High-order harmonic generation from solids using Houston States

Abstract

The Houston states (Krieger and Iafrate in Phys Rev B 33: 5494 1986), which are instantaneous eigenstates of the time-dependent Hamiltonian, are known as an appropriate basis function to observe the role of each band in high-order harmonic generation (HHG) in solids. However, due to the sharp blow-up of the transition matrix elements between the Houston states (Wu et al. in Phys Rev A 91: 043839 2015), the Houston state treatment is known to be highly numerically unstable if more than two conduction bands are included. We develop a smooth variable discretization (SVD) method with Houston states by exploiting the fact that the wave function varies smoothly in time to solve the time-dependent Schrödinger equation (TDSE) for a solid exposed to a strong laser field. We demonstrate that the SVD method with the Houston states is numerically stable and can avoid the blow-up. Using the SVD method with the Houston states, we investigate the role of each band for the multiple plateau structure in HHG, and we find that strongly coupled bands should be treated as a whole and that the traditional classification of the currents into interband and intraband currents is not a suitable way to understand the plateaus in the HHG spectrum.

Appendix

In this appendix, we will show that ${\mathbf{T}}^{-1} \varvec{\varepsilon } {\mathbf{T}}$, appearing in Eq. (21), changes smoothly in time t. For simplicity, let us introduce the following notations: $|\bar{u}_m \rangle = | u_{m k(\bar{t})} \rangle$ and $|{u}_n \rangle = | u_{n k(t)} \rangle$. Using the identity $\sum _m | \bar{u}_m \rangle \langle \bar{u}_{m} | = I$ and the relation $\sum _{m} \langle u_m | \bar{u}_m \rangle \langle \bar{u}_{m} | u_{n'} \rangle = \delta _{nn'}$, we find that $\mathbf{T}^{-1} = \bar{\mathbf{T}}$ where $\bar{T}_{mn} = \langle \bar{u}_m| u_n\rangle$. Then,

$$\begin{aligned} \left( {\mathbf{T}}^{-1} \varvec{\varepsilon }(t) {\mathbf{T}}\right) _{mm'}= & {} \left( {{\bar{{\mathbf{T}}}}} \varvec{\varepsilon }(t) {\mathbf{T}}\right) _{mm'} \\= & {} \sum _{n,n'} \langle \bar{u}_m | u_n \rangle \langle u_n | H(t) | u_{n'} \rangle \langle u_{n'} | \bar{u}_{m'} \rangle \\= & {} \langle \bar{u}_m | H(t) | \bar{u}_{m'} \rangle , \end{aligned}$$

where we have used $\sum _n | u_n \rangle \langle u_{n} | = I$ and $\langle u_n|H(t)| u_{n'}\rangle = \varepsilon _n(t) \delta _{nn'}$. Expressing H(t) in terms of $H(\bar{t})$ as $H(t) = H(\bar{t}) + (H(t) - H(\bar{t}))$ and using $H(t) - H(\bar{t}) = \left[ k(t) - k(\bar{t}) \right] p + \frac{1}{2} \left[ k^2(t) - k^2(\bar{t}) \right] ,$ we can obtain

$$\begin{aligned} \left( {\mathbf{T}}^{-1} \varvec{\varepsilon }(t) {\mathbf{T}}\right) _{mm'}= & {} \varepsilon _m(\bar{t}) \delta _{mm'} + + \left[ k(t) - k(\bar{t}) \right] \langle \bar{u}_m | p | \bar{u}_{m'} \rangle \\&+ \frac{1}{2} \big [ k^2(t) - k^2(\bar{t}) \big ] \delta _{mm'}. \end{aligned}$$

Because $\langle \bar{u}_m | p | \bar{u}_{m'} \rangle$ is evaluated at time $\bar{t}$, it is a constant matrix. Therefore, the time dependence of $\left( {\mathbf{T}}^{-1} \varvec{\varepsilon } {\mathbf{T}}\right) _{mm'}$ is only through $\varepsilon _m(\bar{t})$ and k(t). However, $\varepsilon _m(t)$ and k(t) change smoothly in time t, so we can say that ${\mathbf{T}}^{-1} \varvec{\varepsilon }(t) {\mathbf{T}}$ also changes smoothly in time t.