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4 Single-Photon and Entangled-Photon Sources and Photon Detectors, Based on Quantum Dots

4 Single-Photon and Entangled-Photon Sources and Photon Detectors, Based on Quantum Dots

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7 Nanomechanics – Nanophotonics – Nanofluidics

On the other hand, applications in the field of quantum information science require

optical sources with strong quantum correlations between single photons [7.59].

This is particularly true for quantum cryptography, which makes use of the principles of quantum mechanics to provide unconditional security for communication.

An essential element of secure key distribution in quantum cryptography is an optical source emitting a train of signals that contains each one single photon (see

[7.57, 7.60]). In addition, the availability of a single-photon source enables the

implementation of quantum computation using only linear optical elements and

photodetectors (see [7.57]).

An emission behavior where the emission of two or more photons simultaneously

is precluded is called photon antibunching. It represents the ultimate limit in the

quantum control of the photon generation process and has been observed in a variety

of single quantum emitters, including an atom, a stored ion, a molecule, a semiconductor quantum dot, or a single nitrogen-vacancy center in diamond (see references

in [7.57]). In a single-photon emitter based on semiconductor quantum dots (QDs),

the quantum dots with a small band gap, e.g., InAs and an emission wavelength

of ∼ 950 nm are embedded in a semiinsulating GaAs substrate (see Fig. 7.9a). Upon

pulsed (∼ 250 fs) illumination with a Ti: sapphire laser (750 nm), which supplies

the trigger signal for the single-photon sequence, charge carriers are formed in the

GaAs layers and trapped at the QDs where they form confined excitons with discrete energy states. The exciton recombination radiation with an energy of 1.322 eV

is predominantly emitted from the edge of the disk (Fig. 7.9a) due to a coupling to

the whispering gallery modes located there.

In order to assess whether the exciton recombination radiation from the quantum

dots is single-photon emission, one has to consider photon statistics. A classical light

source, such as a laser, can be described by a coherent or Glauber state. The distribution of photon numbers P(n) with a mean photon number N and the measurement

of n photons in this state is given by a Poisson distribution which leads to the broad

distribution shown in Fig. 7.9b. This prohibits the application of the Glauber state

in quantum cryptography. A non-classical state of light is the so-called Fock state.

Fig. 7.9 (a) Schematic representation of a single-photon source based on semiconductor quantum

dots. The InAs quantum dots with a diameter of ∼ 40 nm and a height of ∼ 3 nm are embedded

between two semiinsulating 100 nm thick GaAs layers. (b) Distribution P(n) of photon numbers

for the Fock state (F.) with a mean photon number N = 1 and for Glauber states (Gl.) with N = 1

and N = 0.1. (Reprinted with permission from [7.58]. © 2001 Wiley-VCH)


Single-Photon and Entangled-Photon Sources and Photon Detectors




Fig. 7.10 Measured unnormalized correlation function G(2) (τ ) of (a) a mode-locked Ti: sapphire

laser (FWHM = 250 fs) and (b) a single QD excitonic ground state (1X) emission under pulsed

excitation conditions (82 MHz). (Reprinted with permission from [7.57]. © 2000 AAAS)

In a mode excited in this state, precisely N photons are available with the variance

N = 0, as sketched in Fig. 7.9b. This is the characteristic signature of a singlephoton source. For measuring the emission statistics a single QD can be excited

[7.57]. The unnormalized correlation function G(2) (τ ) = I(t)I(t + τ ) , with I the

time-dependent intensity for the pulsed Ti:sapphire laser, is given in Fig. 7.10 for

the exciton transition radiation of a QD at T = 4 K. Whereas the G(2) (τ ) of the pulsed

Ti:sapphire laser shows integer multiples of the repetition period τrep = 12.27 ns,

the G(2) (τ ) of the QD exciton emission also shows peaks at the same τrep , but in

contrast to the laser, the peak at τ = 0 is not present; that is, the probability of

finding a second photon at τ = 0 vanishes. The absence of a peak at τ = 0 provides

strong evidence for an ideal single-photon source operation.

Quantum dot structures can also be used for single-photon sources which are

electrically operating [7.61], which are operating at elevated temperatures up to

200 K [7.62], or which emit photons with polarization control at high emission

rates [7.63]. In addition, single-photon sources based on single carbon nanotubes

have been demonstrated [7.64].

7.4.2 Entangled-Photon Sources

Entanglement, the intriguing correlation of quantum systems [7.65, 7.66], is an

essential resource of quantum information and communication (see, e.g., [7.67]).

It has been demonstrated that the biexciton–exciton radiative cascade in single InAs

quantum dots on a GaAs substrate is a source for entangled-photon pairs satisfying

the Peres criterion for entanglement (see [7.67]).

Entangled photons are a basic tool in quantum optics, e.g., to demonstrate the violation of Bell’s inequalities, teleportation, or quantum cryptography [7.59]. Making

use of quantum dots, a compact source of entangled photons on demand has become

available [7.68]. In a single InAs quantum dot a single entangled-photon pair can be

generated at a well-defined time [7.68]. Here, initially a biexcition (XX) is generated


7 Nanomechanics – Nanophotonics – Nanofluidics

Fig. 7.11 In the generation of entangled photons in a cascade decay in a symmetric quantum dot

(b) the two degenerate decay channels cannot be distinguished (entanglement). In the case of an

asymmetry a splitting of the exciton state (X) emerges and the two channels can be distinguished

(a). The arrows indicate the polarization of the photons. (Reprinted with permission from [7.69].

© 2006 Wiley-VCH)

by optical excitation. These two electron–hole pairs can subsequently recombine in

a cascade decay, first to the exciton state (X) and then to the “empty” ground state.

In an ideal symmetric quantum dot either first a right circular polarized photon and

then a left circular polarized photon are emitted or the other way round (Fig. 7.11a).

These two decay paths are principally indistinguishable so that the resulting state is

a polarization entangled state

| ψ = | σ+ | σ− + | σ− | σ+


which is generated within a narrow time window of about 1 ns.

Asymmetries in the quantum dot lead via the electron–hole exchange interaction to a splitting of the ideally degenerated exciton state (Fig. 7.11a) giving rise to

the emission of non-entangled linearly polarized photon pairs. This splitting can be

controlled or removed by an external magnetic field so that the entanglement of the

photons can be switched on and off.

The entanglement of two photons can be demonstrated by studying the tomographic reconstruction of the two-photon density matrix (Fig. 7.12). The appearance

of off-diagonal elements in the density matrix (Fig. 7.12b) in accordance with theoretical predictions (Fig. 7.12d) is a clear signature of entanglement which cannot be

observed in asymmetric quantum dots (Fig. 7.12a, c).

7.4.3 Single-Photon Detection

Practical systems for optical quantum information technology require in addition to

single-photon sources also efficient, low noise single-photon detectors. It has been


Quantum Dot Lasers






Fig. 7.12 The experimentally reconstructed density matrix of the two-photon state differs substantially for quantum dots with splitting (a) and without splitting (b, entanglement) of the exciton

state (X). This is in accordance with the theoretically predicted density matrices for a classically

correlated (c) and an entangled (d) photon state. (Reprinted with permission from [7.69]. © 2006


Fig. 7.13 Schematic

for a quantum dot resonant

tunneling diode for efficient

single-photon detection.

(Reprinted with permission

from [7.70]. © 2005

American Physical Society)

demonstrated that the resonant tunnel current through a double-barrier structure

(Fig. 7.13) is sensitive to the capture of single photo-excited holes by an adjacent layer of quantum dots. This is due to the fact that the current flowing between

the emitter and collector contacts in response to an applied voltage is limited by

tunneling through the double-barrier structure. The resonant tunneling process is

now sensitively modified by the electrostatic potential of the nanosized quantum

dots when these quantum dots trap and confine single electrons or holes upon

single-photon absorption [7.70]. This phenomenon can be used for low noise detection of single photons with an efficiency of up to 12.5% and a time resolution of

150 ns, promising a maximum detection rate of 10–100 MHz [7.70]. Single-photon

detection at 800 nm has been reported for a quantum dot transistor [7.71].

7.5 Quantum Dot Lasers

Diode lasers are based on current injection into a semiconductor active medium,

resulting in charge carrier population inversion (i.e., the higher energy level

becomes more populated than the lower energy level) and sufficient modal gain


7 Nanomechanics – Nanophotonics – Nanofluidics

(an exponential increase of the intensity of the amplified emission per unit length

once it propagates through the active medium) to achieve lasing. Thin or ultrathin layers of a narrow-gap semiconductor embedded in a wide-gap barrier are

traditionally used as the active media in such lasers. Later on, self-organization

effects on semiconductor surfaces leading to the formation of coherent semiconductor nanoislands (see [7.72]) made the use of quantum dots (QDs) as active media in

semiconductor lasers possible.

Here, a semiconductor quantum dot is a nanometer-scale coherent insertion of a

narrow-gap semiconductor into a wider-gap semiconductor [7.72]. A QD combines

the properties of a single atom with those of a semiconductor, such as a discrete

energy spectrum of the charge carriers, which is a quantum mechanical effect due to

the confinement of the charge carriers to a distance similar to its de Broglie wavelength (see Fig. 7.14a). Stable levels are formed for energies that correspond to an

integer number of electron wave function half-waves.

It was proposed in 1976 [7.73] to apply size-quantization effects to improving

laser performance. If the number of translational degrees of freedom of charge carriers is reduced to below two, a singularity occurs in the density of states [7.73]

(see Sect. 1.3), which increases light absorption or light amplification. Among the

anticipated advantages of QD lasers are decreased transparency current, which is the

current at which the gain overcomes losses in the active medium; increased material

gain, which is an exponential increase in the intensity of amplified emission; large

characteristic temperature T0 , which is higher the higher the temperature stability

of the threshold current, etc. The fabrication of QDs for GaAs-based QD lasers

employs the self-organized growth of uniform nanometer-scale islands (Fig. 7.14b).

Fig. 7.14 (a) Schematic illustration of the energy diagrams of charge carriers in a bulk semiconductor crystal and a quantum dot (QD). The width of energy bands is close to kT, where k

is Boltzmann’s constant and T the temperature. (b) Plan view transmission electron micrograph

of QDs formed by a two-monolayer InAs deposition followed by overgrowth with a 5 nm thick

In0.15 Ga0.85 As layer prior to GaAs deposition. (Reprinted with permission from [7.72]. © 2002

Materials Research Society)




Fig. 7.15 Schematic representation of a quantum dot vertical-cavity surface-emitting laser (QD

VCSEL) with a distributed Bragg reflector (DBR). (Reprinted with permission from [7.72]. © 2002

Materials Research Society)

Interaction of the islands via the substrate allows for lateral island ordering [7.72].

Due to the interest in the application of QDs in vertical-cavity surface-emitting

lasers (QD VCSELs; see Fig. 7.15), the first surface-emitting QD lasers were

demonstrated [7.74, 7.75] and QD VCSELs with threshold current densities down

to 170 A/cm2 [7.76] and 1.3 μm emission [7.77] were shown.

Using InGaAsN nanodots in a GaAs matrix (see [7.72]) the wavelength range

of QD lasers can be extended to 1.55–1.6 μm. From CdSe QD lasers visible light

[7.78] and from core–shell CdS/ZnS QD lasers blue light [7.79] is emitted. A highspeed QD laser transferring 10 gigabits/s across a temperature range of 20–70◦ C

was predicted to reach the market in July 2008 [7.80]. For future metropolitan area

networks the demand for inexpensive ultrafast amplifiers is probably even larger

than that for lasers and quantum dot semiconductor optical amplifiers (SOAs) are

expected to play a decisive role there [7.81–7.83].

It may be mentioned here that nanoscale lasers can also be fabricated making use

of quantum wires (see, e.g., [7.84, 7.85]) or photonic crystals (see, e.g., [7.86]).

7.6 Plasmonics

The science of plasmonics [7.87–7.89] describes how metals can essentially transmit and manipulate light waves at length scales much smaller than their wavelengths

and therefore enable the diffraction limit imposed on conventional optics to be

overcome. The interaction of light with free electrons in gold or silver nanoparticles or surfaces gives rise to collective oscillations known as surface plasmons


7 Nanomechanics – Nanophotonics – Nanofluidics

(SPs) [7.90]. SPs are essentially composite particles composed of both light and

charge, thereby providing a potential means to unify photonics and electronics.

Plasmonics can generate signals in the soft x-ray range of wavelengths (between

10 and 100 nm) by exciting materials with visible light. The wavelength can, at the

same frequency (resonance), be reduced by more than a factor of 10 relative to its

free-space value, where the fundamental relation between the two – frequency times

wavelength equals the speed of light – is preserved because the electromagnetic

waves slow as they travel along the metal surface [7.87]. For millennia, alchemists

and glassmakers have unwittingly taken advantage of plasmonic effects when they

created stained glass windows or goblets that incorporate small metallic particles in

the glass [7.87].

When light interacts with a metal, the free electrons of the metal can oscillate

as in a plasma relative to the lattice of the positive ions at the resonant plasma


ωp = (Ne2 /ε0 me )1/2

where N is the number density of the electrons, ε0 is the vacuum dielectric constant,

and e and me are the charge and the effective mass of an electron, respectively.

Quantized plasma oscillations are called plasmons. Since an electromagnetic light

wave impinging on a metal surface only has a penetration depth <50 nm for

Ag and Au, just the electrons on the surface are the most significant [7.90].

Their collective oscillations are properly termed surface plasmon polaritons (SPPs),

but are often referred to as propagating surface plasmons (SPs; see Fig. 7.16a).

Another type of resonant SP (localized SPs or LSPs) corresponds to collective

excitations of free electrons confined to the finite volume of a metal nanocrystal

(see Fig. 7.16b). By resonance, one means a condition in which the frequencies

and wave vectors are approximately the same, leading to constructive interference,

Fig. 7.16 Schematic of the collective oscillations of free electrons for (a) a metal–dielectric interface and (b) a spherical gold nanocrystal. Excited by the electric field of the incident light, the free

electrons can be collectively displaced from the lattice of positive ions. While the plasmon shown

in (a) can propagate across the surface as a charge density wave, the plasmon depicted in (b) is

localized to each particle. (Reprinted with permission from [7.90]. © 2005 Materials Research





Fig. 7.17 Experimental and simulated electron-energy-loss spectroscopy (EELS) maps of plasmons in a silver nanotriangle. (a) Distribution of the modes centered at 1.75, 2.70, and 3.20 eV,

respectively. The outer contour of the nanotriangle, deduced from its high-angle annular dark-field

(HAADF) scanning transmission electron micrograph, is shown as a white line. (b) Simulated

amplitude maps of the three main plasmon modes resolved in the simulated EELS of the Ag

nanoprism, calculated by the boundary element method (see [7.91]). The color linear scale is

common to the three maps. The simulated amplitude distributions of the three different modes

qualitatively match the experimental maps in (a). (Reprinted with permission from [7.91]. © 2007

Nature Publishing Group)

a stronger signal, and peaks in the extinction spectra (extinction = scattering +

absorption). Surface plasmon excitation in the infrared/visible/ultraviolet domain

in, e.g., a silver nanotriangle can be directly mapped by measuring resonance

peaks in the energy-loss spectra of a sub-nanometer electron beam probe scanned

on a nanoparticle (Fig. 7.17a). The three modes peak respectively at the corners,

the edges, and the center of the particle, and simulations of the amplitude maps

(Fig. 7.17b) nicely match the experimental results in Fig. 7.17a. Single surface

plasmons exhibit both wave and particle properties [7.92], similar to those of single


Large local-field enhancement near the particle surface is a consequence of the

excitation of plasmon resonances due to illumination. Local-field calculations have

been performed for a range of particle sizes and shapes (see [7.94]) using a variety

of computational tools such as finite-difference time domain (FDTD) [7.96], discrete dipole approximation (DDA) [7.93], or multiple multipole (MMP) techniques

[7.97]. As shown in Fig. 7.18, the magnitude of the field enhancement in resonance

depends strongly on the particle size and shape and the proximity to sharp points and

narrow gaps. The vectorial nature of the electric fields can be captured and mapped


7 Nanomechanics – Nanophotonics – Nanofluidics

Fig. 7.18 (a) Local electric-field enhancement around a silver nanoprism (100 nm sides) calculated for polarized incident light (770 nm wavelength) at the resonance frequency using discrete

dipole approximation (DDA) calculations. At the resonance frequency, the nanoparticle concentrates the incident E field strength ∼ 20-fold [7.93, 7.94]. (b) Electric-field intensity (E∗ E)

enhancement between two triangular prisms (∼ 60 nm side length, 12 nm thickness) showing (c)

a hot spot of more than 50,000 times the incident electric-field intensity [7.94, 7.95]. (Reprinted

with permission from [7.93] (a) and [7.94] (b) (c). © 2003 American Chemical Society (a) and

© 2008 Materials Research Society (b) (c))

down to the nanoscale [7.98]. These highly confined electric fields have been used in

a variety of near-field enhanced spectroscopies, including surface-enhanced Raman

spectroscopy (SERS; see [7.99]) or surface-enhanced fluorescence [7.100] (see


7.6.1 Plasmon-Controlled Synthesis of Metallic Nanoparticles

Metallic nanoparticles can be fabricated by electron-beam lithography, ion-beam

lithography, or chemical synthesis, etc. (see [7.90]). Surfactants are employed for

growing nanowires [7.101]. It is interesting that the growth of, e.g., anisotropic

triangular silver nanoprisms can be controlled by plasmon excitation in order to produce particles with desired edge lengths from solution [7.102]. By using selectively

excitation wavelengths between 450 and 750 nm, nanoprisms with well-defined

sizes between 38 and 120 nm could be grown (see Fig. 7.19). It is assumed that




Fig. 7.19 Light-induced unimodal growth of silver nanoprisms. (a) Excitation wavelengths

between 450 and 750 nm for controlled solution synthesis of unimodal silver nanoprisms with

sizes of, e.g., 38 nm (b) or 120 nm (c). (Reprinted with permission from [7.102]. © 2003 Nature

Publishing Group)

dipole plasmon excitation in the nanoprisms could facilitate the addition of silver

atoms at the corners and edges leading to a growth of the nanoprisms up to a certain

size limit [7.102].

7.6.2 Extinction Behavior of Nanoparticles and Arrays

The extinction behavior of metallic nanoparticles due to plasmon excitation can

be controlled by the particle size, shape, aspect ratio, by the type of material, or

by the medium surrounding the particle, as discussed in the following. The localized surface plasmon resonance extinction spectra can easily be tuned all the way

from the near-UV through the visible spectrum (see [7.103, 7.104]) by changing the

size or shape (triangle or hemisphere) of the nanoparticles (see Fig. 7.20a). This is

demonstrated by the change of colors in solutions containing Ag nanoparticles of

different sizes and shapes (Fig. 7.20b). The red shift from Ag spheres (diameter of

50 nm) to Ag pyramids with the same volume (Fig. 7.20c) is due to the availability

of longer plasmon wavelengths in non-spherical particles. The red shift of the plasmon resonance of Ag nanoparticles in solution increases linearly with the refractive

index of the solvent (Fig. 7.21). This can be used for molecular detection with high

spatial resolution [7.105].

The interaction of localized surface plasmons (LSPs) plays a role in the case of

large-area uniformly oriented nanoparticle arrays which can be prepared, e.g., by

nanoimprint lithography (NIL, see [7.106], Sect. 3.10) or soft interference lithography (SIL; see [7.107]). The tunability of the extinction is demonstrated by the blue

shift with increasing height of the nanoparticles (Fig. 7.22), which may be attributed

to the appearance of additional resonances characteristic of Ag [7.106]. Arrays

of different metallic (Ag, Cu, Au) or dielectric (Si) nanoparticles with the same


7 Nanomechanics – Nanophotonics – Nanofluidics

Fig. 7.20 Extinction due to plasmon resonance on silver nanoparticles. (a) Size- and shape-tunable

localized plasmon resonance extinction spectra of Ag nanoparticles fabricated by nanosphere

lithography (NSL). The extinction is changed by varying the in-plane width a and the out-ofplane height b of the nanoparticles [7.104]. (b) The solutions of Ag nanoparticles show a variety

of colors due to the different sizes and shapes of the Ag nanoparticles within each solution [7.94].

(c) Extinction spectra of Ag nanoparticles in vacuum having the shapes indicated. Each particle

type has the same volume, taken to be that of a sphere with a radius of 50 nm [7.104]. (Reprinted

with permission from [7.104] (a) (c) and [7.94] (b). © 2005 Materials Research Society (a) (c) and

© 2008 Materials Research Society (b))

geometrical parameters (see Fig. 7.23) show dark-field (DF) scattering peaks from

ultraviolet to visible wavelengths which is in good agreement with the calculated

scattering properties [7.107]. In anisotropic arrays of nanoparticles, the extinction

spectra additionally depend on the polarization of the incident light [7.106].

Plasma resonances of hollow gold nanospheres (Fig. 7.24a) in the near-infrared

spectrum are of particular interest for targeted medical diagnosis and therapy (see

Sects. 12.6, 12.7, [7.108]). A wide range of absorption wavelengths can be covered

by varying the Au sphere size or shell thickness (Fig. 7.24b) and the absorbance

can be blue shifted by coating with a Ag layer (Fig. 7.24c). The scattering and

absorption of metallic nanostructures due to localized surface plasmon resonances

can be simulated by employing the discrete dipole approximation (DDA; [7.93]).

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