I. BALSLEV and A. STAHL : Coherent Nonlinear Response of Excitons 413

phys. stat. sol. (b) 160, 413 (1988)

Subject classification: 71.35 and 78.20; 57.12; 57.15

Fysisk Institut, Odense Univereitetl) (a ) and Institut fiir Theoretische Physik der Rheinisch- Westjalischen Technischen Hochschde Aclehen2) (b)

Coherent Nonlinear Response of Excitons in Semiconductors

BY 1. BALSLEV (a) and A. STAHL (b)

The nonlinear optical response of excitons in semiconductors is studied theoretically, particularly the so-callod nonresonant optical Stark effect. Results from a two-band density matrix formalism and its reduction to a two-level approximation are compared. It is found that the two models give similar results except a t very high pumping levels. The calculated results agree well with experi- ments on quantum wells in (AlGa)As-GaAs.

Theoretische Untersuchungen zur nichtlinearen Optik von Exzitonen in Halbleitern, insbesondere zum sogenannten nichtresonanten optischen Stark-Effekt, werden durchgefuhrt. Ergebmsse der Diohtematrixtheorie fur ein 2-Band Modell und das hieraus abgeleitete 2-Niveau Modell werden diskntiert und miteinander verglichen. Es zeigt sich, daB die Besohreibung durch ein 2-Niveau- System bei nicht zu starker Pumpinkmitiit eine gute Approximation darstellt. Das gilt auch fur den Vergleich mit Experimenten an (A1Ga)As-GaAs ,,quantum well"-Systemen.

1. Introduction

The experiments of Mysyrowicz et al. [l] and von Lehmen et a]. [2] on the so-called nonresonant optical Stark effect of excitons in semiconductors ((AlGa)As-GaAs) initiated a series of theoretical studies [3 to 71 of the following question: What are the mechanisms responsible for the changes of the excitonic absorption caused by a light pulse in the transparent spectral region ? The first theories developed by Schmitt- Rink, Haug, and their coworkers [3 to 51 concentrated on models without irreversible dephasing mechanisms of the excitons. Later, the present authors have shown [6] that a T,-like relaxation plays an important role in the sense that observablefrequency shifts induced by a nonresonant pulse can be a t most of the order 1/T, because of bleaching.

After a brief summary of the findings of [6] we present additional calculations ex- ploring the influence of the properties of the pump pulse (amplitude, center frequency, band width, and time before the probe pulse). In addition we explore the validity range of the two-level approximation to the more general two-band dynamics.

2. Two-Level and Two-Band Electrodynamics

The full nonlinear response of band edge excitations is adequately described by a two-band density matrix formalism [6, 8 to lo]. Since this formalism accounts for the coherent coupling between electromagnetic and matter waves, it has been denoted as.

1) DK-5230 Odense M, Denmark. 2) D-5100 Aachen, FRG.

414 I. BUSLEV and A. STAHL

the coherent wave theory of the band edge. Performing a real-time numerical integra- tion of the two-dimensional two-band system (relevant for multiple quantum wells explored in 111 and [2]) we find in [6] the absorptive excitonic response shown in Fig. 1.

We showed in [6] that the Stark-type influence from the pump field on the envelope function can be neglected in the nonresonant case, and so the dominant effect of the pump pulse is the filling of the bands. (This statement is not true for the resonant Stark effect explored experimentally by Frohlich et al. [ll]). If in addition the exciton resonance is spectrally isolated, the two-band dynamics reduces t o that of a two-level system [6]. The relevant equations are then the simple Bloch equations,

81 + ys1 = W082 2

i, + ys, = -wosl - 2Mo __ EI , h

where s, + is, is the complex transition amplitude, I the inversion, wo the resonance frequency, y the dephasing rate (y = l/T,), M , the transition dipole, and E the electric field. The instantaneous dipole moment P is given by

P = ~M,s, . (2) A weak like nonlinear response can be found analytically from ( l a ) t o ( l c ) . The result is 161

+ 2Mtw, - (Wpr i- +)

2MEw, 1

+-- 2 2 1 Epumpl * (3) h3

X(mpr) =- ' ( W E + y2 - W & - 2i~pry)

(wO t. y2 - Wpr - 2i~pry)z (Wpr - pump)

3.4 I * s

2

2

- 2. al

b 0

B -4

3

1 530 7. 535 7.540 7.545 I550 pmbe frequency(eV) -

Fig. 1. Results of a two-pulse numerical integration of the constitutive equations of a two-band system [a] to [S]. Parameters refer to two-dimensional excitons in a GaAs-(A1Ga)As MQW; de- phasing rate y = 2 meV/h, center frequencies: 1.515 and 1.640 eV of pump and probe, respectively, FWHM of Gaussian intensity pulse: 0.6 and 0.06 ps, respectively. The values of the peak pumping power lpump are: (a) 0, (b)0.04, (c) 0.12, (d) 0.25, (e) 1.0, and (f) 4.0 in units where lpump = 1 corresponds to a Rabi frequenoy of the order 1 meV/d

Coherent Nonlinear Response of Excitons in Semiconductors 416

z(mpr) is the effective susceptibility a t wpr in the presence of a strong pump beam with electric field amplitude Epmp and frequency wpump. From (3) one finds the shift A of the peak absorption given by

(4) &abi =~~~

2(w, - WPUrnP) ’ where w m b j = (EpnmpiMg/h) is the Rabi frequency associated with the pump field.

The case with comparable linear and nonlinear contributions must be explored by real-time numerical integration of the Bloch equations. I n this work we present results from such integrations of the two-level equations (1).

3. Two-Pulse Numerical Integration

We insert into (1 a) to (1 c) an electric field consisting of two Gaussian pulses, pump and probe, with center frequencies wpump and wpr, and peak fields Epump and Epr, respectively. I n the frequency domain the FWHM of the pulses are Awpump and Amp:. I n the time domain the probe pulse is delayed Tdelay after the pump pulse. With thls field applied we integrate numerically ( l a ) t o ( l c ) using time steps of the order 0.160,~. We perform a Fourier transformation of the the resulting temporal behavior of the electric field E and the dipole momentP. Subsequently, we derive the absorptive response by calculating the out-of-phase component of P with respect t o E for each frequency .

We only investigate pump pulses which in the frequency domain have negligible overlap with the resonance. I n this strictly nonresonant case the results of the calcula- tions are essentially indepedent of the properties of the probe pulse as long as this pulse itself does not produce nonlinear effects. For the probe pulse we use wpr = w, and Ampr = 0.30, in the present calculations. The remaining parameters are y , wpulnp, Awpump, Tdelay, and the Rabi frequency WRabi associated with the peak pump field.

4. Results and Discussion

In all our calculations presentedgraphically y/wo = 0.01 and A w ~ ~ ~ ~ / w ~ = 0.03 are used. In Fig. 2 we show spectra of the absorptive response for o ~ ~ ~ ~ / w ~ . = 0.9, Tdelay = 0,

and different values of the pump field. The resonance line is sublect t o a blue-shift which is found to be close to the value in (4) valid in the limit of weak nonlinear re- sponse. As seen the line is subject t o a strong bleaching. Calculations with other dephasing rates y (not shown) indicate that the bleaching becomes pronounced when the shift of the resonance exceeds a value of the order y. The features in Fig. 2 are similar to those of the two-band system (Fig. 1) except for the fact that a t high pump fields, the two-level system exhibits oscillatory behavior while the two-band system becomes more effectively bleached. This property of the two-band model is probably due to the presence of higher resonances and a continuum, which act as additional channels serving as sinks for the coherent electronic excitation. So far, no experiments have been reported on pumping levels in the regime relevant for this different behavior of the two-level and two-band models.

Fig. 3 shows spectra of the absorptive response for W R ~ ~ ~ / W , = 0.2, Tdelay = 0, and different values of the pump frequency. I n agreement with (3), (4) the influence of the pump beam reduces with increasing distance between the pump frequency and the resonance. Calculations with a pump frequency above the resonance (not shown) give

I. BALYLEV and A. STAEL

Fig. 2. Results of a two-pulse numerical integration of (1 a to c), corresponding to a nonresonantly pumped two-level eystem. The parameters are: y = O.Olw,, thc center frequency of pump wpnmp = = 0.90,, A U I ~ , , ~ ~ = 0.030,,, and Tdelay = 0. The Rabi frequency WRabi associated with a peak, pump field has the following values: (a) 0, (b) O.lwo, (c) O.2w0, and (d) 0.4~1,. Curve e shows the am- plitude spectrum of the inciderit electric field (with zero on the bottom of the figure for clarity)

09 7.G I. I frequency lev} -

Pig. 3. Results of a two-pulse numorical integration of (1 a to 1 c). The parameters are: y = O.Olruo the Rabi frequency of pump ORabi = 0 . 2 ~ ~ . Awpump = 0.03w0, and Tdeiay = 0. The pump fre- quency Wpump has the following values: (a) 0.600, (b) O.ea),,, and (C) 0.901,

Coherent Xonlinear Response of Excitons in Semiconductors 41 7

t 50~ I

Fig. 4. Results of a two-pulse numerical integration of ( l a ) to ( l o ) . The parameters are: y = = O.lw,, tho center frequency of pump Wpump = o.9w0, Ampump = 0.3~1,. and WRabi = 0.20,. The delay Tdelay has the following values: (a) 0, (b) 0 . 7 6 / A ~ p u m p , (c) 2.26/AopUmp. Curve d shows the amplitude spectrum of the incident electric field (with zero on the bottom of the figure for clarity).

a red-shift. Due to the unavoidable absorption above the lowest exciton line, such a red-shift might be difficult to observe.

Finally, in Fig. 4 we show spectra of the absorptive response for Wpnmp/6J,, = 0.9, c o m ~ i / w o = 0.2, and different values of delay of the probe beam. These results reflect the fact that the optical Stark effect studied here is instantaneous. Accordingly, we find that the distortion of the resonance is absent if the temporal overlap of pump and probe pulses is negligible. As the probe pulse is short, the Stark effect starts disappearing when Tdelay increases beyond the duration of the pump pulse (which is of the order l/Awpump in the Fourier limit).

6. Conclusion The most characteristic features of the nonresonant optical Stark effect are the fol- lowing :

1. An effective blue-shift of the order 2. a bleaching which becomes pronounced when the shift is of the order of the line-

3. a dependence on the temporal overlap of the two pulses.

w2Rabi/(Wo - Opnmp);

width or larger; and

The results discussed in this paper demonstrate that these features are already ac- counted for by a two-level density matrix formalism. This two-level model should be considered as the single resonance approximation to the more general two-band den- sity matrix theory [ S ] . Only for very strong pump pulses the approximation breaks down. We attribute this t o the fact that in the regime of high pumping levels, the extra channels related to the higher resonances and the continuum in the two-band system become important. 27 physicn (b) 160/2

418 I. BALSLEV and A. STAHL: Coherent Nonlinear Response of Excitontl

The successful application of the two-level model for explaining the nonlinear re- sponse of excitons is not unique for the nonresonant optical Stark effect. It is found also in the case of four-wave mixing as demonstrated by Schultheis et al. [12] and by Hvam et al. [13].

One feature not mentioned above is the dependence on the relative polarization of the two pulses. This aspect remains to be explored as successfully done for the reso- nant Stark effect [14].

Acknowledgements

One of us (I. B.) is endebted to J. M. Hvam for frequent discussions. The present work has been supported by the Danish Natural Science Foundation.

References [l] A. MYSYROWICZ, A. D. HULIN, A. ANTONETTI, A. MIGUS, W. T. MASS=-, and H. MORKOC,

[2] A. VON LEHMEN, D. S. CHEMLA, J. E. ZUCKER, and J. P. HERITAGE, Optics Letters 11, 609

[3] S. SCHMITT-RINK and D. S. CHEMLA, Phys. Rev. Letters 67, 2752 (1986). [4] S. SCHMITT-RINK, D. S. CHEMLA, and H. HAUG, Phys. Rev. B 37, 941 (1988). [5] J. B. MULLER, R. MEWS, and H. HAUG, Z. Phys. B 69, 231 (1987). [6] I. BALSLEV and A. STAHL, Solid State Commun. 67, 85 (1988). [7] M. COMBESCOT and R. COMBESCOT, J. Physique 49, Suppl. 6 ,209 (1988). [8] A. STAHL, phys. stat. sol. (b) 94, 221 (1979); 106, 575 (1981). [9] A. STAHL and I. BALSLEV, Electrodynamics ofthe Semiconductor Band Edge, Springer Tracts

Phys. Rev. Letters 66, 2748 (1986).

(1986).

mod. Phys. 110, (1987). [lo] I. BALSLEV and A. STAHL, Optics Commun. 66, 137 (1988). [ll] D. FROHLICH, A. NOTHE, and K. REIMANN, Phys. Rev. Letters 65, 1335 (1985). [12] L. SCHULTHEIS, J. KUHL, A. HONOLD, and C. W. Tu, Phys. Rev. Letters 67, 1635 (1986). [13] J. M. HVAM, I. BALSLEV, and B. HONERLACE, Europhys. Letters 4, 839 (1987). [14] D. FROELICH, R. WILLE, W. SCHLAPP, and G. WEINANN, Phys. Rev. Letters 59, 1748 (1987).

(Received September 6, 1988)