Single Photon Generation in

Epitaxial Quantum Dots

By Phillip K Poon

Institute of Optics

University of Rochester

Rochester, New York

In partial fulfillment of the masters degree in optics

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Introduction

In 2006, a United States patent was approved (1), entitled, “Quantum Computation with

Quantum Dots and Terahertz Cavity Quantum Electrodynamics.” The inventors, claimed the

ability to create a controlled not gate by manipulating quantum dots through a series of gate

pulses. Since any two bit quantum gate is universal (2), the ability to create a quantum controlled

not gate can be used as the basic building block of a quantum computer based on quantum dots.

Why did inventor use quantum dots? The majority of the quantum computing schemes

involve optically and/or magnetically trapped atoms or molecules and involves complicated

vacuum apparatus to prevent decoherence. Fortunately, a quantum dot exhibits many of the same

properties of atoms, such as discrete energy levels (3), yet the ability to control a quantum dot

does not require nearly the complexity of laser cooling or nuclear magnetic resonance. For this

reason, the quantum dot has been called the superatom (4). The quantized energy levels in a

quantum dot allow scientists the ability to tune its optical properties for applications in quantum

information and spectroscopy. The ability to generate single photons from a quantum dot has

been its most important feature. In this paper, a discussion on the creation of quantum dots and

the application, characterization, and generation of single photon emitting epitaxial quantum dots

will be discussed. Key experiments will be used as examples to highlight important concepts in

this area.

Before exploring each concept, it is important to provide the reader with a big picture

point of view, by previewing a common experimental setup currently being used to generate

single photons. One that will be discussed often is the Santori experiment (5). An InAs quantum

dot is buried inside a GaAs matrix and placed inside a cavity consisting of Distributed Bragg

reflectors, located inside a cryostat. The DBR cavity serves to enhance the spontaneous emission

rate and improves the collection efficiency. The quantum dot is optically pumped and the

emission is spectrally separated in order to destroy all but the last emitted photon, thereby

generating a single photon in each pulse. A Hanbury- Brown Twiss interferometer is used to

record whether or not single photons are being created.

Figure 1: A typical experimental setup used (5).

Applications: BB84 Quantum Key Distribution

The energy levels of a quantum dot can be tuned by varying the composition and size (6).

The ability to tune the discrete energy levels of a quantum dot make them ideal candidates for

low threshold lasers and light emitting diodes (6). However, the ability to generate single

photons on demand from quantum dots promises to be their most ground breaking application. In

optical quantum information, a single photon source is required for linear optical quantum

computing (7), quantum networks (8) , and most notably quantum cryptography (9).

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The BB84 protocol, named after Charles Bennett and Gilles Brassard in 1984, is a

Quantum Key Distribution protocol that can directly benefit from a truly single photon source.

Quantum cryptography is only used to produce and distribute a key, not to transmit any message

data. Information is encoded as 1’s and 0’s in two bases. The horizontal and vertical basis or 45

or a 135 basis. In our example, 1 is represented by a horizontal polarization or a 135

polarization. While a 0 is represented by a vertically polarization or a polarization.

Basis 0 1

+

X Figure 2: The polarization bases used to encode a 1 or 0 bit in the BB84 Quantum Key Distribution protocol

If the sender, Alice, randomly chooses a basis (either + or x) then randomly chooses a 1 or 0 to send, she will record the basis and whether it was 1 or 0. The receiver, Bob, randomly

chooses a basis and then measures a 1 or 0. The beauty of Quantum Key Distribution is that Bob

does not need to be in the same basis as Alice. If Bob is in the right basis, he will measure the

same bit at Alice. If Bob is in the wrong basis, he will have a 50% chance of measuring the right

bit. This process is repeated over and over until Alice and Bob call each other and compared

what basis they used for each step. If they didn’t used the same basis, then they throw away

those bits. The remaining bits comprise their shared key.

Notice that they never communicated whether they have the same bits (0 or 1). While

single photon sources are not required to some encryption schemes such as BB84(5), single

photon sources in Quantum Key Distribution schemes have inherent advantages over others in

that they prevent an eavesdropper from receiving a second photon and thus gaining information

regarding the key (3; 9).

Creation of Epitaxial Quantum Dots

A quantum dot consists of a semiconductor confined inside another semiconductor of a

larger bandgap. Typically ranging in size between 10-50 nm, quantum dots are often thought of

as zero-dimensional analogues to quantum wells. Unlike quantum wells, there are energy bands

for excited charge carriers. Due to their size, carriers are confined and the energy levels of a

quantum dot are discrete, similar to an atom or molecule (6). For this reason, quantum dots are

often called artificial atoms, macroatoms, or superatoms (10). The “bandgap” can still be thought

to be the energy difference between the top of the valence levels and the bottom of the

conduction levels.

Quantum dots are often created by using the Stranski-Krastanov growth technique.

Starting with a substrate, such as GaAs, an epitaxial layer of smaller bandgap semiconductor

materials, such as InAs, is grown using Molecular Beam Epitaxy (MBE) or another epitaxial

process. At particular thickness, due to a lattice mismatch (approximately 7% for InAs/GaAs) the

expitaxial layer breaks up into islands. These islands are then covered by GaAs, creating a small

layer of quantum dots. Underneath the islands a “wetting” layer is formed, approximately 0.28

nm thick in InAs/GaAs. The electrons moving in the wetting layers are close to the dots and have

intermediate energy between bound electron states of the dots and the three-dimensional barrier

like states (10). This report will focus on quantum dots created by the Stranski-Krastanov growth

technique; however the reader should note there are many types of quantum dot growth

techniques for creating many kinds of quantum dots (4).

http://en.wikipedia.org/wiki/Charles_H._Bennett_%28computer_scientist%29http://en.wikipedia.org/wiki/Gilles_Brassardhttp://en.wikipedia.org/wiki/1984

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Ideal Single Photon Sources: The Second Order Correlation Function

In order to measure how close an optical source is to being an ideal single-photon source,

a Hanbury Brown and Twiss interferometer (11) is used to measure the second order correlation

function, (5).

Where can be written as (12)

2

(2)

2( ) 1

n ng

n (1.1)

where is the mean photon number per pulse, and is the variance of the distribution,

which can be found from . A more practical way of calculating can

be obtained by measuring emission intensity

(2)

2

: ( ) ( ) :( )

( )

I t I tg

I t (1.2)

where is the mission intensity at time (13).

An ideal Photon Source has a second order correlation, (3). Where is

the probability of generating two photons in the same pulse, normalized by an equally bright

Poisson-distributed source. In reality getting a is impossible (3).

The Hanbury Brown and Twiss interferometer consists of a beam-splitter and two photon

counters (normally a photo-multiplier tube or avalanche photodiode) and a time interval counter.

The electronic pulses from the photon counters are used as start ( and stop signals for the

time interval counter, which records these intervals as a histogram over many pulses.

The peak corresponds to events in which two photons were detected in the same

pulse, and thus a truly single photon source will have a zero area at . The other peaks

correspond to , when n is a non-zero integer and is the rep rate of the pump,

indicate measurements in which one photon was detected from of two seperate pulses. is

then measured by comparing the area to the area of the more distance peaks (5)

Figure 3: A simple Hanbury Brown and Twiss

interferometer used to measure photon correlation

Figure 4: A typical second order correlation function

from a single photon emitting InAs quantum dot(5)

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The Second Order Correlation Function as a Measure of Photon Bunching/Anti-Bunching

Photon bunching is the tendency for photons to arrive at the detector at the same time. In

“classical” source, the photon statistics approximate a Bose-Einstein distribution (14). Photon

bunching is a quantum mechanical property. It cannot be understood by regarding photons as

independent particles, ignoring the wave properties of the light. The wave picture, predicts that

the instantaneous intensity on the detector is influenced by all atoms in the source. Thus both

detectors see related contributions where light is originated from one atom in the source. For a

Bose-Einstein distribution, where photons are maximally bunched,

Since the laser pulse photon distribution is approximately Bose-Einstein, in the low

expectation value limit, it mimics a Poisson distribution. This effect has been proven in highly

attenuated laser pulses (15). A notable feature of a Poisson distribution is that the variance equals

their mean. From (1.1), it is shown that faint laser pulses have a normalized second order

correlation of . This indicates that the pulses are not maximally bunches but they are

not maximally separated as in the case of photon anti-bunching.

Photon Anti-bunching occurs when the photons are maximally separated from each other

in time. If of if anti-bunching is occurring and at the

source in completely anti-bunched. In a driven anharmonic quantum system, such an atom or

molecule or quantum dot, photon anti-bunching can occur (16; 17).

It should be noted that Zou and Mandel have shown that sub-Poissonian statistics do not

necessarily imply photon anti-bunching (18). Sub-Poissonsian statistics and anti-bunching often

occur together and so many have come to believe that they are one in the same. However, there

are instances in which sub-Poissonian distributions accompany bunching. This is more the

exception than the rule. Thus, authors continue to associate sub-Poisssonian statistics with

photon anti-bunching.

Cavity Quantum Electrodynamics: The Purcell Effect

The spontaneous emission rate depends to a certain extent on the surroundings of a

radiant source. By placing the light source in a cavity, the rate of spontaneous emission can be

modified. Purcell discovered the enhancement of spontaneous emission rates of atoms when

they are matched in a resonant cavity, which ensures that radiative recombination dominates over

nonradiative transitions in the atom (19).This implies that the ground state transition lifetime

in drastically reduced.

In an experiment with electrical pumping of a single InAs/GaAs quantum dot, the high

refractive index of GaAs leads to a small critical angle and only a small fraction of emitted

photons escape. Refraction further reduces the number of photons collected by a lens of finite

numerical aperture. Only 0.5% of photons can be collected into a lens of numerical aperture 0.5

(20). Use of a planar microcavity embedded inside a pillar increased the efficiency with which

photons are collected by a factor of ten (20).

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Figure 5: A simple pillar cavity using

distributed Bragg reflectors (5).

Figure 6: A InAs quantum dot

surrounded by a GaAs microdisk

cavity the employs whispering

gallery modes (19).

Figure 7: A quantum dot and a

planar cavity used with electrical

injection (20).

Figure 8: Measured unnormalized correlation function

of a single quantum dot excitonic ground state

emission out of resonance with a cavity mode.

Figure 9: Measured unnormalized correlation function

of a single quantum dot excitonic ground state

emission in resonance with a cavity mode .

The effect of on resonance and off resonance cavity tuning can be observed in the

unnormalized second order correlation function (19). In an experiment conducted by Michler, a

quantum dot was tuned by varying the temperature. The sample consisted of a InAs quantum dot

within a microdisk cavity made of GaAs, Figure 6. As the temperature varied the single excitonic

transition showed a remarkable difference in the observed second order correlation behavior

between the on and off cavity resonance emission. Figure 8 shows the unnormalized correlation

function for the 1X transition out of resonance and Figure 9 shows it on resonance with

the Whispering Gallery Mode of the microdisk cavity. When the quantum dot is on resonance

with the Whispering Gallery Mode the time jitter between successive photon generation events is

reduced. Thus, the full width at half maximum (FWHM) values of the correlation peaks are

narrower than the out-of-resonance case. This is a direct consequence of the Purcell effect and

the lifetime was reduced by a factor of 6.

When the quantum dot and the cavity are resonant, . Michler suggests that

the Purcell effect could play a role in causing a non-zero . The Purcell effect increases the

probability of capturing a second electron-hole pair from the wetting layer after the single

exciton recombination. They also explain that background light generated by the wetting layer or

by other quantum dots in their sample could be creating a non-zero .

In later experiment by Santori(21), the InAs quantum dot was located inside a planar

cavity that consisted of two distributed Bragg reflectors (DBR). The DBR mirrors were made of

alternating layers of GaAs and AlAs, approximately one quarter wavelength optical thickness.

In Figure 12, the specific emission marks the primary optical cavity modes of a pillar, since

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emission on resonance with these modes is improved through the Purcell effect. This broadband

emission presents a way of locating the cavity modes and determining their quality factors.

Optically Excited Single Photon Emission in Quantum Dots

In the beginning, generating anti-bunched emission from a quantum dot was

accomplished by optical excitation. Two processes have been mentioned in the literature that

could lead to photon anti-bunching through optical excitation.

The first process, initially suggested by Santori et al (21), is not well published. When a

photon is absorbed by a quantum dot, the absorption creates a single exciton. An exciton is a

bound electron-hole pair in a quantum system. When the exciton is created, an electrostatic

interaction between the electron and hole changes the energy level spacing. The fast shifting of

the energy levels suppresses the probability for a second photon to be absorbed and preventing a

second exciton from being created. Since there is only one electron-hole pair, only one photon

can be emitted, therefore creating an anti-bunching emission. However Santori neglects to pursue

investigating this process any further than merely suggesting it.

The second process, which is well accepted (19; 21), uses the Coulomb interaction of

excitons generated by the optical pumping. When electrons and holes are being created through

optical excitation, they will approximately obey the Pauli Exclusion Principle. However the

electrostatic interactions between the electron-hole pairs modify the potential, which in turn

modifies the emitted spectrum. This creates an anharmonicity in the quantum dot multiexciton

transition (19), which leads to photon anti-bunching. The photon emission from each exciton

recombination is unique. Additionally, the wavelength at which the last photon is emitted, which

is associated with the single exciton state is always the same, see Figure 10. Therefore one can

use a notch filter or a grating to allow the last photon to pass to the detector, ensuring sub-

Poissonian statistics.

Figure 10: The optical excitation

scheme of a single quantum dot

(5).

Figure 11: Photoluminescence

spectrum of a quantum dot at

above band excitation (5).

Figure 12: Photoluminescence

spectrum of a quantum dot at

resonant excitation (5).

Optical excitation of the quantum dot can occur in two ways. The quantum dot can be

excited through resonant or above-band excitation. For example, in the Santori experiment with

InAs/GaAs quantum dots, in above band excitation the pump laser is tuned above the bandgap of

the GaAs that surrounds the InAs quantum dot. Free electrons and holes are created within the

conduction and valence bands of GaAs. Some of the electrons and holes relax into the lower

energy levels of the wetting layer, and then finally relax into the energy levels of the quantum

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dot. In resonant excitation, the laser is tuned to directly excite the quantum dot creating excitons

(5).

The difference in above band and resonant excitation are evident in the

photoluminescence spectra. In Figure 11, multiple wavelengths can be seen in the above band

excitation of the InAs quantum dot, named “Dot A”. This specific quantum dot was grown inside

a GaAs matrix and then embedded in a micropillar optical cavity. The researchers used a 750 nm

pump wavelength, which is above the bandgap of GaAs. In

Figure 12, the quantum dot is excited on resonance at 909 nm. The reason why additional

spectral lines are observed in the above band excitation is that electrons and holes are added to

the quantum dot separately. In this way, charged-exciton (trion) states are frequently produced,

which emit at unique wavelengths. In addition, the spectral lines in

Figure 11 could originate from more than one quantum dot. By exciting on resonance, it

is easy to selectively excite a single quantum dot.

Figure 13: Photon correlation measurements with (a) above band excitation and (b) resonant excitation (5).

Differences between above band and resonant excitation also manifest in the second

order correlation functions(5). In above band excitation the probability for two photon emission

is higher compared to resonant excitation, Figure 13. There are several issues related to above

band excitation that increases probability for photon bunching. First, the dynamics of the free

electrons and holes within the GaAs matrix and wetting layers are more chaotic than the resonant

case. The capture process can last longer and if charge carriers are injected into the quantum dot

over a period which is large compared to the exciton recombination then two photons can be

emitted from the exciton state in the same pump pulse. Secondly, the broadband emission around

the main line contaminates the signal. Finally, the background floor that shows up throughout the

second order correlation suggests there is a long lasting component of the carrier capture process,

because it shows up between excitation pulses. At resonant excitation, the performance improves

considerably. In the Santori experiment, the background floor disappears and (5).

This is associated with shorter relaxation dynamics proceeding an excitation pulse.

During pulsed resonant excitation the emission from the ground exciton state exhibits

saturation behavior, Figure 14(21). The biexcitonic and multiexciton peaks continue to grow

with pump power, however the emission of the ground exciton, plateaus. This is because only the

last exciton recombines to emit at this particular wavelength, Figure 15. In this particular

experiment (21), a photon counter was used to measured the emission rate versus pump power,

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and a spectral filters were used to allow only the single exciton peak to pass. The number of

photon counts associated with the first exciton emission reaches a maximum described by

/

0 (1 )satP PI I e (1.3)

where I is the measured intensity for single-exciton emission, P is the pump power, and and

are fitting parameters that characterizes the total collection efficiency and the absorption

rate (21). In the same experiment, it has also been reported that under above band excitation with

continuous wave pumping, the intensity of the exciton line grows linearly and the intensity of the

biexciton line grows quadratically (21).

Figure 14: Pulsed resonant

emission spectra of a InAs/GaAs

epitaxial quantum dot at 0.22,

0.44, 0.88, 1.32 and 2.53 mW

respectively (21).

Figure 15: Emission intensity of

the single exciton line depicts

saturation with increasing pulsed

pump power at resonant

excitation of an InAs/GaAs

epitaxial quantum dot (21).

Figure 16: A log-log plot of

emission line intensity versus

above-band continuous wave

pump power, showing linear

growth of the exciton emission

(circles) and the biexciton

emission (diamonds) (21).

Electrically Excited Single Photon Emission in Quantum Dots

While researchers have obtained excellent results towards photon anti-bunching in

quantum dots with optical excitation, any practical or commercial implementation will require

electrical excitation. Therefore research techniques in electrical pumping of quantum dots, so

called single-photon emitting diodes (SPED), are extremely important for the future (22).

Research is advancing at a rapid pace, producing SPEDs that emit at telecommunication

wavelengths (23; 24) and “plug and play” SPEDs that can be incorporated directly into

Wavelength Division Multiplexed fiber optic networks (23).

In an early experiment, a simple SPED consisted of a low density InAs self-organized

quantum dot layer enclosed within the intrinsic region of a vertical p-i-n junction consisting of

GaAs for the p, i, and n regions (25), see Figure 7 . A single aperture was placed over the layer

of quantum dots to capture the emission from just one. The quantum dot was excited in a method

analogous to above-band excitation, the free electrons are being deposited into the . The p-i-n

diodes were found to display nearly ideal current-voltage characteristics, the injected current

increases rapidly with a forward bias, in this case 1.5 V, Figure 17.

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Figure 17: Current versus voltage plot is distinctive of

ideal diode-like performance(25).

Figure 18: Intensity of the exciton (X) and biexciton line

(X2) with electrical pump current (25). The intensity of

X increases linearly while the X2 increases

quadractically .

Figure 19: Electroluminescence spectra of the single

photon emitting diode, showing line emission

characteristics of individual quantum dot for the

exciton (X) and biexciton line (X2) (25).

Most importantly, the emission time of the single photons from the diode can be

regulated through pulsing of the injection current. Pulsed electrical injection leads to pulsed

emission from the dot, provided that the pulse width is much less than the exciton lifetime. The

rate of multi-photon emission was observed to decline by using shorter electrical pulses (25).

There are several commonalities between electrical and optical driven quantum dots

engineered for single photon emission. By creating different cavity structures such as planar

cavities or microcavities it is possible to increase the quantum efficiency of a SPED. Without a

cavity, the aforementioned simple SPED can require 104 electrons to generate a single photon (22). Also, at 5K, with low injection currents, a sharp electroluminescence line appears at 1.3942

eV. This line is associated with the recombination of a single exciton of one electron and one

hole. At higher currents the X line weakens and a second line at 4.7 meV higher. Another

similarity is the dependence on intensity. The intensity of the second line increases quadratically

with current, this is the biexciton line (25), Figure 18. The quadratic increase in intensity with

pump is consistent with the photoluminescence experiments (21). A key observation is that the

single exciton recombination occurs after the biexcitonic recombination, which is also consistent

with previous quantum dot single photon experiments using optical pumping. While the lifetimes

depend on the size, shape, and construction of the quantum dot, in an example, it has been shown

that for a simple SPED the exciton and biexciton lifetime are 1.02 and .47 ns (25). Unlike the

photoluminescence experiments (21), the biexcitonic state emits at a higher energy than the

excitonic line. The reason for this is unexplained.

Despite the commonalities with optical excitation, there are several areas in which

electrical pumping has yet to achieve. Resonant electrical excitation of quantum dots have been

reported (22; 26). The only technique used a Coloumb blockade, which injects single carriers

into etched double-barrier mesoscopic hetrojunctions, required millikelvin temperatures and the

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collected photon rate was too weak to generate usable date in correlation measurements.

Fortunately, resonant excitation is not necessary for single photon generation (22). As yet, there

have been no reports of generating

Photon State Purity

Conclusion

Researching single photon generation from epitaxial quantum dots is incredibly

important. Anti-bunched photon generation has required ingenuity and the ability to combine

many different areas of science, quantum optics, semi-conductor physics, cavity quantum

electrodynamics, quantum information theory, into a feasible result.

This paper explored the most applicable need, Quantum Key Distribution. This paper

then discussed how to created epitaxial quantum dots using the Stranski-Krastanov growth

technique and how resonant and non-resonant coupling of a quantum dot to a high Q cavity can

increase collection efficiency and the spontaneous emission rate through the Purcell effect.

Experiments with optical excitation were showcased and behaviors in the scenario of above-band

and resonant, pulsed and continuous wave excitation were discussed. Single photon emitted

diodes, the electrical analogy to optical excitation was mentioned and compared to optical

excitation methods. The problem of photon state purity

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