High-harmonic generation in a quantum electron gas trapped in a nonparabolic and anisotropic well

An effective self-consistent model is derived and used to study the dynamics of an electron gas confined in a nonparabolic and anisotropic quantum well. This approach is based on the equations of quantum hydrodynamics, which incorporate quantum and nonlinear effects in an approximate fashion. The effective model consists of a set of six coupled differential equations (dynamical system) for the electric dipole and the size of the electron gas. Using this model we show that: (i) high harmonic generation is related to the appearance of chaos in the phase space, as attested to by related Poincaré sections; (ii) higher order harmonics can be excited efficiently and with relatively weak driving fields by making use of chirped electromagnetic waves.

Figure 1

Volume $V={\sigma }_x{\sigma }_y{\sigma }_z={\sigma }^{\left(0\right)3}$ of the electron gas as a function of the number of electrons $N$ for an isotropic confinement and different values of the anharmonicity parameter $\zeta$. The solid lines are solutions of Eq. (18), whereas the dashed lines are analytical solutions obtained in the $N\to \infty$ limit.

Figure 2

Simulations of the two breathing modes in the linear regime for an isotropic case $k_x=k_y=k_z=1$ with $N=50$ and $\zeta =0.1$. Top panel (a): degenerate mode with constant volume $V\left(t\right)$ (red dashed curve) and varying aspect ratio ${\sigma }_x/{\sigma }_z$ (blue continuous curve); bottom panel (b): nondegenerate mode with constant aspect ratio and oscillating volume. The insets show the Fourier transforms of the oscillating quantities, which peak at the expected degenerate (${\mathrm{\Omega }}_{\sigma }^{\left(d\right)}$) and nondegenerate (${\mathrm{\Omega }}_{\sigma }^{\left(nd\right)}$) frequencies.

Figure 3

Dipole frequencies ${\mathrm{\Omega }}_{d_z}$ (solid lines) and ${\mathrm{\Omega }}_{d_{\perp }}$ (dashed lines) as a function of the number of electrons $N$ for two values of the anharmonicity parameter, $\zeta =0.1$ (blue) and $\zeta =1$ (red). The left panel (a) corresponds to a cigar-shaped trap with $k_{\perp }>k_z$; the middle panel (b) to an isotropic trap ($k_{\perp }=k_z$); and the right panel (c) to a pancake shaped trap ($k_{\perp }<k_z$). In the isotropic case (b) the longitudinal ($z$) and transverse ($\perp$) dipoles coincide and the dotted lines represent the analytical expressions of Eq. (21).

Figure 4

Poincaré sections in the plane $(d_x,d_y)$ for different values of the initial excitation: $\delta =0.01$ (a), 0.1 (b), 1 (c), and 3 (d). The simulations were performed for an anisotropic trap with $k_{\perp }=5$, $k_z=1$, $N=50$, and $\zeta =0.01$.

Figure 5

Power spectrum of the dipole $P\left(\omega \right)$ normalized to the square of the initial excitation $\delta$, for the same cases as in Fig. 4.

Figure 6

Poincaré sections in the plane $(d_x,d_y)$ for different values of the anharmonicity parameter $\zeta =0$ (a), 0.001 (b), 0.01 (c), and 0.1 (d). The simulations were performed for an anisotropic trap with $k_{\perp }=5$, $k_z=1$, $N=50$, and initial excitation $\delta =1$.

Figure 7

Power spectrum of the dipole $P\left(\omega \right)$ for different values of the anharmonicity parameter $\zeta$, for the same case as in Fig. 6.

Figure 8

Laser excitation of the dipole response, for a trap with parameters: $k_{\perp }=5$, $k_z=1$, $N=50$, $\zeta =0.01$. The amplitude of the excitation is $E_0=1.05\times {10}^4\mathrm{V}/\mathrm{m}$. We show the dipole amplitude $d_z\left(t\right)$ (a), the absorbed energy (b), and the power spectrum $P\left(\omega \right)$ (c), for three cases: nonresonant excitation at constant frequency (green curves), resonant excitation at constant frequency (black), and chirped excitation (red).

Figure 9

Poincaré sections in the $(d_z,\dot{d_z})$ plane for the three cases of Fig. 8: nonresonant (a), resonant (b), and chirped excitation (c).