Selected Papers

Coherent Nonlinear Optical Properties in
Quantum Dots
Hideki Gotoh† and Hidehiko Kamada
Abstract
This paper discusses optical nonlinear functions in semiconductor quantum dots (QDs) and theirunderlying physics and possible applications. The optical functions involve controlling light with noenergy dissipation. The applications include conventional devices as well as future devices based on
quantum phenomena, so they will be very useful in various technological areas. Theoretical results
explain the physical principle of the nonlinear functions and the concept of the coherent slowdown oflight. Experiments with actual QDs show that their optical responses agree well with the theoreticalresults. Light absorption was successfully controlled in a QD. These results clearly reveal that QDs provide intriguing optical functions and possible innovative applications.

1. Introduction
Semiconductor quantum dots (QDs) have been
attracting considerable attention because they may
have valuable applications in both conventional andfuture devices [1]. A QD is a semiconductor nanostructure, as shown in Fig. 1(a). Here, an electron is
confined in the InGaAs quantum dot because InGaAs
has a smaller band gap than AlGaAs. QDs can confine electrons inside their structures and behave likeatoms, but unlike atoms, their properties can be artificially designed. They can greatly improve the performance of many optical and electronic devices [1]and have been predicted to be good candidates for thefundamental elements in quantum information
processes [2]. They could potentially lead to many
useful technological innovations in many application
areas.
An exciton, which is composite particle consistingof an electron bound to a hole, can be created in a QD
by laser light irradiation. The processes for creating asingle exciton (X) and a biexciton, which is a pair of
excitons (XX), in a QD are shown in Fig. 1(b). X andXX exist stably in the QD and contribute to the quantum information process as fundamental quantum† NTT Basic Research Laboratories
Atsugi-shi, 243-0198 Japan
E-mail: gotoh@will.brl.ntt.co.jp

Vol. 3 No. 12 Dec. 2005

bits (qubits). These fundamental bits can be represented as in Fig. 1(c), where the ground state g is |0〉= |00〉, X is |1〉 = |01〉, and XX is |11〉. X and XX in a
QD provide a one-qubit rotation gate and a two-qubit
correlation gate. These two gates are necessary forquantum computing, which is expected to lead to aparadigm shift in computing and information processing [3]. QDs will contribute to the developmentof next-generation computers.
The qubits in QDs have another characteristic. The
creation processes of X and XX can be coherent
processes; that is, they can be reversible with no energy dissipation. This allows us to control the creationand erasure processes using appropriate lights. These
control processes correspond to the gating processes
with the qubits in computing. QDs are known to preserve coherence in the creation and erasure processes
longer than other semiconductors [4]-[6]. This allows
us to make many gating processes, which are indispensable when performing quantum algorithms.
The system composed of X and XX (Fig. 1(c)) has
three independent states that interact via lights. Since
X and XX are created and erased coherently, thisthree-level system can be called a coherent threelevel system. Several intriguing optical functions canbe expected in a coherent three-level system such as
quantum interference effects in photoabsorption,
inversionless lasing, and lossless light speed control[7]. These phenomena are known as examples of19

Selected Papers(a)
AlGaAsInGaAs

3 nm
30 nm
Schematic

(b)

SEM

XX

X
hωXX

hωX

g

(c)
|11〉

XX
hωXX

|1〉 = |01〉

X
hωX

g

|0〉 = |00〉

Fig. 1. Characteristics of quantum dots. (a) Schematic
drawing and scanning electron micrograph of aquantum dot. (b) Creation processes of an exciton(X) and a biexciton (XX) in a quantum dot. (c)Energy level structure of (b).

electromagnetically induced transparency (EIT) inatoms and ions. All these functions are optical nonlinear processes that require two kinds of independent
lights such as hωX and hωXX in Fig. 1(c). In particular,
the function that allows us to control the speed oflight is very useful in optical switching or routing asa temporary optical buffer memory and in quantum
computing for tuning many gating pulses. The opticalnonlinear properties in a coherent three-level system
in a QD yield valuable applications in conventionaland future devices.
EIT processes were proposed in a theoretical analysis of atoms interacting with lights [7]. This concept
20

has been experimentally demonstrated in several
actual ions and atoms [8]. The demonstration utilized
several states of atoms; lights corresponding to hωXand hωXX in Fig. 1(c) were chosen so that their energies were resonant with the atomic states. Since theresonant energies were determined by atoms andions, the optical functions were not controllable interms of light wavelengths and corresponding energies. In contrast, QD state energies change according
to the semiconductors used to make them and the QD
size. These properties allow us to achieve the optical
functions with controllable light wavelengths, whichmakes them very useful for actual applications.
In this paper, we present the fundamental physics ofand corresponding experimental results for the optical nonlinear properties of QDs. The coherent controlof light is highlighted in several optical processes.
Possible controllability is discussed based on the
results of a theoretical analysis. We present the resultsof an experimental demonstration of the nonlinear
properties of actual QDs and discuss the present
experimental status and future prospects. Theseexperimental results are an important step towards
coherent control of light.
Section 2 describes optical responses such as theabsorption and refractive index spectra and presents
the results of a theoretical calculation for a coherent
three-level system. Characteristic shapes appear in
the spectra as a result of coherent quantum interference effects. Processes for slowing light are induced
by using the large derivative value in the refractive
index spectrum. Section 3 describes experiments on
actual QDs. It also describes a method for evaluating
the nonlinear coherent process. Experimentally measured absorption spectra were similar to theoretically
predicted ones. Preliminary results for controlling theabsorption properties of a light (probe light) by using
another light (pump light) are also reported.
2. Coherent control of light by using opticalnonlinear effects
Let us consider a coherent three-level system with
two lights called the pump and probe in Fig. 2(a). The
intensity of the probe light is assumed to be much
weaker than that of the pump beam. The pump beamenergy is fixed so that it is resonant with the energy
difference between states 2 and 3. The opticalresponses of this system can be analyzed using asemi-classical theory with the density matrix method[9]. The density matrix method can include coherent
effects between three levels and yields important
NTT Technical Review

Selected Papers

(a)

(b)
3

Dip structure

Pump

Absorption
2

Probe
1

A

B

EnergyRefractive
index: nPositive slope

Fig. 2. (a) Coherent three-level system used in a theoretical analysis.
(b) Calculated optical responses for the three-level system.

optical properties such as the absorption and refractive index spectra. These optical properties play a crucial role in determining the performance of various
optical devices. The rate equation method is often
used to analyze the characteristics of laser diodes.
Although it can simulate the main features of thesedevices, it assumes that the coherent effects are negligible, so it is unsuitable for analyzing the system
shown in Fig. 2(a).
Examples of calculated absorption and refractive
index spectra for the three level-system are shown in
Fig. 2(b) as a function of probe light energy. These
spectra have several characteristic features. A dipshaped structure can be seen in the absorption spectrum. This results from coherent interference effects.
In Fig. 2(a), there are two possible ways for the probe
beam to be absorbed. One involves a simple opticalabsorption process from state 1 directly to state 2(process A). The other involves optical absorption
from state 1 to state 2 via state 3 mediated by thepump beam (process B). The latter process occurs
when the pump beam intensity is strong, as we
assumed for our calculation. Since the optical absorption of process B interferes destructively with that ofprocess A, the absorption disappears when the probe
beam energy is resonant with the energy difference
between states 1 and 2. On the other hand, when theprobe beam energy is not in a resonant condition, thedestructive interference effect decreases and strong
photoabsorption can be seen. As a result, a dipshaped structure is seen in the absorption spectrum.
Since a region that is transparent to the probe light iscreated by another light (pump light), which is anVol. 3 No. 12 Dec. 2005

electromagnetic wave, the phenomenon is called
electromagnetically induced transparency (EIT).
The dip-structure in the absorption spectrum produces a region in the refractive index spectrum thathas a steep positive slope, as seen in Fig. 2(b). In thetheoretical analysis, the density matrix method yields
these two spectra (absorption and refractive index).
Alternatively, a spectrum can be calculated from theother spectra using the Kramers-Kronig relation.
According to this relation, the dip structure in theabsorption spectrum leads to a positive-slope region
in the refractive index, which yields a function forslowing light. When the probe light is a pulsed light,
two kinds of velocity can be determined: the phasevelocity and the group velocity. The group velocity is
the effective speed of the pulse envelope and it determines the transmission time of information carried byseveral optical pulses. The group velocity vg may be
expressed asvg =cn(hω ) + hω∂n(hω )
∂hω

,

(1)

where c is the velocity of light in a vacuum and n(hω)
is the refractive index of the medium. The large positive slope such as that seen in Fig. 2(b) results in thelarge denominator in Eq. (1). This in turn results in asmall vg value and a reduction in the speed of theprobe light induced by quantum interference effects.
This slowdown effect depends strongly on the slopeof the refractive index and the shape of the absorption
spectrum. If the pump light can control the shapes ofthese spectra, then the slowdown function is control21

lable.
The slowdown function can be characterized by thedenominator in Eq. (1). We define this denominator
as slowdown factor S; namely,(a)∂n(hω )
(2).
∂hωThe slowdown factor S is shown as a function ofpump intensity in Fig. 3(a). This calculated result
corresponds to a three-level system in actual QDs. In
this calculation, we used reported physical parameters [10]. The slowdown factor changes with pumpintensity, meaning that QDs can provide a controllable slowdown function. The maximum slowdown
factor is about 1.7 × 105, which enables us to delay
optical pulses by about 567 ns using a device 1 mm
long. This delay time may be increased by modifying
the physical parameters in QDs. A schematic diagramof an actual delay device is shown in Fig. 3(c).
Some slowdown factors that correspond to familiar
speeds are shown in Fig. 3(b) to provide a more intuitive understanding of the slowdown effects. The
velocity of light in a vacuum corresponds to S = 1,
i.e., no slowdown. The walking speed of a human,
which is about 1 m/s, corresponds to S = 3 × 108. The
present slowdown factor in QDs of 1.7 × 105 falls
between the speed of the space shuttle and the speedof a bullet shot from a gun. If the speed of light canbe reduced to human walking pace, then this function
will be very useful for optical routing, switching, andstorage. Moreover, the slowdown function does not
involve any energy dissipation. The ability to slow
light will enable us to develop valuable and innovative optical devices that will find many applications.

Slowdown factor (×105)

Selected Papers

S = n(hω ) + hω

2.0

1.5

1.0

0.5

0.0
0.01

0.1

1
2

Pump intensity (kW/cm )(b)S1
4 × 104
1.7 × 105
6 × 105
3 × 108

Velocity of light in vacuum
Velocity of space shuttle
Velocity of light in QDs
Speed of bullet from a gun
Walking speed of human(c)With device
In

Out

Pump

Delay

3. Fundamental experiments in actual QDs anddiscussion
We examined the optical nonlinearity induced by
an exciton and a biexciton in a single QD [11], [12].
This requires a well-defined exciton-biexciton system and large dipole moments, which characterize
the coupling strength between exciton and lights. Our
QD sample met both these requirements. A demonstration of the slowing of light requires many QDs
where the sum of their cross-sections is larger thanthe cross-section of the probe beam. However, thefundamental properties of the slowing processes are
determined by optical nonlinearities in a single QD,
so results of experiments on a single QD will provide
important reference data for slow light demonstrations with many QDs. We observed strong optical
22

Without device

Fig. 3. (a) Calculated slowdown factor as a function ofpump intensity. (b) Slowdown factors for various
familiar speeds. (c) Schematic of an optical delay
device.

nonlinearity in the exciton absorption of an InGaAs
QD by using single-dot photoluminescence excitation (PLE) spectroscopy. We measured the absorption
spectrum of an exciton interacting with a biexciton in
an isolated QD. In a QD with a weak confinement
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Selected Papers

Vol. 3 No. 12 Dec. 2005

(a)
Laser

Objective lensMetal mask

QD sample

(b)
PLE

PL

X

Signal (arb. unit)

effect, the exciton absorption spectrum had a dipshaped structure at the exciton resonant energy. A
large biexciton population induced large coherent
effects between an exciton and a biexciton, which
resulted in the observed absorption spectrum. A
pump-probe measurement ensured that the exciton
state almost disappeared as a result of the quantum
interference effects in the exciton and biexciton
states. These results constitute an important milestone for demonstrations of the slowing down oflight.
In0.4Ga0.6As QDs were grown by metalorganic
vapor phase epitaxy (MOVPE) on a GaAs (311) B
substrate [13]. The QDs were self-assembled on an
Al0.4Ga0.6As barrier layer and covered with another
Al0.4Ga0.6As layer followed by a thin GaAs capping
layer. The QDs had an average lateral width of 30 nm
and were 3 nm thick. This fabrication method is akind of self-assembly. Although such methods havebeen reported by several groups [4]-[6], the use of a
GaAs (311) B substrate is the key originality in our
technique. This yields a well-defined exciton-biexciton system with large dipole moments, which arerequired to achieve large optical nonlinearities. The
photoluminescence (PL) and PLE of the single QDs
were measured with the micro-PL technique [14].
Our experimental setup is shown in Fig. 4(a). The
QDs were illuminated by a continuous wave Ti:sapphire laser through a microscope objective lens. The
optical spot was about 2 µm in diameter. A metalmask with many holes (0.5 × 0.5 µm2) was placed on
top of the QD sample to detect the PL signal from asingle QD. The pump-probe measurement techniquewith an unbalanced Mach-Zehnder interferometer
was used to evaluate the optical nonlinearity of the
QDs.
Measured PL and PLE spectra for a single QD are
shown in Fig. 4(b). The PL spectrum contains sharp
clearly separated PL peaks for the lowest states of an
exciton and a biexciton. The energy separation
between the peaks is the biexciton binding energy.
The binding energy is about 2.6 meV, which means arelatively large QD with weak confinement effects.
This indicates that the dipole moments of the excitonand biexciton are much larger than those of the quantum wells and bulk.
The PLE spectra (Fig. 4(b)) for an exciton (X) andbiexciton (XX) correspond to the absorption spectra.
Several PLE peaks reveal the discrete density-ofstates achieved in higher states. We focused on level
A in the PLE spectra. Large signals can be seen inboth the X and XX spectra. When the laser energy

X

XX

XX

1.655

XA

1.660

XXA

1.665
1.670
Energy (eV)Level A

1.675

(c)
XXA

hωe

XA

hωe

g

Fig. 4. (a) Schematic of measurement setup. (b) PL and
PLE spectra for an exciton and a biexciton in a
QD. (c) Energy level structure for XA and XXA.

23

Selected Papers

was set to level A, an exciton and a biexciton were
simultaneously created by a single excitation laser.
The corresponding energy level structure is shown in
Fig. 4(c). It consists of three levels (g, XA, and XXA)
that interact coherently. This energy structure is thesame as that in the previous section. Therefore, we
evaluated the coherent effects in XA and XXA.
PLE spectra of X and XX with a fine energy resolution are shown for different excitation intensities inFig. 5. For the lowest excitation intensity (P0), both X
and XX have peak-shaped spectra with peaks at thesame energy values. However, the spectral shapes arevery different for higher excitation. For a high excitation intensity (5 P0 and 20 P0), the XX PLE spectrum
is peak-shaped with a broader linewidth than that at
P0. In contrast, the X PLE spectrum has a completely
different shape with a dip structure. The width of the
dip-shaped absorption increases with excitation
intensity and the absorption of X almost disappears at
the exciton resonance energy of 20 P0. This absorption spectrum is very similar to that obtained from thetheoretical calculation (shown in Fig. 2(b)). The dipshaped absorption spectrum means that there was alarge coherent effect between XA and XXA. This
spectrum leads to the steep-positive-slope region in
the refractive index spectrum, resulting in the slowdown of light. Changing the dip width by means of
the excitation intensity enables us to adjust the slope
region and thereby gives a controllable slowdown
function. The QD is a good candidate for a way toslow light. In the present QD, the slowdown function

Intensity (arb.unit)

P0

can be applied to light of 1.670 eV. This energy canbe tuned by changing the QD structure. This type ofcontrollability cannot be achieved in atom and ion
systems.
We performed a pump-and-probe measurement toexamine the coherent effects as optical nonlinear
effects. This measurement corresponded to controlling the absorption properties of the probe beam byusing the pump beam. In the measurement, PL signals from an exciton and a biexciton were detectedwith two excitation beams that passed through a
Mach-Zehnder interferometer as shown in Fig. 6(a).
The difference between the two paths was chosen to
be several centimeters to make the optical interference effects between the two beams almost negligible. Although the two beams were supplied from asingle laser source, they acted as independent pumpand probe beams. When the XA state is eliminated asa result of coherent effects by the pump beam, no
exciton can be created by the probe beam so there is
no exciton PL emission. Measurements of this typemay provide the fundamental properties of an optical
function using two beams.
X and XX PL intensities are shown as a function of
excitation power density in Fig. 6(b). The probe
beam power was about 0.05 kW/cm2. All the points
indicate nonlinear portions that are the changes in PL
intensity induced by the pump beam. Although the
XX PL increased with pump power density and saturated above 0.3 kW/cm 2 in Fig. 6(b), the X PL
increased up to 0.1 kW/cm2 and then decreased with

5 P0

20 P0

X
X

XX

X
XX

XX
1.665

1.670
Energy (eV)

1.665

1.670
Energy (eV)

1.665

1.670

1.675

Energy (eV)Fig. 5. PLE spectra for X and XX measured with high energy resolution. The two arrows in each panel indicate thepositions of the X absorption peaks at P0. P0 is about 0.05 kW/cm2.

24

NTT Technical Review

Selected Papers

(a)

(b)
200

Probe

Sample∆PL intensity (arb. unit)

Pump

100

XX

0

–100

0.0

X
0.5

1.0

1.5

Pump power density (kW/cm2)

Fig. 6. (a) Schematic drawing of pump and probe measurement.
(b) Dependence of nonlinear portion of PL intensity on pump power density.

increasing pump power density. This decrease resulted from the large coherent effects that occurred
between the XA and XXA states. The pump beam created a dip-shaped absorption spectrum as seen in Fig.
5. The coherent effects erased the X A state andinduced a large optical nonlinear effect in the X PL,
as seen in Fig. 6(b). This result clearly shows that thepump beam could control the probe beam absorption.
This is a fundamental requirement for the demonstration of the slowing down of light.
In this measurement, we used a conventional laser
source for the excitation. In the ideal case, this nonlinear effect needs only an exciton and a biexciton.
This means that only two photons are required for the
optical nonlinear effects. The results clearly showthat a large optical nonlinearity is feasible with several excitation photons in QDs.
Our results reveal that large optical nonlinear
effects were achieved in a QD as a result of the coherent interaction between X and XX states and this
could be evaluated by using another probe beam. Thisnonlinearity corresponds to optical control of probe
beam absorption. To measure the slowing down of aprobe beam, we need a technique for detecting aprobe light transmitted from a QD. At the moment,
there is no efficient detection method because the QD
size (about 30 nm) is much smaller than the probe
beam wavelength (about 750 nm). To detect the transmitted probe beam, we would have to make a sample
that included many QDs. We have already fabricatedsuch QD samples [15], [16]. They will enable us tocalculate the slowdown factor in QDs from the transVol. 3 No. 12 Dec. 2005

mission measurement results. Such measurements ofthe slowdown of light may be demonstrated with QD
samples in the near future. Our current achievement
in a QD constitutes an important step towards obtaining coherent optical device functions with many QDs.
4. Conclusion
We have described an optical nonlinear function in
QDs produced by coherent processes in several quantum states such as exciton and biexciton states. These
optical functions include coherent trapping, lightslowing, and inversionless laser emission as in EIT.
These functions can be used in a conventional opticaldevice to demonstrate coherent optical delay, which
is very useful in the field of optical routing in opticaltelecommunications. Moreover, these functions may
provide a temporary buffer for quantum bit coherence. Such a buffer is crucial for quantum gate control because it can adjust the timing of many opticalpulses for gating processes. As a fundamental step
towards achieving these optical functions, we examined the optical nonlinearity induced by the coherent
process between an exciton and a biexciton in an
InGaAs QD. The characteristic absorption spectrathat were obtained are evidence of coherent effects
mediated by quantum interference effects. The resultsof the pump and probe measurements reveal thatprobe beam absorption was controlled in a QD. These
results constitute an important milestone in terms ofcontrolling the speed of light without any dissipation.
A sample with QD ensembles, which is currently in25

Selected Papers

preparation, may enable us to estimate actual controllability. Appropriate designs of QDs for structuresand semiconductors are also needed to clarify possible application areas. The fundamental information
will enable the implementation of actual QD devices.
Coherent optical nonlinearity in QDs promises valuable applications that will provide many technicalinnovations in various industrial fields.

Hideki Gotoh
Senior Research Scientist, NTT Basic
Research Laboratories.
He received the B.S., M.S., and Ph.D. degrees
in engineering from Hiroshima University,
Hiroshima in 1991, 1993, and 2000, respectively.
Since joining NTT Basic Research Laboratories
in 1993, he has been working on optical physics
and device applications of semiconductor nanostructures. He is the member of the Japan Societyof Applied Physics (JSAP) and Optical Societyof America.

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[3] H. Kamada, “Quantum computing with QD excitons,” NTT Technical Review, Vol. 1, No. 3, pp. 31-40, 2003.
[4] P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D.
Ouyang, and D. Bimberg, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett., Vol. 87, pp. 157401-157404, 2001.
[5] D. Birkedal, K. Leosson, and J. M. Hvam, “Long lived coherence inself-assembled quantum dots,” Phys. Rev. Lett., Vol. 87, Art. No.
227401, 2001.
[6] M. Bayer and A. Forchel, “Temperature dependence of the exciton
homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B 65, Art. No. 041308 (R), 2002.
[7] S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett.,Vol. 64, pp. 1107-1110, 1990.
[8] M. D. Lukin, “Trapping and manipulating photon states in atomic
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[9] M. O. Scully and M. S. Zubairy, “Quantum Optics,” Cambridge,
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[10] H. Gotoh, H. Kamada, T. Saitoh, H. Ando, and J. Temmyo, “Exciton
absorption properties of coherently coupled exciton-biexciton systems in quantum dots,” Phys. Rev. B 71, article No. 195334, 2005.
[11] H. Gotoh, H. Kamada, T. Saitoh, H. Ando, and J. Temmyo, “Effects
of exciton-biexciton coherent coupling on exciton absorption in quantum dots,” Appl. Phys. Lett., Vol. 85, pp. 3480-3482, 2004.
[12] H. Gotoh, H. Kamada, T. Saitoh, H. Ando, and J. Temmyo, “Opticalnonlinearity induced by exciton-biexciton coherent effects in InGaAsquantum dots,” Appl. Phys. Lett., Vol. 87, Art. No. 102101, 2005.
[13] R. Nötzel, J. Temmyo, and T. Tamamura, “Self-organized growth ofstrained InGaAs quantum disks,” Nature Vol. 369, pp. 131-133, 1994.
[14] H. Kamada, H. Ando, J. Temmyo, and T. Tamamura, “Excited-state
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[15] H. Gotoh, H. Kamada, T. Saitoh, H. Ando, and J. Temmyo, “Spatially ordered self-assembled quantum dots with uniform shapes fabricated by patterning nanoscale SiN islands,” Jpn. J. Appl. Phys. Part 1,
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[16] H. Gotoh, H. Kamada, T. Saitoh, H. Ando, and J. Temmyo, “Positional control of self-assembled quantum dots by patterning nanoscale
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26

Hidehiko Kamada
Senior Research Scientist, NTT Basic
Research Laboratories.
He received the B.S. degree in physics from
Sophia University, Tokyo in 1979 and the M.S.
degree in physics from Tokyo Institute of Technology, Tokyo in 1981. He joined the Ibaraki
Electrical Communication Laboratory of Nippon
Telegraph and Telephone Public Corporation(now NTT) in 1981. Since 1997, he has worked
in NTT Basic Research Laboratories. He is themember of JSAP and the Physical Society ofJapan. He received one of the NTT Technical
Review 2003 Awards.

NTT Technical Review