrix ρk(t1) and continuing the time-evolution with the same master equation until the time t2 at which we act on the resulting density with the operator \({c}_{{{{{{{{\bf{k}}}}}}}},{{{{{\mathrm{i}}}}}}}^{{{{\dagger}}} }\). We do this for all pairs of times t1 and t2 in the interval [τ1, τ2] and calculate the electron distribution as detailed in Eq. (5).