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Optical Properties of In(Ga)As/GaAs Quantum Dots for Optoelectronic Devices 129
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Cavity Quantum Electrodynamics with
Semiconductor Quantum Dots
P. Senellart and I. Robert-Philip
CNRS – Laboratoire de Photonique et Nanostructures,
Route de Nozay, F-91460 Marcoussis, France
4.1 Introduction
The fundamental transition of single quantum dots is well described in terms of a two-level sys-
tem, very much like an atom. Its spontaneous emission, which arises from the relaxation of the
emitting system placed on its excited state, is not an intrinsic property of the emitter itself but is
rather a property of the two-level system coupled to its electromagnetic vacuum environment.
Since the spatial and spectral properties of the electromagnetic fi eld are determined by all the
boundary conditions present in the local surroundings (for instance, mirrors, interfaces, etc.),
matter–fi eld interactions in general and in particular spontaneous emission decay are expected to
exhibit signifi cant changes within the environment. For example, the presence of a microcavity
around the emitting two-level system can accelerate or inhibit its spontaneous emission or even
make it reversible. This research fi eld, aimed at controlling spontaneous emission with the use
of wavelength-sized cavities, is called cavity quantum electrodynamics (CQED) [1] . Two regimes
can occur. First, the spontaneous emission rate can be enhanced or reduced compared with its
vacuum level by the presence of the microresonator (this is called the weak coupling regime);
second, the usual irreversible spontaneous emission can be changed to a reversible exchange of
energy between the emitter and the cavity mode with a frequency given by the Rabi frequency
(the system, composed of the single emitter and the cavity modes, is then in the strong coupling
regime).
First observed in atom physics [2–5] , the transposition of these effects to solid-state systems
was inaugurated in 1992 with the demonstration of exciton-photon strong coupling in a pla-
nar microcavity embedding quantum wells [6] . Another conceptual and technological break-
through was the observation of spontaneous emission exaltation that occurs when dissipation
overwhelms the fundamental Rabi dynamic [7] . In this context, the ability to create semiconduc-
tor quantum dots was a turning point in the recent history of cavity effects. These nanoemitters
can effi ciently capture and confi ne electrons and holes in the three directions of space, making
them less sensitive to surrounding defects (for example, those induced by semiconductor surfaces
whose impact becomes signifi cant when cavity dimension decreases). This three-dimensional
confi nement of carriers in individual quantum dots also makes them valuable candidates for the
study of cavity quantum electrodynamics effects applied to semiconductor materials, because,
unlike bulk or quantum-well materials, these nanoemitters exhibit sharper optical features in the
spectrum as determined by the density of states.
CHAPTER 4
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Cavity Quantum Electrodynamics with Semiconductor Quantum Dots 133
In their pioneering work [7] , J.M. Gérard et al. used micropillar cavities loaded with an ensem-
ble of quantum dots. Since then, weak coupling and strong coupling from a single quantum dot
have been experimentally demonstrated using a wide variety of cavity geometries. In this chap-
ter, we describe these recent results on single quantum dots, stressing the cavity and quantum
dot requirements for the implementation of such CQED effects. We consider some representative
applications and outline the challenges for the future.
4.2 Basics of cavity quantum electrodynamics
4.2.1 Optical confi nement and light–matter interaction
Electromagnetic fi eld properties are strongly dependent on the presence of boundaries in the
local environment, such as the presence of a microcavity. For example, the presence of the reso-
nator alters the optical density of electromagnetic states: like for electron motion quantization
in quantum dots, the density of optical states is modifi ed with respect to its value in free space,
when reducing the optical cavity dimensionality to two, one or zero dimensions. In the ideal zero-
dimensional limit, the density of optical modes, which is related to the capacity of the cavity to
confi ne light in time, is represented by a sequence of infi nitely narrow impulse functions at pre-
cise, discrete, allowed energies: the resonator sustains discrete optical modes and confi nes light
at these fi xed resonant frequency values indefi nitely (i.e. without loss). Moreover, wavelength-
size spatial confi nement ensures that these precise resonant frequencies are well separated and
sparsely distributed throughout the spectrum.
Deviation from these ideal conditions is described by the quality factor Q and the modal volume
V of each mode sustained by the cavity. The quality factor describes the coupling of the cavity
mode to the outside continuum of optical modes (i.e. losses). It is proportional to the light con-
fi nement time: the decay of the energy stored in the optical mode at pulsation ω is proportional to
e
ω t
/
Q
. The decay rate 1/ Q is thus the bandwidth at half-maximum of the cavity mode and Q can
be expressed as a function of the mode wavelength λ and its spectral linewidth Δλ as follows:
Q
λ
λ
.
(4.1)
If the quality factor refl ects the ability of the cavity to store light in time, the modal volume V rep-
resents its ability to store light in space. It is usually defi ned in terms of the electrical fi eld energy
in the cavity mode as follows:
V
rEr dr
rEr
ε
ε
() ()|
max( ( )| ( )| )
.
SPACE
∫
|
23
2
(4.2)
Usually, this modal volume is expressed in units of effective optical wavelengh ( λ / n )
3
, where n is
the refractive index of the cavity medium.
Consider now the system formed by a single quantum dot inserted in an optical microcavity.
The energy of the fi rst excited state of the quantum dot is supposed to be resonant to the cav-
ity mode energy. Two states of the cavity–quantum dot system are coupled through ligth–matter
interaction: the state with the quantum dot in its excited state with no occupation of the photon
mode | e , 0 , and the state with the quantum dot in its ground state and one photon occupancy
in the cavity mode |g,1 . In the dipolar approximation, the coupling constant (referred to as ω
1
in Fig. 4.1 ) is proportional to the
eEr dg
em
,
|
() ,01
|
where the electron-hole pair trapped in
the quantum dot is described by a point dipole
d
and
Er
em
(
)
is the elecetric fi eld evaluated at the
quantum dot location. Because the quantum dot is inserted in a semiconductor matrix, the elec-
tron–hole pair trapped in the quantum dot experiences phonon coupling as well Coulomb inter-
action with carriers in the surrounding quantum dot. Moreover, because the cavity is not perfect,
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134 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
the optical mode of the cavity is coupled to the optical mode of the free space through a constant
ω
2
proportional to Q
1
. For the sake of simplicity, we only consider the coupling to the free space
optical mode and neglect Coulomb and phonon interaction.
If ω
1
ω
2
, when the electron–hole pair radiatively recombines, the emitted photon will rap-
idly escape from the cavity. To describe this regime correctly, one needs to consider that the quan-
tum dot in its excited state is coupled to a continuum of optical mode. The effect of the cavity is
to change the density of the fi nal state for the photon emission and to enhance or decrease the
intensity of electric fi eld at the quantum dot position. The quantum dot spontaneous emission
remains irreversible but it can be accelerated or inhibited. The system is in the weak coupling
regime, in the so-called Purcell effect [8] .
If ω
1
ω
2
, i.e. if the cavity losses are minimal, when the electron–hole pair radiatively
recombines, the emitted photon will oscillate inside the cavity before escaping outside. During
these oscillations, it will be periodically reabsorbed and re-emitted by the quantum dot with a
period given by the Rabi frequency. In this so-called strong coupling regime, one can consider
that the cavity–quantum dot system is isolated from any other interaction, so that | e ,0 and
| g ,1 are no longer the eigenstates of the system. At exact resonance, the eigenstates are entan-
gled light–matter states:
eg,,
,
01
2
split by the Rabi frequency
Ω/
( Fig. 4.1 , right). In the following, we detail the description of
these two regimes and discuss the key parameters that control the establishment of exaltation,
inhibition of spontaneous emission as well as strong coupling regime.
4.2.2 Spontaneous emission control – Purcell effect
Hereafter, we shall focus on the weak coupling regime, which occurs when dissipation due to
cavity losses overwhelms the fundamental Rabi dynamic.
In the electric dipolar approximation, the spontaneous emission decay of a monochromatic
two-level system, coupled to a continuum of optical modes, is given by Fermi’s golden rule [1] :
Γ
2
01
2
0
π
ρω
eEr dg E
em
,() ,
()
(4.3)
/e, 0>
e
photon
+
e
/g, 1>
free space
optical modes
light–matter
coupling
ω
1
ω
2
ω
1
<< ω
2
/e, 0>
e-
Ω
/g, 1> /e, 0>
√2
/g, 1> /e, 0>
√2
Weak coupling regime
Purcell effect
Strong coupling regime
Light–matter entangled states
Rabi
splitting
ω
1
>> ω
2
Figure 4.1 Schematics of cavity quantum electrodynamics for a single quantum dot in an optical microcavity.
Top : General case. Left : Weak coupling regime. Right : Strong coupling regime.
CH004-I046325.indd 134CH004-I046325.indd 134 7/1/2008 10:50:18 AM7/1/2008 10:50:18 AM
Cavity Quantum Electrodynamics with Semiconductor Quantum Dots 135
where the emitter is described by a point dipole
d
undergoing harmonic oscillation at at angular
frequency ω
0
around a point
r
em
ρ ( E ω ) is the density of states of the electromagnetic fi eld
at energy E . For a single cavity mode oscillating at ω
c
and exhibiting a quality factor Q , this den-
sity of optical modes can be expressed as follows:
ρω
π
ω
ω
ωω
E
Q
Q
c
c
c
()
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
()
1
2
1
2
2
2
.
(4.4)
The fi eld operator for the cavity eigenmode reads:
Er i
V
ura u ra
c
() () ()
*
ω
ε2
†
()
(4.5)
where
ur()
is the normalized mode spatial distribution and describes the polarization and the
relative amplitude of the electric fi eld at location
r
(
ur()
is normalized to 1 at its maximum). a
and a
†
are the photon annihilation and creation operators in the mode and V is its effective modal
volume. Taking equations 4.4 and 4.5 into account, the spontaneous emission decay given by
4.3 can be expressed as:
Γ
21
1
2
2
2
0
2
Q
V
ur d
Q
em
c
ε
ω
ωω
()
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
()
c
.
(4.6)
The spontaneous emission rate of the same emitter in bulk material of permittivity ε ε
0
n
2
is:
Γ
0
0
32
0
3
3
nd
c
ω
πε
.
(4.7)
We can therefore deduce the modifi cation of the spontaneous emission rate induced by the inter-
action of the dipole with a single cavity mode as follows:
FF
ur d
d
Q
p
em
c
Γ
Γ
0
2
22
0
2
1
1
2
()
ω
ωω
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
()
c
(4.8)
where F
p
, the so-called Purcell factor, is [8] :
F
Q
Vn
p
3
4
2
0
3
π
λ /
()
(4.9)
where λ
0
is the dipole wavelength in free space.
The modifi cation of the spontaneous emission rate by the presence of the cavity, given by
equation 4.8, is therefore the product of three contributions. The fi rst one F
p
is the largest spon-
taneous emission enhancement induced by the cavity mode that can be obtained; it depends only
on optical characteristics of the optical mode, more precisely its quality factor and its effective
volume. In other words, the Purcell factor is a fi gure of merit for the cavity alone and describes
its ability to increase the coupling of an ideal emitter with the vacuum fi eld, via a local enhance-
ment of its intensity (small Vs) and/or of the effective mode density (high Q s). Notice that the
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136 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
amplitude of this acceleration (or Purcell effect) depends on the ratio Q / V whereas the visibility
of the strong coupling regime between a single quantum dot and the cavity fi eld depends on the
ratio Q
2
/ V , as we shall see in the next paragraph. The second and third terms in equation 4.8
highlight the resonant character of the Purcell effect. The fi rst one, proportional to
ur d
em
()
,
takes into account the relative position of the emitter to the spatial maximum of the mode elec-
tric fi eld and the relative orientation of the emitting dipole to the electric fi eld. If the emitting
dipole is located at a node of the electric fi eld, the spontaneous emission will be inhibited by the
presence of the cavity. The last term describes the spectral detuning between the emitter and the
cavity mode. For a dipole off spectral resonance with the cavity mode, the spontaneous emission
will be suppressed. In conclusion, the interaction of a single dipole with a single cavity mode can
give rise to two main effects:
● The inhibition of spontaneous emission, which occurs when the emitting dipole is located
at a node of the electric fi eld and/or is far off resonance with the cavity mode. In the latter
case, the spontaneous emission suppression will be proportional to 1/( Q/V ).
● The acceleration of spontaneous emission, whose largest amplitude is reached for a punc-
tual monochromatic dipole on spectral and spatial resonance with a high- Q / V cavity mode
and parallel to the mode electric fi eld. In this context, the spontaneous emission acceleration
is proportional to Q / V .
In practice, the emitting dipole may be randomly orientated, so that the modifi cation of the
spontaneous emission rate should be averaged over the dipole orientation, which reduces the
amplitude of the modifi cation by a factor of 3 (this factor would be equal to 2 if the dipole were
randomly orientated in the plane of the electric fi eld). Furthermore, the cavity confi ned mode
might be g -fold degenerated (for instance g 2 for the fundamental mode of pillars or disks with
circular cross-section whereas g 1 for the fundamental mode of a post with elliptical cross-sec-
tion), which increases the density of states of the cavity fi eld by a factor of g . Eventually, for a
single emitter with an in-plane dipole randomly orientated in the cavity electric fi eld plane and
coupled to a single cavity mode with degeneracy g , the modifi cation of the spontaneous emission
rate should be expressed as:
F
g
Fur
Q
pem
c
c
Γ
Γ
0
2
2
0
2
2
1
1
2
|( )|
ω
ωω
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
()
.
(4.10)
More generally, the radiative spontaneous recombination can occur either in the single cavity
mode or in other “ leaky ” optical modes. In Fermi’s golden rule formalism, the probabilities of
emitting in the various fi nal states add on, so that the global modifi cation of the spontaneous
emission rate reads:
F
g
Fur
Q
pem
c
c
leak
Γ
Γ
Γ
0
2
2
0
2
2
1
1
2
|( )|
ω
ωω
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
()
ΓΓ
0
.
(4.11)
where Γ
leak
is the spontaneous emission rate in the leaky modes.
Let’s now come back to the validity of this approach. We have supposed here that the the pho-
ton lifetime in the cavity mode (related to the incoherent fi eld damping from the cavity into the
continuum of external modes) is much shorter than the period of the Rabi oscillation (related to
the coherent coupling of the dipole with the cavity fi eld): in other words, the emitted photon is
dissipated before being reabsorbed by the emitter. This condition reads:
Ω
Q
c
ω
1.
(4.12)
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Cavity Quantum Electrodynamics with Semiconductor Quantum Dots 137
Furthermore, we have considered up to this point a punctual monochromatic emitter. This comes
into play in the use of Fermi’s golden rule, which holds only if a discrete state is coupled to a con-
tinuum of states. In practice, emitters are not perfectly monochromatic and the validity of such
approach will depend on the relative amplitude between the natural homogeneous width of the
optical transition and the spectral width of the cavity mode. Fermi’s golden rule can only apply if:
Γ
0
1
Q
c
ω
(4.13)
The large spectral width of commonly used solid-state emitters, for instance semiconductor
quantum wells ( 1 nm, even at low temperature), is a major hindrance in this respect, leading
to spontaneous modifi cation factors of the order of a few units in the weak coupling regime [64,
65] . From this point of view, semiconductor quantum dots appear as valuable candidates for the
implementation of CQED effects in the solid state, since they exhibit a very small spectral broad-
ening at low temperature. For instance, the homogeneous linewidth of the fundamental tran-
sition in InAs/GaAs quantum dots at low temperature ( 10 K) ranges between a few μ eV (for
resonant pumping [66, 67] and a few tens of μ eV when increasing the energy detuning between
the optical excitation and the optical fundamental transition [68] .
4.2.3 Strong coupling regime
When the interaction between the electron–hole pair trapped in the quantum dot and the optical
mode is large compared to any dephasing mechanism, the eigenstates of the system are mixed
exciton–photon states split at resonance by the Rabi splitting
Ω 2g
with
gdE
.
Assuming that the dipole is parallel to the electric fi eld, the coupling constant g can be expressed
in terms of the oscillator strength of the optical transition of the quantum dot
fmde 2
0
22
ω /( )
(4.14)
using:
g
ef
mV
R
1
4
0
2
πε ε
π
.
(4.15)
In this equation, m is the free electron mass.
As discussed below, dephasing mechanisms play a key role in the implementation of CQED in
solid-state systems: cavity losses as well as interaction between the carriers trapped in the quan-
tum dot and their surroundings compete with the light–matter interaction. To include the dephas-
ing mechanism in the system description, one can consider the coupling to these reservoirs by
introducing the linewidth of both the photon ( γ
C
) and the exciton transition ( γ
X
). Considering the
density matrix for the quantum dot–cavity system, the luminescence spectrum can be calculated
in the weak excitation limit [69, 10] . For
g
XC
γγ/4
and at resonance between the cavity
mode and the exciton energy, the spontaneous emission spectrum consists of two lines split by
Ω
RCX
g216
2
2
γγ
()
/
with the average linewidth
()γγ
XC
/.2
For an InAs self-assembled quantum dot, the linewidth of the exciton line stands around a few
tens of μ eV whereas the typical linewidth for the mode resonance is around a few hundreds of μ eV.
In a fi rst approximation, the exciton linewidth can be neglected so that the strong coupling regime
is reached for
g
C
γ /4
. Since the photon mode linewidth is given by
γω
C
0
/Q
(see Eq. 4.1), the
strong coupling regime occurs for:
4
1
0
2
0
εε ω
R
ef
mV
Q
.
(4.16)
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138 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics
This condition is necessary but obsviously not suffi cient. First, without any preferential orientation
of the quantum dot the orientation of the quantum dot emission dipole is random: the strength
of the cavity–quantum dot coupling must be corrected by the cosinus of the angle between the
quantum dot exciton dipole and the electric fi eld of the optical mode. However, this calculation
stands only if the quantum dot is located at the exact maximum of the electromagnetic fi eld.
Finally, when increasing temperature or reducing the radiative lifetime of the excitonic transition,
the broadening of the exciton line can become comparable to the cavity mode linewidth.
4.3 Implementation of cavity quantum electrodynamics in the solid state
4.3.1 The resonator: a semiconductor microcavity
In solid-state physics, two main mechanisms for light confi nement are used. The fi rst one exploits
the total internal refl ection at the interface between two media with different refractive index:
total internal refl ection bounces light to the region of high refractive index when light strikes
the interface at a steep angle. The second one exploits the destructive interference between opti-
cal waves impinging on a medium with a periodically modulated index of refraction, so-called
photonic crystals: for waves with a wavelength close to the period of modulation, the scatter-
ing of light on the photonic crystal combines with destructive interference. This interference
cancels out light of certain wavelengths and the photonic crystal acts as a high-quality refl ec-
tor: Analogous to the electronic band gap in semiconductors, such periodic structures exhibit
a certain frequency range where light cannot propagate through the structure. This frequency
range is known as the photonic band gap. Photonic crystals can be divided into three catego-
ries, namely one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) crystals
according to the dimension of the periodicity. A common one-dimensional photonic crystal is the
Bragg mirror composed of multiple layers of dielectric fi lms. 2D and 3D photonic crystals employ
the Bragg refl ection in more than one spatial direction.
One can use total internal refl ection to make three-dimensionally confi ned resonators, pro-
vided one can force wavefronts inside the cavity to interfere with themselves. This is achieved
with the “ whispering gallery ” resonators. They are essentially circular disks [45] , toroids [46] or
spheres [47] in which the light circulates around close to the dielectric interface. Such modes are
especially low in losses and can reach quality factors far exceeding what can be obtained in other
solid-state microcavities: experimental quality factors of the whispering gallery modes sustained
by silica microspheres [48–50] or toroids [51] reach values up to 10
8
–10
9
, while keeping a rela-
tively small modal volume: V 10
2
–10
4
( λ / n )
3
. Microdisk resonators with diameters as small as
1–10 μ m have also been processed in III–V semiconductor materials (see Fig. 4.2 left) and sup-
port modes with a high-quality factor (up to 10
4
–10
5
[52, 53] ) with modal volume as low as a
few ( λ / n )
3
.
Another strategy to confi ne light in the three directions of space would be to isolate a point
defect, such as one or more missing periods, into a three-dimensional photonic crystal [54, 55] .
A photonic crystal with a complete band gap in the microwave range was created in 1991 [56] .
However, it appeared to be far from trivial to reach a full photonic band gap in the near-infrared
and visible spectral range. This was achieved only recently [57–60] . Until then, control over light
emission in 3D photonic crystals was demonstrated experimentally [61–63] .
While various techniques for the fabrication of photonic crystals with full three-dimensional
band gaps have been proposed and demonstrated with different levels of success, the incorpo-
ration of microcavities in a controllable way is a critical step. In view of the diffi culties in fab-
ricating deterministic photonic defects into 3D photonic crystal structures, a lot of effort has
been devoted to less demanding structures involving a hybrid confi nement: in this context, light
confi nement combines total internal refl ection in one or two spatial directions and interference
effects in the other directions of space. Such cavities are more amenable to fabrication than
three-dimensional photonic crystal resonators, but retain or approximate many of the latter’s
desirable properties, such as 3D confi nement of light.
One example is the micropillar cavity (see Fig. 4.2 centre). In micropost resonators, the
transverse mode confi nement results from total internal refl ection at the semiconductor–air
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