Phase-Resolved Nonlinear Response of a Two-Dimensional Electron Gas under Femtosecond Intersubband Excitation

Strong electric-field transients resonant to intersubband transitions in $n$-type modulation-doped $\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ quantum wells induce coherent Rabi oscillations, which are demonstrated by a phase-resolved measurement of the light emitted by the sample. The time evolution of the intersubband polarization is influenced by Coulomb-mediated many-body effects. The subpicosecond period and the phase of the Rabi oscillations are controlled by the properties of the midinfrared driving pulse.

Figure 1

(a) Envelope wave functions (simplified) of an electron confined in the potential well made by the different conduction band edges of GaAs (wells) and ${\mathrm{A}\mathrm{l}}_{0.35}{\mathrm{G}\mathrm{a}}_{0.65}\mathrm{A}\mathrm{s}$ (barriers). (b) The dispersions of the subbands are almost parallel along the in-plane wave vector $k_{\parallel }$. (c) Stationary IS absorption. (d) Schematic of the time-resolved nonlinear propagation experiment: An intense MIR pulse is sent through the prism-shaped sample. The electric field of the transmitted light induces in a thin ZnTe crystal a time-dependent birefringence, which is temporally sampled using a 12-fs probe pulse and a polarization sensitive detector (PS). (e) Measured coherent linear response of the IS transition. Thin line: incident field $E_{\mathrm{i}\mathrm{n}}(t)$; thick line: reemitted transient $E_{\mathrm{e}\mathrm{m}}(t)$ from the sample.

Figure 2

(a)–(c) Ultrafast Rabi oscillations of the $1\leftrightarrow 2$ IS transition measured as the coherent nonlinear response of the sample [reemitted field: $E_{\mathrm{e}\mathrm{m}}(t)$ (thick lines)] to different strong incident electric fields $E_{\mathrm{i}\mathrm{n}}(t)$ (thin lines). (d) Dashed line: pulse area $\Theta$ calculated from the incident electric field in (c). Dots: progress of the Rabi oscillation $\Xi$ obtained from the envelope of the reemitted field.

Figure 3

(a),(b) Calculated time evolution of the reemitted field (thick lines) and the excited-state population (shaded areas) for an applied field (thin lines) using the Maxwell-Bloch equations [Eq. (1)] for uncoupled two-level systems. (c) For the same field amplitude as in (b), two coupled two-level systems lead to an increase of the Rabi frequency and to a decrease of the reemitted amplitude.