Transient four-wave mixing in intersubband transitions of semiconductor quantum wells Spyridon G. Kosionis a , Andreas F. Terzis a , Emmanuel Paspalakis b,n a Physics Department, School of Natural Sciences, University of Patras, Patras 265 04, Greece b Materials Science Department, School of Natural Sciences, University of Patras, Patras 265 04, Greece article info Article history: Received 20 November 2012 Received in revised form 13 February 2013 Accepted 28 February 2013 Available online 15 March 2013 Keywords: Four-wave mixing Semiconductor quantum well Two-subband system Intersubband transitions Transient spectra Electron sheet density abstract In this work we theoretically study the transient four-wave mixing (FWM) spectrum in a symmetric double semiconductor quantum well structure. The system interacts with a near resonant strong pump field and is probed with a weak field. We assume that the rectangular pump and probe fields interact with two quantum well subbands. In order to describe the system, we use the modified density matrix equations containing terms which owe their presence to the electron–electron interactions. The density matrix equations are solved numerically, for a GaAs/AlGaAs semiconductor quantum well structure, for different moments in the time interval, until the system reaches its steady state. We consider the effects of electron sheet density and pump field frequency on the transient FWM spectrum. A physical interpretation of some spectral characteristics is attempted, via a dressed-state analysis of the semiconductor quantum well coupled to the strong pump field. & 2013 Elsevier B.V. All rights reserved. 1. Introduction In the area of semiconductor quantum wells, significant interest has been shown, for several years, in nonlinear optical effects in intersubband transitions. In several of these studies the effects of electron–electron interactions have been investigated and it has been shown that they strongly influence the nonlinear optical response. Some of the effects that have been considered in theoretical studies are saturation and optical rectification [1,2], nonlinear optical effects up to fifth order [3], four-wave mixing (FWM) [4–7], pump–probe optical response [7–9], Kerr nonlinearity [9,10], optical bistability [11–13], gain without inversion [14], ultra- short pulse propagation [15,16], high-order harmonic generation [17,18] and controlled population transfer [18–25] in a two-subband system. Experimental results also exist in this area [26–29]. In this work, we study the transient FWM spectrum in a GaAs/ AlGaAs double quantum well structure driven by a strong pump field and probed with a weak field. For the analysis of the system we use the modified density matrix equations containing terms due to the electron–electron interactions. We find that the frequency of the pump field and the electron sheet density of the quantum well structure play an important role on the temporal evolution of the FWM spectrum. Even in the case in which the spectral form of the FWM spectra in the steady state is almost identical for two different sets of parameters, their evolution can be quite different according to the precise value of these parameters. We note that recently analogous behavior has been found in the pump–probe optical response and the Kerr nonlinearity in a two- subband system taking into account the effects of electron–electron interactions [9]. In addition, the transient response of the absorp- tion and dispersion spectra has already been analyzed in previous papers [30–32], without considering the effects of electron–electron interactions, in intersubband transitions of asymmetric quantum well structures. Furthermore, recent experimental studies have used transient absorption for the study of optical effects in inter- subband transitions in semiconductor quantum wells [33,34]. Our paper is organized as follows: in the next section, we present the theoretical model and establish the corresponding dynamic equations. The numerical results and physical analysis are shown in Section 3. In the final section, a brief summary is given. 2. Theory In this work, we study a symmetric double quantum well GaAs /AlGaAs structure and we assume that the ground state subband (n ¼ 0) and the first excited subband (n ¼ 1) are the only subbands that contribute to the system dynamics. By choosing a proper value for the electron sheet density, we may adjust the Fermi energy, such that the Fermi level is found below the first excited subband minimum. Thus, practically, the first excited subband is considered to be initially empty. We suppose that the quantum Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/jlumin Journal of Luminescence 0022-2313/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jlumin.2013.02.054 n Corresponding author. Tel.: þ30 2610 969346. E-mail address: paspalak@upatras.gr (E. Paspalakis). Journal of Luminescence 140 (2013) 130–134