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Photons and (artificial) atoms: an overview of optical
spectroscopytechniques on quantum dotsA. N. Vamivakas a; M. Atatüre
aa Cavendish Laboratory, University of Cambridge, Cambridge, UK
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Photons and (artificial) atoms: an overview of optical
spectroscopy techniques on quantum dots
A.N. Vamivakas and M. Atatüre*
Cavendish Laboratory, University of Cambridge, JJ Thomson
Avenue, Cambridge CB3 0HE, UK
(Received 27 March 2009; final version received 6 August
2009)
In most branches within experimental physics technical prowess
lies at the heart of many seminal works. From theobservation of the
photoelectric effect and the ultraviolet catastrophe that led to
the development of quantummechanicsto the first transistor that
shaped the modern age of electronics, significant physical insight
has been achieved on theshoulders of technical advances and
progress. Research on self-assembled quantum dots may be a drop in
the sea ofphysics, but it still is no exception to this trend, and
more physical insight continues to be revealed as the tools of
thetrade get increasingly more complex and advanced. This article
is written primarily for senior undergraduate studentsand first
year graduate students of experimental physics involving optically
active quantum dots. More often than not,we have seen students
shuffling through journal articles trying to relate the reported
physics to the used experimentaltechniques. What we want to cover
here is not in any way the history or the recent progress in
quantum dot research –there are an ample number of topical books
and review articles for that – but rather to highlight a selection
of optics-based measurement techniques that have led to significant
progress in our understanding of quantum dot physics as wellas
their applications in the last two decades. We hope a basic survey
of the relevant optical spectroscopy techniques willhelp the
newcomers in connecting the dots between measurements and
physics.
Keywords: quantum dots; excitons; spectroscopy; quantum optics;
quantum information science
1. Introduction
Quantum dots have become a system of study in a broadrange of
disciplines in a relatively short time. Theincredible progress in
synthesis, growth and fabricationquality fed further the advances
in optical investigationsin physics, biology and chemistry.
Particular to quantumphysics, quantum dots allow optical studies of
confinedcharge and spin systems and in parallel studies
onengineering light–matter interaction and even thesuppression of
spontaneous emission. We start belowwith an introduction to the
growth of quantum dots andthen follow with a discussion of the
various opticallyactive charge complexes that a quantum dot
cansupport. We then proceed with a handful of opticaltechniques
lined up vaguely with increasing technicaldifficulty and
chronological appearance. We finish witha selection of
applications, mainly driven by quantuminformation science, in order
to highlight how muchexperimental progress in quantum information
science isindeed driven by a reinterpretation of results
obtainedvia conventional optical spectroscopy.
2. Quantum dots: from growth to energy levels
Advances in material science have enabled the growthof
heterostructures exhibiting inhomogeneity on a
length scale relevant for influencing the spectrum of
amaterial’s excitations. Quantum dots (QDs) areheterostructures
engineered to provide three-dimen-sional spatial confinement for
electronic excitations.The confinement yields a discrete spectrum
of QDenergy eigenstates and it is not uncommon to refer toQDs as
artificial atoms. In practice, there exist anumber of quantum
confined physical systems thatexhibit a discrete electronic
spectrum – interfacefluctuation QDs that form at the gallium
arsenide/aluminum gallium arsenide (GaAs/AlGaAs) barrierboundary in
a quantum well, core–shell cadmiumselenide/zinc sulphide (CdSe/ZnS)
nanocrystalsformed through both colloidal methods and quantumdots
grown by Metallic Organic Vapour PhaseEpitaxy (MOVPE), electrically
defined QDs via gateelectrodes patterned on two-dimensional
electron gasyielding precise control over the local
electrostaticpotential and self-assembled QDs grown by Molecu-lar
Beam Epitaxy (MBE). Here, we will focus mainly,but not exclusively,
on indium arsenide (InAs)/GaAsself-assembled QDs in portraying the
optical techni-ques used to date. In this section, before looking
intothe characteristic electronic structure of QDs, we willdiscuss
the material science advances that haveresulted in the growth of
self-assembled QDs byMBE.
*Corresponding author. Email: [email protected]
Contemporary Physics
Vol. 51, No. 1, January–February 2010, 17–36
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Epitaxial growth is a process where a new crystal isgrown over a
host crystal surface via layer-by-layeratomic deposition [1].
Epitaxial techniques are capableof depositing high quality
semiconductors with anabrupt change in material composition having
mono-layer (*3 Å) accuracy. The formation of InAs/GaAsQDs is a
natural process and is the manifestation of astrain-driven phase
transition that occurs when com-bining two materials of different
lattice constantsduring one material growth cycle. Every material
hasits own lattice constant and this commonly leads toformation of
strain on two layers constituting anabrupt interface. Figure 1(a)
illustrates the two typicalcases of strain release: a
monolayer-thick materialembraces a lattice constant dictated by the
hostmaterial or a sufficiently thick material recovers itsown
lattice constant resulting in strain release viadislocations and
defects at the interface. The formationoccurs exactly during the
transitional period linkingthe two regimes of strain release. If
the latticeconstants are significantly different (e.g. 7%
mismatchbetween la and lb, as is the case for GaAs and
InAslattices), the epitaxial growth of InAs with the GaAslattice
can not be sustained for more than twomonolayers of growth. At one
point, the newly formedlayer goes through a phase transition
forming minia-ture islands, very much like mercury droplets do on
asmooth flat surface. Further growth with the samematerial as the
handle wafer, in this case GaAs, caps
the QDs and protects them from the surroundingenvironment. After
growth, the height of the QDs istypically 4–5 nm, as determined by
cross-sectionalscanning tunnelling microscopy image of Figure
1(b)[2]. We emphasise that, although the self-assembledQDs exhibit
pristine optical properties, the in-planeQD distribution is
disordered and extensive efforts arestill made today in this field
in order to achieve bettercontrol over the island size distribution
and location ofnucleation.
The MBE growth process results in strong three-dimensional
carrier confinement for electrons in QDsresulting in quantisation
of energy states. However,QDs are composed of around 105 atoms, and
thus forma mesoscopic system with arbitrary shape and composi-tion
which differ from QD to QD. The distribution inshape and
composition combined with the strain profileexperienced by the QD
all influence the single particleQD energy levels in the form of
inhomogeneousbroadening. In addition to material properties,
ifmultiple charges are confined in the QD the Coulombinteraction
between the quantum confined carriers hasto be taken into account
when calculating the multi-particle energy levels. All the previous
complicationsmake an analytical determination of QD
propertiespractically impossible and modelling typically relies
onperturbative or numerical methods. Even with all
thesecomplications it is striking that the roughly 105 InAsatoms in
the GaAs matrix conspire to exhibit a discreteatomic-like energy
spectrum.
The InAs/GaAs QDs covered in this work aresemiconductors in bulk
(three-dimensional) form.Therefore, to solve for the energy levels
of QDs, it isnatural to start from the bulk material properties
anddetermine the consequences of reducing the
system’sdimensionality. For a phenomenological, but satisfac-tory,
prediction of bulk semiconductor band struc-tures, we resort to a
perturbative k � p model. In k � psingle-particle wavefunctions and
energy eigenvaluesare assumed to be known at k ¼ 0 and the
banddispersion is obtained in the small k approximationaround the �
-point [3]. These perturbative methodscan also be applied to
quantum dots since the k-vectordistribution of confined charges is
concentratedaround k ¼ 0. Figure 2 shows a schematic of theband
structure of bulk GaAs with relevant parametervalues at room
temperature. The band structure ofInAs looks esentially identical,
but, the values of theindicated parameters differ significantly
from GaAs.
Excitation of an electron across the bandgap leavesan empty
electronic state in the otherwise electron-filled valence band.
These holes can equally be treatedas positively charged particles
with modified mass andg-factor. The lowest conduction band has to a
verygood approximation parabolic dispersion around the
Figure 1. (a) An illustration of lattice constant mismatchfor
two materials grown by MBE for strained thin layers anddislocated
thick layers. (b) Cross-sectional scanningtunnelling microscopy of
a self-assembled InAs QD grownby MBE [2].
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� -point, as indicated by the red curve in Figure 2.
Thewavefunctions for this band have s-wave charactersustaining a
twofold spin-degeneracy with [SjjSz] ¼[1/2jj + 1/2]. The valence
band wavefunctions havep-wave character that would normally sustain
a sixfoldspin-degeneracy forming a [3/2jj + 3/2, + 1/2] quad-ruplet
and [1/2jj + 1/2] doublet. However, spin-orbit coupling in these
semiconductors causes the[1/2jj + 1/2] doublet to be separated in
energy formingwhat is referred to as the split-off band (Figure
2).Further, upon including the influence of other bands,even the
fourfold degeneracy of the [3/2jj + 3/2,+1/2] states is lifted for
k 6¼ 0 forming the heavy-hole and the light-hole bands with
near-parabolicnegative curvature dispersion as seen in Figure
2.
When the dimensionality of the system is reducedsuch that the
effective Bohr radius becomes compar-able to the physical extent of
the confining material,quantum confinement strongly influences the
densityof states, band dispersion and degeneracies. In the caseof
QDs, the dimensionality is zero resulting inmotional confinement
along all three directions.Therefore, a set of discrete energy
levels arise withlevel spacings determined by the, not necessarily
equal,confinement strength along each direction. In fact, dueto
their particular lens-like topology (see Figure 1(b)),the QDs
considered here display strongest motionalconfinement along the
growth (z) axis. Therefore, themain features of the energy levels
of these QDs can beseen by simply considering a strong confinement
alongthe z direction with a two-dimensional
quasi-parabolicconfinement in the two remaining directions.
Agenerally accepted approach to quantifying the QDenergy levels and
the corresponding wavefunctionsrelies on pseudopotential theory
[4]. A nice tutorial
discussion of pseudopotential theory, with illustrationsof the
QD electronic excitation wavefunctions, can befound in the review
by Zunger [5].
From the optics perspective an important featureof the quantum
confinement is that although theenergy spectrum of the QD is
altered when comparedto the bulk semiconductor, the electrons and
holes thatbecome trapped in the QD inherit the spin structure ofthe
bulk semiconductor. This determines the optical(polarisation)
selection rules for transitions betweenQD electron and hole states
mediated by a photon.Explicitly, focusing on the conduction band
and heavyhole valence band, we can specify the QD electron andhole
spin. The QD levels derived from the conductionband levels sustain
their twofold spin degeneracy,while the QD levels derived from the
valence bandstates display a confinement-induced splitting
intoheavy-hole and light-hole doublets.
We qualitatively established the energy levels ofelectrons and
holes confined in all three dimensions insemiconductor QDs. We now
identify the energy scalesof common InAs/GaAs QD charge
configurations thatare probed optically. The simplest charge
configura-tion linked to an optical emission is the neutralexciton
X0 (see Figure 3), i.e. a single electron–holepair occupying the
lowest discretised energy levels
Figure 2. A simplified band structure illustration for
III–Vsemiconductors such as GaAs and InAs with the
typicallyaccepted values for key energy scales.
Figure 3. Neutral exciton (X0), biexciton (XX) transitionsunder
excitonic level splitting and negatively charged trion(X17)
transitions under magnetic field along the growth axisfor a typical
quantum dot. The wavy arrows indicate photonmediated transitions
between the states. For the negativelycharged trion (right
illustration) the up (down) arrowrepresents the electron spin
projection of þ1/2 (71/2)along the growth direction and the solid
up (down) triangleis the hole projection of þ3/2 (73/2). The ground
state of thetrion transition is a single electron with its spin
projection upor down. Each transition is decorated with a
symbolindicating the emitted photons polarisation – pX (pY)
forhorizontal (vertical) and sþ (s7) for right (left)
circularlypolarised photons. The direction of linear
polarisation(horizontal and vertical) is defined with respect to
themajor and minor axis of the elliptical QD base (as opposedto
circular) due to strain-induced anisotropy of the dotgeometry.
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within the original conduction and valence bands. Theelectron in
the conduction band can have spinquantum number [SjjSz] ¼ [1/2jj +
1/2]. The heavyhole in the valence band has spin [JjjJz] ¼ [3/2jj +
3/2].By addition of angular momentum, a single electron–hole pair
in the QD can end up in any one of fourspin-state combinations. The
total angular momentumof these combinations being DJ ¼ +1 or DJ ¼
+2,each doublet is degenerate. In an optical transitionangular
momentum must be conserved, and this isreflected in the
polarisation of the emitted photons.Recombination via a
single-photon emission processcan only occur for the DJ ¼ +1
exciton doublet, sincesingle photons carry angular momentum +1.
Theangular momentum conservation is reflected in theemitted
photon’s polarisation. Specifically, photonscarrying þ1 (71)
angular momentum are left (right)hand circularly polarised and are
denoted with thesymbol sþ (s7). The exciton doublet DJ ¼ +1 that
islinked to photon emission is called bright, while theremaining
optically inactive doublet is called dark (theDJ ¼ +2 excitons).
The polarisation selection rulesalso constrain the set of excitons
that may be createdoptically to the DJ ¼ +1 doublet.
Of course, the real world is not simple! Thepreviously mentioned
shape nonuniformity and strainact to coherently mix the bright DJ ¼
+1 excitondoublet via the electron–hole exchange interaction.This
interaction couples the spins of the electron andhole confined in
the QD and depends sensitively on thestructural symmetry of the QD.
The electron–holeexchange serves to both break the DJ ¼ +1
excitondoublet’s degeneracy and alter the polarisation of
theemitted photons from circular to linear, indicated bypX/pY in
Figure 3. This new polarisation basis, which isdefined along the
major and minor axis of the ellipticalQD base, led to the phrase X
– Y splitting to denotethis effect. Due to exchange interaction, an
electron–hole pair once created in DJ ¼ þ1 state will
precesscoherently between DJ ¼ +1 spin configurations.
Are-diagonalised Hamiltonian after including this inter-action
leads to new eigenstates with the degeneracy oftheir energies
lifted in proportion to the interactionstrength. Typical energy
scale for the DJ ¼ +1 excitondoublet fine structure splitting is
*10 meV for self-assembled InAs/GaAs QDs. We will see this
fine-structure splitting has consequences for applicationsinvolving
photon emission in later sections of thisarticle. We direct the
reader to [6] for a completediscussion of electron–hole exchange
interaction.
The next QD charge complex we discuss is twoelectrons and one
hole. We call this singly chargedexcitonic QD excited state a
trion, see Figure 3 (rightdiagram), and label it as X17. In forming
the trioncomplex, Pauli’s principle forces the electron pair to
form a spin singlet state where the closest triplet state
hasenergy much higher than typical ambient temperature(4 K). Since
the resident hole can have either spin up orspin down, each QD has
two trionic transitions that areenergetically degenerate. Due to
Coulomb interactionsin this three-body problem, the recombination
energy ismodified with respect to the original neutral X0
excitonic transition energy (ignoring fine structure) byDE ¼
Eee7Eeh – the direct energy due to electron–electron and
electron–hole Coulomb interaction [7] asdictated by the
wavefunctions via the form
Emn /ð�1Þð1�dm;nÞe2
4pe0er
ZZdrdr0
jcmðrÞj2jcnðr0Þj
2
jr� r0j : ð1Þ
In the InAs/GaAs QDs considered here the result is atotal shift
of DE ¼ 6 meV to lower energy for the trionictransitions. In
contrast to the neutral exciton where theelectron–hole spin
exchange breaks the twofold degen-eracy, the electronic spin
singlet is immune to electron–hole exchange and the two trion
states remain degen-erate. In this case, the polarisation of the
emitted photonis in the circular basis and the handedness is
determinedthe direction of the resident hole spin.
The situation is conceptually similar when there aretwo
electron–hole pairs present in the QD referred toas the biexciton
(XX) shown in Figure 3 (middlediagram). The shift in the transition
energy for abiexcitonic transition can once more be determined
bythe energy difference between the initial and finalstates, DE ¼
2Eee þ2Ehh7 Eeh, and is on the order of2 meV for InAs QDs.
Ultimately, every chargecombination results in a distinct spectral
signaturedue to the Coulomb interaction, and we refer thereader to
[7] for a detailed explanation of this approachfor direct and
exchange type interactions.
It is good to note here that the relevant energyscales for each
mechanism considered are well defined.The optical transitions occur
at eV range while directCoulomb interactions within a QD are at
tens of meV.The fine structure such as X–Y splitting as we will
seelater is on the order of tens of meV, which is still muchlarger
than the characteristic transition linewidth of1 meV. While each
quantum dot can have vastlydifferent emission energies due to
inhomogeneity inthe quantum dot ensemble, the relative energy
shiftsare conveniently rather robust. With an understandingof
common QD charge complexes, we can begin toaddress how the tools of
optical spectroscopy revealphysical properties of the QD
states.
3. Optical spectroscopy techniques
In the previous section we highlighted the mostrelevant
excitonic complexes in QD optics and their
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relevant energy scales. How excitons and photonscouple to each
other can be presented in many ways.We will make a distinction
between two methods ofoptical excitation. The first approach, which
we referto as nonresonant excitation, is to use a light
source,typically a laser, with an energy that is larger than
theenergy of the relevant excitonic transition. The lasercreates
exciton population in either higher energy QDexcitons or in the
bulk of the host semiconductormatrix. The higher energy excitons
then nonradiativelyrelax, giving off energy through carrier
scattering andphonons, and populate the lowest energy
availableexciton state. The second approach is to use a laserwith
energy equal to the excitonic state of interestmuch like in atomic
physics. In the latter case we willsay the laser is resonant to
conform to the languageused in the field. Of course either method
of excitationlinks real states of the system and is resonant; but
werather use the term resonant to indicate a laser that hasenergy
commensurate with the relevant excitontransition.
3.1. Optics of quantum dots: nonresonant excitation
3.1.1. Exciton spectrum – photoluminescence
A quantum dot may host discretised electronic levels,but the
surrounding semiconductor matrix introducesa continuum of filled
valence and available conductionband states. Therefore, an optical
field can generate anensemble of electron–hole pairs in the
vicinity of aquantum dot. Typically, most of these pairs
recombinequickly to yield photon generation at the bandgapenergy of
the semiconductor, while occasionally, acombination of electrons
and holes may be capturedinto the discrete exciton levels of the
quantum dot. Forthis to happen, the excess energy of the excitons
has tobe taken away by carrier scattering or phonons.Exactly how
this relaxation occurs and how it dependson external parameters
such as temperature andmagnetic field has formed a whole branch of
researchon quantum dots, especially in the 1990s. Thisrelaxation
process happens at relatively fast timescales,i.e. tens of
picoseconds and removes any coherencewith the excitation laser.
Therefore, generation ofexcitons in a quantum dot via bandgap
excitation canbe treated typically by an incoherent driving
fieldbetween quantum dot states. In addition, the spinorientations
of the optically excited electrons and holesare affected during the
relaxation processes yieldingonly a residual correlation to the
original excitationlaser polarisation. Radiative recombination of
anelectron–hole pair in the quantum dot consequentlyreveals
information on the quantised energy levelsand the optical selection
rules. Therefore, micro-
photoluminescence (mPL), i.e. measuring the spectrumof QD light
emission under continuum excitation, hasbeen an essential workhorse
of quantum dot researchover the years.
In a typical mPL setup, a schematic can be found inFigure 6(a),
Section 3.1.3, laser light is directed by abeamsplitter to an
objective which focuses the lightonto the QD sample (the triangle
in the illustration).The numerical aperture of the objective and
the laserwavelength determine the focal volume the laserexcites.
The luminescence emitted from the sample iscollected by the same
objective and is directed througha pinhole (RP in the
illustration). The function of thepinhole is to limit the sample
volume from whichluminescence is collected. Selection of the
objectivenumerical aperture and pinhole diameter result
indiffraction limited focal volumes of less than 1 mm3
which, provided the sample density is low enough(1–10 dots per
mm2), can enable single QD spectro-scopy. After the pinhole, a flip
mirror (FM in theillustration) directs the luminescence to an
imagingspectrometer that is able to resolve the spectral contentof
the luminescence.
We present in Figures 4(a) and (b) two of the firstreported
ensemble and d-function-like spectra ofphotoluminescence from
InAs/GaAs quantum dots[8,9]. Figure 4(a) displays the broad
emission of a QDensemble at the low energy tail of the spectrum
alongwith the sharper bulk luminescence at 1.38 eV. In
themeasurements presented in Figure 4(b), to probe
Figure 4. (a) Photoluminescence from ensemble of InGaAsQDs [8].
(b) Top three traces are mPL spectra recorded fromthree different
sample locations at 10 K. The bottom trace isthe sum of 20 spectra
[9]. (c) mPL from a single well-isolatedInAs QD showing the
individual transitions. The solid circlesrepresent electrons and
the open circles are holes.
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individual QDs the sample surface was etched tocontain small
mesas separated by 15 m m. The mesaslimited the spatial region of
excitation and collectionrevealing signatures of single QD emission
althoughthe sample density resulted in *250 QDs within
thediffraction limited collection spot size. The etched mesais
similar in effect to an appropriately selected pinholeplaced in a
conjugate image plane of the microscope.Figure 4(c) is a similar
mPL measurement on a well-isolated single InAs QD displaying the
individualoptical transitions with the corresponding
chargeconfigurations.
3.1.2. Exciton lifetime – time correlated photoncounting
Although photoluminescence is able to determine thetransition
energies between optically active quantumdot electronic states, it
is not able to associate a timescale with these transitions.
Assuming a stable groundstate, it is natural to invert the
linewidth measured withthe spectrometer to obtain the transition
timescale.Unfortunately, for most QDs and a single spectro-meter,
the transition linewidth is narrower than theinstrument’s spectral
resolution (a typical resolution is30 meV), hence numerous
publications historicallyhave used the term resolution-limited
emission overthe years. It is clear from these considerations that
inorder to assess the dynamics of the light emissionprocess a new
technique is needed – one with temporalresolution.
One approach with temporal resolution is togenerate an ensemble
of electron–hole pairs in thesurrounding matrix by a short
nonresonant laser pulse,typically on the order of a few hundred
femtosecond toa few picosecond duration. In bulk semiconductorswith
direct bandgap, excitonic lifetime is on the orderof a few
picoseconds. When confined in all three spatialdimensions, as in a
quantum dot, the lifetime ofexcitonic complexes is enhanced by
about three ordersof magnitude. Due to the unequal recombination
ratesof excitons in extended versus confined systems, onlythe
excitons captured by the quantum dot remain evenafter a mere tens
of picoseconds following the initialexcitation pulse. This allows
the quantum dot excitonto decay in the absence of any other
excitations, andthus free from excited carrier induced effects.
QDemission, spectrally filtered from all other
excitonicrecombinations, can then be detected by photoncounting
photodetectors. A registry of photon arrivaltime delays with
respect to each excitation laser pulsethus builds up a temporal
histogram of detectionevents. This histogram can be interpreted as
theprobability that a QD exciton remains alive (or QDremains in the
excited state) over a timescale.
Figure 5 presents data from a single InAs QDmeasured as a
function of average pulse power for a 3ps wide excitation pulse
[10]. At the lowest pumppower (bottom curve), the exciton (or
excited state)lifetime is revealed, and the fit decay time is 0.8
ns. Foran ideal two-level system, such a measurement willproduce a
single exponential decay of width directlyrelated to the excited
state lifetime, and thus thenatural linewidth of the transition
through a Fouriertransform. As the average pulse power is
increased,there is a pronounced shift in the start time for
theexciton decay. This is in fact a very nice signature
ofmulti-excitonic effects taking place at high pumppowers. The
probability of capturing one exciton isalready high, so an
additional exciton capture withinthe recombination time becomes
considerable. Conse-quently, exciton decay is necessarily delayed
by thecharacteristic biexciton decay time. It is important tonote
here that not only the radiative decay, but alsoany other process
that results in exciton populationloss will affect this histogram.
For example, if there is anonradiative channel for exciton decay
that is largerthan the radiative lifetime, the measured
histogrammay exhibit a double exponential where the decay ofeach
exponential is dictated by the radiative andnonradiative decay
rates. Quantum dot systems, suchas CdSe/ZnS core/shell colloidal
QDs, do possess finitenonradiative decay rates [11], where one
mechanismleading to nonradiative decay in CdSe/ZnS is
Augerrecombination – a process whereby an exciton pairrecombines to
form a more energetic electron. In thecase of InAs QDs with
effectively no appreciable
Figure 5. The photoluminescence intensity from a singleInAs QD,
at 20 K, as a function of time after excitation witha 3 ps laser
pulse. The inset indcates the average power forthe excitation
pulse. The lowest power decay curve fits to anexcited state
lifetime of 800 ps [10].
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nonradiative decay channels, another mechanism doeslead to
biexponential decay in lifetime measurements –bright and dark
exciton mixing. A clear study of thiseffect can be found in
[12,13].
3.1.3. Two-time field and intensity correlations
Time-correlated photon counting allows us to measurethe emission
timescale of the photons, but reveals verylittle about their
coherence properties. In order tounderstand the extent of excitonic
and carrier induceddecoherence mechanisms taking place prior to
orduring the exciton recombination, we need to use atechnique where
the signal is directly linked with thedegree of coherence of the QD
emission. A first step toquantify the degree of coherence in the
emitted photonstream is to interferometrically measure two-time
fieldcorrelations, where temporal correlations in the lightbeam are
revealed by interfering the field with itstime delayed replica. One
approach, illustrated inFigure 6(a), is to use a Michelson
interferometer whichmaps path length difference between the two
inter-ferometer arms to time delay. Mathematically, this canbe
represented as
ID Dxð Þ ¼ 2� Iavg 1þ< g Dxð Þ½ �ð Þ; ð2Þ
where
g Dxð Þ ¼ exp �i2kDx� 2g Dxj jð Þ: ð3Þ
Dx is path length difference between the two arms ofthe
Michelson interferometer and g is the totaldamping rate of the
transition (we have assumed thetransition lineshape is Lorentzian).
In Equation (2),Iavg is the average intensity that reaches the
detector.While the first term in the parentheses of Equation
(2)reveals the average intensity of light and does notexhibit
relative optical path length dependence, thesecond term, in
Equation (2), modifies the detectedsignal to the extent that the
parts of the optical fieldat two different times remain correlated.
Figure 6presents this measurement on photons emitted from asingle
self-assembled indium phosphide QD [14]. InFigure 6(b) widefield
images of QD emission arepresented for the open circles decorating
the interfer-ogram in Figure 6(c). The decay of the
interferogramenvelope for large path length differences, Figure
6(d),reveals the damping rate g defined in Equation (2) willresult
in an exciton with an emission linewidth of186 meV. The inverse of
this quantity yields thecoherence time of the emitted photons.
Zwiller et al.indeed extract a coherence time of 140 ps,
muchshorter than the measured excited state lifetime of1.2 ns,
allowing for an accurate quantification of the
dephasing mechanisms. One notable source of dephas-ing in this
measurement is the optical charging of theQD environment that
results from above bandgapnonresonant excitation. Specifically,
photoexcited elec-tron hole pairs, a portion of which relax to
occupy theQD exciton states, may also become trapped at defect
Figure 6. (a) Experimental setup. Flip mirrors (FM) areused to
direct the emitted photons to either to aspectrometer, to a
Michelson interferometer, or to aHanbury-Brown and Twiss
correlator. The latter two arebuilt around nonpolarising beam
splitters (BS). A removablepinhole (RP) can be used to select a
single dot. A narrowbandpass filter (F) is tilted to transmit
single spectral lines.(b) Images taken through the Michelson
interferometershowing several dots with varying intensity as the
mirror isscanned. (c) Single-dot photoluminescence intensity for
thedot marked by an arrow in (b) as a function of mirrorposition.
The circles indicate the three positions where theimages were
taken. (d) The envelope of the exciton emissioninterferogram
measured at 6 K with a 5 mm relative pathdifference between the two
arms. The inset is the intensitycorrelation, g(2)(t), measurement
on the same exciton [14].
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sites in the vicinity of the QD. These local chargescreate a
varying Stark field (uncontrolled electric fieldat the location of
the QD) which acts to shift the QDtransition energy. If the
time-scale of the measurementsamples a number of local defect
charge configurationsthen the QD transition energy will shift
throughout thecourse of the measurement. The shift in the energy
willact to smear out the single-photon interference fringesand will
lead to a reduction in the measured contrast,i.e. extracted photon
coherence time.
Two-time measurements can also be carried oneorder higher, where
the correlations in optical intensityrather than the field are
considered at two differenttimes (in Figure 6(a) the HBT box). In
this case, thesignal arises due to intensity fluctuations and does
notdepend on interference or phase relations within thefields.
Therefore, historically second-order correlationsof strong light
beams were obtained from correlatingphotodetector current outputs
i(t) after detection, inthe form of
gð2ÞðtÞ ¼ iðtÞiðtþ tÞh i= iðtÞh i iðtþ tÞh i : ð4Þ
A truly uncorrelated pair of photocurrent measure-ments gives
unity value for this function, and anydeparture from unity value
indicates correlation (oranti-correlation) within a characteristic
memory time-scale of the source. Consequently, Equation (4) can
bewritten in the form g(2)(t) ¼ 1 þ ZC(t). For intenseoptical beams
detected by photodiodes, the registeredcurrent is proportional to
the intensity, therefore thesecond-order optical correlations are
mapped conve-niently to the current. The scenario is quite similar
forsingle photon light levels where a temporal registry
ofindividual detection events carries similar correlationsto the
incident photon stream. For example, coherentlight (such as laser
emission) exhibits no correlation(Z ¼ 0) and g(2)(t) ¼ 1 for all
delay times whereas thememory present in single photon emission
from a two-level system results in anti-correlation, g(2)(t) 5 1
withZ ¼ (71) and C(t) ¼ exp(7gjtj). In order to performsecond-order
correlation measurements on single QDemission lines, the
Hanbury-Brown Twiss (HBT)experimental arrangement, shown in Figure
6(a), isused. The collected QD emission is split to be detectedby
two photon counting detectors. Photon detectiontimes are then
registered per detector. A count isrecorded conditioned on the
previous detection of aphoton – a coincidence count. A histrogram
of countsis plotted as a function of the delay time
betweensuccessive detection events using time–amplitude
con-version. Figure 7 shows intensity (photon)
correlationmeasurements performed on the emission undernonresonant
excitation for an ensemble of InAs QDs(panel 1) and two individual
InAs QDs (panels 2/3)
[15]. The pump power in panel 2 (3) is at the onset(below) the
power necessary to saturate the QDemission. In panel 1, there is no
correlation in thephoton emission from the ensemble of QDs, and
thereis no deviation from unity as a function of delay time.Panels
2/3 show that the detector outputs areuncorrelated in all time
scales except around zerotime difference, where detection events
are anti-correlated. The QD emission therefore is antibunched,or
has a degree of temporal order. This is aconsequence of the
anharmonicity of the energy levelsinvolved in the QD emission. In
these systems only onephoton can be generated within a radiative
lifetime.Therefore, the two-time dependence of coincidencecounts
also reveals the temporal profile of photonemission. For a
two-level system with Lorentzianspectrum, with a linewidth of g, we
expect to see anexponential decay in the time domain, and
photoncorrelation measurements indeed reveal a symmetricdouble
exponential profile. It is important to note herethat this
technique measures the probability of two-photons being generated
sequentially, therefore it isimmune to deviations from ideal
configurations. Forexample, if the experimental setup includes
imperfectelements or has mechanical instability, parameters
Figure 7. The data presents the measured distribution
ofcoincidence counts and a fit of the correlation function. Panel1
is for a high density QD sample with more than one QDemitting
within the focal volume of the collection optic. Themeasured
g(2)(t) does not exhibit antibunching. Panels 2 and3 are for a
single InAs QD at two different pump powers.Panel 2 (3) is taken at
125 W cm72 (66 W cm72). Panel 3exhibits a pronounced antibunching
dip [15].
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extracted from an interference-based measurementwould be
significantly affected, but not the observedphoton correlation
function. Alternatively, we mightbe dealing with less than ideal
atomic-like states wherepure (elastic) dephasing and nonradiative
decaymechanisms of the excited state may be non-negligible.While
previously discussed lifetime measurementswould only provide the
total excited state decaycharacteristics, photon correlation
measurements, pro-vided the excitation method can be classified
asincoherent, will reveal only the radiative part andremain immune
to all other decay and dephasingchannels [16]. In experimental
physics such powercomes rarely from such a simple arrangement.
Anotherpoint we wish to emphasise here is that while
two-timecorrelation measurements to date have been performedusing
two independent detectors, identical measure-ments could also be
performed using a single detectorwith sufficiently fast response
and timing capability.Therefore, there still is a lot of attention
on thedevelopment of single-photon sensitive, photon-num-ber
resolving and high temporal resolution detectortechnology [17],
where superconducting detectors seemto offer promising capabilities
for this direction [18].
The power of photon correlation measurementsdoes not stop here
at single photon generation, and canreveal a lot more about the
internal dynamics of multi-exciton recombination. We have seen
previously thatthe individual recombination processes of neutral
andcharged excitons as well as biexcitons have welldistinguished
energy shifts (see Figure 4(c)) and havealready seen such
consequences on lifetime measure-ments (see Figure 5, top panel).
Therefore, the intensitycorrelations between any two QD emission
lines can bemeasured as well. Figure 8 shows such measurementsfor
biexciton–exciton cascade decay [19]. The break-down of symmetry
around the zero time delay betweencoincident detection events
reveals the one-sidedordering of the cascade process. The
probability todetect exciton decay is increased on the condition
thata photon from biexciton decay has been detected, whilethe
probability to detect a photon from biexcitonicdecay is essentially
nil upon detection of a photon fromexciton decay. This technique
therefore gives directinformation on the emission time ordering,
plays acrucial part in identification of the observed spectrallines
and allows us to identify metastable intermediatestates during the
cascade.
3.2. Optics of quantum dots: resonant excitation
Techniques presented in the previous section can givevaluable
information on exciton capture and decaydynamics as well as
properties of the generated opticalfields. A common feature in all
of these studies is the
incoherent pumping of the quantum-dot transitionsthrough carrier
generation in either the host matrixsuch as GaAs or the
quasi-continuum states above thehigher-lying confined states of the
quantum dot – whatwe have termed nonresonant excitation. This
excita-tion method leads to photon-emission-time jitter, sinceit
relies on an uncontrolled relaxation step to populatethe excited
exciton state, and spectral wandering of thequantum-dot transition
larger than the transition’slinewidth due to optical charging of
the host semi-conductor matrix. Both effects reduce the usefulness
ofnon-resonantly generated single photons in linear-optics quantum
computing algorithms, even if thequantum dot is coupled to a cavity
[20]. In an attemptto both address this previous shortcoming and
providespectrally selective access to the quantum-dot electro-nic
transitions, increasing attention has turned toresonant optical
excitation. Noting all successfulquantum-information science (QIS)
implementationson well-developed qubit candidates, such as
trappedions, have relied on resonance scattering, it is clear
thatresonant optical control of QD transitions is desirable.In the
realm of resonant spectroscopy, we make afurther distinction
between temporal and spectralmeasurements.
3.2.1. Temporal measurements
A single ultrafast pulse propagating through a materialmay be
affected by both absorption and dispersion.Transient nonlinear
optical spectroscopy, also knownas pump–probe spectroscopy,
involves a sequence ofultrafast optical pulses, which are separated
in time
Figure 8. Cross-correlation of the biexciton–excitoncascade
emission. The experimental apparatus is identicalto Figure 6(a)
except the detection of the exciton emission isconditioned on the
measurement of biexciton emission. Torealise this experimentally,
narrow band spectral filters(1–2 nm FWHM centred on either the
exciton or biexcitonemission line) are placed in front of each APD
[19].
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and tuned to the spectral vicinity of an absorptionband in a
material. Conditional on the opticalexcitations or an induced
polarisation field due to thefirst pulse (usually called the pump)
the propagation ofthe second pulse (i.e. the probe) may show a
deviationfrom what one would expect from the single pulse
case.Therefore, a dynamical response from the material canbe mapped
out based on the time delay between thepump and probe pulses. The
selection of the laser pulsespectral/temporal width depends on the
desired experi-ment, but typical considerations for QD excitons
arethe timescale of the dynamics limiting the pump–probedelay and
the spectral separation of other excitonicresonances limiting the
pulse bandwidth. The pulsespectral width (via Fourier
transformation) dictates theshortest temporal separation between
two pulses andconsequently sets the experiment’s temporal
resolu-tion. Before continuing we highlight that
nonlinearspectroscopy techniques result in weak signals (typi-cally
15 orders of magnitude with respect to theexcitation pulses) and
the measurement of these smallsignals presents an experimental
challenge [21]. A firststep toward signal recovery is to interfere
the signalfield on the detector with a local oscillator field that
iseither derived from a reference pulse that does notinteract with
the QD or with one of the excitationpulses. Mixing the signal field
with a second field on aphotodiode is referred to as heterodyne
detection, aterm borrowed from radio wave engineering. Theadvantage
of heterodyne detection is that the inter-ference term depends on
the signal amplitude multi-plied by the conjugated strong local
oscillator fieldamplitude and this acts to amplify the measured
signal.In addition to interference it is common to modulatethe
excitation pulses so that the signal is carried bydistinct spectral
components in the measured photo-current which can then be accessed
with a phase-sensitive lock-in detection system.
Within the family of pump–probe techniques usedfor ensemble and
single QD spectroscopy, there is aparticularly elegant and powerful
modality, where thepulse sequence itself generates a third-order
nonlinearpolarisation in the QD which acts as a source of
anadditional field that carries information related to thedynamics
of the QD excitons. This polarisation fieldoscillates at a set of
frequencies determined by centralfrequencies of the two laser
pulses including a beatfrequency, and, if the pulses are
non-collinear, thegenerated field may even propagate in a
distinctdirection due to the phase matching requirement ofthe
involved k -vectors. This counter-intuitive responseto a two-pulse
sequence can be seen in the relevantdensity matrix equations of
motion coupling popula-tions to coherences for spectrally shifted
excitationpulses [22].
In the context of QD spectroscopy the initialmotivation for
two-pulse degenerate transient FWMwas to circumvent the
inhomogeneous broadeninginherent in ensemble QD measurements, and
directlyaccess the dephasing time of a single QD exciton. Inthis
spectroscopy modality the central frequency o0 ofa laser pulse is
tuned to the mean frequency of theensemble X0 exciton resonance. As
can be seen inFigure 9(a), the exciting laser pulse is split into
threeparts – a pump, a probe and a reference pulse. Acontrollable
time delay, td, is established between thepump and probe pulses.
The pump (indexed with a 1)and probe (indexed with a 2) pulse each
receive a small,but distinct, frequency up-shift typically with
acousticoptic modulators to o1 ¼ o0 þ oRF1 and o2 ¼ o0 þoRF2. The
pulse pair is subsequently directed to theQD sample exciting a
third-order polarisation. It is atthis point, when the pulses
excite the sample, wherethere are two variations in implementing
FWM. In [23]the time-delayed pump pulse and probe pulseilluminate
the sample with well-defined non-collineark-vectors, k1 and k2,
where the general scheme is nicelyillustrated in Figure 9 [24]. The
resultant polarisationradiates in a direction determined by the
wavevectorcombination 2k27k1 and oscillates with a frequency2o27o1.
The subsequent source field (resulting fromthe QD nonlinear
polarisation) is mixed with areference laser pulse still at o0, but
delayed by timetr, on a photodiode. A phase-sensitive lock-in
amplifierdemodulates the FWM signal in the measured photo-current
at 2oRF27oRF1, which is typically in the tensof MHz frequency
range.
Another approach to FWM, which does notexploit the additional
wavevector selectivity affordedby the previous approach, is
presented in [25]. Againthe pump and probe pulse are time delayed,
butinstead of illuminating along fixed, non-collinear,directions
the collinear pump and probe pulses arefocused by a high numerical
aperture objective ontothe sample as illustrated in Figure 9(b).
The inducednonlinear polarisation radiation is collected by
asimilar objective and mixed with the reference pulsedirectly on a
photodiode. Unlike the previous examplewhere the signal was free
from pump and probe pulsebackground, this second approach relies
entirely on thelock-in amplifier to distinguish the signal of
interestfrom the background pulses, which still modulatesthe
photocurrent at a frequency 2oRF27oRF1. Thestringent k -vector
considerations are not an issue forthis modified version, at the
expense of reducedsensitivity in detection. Although these two
modalitiesare slightly different with respect to signal
acquisition,both benefit from the essential advantage of
thetransient FWM signal; its immunity to inhomogeneousbroadening.
Inhomogeneous broadening is an inherent
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feature of QD emission spectra in high density QDsamples (when
there are a number of optically activeQDs within the microscope
focal volume) as a result ofthe QD size distribution. Specifically,
when theinhomogeneous spectral broadening of the ensembleis
considerably larger than the homogeneous dephas-ing rate of a
single QD exciton, the polarisation willradiate a light pulse at
exactly 2 times the pump–probedelay; the photon echo. Figure 10(a)
presents themeasured echo pulse as a function of the probe delay
tdfrom an ensemble of InAs QDs at a temperature of50 K. The
important information is obtained fromplotting the integrated area
of this light echo pulse as afunction of pump–probe delay. The
strength of thephoton echo generated depends on the coherence ofthe
ensemble, so the decay of the integrated area isdetermined by the
total dephasing rate of a single QDexciton. Figure 10(b) is an
example of the integratedecho pulse area FWM signal. The extracted
dephasingrate of 630 ps when this measurement was repeated at7 K
was the first of its kind to suggest the excited statedecay was
predominantly the result of radiationbroadening.
The second transient nonlinear spectroscopy tech-nique we
describe is degenerate pump–probe
Figure 9. Schematics of the setups for the two four-wave mixing
modalities. (a) A pump and a probe pulse, at two
distinctfrequencies, illuminate the sample along two directions.
The phase matched signal, in a direction distinct from the pump
andprobe, is mixed with a reference pulse on a photodiode. A
lock-in amplifier demodulates the signal, at a frequency
2oRF27oRF1,from the photocurrent. (b) Same as (a) except the pump
and the probe pulses are now focused and collected by high
numericalaperture objectives.
Figure 10. Time resolved four-wave mixing data from anensemble
of InAs QDs at 50 K. (a) The echo pulse as afunction of the
reference pulse time delay tr. The time delaybetween the exciting
pulse pair is varied in 400 fs steps from 0to 3.2 ps. (b)
Time-integrated four-wave mixing obtained byintegrating the echo
pulse area in (a) at different excitationintensities. (a)
Corresponds to the 2Io trace in (b).Exponential fits to the traces
in (b) yield the dephasing timeof a single QD exciton transition
[25].
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spectroscopy. Just like FWM, pump–probe alsointerrogates the
induced third-order polarisation, but,in contrast to FWM, the
pump–probe signal is derivedfrom the polarisation’s response at the
probe field’sfrequency. Consequently, this alternative
techniqueprobes essentially the occupation probabilities of
theexcitonic states rather than being limited by theircoherences.
In other words, the previous FWM signal(at the beat frequency of
2oRF27oRF1) will vanish ifthe coherence between the two states of
the transitionis lost, while the probe response (at the
probemodulation frequency oRF2) will still be visible aslong as the
excitations are present, much like thelifetime measurements
discussed before. The twopulses are delayed in time by td, but now
the signalfield is mixed with the probe field on the photodiode.
Alock-in amplifier filters the measured photocurrentisolating the
signal at a frequency oRF2. In a pump–probe experiment the probe
pulse transmission as afunction of delay time td is a direct
measure of theexcited state lifetime: both radiative and
nonradiativecontributions. A further twist on the
pump–probetechnique results from plotting the probe transmissionat
a fixed delay time td as the pump and probe centrefrequency are
varied in unison. In this case the time-domain technique is able to
unmask frequency domaininformation and probe the absorptive
resonances ofthe QD. An example of the measured data in these
twoapproaches is in Figure 11 for a single GaAs
interfacial(fluctuation) QD [22]. In the top panel in Figure
11(a)two co-polarised 6 ps wide pulses, with a fixed 6 psdelay
between the pulses, illuminate the sample. Theabsorption of the
probe pulse, the transient differentialtransmission, is plotted as
a function of the two pulses’centre frequency. The peaks in the
absorption datareveal electronic transition energies in the
QD.Figure 11(b) is the measured probe pulse area as thedelay
between the pump and probe pulse is varied. Themeasured excited
state decay time is 41 + 2 ps.Finally, the lower panel in Figure
11(b) presentsdifferential transmission with a continuous wave(CW)
laser. The narrowband CW laser greatlyimproves the spectral
resolution of the absorptionmeasurement, and as we will see in the
coming section,has become an indispensable tool for selective
excita-tion of QD transitions.
3.2.2. Spectral measurements
We introduced fundamental transitions per chargeconfiguration in
the quantum dot in the first section(see Figure 3). The purpose of
photoluminescencemeasurements was to map out the spectrum of
allallowed transitions due to various charge combina-tions. Given a
typical resolution of 30 meV, this
appears insufficient for resolving spin and anisotropyinduced
fine structures. Likewise, even when theground state of the quantum
dot is controlled to bein a particular charging configuration, we
introducednothing to prevent the capture of uncontrolled
chargecombinations under above-band-gap excitation.Therefore, while
this particular type of excitationallows for mapping out the
optical transitions, it stillrenders any systematic access to
individual spin statesimpossible. Pump–probe techniques allow for
precisemeasurement of the excitonic dynamics, but due to
theextended bandwidth of the optical pulses, frequencyselectivity
of individual transitions within the finestructure is limited. In
order to address this short-coming, an alternative technique is
utilised wheretemporal resolution is sacrificed for such
spectralselectivity. Ultrafast pulse pairs are replaced by oneor
more highly monochromatic single transverse andlongitudinal mode
lasers with tuneable optical fre-quencies. We can now address a
transition of interestdirectly and selectively among many allowing
us to
Figure 11. Degenerate pump–probe spectroscopy data.
(a)Copolarised pump and probe pulses with widths of 6 ps aredelayed
by 6 ps and the transmission is measured as the pulsecentre
frequency is varied. The dashed line is the zero signallevel. In
the lower panel, a cw laser is tuned through the QDresonance. We
will focus on the merits of this technique inthe next section. (b)
The integrated probe pulse signal as afunction of its delay from
the pump pulse. The fit exponentialdecay reveals an excited state
lifetime of 41 + 2 ps [22].
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study features such as optical selection rules andoscillator
strengths by observing the scattered laserlight.
The signal in this case is still due to the interferenceof the
background laser and the scattered dipoleradiation, where the
transmitted laser field is measuredby photodetectors rather than
the emitted photonsbeing detected by a spectrometer. If the laser
field is inresonance with a quantum dot transition, the
opticalfield scattered by the quantum dot interferes with
thebackground optical field. The total field observed
upontransmission through the quantum dot includes thesignature of
the quantum dot response to the lightfield. All measurements are
performed in the far-fieldso the phase difference is a result
equivalent to theoverall Guoy’s phase acquired by the
backgroundlaser. Consequently, we typically measure the absorp-tive
response directly in intensity change. In short, themeasured
optical field intensity will trace out thefollowing response:
IðDÞ=Ið1Þ ¼ 1� ao ð�2=4Þ=ðd2 þ �2=4Þ� �
; ð5Þ
where ao is the effective absorption strength deter-mined by the
laser focus area and the quantumdot oscillator strength and is
typically in the range of107371072. In order to see this response
one has tohave control over the detuning d and this can beachieved
in two ways: either by sweeping the opticalfrequency of the
excitation laser with respect to the QDtransition or by shifting
the quantum dot emissionenergy via DC Stark effect through an
external electricfield.
Unlike photoluminescence measurements, the sig-nal strength here
with respect to the background laserintensity is usually on the
order of 1073 as dictated byao and lock-in based detection is
required in order toeliminate this background. The modulation
requiredfor the lock-in scheme is obtained by
electronicallymodulating the external electric field with a
squarewave signal. Normalising by the total power at
thephotodetector one can calculate the deviation of
thetransmissivity from the off-resonant value DT/T. Thisis called
Stark-shift modulation spectroscopy andfurther experimental details
of this technique can befound in [26]. Given that we are no longer
limited bythe spectrometer spectral resolution of 30 meV, butrather
the uncertainty of the applied gate voltage andthe spectral
bandwidth of the applied laser, resonantscattering has
significantly higher resolution, on theorder of 0.04 meV. The
previously predicted fine struc-ture of each optical transition can
now be observedclearly, as displayed in Figure 12(a). The black
(grey)data set is obtained with a pX (pY) linearly polarised
laser. Figure 12(b) is the same measurement onX17 displaying a
single peak due to degeneracy [27].Taking advantage of this high
spectral resolution, QDtransition linewidths on the order of 1.3
meV have beenobserved [28]. In addition, this response can
inaddition be controlled to yield both absorptive anddispersive
lineshapes [29] depending on the relativephase between the laser
and the QD scattering field.Nevertheless, the relatively modest
signal to back-ground level limits how fast this measurement can
becompleted. However, recent advances incorporatingsolid immersion
lens technology [30] showed that themeasured signal strength can be
increased furtherleading to higher detection bandwidth [31,32].
4. Recent achievements utilising these techniques
What we have presented so far is a survey of opticaltechniques
utilised in the quantum dot research in thebroad sense. While each
technique presents a com-plementing side to complete spectroscopy
of quantumdots, some have in parallel been utilised to performsome
key achievements. In this context, quantuminformation processing
has benefited significantly fromthe re-interpretation of
spectroscopic concepts andmeasurements. We give below a handful of
examplesalong these lines, but emphasise that we are leaving outa
vast area of quantum dot research on conventionaloptoelectronics
technology, such as lasers and opticalswitches.
4.1. Deterministic source of single photons andquantum
cryptography
One of the earliest applications of quantum informa-tion science
was quantum cryptography, which drewits power of security from the
indivisibility of a singlephoton. The information (the bit) is
encoded on single
Figure 12. Differential transmission data for the neutralexciton
(left panel) exhibiting the fine structure x – y splitting,in this
case 27 meV, due to the electron–hole exchange.Differential
transmission data for the trion exciton (rightpanel). The exchange
splitting observed in panel (a) is zerosince the electrons are in a
spin singlet [27].
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photons thus ‘tapping’ the quantum channel wasquantum
mechanically forbidden. The lack of a truesingle photon source
prevented the use of the no-cloning principle, and had to revert
once again to the‘difficulty’ of eavesdropping by using either
heavilyattenuated laser beams with average photon numbermuch less
than one, or down-conversion processes thatgenerate photon pairs
with a small probability ofgenerating multiple photon pairs in well
determineddirections. Sacrificing one of the photon pairs in
down-conversion for timing gives a sense of heralding to thephotons
in the other direction, but the probability tohave more than one
photon in a pulse still remainsfinite. A compact, stable, reliable
source of singlephotons was the missing link for relying on
quantummechanical principles for ultimate cryptographic se-curity
rather than assumed difficulty in practice ofbreaking the
cryptographic code.
We have discussed how intensity (photon) correla-tion
measurements reveal the decay dynamics of theexcited states in
quantum dots. In 2000, two groupsreported one of the first
applications of this techniquein the realm of quantum information
for the realisationof deterministic heralded single photon
generation[33,34]. In these works, the above-bandgap nonreso-nant
continuous-wave excitation laser was replaced byan ultrafast laser
pulse train which generated excitons,in the vicinity of a quantum
dot, at well known times.After each excitation pulse, a high
density of photo-excited excitons was to ensure the probability
tocapture at least one exciton inside the quantum dotapproached
unity. While the number of excitonscaptured inside a quantum dot is
statistically varying,the distinct exciton emission wavelengths due
toCoulombic interactions allows one to spectrally filterout only
the neutral exciton recombination. The resultis a single photon
generated with unity probability at awell-defined wavelength, at
discrete times manifested
by the excitation laser pulses. Figure 13(a) displays thephoton
correlation measurement under these condi-tions. Unlike the
continuous-wave excitation profile,we see uniform coincidences only
at discrete timedelays indicating that the emission comes as a
pulsetrain following the excitation laser pulse train. Theabsence
of the central coincidence peak is due tothe lack of another photon
emitted simultaneously at
Figure 13. (a) The top panel is the autocorrelation of a250 fs
Ti:sapphire laser and the bottom panel is theautocorrelation of
single photons emitted from the groundstate exciton of a QD
embedded in a microdisc cavity underpulsed excitation. In the
bottom panel (in contrast to the toppanel), the zero delay time is
nearly zero. This is a result ofthe QD emitting one photon at a
time. More importantly,upon pulsed excitation, the QD emits a
single photon in well-defined time bins – the so-called photon
turnstile [33]. (b) Thepolarisation of single photons emitted by a
QD based photonturnstile is used exchange a secure key between
Alice (thetransmitter) and Bob (the receiver). The key is used to
encodeand decode a 140 6 141, 256 pixel colour bitmap of
theStanford University’s memorial church (the top image).
Theencoded message appears as white noise to all parties thatdetect
the encoded photons without the key except for Bobwho has the
correct key [35].
"
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the same wavelength, verifying the single photonnature of this
heralded emission process.
In 2002, Santori et al. implemented the firstquantum key
distribution protocol using a singlequantum dot as a true single
photon source [35].Figure 13(b) shows the original and decoded
imagewhen the polarisation state of individual photonsemitted from
a quantum dot are used to exchangethe key necessary to decode the
image.
4.2. Going beyond single photons: polarisationentangled photon
pairs
A cascade two-photon emission process with inter-mediate level
degeneracy was the source of entangledphotons for the seminal work
by Aspect et al., whereJ ¼ 0! J ¼ 1! J ¼ 0 type cascade decay of
Calciumatoms was used for the first experimental demonstra-tion of
bipartite entanglement and violation of localrealism models for
quantum theory [36]. Once theanalogous cascade nature of the
biexciton-excitondecay in quantum dots was revealed, the route
togeneration of frequency and polarisation correlated(or entangled)
two-photon states was established. Thetwo-photon field emitted via
the biexciton–excitoncascade decay process, illustrated in the
inset ofFigure 14(a), can be written as
ð o1HXX;o2HXj i þ o3VXX;o4VXj iÞ=21=2: ð6Þ
This is a maximally entangled state in the strict senseof the
word. However, revealing the degree of, forexample, polarisation
entanglement would be bur-dened by the simultaneous frequency
entanglement.The frequency tag on the decay channel comes fromthe
electron–hole exchange (discussed previously inrelation to Figure
12), labelled S in the inset ofFigure 14(a), which lifts the double
degeneracy ofthe intermediate X0 excitonic energy levels.
There-fore, removal of the which-path information for thetwo decay
probabilities is necessary to revealpolarisation entanglement. We
can list three ap-proaches to this problem two of which have
shownsuccess.
The first method is to be oblivious! That is, let usfocus only
on the subset of emission events that cannotreveal the path
information. The finite spectral widthof excitons still allows a
region of spectral overlap (andthus indistinguishability) where
spectral pre-selectionof excitons in this overlap region, roughly
in the middleof split X0 energy levels, will remove the
informationon decay paths albeit the heavy cost of detection
rates.Akopian and co-workers have done just that to
showcorrelations that reach beyond the classical bounds for
the two photons generated by such a biexciton–excitondecay from
a single quantum dot [37].
The second method approached the problem fromthe materials side:
the excitonic exchange splitting to acertain extent originates from
the shape anisotropy ofthe quantum dots. Therefore, physically
altering theshape of the quantum dots after growth process has
beencompleted will also alter this undesired level
splitting.Stevenson et al. used post-growth annealing of the
QDsample which led to a systematic reduction of theexchange
splitting as a function of annealing time, as canbe seen in Figure
14(a) for two different QDs [38,39].Since every QD starts from a
different splitting valuewithin a statistical distribution, a
certain annealing rangeprovides a subset of QDs where exchange
splitting iseffectively removed, and thus results in
polarisationcorrelations that reach beyond the classical
boundswithout sub-selecting the emission spectrum. To
verifyentanglement in the emitted photon pairs, Figure
14(b)presents full state tomography of the emitted photons inthe
linear polarisation basis. The inset of the figure is theresults of
various entanglement tests performed on themeasured density matrix
[40].
The final method we mention was in fact the firstone proposed,
but is still yet to be realised: coupling anoptical transition to a
cavity mode coherently broadensthe spectrum of the emitted photons
due to a reductionof the radiative lifetime, known as the Purcell
effect.We will not discuss emitter–cavity coupling in thisarticle;
however, it suffices to note that the pathdistinguishability can be
removed by increasing thespectral overlap of the two excitonic
transitions viacavity-induced broadening [41]. This method not
onlyrecovers polarisation-only entanglement, but furtherincreases
the photon-pair generation rate – bothfeatures desirable for QIS
applications.
4.3. A quantum gate with excitonic qubits
Another application we would like to highlight is thepossibility
to use the QD exciton as the physicalrepresentation of a qubit. In
2003, the two-colourversion of the pump–probe transient nonlinear
spec-troscopy technique was utilised along with the
linearpolarisation selection rules of a quantum dot excitonand
biexciton transitions in order to operate aquantum gate. We saw in
the previous section thatthe cascaded decay process of a biexciton
follows eitherof the two decay paths into the ground state.
Here,Gammon et al. approached from the other directionwhere a large
value for S is favoured and demonstratedthe conditionality of the
biexciton generation onthe initial exciton polarisation [42]. The
excitonicstates of Figure 15(a) can be interpreted via two-qubit
logic states. A laser pulse resonant with the
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exciton–biexciton transition was labelled as the ‘opera-tion’
and had an effect on the quantum dot excitationthat is conditional
on the initial condition. Forexample, if the quantum dot was in the
j00i or j01istate, the operation left the system unaltered due
toeither frequency or polarisation selectivity. If, how-ever, the
initial state was j10i the ‘operation’ pulseresulted in a p
-excitation to the biexciton state eipj11i,while the initial state
of j11i dropped to j10i throughstimulated emission. Consequently,
one could con-struct a truth table for this effective
two-qubitoperation that is analogous to a controlled rotation(CROT)
gate. Verification of this operation andquantifying the gate’s
fidelity still relied directly on
the transient pump–probe technique discussed earlier.In Figure
15(b) the reconstructed physical truth tablefor this excitonic gate
is presented. The system isprepared in the various input states,
the operationalpulse is applied, and the final state ofthe QD
isdetermined. Ideal operation corresponds to the fourwhite bars
having a value of 1 and the rest 0.
4.4. Initialisation and coherent rotation of quantumdot spin
qubits
All nonresonant excitation mechanisms introducemultiple carriers
into the quantum dot levels, whereasresonant excitations only
create single excitons within
Figure 14. (a) In the inset, S depicts the fine structure
splitting of the neutral exciton X0 (see Figure 3(a)) as a result
of theelectron–hole exchange interaction. The data points represent
how the fine structure splitting can be tuned through annealing
twodifferent InAs QD samples for 5 min intervals at 6758C. For
generating polarisation entangled photons from QD emission thethird
data point for DOT B is of interest [39]. (b) The real and
imaginary components of the measured density matrix for the
two-photon state emitted from a QD with an exchange splitting less
than 1.3 meV. Inset: The results of various entanglement
testsperformed on the measured density matrix [40].
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the quantum dot. The unprecedented spectral resolu-tion provided
by the differential transmission techni-ques highlighted above
allow direct access to thetransitions between the individual fine
structure andZeeman levels. We showed in Figure 12 how this
accessibility can be used for spectroscopic measure-ments to
reveal the strength of electron–hole ex-change interaction and
quantum dot anisotropy. Inthe case of a single excess electron
trapped in aquantum dot, we do not have a priori control, over
or
Figure 16. (a) Experimental setup for pulsed optical rotation of
QD spin. During each experimental cycle, one or two rotationpulses
may be sent to the sample to observe Rabi oscillations or Ramsey
interference, respectively. The time delay, t, betweenpairs of
pulses is controlled by a retroreflector mounted on a
computer-controlled translation stage. CW, continuous wave;QWP,
quarter-wave plate; PBS, polarising beam splitter; SPCM,
single-photon counting module; CCD, charge-coupled device; c,speed
of light. (b) Reconstructed evolution of the Bloch vector. The
curves trace out the tip of the Bloch vector in the one-pulse(Rabi
oscillation) experiment. The colour scale indicates the length of
the Bloch vector, which shrinks exponentially as the systemis
evolved for longer times. Views are from the perspective of the x
axis [44].
Figure 15. Controlled Rotation (CROT) gate based on QD biexciton
and exciton transitions. (a) Panel A is a schematic of theQD states
relevant to the CROT gate. Panel B is the excitation energy level
diagram and Panel C is the CROT gatetransformation matrix (note
j10i is transformed to 7 j11i). (b) The reconstructed physical
truth table for the CROT gate.The operational pulse is a p -pulse
tuned to the j10i 7 j11i. The system is prepared in the various
input states, the operationalpulsed is applied, and the final state
of the QD is determined. Ideal operation corresponds to the four
white bars having a value of1 and the rest 0 [42].
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knowledge of, its spin. In atomic physics, the conceptof optical
pumping is used to create an imbalance inthe spin projection of the
ground states of an atomicensemble. This process relies on the fact
that uponexcitation to a higher electronic state, the atoms
willdecay randomly into a set of ground states inaccordance with
the strength of the correspondingtransition matrix elements. In
charged quantum dots,such as the trion in Figure 3, the process can
notwork in ideal conditions, but in the presence of statemixing
mechanisms such as nuclear spin coupling andhole mixing, there is
small but finite rate of decay intothe other electron ground state.
Specifically, if aresonant laser drives the sþ transition, there is
asmall, but finite probability for a spontaneousRaman spin flip
process to take place andshelve the electron in the spin down
ground state.The system then goes dark and no more light
isscattered due to both frequency and polarisationselection rules.
This exact idea, an outgrowth of thepreviously discussed DT
technique, has been demon-strated in [43], where a QD electron was
opticallypumped and thus prepared a given spin state with99.8%
fidelity. This is the initial step of any QIPprotocol utilising
single QD spins.
Finally, we briefly touch on a recent work thatdemonstrates
coherent control of individual electronspin states using ultrafast
optical pulses and opticallyinduced spin rotations. The spin of a
singly chargedQD serves as the qubit. The sample is placed in
astatic magnetic field perpendicular to the growthdirection (the
Voigt geometry). If a single electron isloaded into the QD, it will
coherently precess aroundthe applied perpendicular field at the
Larmorfrequency. Visualised on the Bloch sphere the stateof the
electron spin is traversing a great circle of thesphere ignoring
any spin dephasing effects. Applica-tion of a circularly polarised
ultrafast optical pulsered-detuned from the electron transition
Stark shiftsthe two ground state electronic levels and results
inspin precession around the direction parallel to theQD growth
direction (perpendicular to the externallyapplied magnetic field).
The key point is that theoptical pulse, via the dynamic Stark
effect, acts as anultrafast effective magnetic field and induces
coherentspin rotation. Figure 16(a) is the experimental
setupemployed to optically evolve the electron spin andFigure 16(b)
is the reconstructed evolution of theBloch vector. We direct the
reader to [44,45] for amore detailed discussion of the two
approaches, butthe main message here is that coherent
opticalrotation of a single QD spin has now been observed10 years
after the first photon correlation measure-ments indicating
nonclassical light emission fromQDs.
5. Conclusion
Given that the word spectroscopy comprises two roots(one Latin
and one Greek) meaning appearance-watching, one may be led to
believe that spectroscopyindeed goes all the way back to the Greeks
andRomans. Surprisingly, this powerful word itself wasnot coined
until 1882 by Arthur Schuster in order toclassify studies focused
on the spectral properties oflight–matter interaction. With this
particular defini-tion, the first systematic spectroscopy
experimentswere performed two centuries earlier. In 1666 Sir
IsaacNewton demonstrated the multi-colour nature ofsunlight by
using a glass prism which was readilyavailable from the local
market as a ‘Fool’s Paradise’.His simple but ground-breaking
findings initiated 350years of investigations performed on light
absorbed,scattered, and emitted by matter. Today, spectroscopyhas
generated a sea of knowledge on the nature of lightand its
interaction with matter, where QD researchconstitutes a modest
component. We tried to give abrief survey of the optical
measurement techniquesutilised to investigate the physics of
quantum dots.Today, thanks to these studies we now have a
high-level of control over the physics of QDs and we canexpect to
see a continuation of productivity inquantum dot research in the
coming years with lessemphasis on observation and more on control.
Theoptical techniques used for this progress, however, willlikely
be a combination of what we discussed here,as the general
principles of spectroscopy are here tostay.
Notes on contributors
Nick Vamivakas is a postdoctoralfellow in the Cavendish
Laboratory atthe University of Cambridge since 2007supported by QIP
IRC (GR/S82176/01)and EPSRC Grant no. EP/G000883/1.He received his
Bachelor of Sciencedegree in 2001 from Boston UniversityECE
Department and received hisPh.D. from the OCN Laboratory atBoston
University in 2007 under the
supervision of Professors Swan, Unlu and Goldberg.
Mete Atatüre is a Lecturer in theCavendish Laboratory at the
Univer-sity of Cambridge. He received hisBachelor of Science degree
in 1996from Bilkent University Physics De-partment in Turkey. Then,
he joinedthe Quantum Imaging Laboratory atBoston University for his
Ph.D. studiesuntil 2002. From 2002 to 2007, heworked as a
Postdoctoral Fellow inthe Quantum Photonics Group at ETH
Zurich. Current research efforts of his research group
include
34 A.N. Vamivakas and M. Atatüre
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optical control of single and multiple quantum-dot
spins,high-resolution spectroscopy of diamond-based emitters,
andnanoplasmonics.
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Electronic Properties of Semiconductors, 3rded., World Scientific,
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