High-harmonic generation in quantum spin systems

We theoretically study the high-harmonic generation (HHG) in one-dimensional spin systems. While in electronic systems the driving by ac electric fields produces radiation from the dynamics of excited charges, we consider here the situation where spin systems excited by a magnetic field pulse generate radiation via a time-dependent magnetization. Specifically, we study the magnetic dipole radiation in two types of ferromagnetic spin chain models, the Ising model with static longitudinal field and the XXZ model, and reveal the structure of the spin HHG and its relation to spin excitations. For weak laser amplitude, a peak structure appears which can be explained by time-dependent perturbation theory. With increasing amplitude, plateaus with well-defined cutoff energies emerge. In the Ising model with longitudinal field, the thresholds of the multiple plateaus in the radiation spectra can be explained by the annihilation of multiple magnons. In the XXZ model, which retains the ${\mathbf{Z}}_2$ symmetry, the laser magnetic field can induce a phase transition of the ground state when it exceeds a critical value, which results in a drastic change of the spin excitation character. As a consequence, the first cutoff energy in the HHG spectrum changes from a single-magnon to a two-magnon energy at this transition. Our results demonstrate the possibility of generating high-harmonic radiation from magnetically ordered materials and the usefulness of high-harmonic signals for extracting information on the spin excitation spectrum.

Figure 1

(a) Schematic picture of the HHG from quantum magnets discussed in this paper. The spins are excited by a magnetic field pulse, and the induced magnetization dynamics results in electromagnetic radiation with high frequency components. (b) Example of the correspondence between the spin HHG intensity and the spin excitation spectrum (XXZ model with $J_{xy}=2$ and $J_z=10$, pulse with $\mathrm{\Omega }=1$ and $B=4$). The thresholds (vertical dashed lines) of the multiple plateaus in the HHG signal correspond to multiples of the magnon excitation energy at $q=0$, see horizontal arrow. (Inset: The shape of the magnetic field for the linearly polarized pulse laser with $B=4$ and $N_{\mathrm{cyc}}=9$.) (c) Comparison between the HHG in electronic systems and that in spin systems.

Figure 2

DSFs (a)–(d) $|\mathrm{Im}{\chi }^{xx}(q,\omega )|$ and (e)–(h) $|\mathrm{Im}{\chi }^{zz}(q,\omega )|$ for the Ising model with $J=2$ and $H=6$.

Figure 3

Time evolution of (a) $m^x$ and (b) $m^z$ calculated by iTEBD for the Ising model with $J=2$ and $H=6$.

Figure 4

The radiation power from (a)–(d) $m^x$ and (e)–(h) $m^z$ in the Ising model with $J=2$ and $H=6$. The dashed lines correspond to the mass of a magnon at $q=0$ (times integer).

Figure 5

Scaling of the magnitude of the Fourier components for the magnetizations (a) $|m^x\left(\omega \right)|$ and (b) $|m^z\left(\omega \right)|$ in the region of small $B$, in the Ising model with $J=2$ and $H=6$. The power of $B$ agrees well with the prediction from the time-dependent perturbation theory.

Figure 6

Color map of the subcycle radiation spectrum (a) ${log}_{10}{\left|{\omega }^2{m}_{\mathrm{sub}}^x(\omega ;t_{*})\right|}^2$ and (b) ${log}_{10}{\left|{\omega }^2{m}_{\mathrm{sub}}^z(\omega ;t_{*})\right|}^2$ for the Ising model with $J=2$, $H=6$, and $B=8$. The solid lines indicate the energies of one, two, and three magnons at the corresponding $B\left(t_{*}\right)$.

Figure 7

The Floquet DSF (a) $|\mathrm{Im}{\chi }_{\mathrm{F}}^{xx}(q,\omega )|$ and (b) $|\mathrm{Im}{\chi }_{\mathrm{F}}^{zz}(q,\omega )|$ for the Ising model with $J=2$, $H=6$, and $B=8$.

Figure 8

DSFs (a)–(e) $|\mathrm{Im}{\chi }^{xx}(q,\omega )|$ and (f)–(j) $|\mathrm{Im}{\chi }^{zz}(q,\omega )|$ for the XXZ model with $J_{xy}=2$ and $J_z=10$.

Figure 9

Time evolution of (a) $m^x$ and (b) $m^z$ for the XXZ model with $J_{xy}=2$ and $J_z=10$.

Figure 10

HHG from (a)–(e) $m^x$ and (f)–(j) $m^z$ for the XXZ model with $J_{xy}=2$ and $J_z=10$. Dashed lines correspond to ${\mathrm{\Delta }}_{q=0}$ (times integer) and dashed-dotted lines correspond to $2{\tilde{\mathrm{\Delta }}}_{q=\pi }$.

Figure 11

Color map of the subcycle radiation spectrum ${log}_{10}{\left|{\omega }^2{m}_{\mathrm{sub}}^x(\omega ;t_{*})\right|}^2$ for the XXZ model with $J_{xy}=2$ and $J_z=10$ under the laser field (a) $B=4$ and (b) $B=10$. The solid lines show the single-magnon and two-magnon modes of the snapshot Hamiltonian at time $t_{*}$ in (a) and the two-magnon mode in (b).

Figure 12

(a) $\langle {\mathrm{\Phi }}_n\left|S_{\mathrm{tot}}^x\right|{\mathrm{\Phi }}_n\rangle$ and (b) $A_n$ [Eq. (17)] calculated by ED for the model (16) with $J_{xy}=2$, $J_z=10$, and $B=10$. The arrows show the energy $E_n-E_0=24.5$. Cross marks are used for the states in the $\langle {\mathrm{\Phi }}_n\left|S_{\mathrm{tot}}^x\right|{\mathrm{\Phi }}_n\rangle \simeq -2$ sector to demonstrate that the contribution to the HHG signal comes mainly from this sector.

Figure 13

The magnon band structure from the spin wave theory for the XXZ model with (a) $J_{xy}=2$, $J_z=10$ and (b) $J_{xy}=8$, $J_z=10$.

Figure 14

DSFs (a)–(d) $|\mathrm{Im}{\chi }^{xx}(q,\omega )|$ and (e)–(h) $|\mathrm{Im}{\chi }^{zz}(q,\omega )|$ for the Ising model with $J=4$ and $H=2$.

Figure 15

Time evolution of (a) $m^x$ and (b) $m^z$ for the Ising model with $J=4$ and $H=2$ calculated by iTEBD.

Figure 16

Radiation power from (a)–(d) $m^x$ and (e)–(h) $m^z$ for the Ising model with $J=4$ and $H=2$. The dashed and dotted lines represent the mass of the single magnon at $q=0$ and that of the two-magnon bound state, respectively. The dashed-dotted line corresponds to the energy of two magnons at $q=\pi$. The crossover of the threshold energy from the former to the latter occurs with increasing $B$.

Figure 17

Color map of the subcycle radiation spectrum ${log}_{10}{\left|{\omega }^2{m}_{\mathrm{sub}}^x(\omega ;t_{*})\right|}^2$ for the Ising model with $J=4$ and $H=2$ under the (a) weak transverse field $B=2$ and (b) strong transverse field $B=6$. The solid lines show the single-magnon, two-magnon-bound state, and two-magnon (with $q=\pi$) state from bottom to top, respectively, of the snapshot Hamiltonian at time $t_{*}$.

Figure 18

DSFs (a)–(e) $|\mathrm{Im}{\chi }^{xx}(q,\omega )|$ and (f)–(j) $|\mathrm{Im}{\chi }^{zz}(q,\omega )|$ for the XXZ model with $J_{xy}=8$ and $J_z=10$.

Figure 19

Time evolution of (a) $m^x$ and (b) $m^z$ for the XXZ model with $J_{xy}=8$ and $J_z=10$.

Figure 20

HHG from (a)–(e) $m^x$ and (f)–(j) $m^z$ for the XXZ model with $J_{xy}=8$ and $J_z=10$. Dashed lines correspond to $m_{q=0}$ (times integer).

Figure 21

Color map of the subcycle radiation spectrum ${log}_{10}{\left|{\omega }^2{m}_{\mathrm{sub}}^x(\omega ;t_{*})\right|}^2$ for the XXZ model with $J_{xy}=8$ and $J_z=10$ under the laser field (a) $B=4$ and (b) $B=10$. The solid lines show the single-magnon and two-magnon modes of the snapshot Hamiltonian at time $t_{*}$.

Figure 22

(a) $\langle {\mathrm{\Phi }}_n\left|S_{\mathrm{tot}}^x\right|{\mathrm{\Phi }}_n\rangle$ and (b) $A_n$ [Eq. (17)] calculated by ED for the model Eq. (16) with $J_{xy}=8$, $J_z=10$, and $B=10$. The arrows show the energy $E_n-E_0=20.9$. Cross marks are used for the states in the $\langle {\mathrm{\Phi }}_n\left|S_{\mathrm{tot}}^x\right|{\mathrm{\Phi }}_n\rangle \simeq -1.5$ sector to demonstrate that the contribution to the HHG signal comes mainly from this sector.

Figure 23

Time evolution of $m^x$ for (a) the Ising model with $J=2$ and $H=6$ and (b) the XXZ model with $J_{xy}=2$ and $J_z=10$ after the subtraction of the linear response component $(B/B_0)m_0\left(t\right)$.

Figure 24

Color map of the wavelet radiation spectrum ${log}_{10}{\left|{m^{″}}_{\mathrm{W}}^x(\omega ;t_{*})\right|}^2$ for the Ising model with $J=4$, $H=2$, and $B=8$ (a) and the XXZ model with $J_{xy}=2$, $J_z=10$, and $B=10$ (b). The solid lines show (a) the (multiple) single-magnon modes and (b) the two-magnon (with $q=\pi$) mode of the snapshot Hamiltonian at time $t_{*}$.