theory of coherent transients in semiconductor pump—probe spectroscopy

Vol. 5, No. 1/January 1988/J. Opt. Soc. Am. B 139

Theory of coherent transients in semiconductor pump-probespectroscopy

M. Lindberg and S. W. Koch

Optical Sciences Center and Department of Physics, University of Arizona, Tucson, Arizona 85721

Received June 5, 1987; accepted August 26, 1987

Probe-transmission spectra are calculated for a semiconductor that is excited by ultrashort pulses well above itsband edge. Coherent coupling of the light field to the band-to-band transitions is assumed, neglecting the mixing ofdifferent momentum states. The transmission spectra exhibit pronounced oscillatory structures caused by theinteraction of the probe pulse with the medium polarization, which is driven by the pump pulse. The details of theoscillations are determined by the temporal separation between pump and probe pulses and by the total temporalwidth of the pump pulse. Characteristic modifications of the spectra are obtained for strong pump fields (ir and 27rpulses). Possible applications of the theoretical results for the determination of intraband scattering times insemiconductors are discussed.

1. INTRODUCTION

Until recently, time-resolved optical studies of electron-holeinterband transitions in semiconductors were restricted toincoherent processes because of the rapid intraband scatter-ing. This situation has changed because of the developmentof techniques to produce ultrashort pulses with durations ofonly a few femtoseconds. Spectroscopy with these pulsesprovides a possibility to investigate semiconductors in thevery early states after the excitation, when the coherentproperties are still at least partially present.'

The dynamic evolution of laser-excited band-to-bandtransitions takes place in essentially three stages. In thefirst stage-the collision-free or coherent regime-differentk states react independently to the exciting light. The lightfield introduces a coherent coupling between valence-bandstates and conduction-band states, thus driving an oscillat-ing polarization in the matter. The semiconductor elec-trons exhibit Rabi flops between the respective states in thevalence and the conduction bands not unlike the coherentelectronic excitations in two-level-atom spectroscopy. 2 Nok mixing occurs on time scales shorter than the Coulombscattering time.

In the second stage of the band-to-band-transition dy-namics, phase relaxation has occurred, and the electrons andholes are described by nonthermal distributions, which arelocally created in k space (spectral hole around the centerfrequency of the excitation pulse). Electrons and holes aresubject to Coulomb scattering and to the interaction withphonons. Both interaction mechanisms redistribute theelectronic excitations toward the extrema of their respectivebands. The optical excitation process acts as a local sourceof new electron-hole pairs. When the bands are filled up tothe resonant k states, the transition is bleached, and no morelight is absorbed.

In the third stage, a quasi-thermal equilibrium is reachedwhen electron-hole pairs are created by the pump beam atthe same rate as they recombine radiatively or nonradiative-ly. This hydrodynamic quasi-equilibrium regime has beenstudied in great detail, both experimentally and theoretical-

ly. 3,4 The optical semiconductor spectra in the band-gap

region are dominated by Coulomb and phase-space-fillingeffects that cause a variety of optical nonlinearities.

Experimentally, the dynamics of the band-to-band-tran-sition processes are often investigated using pump-probespectroscopy. By using femtosecond pulses, we can nowstudy the first coherent regime. To obtain a clear under-standing of the anticipated features and to extract informa-tion on decay times, one needs a theory for the coherentlight-matter interaction that is capable of separating theinfluence of the different relaxation mechanisms. In thispaper, we develop a description of the modifications oftransmission spectra that are expected in an ultrafastpump-probe experiment. In such experiments, the changesof the pulse envelopes are so fast that their spectral charac-teristics become important. In our theory, we assume thatpump and probe pulses are mutually coherent.

In Section 2 of this paper we introduce a model withcoupled equations for the classical, slowly varying envelopesof the fields and quantum-mechanical density-matrix equa-tions to determine the material polarization. The suscepti-bility seen by the probe is derived in Section 3 for arbitrarypump-pulse shapes by using linear response to the probefield. In Section 4 we discuss the small-pump-intensitylimit and give analytic results using two different shapes forthe pulse envelope. Transmission spectra and so-called dif-ferential transmission spectra are evaluated and plotted.When the pulses overlap in time, but the maximum of theprobe pulse precedes that of the pump pulse, our theorypredicts an oscillatory structure in the transmission spectrathat has a period inversely proportional to the time delaybetween the pulses. Reversing the ordering between pumpand probe, that is, having the pump preceding the probe,eliminates the oscillations. Structures strongly resemblingour theoretical results have already been seen in experi-ments with organic materials5 6 and, more recently, also withthe direct-gap semiconductor CdSe7 and semiconductor mi-crocrystallite-doped glasses.

For strong pump pulses our theory predicts interestingmodifications of the transmission spectra. These modifica-

0740-3224/88/010139-08$02.00 © 1988 Optical Society of America


140 J. Opt. Soc. Am. B/Vol. 5, No. 1/January 1988

tions are discussed in Section 5 of this paper. An exactanalytic solution for the hyperbolic-secant pulse is used.We then proceed to evaluate transmission spectra for caseswhere the pulse area is equal to r or 27r. The paper closeswith a discussion of our results in Section 6. Here we alsoanalyze how deviations from the ideal coherent spectra maybe used to obtain estimates of the k-state-relaxation rates.

2. MODEL

In this section, we develop a simple model to calculate thetransmission spectrum of a weak probe pulse after it hasinteracted with the polarization of the medium that is excit-ed by the pump pulse. Our treatment is semiclassical.Pump and probe fields are determined by using Maxwell'sequation, and the polarization induced in the matter is de-scribed quantum mechanically.

We use the macroscopic Maxwell equations without exter-nal currents or charges in the form

V X E =- - B,at

V D 0,

VXB= 2-D,EO2at

V B=, (1)

where B is the magnetic field and 0 is the dielectric constantof the vacuum. The polarization field P and the electricfield E are related to the displacement field D by

D = OE + P. (2)

Neglecting all surface charges, we set V P = 0 and obtainthe inhomogeneous wave equation

V2E- 2 E = 2 P (3)c 2

EOC2 t

Because Eq. (3) is explicitly linear in the fields, we candescribe each spatial oscillation component independently.We make the usual approximation that E and P can bedecomposed into rapidly oscillating terms with slowly vary-ing envelopes

as the equation for the amplitudes. Furthermore, we ne-glect all pulse-propagation effects by ignoring the secondterm on the left-hand side. Denoting by xp the coordinate inthe direction of the wave vector kp, we obtain the familiarfield equation

a Ep (t) = p Pp(t),axp 2 c0 0

(7)

which we use to determine the transmission spectrum of thetest beam.

Because of the extremely fast phase-destroying interac-tion processes in laser-excited semiconductors, we limit ourtheoretical treatment to the case of ultrafast spectroscopy,in which experiments are done using subpicosecond pulses.In this case the theoretical treatment is simplified by assum-ing that the many-body effects-in contrast to a quasi-equi-librium situation-play only a minor role in determining theoptical properties. Under these conditions an independentelectron-hole model becomes suitable to describe the inter-action between semiconductor and light field. The totalinduced polarization is then given by

Ptot = E P, (8)

where the sum runs over k and P is the polarization of theelectron-hole pair with wave vector k. We use the semiclas-sical description, in which the material part is calculatedquantum mechanically with classical external time-depen-dent fields. The induced polarization of a single electron-hole pair is given by the quantum-mechanical expectationvalue

P(t) = (er). (9)

Each k state is described by a two-level system that is dipole-coupled to the external scalar field. Polarization effects ofthe field are neglected. Denoting the valence-band statewith 1 and the conduction-band state with 2, we can writethe expectation value as

P(t) = At Tr[(P12 + P2 )p(t)]- (10)

P12 and P21 are the quantum-mechanical projectors (in Diracnotation)

Ep(t) = EP(t)exp[i(kp r - Opt)] + c.c.

and

Pp(t) = Pp(t)exp[i(kp r - pt)] + c.c. (5) P2 1 = 12) (11 = P12 t.

The scalar fields are projections into the polarization direc-tion of the incoming probe beam. The slowly varying enve-lope approximation approaches the limits of its validity forthe case of femtosecond pulses with optical frequencies.

For this paper, we are interested only in the absorption ofa weak probe beam in the presence of a stronger pump beam.The attenuation of the pump beam is neglected. In theslowly varying amplitude approximation we obtain, fromEq. (3),

(kp V)Ep + 2P Ep - 2 PP (6)c 2 at - 2eoc 2 p

In the Schrodinger picture, the equation of motion for thedensity matrix has the form (we use the convention h = 1throughout this paper)

i ap = 00, PM]. (13)

Generally, for pump-probe experiments one can subdividethe Hamiltonian into two parts, one of which is small (-probe field). Making the rotating-wave approximation,that is, transforming to the rotating frame and neglectingthe nonresonant terms, we obtain for the nonperturbativepart of the Hamiltonian

(4)

and

P 1 2 = 1)(21 (11)

(12)



Fo (t) = 2 - PO

- A4EL(t)exp[i(kL- r - Lt)]P21 + h c.. (14)

The k dependence of E is

E Eg + 2m+ 2Mh k (15)

Unless explicitly necessary, we always suppress the explicitk dependence in the list of arguments. The operators P1and P2 are the diagonal projectors into the states 1 and 2.The perturbative part of the Hamiltonian is given by

Hl(t) = -{Ep(t)exp[i(kP r - Qpt)]P21 + h.c4.) (16)

3. SUSCEPTIBILITY

We assume that our system is prepared at the time to into thestate p(to). We denote by U(t, to) the matrix that is thesolution of the equation

i a U(t, to) = 170(t)U(t, to), (17)at

an integrodifferential equation. However, if we assume avery short probe pulse with a broad spectrum, we can simpli-fy the resulting equations considerably. Under the integralon the right-hand side of Eq. (20), we assume that Ep(t, xp) issharply peaked around t tp, where tp is the temporalseparation between the peaks of the probe and pump pulses.(We always take the peak of the pump pulse as origin of time,t = 0.) Because we have neglected the attenuation of thepump beam, we can define the probe susceptibility xthrough the relation

Pp(o, x,) = x(w)E(w, x,),

yielding

X((o) = i 2J| dt exp[iw(t - t)](t - tp)o(tp - to)

(23)

(24)

This formula describes the response of one single electron-hole pair. The total response of the semiconductor is ob-tained by summing over all pairs. Assuming isotropy in kspace, we obtain for the susceptibility (x) of the total system

with the initial condition

U(to, to) = 1. (18)

Note that 17o(t) is independent of to. With the matrix U(t,to) we can solve Eq. (13) in first order in Al. The result is

p(t) = U(t, t)p(to)U(t, to)-'t

-iU(t, to) | dt'[U(t', to)-i~qW)U4ttS to) (to)]

X U(t, to)-' + O(H12). (19)

The first term on the right-hand side of Eq. (19) is indepen-dent of the probe field. It does not contribute to the probesusceptibility. For simplicity of notation, we therefore sup-press this term from now on. Inserting fH(t) into Eq. (19)and neglecting the backward-traveling wave, we obtain forPp(t) the expression

Pp(t) = i 2 exp(iQpt)Tr{U(t t)f 1P12U(t, to)

X | dt'EpWt)exp(-iQPt')to

x [U(t', t 0)YP 21 U(t', to), P(to)] 0(t - to).

(20)

To obtain the probe-transmission spectrum we take theFourier transforms

Ep(w, xp) = J dt eiwtEp(t, xp)exp(-ipt), (21)

P(o, xp) = dteiwtPp(t, xp)exp(-iQpt)_ f

(22)

(x(Wo)) = f dkk2XW. (25)

Using Eqs. (25) and (7), we see that the absorption coeffi-cient at any probe frequency c is given by

a = Im[P (xO) (26)

At this point we could introduce incoherent damping forthe electron-hole pairs. We would then assume that eachpair is generated at some time to and that it has only a finiteprobability to survive from to to t. Choosing an exponentialdecay rate characterized by the decay constant -y and averag-ing over all generation times to, we obtain an expression forthe averaged response in the form

X (')ave = iA2' | dto exp[--y(tp - to)]

X dt exp[(iw - y)(t - tp)]Tr1U(t, t)Y1 P12tp

X U(t, to) [U(tp, to) - P21 U(tp, to) p (to)]} (27)

Because we want to keep the discussion of damping effects aseparate issue, in the following expression we take the limit 'y- 0. One can verify that

r~p

lim J dto exp[--y(tp - to)]F(to) = F(--) (28)

if the limit is finite. Hence y - 0 for (X(°))ave, obtainedfrom Eq. (27), is equivalent to taking the limit to - - in(x(o)), Eq. (25). This way we rather naturally obtain aconvergence-generating factor

%X(W) = i 2 j dt exp[(iw - y)(t - tp)]tp

and insert these expressions into Eq. (20). From Eqs. (20)and (7) we see that the most general situation is described by (29)


X TrJU(t, t0)_1P12U(t1 tO)[U(tpl t0)_1P21

X U(tp) to), P(to)II.

X Mt, - -), P (- -) I 1,


where y - 0 after the integration. Note, however, that ingoing from Eq. (27) to (29) the limit to -O - must be takenafter possible to-dependent phase factors are canceled.

We can use some general properties of U(t, to) to simplifyexpression (29). In our problem, U(t, to) is a 2 X 2 matrix.Both columns in U(t, to) must be solutions of the vectorequation

i a u = Ro(t)u,at (30)

where u is a two-component vector. Because of the hermiti-city of fo(t), the norm of the vector is constant in time and istaken as unity. Now let u be a solution with the initialconditions

uj(to, to) = 1,

u2(to, to) = 0.

Ca , P11() - P22(')IE=@,-

This result is sensible because the pump pulse alone does notcreate polarization components traveling in the probe direc-tion. After the pump is gone, the population inversion doesnot change any more, and the probe sees the final distribu-tion created at the respective probe frequency w.

4. LIMIT OF SMALL PUMP INTENSITIES

The pump intensity can be taken as small if the changes of uand v during the interaction time are small. We then canestimate

Au= dtu = J_ dtiEL*(t)v(t)

(31)

Because we have chosen the zero of the energy so thatTr[Ro(t)] = 0, the matrix U(t, to) is given by

(t 0t) -u 2*(t, t) 1U(t [U2 (t t0) u 1 *(tt 0) (32)

For convenience we write

ul(t, to) = exp[iE(t - to)/2]u(t, to),

u2 (t, to) = exp[iE(t - t)/2exp[i(kL - r - QLt)]v(t, to).

(33)

Inserting Eq. (32) into Eq. (31) and assuming that there areno initial coherences, we obtain

x(W) = i 2 j dt exp[(iw - y)(t - t)]p

X exp[-iE(t - tp)]u*(t)2u(tp) 2

- exp[i(E - 2QL)(t - tP)]V(t)2V*(tP)2l

(37)

This expression means that the pump-pulse area must besmall compared to unity. A crude estimate of the area of thepulse would be

(38)JdtIEL(t)I ELmax X AtpuIse,

In the small-intensity regime we can drop the term v4 inEq. (34). For convenience we assume that p1 1(-o) = 1 andP22(-c) = 0, that is, only the valence band is populated withelectrons at t = -. We then solve Eqs. (35) and (36) insecond order in the pump amplitude. As a result, we obtainthe changes in x(X) caused by the presence of the pump inthe form

6 X(v) = X(°) - X()no pump

_-i2y' f dt exp[i(w - E) - y]tlX [P1 1(-o) - P22( -m)].

The functions u(t) and v(t) obey the equations of motion

i d u = -EL*(t)v,at

i v = (E -QL)v -AtELMtU,atLX {| dt' | dt" gEL(t')AEL*(t')

(35)

(36)

with the initial conditions

u(--) = 1,v(-c) = 0.

The calculation of the probe absorption is hence reduced tosolving the coupled differential equations (35) and (36) for agiven excitation pulse. Because x(w) [Eq. (34)] is not givenin terms of 1u12 and Iv12, not only population effects deter-mine the spectra. From the formal solution of Eqs. (35) and(36) we see that the second term in Eq. (34) is essentiallyproportional to the fourth power of the pump-field ampli-tude and therefore will be suppressed for low pump intensi-ties. Also, this term is small for short interaction timesbecause v(-c) = 0.

When the two pulses do not overlap, that is, E(t) = 0 for t> tp, we can use Eq. (34) to show that the absorption a [Eq.(26)] is given by

X exp[i(E - QL)(t'- ti)

+ | dt' dt" /AEL*(t'),4EL(t')

X exp[-i(E - QL)(t' - t")]}- (39)

Now we evaluate the summation over the k states. Becausewe are interested mainly in the coherent coupling of the lightto interband transitions where the excitation frequency QL ishigh above the band edge, that is, QL - Eg >> the bandwidthof the pump laser, we can simplify the k integration byassuming a frequency-independent density of states, D(E) D.

X k2 5X(.) 27 -2 d @ - D dEx(w).7r20 2r2 - (40)

This approximation is quite realistic for most semiconduc-tors well above the band edge, and it allows us to evaluatemost parts of the integrations analytically. We obtain fromEq. (31)

(34)


<,v f - dt1EL(01 << 1-


5(x(c)) = -i2Df 2 2r2 J dt expl[i(cto - QL) -

X f dt',tEt(t + tp- t')pEL*(tp -t'). (41)

Hence, for not-too-strong pump pulses, the change of X isgiven by the Laplace transform of the autocorrelation func-tion of the pump-laser field.

There are some general features of b(X(a)) in the weak-field limit, which follow directly from Eq. (41). First consid-er the case that the probe and pump pulses do not overlap intime. If the probe comes before the pump, that is, EL(t) = 0for t < tp, we see that 6(x(') ) is zero. The probe transmis-sion is not influenced by the changes caused by the pump.If, however, the pump comes first, that is, EL(t) = 0 for t > tp,we obtain the change in absorption (TIm x) as

5e(co) -IF(,UEL)(QL -)12,

where F denotes the Fourier transform. Hence &a is propor-tional to the intensity of the pump pulse at the particularprobe frequency. The probe is also affected by the popula-tion changes caused by the pump field (see the discussion atthe end of Section 3).

Temporal overlap between the two pulses gives rise to newfeatures. The autocorrelation function is essentially mono-tonically decreasing with time when the probe comes afterthe pump-field maximum, tp > 0. If the decrease is smoothenough, the Laplace transform is a smooth function of theprobe frequency peaked at c = QL. However, when theprobe precedes the maximum of the pump field, tp < 0, theautocorrelation function has a clear maximum at t - ItpI.This maximum means that the Laplace transform will havean oscillatory component that behaves like

exp[-i(co -Qdtp].

Moreover, the autocorrelation function decreases with de-creasing pulse separation.

An additional property of 5(x(co)) is that if the pump-pulse amplitude is real, then the change of absorption is asymmetric function of X - L. Remember that a constantphase does not change the situation, and a linearly changingphase would only shift the carrier frequency QL. However, afrequency drift in the pump field during the interactioncauses the spectrum of the probe to be asymmetric. Ofcourse other effects (such as incoherent perturbations) mayalso change details of the absorption spectrum.

To obtain explicit results we study pulse shapes that aresmooth and symmetric in time. As an example we take thehyperbolic-secant pulse, which actually is a good representa-tion of experimental femtosecond pulses when possible fre-quency shifts are neglected. By using

EL(t) = o Ih(t)

Now it is easy to evaluate the Laplace transform numerical-ly. For various values of ot,, we show in Fig. 1 the normal-ized differential transmission spectra, which are simply thenegative of the respective normalized absorption changes.The spectra show clear oscillatory behavior when at, is nega-tive. For positive values of o-tp we see only one overswing.

To gain a better understanding of the parameter depen-dence of the spectral features we analyzed the case of anexponential pulse shape

MEL(t) = Ke (ltl (44)

where we obtain analytical results. The pulse shape, Eq.(44), is a continuous function of time, but its derivative isdiscontinuous at t = 0. However, because the autocorrela-tion function is an integral over the pulse amplitudes, itsderivative is continuous. We therefore expect the unphysi-

aI

FS

z5

5

(C

-5 0 t

(d)

VO 4~~~~~~~~~

(42)

the autocorrelation function may be calculated analytically:

dt'ptEL(t + t- t')pEL*(t - t')

2K2 e- ot ( + I{+ exp[-2of(t + tp)]f 4

-' 1 -e- I2 r~n + exp(-2ortp) J

Normalized Probe Detuning

Fig. 1. Normalized differential transmission spectra for a weakpump pulse having the temporal shape of a hyperbolic secant [Eq.(42)]. The respective temporal separation tp between the maximaof the pump pulse and the probe pulse is (a) t, = 0, (b)tp = -2/6, (c)tp = -4/3, and (d) tp =-6/a. The probe detuning is defined withrespect to the central frequency QL of the pump pulse, i.e., (w -QLV


5

- U J


cal cusp of the pulse shape near zero not to cause dramaticdistortions of the results. Evaluating the integrals, we ob-tain the change of absorption as

a 4 + ( L 2 (o [42 + ( - QL )2 xp(-2tp)]

+ (-tp){exp(4tp) 42 ) [4 Cos( QL)tp~2+(Cc - QL)

+ ( - QL)sW(cc - QL)tp] - exp(24tp)}). (45)

Here the oscillations are seen analytically for the situationwhen the probe pulse precedes the pump, tp < 0. Thefrequency of the oscillations is t; their amplitude decreasesexponentially with decreasing t. When tp increases towardzero, the number of oscillations decreases. They disappearaltogether for positive tp; only one overswing remains. Theamplitude of this overswing decreases exponentially withincreasing pulse separation.

For a square pulse, additional oscillations appear becauseof the discontinuity in the field amplitude. The Fouriertransforms both of the hyperbolic-secant pulse and of theexponential pulse are positive and nonoscillatory functions,whereas the Fourier transform of the square pulse exhibitsoscillations, which are superimposed to the coherent oscilla-tions discussed in this paper.

5. HIGH FIELD EFFECTS

So far we have been studying the case of only relatively smallpump fields. However, two-level systems under strong co-herent excitation can show interesting deviations from thelow-field behavior. Complete inversion and transparency(so-called self-induced transparency) are both possible.2Recent calculations show that pulse excitation might alsocause interesting modifications in the fluorescence spectra. 8

To analyze coherent effects caused by strong fields, we haveto solve Eqs. (35) and (36). The equations can be written inthe form of a single second-order linear differential equationfor u(t):

d u + [i(E - L)- E* ( EL*)] dt + / 2IELI u = 0.

(45)

There are very few pulse shapes EL(t) for which an analyticsolution of Eq. (45) is known. Fortunately, however, thereexists the solution by Rosen and Zener9 for a pump pulsethat has the shape of a hyperbolic secant [Eq. (42)]. Nowinserting Eq. (42)-with K a complex constant-into Eq. (45)allows the solution to be written as

u(t) = 2F1 F[J - H. + ; z(t) Lu 2 2a

and

v(t) = K + Z(t)11 z(t)12F{1 + , 1 a;

where z(t) denotes the function

z(t) = + tanh(ot)2

(47)

and = E - L. The function 2F1 is the hypergeometricfunction.1 l

An analytic evaluation of x(U), Eq. (34), and the integra-tion over the band energies is now possible. The resultingexpressions in general are difficult to evaluate even numeri-cally, and they allow little physical insight. Therefore wehave chosen to restrict ourselves to cases in which the solu-tions [Eq. (46)] can be expressed as a finite series in terms ofz. This is possible for so-called Nr pulses, where the ampli-tude of the pulse fulfills the condition

I _=-N, N= 1, 2,3, ... (48)

The names r pulse, 27r pulse, etc. are used for historicalreasons.2 The only important feature of these pulses is that

or Pulse

.gE

-aZ

T

is

!E

(b)

-5 0 5

-5 0 15W~~~~~~

(_d \C~~~~

-J V

Normalized Probe Detuning

3 43f; 1 )2 2 cr'

Fig. 2. Normalized differential transmission spectra for the case of(46) a r pulse, KI = a; see Eq. (42). The other parameters are the same as

in Fig. 1.


i

i

U


°2

.E

F,

-

._

'M

EP

;Z

2vr Pulse

(a)

-l0 -5 0 5 10

Ib)

-10 -5 0 5 IC

(c)

-lo -5 0 5 l0

(d)

C -AVA ~ ~ .~~ A,,

q V V - -D

I If --- ---

-1_ -5 0 5 10Normalized Probe Detuning

Fig. 3. Normalized differential transmission spectra for the case ofa 27r pulse, K1 = 2a; see Eq. (42). The other parameters are the sameas in Pig. 1.

the population, which is initially in the ground state, returnsinto the ground state after the pulse is over. For the exam-ple of N = 1 ( pulse), the solutions are

u(t) = 1 - + z(t),ar + 43

v(t) = i 26i Vz(t)[1 - z(t)]. (49)a. + 43

Now the integration over the band energies can be done.The Laplace integral may also be evaluated analytically, butfor convenience we chose to evaluate it numerically. In Fig.2 we plotted the differential transmission spectra for thecase N = 1 for the same values of pulse separation tp as in thelow-field case (Fig. 1). In comparison with the low-fieldresults, the spectra for N = 1 show no dramatic changes forpositive values of tp. However, we see an interesting addi-tional feature arising in the spectrum when tp decreasesthrough zero to negative values. A hole appears in the

middle of the central peak. This hole is an indicator of thepump saturating the transitions. An overall feature is thebroadening of the spectrum. We did not study extensivelyhow the spectrum changes as a function of 1KI. However, byevaluating the case N = 2 (27r pulse), we see that additionalstructures appear in the spectrum when the field is in-creased. In Fig. 3 we show an example of this behavior.The details of the oscillation pattern become quite complex.

6. DISCUSSION AND CONCLUSIONS

In conclusion, we have developed a model to calculate theprobe-transmission spectrum, which should be observed in acompletely coherent, nondegenerate femtosecond pump-probe experiment in semiconductors. Our model is similarin some features to that used by Wherrett et al.11 to analyzetransient gratings in picosecond four-wave-mixing. We findthat the transmission spectrum for weak pump fields can beexpressed in terms of the Laplace transform of the autocor-relation function of the pump amplitude. Examples aresolved analytically. The computed differential-transmis-sion spectra show oscillatory features when the probe pre-cedes the pump maximum. In the strong-field case, wepredict characteristic modifications of the spectra.

The oscillatory structures in our computed spectra foroptical pump-probe experiments are similar to the so-calledRamsey fringes'2 observed in nuclear magnetic resonance.Ramsey fringes occur if an atomic beam, whose atoms ini-tially are in the lower of two possible states, is directedthrough two spatially separated oscillating magnetic fields.The transition probability as function of detuning is record-ed after the interaction with the second beam. Characteris-tic oscillations, the so-called Ramsey fringes, are observed.Both effects, the Ramsey fringes in nuclear magnetic reso-nance and the transmission oscillations in optical ultrashortpump-probe excitations, are results of quantum-mechanicalinterference. However, the two effects do have some sys-tematic differences. The semiconductor transitions can beconsidered inhomogeneously broadened, since each k statecorresponds to a different frequency. Nevertheless the os-cillatory structures in the probe-transmission spectrum arestill present under the discussed conditions. On the otherhand, a similar inhomogeneous (Doppler) broadening in theatomic beam would cause a distribution of transit times forthe atoms that would wash out the Ramsey fringes. Incontrast to the atomic beam in the semiconductor system,the inhomogeneous broadening is an intrinsic feature causedby the band structure, but still all transitions are subject tothe same pulse delay.

To obtain an estimate for a typical wavelength of theexpected oscillatory pattern, we assume the pump wave-length to be 7000 A and the pulse separation to be 100 fsec.The resulting oscillation period is then 160 A, whichshould be detectable experimentally. The oscillatory fea-tures begin to disappear as soon as the different k statesstart to mix. Hence one could use the disappearance of theoscillatory structures in experiments as a sensitive measurefor the characteristic times of coherence-relaxation process-es in semiconductors. As shown by our theory, the shapes ofthe spectra without incoherent processes are determined bythe pulse duration and by the temporal separation betweenpump and probe pulses. Thus, by measuring differential-

M. Lindberg and 8. W. Koch

.

I

- I


transmission spectra for different pump-pulse durations,one can directly relate the deviations from the completelycoherent results to the influence of incoherent processes.However, to obtain quantitative values for the coherence-decay rates, one needs more detailed theoretical models inwhich incoherent processes are systematically included.'3

Nevertheless, even without the explicit treatment of inco-herent processes, our computed transmission spectra closelyresemble experimental spectra obtained by femtosecond ex-citation of bulk CdSe.7 Similar features have also beenobserved in organic materials. 6 The close resemblance be-tween the spectra in physically quite-different systems is notat all surprising if one considers the rather general form ofthe involved interaction processes. This is also reflected inour theory, which should be applicable to all systems thathave broadband transition features.

The additional spectral structures that our theory pre-dicts for excitations with strong pump fields may also havesome useful applications. For the 7r-pulse example, ourresults show increasing absorption directly at the resonancefrequency. We believe that this effect is possible only whenthe process is completely coherent. The observation of thisfeature and the monitoring of its disappearance could againbe related to the coherence-decay time. However, to studythe dependence of the details of the transmission structureson the explicit pulse shapes, one has to find exact solutionsto Eq. (45). There are some known solutions in addition tothe case of a hyperbolic-secant pulse,'4 and numerical meth-ods for arbitrary symmetric pulse shapes have been devel-oped.15 As a next step in our theory, we plan to model theincoherent-damping mechanisms to see how details of thespectra are changed by the different relaxation processes.We will extend our theory to include intraband Coulombeffects as well as other mechanisms causing coherence decay.

ACKNOWLEDGMENTS

We want to thank N. Peyghambarian and G. R. Olbright,Optical Sciences Center, University of Arizona, for manydiscussions on femtosecond experiments as well as J. D.McCullen, Department of Physics, University of Arizona,

for a critical reading of the manuscript and for pointing outthe similarity of our computed structures to the Ramseyfringes in atomic spectroscopy. Our work is financially sup-ported through the Optical Circuitry Cooperative, Universi-ty of Arizona. A grant for computer time at the John vonNeuman Computer Center, Princeton, is gratefully acknowl-edged.

REFERENCES AND NOTES

1. See, e.g., the articles in S. L. Shapiro, ed., Ultrashort LightPulses, Vol. 18 of Topics in Applied Physics (Springer-Verlag,Berlin, 1977); M. H. Pilkhun, ed., Feature on High Excitationand Short Pulse Phenomena, J. Lumin. 30 (1) (1984); T. W.Mossberg, ed., Feature on Optical Coherent Transfers, J. Opt.Soc. Am. B 3, 474-632 (1986).

2. L. Allen and J. H. Eberly, Optical Resonance and Two-LevelAtoms (Wiley, New York, 1975).

3. See, e.g., the articles in H. Haug, ed., Nonlinear Optical Proper-ties of Semiconductors (Academic, New York, to be published);H. Haug and S. Schmitt-Rink, Progr. Quantum Electron. 9, 3(1984).

4. H. M. Gibbs, Optical Bistability-Controlling Light with Light(Academic, New York, 1985).

5. T. F. Heinz, S. L. Palfrey, and K. B. Eisenthal, Opt. Lett. 9, 359(1984); S. L. Palfrey and T. F. Heinz, J. Opt. Soc. Am. B 2, 674(1985).

6. C. H. Brito Cruz, R. L. Fork, W. H. Knox, and C. V. Shank,Chem. Phys. Lett. 132, 341 (1986).

7. G. R. Olbright, M. Lindberg, B. D. Fluegel, S. W. Koch, F. Jarka,and N. Peyghambarian, in Digest of XV International Confer-ence on Quantum Electronics (Optical Society of America,Washington, D.C., 1987), paper PD2.

8. M. Lewenstein, J. Zakrzewski, and K. Rzazewski, J. Opt. Soc.Am. B 3, 22 (1986).

9. N. Rosen and C. Zener, Phys. Rev. 40, 502 (1932).10. M. Abramowitz and I. A. Stegun, Handbook of Mathematical

Functions (Dover, New York, 1972), p. 555.11. B. S. Wherrett, A. L. Smirl, and T. F. Boggess, IEEE J. Quan-

tum Electron. QE-19, 680 (1983).12. N. F. Ramsey, Molecular Beams (Clarendon, Oxford, 1956); P.

Knight, Comm. At. Mol. Phys. 10, 241 (1981).13. M. Lindberg and S. W. Koch, submitted to Phys. Rev. B.14. A. Bambini and P. R. Berman, Phys. Rev. A 23, 2496 (1981); A.

Kujawski and J. Mostowski, J. Opt. Soc. Am. B 3,1700 (1986).15. A. Bambini and M. Lindberg, Phys. Rev. A 30, 794 (1984).


theory of coherent transients in semiconductor pump—probe spectroscopy

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