coherent pump-probe response of quantum wells: polarization dependence

Journal of Russian Laser Research, Volume 34, Number 6, November, 2013

COHERENT PUMP-PROBE RESPONSE

OF QUANTUM WELLS:

POLARIZATION DEPENDENCE

Hoang Ngoc Cam

Institute of Physics

Vietnam Academy of Science and Technology

Dao Tan 10, Ba Dinh, Hanoi, Vietnam

E-mail: hncam@ iop.vast.ac.vn

Abstract

We consider the coherent pump–probe response from quantum wells using the exciton–boson formalismapproach. We obtain the mean-field contribution and that of the biexciton as well as of a molecularscattering state in analytic form and show that each of these contributions may completely disappearin one or another polarization configuration of the pump and probe pulses, which makes the responsestrongly dependent on polarization. We apply the derived formulas for calculating the differentialabsorption spectrum of quantum-well samples under excitation conditions supporting the slow exciton–exciton scattering. The obtained results are in good qualitative agreement with available experimentalobservations.

Keywords: coherent pump–probe response, polarization dependence, third-order susceptibility, diffe-

rential absorption spectrum.

1. Introduction

Experimental and theoretical studies in the field of ultrafast semiconductor spectroscopy have shown

that the coherent third-order response in the spectral region near the band gap has its origin in Coulomb-

mediated two electron–hole pair correlations [1]. In a recent paper [2], we presented an approach within

the exciton–boson formalism that allows us to describe the contribution of the bound and unbound corre-

lated two-exciton (molecular) states to the response of quantum wells (QWs) in experiments supporting

the slow exciton–exciton scattering. In the present paper, the core of the approach – the closed system

of dynamical equations for the quasiparticles relevant to a third-order ultrafast process – will be approxi-

mately solved to obtain the pump–probe response. We see that the momentum and polarization selection

rules incorporated in the system make the response strongly dependent on the polarization configuration

of the pump and probe pulses. For each of four common configurations, we obtain the pump-probe

response in the form of a time function depending yet on other excitation parameters. In the frequency

domain, the function gives the third-order susceptibility, and with it the differential absorption spec-

trum of samples. For illustration, the spectrum is demonstrated for two representative cases meeting

slow exciton–exciton scattering: a quasi-resonant excitation of a 4.8 nm width ZnSe-based QW and an

off-resonant excitation of a GaAs QW. It is these QW samples for which the small-momentum small-

energy molecular problem has been approximately solved in [2]. Moreover, their polarization-dependent

pump–probe spectra were observed but have still not received adequate interpretation [3, 4].

Manuscript submitted by the author in English first on November 1, 2012 and in final form on September 24, 2013.

1071-2836/13/3406-0565 c©2013 Springer Science+Business Media New York 565

Journal of Russian Laser Research Volume 34, Number 6, November, 2013

This paper is organized as follows.

In Sec. 2, we obtain general formulas for the pump–probe response and its components in the time

domain from an approximate solution of the system of dynamical equations. In Sec. 3, the third-order

susceptibility of QW samples under the condition of slow exciton–exciton scattering is calculated and

their differential absorption spectra are considered. In Sec. 4, conclusions are given.

2. Coherent Pump–Probe Response of a QW in Time Domain

We consider a two-pulse coherent nonlinear optical experiment in the third-order regime in the close-

to-normal-incidence excitation geometry. The QW sample is subjected to two ultrashort laser pulses,

the probe and the pump, with polarizations λ1 and λ2 and wave vectors K1 and K2, respectively. The

central frequencies ω1, ω2 of the pulses are assumed to be near but below the exciton resonance, and their

durations τ1 and τ2 are short compared to the dephasing time T2 of the electron excitation in the medium.

Containing a huge number Ni = 〈C+λiKi

CλiKi〉 of coherent photons in one single quantum state λiKi, the

pulse i (i = 1, 2) is a macroscopically occupied mode with a macroscopic amplitude, CλiKi∝ √

Ni. At

the first moment of its entering the sample, the pulse creates Ni coherent excitons in the state λiki, ki

being the in-plane component of Ki (Ki = ki + kiz), so in the first approximation the induced exciton

state is also macroscopically occupied, A(1)λiki

∝ √Ni [5, 6].

Since commutators[CλiKi

, C+λiKi

]= 1 and

[Aλiki

, A+λiki

]≈ 1 are negligibly small compared to the

macroscopically large number√Ni ∝ √S, the corresponding photon and exciton operators may be

treated as ordinary numbers (C-numbers) [7]. The photon field of a laser pulse can be presented as

follows:

CλiKi=

√Seλi

Ei(t) exp[i(KiR− ωit)], i = 1, 2, (1)

where R is the coordinate vector in the three-dimensional (3D) space. The total electromagnetic field is

just the sum of the fields of two pulses

CλK = δλλ1δKK1Cλ1K1 + δλλ2δKK2Cλ2K2 . (2)

In the lowest-order nonlinear regime, the coherent response of the quasi-2D sample to the field can

be expressed as the sum of the linear and third-order responses

Aλk = δλλ1δkk1A(1)λ1k1

+ δλλ2δkk2A(1)λ2k2

+A(3)λk . (3)

The combination of Eqs. (2) and (3) along with Eq. (2) of [2] gives{∂

∂t+ i Eγ(Ki)

}Cλi,Ki

= ΩR A(1)λiki

, i = 1, 2. (4)

From here, it follows that the in-plane matter state induced by pulse i having the wave vector Ki, in

turn, acts as a source for an outside light field propagating in the same direction Ki. This is the reason

for the state’s radiative decay [8], but this question is beyond the scope of this paper. We assume here

that the radiative decay rate is small compared to that of the intrinsic semiconductor decay. Hence the

linear response may be expressed in the same form as (1),

A(1)λki

=√Seλi

P(1)i (t) exp[i(KiR− ωit)], i = 1, 2, (5)

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Volume 34, Number 6, November, 2013 Journal of Russian Laser Research

where its amplitude envelope is related to that of the driving field as follows:

P(1)i (t) = −ΩR exp[−(iΔi + γx)t]

t∫−∞

dt′ exp[(iΔi + γx)t′] Ei(t′), i = 1, 2. (6)

The equation for the pump–probe response A(3)λk1

∝ E2pEt (Ep and Et denote the pump and probe am-

plitudes, respectively) emitting the signal in the propagation direction of the probe pulse (K = K1) is

obtained by inserting (3) into Eq. (18) of [2] with replacement of Aλk and B(J,k) in the nonlinear part

on the right-hand side by their leading order A(1)λk and B(2)(J,k), respectively. The time evolution of the

latter is described by the equations{∂

∂t+ i[E0

xx − iγxx]

}B

(2)b/s(0,k) = δλ2,−λ1δk,k1+k2

Ψ0,b/s√2S

[A

(1)λ1k1

Cλ2K2 +A(1)λ2k2

Cλ1K1

], (7){

∂

∂t+ i[E2

xx − iγxx]

}B

(2)s (2,k) = δλ2,λ1

Φ2,s√2S

{δk,2k1A

(1)λ1k1

Cλ1K1 + δk,2k2A(1)λ2k2

Cλ2K2

+δk,k1+k2

[A

(1)λ1k1

Cλ2K2 +A(1)λ2k2

Cλ1K1

]}, (8)

as follows from Eqs. (15) and (16) of [2] and Eqs. (2) and (3). As a consequence of the wave

vector conservation law, A(3)λk1

arises from the nonlinear sources associated with A(1)+λ2k2

A(1)λ2k2

A(1)λ1k1

and

A(1)+λ2k2

B(2)(J,k1 + k2), J = 1, 2. On the other hand, the spin-polarization selection rule incorporated in

the system of dynamical equations (2), (15), and (16) along with (19) of [2]) excludes the contribution of

the spin-0 molecule in the cocircular configuration and of the spin-2 molecule along with the mean-field

term in the counter-circular configuration. In this way, we find the equation for the component of the

third-order response in the form{∂

∂t+ i[Ex − iγx]

}A

(3)λk1

=

−iδλλ1

{δλ1λ2A

(1)+λ2k2

[9

8SU exA(1)λ2k2

A(1)λ1k1

+1√2S

∑s

εsΦ∗2,sB

(2)s (2,k1 + k2)

]

+δλ1,−λ2

A(1)+λ2k2√2S

[∑s

εsΦ∗0,sB

(2)s (0,k1 + k2)− |εb|ΦbB

(2)b (0,k1 + k2)

]}. (9)

Equation (9) shows that the pump–probe response in the cocircular configuration is generated jointly

by the mean-field effect and the coupling between the pump-induced excitons with the states of spin-2

molecule, while in the counter-circular configuration it is generated entirely by the coupling between

excitons and the spin-0 molecule states. For linearly polarized incident pulses, nonlinear sources of the

pump–probe response can be found also just by the selection rules. In fact, due to the spin-polarization

selection rule, the molecular states B(2)(J,k1 + k2), J = 0, 1 are absent in the cross-linear configuration.

In this linear polarization configuration with λ2 �= λ1, A(3)λk1

is generated completely by the mean-field

effect, {∂


}A

(3)⊥λk1

= −iδλλ1

9

16SU exA(1)+λ2k2

A(1)λ2k2

A(1)λ1k1

. (10)

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In contrast, in the collinear configuration (λ2 = λ1) both type of molecular states exist. The pump–probe

response is hence the joint effect of the nonlinear sources of all types,

{∂


}A

(3)‖λk1

= −iδλλ1A(1)+λ1k2

{9

16SU exA(1)λ1k2

A(1)λ1k1

+1√2S

∑s

εsΨ∗2,sB

(2)s (2,k1 + k2)

+1√2S

[∑s

εsΨ∗0,sB

(2)s (0,k1 + k2)− |εb|Ψ∗

bB(2)b (0,k1 + k2)

]}, (11)

where the equation for B(2)(J,k) has the same form as Eq. (8) for both J = 0 and J = 2. It is convenient

therefore to classify a molecular state by its correlation energy ε (ε < 0 for the biexciton and ε ≥ 0 for

continuum states) and present it in the following form:

B(2)ε (J,k1 + k2) =

√2SΨJ,ε eλ1eλ2Bε(t) exp[i(K1 +K2)R− (ω1 + ω2)t)]. (12)

The equation for the amplitude envelope Bε(t), following from Eqs. (1), (5), (7), and (8), shows its

independence of J . It might be seen that the solution for the amplitude has the form of a sum of two

superpositions of a field and a linear exciton amplitude of Eq. (6) type.

As to the third-order response A(3)λk1

, oscillating in phase with the probe field, it has the same form

as Eq. (5) with i = 1. The first-order differential equation for its amplitude envelope P(3) in four

configurations is readily apparent from Eqs. (9)–(11). The solution for P(3), which is a polarization-

dependent function of the real time t after arrival of the later of the pulses and of their time delay T ,

can be presented in the following form:

P(3)σσ(t, T ) = 2P(3)x (t, T ) +

μx

4π

∫dεP(3)

2,ε (t, T ),

P(3)σ+σ−(t, T ) = P(3)

b (t, T ) +μx

4π

∫dεP(3)

0,ε (t, T ), (13)

P(3)⊥(t, T ) = P(3)x (t, T ), P(3)‖(t, T ) = −P(3)⊥(t, T ) + P(3)σσ(t, T ) + P(3)σ+σ−

(t, T ),

where the sum over s in Eqs. (9) and (11) is put in the form of the integral over ε. The mean-field

contribution of single excitons is presented by a superposition of three linear amplitudes of (6) type,

P(3)x (t, T ) = iΩ3

R

9

16U ex exp[−(iΔ1 + γx)t]

t∫−∞

dt′ exp[−2γxt′]

×t′∫

−∞dt′′ exp[(iΔ1 + γx)t

′′]E1(t′′)∣∣∣∣∣∣

t′∫−∞

dt′′ exp[(iΔ2 + γx)t′′]E2(t′′)

∣∣∣∣∣∣2

, (14)

while the contribution of the state of the spin-J molecule having correlation energy ε is presented by

the product of the state weight, ε|ΨJ,ε|2, on the superposition of a linear amplitude of (6) type and the

568


molecular amplitude. It looks as follows:

P(3)J,ε (t, T ) = iΩ3

R ε|ΨJ,ε|2 exp[−(iΔ1 + γx)t]

t∫−∞

dt′ exp[−i(ε− iγxx)t′]

t′∫−∞

dt′′ exp[(−iΔ2 + γx)t′′]E2(t′′)

×⎧⎨⎩

t′∫−∞

dt′′ exp{i[Δ1 + iγx + ε− iγxx]t′′}E1(t′′)

t′′∫−∞

dt′′′ exp[(iΔ2 + γx)t′′′]E2(t′′′)

+

t′∫−∞

dt′′ exp{i[Δ2 + iγx + ε− iγxx)]t′′}E2(t′′)

t′′∫−∞

dt′′′ exp[(iΔ1 + γx)t′′′]E1(t′′′)

⎫⎬⎭ . (15)

Finally, function P(3)b (t, T ) in Eq. (13) stands for the particular case of P(3)

0,ε (t, T ) with ε = −|εb|.

3. Third-Order Susceptibility and Differential Absorption Spectra of

QW Samples

In a typical coherent pump-probe experiment conducted in the exciton spectral range, the probe

is too weak and quasi-resonant with the exciton resonance (Δ1 δω1) to monitor the changes in the

sample optical property, e.g., absorption induced by the pump. Going to the frequency domain by the

ω1-centered Fourier transform [9]

P(3)(ω, T ) =

∞∫−∞

dt exp [i(ω − ω1)t] P(3)(t, T ), E1(ω) =∞∫

−∞dt exp [i(ω − ω1)t] E1(t), (16)

we get for the third-order susceptibility X (3)(ω, T ) = P(3)(ω, T )�E1(ω), and with it the differential

absorption spectrum ΔK(ω, T ) ∝ [X (3)(ω, T )], in four polarization configurations, which is the same

formula as (13), where X (3) or ΔK appears instead of P(3). The formula provides general features of the

pump–probe spectra of QWs, which might be observed in experiments with pump and probe spectral

characteristics meeting the condition of slow exciton–exciton scattering [2],

δω1 −Δ1 + δω2 −Δ2 Ry∗. (17)

In this case, the wave function of the biexciton and of molecular scattering states is available (see Eq. (22)

of [2]). Condition (17) is certainly more strict than that of selective excitation of the ground-state heavy-

hole excitons.

For illustration, we present below the polarization-dependent third-order susceptibility and the diffe-

rential absorption spectrum ΔK(ω, T ≥ 0) of QW samples considered in [2] for two excitation situations.

3.1. Quasi-Resonant Excitation

On ZnSe-based QWs (Ry∗ ≈ 20 meV) having large exciton dephasing time, experiments meeting

condition (17) with a quasi-resonant pump Δ2 δω2 may be performed using spectrally narrow pulses,

569


δω2 < δω1 10 meV. In this case, the pump produces real excitons, resulting in persistence of nonlinear

effects after the pump action.

Under the condition Δ2 δω2, or τ2 1/Δ2, in the time integral of Eq. (6) with i = 2 the

pump envelope may be treated as a delta function in comparison with the slowly varying exponent.

Consequently, Eqs. (14) and (15) are reduced to elementary functions, from which after some algebra we

arrive at

X (3)x (ω, T ≥ 0) |Δ2�δω2 ∝ −U ex(ΩRτ2Ep)2 exp [−2γxT ]

[ω − Ex + iγx] [ω − Ex + 3iγx], (18)

X (3)J,ε (ω, T ≥ 0) |Δ2�δω2 ∝ −ε|ΨJ,ε|2 (ΩRτ2Ep)2 exp[−2γxT ]

[ω − Ex + iγx] [ω − Ex − ε+ i(γx + γxx)]. (19)

It is obvious from here and Eq. (13) that excitation parameters other than the polarization configuration,

such as the pump intensity or the time delay, affect the amount of pump–probe spectra in the same

way in all the polarization configurations. In the following, they are assumed to be the same while the

a) b)

c) d)

Fig. 1. Differential absorption spectra of a 4.8 nm width ZnSe QW under quasi-resonant excitation meeting thecondition (17) (a, b, and c), the spectrum (bold solid line) and its components for the cocircular, counter-circular,and collinear polarizations, respectively, and relative quantities of the spectrum in four polarization configura-tions (d). The sample coherent parameters are T2 = 10 ps, γxx = 2γx.

570


differential absorption spectrum with its components in different configurations is studied. As can be

seen from (18), the mean-field contribution ΔKx brings about the absorption modification in the range of

width γx round the exciton resonance (Fig. 1 a), which is known to correspond to a blue shift with a slight

bleaching of the resonance in the total absorption spectrum [9]. Unlike this, the biexciton contribution

ΔKb (derived from X (3)J,−|εb|) includes a minor decrease of absorption at the exciton resonance (Fig. 1 b)

and its increase in the range of width γx + γxx around the biexciton resonance at ω ≈ Ex − |εb| (see the

inset in Fig. 1 b). The increase, which is associated with the transition of real excitons created by the

quasi-resonant pump to the biexciton level under the probe action, however, cannot be seen by a narrow

probe pulse. In regard to the total contribution of a molecular continuum ΔKJcont ≡

∫dεΔKJ,ε μx/(4π),

as the dashed line in Fig. 1 a and b shows, it corresponds also to a blue shift with bleaching of the exciton

resonance. While the contribution of the spin/2 molecule’s continuum in the cocircular configuration is

negligible in comparison with the mean-field effect (Fig. 1 a), that of the spin/0 molecule’s continuum in

the counter-circular configuration is decisive (Fig. 1 b). As the biexciton contribution is negligible, the

combined contribution of the two continua, ΔKcont = ΔK0cont + ΔK2

cont, defines the difference between

differential spectra in the cross-linear and collinear configurations (Fig. 1 c).

Differential absorption spectra, obtained for a 4.8 nm width ZnSe-based QW in four configurations

according to Eqs. (13), (18), and (19), are shown together in Fig. 1 d. The figure shows that, under a

quasi-resonant excitation, a blue shift with a slight bleaching of the exciton resonance takes place in all

polarization configurations, but with different quantity. The largest shift with the most bleaching occurs

in the cocircular configurations; the second place is occupied by the collinear configuration. This agrees

with observations of [4], but the smallest shift with least bleaching is obtained here for the counter-circular

polarization, while in [4] it is found in the cross-linear configuration. The comparison, however, is not

entirely justified, because under the excitation condition of [4] not only slow molecular scattering states

are excited. As a result, the share of the molecular continuum contribution in the spectral renormalization

in the counter-circular and collinear configurations is likely to be larger.

3.2. Off-Resonant Excitation

In the limit Δ2 � δω2, the pump produces exclusively virtual excitons with effective lifetime τeff ∼1/Δ2 τ2. As follows from Eq. (6) for i = 2, in this case the linear response of the sample to the

pump field is proportional to the instantaneous pump amplitude, P(1)2 ≈ iωcEp/Δ2. Then only a probe

overlapping with the pump (T = 0) may see the pump action. Under these pump–probe quasi-stationary

conditions [10], contributions to the third-order susceptibility of single excitons and of a molecular state

are approximately described by the functions

X (3)x (ω, 0) |Δ2�δω2 ∝ −U ex (ΩREp)2

Δ22

1

[ω − Ex + iγx]2 , (20)

X (3)J,ε (ω, 0) |Δ2�δω2 ∝ ε|ΨJ,ε|2 (ΩREp)2

Δ2

[1

ω − Ex + iγx− 1

2Δ2

]

× 1

[ω − Ex + iγx] [ω − Ex − (Δ2 + ε) + iγxx]. (21)

Comparing (20) with (18), we see that the mean-field contribution to ΔKx has the same shape as

under the quasi-resonant excitation (Fig. 2 a). It is connected with a blue shift of the exciton resonance,

571


a) b)

c) d)

Fig. 2. The same as in Fig. 1, but for a GaAs QW with T2 = 2 ps under off-resonant excitation with detuningΔ2 = 4.5 meV.

known as the exciton optical Stark effect (OSE) [11, 12]. In experiments on the observation of the

exciton OSE, however, pumping was performed with Δ2 � Ry∗, where the anharmonic exciton–photon

interaction (phase-space filling effect) introduces the main contribution, ∝ 1/Δ2. In the exciton spectral

region, the contribution coming from the interaction among excitons dominates with ΔKx ∝ 1/Δ22, as

seen from Eq. (20). As discussed earlier, here effects of pure Coulomb two-exciton correlations, which

depend strongly on the polarization configuration, are decisive.

In QWs with relatively large biexciton binding energy, the most prominent manifestation of Coulomb

two-exciton correlation effects under off-resonant pump is that of the biexciton. That is because, in this

case, the pump often lies in the range nearer to the biexciton resonance than to the exciton one and the

molecular continuum. In particular, under the pump with Δ2 > δω1 + δω2, the continuum states are not

excited, so the whole molecular contribution is reduced to that of the biexciton. As can be seen from

X (3)0,−|εb| in Eq. (21), ΔKb includes an additional peak of absorption of width γxx at ω − Ex = Δ2 − |εb|,

which is associated with the two-photon transition to the biexciton level. Further, as Δ2 � 1/τ2 � γx,

in the vicinity of the exciton resonance the last term of Eq. (21) is negligible in comparison with the first

one. Detailed analysis of the first term shows that in this range the shape of ΔKb depends substantially

on the sign and quantity of the pump detuning from the biexciton resonance, D = Δ2−|εb|. Putting, for

572


simplicity, γxx = 2γx, we see that under a pump resonant with the biexciton resonance (D = 0), a drastic

decrease of absorption ∝ 1/γ3x is obtained at ω = Ex. It can be shown to correspond to a splitting of the

exciton resonance. As to D �= 0, the biexciton contribution yields a red shift of the exciton resonance

for D > 0, and a blue shift for D < 0. The theoretical result confirms the long-known observations [13],

since the spin-dependent exciton Hamiltonian of the CuCl bulk crystal has the same structure as that of

the system under consideration.

With regard to GaAs QWs having biexciton binding energy of order 2 meV, the off-resonant ultrashort

pump supporting (17) is always below the biexciton resonance, i.e., D > 0. The biexciton effect, in

this case, corresponds to a red shift, as can be seen from the shape of ΔKb in Fig. 2 b. The effect

of the continuum is inverse, and it fairly reduces the total effect, which has the biexciton character.

Concerning the relation between the mean-field and molecular contributions in the spectrum for the

cocircular polarization, as under quasi-resonant pump, the former dominates (Fig. 2 a). The mean-field

effect also governs the character of the spectrum in the collinear configuration, though the magnitude of

the effect is significantly reduced by that of the biexciton, as seen in Fig. 2 c. The relative quantities of

the differential absorption in four polarization configurations shown in Fig. 2d agree very well with the

experimental observation of [3]. We find it necessary to stress, however, that according to our results the

red shift character of the spectrum in the counter-circular configuration is definitely connected with the

biexciton.

4. Conclusions

In this paper, we considered the coherent pump–probe response from QWs using the exciton–boson

approach developed in our previous paper [2]. Besides the mean-field contribution of single excitons,

the response includes those of the spin/0 and spin/2 molecules, whose existence is defined essentially by

the polarization selection rule. We showed that, because of the polarization and wave-vector selection

rules, each of three mentioned contributions may disappear in one or another polarization configuration

of incident pulses. As a result, the pump–probe response is polarization sensitive. We derived general

formulas for the response in cocircular, counter-circular, cross-linear, and collinear polarizations, which

can serve as a basis for calculations of pump–probe spectra of different QWs under excitation conditions

supporting the slow exciton–exciton scattering. Representative cases of quasi-resonant and off-resonant

excitations have been chosen for illustration. In both cases, the contributions of single excitons and of

the molecular state to the third-order susceptibility were obtained in the form of analytical functions.

Further, we obtained differential absorption spectra for a 4.8 nm width ZnSe- and a GaAs-based QWs

in four common polarization configurations. The results reproduce well the polarization dependence as

well as many details of the available experimentally observed spectra.

Acknowledgments

The author acknowledges financial support from the Vietnam National Foundation for Science and

Technology Development (NAFOSTED) under Grant No. 103.02.53.09.

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coherent pump-probe response of quantum wells: polarization dependence

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