Physica E 12 (2002) 558–561 www.elsevier.com/locate/physe Coherent control and four wave-mixing of Fermi edge singularities in doped quantum wells D. Porras, J. Fern andez-Rossier, C. Tejedor * Departamento de Fsica Te orica de la Materia Condensada, Universidad Aut onoma de Madrid, Cantoblanco, 28049 Madrid, Spain Abstract We discuss Fermi-edge singularities in the linear and nonlinear transient response of a doped quantum well at nite temperatures. In the linear response, we compute the coherent control of the energy absorption. A phase-shift appears in the coherent control oscillations. For the nonlinear response, we show that four wave-mixing experiments with varying energy presents a Fermi-edge singularity. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 78.47.+p Keywords: Fermi edge singularity; Coherent control; Four wave mixing The enhancement of the optical transition probabil- ity of an electron from a localized state in the valence band to a conduction band state just above the Fermi level of a doped quantum well (QW) is known as the Fermi edge singularity (FES). FES has been observed in continuous wave spectroscopy in a variety of doped semiconductor heterostructures [1]. FES results from the interplay between the sudden appearance of a hole potential and the presence of an extra electron at the conduction band. This produces a set of coherent low energy electron–hole pairs that can be described as Tomonaga bosons [2]. In this work, we study the coherent ultrafast spec- troscopy of FES in a doped QW. We analyze both linear (coherent control (CC)) and nonlinear (four wave-mixing (FWM)) regimes in which just a few ex- periments [3] have been performed. We concentrate in a doped QW, in zero magnetic eld, excited by laser * Corresponding author. Tel.: +349-1-397-4908; fax: 349-9- 1397-4905. E-mail address: carlos.tejedor@uam.es (C. Tejedor). pulses spectrally peaked around the absorption thresh- old. The laser pulses are spectrally narrow compared to the Fermi energy  F so that the photoexcited elec- trons have energies close to the Fermi level, but the pulses are shorter enough so that transient coherent eects can be observed. Our main ndings are: (i) CC of the energy absorbed by doped QW presents oscillations showing a characteristic phase- shift which depends on the exponent of the continuous wave Fermi edge singularity. This is a remarkable dif- ference with undoped systems where coherent control of the exciton density does not show any phase shift. (ii) The intensity of the FWM signal shows a sin- gularity when the exciting frequency is varied near the Fermi edge. Such singularity is characterized by the same parameter than the one appearing in the linear response. (iii) The optical coherence induced by the laser, both in the CC and FWM situations, has an intrinsic exponential decay roughly proportional to the temper- ature T . 1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII:S1386-9477(01)00471-4