Modifications of laser field assisted intersubband transitions in the coupled quantum wells due to static electric field

Phys. Status Solidi B 248, No. 7, 1727–1734 (2011) / DOI 10.1002/pssb.201046582 p s sb

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basic solid state physics

Modifications of laser field
assisted intersubband transitions in thecoupled quantum wells due to static electric field
Vinod Prasad* and Brijender Dahiya

Department of Physics, Swami Shraddhanand College, University of Delhi, Delhi 110036, India

Received 15 November 2010, revised 17 January 2011, accepted 14 February 2011

Published online 9 March 2011

Keywords Floquet theory, intersubband transitions, quantum wells, terahertz fields effects

*Corresponding author: e-mail [email protected], Phone: þ91-9871365843, Fax: þ91-11-27206722

We consider the electron bound in the conduction band of a

coupled asymmetric double quantumwell (ADQW) interacting

with static as well as strong monochromatic field in terahertz

(THz) region. We use non-perturbative Floquet theory to study

the static field and quantumwell (QW) system in the presence of

strong laser fields. We show that the intersubband dynamics

getsmodified due to the static field. The sensitivity to the ratio of

strengths of static and laser field (Es/E0) of the state populations

can be used in various optoelectronic device applications.

� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Recently, coherent effect induced bystatic electric fields in the electromagnetic field excitedsemiconductors has received considerable interest. Theinteraction of oscillating electric field with intersubbandelectrons in semi-conductor quantumwells (QWs) has led tothe prediction of several interesting and potentially usefuleffects, e.g., enhanced nonlinear-optics [1, 2], gain withoutinversion [3], and electromagnetically induced transparency(EIT) [4]. There have been few experimental studies in thefield of QW intersubband transitions [5–7]. Several usefuldevices such as lasers, photo detectors, modulators, optical,and quantum switches are based on intersubband transitionsin QWs [8–11]. The design of these efficient opto-electronicdevices depends on understanding the basic physics involvedin this interaction process [12]. The various properties ofQWs, such as, photoionization [13–16], depolarization shift[17, 18], optical absorption [19], free-free transition [20], etc.have been intensively studied. In most of these studies, theoptical properties are studied and the atomic(molecular) likemultilevel theoretical approaches have been used for thedescription of the dynamics of intersubband transitions. Forthese devices, large intersubband oscillator strength is oftenfavorable. In this contest, an applicable QW structure mustpossess an intersubband transition of desired energy andoscillator strength as large as possible [21]. It is expected thatthese properties have to be strongly modified by theconfining potential shape [22]. Many alternative approaches

have been suggested to modify the properties of QWs, e.g.,changing the shape of confining potential, changing theconcentration of atoms in barrier, applying external field thateffect the energy levels and wave functions in low-dimensional structures [23–25]. The external static fieldcan strongly modify the quantum states of the QW structuresand hence influence their electronic and optical properties.When the subband energy separation is close to the photonenergy, the electric field can couple with the quantum statesto form dressed states. Quantum interferences in theintersubband transitions can lead to various interestingphenomenon including EIT [26], coherent populationtrapping (CPT) [27], etc. In present work, we study theelectronic properties of GaAs/AlGaAs asymmetric doublequantum well (ADQW) (rather than the optical properties)under an intense high frequency resonant laser field.ADQWs are good semiconductor structures for realizingthree energy level or four energy level system. However, inthis study, we have taken confining potential such that thesystem supports at least five or six bound and manycontinuum states. We have performed all calculationsincluding all levels into account. Calculations included theeffect of the static electric field on various intersubbandtransition energies and oscillator strengths. GaAs/AlGaAsDQWs are good semiconductor structures for realizing threeenergy level or four energy level system and have beenstudied extensively [28, 29]. The non-perturbative Floquet


1728 V. Prasad and B. Dahiya: Modifications of laser field assisted intersubband transitions in coupled QWsp

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technique is used to study the static field modified QWinteraction with the monochromatic laser field. When theexternal field is very strong, it is very convenient to use thequantum Floquet theory formalization for treating period-ically driven quantum systems in the semiclassical approxi-mation for the driving field. The method allows thetransformation of the periodically time dependentSchrodinger equation into an equivalent infinite-dimen-sional time independent Floquet matrix eigenvalue problem[30–32]. In Sections 2 and 3, we present theoretical analysisof the periodically drivenQWsystem and the computation ofthe optical properties for the various states of ADQW. Theresults are discussed in Section 4.

2 Theory The approach used in this work is based on anon-perturbative theory that was originally developed todescribe the atomic behavior in the presence of the intenselaser fields [33]. We assume that the radiation field can berepresented by a monochromatic plane wave of angularfrequency, v. We consider conduction band electronconfined in a typical AlxGa1�xAs based ADQW grown inthe Z-direction subjected to static electric field Es and aperiodic perturbation in the term of a spatially homogeneouslaser field Ef ðtÞ. The laser field is characterized by a planemonochromatic electric field (with polarization vector z)along the Z-direction and in the dipole approximation:

� 20

Ef ðtÞ ¼ zE0cosðvtÞ; (1)

where E0 is the amplitude of the electric field and v is thefrequency of the laser field. The time dependent Schrodingerequation for such a system is given by

HðtÞcðtÞ ¼ 0; (2)

where

H ðtÞ ¼ H0 þ ezEs þ Wðz; tÞ� i@

@t: (3)

The Hamiltonian of the quantum system is periodic intime with period T such that Hðt þ TÞ ¼ HðtÞ, wherevT¼ 2p and �h ¼ 1. Here, H0 is the basis Hamiltonian, Es

the static electric field strength, andWðz; tÞ is the interactionpotential energy of the electronwith the laser field. Thewavefunction cðtÞ, called the quasi energy state (QES) or theFloquet state, can be written as

cðtÞ ¼ expð�ietÞfðtÞ; (4)

where e is the real parameter, unique to multiplier of 2pn/T,called the Floquet characteristic exponent or the quasienergy. Equation (2) reduces to the Schrodinger equation ofADQW

i@f

@tr; tð Þ ¼ H0 þ ezEs þ W z; tð Þ½ �f r; tð Þ; (5)

where f r; tð Þ are the dressed states of the ADQW in thepresence of electric field. The total wave function ofthe system c r; tð Þ in terms of dressed state of ADQW in the

11 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

presence of laser field can be expanded as:

cp r; tð Þ ¼Xq

apq ðtÞfqðrÞ: (6)

Here ‘‘q’’ stands for the all the quantum states of thesystem. Substituting Eq. (6) into Eq. (2) and using theorthogonality condition for the dressed states. We obtain thefollowing set of coupled differential equation:

i@apq ðtÞ

@t¼ Eqaqp tð Þ þ

Xs

Vqsasp ðtÞ; (7)

where

Vqs ¼ fqjW z; tð Þjfs

� �(8)

denotes the transition matrix elements for the electroninteracting with the laser field. The above coupleddifferential equation can be written in matrix notation, like

i@FðtÞ@t

¼ HðtÞFðtÞ; (9)

where H(t) is the time dependent periodic matrix withperiodicity T¼ 2p/v. It is well known that the conventionalmethod fails at resonance frequency as well as highintensities. Hence, we have used the non-pertubativeFloquet theorem for the solution of such problem. TheFloquet wave functionF(t) is also periodic in time with timeperiod T. The Fourier expansion of F(t) and HðtÞ ¼H0 þ ezEs þ Wðz; tÞ, in terms of Floquet state basis gives

FðtÞ ¼Xþ1

n¼�1fnab einvt; (10)

HðtÞ ¼Xþ1

n¼�1Hn

ab einvt;

where fnab and Hn

ab are the Fourier amplitudes correspond-ing to a particular value of n (photon number). Substitutingthese expansions in the Schrodinger equation, one obtainsan infinite set of recursion relations for fn

ab:
Xgk
Hn�kag þ nv dag dnk

h iFk

gb ¼ eb fnab: (11)

The term in the bracket represents time independentFloquet HamiltonianHf. It follows from the Eq. (11) that thequasi energies are the eigenvalues of the Floquet equation.

detjHf :ebj ¼ 0: (12)

By diagonalizing the Floquet matrix Hf, one can obtainthe eigenvectors corresponding to eigenvalues eb. Thetransition probability for the excitation of the final state qfrom the initial state p summed over all the field states can bewritten as

Ptp!q ¼ qh jU t; t0ð Þ pj i

�� 2 ; (13)

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Phys. Status Solidi B 248, No. 7 (2011) 1729

Original

Paper

where U t; t0ð Þ is the evolution operator, however thequantity of experimental interest is the transition probabilityaveraged over the initial time (t0) keeping the elapsed time(t, t0) fixed, is given by

|Wel

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entia

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www

Ptt!t0

¼Xn

Xeb

Fnpb

D ��exp �ieb t�t0ð Þ� �

F0qb

�� E�� 2:(14)

Finally the continuous operation of the laser yields a longtime averaged transition probability given by

Pp!q ¼ limT!1

1

T

� � Z T

0

Ptp!qðtÞdt

¼Xn

Xeb

jFnpb F0

qbj2: (15)

Hf ¼
3 Computation We study the interaction of the laser
field with an ADQW in static electric field. The ADQWsystem consists of two GaAs coupled QWs separated byAlxGa1�xAs barrier. The potential function V(z) required forthe numerical solution is the conduction band edge. Theeigenenergies and the wavefunctions of the conduction bandof an ADQW in static field are obtained by solving the timeindependent Schrodinger equation for the system, usingfinite difference method. This is a numerical method forsolving the partial differential equation(PDE) based ondiscretization of the Hamiltonian on a spatial grid. TheSchrodinger equation in presence of static electric field,under finite difference method can be written as follows:

H0cþ ez Esc ¼ � 1

2m�e

c zjþ1

� ��2c zj

� �þ c zj�1

� �D2

�

þ V zj� �

c zj� �

þ ezj Esc zj� �

¼ E c zj� �

;

(16)

whereH0 is the bare Hamiltonian,m�e the effective mass of an

electron, E the eigenvalue in presence of static field,cðzjÞ theeigenvector of the Schrodinger equation, and D ¼ zjþ1�zj is

0

0.04

0.08

0.12

0.16

0.2

0.24

0.28

-30 -20 -10 0 10 20 30Well Width in nm

(a)

n1

n2

n3

n4

n5

n6QW

2nm

0

0.04

0.08

0.12

0.16

0.2

0.24

0.28

-30

(b

n3

n4

n5

n6

re 1 (onlinecolourat:www.pss-b.com)Thewavefunctionsand t

s¼ 10.0V/cm. BW in (a) is 2 nm and for (b) 4 nm.

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the spacing between the two neighboring discrete points. Inthe limit Es! 0 the above equation gives the eigenvalues andwavefunctions for the unperturbed system. The HamiltonianH0 þ ez Esð Þ is reduced to tridiagonal matrix and isdiagonalized using standard Fortran subroutines to obtainthe eigenvalues and the wavefunctions of an ADQW that oneconfined in the QW. This method has been implemented invarious semiconductor heterostructures [34–36] to obtain theunperturbed eigenvalues and wavefunctions. We thenimplement the Floquet theory to obtain the FloquetHamiltonian in the form of infinite matrix Hf. The FloquetHamiltonian so obtained is shown below

-20

)

hew

. ..

. ..

0 B 0 0 0 0

B Aþ 2Iv B 0 0 0

0 B Aþ Iv B 0 0

0 0 B A B 0

0 0 0 B A�Iv B

0 0 0 0 B A�2Iv. ..

. ..

0BBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCA

;

(17)

where

A ¼

0 0 0 0 0

0 Es1 0 0 0

0 0 Es2 0 0

0 0 0 Es3 0

0 0 0 0 Es4

0BBBB@

1CCCCA; (18)

s0 1

B ¼

0 Z12 0 0

Zs21 0 Zs

23 0

0 Zs32 0 Zs

34

0 0 Zs43 0

BB@ CCA: (19)

The Floquet Hamiltonian matrix is an infinite dimen-sional matrix (n ¼ 0;�1;�2; . . .) and possesses a block

-10 0 10 20 30Well Width in nm

n1

n2

QW

4nm

ellpotential asafunctionofwellwidthhavingstaticfieldstrength



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0 5 10 15 20 25 30 35

|D.M

.E.(i

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Static Field Strength in a.u.

P.B.W.=2 nm

(a)0-10-20-30-40-51-21-31-41-5

0

40

80

120

160

0 5 10 15 20 25 30 35Static Field Strength in a.u.

P.B.W.=4 nm

(b) 0-10-20-30-40-51-21-31-41-5

Figure 2 (online colour at: www.pss-b.com) Variation of dipole matrix elements (absolute value) as a function of static field strength.Potential barrier width (PBW) in (a) is 2 nm and for (b) 4 nm.

0

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0 2 4 6 8 10 12 14 16

Tra

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0

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1

0 2 4 6 8 10 12 14 16

Freq. in THz

(g)

0

0.2

0.4

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1

0 2 4 6 8 10 12 14 16

Es = 0.0 Kv/cm

(a)

0

0.2

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1

0 2 4 6 8 10 12 14 16

(e)

0

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1

0 2 4 6 8 10 12 14 16

Freq. in THz

(h)

0

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0.6

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1

0 2 4 6 8 10 12 14 16

Es = 5.0 Kv/cm

(b)

0

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1

0 2 4 6 8 10 12 14 16

Freq. in THz

(i)

G S0-10-20-30-40-5

0

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1

0 2 4 6 8 10 12 14 16

(f)

G S0-10-20-30-40-5

0

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0.6

0.8

1

0 2 4 6 8 10 12 14 16

Es = 10.0 Kv/cm

(c)

G S0-10-20-30-40-5

Figure 3 (online colour at:www.pss-b.com)Transition probability of various states as a functionof frequencyof laser field.Here, the laserfield strength isE0¼ 1.0� 105W/cm2 for subparts a,b,c,g,h,i andE0¼ 1.0� 104W/cm2 for subparts d,e,f. Static field strength is written asper column.TheLWisof15 nmwidthwhileRWis of10 nmwidth. Subparts a,b,c,d,e,f have2 nmBWand subparts g,h,i have4 nmBW.Thesymbol ‘GS’ in the figure shows the ground state and other symbols shows the transition probability of corresponding excited state fromground state.

� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com


Original

Paper

tridiagonal form. It has periodic structure with only numberofv varying in the diagonal elements from block to block. Inthe matrix Hf, E

sn are the energy eigenvalues of the DQW

system in presence of static fields and Zsmn ¼ � 1

2E0:m

smx

where Zsmn is off diagonal matrix and ms

mx denotes the dipolematrix element in presence of static electric fields. Theoptical behavior of the heterostructure is described by theoscillator strength of the transition between various energylevels of the QW defined as

Eig

enva

lues

in a

.u.(

x 1

0-3 )

Eig

enva

lues

in a

.u.(

x 1

0-3

)

FiguE0¼freq

www

fi!f ¼2m0

�h2ðef�eiÞðZijÞ2

¼ 2m0

e2�h2ðef�eiÞðmijÞ2;

(20)

m0 is the free electron mass (taken as m0¼ 1), ei theenergy of ith level, andmij is the dipole matrix element givenby

mij ¼Z

c�f ðzÞez ciðzÞdz: (21)

We obtain the quasi energies eb and correspondingFourier components fn

ab by diagonalizing the Floquet

-40

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40

80

120

0 5 10 15 20 25 30 35

Static Field Strength in kV/cm

(c)

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0

40

80

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0 5 10 15 20 25 30 35


(a)

re 4 (online colour at: www.pss-b.com) Eigen values are plo1.0� 104W/cm2 for subparts a,b and E0¼ 1.0� 105W/cm2 fouency of laser field is v¼ 4.317 THz.

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Hamiltonian (17). In our calculations, Hf (17) is truncatedto contain seven photon blocks, i.e., n ¼ 0;�1;�2;�3),although the results were found to converge after truncatingthe Floquet Hamiltonian by including only five photonblocks, i.e., n ¼ 0;�1;�2).

4 Results and discussion In this paper, we havetheoretically investigated the intersubband transitions inADQW assuming that the conduction band electron issubjected to the action from external static and anelectromagnetic field. The ADQW under study consists oftwowells, as left well (LW) and rightwell (RW) separated bya barrier width (BW). The well and the barrier have differenteffective masses; effective mass of the well materialmw ¼ 0:068m�

e and the effective mass of the barriermaterial mb ¼ 0:068m�

e þ x� 0:083m�e and barrier height

V0¼ 228meV. These parameters are suitable forGaAs=AlxGa1�xAs DQW with aluminum concentration ofx ’ 0:025. The parameters ensure presence of six conduc-tion band levels. We show in Fig. 1 the well structure and thewavefunctions for two different BWswith static electric field(Es¼ 10 kV/cm). The ADQWunder study consists of LW of15 nm and RW of 10 nm widths separated by a BW.

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(d)

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(b) n1n2n3n4n5n6

tted as a function of static field strength. Laser field strength isr subparts c,d. BW is 2 nm for a,c and 4 nm for subparts b,d. The



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Figure 1(a) has a BW of 2 nm and Fig. 1(b) has BW of 4 nm.The change in the wavefunctions are clearly visible andenergy levels are also shifted.

It can be observed that, with increase in static field andwell width, the wavefunctions change and this change inwavefunctions further leads to change in the dipole matrixelements (Fig. 2).

In Fig. 2, we present the variation of magnitude of dipolematrix elements between different well levels as a functionof static field strength. Figure 2(a) has a BW of 2 nm andFig. 2(b) has BW of 4 nm. The plot shows that the somedipole matrix elements mif decrease with Es and someincreases withEs. This change inmifwith theEs is because ofthe change in wave functions with the static electric field Es.This kind of effect is also observed by others [23, 37].

In Fig. 3, we show the variation of time averagedtransition probabilities for an ADQW for various transitionsfrom the ground state, with frequency of the laser field. Theresults are plotted for laser field intensity of 105W/cm2. Theresults are plotted for LW¼ 15 nm, RW¼ 10 nm, andtwo different BWs BW¼ 2 and 4 nm. The variation intransition probabilitieswith change in static electric field andBW is compared. The GS curve in the figure shows the

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0 2 4 6 8 10 12 14 16 18

Eig

en V

alue

s in

me

V

Frequency of Laser in THz

(c)

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0 2 4 6 8 10 12 14 16 18

Eig

en V

alue

s in

me

V


(a)

Figure 5 (online colour at: www.pss-b.com) Eigen values are plottE0¼ 1.0� 105W/cm2. Static field strength is 0.0 kV/cm for subparts asubparts c,d.


ground state probability (i.e., survival probability of groundstate).

Figures 3(a–c) show results for static field strengthsEs¼ 0.0, Es¼ 5 kV/cm, and Es¼ 10 kV/cm, respectively. Itis observed that the states are resonantly coupled with thefield, when the field frequency is close to one-photon, two-photon, three-photon, four-photon, and five photon tran-sitions between the ground and the excited states. The energydifference [for Fig. 3(a)] between the states are 3.43 (for0! 1), 11.90 (for 0! 2), 23.74 (for 0! 3), 32.73 (for0! 4), and 50.70 (for 0! 5) in terahertz (THz) unit. Similarenergy differences are for every subfigure. Hence, it shows alinear (resonant) absorptions corresponding to the intersub-band transitions between ground and excited states. With theincrease in static field strength, the resonance shows a redshift. It is also observed that with the increase in the staticfield strength, the higher photon transitions get enhanced i.e.,the static field helps in multiphoton transitions. The timeaveraged transition probabilities can be physically inter-preted as the character computation of the Floquet states. Aswithin resonance condition the Floquet states get mixed i.e.,curve crossing takes place. It is interesting to note thatclosely related phenomena have been predicted and observed

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(b) n1n2n3n4n5n6

ed as a function of frequency of laser field. Laser field strength is,c and 5.0 kV/cm for b,d subparts. BW is 2 nm for a,b and 4 nm for

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Original

Paper

in atomic and molecular physics [38]. Very recently, anexperimental observation of coherent control of singleparticle has been reported in strongly driven DQW systems,and good agreement with the experiment has been achievedusing Floquet theory [39].

The subband energy levels as a function of static electricfield strength (Es) for different laser parameters andconfinement geometries are shown in Fig. 4. The parametersfor the figure are explained in caption of the figure. It is clearfrom the Fig. 4 that ground state energy first decreases withEs, then increases and finally becomes constant.

Figure 5 shows the variation of QES eigenvalues ofFloquetHamiltonianHf for the states that are confinedwithinthe ADQW as a function of laser frequency, for Es¼ 0.0 and5.0 kV/cm. We have taken E0¼ 1.0� 105W/cm2 andADQW system under study consists of LW of 15 nm andRWof 10 nmwidths separated by a BWdefined in caption ofFig. 5. The anticrossing in the eigen values show a transitionat that particular frequency from one state to another. Thequasi energies reveal a change of character with thefrequency of the field. It is observed that the changes occur,even when, the quasi energies approach each other.

Such results have been obtained and explained by others[40–42].

5 Summary and conclusions In this work, the effectof an intense resonant laser field in the excitation process in aGaAs=AlxGa1�xAs ADQW in an applied electric field ispresented. Our work shows that intersubband transitions canbe controlled not only byBWbut also by laser field and staticelectric field. The intersubband transition probabilities canbe controlled with the help of laser field intensity as well asby the static electric field(as shown in Fig. 3). Also resonanceof the intersubband transitions can be shifted by the staticelectric field strength. Our calculations show some signifi-cant modifications of the intersubband dynamics due tocompeting effects of the well confinement and the externalapplied fields. The dependence of optical intersubbandtransitions in asymmetric double quantum structure geome-try and parameters shown in this work suggests a way inwhich experiments may be carried out to examine physicalphenomena related to modifications of optical intersubbandtransitions.

Acknowledgements The authors are grateful to theunknown referee for valuable suggestions for the improvementof the paper.

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Modifications of laser field assisted intersubband transitions in the coupled quantum wells due to static electric field