Coherent Control of Light Absorption and Dynamics in …downloads.hindawi.com/journals/vlsi/1998/032057.pdf · 2018. 11. 13. · 206 W.PITZANDX. HU duration ofthe MWfield are adjusted

VLSI DESIGN1998, Vol. 8, Nos. (1-4), pp. 203-207Reprints available directly frem the publisherPhotocopying permitted by license only

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Coherent Control of Light Absorption and CarrierDynamics in Semiconductor Nanostructures

WALTER P(TZ* and XUEDONG HU

University of Illinois at Chicago, Physics Department, Chicago, IL 60607, USA

We present two examples of coherent control of inter(sub)band transitions in asemiconductor double well by coherent light sources. Accounting for the upper holesubband and two lowest electron subbands, a microscopic theoretical analysis showsthat electron-hole pair generation by a sub-picosecond pump pulse can be controlled bythe intensity and the phase of a dc microwave field which resonantly couples the twoelectron subbands. Light absorption can be either enhanced or reduced. Secondly, it isshown that proper combination of two pulsed laser fields allows control of electroninter(sub)band transitions and final-state population, i.e., the formation of indirectversus direct excitons.

Keywords: Coherent control, semiconductor, quantum well, optics, theory

Coherent control of final-state population andchemical reactions has long been pursued inatomic and molecular physics [1-5]. More re-cently, improvements in ultrafast spectroscopyhave allowed induction and observation of coher-ent phenomena in semiconductors, in form ofcoherent charge oscillations in double wells, Blochoscillations, and coherent control of photocurrent,to name some of the highlights of progress inrecent years [6-10]. In this paper we investigatecoherent control of inter(sub)band transitions,absorption, and final-state population theoreti-cally. Specifically we discuss a scheme whichallows phase controlled light absorption and

control of final-state population in semiconductorheterostructures by the interplay of two coherentlight sources. As a specific example we give resultsfor an asymmetric 145 /25 /100, GaAs-A1-GaAs-GaAs double well whose electron subbandsplitting is controlled by a static electric field andwhich is exposed to a tunable sub-picosecondpump pulse which generates electron-hole pairs(excitons) across its main energy gap. Calculationsare done within the framework of a microscopictheory in form of Boltzmann-Bloch equationswhich account for the carrier-carrier Coulombinteraction [11]. In the present study, subbandsplittings and time scales are such that LO phonon

* Corresponding author.

203

204 W. P(3TZ AND X. HU

scattering may be neglected. Light pulses are ofGaussian shape and the light-matter coupling istreated for a classical light field including thecounter-rotating part. The peak of the pump pulsearrives at time zero, relative to which we define thephase q of the microwave (MW) field /MW(t)=ffo(t)COS(WMWt + b). A comparison to the rotat-ing-wave approximation (RWA), which has beenused in an earlier study, is made [12].When a coherent dc MW field resonantly

couples the two electron subbands, the systemundergoes Rabi oscillations between its uncoupledeigenstates when originally prepared in one of thelatter. In the situation depicted in Figure 1, thedipole matrix element between hole I1) and upperelectron subband 12) is much stronger than holeand lower electron subband 13). Hence, if a pumppulse is applied at resonance with the directexciton maximum absorption is obtained. How-ever, application of a MW field reduces theadmixture of left-well eigenfunction ]W) in 12)and transfers it to 13). Hence, the MW fieldreduces absorption at the direct exciton peak [13].Conversely, if the pump pulse is tuned near theindirect exciton, application ofaMWfield enhances

absorption. The phase of the coherent dc MW fieldenters the coupling between the electron subbandsand hence the complex electron interband polariza-tion. When the pump pulse duration is shorter thanthe inverse of the MW-induced Rabi frequency, thephase influences the absorption process from a thirdlevel, here, the top hole subband I1). This isdemonstrated in Figure 2 for the case where a 80fs pump pulse of Gaussian profile is tuned near

resonance with the indirect excitons of the DW witha subband splitting of 20 meV. The MW intensity isabout 2 MWcm-2, corresponding to a Rabi periodnear 100 fs. Clearly, the presence of the MW fieldincreases absorption by about 15 percent for phase7r/2. This effect is more pronounced at the directexciton peak [13] The RWA underestimates theimportance of second harmonics in the carrierdynamics induced by the MW field. Therefore,shorter pump pulses than predicted by within theRWA may be required to display the phasedependence of absorption.For the present three-level system and within the

RWA for transitions between conduction andvalence bands, the pump pulse photon Boltzmannequation in the presence of coherence in the carrier

Iw>?’11._.___ MWfield

13>

pulsepump

I1 >

1

FIGURE Illustration of a biased asymmetric GaAIAs-GaAs double well. I1), 12) and 13) denote double-welleigenstates; [W) and IN) are the lowest wide-well andnarrow-well eigenstates, respectively. The shape of wavefunctions is indicated by thin solid lines on top of thecorresponding energy levels.

1.4

1.2

o 1.0

0.

0.6

0.4

0.2

Number of e-h Pairs versus Time(subbands 20 mev off resonance, laser 20meV below direct exciton line)

0=90

No

0.0 200.0Time (fs)

0.0-200.0 400.0

FIGURE 2 Density of photo-generated electron-hole pairs asa function of time and MW phase for a pump pulse photonenergy which is 20 meV below the direct exciton peak.

COHERENT CONTROL IN SEMICONDUCTOR NANOSTRUCTURES 205

system is of the structure [11].

dN(q, t) 27rdt h- Z M,(q)M

k,a,a 2,3

{f, (k, t)(1-fl(k, t))(N(q, t)+ 1)

-fl(k, t)(1-f,(k,t))N(q, t)},

27rN(q, t) Z MI4,(q)MH(q)6F

k,a,a 2,3

{f,(k, t)-f(k, t)), for N(q, t)>> 1,

where f,(k, t) are carrier distribution functions,for a a, and interband polarizations, for a a.N(q,t) is the photon occupation number andM,(q) is the matrix element for coupling sub-band a to subband a via photon q. 6e is theappropriate energy-conserving delta function. Thisequation shows that electron interband polariza-tion influences photon absorption. Figure 3, inturn, shows how the phase of the MW fielddetermines the sign of the real part of the electroninterband polarization during the presence of thepump pulse to allow coherent control of theabsorption process.A possible scheme to control the final-state

population in semiconductors resembles previouswork on atoms for which adiabatic switching hasbeen demonstrated to allow transitions betweenmolecular levels which are dipole-forbidden by

0.5

-o.1

-0.13

"’.4o.o .2o.o 0.0 20.0 40.0Time (fs)

FIGURE 3 Real part of lW IN) interband polarization asa function of time and MW phase.

means of optical coupling to a third level [4, 5].In this scheme, which is sketched in Figure 1, twotemporally and spatially overlapping light pulsescouple three levels of the system. The initial stateof the system is I1), the desired final state is 13),and the intermediate state is 12). If the light pulsecoupling 12) and 13), with amplitude a(t) andduration TMW, arrives and ends before arrival and,respectively, end of the pulse coupling I1) andwith amplitude b(t) and duration Tp, and the pulseamplitudes change sufficiently slowly (relative tothe characteristic frequencies of the driven three-level system), stimulated Raman adiabatic passage(STIRAP) occurs and the system undergoes anadiabatic transition from level I1) to level 13). [4]The condition for an adiabatic transition forT= Tp= TMwis

where 6 is the light field detuning.However, in semiconductors resonant coupling

between subbands calls for MW pulses, which maybe difficult to generate at sufficiently high intensity.Moreover, due to characteristic decoherencingtimes for (free) carriers in semiconductors passagetimes can not be much longer than a picosecond.Therefore, the STIRAP process may be imprac-tical, if not impossible, to be adopted to semi-conductor nanostructures. Hence, we haveinvestigated sub-picosecond nonadiabiatic transferbased on the three-subband scheme in Figure 1,where subband I1) is the top hole subband of aheterostructure, [2) and [3) are the two lowestelectron subbands separated by 25 meV, in thepresent case. A 100fs subpicosecond pump pulse isused to generate electron-hole pairs (direct ex-

citons) associated with levels 11) and 12). Itsduration must be shorter than the inverse Rabifrequency between subband 12) and 13) which isestablished by a concurrent MW pulse. Forexcitation densities of about 101 carriers per cm2

this ensures MW-induced charge oscillationsbetween subband 12) and 13). Intensity and

206 W. PITZ AND X. HU

duration of the MW field are adjusted so that thelatter permits one half of a Rabi oscillationbetween 12) and [3), such that the electrons gettrapped in state 13) ( IN)) after a single tunnelingprocess. Here, the MW pulse resonantly couplesthe two electron subbands ( 0) and has aduration of TMW 320 fs, corresponding to abouttwo MW periods, and a peak intensity of aboutMW cm-2. Zero time delay between the two pulsesand phase r of the MW field relative to the peak ofthe pump pulse were found to give best results.

In Figure 4 we show the number of electrons inthe left well versus the number of electrons in theright well, and the total number of carriers (holes).Calculations including the counter-rotating fieldcontributions, thick lines, are compared to thosewithin the RWA, thin lines. It is clearly evidentthat the presence of the MW field reverses thetendency for direct (solid lines) versus indirectexciton (dot-dashed lines) formation. WithoutMW field and owing to the shape of wave

functions, predominantly direct excitons areformed initially. Figure 4 also shows that theRWA gives almost quantitatively correct resultsfor the final state population in the present case.However, it predicts simple (damped) harmoniccharge oscillations, whereas the full calculationshows a more complicated dynamics.

2.0

1.0(D

0.5o

0.0-200.0 300.0

Time (fs)

FIGURE 4 Carrier densities in the double well versus time.Solid lines: wide well (direct excitons); dot-dashed lines: narrowwell (indirect excitons); dashed lines: holes (total number ofexcitons); thick lines: with microwave field (phase r) regularlines: no MW field; thin lines: MW field in RWA approxima-tion.

In summary, we have given theoretical resultswhich indicate that coherent control of intersub-band transitions on a subpicosecond time scale ispossible by means of coherent light sources. Inparticular, we have investigated coherent controlof light absorption and final-state population insemiconductor double wells. It should be pointedout that the latter should also be achievable byinterference between single and triple photonabsorption, in analogy to experiments on mole-cules [3], and coherent current control via inter-ference between single and two-photon absorption[8]. Details of these findings will be publishedelsewhere [12]. The rotating-wave approximation,which allows analytic solution of the coupledthree-level system, is found to give nearly quanti-tatively correct results for the non-adiabatic final-state-control processes. However, for the coherentcontrol process of absorption discussed here, itmerely gives qualitatively correct results. Specifi-cally, it fails to give an accurate account of chargeoscillations induced by the MW field and, conse-quently, the correct value for the phase of optimalcoupling.

Acknowledgements

We thank Prof. R. J. Gordon and Prof. W. A.Schroeder for helpful discussions. This work hasbeen supported by the US Army Research Office.

References

[1] Tannor, D. J. and Rice, S. A. (1988). Adv. Chem. Phys.,70, 441.

[2] Brumer, P. and Shapiro, M. (1989). Aeets. Chem. Res.,22, 407.

[3] Lu, S., Park, S. M., Xie, Y. and Gordon, R. J. (1992).J. Chem. Phys., 96, 6613.

[4] Oreg, J., Hioe, F. T. and Eberly, J. H. (1994). Phys. Rev.A, 29, 690.

[5] Gaubatz, U., Rudecki, P., Schiemann, S. and Bergmann,K. (1990). J. Chem. Phys., 92, 5363; Bergmann, K. andShore, B. W. (1995). in Molecular Dynamics and Spectro-scopy by Stimulated Emission Pumping, Eds. H.-L. Daiand R. W. Field (World Scientific, Singapore), p. 315.

[6] Heberle, A. P., Baumberg, J. J. and K6hler, K. (1995).Phys. Rev. Lett., 75, 2598.

COHERENT CONTROL IN SEMICONDUCTOR NANOSTRUCTURES 207

[7] Citrin, D. S. (1996). Phys. Rev. Lett., 77, 4596.[8] Hach, A., Kostoulas, Y., Atanasov, R., Hughes, J. L. P.,

Sipe, J. E. and van Driel, H. M. (1997). Phys. Rev. Lett.,78, 306.

[9] Leo, K., Shah, J., G6bel, E. O., Damen, T. C., Schmitt-Rink, S., Sch/ifer, W. and K6hler, K. (1991). Phys. Rev.Lett., 66, 201.

[10] Waschke, C., Roskos, H. G., Schwedler, R., Leo, K.,Kurz, H. and K6hler, K. (1993). Phys. Rev. Lett., 70,3319.

[11] P/Stz, W. (1996). Phys. Rev. B, 54, 5647; P6tz, W. (1996).Appl. Phys. Lett., 68, 2553; P6tz, W. and Hohenester, U.,unpublished.

[12] P6tz, W. (1997). Appl. Phys. Lett., 71, 395.[13] P6tz, W. (1997). Phys. Rev. Lett., 79, 3262.

Authors’ Biographies

Walter P6tz got his Ph.D. from the University ofGraz, Austria, in 1982. His research area is thetheory of semiconductors, with emphasis on opticaland transport phenomena in nanostructures.Xuedong Hu received his Ph.D. from the

University of Michigan, Ann Arbor, in 1996. Hisresearch is in the area of squeezed phonon statesand, more recently, coherent phenomena insemiconductors.

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Coherent Control of Light Absorption and Dynamics in …downloads.hindawi.com/journals/vlsi/1998/032057.pdf · 2018. 11. 13. · 206 W.PITZANDX. HU duration ofthe MWfield are adjusted