Coherent ultrafast nonlinear optical processes in semiconductor quantum wells

Coherent ultrafast nonlinear optical processes in semiconductor quantum wells

Pergamon Solid State Communications, Vol. 92, Nos 1-2, pp. 37-43, 1994 Elsevier Science Ltd Printed in Great Britain 0038-1098(94)00487-0 0038-1098/9...

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Pergamon

Solid State Communications, Vol. 92, Nos 1-2, pp. 37-43, 1994 Elsevier Science Ltd Printed in Great Britain 0038-1098(94)00487-0 0038-1098/94 $7.00+.00

COHERENT ULTRAFAST NONLINEAR OPTICAL PROCESSES IN SEMICONDUCTOR QUANTUM WELLS D.S. Chemla l ~ y a e s Delmrm~nt, UniverMty of California m Berkeley, Materials Sciences Division, Lawrence Berkeley ~ t o r y

l This article reviews recent experiments where, for the fh-st time, the dynamics of both the amplitude and the phase of ultrafast coherent emission from semiconductor quantum well structures have been investigated. Keywords : A. quantum wells. E. nonlinear optics,time- resolved optical spectroscopies

are generated in the continuum. The experimental results are compared to the most advanced theory, the limitations of which are pointed ouL Finally in Section IV we conclude.

I) Introduction In the last decade, quantum confined semiconductor structures have attracted much attention because of their novel properties and potential applications. A wealth of new electronic and optical effects have been observed and investigated in these heterostmctures. 1 In semiconductors, optical properties near the fundamental absorption edge are dominated by elementary excitations of electronic nature, which are complex, composite and delocalized qu~iparticles. Their properties are determined by a delicate balance between quantum statistics and Coulomb correlation. These many-body mechanisms are modified in a non-trivial manner by quantum confinement.2, 3 The typical values of the group and phase velocities of quasi-particles are such that the length scale for quantum size effects, 50~k
U) Optical nonlinearities in semiconductors When a semiconductor is excited close to the fundamental bandgap, it is brought into an admixture of excited states that is described by the density matrix 6(t).The theory of the time evolution of fi(t) has attracted much attention recently.4,5 In this article we base our discussion on the so called "Semiconductor Bloch Equation" (SBE) formalism which allows the most intuitive description of the underlying physics~ It is convenient to express fi(t) in k-space where the powerful arsenal of effective mass approximation (EMA) tools and concepts can be readily exploited. In the simple two-band EMA model of a semiconductor the density matrix, fi(t), breaks into 2 x 2 blocks. The diagonal elements, nc,v(k), represent the population in the two bands which, for ultrashort pulse excitation, can be off-equilibrium and transienL The offdiagonal terms, Pk and its c.c., represent interband transition amplitudes which determine the polarization and hence the optical response. It is important to note that populations and polarizations are intimately coupled and are, in facL two aspects of a single reality: the density matrix. It satisfw.s

the Liouville equation;

~'~nk t (t)= --Jig k (t), nk (t)]+~-~ Ot fik (t)[m~x .

(I)

Treating the Coulomb interaction on the same footing as the interaction with the applied EM-field, the Hamiltonian matrix of Eq.(1) is: 0

0

within the unrestricted Hartree-Fock approximation. It comprises of three contributions. The first gives the bareband energies, e¢,v(k)-- ec,v(0)+k 2/2mc,v" The second 37

38

SEMICONDUCTOR QUANTUM WELLS

expresses the coupling of the interband dipole matrix element, t~k, with the EM-field E(t). Fmally, the third describes how the Coulomb potential, Vk,k,, couples states at different wavevectors. Dephasing and relaxation are accounted for pbenomenologically by the last term of Eq. (1). The Coulomb interaction renormalizes the energies and the Rahi frequency, ei(k)-~ei(k)-Y.Vk,k.ni(k'),and k'

ttkE --~ ~kE + ~'~ Vt.t.pt. respectively These renormalizations k'

depend directly, but also implicitly through the screened potential Vk,k', on the populations and polarizations. 2"4 In the zeroth order and for undoped materials this results in the "observed" band gap and exciton structure. Excited states are strongly interacting electron (e) hole (h) pairs. The bound states correspond to sharp resonances, but e-h correlation influences strongly the scattering (unbound) states as well. In doped materials, the situation is more complicated. When an electron is promoted in a conduction band, the hole left behind interacts not only with its companion electron but with all the electrons of the Fermi sea which were introduced by doping. This complicated many-body interaction also produces a resonance, called the "Fermi edge singnlarity".6 It has not been observed in bulk materials because the coherence of the Fermi-sea pair excitations, which are involved, is easily destroyed by impurity scattering. In 2D and ID modulation doped semiconductor heterostructures, however, it is the dominant features of the linear optical response. 7 The pair amplitude, Pk, obeys a nonlinear Schr0dinger equation which, for no applied EM field, reduces to the Wannier equation. It includes two sources of optical nonlinearities. The first one originates from the Panli blocking and appears as a reduction of the coupling to the EM-field proportional to the population, I.tkE(t) ~ {1-n e ( k ) - n h (k)}l~kE(t).

(2a)

This nonlinearity is present in all material systems, including atoms and semiconductors, and is essentially instantaneous. The other source of nonlinearity, v,x [prin, ( t ' ) + n~ (k')}-p,, In. (k) + n, (k)}]

(2b)

t,'

appears as a coupling between populations and populations mediated by the Coulomb potential. It is clear that this many-body nonlinearity is qualitatively different from that of isolated atomic-like systems. Fwst it should be noted that its dependence on ne,h(k) and Pk is quite complicated since the screened potential Vkk, itself is an implicit function of the populations.2, 3 It becomes visible only when the excitation has produced signiFw.ant population and polarization densities. Furthermore, in the case of excitation by ultrashort pulses for which the temporal proftle of ne h(k) and Pk is step-like, its contribution to the nonlinear optical response is delayed and dephased with respect to that due to the Pauli blocking. Therefore the nonlinear optical response of semiconductors is qualitatively different from that of atomic systems. And, many of the specificities of dynamics of many-bedy effects appear in ultrashort pulse timeresolved nonlinear optical experiments. The exciton binding

Voi.92, Nos I-2

energy, Ry, and Bohr radius, ao, determine the natural energy and length scales for all near band gap optical processes. Quantum size effects thus appear when the dimension of the sample in one or several directions is such that L~ ao. In this regime the density of states, the energy levels and the e-h pair structures are strongly modified as compared to the bulk2 Correspondingly the relative strength of the two nonlinearities, Eq. (2a) and (2b), is affected by quantum com"mement Nonlinear spectroscopy techniques were applied early to quantum confined heterostructures and a number of interesting processes were observed and investigated. They have been reviewed recentiy.2,3, 8 In this article we shall concentrate on ultrafast coherent nonlinear optical processes near the fundamental absorption edge.

HI) Coherent Amplitude and Phase Measurements

The simplest coherent wave-mixing configuration is the so called four-wave-mixing (FWM), in which two laser pulses, labeled I and 2, separated by a time delay, At= t2-tI, and propagating in the directions k 2 and kl, interferein a sample. They generate a nonlinear polarization that contains a contribution Ps(t), which emits photons in the background-free direction ks = 2k2-kl. In the case of homogeneously broadened two level systems the FWMsignal is emitted immedia~ly after the second pulse and corresponds to free induction decay.9 For inhomogeneously broadened fines, the FWM-signal is delayed by At after the second pulse and corresponds to a "photon echo". FWM techniques have been applied extensively to atomic and molecular systems to study their dephasing. I0 The first investigations of FWM in semiconductor QW also concentrated on the origin of dephasing of resonances and were analyzed using theory borrowed from atomic physics.I 1-12 The easiest and most commonly used measurement technique is to time-integrate the FWM signal, with a slow detector, as a function of At. In the case of two level systems this reproduces the temporal behavior of the free induction decay i.e.no signal seen for At<0 and exponentially decaying signal for At>0. I0 In careful ultrashort pulse experiments performed on QWs, signals were observed for At<0 with a decay constant twice that of the "normal" signal seen for At>0.13,14 These observations are in qualitative contradiction with the independent level model. They provided the first direct evidence of the importance of Coulomb many-body nonfineanty, and triggered intensive research on the origin of optical nonlinearitiesin semiconductors. 14-18 The observation of a FWM signal for At<0 is readily explained by Eq.(2b) because, as opposed to the Panli blocking Eq.(2a), the many-body nonfinearity contributes for the two pulse orderings.13-15 Since many-hody nonlinearities change the time-integrated signal so much, it is natural to investigate ~ek influence on the true temporal profile of the FWM signal. This is more difficult to measure. For a fixed At, one has to cross-conelate the FWM emission with a reference laser pulse in a highly transparent nonlinear crystal. Such experiments on homogeneously

Vol.92,

Nos 1-2

SEMICONDUCTOR QUANTUM WELLS

Time-Integrated Amplitude (vs At) II~ Time-Resolved Amplitude (vst atAt=0)

0.8

"

~

4 dVm,.h

!"

6oo

~o

\

1,21

"lf'V',"

200

O

,

,

o

N~ 1011cm "2

,,,,-:=,s A,

" ,.Ir't

-2oo

~me,(fs)

a)

E

1

0.6

)~ ~"~ "

39

200

400

0.4

I

600

AL I4

1.44

1.48

Energy(eV)

0.2 0

-4oo

i

(b)[ ( ~ ~ I ~"c s), 0

Figure h Comparison of the temporal profdes of the timeresolved FWM intensity (vs. absolute time t) and of the time-integrated intensity (vs. time delay At), measured in a Gabs QW at an exciton density Nx=101 lcm-2.

broadened samples, show that although the FWM-signal is emitted immediately after the second pulse, it comprises of two contributions. I6"18 The first one is instantaneous and originates from Paul blocking, the second is delayed and originates from the Coulomb interaction. Surprisingly the latter can he larger than the former, and in high quality samples, its delay can be so large that it can appear as a separate pulse.18Figure 1 displays the temporal profdes of the time-resolved FWM intensity (vs. absolute time t) and of the time-integrated intensity (vs. At), measured in a GaAs QW structure for an exciton density Nx=1011cm -2 , with laser pulses of duration (98 ± 2)fs. The sample was at room temperature to homogenize the exciton resonance by collision with thermal phonons. The two curves have been normalized to unity, and the unrelated time axes have been shifted to bring the maxima into coincidence to compare their lineshapes. The difference is evident, the time-resolved FWM amplitude is clearly broader than the time-integrated FWM amplitude, with a slower rising edge and a significantly non-exponential trailing edge. This difference is density dependent. When the total excitation density is low enough the photo-generated excitons are in a bound state and are spatially well separated. They interact effectively via the Coulomb potential. As the exciton density increases, however, they ionize generating e-h pairs in scattering states. The charged carders, in turn, screen the Coulomb interaction to the point that the nonlinearity it mediates, F_,q.(2b), is eliminated, and only the nonlinearity due to Panli blocking survives. 17 Thus the time-resolved and time-integrated profiles become more s~nilar, and at very high densities the two profiles follow closely that of the laser pulse. So far however, only the amplitude of the signal has been measured. As mentioned in the introduction, the phase of the nonlinear polarization also contains information on the physics involved in the semiconductor-light interaction. The problem of phase recovery is very difficult, especially in the case of spectroscopic signals containing only a few thousands photons per pulse. It has been addressed in the context of laser diagnostics, where the notion of

200

400

600

0

200

400

~(fs)

Figure 2: For excitation just below the hh-resonance, (a) Power spectrum the FWM emission and of the laser. (2b) and (2b') intefferometric autocorrelations of the FWM and of the laser. (2c) Difference in fringe spacing between 2b and 2b'.

"spectrogram" or "time-energy" signal-representation, well known in the audio domain, has generalized to optics. 19,20 In the case of the very weak FWM signals we have found that by complementing the two amplitude measurements, discussed above, by one frequency and two correlation measurements it is possible to obtain a "timeenergy" picture of the FWM emission.21,22 The power spectrum (PS), measured at fixed At, gives all the frequencies contained in the signal, but contains no information on their time ordering. The interferometric autocorrelation (IAC), also measured at luted At, is in principle the Fourier transform of the PS. It turns out, however, that experimentally the IAC and PS are not equivalent, because of the finite windows of the measurement apparatus and the limited accuracy of the Fourier transform algorithms. The experimental PS is very accurate for the frequencies close to the resonances but inaccurate for short times, whereas the experimental IAC exhibits the opposite behavior. Furthermore IAC is an even functions of time, therefore the "time ordering" must be determined by intefferometric crosscorrelation (ICC) with a reference laser.22,22 The IAC does not give directly the phase of the FWM signal. However, by comparing the intefferometric fringe spacing of the IAC of the FWM signal to that of the IAC of a reference laser pulse (passing through the same experimental setup but just missing the sample itself) one can recover approximately the phase difference between the FWM signal and the laser. For future reference we call this the differential fringe spacing (DFS). An example of such measurement obtained at low excitation (Nx,,,3xl09cm -2) just below the lowest hh-exciton resonance, is shown in Figure 2. Figure (2a) shows the PS(w) of the FWM emission and of the laser. The FWM signal appears as a peak centered at the hh-exciton. It has, however, an asymmetric profile with a low energy tail extending well into the laser spectrum. Figure 2b and 2b', present the IAC of the FWM and of the laser as described

been

40

SEMICONDUCTOR QUANTUM WELLS

Vol.92, Nos 1-2

-~_

[o:1 . . . . . -180

At

(fs) 380

"v

~05 210

above. The FWM-IAC shows the typical "free decay" of the induced polarization significantly longer than that of the exciting laser. The DFS is shown in Figure 2c. It shows first a linear part which slope corresponds exactly to the energy difference between the main peak of the FWM emission and the central laser frequency of Fig. 2a. Then after about 250fs, the slope changes and finally flattens. This translates the fact that, within the ultrashort pulsed emission, the "instantaneous" frequency shifts toward that of the laser and finally becomes very close to iL The direction of the "time axis" is determined by an independent ICC between the laser and the FWM signal. From the combination of all this information a "time-energy" picture of the emission can be formed. At the beginning of the pulse the FWM emission is centered at the hh-exciton, then beginning around 250fs its frequency experiences a "red shift" which, brings it in coincidence with that of the laser at about 350fs and remains at this frequency for about 100fs. We have observed a hint of a further increase of frequency at late times (>450fs), but the signal is then so faint that this last tendency is unclear. Of course this measurement can be repeated at various Dt for the same total excitation. As shown in Figure 3, the results are quite surprising. The left curve gives the log of the time integrated FWM signal amplitude vs Dr. The three arrows mark the Dt at which the PS and DFS, shown on the righL were obtained. Let us note first that, clearly the time integrated amplitude contains very little information as compared to the PS and DFS. For Dt=-80fs the emission is essentially at the hh-exciton and the DFS shows a constant slope corresponding to a constant phase difference with the laser central frequency. For Dt=0 one recovers the behavior discussed above. At Dt=160fs two separate contributions appear in the PS, one centered at the hh-exciton and the other following the laser spectrum. Correspondingly the DFS takes a pronounced S-shape indicating that the emission is first centered at the laser, then at the hh-exciton, and then moves back again to the laser. In order to understand this behavior, the generalization of the SBE

i

.....

~-2x 1 0

Figure 3: Left, log of the time-integrated FWM amplitude vs At, FWM power spectra (center:.) and differential fringe spacing (right:) for the three At marked by the arrows.

: ; ,

-~ ---~0s)

~

160

240

Figure 4: Excitation of the hh and lh resonances adjusted for quantum beats, (a) power spectrum the ~ emission and of the laser, (2b) FWM interferometric autocorrelation showing the interference patterns, (2c) difference in fringe spacing showing an emission starting at the lh-exciton and experiencing a g-shift around 12Ors, (d) blow-up of fringes at the center and at the first node of the autocorrelation pattern.

described in Section II to the six spin-degenerate band structure adequate for describing the GaAs QW was solved numerically.22 These SBE represent the most sophisticated description of interactions of ultrashort pulsed fight with semiconductors presently available5. In their present form they involve, however, two approximations. Screening by eh is treated in the static single plasmon pole approximation5 and dephasing is accounted for by a constant rate, corresponding to "Lorentzlan" broadening. They account very well for the amplitude measurements and, in fact, reproduce the salient features of the PS. 22 However, a careful comparison with the experiments point out s~gnificant limitations of the theory. This is shown in Figure 3, where the calculated PS (gray curves) are compared to the experimental ones. Clearly it is impossible to account for the PS lineshape with a constant dephasing rate and that a frequency dependent rate should be used. This corresponds, in the time domain, to non-Markovian processes. Furthermore at a fixed Dt the density dependence of the PS fineshape and position with respect to the laser also shows the deficiency of the theory. This can be attributed to the statistical treatment of e-h screening.22 Furthermore, this treatment does not account for the screening by nonequilibrium exciton populations, which in the conditions of the experiments are the first to be created. Consistent theories able to describe Coulomb scattering and collisions with phonons in non-equilibrium populations have been proposed. 23,24 How eve r, applications to complicated FWM experiments are only beginning. The phase measurements are sensitive to the evolution of events occurring within a few optical cycles (in our case 2.81fs!). It is therefore not surprising that these delicate experiments reveal important informations on the scattering processes that the "amplitude only" experiments miss completely.

41

SEMICONDUCTOR QUANTUM WELLS

Vol.92, Nos 1-2

0.1 Laser

0 0.1 0

~

,

---....,_j, ,! At=5k~LS

0.I

0 At=250fs

0.1 0

,^

.-----.__

II

Ener~/(eV) 1.48

1.5

1.52

1.54

1.46

Energy (eV)

.56

Figure 5:. Normalized FWM power spectra for excitation in the continuum, 44meV above the hh-resonance, showing the blue shift relative to the laser spectrum.

Figure 6: Change in transmission induced by the laser, measured simultaneously with the FWM of Fig. (5), showing the complementary red shiftin the absorption.

The nonlinear dynamics of the coherent emission instantaneous frequency is also seen when the laser is exciting both hh and lh excitons. In this case the emission comprises of two unequal contributions centered at the two resonances, the weight of which is very density dependent. Using the procedure described above, the "t~ne-energy" picture of the emission can be reconstructed for this case as well. For excitation close to the lh-exciton and at low density the IAC envelope is substantially longer than that of the laser and exhibits a modulated profile. At the beginning, the slope of the DFS is fiat indicating that the emission occurs mainly at the lh-exciton almost in coincidence with the laser, then it becomes negative showing an increase of low frequency contribution originating from the hh-exciton. In this case, however, because of the delicate balance between the unequal contributions of the two resonances, the frequency variation is quite complicated, and does not correspond to a simple chirp. 22 The comparison of the experimental results with theory shows overall agreement. But, again, there are some signifw~aat discrepancies in the fine features. For example the theory does not reproduce correctly the llneshape, separation and relative weights of the hh and lh contributions vs Dr. Again,. these differences can be traced to the way screening and dephasing are accounted for in the theory and stress the need for an improved description. The configuration where both excitons contribute to the FWM provides an opportunity to investigate the speed of the frequency modulation. By adjusting the excitation parameters to get hh and lh contributions of roughly equal weight, full quantum beats are expected.25, 26 As shown in Figure 4b in these conditions the IAC exhibits beautiful interference patterns, with a 230fs beat period corresponding to the hh-lh splitting shown in the PS of Figure 4a. The DFS, Figure 5c, shows that the emission starts approximately at coincidence with the laser, which in this case coincides with the lh-exciton.

Then around 120fs it experiences an abrupt p-shift when it moves suddenly to the hh-exciton, where it remains until the next beat. The p-shift occurs in about 50fs. The beat "duration" is more precisely determined in Figure 4d, where a set of fringes at the center of the IAC are compared to a set of fringes close to the first node. One can actually count the number of fringes it takes to complete the p-shift. The frequency modulation is very fast, AFAr, 1.4h, juSt above the fundamental quantum limit. This result is consistent with the SBE calculation.22 The phase dynamics of the emission by unbound e-h pairs in the continuum-states, well above the band gap, is in fact easier to study because of their ultrafast dephasing. In such conditions, no interesting information is contained in the time integrated FWM signal, when the pulse width exceeds the dephasing time T 2. In contrast, the variation of the power spectrum vs time delay, PS(w,Dt) reproduces the spectrogram,19,20 and is a direct visualization of the phase dynamics of the emission frequency. This was demonstrated with quasi-instantaneous Kerr-medlafl0 For continuum excitation with densities in the range N=I010-1012 cm -2 the dephasing times in GaAs QW is of the order of a few tens of fs,27,28 and therefore this method can be readily applied to our samples. A series of PS(w,Dt) obtained for excitation 44meV above the lowest exciton and with a density N=3xl012cm -2 is shown in Figure 5. Each spectrum has been normalized to unity in order to display the dynamical behavior. The laser spectrum is shown at the top of the figure. A clear dynamical shift of PS(w,Dt) is observed. The maximum is shifted to higher energies relative to the laser spectrum at early delays (Dt<0), and it shifts to lower energies as Dt increases. The shift can be as large as A E - S m e V . This temporal behavior is density dependent. We attribute the above observations to manybody effects that renormalize the optical response of the non-equilibrium e-h Fermi-sea created by the intense

42

SEMICONDUCTOR QUANTUM WELLS

pulse. 29 This renormalization originates mostly from the Fermi-sea excitations with very small energy and is therefore concentrated at its two edges. For equilibrium distributions of electrons this corresponds to the well known Fermi-edge singularity. 6 In the case of optically generated non-equilibrium distributions, it produces an enhanced emission at the high energy edge and a reduced emission at the low energy one, resulting in the observed dynamical shift. This "blue shifted" emission should be the counterpart of the "red shifted" spectral hole burning observed in pumpprobe experiments. 30,31 In order to check this we have measured the change of transmission induced on the weak beam-1 by the beam-2. As seen on Figure 6, simultaneously with the FWM emission, the pulses-I experience an increased absorption on the high energy edge and a reduced absorption on the low energy edge, induced by the pulses-2, thus confLrming the interpretation. IV. Conclusion

We have shown that retrieving the phase of the electro-magnetic field emitted by semiconductor QWs

Voi.92, Nos 1-2

provides important and new information on the behavior of electronic processes on a very short time scale. We have compared our experiments to a theoretical analysis of the experiments based on the Semiconductor Bloch Equations and a six band model of a GaAs QW. We have found that although the theory reproduces the salient fealares of the experimental data, it fails to account for some of its important aspects. These discrepancies can be traced to two approximations in the theory: the statistical treatment of screening, and the Lorentaiao description of dephasing. In both cases this indicates a clear limitation of the SBE approach and the need for theoretical refinements requiring a microscopic description of relaxation and corresponding dephasing processes in a non-Markovian theory. Acknowledgements: This work was performed in collaboration with J.Y. Bigot, M-A Mycek, S. Weiss, and W. Schiller. It was supported by the Director, Office of EnergyReseatch, Office of Basic Energy Sciences, Division of Materials Sciencesof the US Department of Energy, under Contract No. DE-AC03-76SF00098.

REFERENCES

1. See for example several articles of this special issue. 2. S. Schmitt-Rink, D.S. Chemla, D.A.B. Miller, Adv. Phys., 38, 89 (1989). 3. See for example: "Optics of Semiconductor Nanostructures" F. Hermeberger, S. Schmitt-Rink E.O. G0bel, ed, Akademie Verlag, Berlin (1993), "Quantum Theory of the Optical and electronic Properties of Semiconductors" H. Haug and S.W. Koch, World Scientific Singapore (1990), and "Optical Nonlinearities and Instabilities in Semiconductors" H. Haug od, Academic Press N.Y. (1988). 4. Schmitt-Rink, D.S. Chemla, Phys. Rev. LetL 57, 2752 (1986); andS. Sehmiu-Rink, D.S. Chemla, H. Haug, Phys. Rev. B37, 941 (1988). 5. For a recent review and updated references list see: W. Schafer,"Manybody Effetcs in Nonlinear Optics of Semiconductor Structures" in "Optics of Semiconductor Nanostructures" F. Henneberger, S. Sehmitt-Rink E.O. GObel, ed., Akademie Verlag, Berlin (1993). 6. I.E. Ferakis and Y.-C Chang, Phys. Rev. B 43, 12556 (1991), B 4,1, 5877 (1991) and B 47, 6573 (1993). 7. M.S. Skoinick, J.M. Rorison, K.J. Nash, D.J. Mowbray, P.R. Tapster, S.J. Bass, and A.D. Pitt, Phys. Rev. Lett. 58, 2130 (1987). and, G. Livescu, D.A.B. Miller, D.S. Chemla, M. Ramaswany, T.Y. Chang, N. Sauer, A.C. Gossard and J.H. English, IEEE J. Quant. Electron. 24, 1677 (1988). 8. See for example, D.S. Chemla Physics Today 46, 46 (1993). 9. See for example: L. Allen and J.H. Eberly, "Optical Resonancesand Two Level Atoms" (Wiley, New York 1975). 10. T. Yajima,and Yoichi Talra, J. Phys. Soc. Jap. 47, 1620 (1979).

11. L. Schultheis, M.D. Sturge, and J. Hegarty,Appl. Phys. LetL 47, 995 (1985). 12. L. sehultheis, J. Kuhl, A. Honold, C.W. Tu, Phys. Lett. 55, 1635, (1986) and Phys. LetL 55, 1797 (1986). 13. K. Leo, M. Wegener, J. Shah, D.S. Chemla, E.O. G0bel, T.C. Damen, S. sehmiu-Rink, W. sehafer, Phys. Rev. Lett. 65, 1340 (1990). 14. M. Wegener, D.S. Chemla, S. sehmitt-Rink, W. sehllfer Phys. Rev A42, 5675 (1990). 15. S. Schmitt-Rink, S. Mukamel, K. Leo, J. Shah, and D.S. Chemla, Phys. Rev. A 44, 2124 (1991). 16. M-A Mycek, S. Weiss, J-Y. Bigot, S. sehmitt-Rink, D.S. Chemla and W. Sch~er, Appl. Phys. Lett. 60, 2666 (1992). 17. S. Weiss, M-A Mycek, J-Y. Bigot, S. sehmitt-Rink, D.S. Chemla Phys. Rev. Lett. 69, 2685 (1992). 18. D-S. Kim, J. Shah, T.C. Damen, W. sehafer, F. Jahnke, S. sehmitt-Rink, K. Kohler, Phys. Rev. LetL 69, 2725 (1992). 19. J. Paye, IEEE Journ. Quant. Elect. 28, 2262 (1992). 20. D.J. Kane, R. Trebino, IEEE Joum. Quantum Elect., 29, 571, (1993). 21. J.-Y. Bigot, M.-A. Mycek, S. Weiss, R. G. 01brich, D.S. Chemla, Phys. Rev. Lett. 70, 3307 (1993). 22. D.S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, and W. Schiller submitted to Phys. Rev (1994). 23. M. Hartman and W. Schafer, Phys. Sial Sol. b173, 165 (1992) 24. D.B. Tran Thoi and H. Haug, Z. Phys. B91,199 (1993) 25. K. Leo, J. Shah, E.O. G0bel, T.C. Damen, S. sehmittRink, W. seh~er, J. Muller, K. KC~hler,Phys. Rev. LetL 66, 201 (1991) 26. M. Koch, J. Feldman, G. yon Plessen, E.O.GObel, P. Thomas, K. K0hler, Phys. Rev. Lett. 69, 3631 (1992)

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27. P.C. Becker, H.L. Fragnito, C.H. Brito-Cruz, R.L. Fork, J.E. Omningham, I.E. Henry, C.V. Shank, Phys. Rev. l~tL 61, 1647 (1988). 28. J.-Y. Bigot, M.T. Port~lla, R.W. Schocnlein, J.EI Cunningham, C.V. Shank, Phys. Rev. Lett. 67, 636 (1991) 29. C. Tanguy, M. Combescot, Phys. Rev. LetL 68, 1935 (1992)

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30. J.-P. Foing, D. Hulin, M. Joffre,M.K. Jackson, J.-L. Oudar, C. Tanguy, M. Combe,scot,Phys. Rcv. LeU. 68, II0 (1992) 31. W.H. Knox, C. Hirlimann, D.A.B. Miller, J. Shah, D.S. Chemla, C.V. Shank, Phys. Rev. Lett. $6, 1191 0986)