physia pss urrent topis in solid state physis Eletromagnetially indued transpareny in asymmetri double quantum wells in the transient regime Leonardo Silvestri1 and Gerard Czajkowski2 1 2 Dipartimento di Sienza dei Materiali, Universita di Milano Bioa, via Cozzi 53, Milano, Italy University of Tehnology and Life Sienes, Kaliskiego 7, 85789 Bydgoszz, Poland Reeived 10 September 2007, revised 10 Otober 2007, aepted 3 Deember 2007 Published online 22 February 2008 PACS 42.65.-k, 72.10.Di, 78.67.De We report omputation of the nonlinear response of a probe light in the presene of a driving field satisfying the Eletromagnetially Indued Transpareny (EIT) ondition in the Λ onfiguration appropriate to an Asymmetri Double Quantum Well (ADQW). The novelty of the present results with respet to those of previous alulations is to onsider the transient regime in whih both the pump and the probe radiation are pulses of pioseond duration. We investigate the dependene of the transpareny on pump and probe duration and also on the delay of the latter with respet to the former. We show the advantages of properly hoosing the duration of the pump and the probe light and the asymmetry in the delay time of the probe. phys. stat. sol. () 5, No. 7, 2412 2415 (2008) / DOI 10.1002/pss.200777649 T N I R P E R
phys. stat. sol. () 5, No. 7, 2412 2415 (2008) / DOI 10.1002/pss.200777649 Eletromagnetially indued transpareny in asymmetri double quantum wells in the transient regime physia pss urrent topis in solid state physis Leonardo Silvestri 1 and Gerard Czajkowski 2,* 1 Dipartimento di Sienza dei Materiali, Università di Milano Bioa, via Cozzi 53, Milano, Italy 2 University of Tehnology and Life Sienes, Kaliskiego 7, 85789 Bydgoszz, Poland Reeived 10 September 2007, revised 10 Otober 2007, aepted 3 Deember 2007 Published online 22 February 2008 PACS 42.65.-k, 72.10.Di, 78.67.De Corresponding author: e-mail zajk@utp.edu.pl We report omputation of the nonlinear response of a probe light in the presene of a driving field satisfying the Eletromagnetially Indued Transpareny (EIT) ondition in the Λ onfiguration appropriate to an Asymmetri Double Quantum Well (ADQW). The novelty of the present results with respet to those of previous alulations is to onsider the transient regime in whih both the pump and the probe radiation are pulses of pioseond duration. We investigate the dependene of the transpareny on pump and probe duration and also on the delay of the latter with respet to the former. We show the advantages of properly hoosing the duration of the pump and the probe light and the asymmetry in the delay time of the probe. 1 Introdution The EIT has been studied in the steady state limit in dense atomi systems where a large number of effets have been observed, inluding in partiular lasing without inversion and slow light. The phenomenon has also been studied in the ase of solid materials, like exitons in bulk materials and in the ase of exiton states in QW or intersubband transitions in ADQWs [1]. Beause of the large dephasing proesses whih take plae in solid materials due to eletron-phonon interation and eletron-eletron sattering, the steady state ondition for EIT is more diffiult to obtain and is of interest instead to analyze the results of transient spetrosopy where both the driving field and the probe field have a short duration and the latter an be delayed with respet to the former. Experiments of this type on exiton transitions in a QW have been reently arried out by Phillips and Wang [2] for a three level Λ situation on a transition from exiton states with opposite spin to a biexiton state. We onsider the ase of intersubband transitions in an ADQW whose ground state is populated by modulation doping. We show that the amount of EIT depends ritially on the duration of the pump and probe fields in relation to the values of the deoherene times. Moreover for an appropriate hoie of the pulses duration and of the pump intensity it is possible to observe a better transpareny with respet to the stationary ase. We make use of the density matrix approah [3]-[4], and onsider the dephasing rates due to eletron-eletron and eletron-phonon interation. This enables us to study the effets of the oherene between the external eletromagneti fields and the motion of eletrons in the ondution subbands, and to larify the role of speifi sattering proesses in the EIT. 2 Theoretial model We onsider the ondution band of an ADQW split in three subbands orresponding to the energies hω 1, hω 2 and hω 3. The system is driven by an external eletromagneti field. A nonlinear optial response to a probe field will be obtained by solving a system of equations for the mean values of the density matrix elements ρ nm k obtained from the Heisenberg
Contributed Artile phys. stat. sol. () 5, No. 7 (2008) 2413-8 -6-4 -2 0 2 4 6 8 Figure 1 Absorption of a weak probe beam in the presene of a resonant oupling field with Ω =2ps 1, in the steady-state (dashed line) and in the transient regime under optimal onditions (dotted line). Solid line represents the absorption without the pump. 0,9 0,7 0,5 0,3 τ = 0 ps τ = -2 ps τ = 2 ps τ = 5 ps τ = -5 ps τ = 10 ps 0,1-5 -4-3 -2-1 0 1 2 3 4 5 Figure 2 Absorption of the probe beam in the presene of a pump pulse of duration 10 ps and Ω =2ps 1. The probe pulse has a 1 ps duration and a delay τ given in the figure. equation for the density operator with an inoherent relaxation ontribution. The Hamiltonian used onsists of an unperturbed part, from whih the three subbands arise, and an interation term. The inoherent ontributions are the eletron-eletron sattering and the phonon-eletron interation. Sine we intend to onsider QW in III-V ompounds, we take into aount only LO phonons (frequeny ω LO ) beause their oupling onstants are muh larger than those of aousti phonons. For onveniene we perform a low temperature alulation, sine in the ase of nanostru
physia p s s 2414 L. Silvestri and G. Czajkowski: Transient EIT in quantum wells 1,2 = 2 ps = 5 ps = 10 ps -5-4 -3-2 -1 0 1 2 3 4 5 Figure 3 Absorption of a weak probe field of duration 1 ps in the presene of a pump of various durations () and Ω =2ps 1. τ p = 1 ps τ p = 0.1 ps -10-5 0 5 10 Figure 4 Absorption of a weak probe of various durations (τ p) in the presene of a 10 ps pump with Ω =2ps 1. tures the temperature an be fixed at will. Consequently, we onsider only the possibility of phonon emission with eletrons deaying from higher subbands into the lower subbands and within different states of the same band. To obtain the equations for the density matrix elements ρ ij k, we use the appropriate expressions for the irreversible dephasing terms. The final form of the onstitutive equations an be found expliitly in our previous work [1]. We only mention here that the intensities of the probe and of
Contributed Artile phys. stat. sol. () 5, No. 7 (2008) 2415 the pump are expressed in terms of their Rabi frequenies, denoted respetively by Ω p and Ω. 3 Results A system of oupled equations for eletron density matries are derived, inluding all the dephasing rates. In the stationary ase we assume that the laser operates in ontinuous wave, so that the Rabi frequenies do not depend on time. We an therefore easily solve the system of equations in the stationary ase ( ρij k t =0) at first order in the probe field and ompute absorption by the relation I[χ(ω p )] I[ρ 13 dk k ] (2π) 2, where χ is the suseptibility and ω p is the probe frequeny. In the ase of transient regime instead Rabi frequenies are assumed to be time dependent and it is therefore neessary to integrate the system of equations. We used a fourth order Runge-Kutta method and assumed a Gaussian profile for the laser pulses. The absorption in this ase an be found as I[χ(ω ω 31 )] [ ρ 13 I k (ω ω 31) Ω p (ω ω 31 ) ] dk (2π) 2, where ω ij = ω i ω j. The alulations are performed onsidering an ADQW onsisting of a Al 0.4 Ga 0.6 As barrier followed by a Al 0.16 Ga 0.84 As shallow well (6 nm). This well is separated from the GaAs deep well (6 nm) byaal 0.4 Ga 0.6 As barrier (3 nm). This potential profile gives rise to subbands separations of energy hω 31 = 184.9 mev and hω 32 = 64.3 mev. We onsider a low temperature ondition (T = 4K) and low eletron density (n e =1 10 11 m 2 ) in order to minimize the dephasing rates. The population of the subbands is pratially unhanged at low eletron densities and we find a well defined EIT for appropriate intensities of the ontrol beam. The EIT in the transient regime an be more pronouned than in the stationary one (Fig. 1) depending on the pulses durations and delay (Fig. 2-4). Pulse durations reported in the figures have been defined as the full width at half maximum of the Gaussian profile. We note that the transpareny is larger when 1) the probe pulse duration is muh smaller than the pump pulse duration and 2) the probe delay is small ompared to the pump pulse duration. It is interesting to note that the dependene on the probe delay is not symmetrial (Fig. 2), and that the transpareny is more evident when the probe arrives before the pump. The present results onfirm the possibility of obtaining EIT in solids, using asymmetri double quantum wells. Referenes [1] L. Silvestri, F. Bassani, G. Czajkowski, and B. Davoudi, EPJ B 27, 89 (2002). [2] M. C. Phillips and H. Wang, Phys. Rev. B 69, 115337 (2004). [3] A. Stahl, and I. Balslev, Eletrodynamis of the Semiondutor Band Edge (Springer-Verlag, Berlin/Heidelberg/New York, 1987). [4] G. Czajkowski, F. Bassani, and L. Silvestri, La Rivista del Nuovo Cimento 26(5-6), 1-150 (2003). Aknowledgements This researh is based on work supported by I.N.F.M. (Istituto Nazionale per la Fisia della Materia). One of us (GC) wishes to aknowledge the support from the grant MIUR-PRIN-2006021037, and the Suola Normale Superiore for the invitation and hospitality. We thank A. Trediui, G. La Roa, and F. Bassani for useful disussions.