High-harmonic generation in one-dimensional Mott insulators

We study high-harmonic generation (HHG) in the one-dimensional Hubbard model in order to understand its relation to elementary excitations as well as the similarities and differences to semiconductors. The simulations are based on the infinite time-evolving block decimation (iTEBD) method and exact diagonalization. We clarify that the HHG originates from the doublon-holon recombination, and the scaling of the cutoff frequency is consistent with a linear dependence on the external field. We demonstrate that the subcycle features of the HHG can be reasonably described by a phenomenological three step model for a doublon-holon pair. We argue that the HHG in the one-dimensional Mott insulator is closely related to the dispersion of the doublon-holon pair with respect to its relative momentum, which is not necessarily captured by the single-particle spectrum due to the many-body nature of the elementary excitations. For the comparison to semiconductors, we introduce effective models obtained from the Schrieffer-Wolff transformation, i.e., a strong-coupling expansion, which allows us to disentangle the different processes involved in the Hubbard model: intraband dynamics of doublons and holons, interband dipole excitations, and spin exchanges. These demonstrate the formal similarity of the Mott system to the semiconductor models in the dipole gauge, and reveal that the spin dynamics, which does not directly affect the charge dynamics, can reduce the HHG intensity. We also show that the long-range component of the intraband dipole moment has a substantial effect on the HHG intensity, while the correlated hopping terms for the doublons and holons essentially determine the shape of the HHG spectrum. A new numerical method to evaluate single-particle spectra within the iTEBD method is also introduced.

Figure 1

Single-particle spectrum $A(p,\omega )$ in equilibrium calculated by iTEBD for $U=10$ and $h_z=0.001$. The red dashed lines indicate the peak positions of the spectrum at each $p$. The black dot-dashed line indicates ${\epsilon }_d\left(p_d\right)$ [Eq. (16)]. To obtain $A(p,\omega )$ from $G^R(p,t)$, we apply a Gaussian window $exp(-\frac{t^2}{2{\sigma }^{\prime 2}})$ with ${\sigma }^{\prime }=5.0$ in the Fourier transformation.

Figure 2

HHG spectra of the Mott insulator for (a) $\mathrm{\Omega }=0.75$ and (b) $0.5$ at field strengths $E_0=0.5$ and 0.7. Solid lines indicate the full HHG spectra, while the dashed lines indicate the contribution from the doublon-holon hopping $I_{\mathrm{hop}}$. Here, the Mott gap and the maximum band-energy difference extracted from the single-particle spectrum are $6.7$ and $15.0$, respectively. Inverted triangles indicate the cutoff frequency determined by the criterion [72]. The solid vertical lines correspond to even harmonics of $\mathrm{\Omega }$, while the dashed ones show odd harmonics. The model parameters are $U=10$ and $h_z=0.001$. The pulse parameters are $t_0=60$, and $\sigma =15$, while ${\sigma }^{\prime }=20$ is used for the Fourier transformation of the current.

Figure 3

HHG spectra in the plane of frequency $\omega$ and field strength $E_0$ for (a) $\mathrm{\Omega }=0.75$ and (b) $\mathrm{\Omega }=0.5$. The red circles indicate the cutoff frequencies and the filled red triangle shows the Mott gap. The solid vertical lines correspond to even harmonics of $\mathrm{\Omega }$, while the dashed ones show odd harmonics. The model parameters are $U=10$ and $h_z=0.001$. The pulse parameters are $t_0=60$, and $\sigma =15$, while ${\sigma }^{\prime }=20$ is used for the Fourier transformation of the current.

Figure 4

[(a) and (b)] Applied electric field and induced current for (a) $\mathrm{\Omega }=0.75$ and (b) $0.5$. [(c) and (d)] Subcycle analysis of the HHG, $I_{\mathrm{HHG}}(\omega ,t_p)$, for (c) $\mathrm{\Omega }=0.75$ and (d) $0.5$. The blue markers show the results of the semiclassical analysis (${\omega }_{\mathrm{emit}}\left(t_{\mathrm{rec}}\right)$) with ${\epsilon }_g\left(p\right)$ obtained from the Bethe ansatz results Eq. (16), while the red markers correspond to those for the unrenormalized dispersion, $U-4vsin\left(p\right)$. The pink markers are the results obtained using the dispersion extracted from the single-particle spectrum, i.e., Fig. 1, and they almost overlap with the blue markers. The vertical dashed lines indicate the times when $E\left(t\right)=0$. The parameters of the system and the pump field are the same as in Fig. 2.

Figure 5

(a) Single-particle spectrum $A(p,\omega )$ in equilibrium calculated by iTEBD for $U=7$ and $h_z=0.001$. The red dashed lines indicate the peak positions of the spectrum at each $p$. The black dot-dashed line indicates ${\epsilon }_d\left(p_d\right)$ [Eq. (16)]. To obtain $A(p,\omega )$ from $G^R(p,t)$, we apply a Gaussian window $exp(-\frac{t^2}{2{\sigma }^{\prime 2}})$ with ${\sigma }^{\prime }=5.0$ for the Fourier transformation. (b) Subcycle analysis of the HHG, $I_{\mathrm{HHG}}(\omega ,t_p)$ for $U=7$ and $h_z=0.001$. The parameters of the pump are $\mathrm{\Omega }=0.3,E_0=0.4,t_0=100$ and $\sigma =30$. The blue markers show the results of the semiclassical analysis $\left[{\omega }_{\mathrm{emit}}\left(t_{\mathrm{rec}}\right)\right]$ with ${\epsilon }_g\left(p\right)$ obtained from the Bethe ansatz results Eq. (16), while the red markers correspond to those for the unrenormalized dispersion, $U-4vsin\left(p\right)$. The pink markers are the results obtained using the dispersion extracted from the single-particle spectrum in (a), and they almost overlap with the blue markers. The vertical dashed lines indicate the times when $E\left(t\right)=0$.

Figure 6

(a) Single-particle spectrum $A(p,\omega )$ of the effective model ${\widehat{H}}_{\mathrm{Mott},1}$ in equilibrium evaluated with iTEBD for $U=10$. Here, the spin configuration of the equilibrium state is determined by ${\widehat{H}}_{\mathrm{spin}}={\widehat{H}}_{\mathrm{spin},\mathrm{ex}}+h_z{\sum }_i{(-)}^i{\widehat{s}}_{z,i}$ with $h_z=0.001$. The red lines indicate $\pm \frac{U}{2}\pm 2vsin\left(p\right)$. (b) Comparison of the HHG spectra of the Hubbard model and the effective model ${\widehat{H}}_{\mathrm{eff},1,1}$ for $\mathrm{\Omega }=0.5$. (c) Subcycle analysis of the HHG, $I_{\mathrm{HHG}}(\omega ,t_p)$, for ${\widehat{H}}_{\mathrm{eff},1,1}$. Here, the excitation parameters are $\mathrm{\Omega }=0.5$ and $E_0=0.7$. The red markers indicate the result of the semiclassical theory (${\omega }_{\mathrm{emit}}\left(t_{\mathrm{rec}}\right)$) with the unrenormalized dispersion, $U-4vsin\left(p\right)$. (d) Comparison between the HHG spectra from ${\widehat{H}}_{\mathrm{eff},1,1}, {\widehat{H}}_{\mathrm{eff},1,1}+{\widehat{H}}_{\mathrm{spin}}$ and ${\widehat{H}}_{\mathrm{eff},1,1}+{\widehat{H}}_{\mathrm{dh},\mathrm{ex}}$. In (b)–(d), we use $U=10, h_z=0.001, \mathrm{\Omega }=0.5, t_0=60$ and $\sigma =15$. Here, the Mott gap and the maximum band-energy difference extracted from the single-particle spectrum are 6.0 and 14.0, respectively.

Figure 7

HHG spectra for the effective models evaluated with ED for $U=10,h_z=0$ and $L=10$. The corresponding Hamiltonians are indicated in the labels. In particular, the labels for the solid lines in panel (b) indicate the correction added to $H_{\mathrm{eff},1,1}$. For example, “$H_{\mathrm{hop}}^{\left(2\right)}$” means that ${\widehat{H}}_{\mathrm{eff},1,1}+{\widehat{H}}_{\mathrm{hop}}^{\left(2\right)}$ is used in the simulation. The parameters for the pump pulse are $\mathrm{\Omega }=0.5, E_0=0.7, t_0=60$, and $\sigma =15$.

Figure 8

(a) Single-particle spectrum $A(p,\omega )$ of the effective model ${\widehat{H}}_{\mathrm{Mott},1}$ in equilibrium evaluated with iTEBD for $U=10$. Here, the spin configuration of the equilibrium state is antiferromagnetic. (b)(c) Comparison of the HHG spectra of the effective model, ${\widehat{H}}_{\mathrm{eff},1,1}$ with $U=10$, for the ground state of ${\widehat{H}}_{\mathrm{spin}}+h_z{\sum }_i{(-)}^i{\widehat{s}}_{z,i}$ and the antiferromagnetic state. Here, (b) $\mathrm{\Omega }=0.75$ and (c) $0.5$. The parameters of the pump are $t_0=60$ and $\sigma =15$.

Figure 9

[(a) and (b)] HHG spectra of the semiconductor with $E_g=6.0$ for (a) $\mathrm{\Omega }=0.75$ and (b) $0.5$ for the specified field strengths. Here, we use $t_0=60$ and $\sigma =15$ for the pulse and ${\sigma }^{\prime }=20$ for the Fourier transformation. (c) HHG spectra of the semiconductor with $E_g=6.0$ for $\mathrm{\Omega }=0.5,E_0=0.7,t_0=160,\sigma =40$ with and without the phenomenological dephasing term ($T_2$). Here, ${\sigma }^{\prime }=60$ is used for the Fourier transformation. [(d) and (e)] Subcycle analysis of the HHG signal. $I_{\mathrm{HHG}}(\omega ,t_p)$ is plotted for (d) $\mathrm{\Omega }=0.75$ and (e) $\mathrm{\Omega }=0.5$. Here, we use $E_g=6.0, E_0=0.7, t_0=60,$ and $\sigma =15$ for both cases. (f) $I_{\mathrm{HHG}}(\omega ,t_p)$ for $E_g=3.0$. The parameters of the pump are $\mathrm{\Omega }=0.3,E_0=0.4,t_0=100,$ and $\sigma =30$. For the subcycle analysis ${\sigma }^{\prime }=0.8$ is used. The red markers indicate the results of the subcycle analysis (${\omega }_{\mathrm{emit}}\left(t_{\mathrm{rec}}\right)$) with the dispersion ${\epsilon }_a\left(p\right)=\pm \frac{D}{2}\pm 2v_{a}cos\left(p\right)$. The results in (a), (b), (d), (e), and (f) are obtained without phenomenological dephasing term ($T_2=\infty$).