Henini M. Handbook of Self Assembled Semiconductor Nanostructures for Novel devices in Photonics and Electronics

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Cavity Quantum Electrodynamics with Semiconductor Quantum Dots 149

Interesting features are also expected on the bi-exciton line for an exciton experiencing strong

coupling regime with an optical mode analogue to the Autler–townes doublet [91] . Finally, the

strong optical non-linearity resulting from the strong coupling regime could be used to realize

photon blockade [92] as it has been recently reported for an atom-based system [93] .

4.6 Towards a deterministic cavity–dot coupling

As presented in section 4.7, controlling the spontaneous emission rate for an atomic-like emitter

in a solid-state system is very promising for applications in quantum cryptography, and future

quantum computation protocols. The successful demonstrations of Purcell for single quantum

dots as well as a strong coupling regime are important milestones in the new ﬁ elds of solid-state

quantum optics. However, to be able to investigate this new system, technological challenges

need to be addressed.

To experience cavity quantum electrodynamics effects, the quantum dot has to be located pre-

cisely with respect to the electric ﬁ eld: either at an antinode (spontaneous emission exaltation,

strong coupling regime) or at a node (spontaneous emission inhibition). Up to now, in all the

experimental works except one [76] , no control of the quantum dot position within the cavity is

demonstrated. Many cavities are investigated to ﬁ nd a well-positioned quantum dot.

Finally, temperature tuning which has been used to reach the spectral resonance between

the exciton and the photon mode has its limitations: ﬁ rst, spectral tuning is only possible for an

initial detuning around 1–2 meV. Moreover, when increasing the temperature, the exciton line

gets thermally broadened, which can prevent the establishment of a strong coupling regime.

As a result, a more ﬂ exible way to tune the spectral resonance between the cavity mode and the

excitonic transition is needed. In the present section, we report on various techniques that could

allow these spatial and spectral tunings.

4.6.1 Spatial tuning

Most cavity effects on single quantum dots have been reported either on self-assembled quantum

dots or natural quantum dots. In both cases, the quantum dot position is not controlled by the

growth techniques.

However, techniques to control the growth position of single quantum dots have been investi-

gated. One technique consists of patterning GaAs substrates using wet etching yielding to the for-

mation of tetrahedral pyramids. The sample is then re-introduced in the growth chamber to grow

AlGaAs barriers and a thin GaAs layer. The pyramidal geometry leads a conﬁ nement of the car-

riers in all space directions at the top of the pyramid [94, 95] . Single photon emission from these

quantum dots ’ has been evidenced [96] showing the good optical quality of these dots’ emission.

However, up to now, these quantum dots have not been inserted in an optical microcavity.

More recently, InAs site controlled growth has been demonstrated. After substrate patterning

through electron-beam lithography and reactive ion etching, GaAs/InAs re-growth was per-

formed [97] . Photoluminescence on such a quantum dot ensemble shows non-radiative proc-

esses for the ﬁ rst quantum dot layer. However, using quantum dot vertical stacking, the optical

properties of the dot layer improve when growing several layers of quantum dots. Power depend-

ent measurements show that non-radiative processes are negligible after the sixth stack, with

an exciton radiative lifetime around 1 ns [97] . Another work from the same team shows that

depending on the growth conditions, self-aligned quantum dot pairs could be realized with the

same technique [98] . These quantum dot pairs could be interesting in the context on quantum

information processing as we explain in section 4.7.5. Finally, the same technique was used to

realize organized GaAs quantum dots: good results were achieved on single quantum dot photo-

luminescence and single photon emission [99] .

Controlled strain has also been used to grow quantum dot lattices. However, up to now, such

a technique yields the formation of high-density quantum dot arrays, useful for laser devices but

not for cavity quantum electrodynamics effects [100–102] .

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150 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

Last but not least, we would like to mention the work of Xie and Solomon on spatial order-

ing of quantum dots in microdisks [103] . In this work, an Al-rich layer is grown below an GaAs

layer, in a ﬁ rst growth step. Then microdisks are realized through optical lithography and wet

etching. Surface deoxidation is performed before introducing the microdisk sample into the MBE

chamber for quantum dot growth. The authors show that the microdisk geometry leads to pref-

erential In adatom surface diffusion so that InAs quantum dots are grown within 100 nm from

the microdisk edges. The edge linear quantum dot density is shown to depend strongly on the

disk diameter, so that a 3 μ m microdisk with one or two dots located at the edges are realized.

Even though cavity effects have not yet been reported in this system, single spectroscopy and

high-quality factor indicate that this system could be very promising.

Another approach has been developped by the group of Professor Imamog

lu (Zurich,

Switzerland) to control the quantum dot position within the cavity. In a ﬁ rst demonstration

[76] , vertically aligned quantum dot stacks were grown up to the sample surface. The surface

quantum dot acted as a tracer for the buried quantum dot, whose optical transition is at a higher

energy than the upper dots. A matrix of gold markers and scanning electron microscopy was

used to map the isolated quantum dot position. Controlled Purcell effect was ﬁ rst demonstrated

with this technique. However, the lower energy absorption of the upper dots resulted in quality

factor limitation for the optical mode. More recently, the same technique was realized without

the tracer quantum dots. The buried quantum dot position was measured using AFM techniques.

In this last report a strong coupling regime was observed [87] .

4.6.2 Spectral tuning

Up to now, no real control of the quantum dot emission energy has been reached with standard

growth techniques. To tune the exciton–photon mode resonance, several techniques to control

the mode energy after the cavity realization have been investigated.

The ﬁ rst technique relies on chemical etching [85] . After the realization of a photonic crys-

tal microcavity, a few monolayers of material are removed from the crystal surface by etching a

self-formed native oxide. Doing so, the mode resonance wavelength can be tuned in a range of

80 nm, by 2–3 nm steps. This technique has been successfully used to demonstrate a strong cou-

pling regime [87] .

This ﬁ rst technique reveals the extreme dependence on the hole size of the mode energy in

photonics crystal cavities. Thin ﬁ lm deposition in a helium ﬂ ow cryostat has also been observed

to tune the cavity mode energy [104] . This extreme dependence of the photon mode energy can

be an advantage but also a critical drawback, the energy of the optical mode being determined

by the quality of the chamber vacuum. Note that this effect is much more sensitive for photonic

crystal cavities than for microdisk cavities and presumably micropillar cavities. However, depos-

ing a thin glass slide seems to prevent this unwanted effect [104] .

Nonetheless, when introducing a controlled ﬂ ow of Xe in the cryostat, forming an epilayer on

the photonic crystal cavity, the energy of the optical mode has been tuned in resonance with two

quantum dot exciton lines, showing a strong coupling regime [105] .

Finally, in situ laser microprocessing has been used to tune either the exciton or the photon res-

onance [106] . Using high excitation power, the temperature of a microdisk microcavity embed-

ding InAs quantum dots can be raised during a low-temperature photoluminescence experiment.

Increasing the disk temperature, controlled desorption is reached, showing red shift of the optical

mode. Increasing the disk temperature further, In and Ga atom interdiffusion leads to blue shift

of the exciton transition. This technique is used to demonstrate exciton–photon resonance.

4.7 Applications of solid-state cavity quantum electrodynamics

Cavity quantum electrodynamics effects can ﬁ nd various potential applications in implement-

ing novel photonic devices. Hereafter, we shall discuss several applications of CQED effects in the

solid state: highly efﬁ cient light sources [27, 107, 108] , generation of indistinguishable single

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Cavity Quantum Electrodynamics with Semiconductor Quantum Dots 151

photons, generation of polarization entangled photons, engineering of ultra-low threshold lasers

and realization of quantum optical gates. We will shortly address the delicate question of quan-

tum computing based on strongly coupled quantum dots in a cavity, stressing our point on prac-

tical ﬁ rst steps one could foresee in a mid-term project. Other interesting topics, such as ultra-fast

nanolasers [109] , will not be reviewed because of space limitations.

4.7.1 Efﬁ cient single photon source: possibilities and limitations

Purcell effect has been proposed to increase the extraction of quantum dot emission [27, 107] .

Because of the high index in III–V compounds (around 3.5), most of the light emitted by a single

quantum dot is reﬂ ected at the air–semiconductor interface so that only a small fraction (around

4%) of the emitted light can be extracted. When a single quantum dot placed in an optical micro-

cavity experiences the Purcell effect, its emission rate into the cavity mode is increased by a factor

as compared to the leaky optical modes. As a result, the fraction of the spontaneous emission

that is emitted into the mode is given by

β FF

/( / ),

leak

ΓΓ

considering a perfectly reso-

nant optical mode, and a perfectly positioned quantum dot. Without inhibition effect (such as

reported in [7]),

β ≈ FF

/( )1

so that 75% of the quantum dot emission is extracted in the

optical mode for a Purcell effect of only 3.

Another possible application of the Purcell effect could be to increase the repetition rate of

the single photon source. Actually, contrary to atoms or molecules, semiconductor quantum

dots present spontaneous emission rates in the nanosecond range. Together with the Purcell

effect, one could expect to reach a 10–100 GHz repetition rate single photon source. Altogether,

semiconductor quantum dots embedded in optical microcavities appear promising to realize a

high repetition rate efﬁ cient single photon source.

However, we would like to stress some limitations for such an application. First of all, the rel-

evant parameter for single photon source efﬁ ciency is not only the β factor: the radiation pattern

of the optical mode is also crucial for optical ﬁ bre coupling, for instance. In that matter, the far-

ﬁ eld radiation pattern of micropillar microcavities has been experimentally measured [32] : for

2 μ m diameter micropillars, the radiation pattern of the fundamental mode spreads over 20° and

broadens when decreasing the pillar diameter. Obviously, since the Purcell effect scales-as Q / V , a

high-quality factor should be preferred to a small effective volume microcavity.

Moreover, the scheme for realistic single photon source using quantum dots [110] relies on

a non-resonant pulsed excitation creating high-energy electron–hole pairs in the wetting layer.

Whenever capture processes into quantum dots are much faster that radiative recombination,

one can consider that excitons are instantaneously trapped in the lowest energy states of the

quantum dots. The system then returns to its ground state by emitting several photons. Only a

single photon is emitted at the lowest energy for each excitation pulse. However, if the exciton

radiative lifetime is shortened through the Purcell effect, excitons may remain in the wetting

layer after the recombination of the low energy exciton. Another exciton can then be captured

into the quantum dot.

Considering capture processes, one can estimate the limitation of the Purcell effect for efﬁ cient

single photon source. A model for calculating the second-order autocorrelation function g

(0) is

developed in [21] to account for photon correlation measurements on fast radiative natural GaAs

quantum dots. This simple model considers a non-resonant excitation and includes capture proc-

esses. The same model is used to calculate g

(0) for various Purcell effects on single InAs quantum

dots. Figure 4.8 presents the expected g

(0) for various Purcell factors. A strong increase in the

probability that the quantum dot emits more than one photon as compared to a weak coherent

pulse increases drastically with increasing F

. Photon correlation measurements have been per-

formed on single quantum dots experiencing the Purcell effect. Most of the time the amplitude of

the zero delay correlation peaks is larger for quantum on resonance with the optical mode than

off resonance [28, 72] . These observations are well accounted for by the present model. From

these simple considerations, one can conclude that the most efﬁ cient single photon source will be

easily obtained with moderate quality factor, 2–3 μ m diameter micropillar microcavities. Neither

a smaller effective volume nor a larger quality factor is needed for such an application.

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152 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

0.00

0 500 1000 1500

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

(0)

Exciton radiative lifetime

 25

Figure 4.8 Calculated second-order correlation function as a function of the exciton radiative lifetime,

corresponding to a Purcell factor indicated as a label on the plot. The exciton radiative lifetime without Purcell effect

is 1.5 ns, the bi-exciton radiative lifetime is 750 ps, the capture time into the quantum dot is 50 ps and the radiative

decay time of the wetting layer exciton population is 150 ps. A mean population of one exciton is created at the

origin of time.

4.7.2 Indistinguishable single photon sources

In quantum mechanics, when two identical photons collide, they cannot be distinguished from

one another, even in principle, either through their relevant intrinsic physical properties (polari-

zation, energy, temporal proﬁ le, etc.) or through their respective trajectories. In this situation, the

description of the two-particle system becomes ambiguous and the so-called exchange degener-

acy appears. The problem of ﬁ nding the correct and unambiguous description for such systems

requires the introduction of a new postulate for quantum mechanics: the symmetrization postu-

late. This postulate distinguishes two classes of particles according to their collective behaviour

in indistinguishable situations. These are bosons and fermions. In some experimental situations,

the results of the experiment can only be deduced from this postulate. This is the case when two

identical particles are incident on a 50/50 beam splitter. In that case, bosons emerge together

along the same output port of the beam splitter (see Fig. 4.9 ), while fermions always follow two

different output pathways. These coalescence and anti-coalescence phenomena result from the

quantum interference between the amplitude probabilities of the four different output conﬁ gura-

tions (either both particles are transmitted or reﬂ ected; or one particle is reﬂ ected and the other

one transmitted) [111] : these amplitude probabilities add coherently and an interference term

appears, between the two paths in which the particles leave by different output ports. In other

1 photon

50–50

beam splitter

 0

Figure 4.9 Output conﬁ gurations when two identical photons are incident on a 50–50 beam splitter and collision

characteristics: the two photons follow the same output port.

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Cavity Quantum Electrodynamics with Semiconductor Quantum Dots 153

words, this is the indistinguishability of the “ alternative amplitudes ” that leads to the “ ﬁ nal

event ” . The interference is destructive for bosons and constructive for fermions.

More generally, two identical single photons can be deﬁ ned as bosonic individual particles

for which the symmetrization postulate has to be applied. In the phenomenon of two-photon

interference, these two single photons arriving simultaneously by two different input ports on

a beam splitter, both leave by the same output port, so that a two-photon state is constructed

out of two distinct one-photon Fock states. This coalescence effect was ﬁ rst observed in para-

metric down conversion (in which a non-linear crystal splits a single photon into two photons

of lower energy) [112] . It also occurs with single independently generated photons, for instance

single photons emitted by single molecules [113] , single atoms [114, 115] or single quantum

dots. However, the usual spontaneous emission lifetimes of excitonic transitions in self-assembled

quantum dots ranges from 500 ps to 1.5 ns, whereas the decoherence time of the excitonic dipole

caused by phonon scattering is lower than a few hundred ps for non-resonant pumping: the nat-

ural width of spontaneous emission is of about the same order as the homogeneous broadening

caused by phonon scattering. This precludes the generation of indistinguishable single photons

from self-assembled quantum dots buried in bulk material, since successive single photons emit-

ted through the radiative recombination of the exciton will be “ marked ” in energy by the random

exciton–phonon interaction which causes the dephasing of the emitting exciton state (with a

characteristic dephasing time

). In order to reduce the impact of dephasing on the emission

process and thus restore the indistinguishability of the emitted photons, the radiative lifetime of

the emitter (denoted by T

) must dominate over the dephasing time

in determining the overall

coherence time of the photon wave train, T

, deﬁ ned by:

111

22 1

TT T



(4.20)

While it is extremely difﬁ cult to eliminate the dephasing processes and lengthen

, it is possible

to reduce the impact of decoherence phenomena by shortening the radiative lifetime of the emit-

ter. This can be achieved by embedding the quantum dot in a microcavity, thus taking advantage

of cavity quantum electrodynamics effects, in particular the Purcell effect. This idea of exploiting

the Purcell effect has been successfully implemented by use of micropillar cavities [78, 116, 117]

or photonic crystal slab resonators [118] .

In these experiments, the microcavity source, cooled below 10 K, is excited twice by a pair of

sequential laser pulses and resonant with an absorption of the quantum dot (typically 30–40 meV

above the exciton transition energy, thus corresponding to an excited state of the exciton). This

quasi-resonant excitation on a trapped excited state of the quantum dot exciton allows the reduc-

tion of the impact of dephasing processes through a relative lengthening of the pure dephasing

time (of the order of a few hundred ps). Upon each excitation cycle, the dot emits at most two

single photon pulses in the microcavity at the exciton wavelength. These emitted photons exit

the microcavity and are combined on a 50–50 beam splitter with a relative time delay τ (see Fig.

4.10 left). They are then detected by single-photon detectors placed along the two output ports

of the beam splitter and a correlation measurement is carried out, in order to infer the probabil-

ity for the two photons to follow two different output pathways as a function of the time delay τ

between the two single photon pulses. This probability is related to the visibility of the two-photon

quantum interference. For distinguishable single particles, this probability is equal to 50% what-

ever the value of the time delay, while for ideal indistinguishable single photons impinging simul-

taneously on the beam splitter ( τ  0), it would cancel out. In this experiment ( Fig. 4.10 right),

the visibility of the quantum interference presents a dip at null delay, which indicates that the

two photons in the majority both go to the same output when they arrive simultaneously on

the beam splitter. This strong reduction of simultaneous photocounting probabilities is strongly

dependent on the overlapping of the two pulses and reaches its maximum for “ perfect ” wavefront

matching ( τ  0). The interference disappears when the delay time τ between the two particles

exceeds the pulse width 2 T

(of about a few hundred ps).

This high visibility of the quantum interference in these two photon experiments (  70%) results

from the high Purcell effect the single quantum dot is submitted to: the spontaneous emission

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154 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

t  τ

0.0

0.1

0.2

0.3

0.4

0.5

Opposite output probability

Time delay τ (ps)

400 200 0 200 400

Figure 4.10 Collision experiment (left) used to observe the quantum interference between two identical single

photons and (right) inﬂ uence of the wavefront matching on the quantum interference visibility as a function of the

time delay τ between both particles, emerging from a single InAs/GaAs quantum dot embedded in a 400 nm diameter

GaAs/AlGaAs Pillar with a Q factor of  1500. The spontaneous emission lifetime T

of the quantum dot exciton is

decreased to 105 ps by the presence of the cavity, while the pure dephasing time is of the order of 660 ps.

lifetime shortening due to the presence of the microcavity restores partially the single photon’s

indistinguishability. The remaining imperfection is likely to arise from several decoherence mech-

anisms. When the quantum dot is excited by a laser pulse, the photogenerated electron–hole pair

is initially in an excited state and relaxed to its lowest excitonic state through phonon emission

before emitting a single photon. The ratio of this relaxation time, which can reach tens of pico-

seconds, to the exciton radiative lifetime T

, may limit the visibility of this quantum interference.

Decoherence by phonon scattering is another possible mechanism, since even at low tempera-

ture ( T  10 K), the pure dephasing time in these experiments is long (of the order of a few hun-

dred ps) but not inﬁ nite, thus contributing to the overall coherence time of the photon wave train

(see Eq. 4.20).

4.7.3 Proposal for entangled photons sources

Recent proposals for quantum communication and quantum information protocols provide a

signiﬁ cant incentive to develop practical entangled two-photon sources. One requirement for

such sources is that the emission time of the photons is periodic with a precisely deﬁ ned clock fre-

quency. Bi-exciton (denoted XX) radiative cascade in quantum dots is a candidate system for such

sources [119] (see Fig. 4.11a ). In the ﬁ rst step of the cascade, a photon is emitted with random

polarization. Conservation of angular momentum requires that the polarization of the photon

emitted in the second step of the cascade is ﬁ xed relative to the ﬁ rst photon: the pair of photons

is in a polarization entangled state. Entanglement requires two decay paths with different polari-

zations which are otherwise indistinguishable. However, in typical quantum dots, the excitonic

relay level (denoted X) is split by the anisotropic exchange interaction, caused by in-plane ani-

sotropy of the exciton wave function [120] (see Fig. 4.11b ). Therefore, the energy splitting of the

excitonic relay level provides information about which pathways the two photons were released

along, via the energy of the emitted photons, and entanglement is destroyed.

In order to erase the “ which path ” information and recover polarization entanglement with

a high ﬁ delity, it is crucial to reduce the excitonic energy splitting within the radiative linew-

idth of the relay levels. First proofs of principle have been recently established, where entangle-

ment between the frequency and polarization degrees of freedom has been measured in photon

pairs generated by the cascade emission from single quantum dots. These experiments use either

strong spectral ﬁ ltering discarding unentangled photons pairs [121] , lateral magnetic ﬁ eld [122]

or thermal annealing [123] , to tune the exciton ﬁ ne splitting close to zero. However, the entan-

gled photon collection is kept rather low, which precludes the practical use of such sources in

quantum information processing platforms. Moreover, the degree of entanglement is still limited

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Cavity Quantum Electrodynamics with Semiconductor Quantum Dots 155

by dot dephasing and residual excitonic ﬁ ne-structure splitting. It is believed that these limitations

can be to a large extent overcome by increasing the photon emission rate, through the embedding

of the quantum dot in a semiconductor microcavity [124, 125] .

Indeed, application of an optical microcavity resonant with the excitonic transition of the cas-

cade increases the exciton decay rate and therefore is another means to enhance the entanglement

visibility. First, a fast emission of the excitonic photon reduces the time during which dephasing

degrades the intermediate relay state in the bi-excitonic cascade. Second, an increased excitonic

decay rate results in an enhanced homogeneous linewidth of the excitonic transitions, and there-

fore increases the overlap between the wave packets corresponding to horizontally and vertically

polarized photons released through exciton recombination: the excitonic photon is released before

quantum beats between the two relay levels occur. Third, the reduction of the exciton transition

dipole lifetime decreases the probability of re-exciting the system before emission of the second pho-

ton, when increasing the excitation rate. For typical quantum dots with excitonic energy splitting

lower than 30 μ eV, Purcell factors as high as 30 should allow for restoration of polarization entan-

glement. However, the requirements for the implementation of such an effect are stringent. In the

ﬁ rst place, the exciton energy splitting must be kept low (  30 μ eV) regarding achievable Purcell

effects with state of the art fabrication techniques. Moreover, the cavity has to support two degen-

erate polarization modes in the transition dipoles plane, in order to preserve the single photon pair’s

polarization correlations. If not, the transition dipoles will have modiﬁ ed coupling constants, which

will favour one recombination path and thus corrupt entanglement.

4.7.4 Proposal for ultra-low threshold lasers

Whether useful or detrimental, spontaneous emission plays a key role in laser devices. Spontaneous

emission into the so-called lasing mode is desirable since it provides photons that are subsequently

ampliﬁ ed by stimulated emission. By contrast, spontaneous emission into other non-lasing modes

is prejudicial, because it tends to increase the threshold power required to initiate lasing. In order

to reduce by several orders of magnitude the minimum threshold power of semiconductor lasers,

one strategy would be to implement the Purcell effect. As mentioned above, the Purcell effect is the

enhancement of the spontaneous emission rate of a single emitting dipole coupled to a high- Q / V

cavity mode. The concomitant effect is the spatial redistribution of spontaneous emission and the

preferential funnelling of light in the cavity mode. Indeed, the ratio of spontaneous emission fun-

nelled into the lasing mode depends on the spontaneous emission decay rate into the lasing modes

Γ and on the spontaneous emission decay rate into the non-lasing modes Γ

leak

as follows:

β 



ΓΓ

leak

(4.21)

This spontaneous emission coupling factor β can approach unity in nanocavity lasers with a

large Purcell effect, with serious implications on the nature of light emission. In conventional

Ground state

(b)

Ground state

(a)

Figure 4.11 Schematic description of the bi-excitonic cascade in a typical quantum dot (a) with zero energy

splitting on the relay excitonic level leading to polarization of maximally entangled photons and (b) with an energy

splitting δ leading to two collinearly polarized photons in a single preferred basis.

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156 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

semiconductor lasers, this β factor is very low, of the order of 10

 5

and the laser threshold

is marked by a sudden increase of output power and an abrupt quenching of the laser’s quan-

tum noise as the laser switches from spontaneous emission to coherent laser-like output. The

switch takes place within a very well-deﬁ ned threshold region with a relative width of β

1/2

[126].

With increasing β , the step-like ‘ ‘ threshold ’ ’ marked by a kink in the output intensity gradually

changes to an s-shaped smooth intensity transition and disappears up to the so-called “ threshold-

less laser ” in the limit β  1 [127] : in this limit, the input–output curve is linear with no abrupt

transition from a non-lasing to a lasing state. Therefore, for these microlasers with high β fac-

tor, the threshold is poorly deﬁ ned both in terms of intensity and ﬂ uctuations. This is not surpris-

ing considering that the relative threshold width β

1/2

becomes of order unity for β  1 and such

lasers are hence commonly referred to as thresholdless lasers.

In the regime of efﬁ cient spontaneous emission funnelling ( β close to one), a more accurate

interpretation of the lasing onset and the nature of the emitted light is given by a intra-cavity

photon statistics analysis. One criterion is the Fano factor as suggested by Rice and Carmichael

[126] . It is deﬁ ned by:

ng

01=

()

(4.22)

where g

(0) is the second-order correlation function, n is the intra-cavity photon number, and

δ n  n  n

, with n

the steady-state photon number. In conventional lasers, the Fano factor

exhibits a sharp peak with a height of roughly 2 β

1/2

on threshold and the peak has a relative

width of β

1/2

. It is equal to one both below and above threshold. The transition to lasing can thus

be determined through second-order correlation measurements, where bunching is observed

below threshold ( g

(0)  1) and Poissonian photon statistics are observed well above threshold

( g

(0)  1). The different photon statistics between conventional lasers and high- β factor lasers

are revealed by the evolution of g

(0) below threshold. In a conventional laser with β   1 and

below threshold, g

(0) reaches the value of 2 for thermal light and decreases abruptly to unity in

the threshold region. For large values of β , this transition becomes broader, while g

(0) deviates

from 2 well below threshold and decreases gradually to the value of 1 for ideal 0D laser cavities

with inﬁ nite Q factor.

The implementation of thresholdless lasers requires therefore the use of nanocavities with

large β factors. As mentioned above, one strategy uses the Purcell effect. However, the bottleneck

blocking the demonstration of thresholdless operation arises from the resonant character of the

Purcell effect. The wavelengths and positions of the quantum dots in the nanocavity are random.

If the cavity mode is resonant with respect to the wavelength and position of the quantum dot,

the dot emission is mainly coupled to the single cavity mode, allowing thresholdless operation

to occur. On the contrary, if the wavelengths and positions of quantum dots are not perfectly

matched with the cavity mode, the amplitude of the Purcell effect is reduced and the ratio of

dot emission coupled to non-lasing modes increases, preventing thresholdless lasing operation.

Nevertheless, nearly thresholdless laser operation in such a system is feasible even in the off-

resonant case and was recently achieved by use of photonic crystal–slab cavities [86, 128] or

micropillars [129] containing an array of self-assembled quantum dots. A smooth transition from

spontaneous emission with signiﬁ cant photon bunching to stimulated emission with Poissonian

photon statistics has been observed and β factors as high as 0.85 have been measured.

The ultimate thresholdless laser would be the single dot laser. The scaling of lasers to a small

number of emitters has been investigated for two decades using atomic emitters, such that

recently a one-atom laser has been demonstrated [130] . Single quantum dots isolated in high- β

factor cavities may be a promising route to solid-state, single emitter lasing. In this regime, a frac-

tion β of spontaneously emitted photons will be captured by the laser mode of the optical cavity.

If the cavity mode is on resonance with emission, the captured photons will remain in the cavity

for an average storage time Q / ω

before leaking out. If the quantum dot is excited again and emits

a second photon in the cavity mode before the ﬁ rst photon leaks out, lasing occurs. In order to

reach threshold, then, it is necessary to implement high-oscillator strength quantum dots and a

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Cavity Quantum Electrodynamics with Semiconductor Quantum Dots 157

cavity with very high Q as well as a very high β that captures a reasonable fraction of the spon-

taneous emission [131] . These single emitter lasers would allow the generation of intensity-

squeezed (or number-squeezed) states [132] .

4.7.5 Possible applications for quantum information processing

With the development of optical studies of single quantum dots, theoretical works have inves-

tigated the possibility of implementing quantum information processing based on single quan-

tum dots. Two different types of qbits are explored: one based on spin, the other based on orbital

degree of freedom. The latter approach uses exciton states with typically 1 ns limited lifetime, so

that ultra-fast quantum computing operations are proposed based on optical pulses. The former

approach relies on the spin states of a resident electron in the quantum dot. This approach ben-

eﬁ ts from much longer coherence times [133, 134] but the implementation of quantum gates is

slower than with excitonic states [135, 136] . Both approaches meet a common difﬁ culty which is

that only neighbouring quantum dots are coupled through dipole–dipole interaction [137, 138] .

Recently, several theoretical approaches have proposed to mediate the coupling between distant

quantum dots using a high-quality factor cavity mode [139–142] . The qbits are based on the

spin state of an excess electron using either an in-plane [139] or off-plane [140] magnetic ﬁ eld.

In these protocols, each quantum dot is individually addressed using near-ﬁ eld optics. Moreover,

the distant quantum dots need to present the same transition energy as well as maximum cou-

pling to the optical mode. The experimental conditions to implement these protocols are far from

being gathered and will not be gathered in the near future either.

However, on the way to gathering these experimental conditions, important milestones have

to be realized. One is the control of the charge state of a single quantum dot while being inserted

in an optical microcavity. The control of the charge state of a single quantum dot has been suc-

cessfully demonstrated using Schottky gates [143] . A ﬁ rst demonstration of such a charge con-

trol has very recently been reported in a VECSEL-like structure [144] with a structure conﬁ ning

photons only in the growth direction. To implement both cavity effects and charge states control,

the same technique could be used, and the optical ﬁ eld conﬁ nement could be obtained through

lateral oxidation [145] . Another promising approach could be inspired by the work of Muller and

co-workers, using high-quality factors and small effective volume on a planar cavity presenting a

local variation of the cavity thickness [146] . This technique, based on regrowth on a pattern 2D

microcavity, could reasonably be modiﬁ ed for contact implementation.

Another key ingredient for these protocols is a non-destructive measurement of the spin state of

the electron trapped in the quantum dot. First demonstrations of the spin state of a single electron

have very recently been reported [144, 147] . The information on the spin state is obtained through

the conditional Faraday rotation of a spectrally detuned laser. Sensitivity is a crucial matter for

these experiments, which have already been shown to be greatly enhanced using cavity effect [143]

and will be even more improved in high-quality factor small effective volume cavities.

Finally, cavity effects with several quantum dots have to be demonstrated. To realize such

a demonstration precise control of the spectral and spatial tuning between the cavity and the

quantum dot is necessary.

4.8 Summary and conclusions

Engineering the interaction of light with matter allows one to tune important properties of sol-

ids like the spontaneous emission rate or the spontaneous emission coupling factor into a laser

mode. Light–matter interaction in semiconductors using quantum dots has been investigated in

various microcavities. Inhibition and exaltation of a single quantum dot spontaneous emission

has been demonstrated in various systems, and has already led to several applications in quan-

tum cryptography. Strong coupling regime has recently been demonstrated reaching the ultimate

limit for light–matter coupling in a solid-state system. Technological challenges now need to be

addressed to reach either Purcell effect or strong coupling regime on demand. Indeed, controlling

solid-state quantum electrodynamics will eventually open new ﬁ elds of investigation, such as

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158 Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics

ultra-low threshold laser or quantum computation based on the coupling between a single spin

and a cavity mode.

Acknowledgements

We would like to thank I. Abram, J. Bloch, A. Beveratos, R. Braive, A. Cavanna, S. Guilet, J. Hours,

S. Laurent, L. Le Gratiet, A. Lemaitre, D. Martrou, A. Miard, G. Patriarche, E. Peter, I. Sagnes, and

S. Varoutsis who largely contributed to this work. We thank P. Lalanne, J.M. Gérard and P. Voisin

for fruitful discussions. This work was partly supported by ANR JCJC Micados, SANDIE European

Network, ACN NanoQub, ACN Polqua, by the ‘ ‘ Région Ile de France ’ ’ and the ‘ ‘ Conseil Général de

l’Essonne ” .

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