Theory of coherent optical nonlinearities of intersubband transitions in semiconductor quantum wells

We theoretically study the coherent nonlinear response of electrons confined in semiconductor quantum wells under the effect of an electromagnetic radiation close to resonance with an intersubband transition. Our approach is based on the time-dependent Schrödinger-Poisson equation stemming from a Hartree description of Coulomb-interacting electrons. This equation is solved by standard numerical tools and the results are interpreted in terms of approximated analytical formulas. For growing intensity, we observe a redshift of the effective resonance frequency due to the reduction of the electric dipole moment and the corresponding suppression of the depolarization shift. The competition between coherent nonlinearities and incoherent saturation effects is discussed. The strength of the resulting optical nonlinearity is estimated across different frequency ranges from mid-IR to THz with an eye to ongoing experiments on Bose-Einstein condensation of intersubband polaritons and to the speculative exploration of quantum optical phenomena such as single-photon emission in the mid-IR and THz windows.

Figure 1

(a) Example of imaginary-time SP evolution of the energy starting from a simply guessed wave function (a Gaussian centered on the well) and converging toward the ground state. The insets show the spatial profile of the total potential including the QW and Coulomb interaction potentials (black lines) and the squared modulus of the electronic wave function (blue lines) at different imaginary times. The top inset refers to an early imaginary time indicated by the arrow, well before the convergence of the algorithm. The bottom inset refers instead to a late imaginary time after convergence. Within each panel, the solid/dashed lines refer to different values of the electron density, respectively, negligible (solid) and substantial (dashed). (b) Results of the calculation for two different locations of the impurities, as indicated by the red arrows. The black solid line shows the total potential experienced by the electrons. The blue lines show the squared modulus of several single-particle eigenstates for an electron density of $3\times {10}^{12}{\mathrm{cm}}^{-2}$. Each wave function is shifted upward by an amount corresponding to its eigenenergy. In all calculations within this paper, we take values $\epsilon =12.9$ for the background dielectric constant and $m^{*}=0.067m_{\text{el}}$ for the effective mass of electrons.

Figure 2

(a) Example of the time dependence of the electric field amplitude of the incident pulse (red) and of the induced electronic polarization (green). The insets show the total potential $V_{\mathrm{tot}}\left(z\right)$ (black) and the squared modulus of the electronic wave function (blue and orange) at two different times, indicated by the stars in the main graph. (b) Numerically evaluated ISB transition frequency in the linear regime of weak excitations as a function of the electron density (solid and dashed lines) compared with the prediction $E_{21}$ of the static Hartree approximation (squares) discussed in the text and in Appendix pp2. The bare transition frequency $E_{21}^o$ is indicated by the horizontal black dotted line. The dopant impurities are located either outside the well (modulation doping, dashed line, and open symbols) or inside the well (solid line and filled symbols).

Figure 3

Oscillation frequency $\hslash {\omega }_{\mathrm{res}}$ as a function of the deposited energy per electron $\mathrm{\Delta }E$. Different curves refer to growing values of the electron density, from a very small ${\sigma }_{\text{el}}\sim 0{\text{cm}}^{-2}$ value (red triangles) to higher ones ${\sigma }_{\text{el}}=1\times {10}^{12}{\text{cm}}^{-2}$ (green diamonds), $2\times {10}^{12}{\text{cm}}^{-2}$ (yellow pentagons), and $3\times {10}^{12}{\text{cm}}^{-2}$ (blue dots). The two (a), (b) panels refer to different location of the dopant impurities outside (a) and inside (b) the QW. Solid lines are guides to the eyes based on a linear regression of the numerical data points. Within each panel, the inset shows (black line) the total potential $V_{\mathrm{tot}}\left(z\right)$ in the ground state and (blue lines) the squared modulus of several single-particle eigenstates in the total potential for the highest value of the electronic density considered in the main panels. Each wave function is shifted upward by an amount corresponding to its eigenenergy. For the chosen QW parameters, the bare ISB transition is in the mid-IR range at an energy around 110 meV. The vertical dashed lines indicate the position of the saturation point $\mathrm{\Delta }E=E_{21}^o/2$.

Figure 4

Normalized squared amplitude of the oscillating electric dipole $\overline{z}$ as a function of the deposited energy $\mathrm{\Delta }E$ for the same configurations shown in Fig. 3. The vertical dashed lines indicate the position of the saturation point $\mathrm{\Delta }E=E_{21}^o/2$.