Optical absorption properties of laser-driven matter

Characterizing and controlling matter driven far from equilibrium represents a major challenge for science and technology. Here, we develop a theory for the optical absorption of electronic materials driven far from equilibrium by resonant and nonresonant lasers. In it, the interaction between matter and the driving light is treated exactly through a Floquet analysis, while the effects of the probing light are captured to first order in perturbation theory. The resulting equations are reminiscent to those for equilibrium absorption but with the Floquet modes playing the role of the pristine eigenstates. The formalism is employed to characterize the optical properties of a model nanoscale semiconductor dressed by nonresonant light of intermediate intensity (nonperturbative, but nonionizing). As shown, nonresonant light can reversibly turn this transparent semiconductor into a broadband absorber and open strong absorption and stimulated emission bands at very low frequencies ($\sim \mathrm{meV}$). Further, the absorption spectra of the driven material exhibit periodic features energetically spaced by the photon energy of the driving light that reflect the periodic structure of the Floquet bands. These developments offer a platform to understand and predict the emergent optical properties of materials dressed by the electric field of light, and catalyze the design of laser-driven materials with desired optical properties.

Figure 1

Simulation of the Autler-Townes splitting using Eq. (46). (a) Energy diagram for a three-level system. The driving pulse resonantly drives the $|2\rangle \to |3\rangle$ transition, while the probe light measures the optical absorption. Here, ${\mu }_0=\langle 1|\mu |2\rangle =\langle 2|\mu |3\rangle$ are the nonzero transition dipoles. (b) The absorption spectrum computed using Eq. (46) (broadened with a Lorentzian function of width $0.01\mathrm{\Delta }$) under different driving amplitudes ${\varepsilon }_d$ naturally exhibits the Autler-Townes splitting. (c) The AT splitting yielded by Eq. (46) is $\hslash {\mathrm{\Omega }}_{\text{R}}$ where ${\mathrm{\Omega }}_{\text{R}}\equiv {\mu }_0{\varepsilon }_d/\hslash$ is the Rabi frequency in quantitative agreement with theoretical predictions using a different method [20] and experiments [36].

Figure 2

Interpretation of the nonequilibrium absorption spectra using Eq. (46) of the three-level system in Fig. 1 initially prepared in $|1\rangle$ dressed by light resonant with the $|2\rangle \to |3\rangle$ transition with ${\mu }_0{\varepsilon }_d/\mathrm{\Delta }=0.04$. (a) The absorption spectrum has two main transitions labeled X and Y. (b) Overlap between the Floquet modes at times $t_0+nT$ and the pristine states ${\left|\langle \alpha |{\varphi }_{\lambda }\left(t_0\right)\rangle \right|}^2$ in the first FBZ. The Floquet modes are ordered by the quasienergy in the first FBZ. The optical transitions X and Y are indicated by green arrows with the number of FBZs that separates them $+n$. (c) Effective dipole between Floquet modes two FBZs away. Other nonzero transition dipoles are not allowed by the population factor. (d) Effective population factor $P_{{\lambda }^{\prime }\lambda }({\lambda }^{\prime }\to \lambda )$ between Floquet modes. The elements in (c) and (d) corresponding to the X, Y transitions are marked.

Figure 3

Linear optical absorption spectrum for a semiconducting nanoparticle with Hamiltonian in Eq. (47) dressed with a continuous wave laser of varying amplitude ${\varepsilon }_d$ (in V/Å). Red solid lines indicate net absorption, blue dashed ones indicate stimulated emission. The gray peaks signal the frequency and amplitude of each absorption transition between Floquet modes. Peaks are broadened by a Lorentzian function of width $\sigma =0.04$ eV.

Figure 4

Inter-FBZ transition dipoles ${\mu }_{{\lambda }^{\prime },\lambda }^{\left(n\right)}$ between Floquet modes with indexes $\lambda$ and ${\lambda }^{\prime }$ separated by $n$ Brillouin zones. The figure contrasts results for weak ${\varepsilon }_d=0.01$ V/Å (a), (b) and stronger ${\varepsilon }_d=0.06$ V/Å (c), (d) driving field amplitudes. Note that all the nonvanishing elements of the transition dipoles for $n=8$ have replicas in $n=9$. They correspond to the transitions between the same Floquet modes but separated by different number of Floquet Brillouin zones.

Figure 5

Interpretation of the absorption spectrum for ${\varepsilon }_d=0.26$ V/Å. The absorption spectrum (a) of the laser-driven material exhibits a strong absorption feature at low frequency (labeled X), and several absorption features in the IR-Visible-UV region that are periodically separated by $\hslash \mathrm{\Omega }$ which make the material a broadband absorber. Two of them are labeled Y and Z. (b) Quasienergies of the Floquet modes in the first FBZ and population ${\left|\langle k|{\varphi }_{\lambda }^0\rangle \right|}^2$ (gray scale) of the Floquet modes in the site basis at times $t_0+nT$. The arrows signal optical transitions responsible for the X, Y, Z spectral features in (a), and the $+n$ the number of FBZs separating the two Floquet modes involved in the transition, e.g., X +0 means this is an intra-FBZ transition. (c) Effective dipole between Floquet modes within the first FBZ $|{\mu }_{\lambda {\lambda }^{\prime }}^{\left(0\right)}{|}^2$. (d) Effective population factor $P_{\lambda {\lambda }^{\prime }}$ (${\lambda }^{\prime }\to \lambda$) between Floquet modes. The numeric labels to the Floquet modes in (c) and (d) are those assigned in (b).

Figure 6

(a) Quasienergy, (b) intra-FBZ transition dipole, and (c) population factors around the avoided crossing occurring at ${\varepsilon }_{d,\text{crossing}}=0.2522$ V/Å in the first FBZ between Floquet modes 6 and 7. The hybridization of the two Floquet modes around the crossing leads to large intra-FBZ effective transition dipole $|{\mu }_{67}^{\left(0\right)}{|}^2$, population mixing between the two states, and absorption and stimulated emission features at meV frequencies.

Figure 7

Net absorption (red solid lines) and stimulated emission (blue dashed lines) spectra at low frequencies for a driving pulse (a) just below the avoided crossing shown in Fig. 6, (b) right at the crossing ${\varepsilon }_d={\varepsilon }_{d,\text{crossing}}$, and (c) just above it. The low-frequency absorption spectra transition from stimulated emission to absorption as the driving electric field amplitude is increased across the avoided crossing.

Figure 8

Schematic of the energy diagram of a semiconductor dressed by nonresonant light of frequency $\mathrm{\Omega }$. The driving creates Floquet replicas of the valence (VB) and conduction (CB) band levels of the pristine material separated by integer $n\hslash \mathrm{\Omega }$, leading to features in the absorption spectrum including below band-gap absorption, low-frequency transitions, and broadband absorption.