Nonlinear response of radiatively coupled intersubband transitions of quasi-two-dimensional electrons

Radiative coupling of resonantly excited intersubband transitions in $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ multiple quantum wells can have a strong impact on the coherent nonlinear optical response, as is shown by phase and amplitude resolved propagation studies of ultrashort electric field transients. Upon increasing the driving field amplitude, strong radiative coupling leads to a pronounced self-induced absorption, followed by a bleaching due to the onset of delayed Rabi oscillations. A many-particle theory including light propagation effects accounts fully for the experimental results.

Figure 1

(Color online) (a) Schematic of the $n$-type modulation-doped $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ multiple quantum well sample containing 51 GaAs quantum wells. (b) Light propagation within our prism-shaped multiple quantum well sample.

Figure 2

(Color online) Measured linear IS absorption spectra of two structurally identical multiple quantum well samples with different electron densities per quantum well $n_s$ for various lattice temperatures. (a) Sample $L$ with $n_s=5\times {10}^{10}{\mathrm{cm}}^{-2}$. (b) Sample $H$ with $n_s=1.2\times {10}^{12}{\mathrm{cm}}^{-2}$. At higher lattice temperatures one observes for $H$ additionally the $2\leftrightarrow 3$ IS transition of thermally excited electrons in the $n=2$ subband.

Figure 3

(Color online) (a) Linear IS absorption of a single QW corresponds to the destructive interference of the incident wave $E_{\mathrm{in}}$ with the wave $E_{\mathrm{em}}$ coherently emitted by the macroscopic IS polarization resulting in an attenuated transmitted wave $E_{\mathrm{tr}}$. The QW also emits a reflected wave $E_{\mathrm{re}}=E_{\mathrm{em}}$. (b) Schematic of the relevant interacting waves in our multiple QW samples showing additionally total reflection of all waves at the base of the prism. (c) Solid lines, transmission, reflection, and absorption of light resonant to the IS transition of a single QW as a function of the radiative coupling parameter $\xi$. (d) Solid line, calculated absorption for situation (b) together with experimental values (symbols) obtained from temperature-dependent linear IS absorption spectra (Fig. 2) (Ref. 18). The dashed lines in (c) and (d) are values according to Beer’s law.

Figure 4

(Color online) (a)-(c) Electric field transients measured after sample $H$ (electron density of $n=1.2\times {10}^{12}{\mathrm{cm}}^{-2}$ per QW) for incident field amplitudes of (a) $5\mathrm{kV}∕\mathrm{cm}$, (b) $45\mathrm{kV}∕\mathrm{cm}$, and (c) $100\mathrm{kV}∕\mathrm{cm}$. Dashed line, field envelopes of the input pulses. (d)–(f) Both the shape and the amplitude of the theoretically calculated transients are in good agreement with the corresponding experimental counterparts (a)–(c).

Figure 5

(Color online) (a) Absorption of a femtosecond infrared pulse resonant to the IS transition in sample $H$ (triangles) and in sample $L$ (dots) as a function of the incident pulse amplitude. (b) Delay of the center of gravity of the pulse transmitted through sample $H$. Solid lines, many-body calculations with excitation-induced dephasing.

Figure 6

(Color online) Calculated transient populations in the $n=2$ subbands of the different QWs in sample $H$ [(a)–(c)] and in sample $L$ [(d)–(f)] for different amplitudes $A$ of the incident field. Note the different max values in the color encodings of the different contour plots. QW number 0 is the one hit first by the incident field [Fig. 2a].