Extended high-harmonic spectra through a cascade resonance in confined quantum systems

The study of high-harmonic generation in confined quantum systems is vital to establishing a complete physical picture of harmonic generation from atoms and molecules to bulk solids. Based on a multilevel approach, we demonstrate how intraband resonances significantly influence the harmonic spectra via charge pumping to the higher subbands and thus redefine the cutoff laws. As a proof of principle, we consider the interaction of graphene nanoribbons, with zigzag as well as armchair terminations, and resonant fields polarized along the cross-ribbon direction. Here, this effect is particularly prominent due to many nearly equiseparated energy levels. In such a scenario, a cascade resonance effect can take place in high-harmonic generation when the field strength is above a critical threshold, which is completely different from the harmonic generation mechanism of atoms, molecules, and bulk solids. We further discuss the implications not only for other systems in a nanoribbon geometry, but also systems where only a few subbands (energy levels) meet this frequency-matching condition by considering a generalized multilevel Hamiltonian. Our study highlights that cascade resonance has a fundamentally distinct influence on the laws of harmonic generation, specifically the cutoff laws based on laser duration, field strength, and wavelength, thus unraveling additional insights in solid-state high-harmonic generation.

Figure 1

(a),(b) Structure of graphene nanoribbons with (a) armchair edges and (b) zigzag edges. Blue and orange cycles represent the two nearest-neighbor carbon atoms. The boundary condition is periodic in the $x$ direction but is open in the $y$ direction. (c),(d) Band structure of (c) 10-AGNR and (d) 10-ZGNR. Red (blue) curves stand for the subbands belonging to the conduction (valence) band. ${\mathrm{\Delta }}_{\mathrm{sg}}$ denotes the subband gap. $K_a=0$ and $K_z=2\pi /3d_z$, $4\pi /3d_z$ are Dirac points, respectively, for the armchair and zigzag GNRs if the boundary condition along the transverse ($y$) direction is periodic. $w_a$ ($w_z$) is the width of the armchair (zigzag) nanoribbon. $d_a$ ($d_z$) is the distance of the armchair (zigzag) unit cell.

Figure 2

(a),(b) Ratios of the driver frequency ${\omega }_0$ and subband gaps ${\mathrm{\Delta }}_{\mathrm{sg}}$ at Dirac points as a function of system size $N$ for (a) AGNRs and (b) ZGNRs. Lines with symbol $\circ$ ($\blacksquare$) represent the variation of the maximum (average) value of subband gaps. (c),(d) Harmonic spectra vs number of sites $N$ in (c) AGNRs and (d) ZGNRs, respectively, using a driver with $n_{\text{cyc}}=16$, ${\omega }_0=0.0152\text{a.u.}(\lambda =3\mu \text{m})$, and $E_0=0.0012\text{a.u.}(I_0=5.04\times {10}^{10}{\text{W/cm}}^2)$. The color bar characterizes the intensity of harmonics. The resonance peaks ($p_1$, $p_2$, and $p_3$) in (c) and (d) are marked in (a) and (b), respectively, in the form of red stars.

Figure 3

(a) Harmonic spectrum for 36-ZGNR interacting with a resonant field. The intensity of the harmonic spectrum is in arbitrary units (arb. units). The field strength is $E_0=0.0012$ a.u. ($I_0=5.04\times {10}^{10}\mathrm{W}/{\mathrm{cm}}^2$), and the field wavelength is $\lambda =3\mu \mathrm{m}$ ($\omega =0.0152$ a.u.). (b) Time-frequency distribution of the harmonic emission. The red curve is the vector potential of the laser field. The cyan line indicates the cutoff frequency. (c1)–(c3) Electron dynamics in the conduction subbands for times $t_i$ marked in (b). $T=2\pi /\omega$ is the optical cycle. The color bar indicates the electron population.

Figure 4

Population dynamics of conduction states in an equispaced model with $N=20$ in (a1) the weak-coupling regime (${\gamma }_R=0.2$) and (a2) the strong-coupling regime (${\gamma }_R=2$). Corresponding plots for the 20-level ZGNR-like model are shown in (b1) and (b2). The color bar identifies the conduction levels, and the black dashed curve shows the total population of $i=11,...,20$ scaled by $1/5$ in all cases except (b1). Parts (a3) and (b3) show the HHG spectrum for the equispaced and the ZGNR-like systems, respectively. The red and blue curves represent harmonic spectra for ${\gamma }_R=0.2$ and 2, respectively.

Figure 5

(a) Harmonic spectrum for 80-ZGNR. The field strength is $E_0=0.0012$ a.u. ($I_0=5.04\times {10}^{10}$ W/${\mathrm{cm}}^2$), and the field wavelength is $\lambda =3\mu \mathrm{m}$ ($\omega =0.0152$ a.u.). (b) Time frequency distribution of the harmonic emission. The red curve is the vector potential of the external field. The cyan line indicates the cutoff frequency. (c1)–(c3) Electron dynamics in the conduction subbands for the times $t_i$ marked in (b).

Figure 6

(a) Harmonic spectra vs number of cycles ($n_{\text{cyc}}$) in 80-AGNR. The field strength is $E_0=5\times {10}^{-4}$ a.u. ($I_0=8.75\times {10}^9$ W/${\mathrm{cm}}^2$), ${\gamma }_R\approx 1.25$, and the frequency is ${\omega }_0=0.0076\text{a.u.}(\lambda =6\mu \text{m})$. (b) The cutoff as a function of $n_{\mathrm{cyc}}$ corresponding to the harmonic spectra in (a). (c) Harmonic spectra vs field strength in 40-AGNR, using a driver with $n_{\text{cyc}}=16$, and ${\omega }_0=0.0152\text{a.u.}(\lambda =3\mu \text{m})$. The color bar indicates the intensity of harmonics. (d) The cutoff as a function of field strength, corresponding to the harmonic spectra in (c). (e) Harmonic spectra extracted from (c) for $E_{\text{sec}}=4\times {10}^{-4}$ a.u. and $E_{\text{cri}}=8\times {10}^{-4}$ a.u. (f) Harmonic spectra vs wavelength in 40-AGNR, using a driver with $n_{\text{cyc}}=16$, and $E_0=0.0012\text{a.u.}(I_0=5.04\times {10}^{10}{\text{W/cm}}^2)$. The white line represents the onset of cascade resonance.

Figure 7

Population dynamics and HHG spectrum depending on energy-level structure. (a1),(a2) Population dynamics for $E_{\mathrm{g}}<{\mathrm{\Delta }}_{\mathrm{sg}}$. (a1) Laser frequency matches fundamental gap ($\omega =E_{\mathrm{g}}=1$ a.u.). (a2) Laser frequency matches sub(band) gap ($\omega ={\mathrm{\Delta }}_{\mathrm{sg}}=4$ a.u.). (a3) Harmonic spectra in red and blue correspond to the electron dynamics in (a1) and (a2), respectively. (b1),(b2) Population dynamics for $E_{\mathrm{g}}>{\mathrm{\Delta }}_{\mathrm{sg}}$. (b1) Laser frequency matches fundamental gap ($\omega =E_{\mathrm{g}}=4$ a.u.). (b2) Laser frequency matches sub(band) gap ($\omega ={\mathrm{\Delta }}_{\mathrm{sg}}=1$ a.u.). (b3) Harmonic spectra in red and blue correspond to the electron dynamics in (b1) and (b2), respectively. (c1),(c2) Population dynamics for a system with random sub(band) gap in (c1) the weak-coupling regime ($E_0=0.2$ a.u.) and (c2) the strong-coupling regime ($E_0=2$ a.u.). The laser frequency matches the average sub(band) gap. (c3) Harmonic spectra in red and blue correspond to the electron dynamics in (c1) and (c2), respectively.

Figure 8

Dipole matrix elements $\langle \left|d_{\mathrm{\Delta }J=1}^y\left(K_l\right)\right|\rangle$ with respect to the ribbon widths $N_l$ for (a) AGNRs $(l=a)$ and (b) ZGNRs $(l=z)$. The open circles are data points, and orange lines are the linear fit of the data points.

Figure 9

(a) Linear rise time as a function of ${\mathrm{\Omega }}_R^{-1}$ for $N=20$. The dipole matrix elements are the same as in Eq. (3). Laser frequency is set to $\omega ={\mathrm{\Delta }}_{\mathrm{sg}}=1$ a.u. (b) Linear rise time as a function of $N$. We fix field strength $E_0$ at 0.4 a.u., and thus ${\mathrm{\Omega }}_R=E_0{d}_{i,i+1}=0.4$ a.u. Laser frequency is set to $\omega ={\mathrm{\Delta }}_{\mathrm{sg}}=1$ a.u. Green open circles represent data points, and blue lines are linear fitting results for data points.

Figure 10

(a),(b) Harmonic spectra vs field strength in 40-AGNR and 40-ZGNR, respectively, when the field is polarized along the ribbon direction. The laser frequency is fixed at ${\omega }_0=0.0152\text{a.u.}(\lambda =3\mu \text{m})$. (c),(d) Harmonic spectra vs wavelength in 40-AGNR and 40-ZGNR, respectively. The field strength is $E_0=0.0012$ a.u. ($I_0=5.04\times {10}^{10}$ W/${\mathrm{cm}}^2$).

Figure 11

(a) Harmonic spectrum for a 36-ZGNR with $E_{\mathrm{g}}=1.0$ eV. The field strength is $E_0=0.0012$ a.u. ($I_0=5.04\times {10}^{10}$ W/${\mathrm{cm}}^2$), and the field frequency matches a subband gap $\omega =0.0152$ a.u. ($\lambda =3\mu \mathrm{m}$). (b) Time-frequency distribution of the harmonic emission. The red curve is the vector potential of the laser field. The cyan line indicates the cutoff frequency. (c1)–(c3) Electron dynamics in the conduction subbands for the times $t_i$ marked in (b).