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Physical Review Letters

Rabi Flopping in a Two-Level System with a Time-Dependent Energy Renormalization: Intersubband Transitions in Quantum Wells

Adriano A. Batista, David S. Citrin

Published Mar 25, 2004• 58 citations
Publisher

Abstract

We obtain pulse-driven Rabi oscillations guided by a generalization of the rotating-wave approximation to include, in the optical-Bloch equations, two-level systems with a time-varying transition energy. We achieve this by using chirped pulses with the central frequency given by the time-varying transition energy. Using this approach, we predict Rabi oscillations in intersubband transitions in a two-subband n-type modulation-doped quantum well by taking into account the time-dependent intersubband energy-gap renormalization due to depolarization-shift effects. We obtain Rabi oscillations for jpi (j=0,1,2, ) pulses in the presence of dephasing.

DOI
10.1103/physrevlett.92.127404

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We obtain pulse-driven Rabi oscillations guided by a generalization of the rotating-wave approximation to include, in the optical-Bloch equations, two-level systems with a time-varying transition energy. We achieve this by using chirped pulses with the central frequency given by the time-varying transition energy. Using this approach, we predict Rabi oscillations in intersubband transitions in a two-subband n-type modulation-doped quantum well by taking into account the time-dependent intersubband energy-gap renormalization due to depolarization-shift effects. We obtain Rabi oscillations for jpi (j=0,1,2, ) pulses in the presence of dephasing.
We obtain pulse-driven population inversions in two-subband quantum wells (QW's) with a time-varying transition energy. We achieve this by using linearly chirped pulses with the center frequency given by a fit of the time-varying transition energy obtained from the effective nonlinear Bloch equations. The coherent population inversions (Rabi oscillations) are given for square QW's and for dc-biased square QW's. We further verify that the time-dependent polarization of the intersubband transitions can also be controlled.
We use chirped pulses to achieve substantial electron population transfer from the wider well to the narrower well of an nnn-type doped asymmetric double quantum well (QW). To realize this goal we are guided by a generalization of the rotating-wave approximation to include, in the optical-Bloch equations for three-subband QW's, a time-varying central frequency equal to the renormalized intersubband transition energy. The energy-gap renormalization in our model is due to depolarization-shift effects. The first pulse central frequency is near the time-varying renormalized ${ϵ}_{2}\ensuremath{-}{ϵ}_{0}$ gap, while the second pulse is near the renormalized ${ϵ}_{2}\ensuremath{-}{ϵ}_{1}$ gap. An upper-bound estimate of the Hartree-Fock corrections to these transition energies is given. We find out that for the QW structure and the sheet density used these corrections are small.
Here we use perturbation techniques based on the averaging method to investigate Rabi oscillations in cw and pulse-driven two-level systems (TLS's). By going beyond the rotating-wave approximation, especifically to second-order in perturbation, we obtain the Bloch-Siegert shift of the TLS resonant frequency, in which the resonant frequency increases with the driving field amplitude. This frequency shift implies that short resonant πππ-pulses in which the Rabi frequency is approximately 40\% or higher of the transition frequency do not achieve complete inversion in TLS's. Hence, guided by analytical results based on the averaging technique, we propose two methods for obtaining population inversions in the TLS driven by short πππ-pulses: one with chirping and the other with pulse shaping and near resonance blue-shifted detuning. Both methods minimize dephasing due to the Bloch-Siegert shift, reduce the dependance of the excitation of the TLS on the pulse phase, and are very effective in achieving complete population inversions.
We investigate the interaction of a multilevel atom with a short femtosecond optical pulse beyond the scope of the rotating-wave approximation. We consider the behavior of a three-level atom under the action of two optical pulses resonant with two allowed transitions and a two-level atom under the action of a resonant optical pulse. It is shown that the imaginary part of the off-diagonal elements of the density matrix oscillates with high frequency due to the counter-rotating terms. It is shown also that an effective time-dependent addition to the Rabi frequency exists, which is of importance for ultrashort optical pulses.
We study the interaction of an ac electric field with a semiconductor quantum well by using the effective nonlinear Bloch equations. Only the first two electron subbands in the well are considered. We apply the rotating wave approximation and derive analytical solutions for the Bloch equations for two different values of the detuning. At exact resonance we find a critical value of the Rabi frequency around which the dynamics of the system changes abruptly. Above this critical value one obtains electron oscillations with complete inversion in the two-subband system, while below this value we obtain electron oscillations without complete inversion and with the majority of the electron population on average in the lower subband. We also present numerical calculations for a specific quantum well structure and assess the limits of validity of the analytical results.
The dependence of multiple quantum well intersubband Rabi oscillations on the radiative coupling between different quantum wells is investigated. We show that in a highly doped multiple quantum well sample, radiative coupling has a large impact on coherent density oscillations, even stronger than the impact of many-body effects. Specifically, radiative coupling can dominate nonlinear optical response, such as Rabi oscillations of subband populations, yielding distinctively different excitation behaviors in the different wells of a multiple quantum well sample. The magnitude of radiative coupling may be controlled by varying the angle of incidence of the external electric field.