Gauge invariance and interpretation of interband and intraband processes in high-order harmonic generation from bulk solids

Abstract

A theoretical model for high-order harmonic generation (HHG) in bulk solids is considered. Our approach treats laser-induced inter-and intraband currents on an equal footing. The sum of these currents is the source of the high-order harmonic radiation, and does not depend on the particular electromagnetic gauge we choose to describe the process. On the other hand, as it is shown using analytic and numerical calculations, the distinction between intra-and interband dynamics is gauge dependent, implying that the interpretation of the process of HHG using these terms requires carefulness.Introduction The idea that solid state targets illuminated by strong, short near infrared pulses can produce coherent X-ray radiation [1] or even attosecond electromagnetic bursts [2] preceded the first experiments that demonstrated the appearance of high-order harmonics (up to order of 25) in the spectra of strongly driven solid state samples. In Ref.[3] a wide bandgap ZnO target (3.2 eV) was excited by a pulse with central wavelength of 3.25 µm, meaning that at least 9 photons are required for an excitation from the valance to the conduction band [see also [4] for more details]. Recently, a direct comparison of high-order harmonic generation in the solid and gas phases of argon and krypton has been reported [5].Theoretically, semiconductor Bloch-equations were applied to describe the problem [6], a closed-form expression were given to the subcycle-resolved transition rate of electrons between bands [7], appearance of attosecond pulses were predicted [8], semiclassical [9] and a saddlepoint [10] analysis were performed and the role of an indirect bandgap was also investigated [11].

Interference-Induced Phenomena in High-Order Harmonic Generation from Bulk Solids

“…In the current work, we focus on the case of bulk solids. As we shall see, a one-dimensional model [40] for the crystalline solid with a bandgap having the order of a few eV can reproduce the qualitative features of the HHG spectra (plateau region and cutoff, including its intensity dependence). Working in single electron picture, we determine the Bloch states and the corresponding energies.…”

“…Note that J is often considered as a sum of "polarization-like" interband and "current-like" intraband components. Since this distinction -at least in the single electron picture we use here -is shown to be gauge dependent [40], we shall not use it in the following. Instead, we consider the gauge independent, entire J as the source of the HH radiation.…”

Density-based one-dimensional model potentials for strong-field simulations in He, H2+ , and H2

“…The parameter d is the internuclear distance for H + 2 and H 2 , while for d = 0 we get the helium atom (with its nucleus in the origin). The zE z (t) term of (1) corresponds to the interaction with the laser field, polarized along the z-axis, using dipole approximation and length gauge [48][49][50]. The E z (t) denotes the electric field of the laser pulse evaluated in the origin, and we assume that it is present only after t > 0, i.e.…”

Finite-system effects on high-order harmonic generation: From atoms to solids

“…[25][26][27]), solving the time-dependent Schrödinger equation for a periodic potential in an effective one-electron approximation [28][29][30][31] or using time-dependent density functional theory (TDDFT) as a many-electron correlated approach in both reduced [32] and full dimension [33,34], and recently including propagation effects [35]. From these studies it emerged that it is helpful to characterize solid-state harmonics as associated with either intra-or interband processes, even though it has been pointed out that there is no clear definition of intraand interband HHG because of gauge dependence [36]. Intraband HHG is produced when electrons are driven in the bandstructure in bands that are not purely sinusodial.…”