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Paspalakis and Boviatsis NanoscaleResearchLetters 2012, 7:478

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NANO EXPRESS Open Access

Ultrashort electromagnetic pulse control of

intersubband quantum well transitions

Emmanuel Paspalakis1*and John Boviatsis2

Abstract

We study the creation of high-efficiency controlled population transfer in intersubband transitions of semiconductor

quantum wells. We give emphasis to the case of interaction of the semiconductor quantum well with

electromagnetic pulses with a duration of few cycles and even a single cycle. We numerically solve the effective

nonlinear Bloch equations for a specific double GaAs/AlGaAs quantum well structure, taking into account the

ultrashort nature of the applied field, and show that high-efficiency population inversion is possible for specific pulse

areas. The dependence of the efficiency of population transfer on the electron sheet density and the carrier envelope

phase of the pulse is also explored. For electromagnetic pulses with a duration of several cycles, we find that the

change in the electron sheet density leads to a very different response of the population in the two subbands to pulse

area. However, for pulses with a duration equal to or shorter than 3 cycles, we show that efficient population transfer

between the two subbands is possible, independent of the value of electron sheet density, if the pulse area is π.

Keywords: Coherent control, Semiconductor quantum well, Intersubband transition, Ultrashort electromagnetic

pulse

Background

The coherent interaction of electromagnetic fields with

intersubband transitions in semiconductor quantum

wells has led to the experimental observation of sev-

eral interesting and potentially useful effects, such as

tunneling-induced transparency [1,2], electromagneti-

cally induced transparency [3], Rabi oscillations [4,5],

self-induced transparency [5], pulsed-induced quan-

tum interference [6], Autler-Townes splitting [7,8],

gain without inversion [9], and Fano signatures in

the optical response [10]. In most of these stud-

ies, atomic-like multi-level theoretical approaches have

been used for the description of the optical proper-

ties and the electron dynamics of the intersubband

transitions.

Many-body effects arising from the macroscopic car-

rier density have also been included in a large number of

theoreticalandexperimentalstudiesofintersubbandexci-

tation in semiconductor quantum wells [6,10-38]. These

*Correspondence: paspalak@upatras.gr

1Materials Science Department, School of Natural Sciences, University of

Patras, Patras, 26504, Greece

Full list of author information is available at the end of the article

studies have shown that the linear and nonlinear opti-

cal responses and the electron dynamics of intersubband

quantum well transitions can be significantly influenced

by changing the electron sheet density.

An interesting problem in this area is the creation

of controlled population transfer between two quan-

tum well subbands [23-27,29,30]. This problem was first

studied by Batista and Citrin [23] including the many-

body effects arising from the macroscopic carrier density

of the system. They showed that the inclusion of the

electron-electron interactions makes the system behave

quite differently from an atomic-like two-level system.

To have a successful high-efficiency population transfer

in a two-subband, n-type, modulation-doped semicon-

ductor quantum well, they used the interaction with a

specific chirped electromagnetic field, i.e., a field with

time-dependent frequency. They showed that a combi-

nation of π pulses with time-dependent frequency that

follow the population inversion can lead to high-efficiency

population inversion. Their method was refined in a fol-

lowing publication where only linearly chirped pulses

were used for high-efficiency population transfer [27]

and was also applied to three-subband quantum well

systems [26].

© 2012 Paspalakis and Boviatsis; licensee Springer. This is an Open Access article distributed under the terms of the Creative

Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

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Different approaches for creating high-efficiency inter-

subband population transfer were also proposed by our

group[24,25,29,30].Usinganalyticalsolutionsoftheeffec-

tive nonlinear Bloch equations [20], under the rotating

wave approximation, we presented closed-form analyti-

cal solutions for the electric field amplitude of the elec-

tromagnetic field that leads to high-efficiency popula-

tion transfer [24,25]. In addition, closed-form conditions

for high-efficiency transfer were also presented [24,29].

Moreover, efficient population transfer is found when a

two-subband system interacts with a strong chirped elec-

tromagnetic pulse, for several values of the chirp rate and

the electric field amplitude [30].

In this article, we continue our work on the creation

of high-efficiency controlled population transfer in inter-

subband transitions of semiconductor quantum wells. We

give emphasis to the case of interaction of the semicon-

ductor quantum well with electromagnetic pulses with

a duration of few cycles and even a single cycle. We

numerically solve the effective nonlinear Bloch equations

[20] for a specific double GaAs/AlGaAs quantum well

structure, taking into account the ultrashort nature of

the applied field, and show that high-efficiency popula-

tion inversion is possible for specific pulse areas. The

dependence of the efficiency of population transfer on

the electron sheet density and the carrier envelope phase

of the pulse is also explored. More specifically, we find

that for electromagnetic pulses with duration of several

cycles, the change in the electron sheet density leads

to a very different response of the population in the

two subbands to pulse area. However, a π pulse with

a duration equal to or shorter than 3 cycles can lead

to efficient population transfer between the two sub-

bands independent of the value of electron sheet density.

We note that the interaction of ultrashort electromag-

netic pulses with atoms has been studied in the past

decade, giving emphasis either to ionization effects [39-

41] or to population dynamics in bound two-level and

multi-level systems [40,42-46]. Also, the interaction of

ultrashortelectromagneticpulseswithintersubbandtran-

sitions of semiconductor quantum wells has been recently

studied[47,48],butwithouttakingintoaccounttheeffects

of electron-electron interactions in the system dynamics.

Methods

The system under study is a symmetric double semicon-

ductor quantum well. We assume that only the two lower

energy subbands, n = 0 for the lowest subband and

n = 1 for the excited subband, contribute to the system

dynamics. The Fermi level is below the n = 1 subband

minimum, so the excited subband is initially empty. This

is succeeded by a proper choice of the electron sheet den-

sity. The two subbands are coupled by a time-dependent

electric field E(t). Olaya-Castro et al. [20] showed that the

system dynamics is described by the following effective

nonlinear Bloch equations:

˙S1(t) = [ω10− γS3(t)]S2(t) −S1(t)

˙S2(t) = −[ω10− γS3(t)]S1(t) + 2[μE(t)

−S2(t)

T2

˙S3(t) = −2[μE(t)

?

Here, S1(t) and S2(t) are, respectively, the mean real

and imaginary parts of polarization, and S3(t) is the mean

population inversion per electron (difference of the occu-

pation probabilities in the upper and lower subbands).

Also, μ = ez01 is the electric dipole matrix element

betweenthetwosubbands,andtheparametersω10,β,and

γ are given by

ω10 =E1− E0

?

γ =πe2

β =πe2

Here, N is the electron sheet density, ε is the relative

dielectric constant, e is the electron charge, E0and E1are

theeigenvaluesofenergyforthegroundandexcitedstates

inthewell,respectively,andLijkl=? ?dzdz?ξi(z)ξj(z?)|z−

wavefunction for the ith subband along the growth direc-

tion (z-axis). Finally, in Equations 1 to 3, the terms con-

taining the population decay time T1and the dephasing

time T2describe relaxation processes in the quantum well

and have been added phenomenologically in the effective

nonlinear Bloch equations. If there is no relaxation in the

system T1,T2→ ∞, then S2

In comparison with the atomic (regular) optical Bloch

equations [49], we note that in the effective nonlinear

Bloch equations, the electron-electron interactions renor-

malize the transition frequency by a time-independent

term (see Equation 4). The parameter γ consists of two

compensating terms: the self-energy term and the vertex

term [20]. In addition, the applied field contribution is

screenedbytheinducedpolarizationtermwithcoefficient

β. The screening is due to exchange correction. Surpris-

ingly, the exchange corrections appear with terms which

are linearly dependent on the electron sheet density, as all

exchange terms which present a nonlinear dependence on

the electron sheet density are exactly canceled out due to

the interplay of self-energy and vertex corrections [20].

For very short electromagnetic pulses, pulses that

include only a few cycles, the field envelope may change

T2

, (1)

?

− βS1(t)]S3(t)

, (2)

− βS1(t)]S2(t) −S3(t) + 1

T1

. (3)

+πe2

L1001−L1111+ L0000

?εNL1111− L0000

2

, (4)

?εN

?

2

?

, (5)

?εNL1100.(6)

z?|ξk(z?)ξl(z), with i,j,k,l = 0,1. Also, ξi(z) is the envelope

1(t) + S2

2(t) + S2

3(t) = 1.

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significantly within a single period. In such a case, one

should first define the vector potential and then use it

to obtain the electric field; otherwise, unphysical results

may be obtained [39-43,47,48]. So, the electric field E(t)

is defined via the vector potential A(t) as E(t) = −∂A/∂t

[39-43,47,48] where

A(t) = A0f(t)cos(ωt + ϕ).

Here, A0is the peak amplitude of the vector potential,

f(t) is the dimensionless field envelope, ω is the angular

frequency, and ϕ is the carrier envelope phase of the field.

The form of the electric field becomes

E(t) = ωA0f(t)sin(ωt + ϕ) − A0∂f

In the above formula, the first term corresponds to an

electromagnetic pulse with a sine-oscillating carrier field,

while the second term arises because of the finite pulse

duration. This second term can be neglected for pulses

with a duration of several cycles, but has an important

effect in the single-cycle regime [39-43,47,48].

If the electron-electron interactions are neglected, then

the nonlinear effective Bloch equations coincide with the

optical Bloch equations of a two-level atom [49]. In this

case, in the limit of no relaxation processes (T1,T2→ ∞),

if the ultrashort pulse effects are neglected and under the

rotating wave approximation, the population inversion,

with the initial population in the lower state, is given by

(7)

∂tcos(ωt + ϕ). (8)

S3(t) = −cos[?(t)],?(t) = −

where ?(t) is the time-dependent pulse area [49]. At the

end of the pulse, ?(t) takes a constant value that is known

as pulse area θ. Equation 9 clearly shows how important

pulse area can be. If θ is an odd multiple of π, then com-

plete inversion between the two states is found at the end

of the pulse, while if θ is an even multiple of π, then the

population returns to the lower state at the end of the

pulse.

?t

0

μωA0f(t?)

?

dt?, (9)

Results and discussion

In the current section, we present numerical results from

the solution of the nonlinear Bloch equations, Equations

1 to 3, for a specific semiconductor quantum well system.

We consider a GaAs/AlGaAs double quantum well. The

structure consists of two GaAs symmetric square wells

with a width of 5.5 nm and a height of 219 meV. The

wells are separated by a AlGaAs barrier with a width of

1.1 nm. The form of the quantum well structure and the

corresponding envelope wavefunctions are presented in

Figure 1.

This system has been studied in several previous works

[20,24,25,28,35-38]. The electron sheet density takes val-

ues between 109and 7 × 1011cm−2. These values ensure

that the system is initially in the lowest subband, so the

10

5

05 10

0

50

100

150

200

z nm

V z

meV ,

nz

Figure 1 Quantum well structure and corresponding envelope

wavefunctions. The confinement potential of the quantum well

structure under study (blue solid curve) and the energies of the lower

(green lower line) and upper states (red upper line). The envelope

wavefunctions for the ground (dotted curve) and first excited (dashed

curve) subbands.

initial conditions can be taken as S1(0) = S2(0) = 0 and

S3(0) = −1. The relevant parameters are calculated to

be E1− E0 = 44.955 meV and z01 = −3.29 nm. Also,

for electron sheet density N = 5 × 1011cm−2, we obtain

πe2N(L1111− L0000)/2ε = 1.03 meV, ?γ = 0.2375 meV,

and ?β = −3.9 meV. In all calculations, we include the

populationdecayanddephasingrateswithvaluesT1= 10

ps and T2= 1 ps. Also, in all calculations, the angular fre-

quency of the field is at exact resonance with the modified

frequency ω10, i.e., ω = ω10.

In Figure 2, we present the time evolution of the inver-

sion S3(t) for different values of the electron sheet density

for a Gaussian-shaped pulse with f(t) = e−4ln2(t−2tp)2/t2

Here, tp= 2πnp/ω is the duration (full width at half max-

imum) of the pulse, where npis the number of cycles of

the pulse and can be a noninteger number. The compu-

tation is in the time period [0,4tp] for pulse area θ = π.

For electron sheet density N = 109cm−2, which is a

small electron sheet density, Equations 1 to 3 are very

well approximated by the atomic optical Bloch equations;

therefore, a π pulse leads to some inversion in the sys-

tem in the case that the pulse contains several cycles.

However, the inversion is not complete as the relaxation

processes are included in the calculation and T2is smaller

than the pulse duration. In Figure 2a, that is for np =

10, we see that the electron sheet densities have a very

strong influence in the inversion dynamics. For example,

for N = 3 × 1011cm−2, the population inversion evolves

to a smaller value, and for larger values of electron sheet

density, the final inversion decreases further and even

becomes nonexistent.

A quite different behavior is found in Figure 2b,c,d for

pulses with smaller number of cycles. In Figure 2b, we

see that essentially the inversion dynamics differs slightly

p.

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0.00.51.01.5

Time ps

2.02.5 3.0 3.5

1.0

0.5

0.0

0.5

1.0

Inversion

ab

cd

1.0

0.5

0.0

0.5

1.0

Inversion

1.0

0.5

0.0

0.5

1.0

Inversion

1.0

0.5

0.0

0.5

1.0

Inversion

0.0

0.20.4 0.60.81.0

Time ps

0.00.10.2 0.3

Time ps

0.4 0.5 0.60.7

0.000.05 0.100.15

Time ps

0.20 0.25 0.300.35

Figure 2 The time evolution of the inversion S3(t) for a Gaussian pulse. The excitation is on-resonance, i.e., ω = ω10, the pulse area is θ = π,

and ϕ = 0. (a) np= 10, (b) np= 3, (c) np= 2, and (d) np= 1. Solid curve: N = 109cm−2, dotted curve: N = 3 × 1011cm−2, dashed curve:

N = 5 × 1011cm−2, and dot-dashed curve: N = 7 × 1011cm−2.

0.00.51.0 1.5

Time ps

Inversion

0.00.1 0.20.30.40.5

Time ps

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Time ps

0.00

0.050.10

0.15

Time ps

1.0

0.5

0.0

0.5

1.0

Inversion

1.0

0.5

0.0

0.5

1.0

Inversion

1.0

0.5

0.0

0.5

1.0

Inversion

1.0

0.5

0.0

0.5

1.0

a

b

c

d

Figure3 The time evolution of the inversion S3(t) for a sin-squared pulse. The excitation is on-resonance, i.e., ω = ω10, the pulse area is θ = π,

and ϕ = 0. (a) np= 10, (b) np= 3, (c) np= 2, and (d) np= 1. Solid curve: N = 109cm−2, dotted curve: N = 3 × 1011cm−2, dashed curve:

N = 5 × 1011cm−2, and dot-dashed curve: N = 7 × 1011cm−2.

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0.0

0.5 1.0

Pulse Area

1.52.0 2.5

3.0

1.0

0.5

0.0

0.5

1.0

Final Inversion

1.0

0.5

0.0

0.5

1.0

Final Inversion

1.0

0.5

0.0

0.5

1.0

Final Inversion

1.0

0.5

0.0

0.5

1.0

Final Inversion

0.0

0.5 1.0

Pulse Area

1.52.02.5

3.0

0.0

0.5 1.0

Pulse Area

1.52.02.5

3.0

0.0

0.5 1.0

Pulse Area

1.5 2.0 2.5

3.0

a

b

c

d

Figure 4 Final inversion S3(2tp) for sin-squared pulse as a function of pulse area θ. The pulse area is in multiples of π. The excitation is

on-resonance and ϕ = 0. (a) np= 10, (b) np= 3, (c) np= 2, and (d) np= 1. Solid curve: N = 109cm−2, dotted curve: N = 3 × 1011cm−2, dashed

curve: N = 5 × 1011cm−2, and dot-dashed curve: N = 7 × 1011cm−2.

for N = 109cm−2, N = 3 × 1011cm−2, and N =

5 × 1011cm−2and all of these values lead to essentially

the same final inversion. There is only a small differ-

ence in the inversion dynamics for the case of N =

7 × 1011cm−2that leads to slightly smaller inversion.

For even smaller number of cycles, Figure 2c,d, the inver-

sion dynamics differs slightly for all the values of electron

sheet density, and the final value of inversion is practi-

cally the same, independent of the value of electron sheet

density. We note that the largest values of inversion are

obtained for np = 2 and np = 3 and not for np = 1,

as one may expect, as in the latter case the influence of

the decay mechanisms will be weaker. However, the sec-

ond term on the right-hand side of the electric field of

Equation 8 influences the dynamics for np= 1, and in this

case, the pulse area θ = π does not lead to the largest

inversion [42].

Similar results to that of Figure 2 are also obtained for

the case of sin-squared pulse shape with f(t) = sin2(πt

that are presented in Figure 3. In this case, the compu-

tation is in the time period [0,2tp] and the pulse area is

again θ = π. We have also found similar results for other

pulse shapes, e.g., for hyperbolic secant pulses. These

results show that the present findings do not depend on

2tp)

0.00.51.0 1.52.0

0.830

0.835

0.840

0.845

0.850

Final Inversion

0.0

0.5 1.01.5

2.0

0.73

0.74

0.75

0.76

0.77

0.78

0.79

0.80

Final Inversion

a

b

Figure 5 Final inversion S3(2tp) for sin-squared pulse as a function of carrier envelope phase ϕ. The carrier envelope phase is in multiples of

π. The excitation is on-resonance and the pulse area is θ = π. (a) np= 2 and (b) np= 1. Solid curve: N = 109cm−2, dotted curve: N = 3 × 1011

cm−2, dashed curve: N = 5 × 1011cm−2, and dot-dashed curve: N = 7 × 1011cm−2.

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the actual pulse shape, as long as a typical smooth pulse

shape is used.

In order to explore further the dependence of the inver-

sion in pulse area, we present in Figure 4 the final inver-

sion, i.e., the value of the inversion at the end of the pulse,

as a function of the pulse area θ for a sin-squared pulse.

We find that for pulses of several cycles, e.g., np= 10, the

pulse areas for maximum inversion can be quite different

than π depending on the value of electron sheet density.

For example, for N = 5 × 1011cm−2, the pulse area is

about 1.5π, and for N = 7 × 1011cm−2, the pulse area

is about 2.1π. Similar results have also been obtained for

other pulse shapes, e.g., Gaussian and hyperbolic secant

pulses. The displayed dependence explains the results of

Figure 3a (and of Figure 2a), as one may see that a π pulse

area leads to some final inversion for N = 109cm−2and

N = 3 × 1011cm−2but gives very small final inversion

for N = 5 × 1011cm−2and N = 7 × 1011cm−2. How-

ever, for pulses with 3 cycles or with a smaller number of

cycles, the maximum inversion occurs for pulse area π or

very close to π (and odd multiples of π if the figures are

extended in higher pulse areas) independent of the value

of electron sheet density.

An interesting effect in the interaction of an ultra-

short electromagnetic pulse with a multi-level system is

the influence of the carrier envelope phase ϕ on the

populations of the quantum states [40,43,45,46,48]. In

Figure 5, we present the dependence of the final inver-

sion on the carrier envelope phase ϕ for a sin-squared

pulse with θ = π for different number of cycles and

electron sheet densities. We find that there is a depen-

dence of the final inversion on the carrier envelope

phase and this dependence is strongest for larger sheet

electron densities and for pulses with smaller number

of cycles.

Conclusions

In this work, we have studied the electron dynamics of

intersubband transitions of a symmetric double quantum

well, in the two-subband approximation, that is coupled

byastrongpulsedelectromagneticfield.Wehaveusedthe

effective nonlinear Bloch equations [20] for the descrip-

tion of the system dynamics, giving specific emphasis

to the interaction of the quantum well structure with

few-cycle pulses. We have found that high-efficiency pop-

ulation inversion is possible for specific pulse areas. The

dependence of the efficiency of population transfer on the

electron sheet density and the carrier envelope phase of

the pulse has also been explored. More specifically, we

have shown that for electromagnetic pulses with a dura-

tion of several cycles, the change in the electron sheet

densityleadstoaverydifferentresponseofthepopulation

in the two subbands to pulse area. However, electromag-

netic pulses with pulse area π or close to π and with

duration equal to or shorter than 3 cycles can lead to

efficient population transfer between the two subbands

independent of the value of electron sheet density.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

EPconceivedanddevelopedtheideaforthestudy,performedthesimulations,

and wrote the main part of the manuscript. JB contributed in the development

of the original idea, in the analysis of the results, and in the writing of the

manuscript. Both authors read and approved the final manuscript.

Authors’ information

EP holds a PhD degree from the Physics Department of Imperial College from

1999. In 2001, he joined the Materials Science Department of the University of

Patras, where he is currently an assistant professor. JB holds a PhD degree from

the Department of Physics of the University of Patras from 1982. From 1991 to

2001, he was an assistant professor of Physics at the Department of Applied

Sciences of the Technological Educational Institute of Chalkis. In 2001, he

joined the Department of Renovation and Restoration of Buildings of the

Technological Educational Institute of Patras, where he is currently a professor

of Physics.

Acknowledgements

This research has been co-financed by the European Union (European Social

Fund - ESF) and Greek national funds through the operational program

‘Education and Lifelong Learning’ of the National Strategic Reference

Framework (NSRF) - Research Funding Program: Archimedes III. Discussions

with AF Terzis and CH Keitel are gratefully acknowledged.

Author details

1Materials Science Department, School of Natural Sciences, University of

Patras, Patras, 26504, Greece.2Technological and Educational Institute of

Patras, Megalou Alexandrou 1, Patras, 26334, Greece.

Received: 18 July 2012 Accepted: 3 August 2012

Published: 23 August 2012

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doi:10.1186/1556-276X-7-478

Cite this article as: Paspalakis and Boviatsis: Ultrashort electromagnetic

pulsecontrolofintersubbandquantumwelltransitions.NanoscaleResearch

Letters 2012 7:478.

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