Quantum-optical influences in optoelectronics—An introduction

This focused review discusses the increasing importance of quantum optics in the physics and engineering of optoelectronic components. Two influences relating to cavity quantum electrodynamics are presented. One involves the development of low threshold lasers, when the channeling of spontaneous emission into the lasing mode becomes so efficient that the concept of lasing needs revisiting. The second involves the quieting of photon statistics to produce single-photon sources for applications such as quantum information processing. An experimental platform, consisting of quantum-dot gain media inside micro- and nanocavities, is used to illustrate these influences of the quantum mechanical aspect of radiation. An overview is also given on cavity quantum electrodynamics models that may be applied to analyze experiments or design devices.


Quantum-optical influences in optoelectronics-An introduction
Weng W. Chow 1  This focused review discusses the increasing importance of quantum optics in the physics and engineering of optoelectronic components. Two influences relating to cavity quantum electrodynamics are presented. One involves the development of low threshold lasers, when the channeling of spontaneous emission into the lasing mode becomes so efficient that the concept of lasing needs revisiting. The second involves the quieting of photon statistics to produce single-photon sources for applications such as quantum information processing. An experimental platform, consisting of quantum-dot gain media inside micro-and nanocavities, is used to illustrate these influences of the quantum mechanical aspect of radiation. An overview is also given on cavity quantum electrodynamics models that may be applied to analyze experiments or design devices. Optoelectronics is entering an exciting period. There are intense device development activities driven by applications ranging from telecommunications and room lighting to the internet and quantum information processing. Significant advances are being made in laser performance, including reducing spatial footprint, consuming and emitting less energy, improving spectral stability, and controlling photon statistics. Common to these activities is the increasing role of quantum optics. The importance of contributions due to the quantum mechanical aspect of radiation is evident when analyzing experiments and enabling new functionalities. While the underlying physics has been established long ago, its implementation to develop components that are a short step from being applicable in optoelectronics represents forefront applied physics and engineering.
Our goal in this focused review is to introduce the reader to how quantum optics, as originally developed for atomic, molecular, and optical (AMO) systems, is being extended for use in semiconductor device research. (For textbook descriptions of AMO quantum optics and semiconductor device engineering, see Refs. 1 and 2, respectively.) It is a growing effort, with important contributions from many groups throughout the world. We hope to give the general audience a flavor of the common interests and challenges, as well as a background understanding for further learning from recent literature covering both science and engineering aspects.
Quantum optics enters optoelectronics in diverse ways. It is used to develop new properties to enable new applications. Other times, it is necessary to explain behaviors in devices we are trying to improve, especially for devices performing close to fundamental limits. This review will illustrate these two functions with examples from cavity quantum electrodynamics (CQED) research involving quantum-dot (QD) micro-or nanocavity light sources.
A QD, micro-or nanocavity light source (see, e.g., Fig.  1) is a good experimental platform for studying laser physics and exploring device applications. Section II describes new questions involving lasing when the channeling of spontaneous emission into a single or very few cavity modes becomes very efficient. This has been demonstrated with the class of micro-and nanocavities, referred to as high-spontaneousemission (or high-b) cavities. For lasers using these high-b cavities, the transition from spontaneous emission to lasing is distinctly different from that of conventional (low-b) lasers. The use of CQED to explain observed new behaviors is demonstrated. Section III describes an example of new functionality in semiconductor light sources arising from emphasizing the quantum optics aspect of active regions, made possible by the maturing of semiconductor QDs. The high optical gain per active material volume enables lightsource operation with very few QDs. This gives rise to a possibility of producing nonclassical light, which exhibit photon number fluctuation below the classical limit established by the Poisson photon statistics for coherent light. We will discuss the development of nonclassical light sources. Of most interest is a single-photon source achieved by having only one QD in the active medium. Also discussed is the challenge to produce single-photons at a high rate while maintaining single-photon purity. Section IV describes how CQED effects in devices may be modeled. An overview of models of diverse levels of sophistication is given. These are microscopically based fully quantized models, where both active medium and radiation field are treated quantum mechanically. Furthermore, device behavior is described in terms of electrons and holes occupying states of a nanostructured semiconductor. Finally, Sec. V summarises the review and provides an outlook.

A. Towards smaller lasers
Lasing is well understood in conventional lasers. For semiconductor lasers, the most in use today and the largest in size are in-plane lasers. They typically measure 1 mm Â10 lm Â 0:1 lm. The longest dimension is the resonator axis, which lies in the plane of the semiconductor substrate. Also called edge-emitting lasers, they are commercially available and can deliver constant wave (cw) power up to tens of watts per laser.
There are many applications requiring much less power, and therefore, much smaller lasers. Also regard as conventional are microlasers, such as vertical-cavity surface-emitting lasers (VCSELs), where size is drastically reduced by rotating the resonator axis perpendicular to the substrate. The result is cavity dimensions decreasing to $1 lm Â 10 lm Â 10 lm. VCSELs are the second most used semiconductor lasers today. Even in second place, there are over 2 Â 10 9 VCSELs in use and roughly 100 Â 10 6 VCSELs are produced each year to keep up with ever increasing demand.
A goal in semiconductor laser development is to continue reducing laser size. A major motivation is to integrate more lasers onto a chip, with each laser using less energy. There is a never satisfied desire for less energy consummation. For example, optical interconnects connecting computers to make the internet and search engines function run at about picojoules per bit. The goal now is to reduce to attojoules per bit to keep up with future user demands. 4 To meet that goal, advances in semiconductor nanofabrication have ushered in a new class of smaller light sources. These devices operate with nanocavities with optical mode volumes Շ k=2pn b h i 3 , where k is the lasing wavelength and n b is the cavity background refractive index. 5 Reducing volume decreases the number of optical modes within a wavelength range. The challenge to size reduction is tighter optical mode control. Most approaches are based on having higher refractive index contrast between intracavity and surrounding regions. This is achieved using semiconductor/air or semiconductor/metal interfaces to form micropillars, photonic-crystal cavities, nanobeam cavities, microdisks, and coaxial cavities. 6,7 Lasers operating with these different cavity configurations have been demonstrated. Emission is normal to the substrate (surface emission) for some of the configurations and in-plane with the substrate for others, thus providing flexibility to serve a broad range of applications. When researching the literature, it is useful to note that some nanocavities are still referred to as "micro," especially when the approaches originated with fabrication of microcavities.

B. Channeling spontaneous emission
Sub-wavelength cavities exhibit unique and useful properties that are not present in larger ones. Most notable is the greater control of spontaneous emission prior to lasing because of larger frequency separation between adjacent cavity modes. The efficiency in channeling of spontaneous emission into desired modes may be quantified by the spontaneous emission factor 8 b Spontaneous emission rate into lasing mode Total spontaneous emission rate ; where the typical value for semiconductor lasers is b % 10 À4 . Having significantly fewer cavity modes within the spontaneous emission spectrum means larger spontaneous emission factor. Presently, the capability exists to approach the ultimate limit, where all spontaneous emission is channeled into a single mode (b ¼ 1). One then has the potential to achieve the lowest lasing threshold and highest efficiency for a given active medium. This makes high-b lasers attractive for many applications. The possibility for low threshold and high efficiency make it a strong candidate for optical interconnects in energy demanding applications such as the internet and search engines. With single-photon generation (which will be discussed in more detail later), QD high-b light sources can help maximize the photon-generation rate while maintaining the quantum nature of emission purity. There is even an idea for versatile and efficient lighting based on combining the high-b cavity's efficient channeling of radiation prior to lasing with the clamping of nonradiative losses during lasing. 9,10 We emphasize the difference between single-mode lasing and b % 1 operation. With large mode volume and resulting small b, single-mode operation arises from mode competition after lasing threshold is reached. Prior to lasing, spontaneous emission is distributed, and therefore lost, among many cavity and free-space modes. With nanocavities possessing b % 1, almost all emission (spontaneous and stimulated) is channeled into the lasing mode, both prior to and after the onset of lasing.
This recently realized ability to control spontaneous emission via b gives new directions to laser research and engineering. 11 There is the excitement of significantly reduced power consumption. Having b % 1 also gives rise to several much-discussed laser physics questions, motivated by experiments indicating changes in the transition from spontaneous emission to lasing. For example, the lasing threshold becomes ambiguous, when one uses the typical criteria of input-output characteristics and linewidth narrowing. There is also a much-debated claim of thresholdless lasing. 12 Subsection II C considers how these issues may be resolved with the help of quantum optics.

C. High-b lasing
We start with a conventional laser cavity, which has many cavity modes besides the lasing mode within the spontaneous emission spectrum. With the resulting small b, the transition from spontaneous emission to lasing appears as a distinct jump in the output versus input power curve [see solid black curve in Fig. 2(a)]. This jump indicates the transition of emission from predominantly spontaneous emission to predominantly stimulated emission. When sufficiently abrupt, it precisely locates the lasing threshold, i.e., the pump power at the onset of lasing.
When b is increased, e.g., by operating with micro-or nanocavities, the mode spacing increases, resulting in fewer cavity modes within the spontaneous emission spectrum. With less spontaneous emission lost to nonlasing modes, the lasing threshold is lower as depicted by the other curves in Fig. 2(a). In the limit of only one mode inside the spontaneous emission spectrum, all emission (spontaneous and stimulated) is channeled into that mode and b becomes unity. A result is that the intensity jump marking the lasing threshold disappears completely (dotted-dashed curve).
To isolate the effects of b, the calculations are performed assuming perfect carrier injection g ¼ 1 and absence of non-radiative carrier losses c nr ¼ 0. Carrier injection losses (g < 1) leads to a rescaling of the pump power. With non-radiative losses, a change in slope appears at the lasing threshold for the b ¼ 1 curve. 13 Further confirmation of lasing usually comes from narrowing of the emission linewidth with the increase in the excitation. Figure 2 Table I. threshold, it settles (gradually or abruptly, depending on b) to an excitation dependence described by the Schawlow-Townes formula. 14 Similar to Fig. 2(a), the drop in linewidth, and consequently, the precision in locating the lasing threshold degrades with increasing b. With b ¼ 1, the drop vanishes all together, making the locating of lasing threshold impossible without knowing how much narrowing is sufficient to claim lasing.
An answer may come from photon statistics. 15 It is well known to the quantum optics community that a fundamental difference between radiation from lasing and nonlasing sources is in the nature of photon number fluctuations. 16,17 Laser or coherent radiation is described by Poisson statistics, while a non-lasing source emits radiation described by a thermal or Planck's statistics. With the measurement or calculation of the full photon statistics being non-trivial, the equal-time second-order photon auto-correlation function is often used as an alternative. In Eq. (2), the numerator is the variance in photon number and n h i is the average photon number. For a Poisson distribution, g 2 ð Þ 0 ð Þ ¼ 1, while Planck's distribution has g (2) (0) ¼ 2. Nonclassical light, which will be discussed in Sec. III, is characterized by g 2 ð Þ 0 ð Þ < 1. Figure 2(c) shows a plot of g 2 ð Þ 0 ð Þ versus pump. For every b, the transition from thermal to coherent emission with increasing excitation is clearly indicated by a drop in g 2 ð Þ 0 ð Þ from 2 to 1. It is interesting to note that unlike the curves for input-output and linewidth narrowing, a transition region exists for b ¼ 1, which locates a finite lasing threshold (when g 2 ð Þ 0 ð Þ % 1) and highlights the fact that a "thresholdless laser" still requires finite pump power to enter the regime of coherent emission of light. A similar result was reported quite some time ago, in an AMO study where CQED calculations were performed assuming a 2-level active medium. 18 Returning to the question of the degree of linewidth narrowing resulting from lasing, we combined Figs. 2(b) and 2(c) to arrive at Fig. 3, where the abscissa is the deviation from the Poisson statistics value of g ð2Þ 0 ð Þ ¼ 1, which is approached only with infinite excitation. The curve suggests a narrowing to less than 2% of the passive cavity linewidth as a conservative indication of onset of lasing. More interesting is that the curve is independent of b, i.e., all the curves in Fig. 2 condense to one curve when plotted as in Fig. 3. We have also modeled sources with different device parameters, such as cavity Q and QD density, and found the results to be all describable by Fig. 3.

D. Lasing and the intracavity photon number
The condensation of results achieved in Fig. 3 motivated a search for a physical parameter or set of physical parameters that controls the transition from spontaneous emission to lasing. A first hint comes from combining Figs. 2(a) and 2(b) to give a plot of linewidth narrowing versus average intracavity photon number. All the curves for different b's overlap, giving the single curve in Fig. 4(a). Figure 4 To see the extent the theoretical prediction of laser behavior is supported by experiment, in Fig. 4, we plotted the measured data from five QD-based light sources, ranging from nonlasing to lasing devices. The device parameters are summarized in Table I. Each pillar consists a single InGaAs QD active layer and a high-Q microcavity with lower and upper AlAs/GaAs DBRs of 25 and 30 layered pairs, respectively (see Fig. 1). Further details on the device configuration and processing may be found in Ref. 19. The measurements were performed with the micropillars in liquid-helium flow cryostat at 10 K temperature and excited with a constant wave (cw) titanium:sapphire laser tuned to resonate with the QD wetting layer. The conversion from detector counts to intracavity photon number is made by fitting to curves such as shown in Fig. 2(a).
In Fig. 4(a), the experimental data overlap very well with the calculated curve, and hence, strongly support the claim of sole dependence of linewidth narrowing on intracavity photon number. In Fig. 4(b), the measured data are more scattered, especially at low photon number. This is caused by the conflicting demands of long collection time to compensate for the low signal and need for sufficient time resolution (or longer coherence time) to pinpoint the zerodelay value. A consequence is an underestimation of g 2 ð Þ 0 ð Þ at thermal light conditions. Nevertheless, the measurement shows a trend that is consistent with the calculated curve.
The consolidation of results depicted by Fig. 4 indicates that laser (or non-laser) action comes solely from achieving a certain photon number inside the optical cavity. Other factors, such as cavity configuration, active medium, and carrier transport, enter only in that they affect the intracavity photon number. This is a helpful result for laser engineers because there is considerably more expertise in maximizing intracavity intensity than in controlling abstract properties such as photon correlation. While the plots in Fig. 4 improve our understanding of lasing, they also expose behaviors that are presently not completely understood. For example, from radiation field quantization, we understand that having more than one photon will cause the onset of stimulated emission, and hence of the start of the lasing transition. However, we have yet to have an answer for the number of photons necessary to completely convert from a thermal distribution to a coherent one, i.e., from g ð2Þ 0 ð Þ ¼ 2 to g ð2Þ 0 ð Þ % 1. There is also the question of whether Fig. 4 shows the fundamental limit. The curves are obtained assuming that inter-QD correlations are negligible, which is true for present lasers. An interesting and useful question is whether these correlations, which have been shown to give rise to interesting quantum optical phenomena, such as superradiance, can be exploited to produce a new class of ultra-low-threshold lasers. Superradiance will be discussed further in relation to inter-QD correlations in Sec. IV.

III. QUIETING PHOTON STATISTICS
In Sec. II, we show how one started with improving laser performance and ended up having to resort to quantum optics to understand the results. In this section, we describe a different situation, where the use of quantum optics is intended right from the beginning to enable a new class of light sources for entirely new applications. Novel to optoelectronics is the generation of nonclassical light for applications that require regularity in spontaneous emission, rather than the naturally occurring randomness. The nonclassicallight source that is a key component in the realization of optical processing of quantum information, such as cryptography, computing, and communication, is the single-photon source. With complete photon antibunching, fluctuations in photon number (photon statistics) are reduced to the stage where only one photon is emitted at a time. This section starts with a review of the types of light (thermal, coherent, and nonclassical), and then move on to a discussion of CQED considerations involved with developing sources of high single-photon flux and purity.
A. Types of light Figure 5 shows ways of thinking about the types of light. 20 For a large signal, the classical picture is that of intensity and its fluctuations in time (first row). When the signal is very weak, one counts photoelectrons emitted by a detector. A one-to-one correspondence between photoelectrons and photons gives the picture depicted in the second row. The actual measurement involves repeatedly counting the number of emitted photoelectrons within some set time interval (time bin). With the information, one is able to plot the probability of finding a certain number of photons versus the number of photons, i.e., the photon statistics (row 3). When a complete picture provided by the photon statistics is unnecessary, one may condense the information to a normalized photon number variance, which is g ð2Þ 0 ð Þ in Eq. (2). The first column in Fig. 5 is for thermal light. It is noisy, with intensity spikes and nulls. In quantum optics language, one sees photons arriving in groups (photon bunching), as illustrated in the second row. The photon statistics is described by an exponential function, which gives a g ð2Þ 0 ð Þ ¼ 2. Until recently, we in optoelectronics think of laser light as the highest quality of light possible and we refer to it as coherent light ð Þ ð0Þ versus intracavity photon number. The curves are calculated using the parameters in first row of Table I. All the curves for b ¼ 10 À3 to b ¼ 1 overlap. The experimental points are for the cavity-enhanced light sources with parameters listed in rows 2-5 of Table I (column 2). However, there is still noise resulting from randomly, spontaneously emitted photons especially in high-b lasers. The resulting photon number fluctuation approaches that described by a Poisson distribution, where g ð2Þ 0 ð Þ ¼ 1. Lastly, the influence of quantum optics in optoelectronics has led to interest in another kind of light source, the nonclassicallight source. As sketched in the third column, the photons are more spread out (antibunched). In the limit of a single-photon source, photons always appear one at time, giving g ð2Þ 0 ð Þ ¼ 0.

B. Single-photon generation
Schemes for generating single photons range from the early experiments involving atomic and ionic beams to parametric down conversion and defect centers in crystals. Of these, using only one emitter (e.g., one InAs QD) gives perhaps conception-wise the cleanest path to a single-photon source. A single 2-level emitter can only deliver one photon at a time. However, the high single-photon purity, usually gauged in terms of a low g 2 ð Þ ð0Þ value, is achieved at the cost of a low repetition rate that is determined by the freespace spontaneous emission rate. For a semiconductor QD in a heterostructure, which is a strong candidate for singlephoton sources because of the advantages of electrical excitation and chip integration, this rate is at best in sub GHz range. Also serious is the lack of directionality in the spontaneous emission, which drastically reduces collection efficiency by the detector optics.

Cavity enhancement
Spontaneous emission rate and directionality have been demonstrated to improve by placing the single emitter in an optical cavity. Figure 6 depicts the cavity enhancement process. Without a cavity, the emitter radiates isotopically at a spontaneous emission rate c sp free , populating photon states with dispersion determined by free space [Fig 6(a)]. Under the influence of an optical cavity, the photon states are those of a combined cavity/free-space system [Fig 6(b)]. There are discrete energy states residing primarily within the cavity and states forming a continuum that reside mostly outside the cavity, i.e., in free-space. In this review, we discuss experiments with micropillars of sufficiently small cavity dimensions so that the combined cavity/free-space system has one discrete mode (the fundamental mode or FM) and a continuum of lossy modes. For these experiments, a gauge of cavity effectiveness may be obtained from a generalized version of the Purcell factor which compares emission rate into the one discrete cavity mode to total spontaneous emission rate without a cavity. Equation (3) is a broader interpretation of the original Purcell definition (customarily denoted by F p ) defined for an optimally matched (spectral, spatial, and polarization) emitter/cavity combination. 21

Single-quantum-dot devices
The single-photon sources we studied each has an InAs QD inside a small 3lm diameter micropillar. Each micropillar has 15 and 25.5 mirror pairs for upper and lower DBRs, which results in moderate cavity quality factors of 2000 < Q < 3000. The device fabrication uses deterministic processing by means of in-situ optical lithography. First, one uses photoluminescence mapping to locate suitable QDs based on emission intensity, wavelength and perhaps other secondary details. Then optical lithography is used to pattern pillar cavities that are spatially and spectrally aligned with the selected QDs. Then, we temperature fine tune as in Fig. 7(a). Using in-situ optical or electron-beam lithography, single-QD devices can be processed with a yield >90%. 22 The standard lithographic process, which lays out the cavities using regular arrays and without as careful alignment to QD, reports yield <1%. Figure 7 shows how the Purcell factor (henceforth, we drop "generalized") may be determined for a device. First, the single-QD resonance is temperature tuned to match that of the micropillar fundamental mode [red trace, Fig. 7(a)]. The decay of spontaneous emission under short-pulsed excitation is recorded [red data points, Fig. 7(b)]. Then, the measurement is repeated with a large detuning between QD and micropillar resonances. The blue data points in Fig. 7(b) are for detuning of 2.7 meV, which is sufficiently large for the spontaneous emission decay to approximate that without a cavity. When the data are fitted to exponential functions (solid curves), decay rates, c sp mode ¼ 6:0 Â 10 9 s À1 and c sp free ¼ 8:8 Â 10 8 s À1 are extracted, which gives a Purcell factor of F ¼ 6:8, or a roughly 7-fold increase in spontaneous emission rate compared to the free-space value.
The evaluation of single-photon source performance is more complicated than that of lasers. For example, source performance is more tightly connected to the rest of the experimental setup because of weaker photon signal. Consequently, in addition to maximizing the spontaneous emission rate, it is important that the spontaneously emitted photon be directed into the cavity mode that can be efficiently collected by detector optics. 24,25 The extraction efficiency is introduced to quantify this requirement 26 FIG. 6. Sketches showing changes in the contributions to the emission rate from (a) without to (b) with cavity. The dark wavily lines represent spontaneous emission channeled into cavity modes, while the grey ones represent spontaneous emission that rapidly leak out of the cavity into free-space. (c) Spontaneous emission rates for evaluating cavity enhancement. Copyright 2016 OSA Publishing. 23 where g t , involving photon transport between source and detector, is the fraction of photons emitted into fundamental mode that ends up collected by the first lens (with given numerical aperture, NA) in the detection path. The spontaneous emission factor is changed from Eq. (1) so that it applies to both lasers and nonlasers such as single-photon sources. One then defines an overall device efficiency, giving the probability of detecting a photon per excitation pulse where g excitation is the probability of emitting one photon with one excitation pulse. Figure 8(a) plots the charged-coupled device (CCD) detector counts measured for different excitation pulse strengths. In the resonance fluorescence experiment, the excitation laser pulse is resonant to the QD transition, its strength is given in terms of the pulse area, and the counts are for photon flux into the first lens (NA ¼ 0.42) of the collection optics. In this particular experiment, the pulse area was used to indicate how effective a laser pulse was able to invert the QD population, with p indicating total inversion. In the optical Bloch picture 27 of coherent transient in lightmatter interaction, a 2-level system may be converted from entirely in the ground state to totally in the excited stated by a laser pulse with area p. In Fig. 8(a), the experimentalist assumed that the peak of the CCD count rate is from a totally inverted system and so assigned the excitation laser pulse area of p. The plot shows a maximum single-photon flux of 1.3 MHz occurs with a p area excitation pulse. On the right y-axis is the conversion to overall device efficiency, showing that the photon flux maximum translates to an overall device efficiency of g sps ¼ 0:7460:04.
There is a trade-off involving maximizing b and g t in Eq. (4) to obtain a high the extraction efficiency g ext . While b may be made to approach unity by lowering mode volume, doing so with a smaller pillar diameter increases optical scattering at the pillar surfaces. This, in turn, degrades g t , the coupling efficiency to collection optics. With careful balancing of these diametrical effects, we are about to routinely achieve g ext տ 0:7, with pillar diameters between 2 lm to 3 lm and moderate pillar Q-factors between 2000 and 3000. 23 As stated earlier, compared to other candidates, a semiconductor QD has the advantage of excitation by electrical injection. This capability is very attractive if not essential for applications. There are several serious challenges, including the need for highly optimized doping schemes, mitigation of optical losses due to free carrier absorption from doping, and precise deposition of electrical contacts because of the small scale of nanophotonics elements. Figure 8(b) shows progress towards mastering these challenges. The plot is for electrically driven single-QD micropillars achieving to date some of the highest single-photon flux. Each device has a InAs QD embedded in an electrically contacted 2 lm diameter pillar cavity with Q ¼ 2100. From device characterization measurements, we extracted Purcell-factor F ¼ 3:2, spontaneousemission factor b ¼ 0:76, and coupling efficiency g t ¼ 0:70. These give a photon extraction efficiency of g ext ¼ 0:53, which is close to the maximum overall device efficiency of 0.61 6 0.10 at 625 MHz pump repetition rate, obtained more directly from the plot in Fig. 8(b). The closeness of extraction and overall efficiencies indicates relatively efficient current injection and conversion of injected carrier to QD population, g excitation . With our present devices, parasitic electrical effects related to a non-ideal impedance matching are responsible for the limits on F SPS and g SPS at excitation rates exceeding 625 MHz.
So far, the presented results do not confirm the generation of single-photon statistics. That was accomplished with the measurement of the second-order photon auto-correlation function, g 2 ð Þ s ð Þ. Figure 9(a) is g 2 ð Þ s ð Þ for the optically pumped device. At zero-time delay, g 2 ð Þ 0 ð Þ ¼ 0:0092 6 0:0004, which confirms that we have a source of very high single-photon purity at the maximum of single-photon flux. Figure 9(b) show the plots of g 2 ð Þ s ð Þ for electrical pulse repetition rates of 373, 625, and 939 MHz. The corresponding values for g 2 ð Þ 0 ð Þ are 0.076 6 0:014, 0.149 6 0:021, and 0.227 6 0:021, respectively. There is some degradation of single-photon purity with increase pump repetition rate. However, even at close to 1 GHz repetition rate, g 2 ð Þ 0 ð Þ remains well below the single-photon criterium of g 2 ð Þ 0 ð Þ < 0:5.

Emission rate versus purity
The plots in Fig. 9(b) suggest a delicate balance between photon production rate and g 2 ð Þ 0 ð Þ. To help understand that balance, one may think of three operational cases for a single-photon source, as sketched in Fig. 10(a). In the absence of a cavity, g 2 ð Þ 0 ð Þ ¼ 0 for all excitation, but the single-photon flux is low [case (i)]. On the other hand, a high-Q cavity may increase photon flux, but g 2 ð Þ 0 ð Þ may suffer from the likelihood of multiple (>1) intracavity photons To explore how that may be accomplished, CQED calculations were performed, resulting in Figs. 10(b) and 10(c). They show the pump rate dependences of the average intracavity photon number and g ð2Þ 0 ð Þ for the three cases. The curves are obtained by applying the basic model described in Sec. IV B 2 to an InAs QD in a nanocavity with light-matter coupling constant and carrier relaxation rate g ¼ c r ¼ 10 11 s À1 , and three cavity loss rates. In curves labeled (i), a large cavity loss c c ¼ 5 Â 10 15 s À1 is used to approximate the absence of a cavity. The results show near-perfect single-photon generation, with g ð2Þ ð0Þ % 0 for the entire excitation range. However, the average photon number saturates at a value much less than unity. The curves for case (ii) are computed using c c ¼ 10 12 s À1 . They indicate increases in photon number (which is good) and g ð2Þ ð0Þ (which is bad). However, one still has a useable single-photon source because g ð2Þ ð0Þ < 0:5 all the way to saturation. (As an aside, an agreed upon singlephoton criterion of g ð2Þ ð0Þ < 0:5 is motivated by for n ¼ 2.) To illustrate case (iii), we use c c ¼ 5 Â 10 10 s À1 . The curves show a device that is totally unusable as a single- photon source regardless of pump rate. Curve (iii) in Fig.  10(c) indicates photon bunching with g ð2Þ ð0Þ $ 1:5 at low excitation, followed by an interesting dip to g ð2Þ ð0Þ < 1:0, indicating nonclassical photon statistics. Then, the curves predict lasing with single emitter, a much researched quantum optical phenomenon within the AMO and semiconductor communities. 30,31 Curve (iii) in Fig. 10(b) clearly has the "S" shape signature for lasing, which is confirmed in Fig.  10(c) by g ð2Þ ð0Þ reaching unity before saturation sets in. Figure 11 shows information that may be used to optimize a cavity for a single-photon source. To obtain such a curve, we repeated the calculations resulting in Figs. 10(b) and 10(c), for a range of cavity losses to continuously trace the progression from operation without cavity to too much cavity. Then, using the values of photon emission rate and g ð2Þ ð0Þ at saturation, we arrived at Fig. 11(a), which indicates the attainable degree of antibunching as a function of the attainable emission rate. For example, it predicts the possibility of single-photon (g ð2Þ ð0Þ < 0:5) generation rates up to 30 GHz with c c ¼ 10 11 s À1 and a pump rate of 50 GHz.
In terms of applications, one may need a more direct evaluation than indicated by g ð2Þ ð0Þ. The process of coming up with a better gauge can be involved, because of considerations involving measurement setup or application. 32,33 For the present discussion, we settle on a single-photon purity, which one can calculate knowing the full photon statistics from, e.g., a density-matrix calculation 34 where p n is the probability of finding n photons and g À1 is the error arising from multiple photon emission. The curves in Fig. 11(b) show informative and interesting features. First, the blue curve indicates that for purity g > 10 3 , it is necessary to ascertain that there is only one QD present, which in turn limits the emission rate to at best 30 GHz. Interesting are the red and green curves, which apply to devices where fabrication imperfections led to more than one QD within the cavity. They show that if the purity requirement is reduced to g < 10 3 , the two or three QD devices may be serviceable. Moreover, they produce higher emission rates than the single-QD device.

IV. MODELING QUANTUM OPTICAL EFFECTS
This section describes the incorporation of quantum optical physics into models for semiconductor light-sources. Using a Hamiltonian where semiconductor medium and radiation field are treated quantum mechanically, equations of motion are derived for carrier and photon populations, as well as for correlations arising from light-matter interaction. Solving these equations gives the intracavity photon number, emission linewidth and 2nd order intensity correlation. The approach improves on the earlier 2-level system treatment used in AMO studies on the effects of CQED on laser behavior. It gives a more detailed description of the semiconductor aspect, such as the presence of QD inhomogeneous broadening and the effects of the embedding quantum well (QW). More importantly, it accounts for the quantum statistics of the charge carriers. Additionally, the approach gives the emission coherence and photon correlation properties to arbitrary order via a systematic extension of the derivation to higher orders in light-matter correlations.

A. Hamiltonian and cluster expansion
The starting point for describing the interaction of electrons and holes in QD states with photons is the Jaynes-Cummings Hamiltonian, 35

modified for a semiconductor
In the Hamiltonian, h is the photon energy, a † and a are the photon creation and annihilation operators, the subscript i labels the QD, e e i and e h i are the single-particle QD electron and hole energies, c † i and c i are creation and annihilation operators for electrons, and b † i and b i are the corresponding operators for holes. Also in Eq. (9), the light-matter coupling coefficient is where } is the bulk material dipole matrix element, V is the optical mode volume, and b is the background permittivity. Most models assume location of the QDs at the laser mode antinode and ignore electronic structure details such as overlap of electron and hole envelope functions. We follow these assumptions in the calculations performed for this paper. The Hamiltonian is written in a second quantized (Fock) representation for two reasons. First is that the carriers and photons are among themselves indistinguishable. Therefore, properly symmetrized (for photons) and antisymmetrized (for electrons and holes) wavefunctions are necessary. The second quantized representation conveniently takes care of the book keeping via the use of commutation relations for the photon operators and anti-commutation relations for the carrier operators, e.g., The second reason is that absorption and emission create or annihilate photons and electron-hole pairs. This can be readily handled with second quantization by particle creation and annihilation operators. There are many directions for proceeding. Some approaches are based closer to first-principles 36 and they lay the foundation for the more phenomenological ones, which are more suitable for device modeling. Figure 12 is a schematic outline of the derivation procedure that is particularly suited for incorporating details useful for device engineering. At the first (singlet) level, are the equations of motion for the . The light-matter interaction couples these equations to expectation values of products of two single-particle operators. For example, in the case of the polarization equation of motion where hx i ¼ e e i þ e h i is the QD transition energy. Equation (14) shows that b † i c † i a D E dynamics depends on the "2- In turn, the equations of motion for the doublets contain "3-particle" expectation values (triplets) and so on. Consequently, the problem is described by an infinite hierarchy of coupled differential equations. The highly nonlinear coupled differential equations can only be solved numerically, using some truncation procedure to approximately deal with the infinite hierarchy. One approach is to first normal order (creation operators to left of annihilation ones), then factorize higher-order expectation values into products of single-particle quantities, n p , n e i , n h i , and p i . For example, Keeping only correlations necessary for the determination of emission linewidth and second order intensity correlation, this Cluster expansion method provides a flexible analytical tool for modeling QD lasers. Properties of the experimental structure, such as details of carrier transport and QD homogeneity, may be included at various levels of details.

B. Levels of detail and comparison
This subsection gives an overview of how one continues with the Cluster expansion approach to arrive at CQED models of diverse levels of sophistication. These models have in common that both active medium and radiation field are treated quantum mechanically. They differ in the microscopic details, i.e., in how the electrons and holes occupying states of a nanostructured semiconductor are treated. In the remainder of this subsection, we will show how these differences are clearly illustrated by the equations of motion for the carrier and photon populations. Other differences, regarding the treatment of the electron-hole polarization and the higher-order correlations contributing to g ð2Þ ð0Þ will be presented in the Appendix.

Basic model
Under certain experimental conditions, such as at sufficiently low temperature, one may decouple the QW from the QDs, and only tracks the QDs interacting with the radiation field. All carrier transport effects are approximated by an injection efficiency g. A further simplification comes from accounting for QD inhomogeneous broadening via a parameter N QD , which is the effective number of QDs interacting with the laser field. This approach is closer to the earlier rate-equation treatments, 5 except that the derivation extends to include the linewidth and intensity correlation.
With the above simplifications, the Cluster expansion gives dn e dt ¼ À2gRe p ð Þ þ gP 1 À n e ð ÞÀ c nr n e À c nl n e n h ; (16) dn h dt ¼ À2gRe p ð Þ þ gP 1 À n h ð ÞÀ c nr n h À c nl n e n h ; (17) where the QD label "i" is redundant and therefore dropped. In Eqs. (18) and (19), the cavity properties enter in terms of the lasing mode frequency and the photon decay rate 2c c ¼ =Q, where Q is the cavity quality factor. To model the complete emitter, contributions from the laser excitation, nonradiative loss and the presence of nonlasing modes are added phenomenologically into Eqs. (16)- (19). For these contributions, c nr accounts for nonradiative carrier loss, c nl is the spontaneous emission rate into nonlasing modes and c is the dephasing rate. For the excitation, P is the rate of carrier injection at the cladding layers, g accounts for carrier transport from the cladding layers to the QDs, and 1 À n r ð Þ accounts for Pauli blocking.

Reduction to 2-rate equation limit
Equations (16)- (19) may be simplified in the rateequation limit, when the polarization changes sufficiently fast to follow any time variation in the photon and carrier populations. As shown in the Appendix, the adiabatic elimination of the polarization then gives þ P 1 À n e i À Á À c nr n e i À c nl n e i n h i : Furthermore, we define a total carrier number N ¼ P i n e i ¼ P i n h i , so that Eqs. (16) and (17) become In the above equation, we make a connection with two quantities traditionally used to characterize a gain medium: the gain (with dimension of inverse length) and the transparency carrier density N tr ¼ N QD = 2V ð Þ, where V is the active region volume. For spontaneous emission, we choose the bimolecular description where B [with dimension of 1/(volume-time)] is the bimolecular carrier recombination coefficient. Equally often the simple expression c sp N is used, where c sp is the spontaneous emission rate. The actual evaluation of P i n e i n h i for QW or bulk active medium and fitting to c sp N n produces n ¼ 2 at low carrier density and progresses to n < 2 at high carrier density because of state filling in the presence of the Exclusion Principle. 37 In addition, P ¼ JV= ed ð Þ, where J is the injection current density, e is the electron charge and d is the active region thickness. Making the same substitutions in Eq. (18) gives Equations (21) and (23) are the semiconductor equivalent of the two rate equations used in AMO CQED studies. They also represent the often used and highly simplified semiconductor laser rate equations. In the semiconductor laser case, the equations are usually expressed in terms of carrier and photon densities. We found that working with photon and carrier numbers makes more convenient the tracking of particle number conservation, especially for single and few QD sources, and when the active region and optical mode volumes are different. When working with photon and carrier densities, a mode confinement factor is typically introduced to account for the volume difference. 2 In the end, the choice is decided based more on familiarity.

QD-QW model
One may proceed in the opposite direction and extend the basic model to more accurately describe the semiconductor active region. Examples are a more detailed account of inhomogeneous broadening in QDs and carrier transport between QDs and embedding QW, where carrier injection typically occurs. These two improvements allow the treatment of nonequilibrium carrier distribution effects, such as spectral hole burning, plasma heating, and population bottleneck, at the level of carrier-carrier and carrier-phonon scattering.
The Cluster expansion now leads to dp i dt dn p dt ¼ 2 X n n inh x n ð ÞRe gp n ð Þ À 2c c n p ; dn QD r;i dt ¼ À2Re gp i ð Þ À c nl n QD e;i n QD h;i À c nr n QD r;i À c cÀc Â n QD r;i À f e QD r;i ;l r ;T h i À c cÀp n QD r;i À f e QD r;i ;l l r ;T l h i ; r f e QW r;k ; l p r ; T p 1 À n QW r;k À c nr n QW r;k À c cÀc n QW r;k À f e QW r;k ; l r ; T h i À c cÀp n QW r;k À f e QW r;k ; l l r ; T l h i ; (27) where r ¼ e (h) indicates the electron (hole) populations, n QW r;k is the QW carrier population, and k is the 2-D carrier momentum. Inhomogeneous broadening due to QD dimension or composition non-uniformity appears in the form of a QD population distribution where N QD is now the actual number (resonant and non-resonant) of QDs in the active region, x 0 is the central transition transition frequency of the inhomogeneous distribution, and D inh is the inhomogeneous width. In the numerical simulations, the QDs are grouped into bins, each with width hc, average transition frequency energy x i and QD number n inh x i ð Þ.
The above equations show the carrier injection occurring at the QW carrier population, where P is the injection rate and N p r ¼ P k f e QW r;k ; l p r ; T p is the steady-state QW carrier population created by the optical or electrical excitation when all radiative processes are switched off. The injected carrier distribution f e QW r;k ; l p r ; T p is assumed to be a Fermi-Dirac function with chemical potential l p r and temperature T p .
The carrier capture and escape are described in terms of effective carrier-carrier and carrier-phonon scattering rates c cÀc and c cÀp , respectively. Scattering terms in the above equations approximate the situation where a carrier distribution is first driven to quasi-equilibrium at some plasma temperature by carrier-carrier scattering. Then, carrier-phonon scattering further relaxes the carriers to quasi-equilibrium distributions at the lattice temperature. The asymptotic quasi-equilibrium distributions are Fermi-Dirac functions f e; l r ; T ð Þand f e; l l r ; T l À Á . For the carrier-carrier scattering, the chemical potential l r and plasma temperature T are computed by conservation of carrier population and energy. In the case of carrier-phonon scattering, the chemical potential l l r is obtained from conservation of carrier population and the lattice temperature T l is an input quantity.

Comparison
All three models have been applied to a wide range of QD active media. When investigating QD laser physics, we often consider a simple shallow QD structure, such as In 0.3 Ga 0.7 As QDs embedded in a GaAs QW, where only the ground state transition is supported. To connect to applications, we typically use the case of InAs QDs embedded in a In x Ga 1-x As QW, which emits in the 1.3lm to 1.5lm wavelength range of interest to telecommunication. Input into the calculations are the QD level energies and transition dipole matrix elements. For the QD-QW model, the QW electron and hole band edge energies and their effective masses are also used.
A good comparison of the different models is the exercise for extracting the spontaneous emission factor b from experimental data. The points in Fig. 13 are measured for Emitter D in Table I. The curves are the calculated results from the three models. The simulation uses carrier-carrier and carrier-phonon scattering rates of c cÀc ¼ 10 12 s À1 and c cÀp ¼ 10 11 s À1 , respectively, estimated for T ¼ 10K. They show that all three models are able to reproduce the shape of the experimental curve. Input parameters for the models change the predicted threshold pump power, which may be made to match the measured value by adjusting the injection efficiency parameter g. Another fitting parameter is the vertical axis scaling, i.e., the detector count to intracavity photon number conversion. On the other hand, the extent of the jump in output at lasing threshold is essentially insensitive to all input parameters except b. For this reason, the fitting of cQED models to log-log input-output plots of experimental data is widely used for determining b.
For the 2-rate-equation and basic models, the fit to experimental data is obtained with b ¼ 0:014 and 0:0085, respectively. The extracted b values are essentially similar considering the preciseness of the shape fitting. More significant is the QD-QW model prediction of b ¼ 0:21 for Emitter D, which is significantly higher than those from the other two models.
C. Caution: Underestimating b

QD inhomogeneous broadening
When fitting with a scaled-down cQED model, e.g., the 2-rate-equation model, a cause for error is from ignoring the presence of QDs that are detuned from the cavity mode. They are present because fabrication imperfection leads to distributions in QD size or composition. The detuned QD nevertheless interact with the cavity field because of transition broadening from interactions between the QD and its environment, such as carrier-carrier and carrier-phonon scattering. From semiclassical laser theory, each stimulated transition is scaled by a lineshape function 38 where D is the detuning frequency and T 2 is the reciprocal of the polarization dephasing rate c in Eq. (19). After the onset of lasing, when intracavity intensity is high, there is further broadening (power broadening) with 27,38,39 where T 1 is the population lifetime and I=I sat is the laser intensity relative to its saturation intensity. As described earlier, the QD-QW model can account for their influences. In Fig. 13, the QD-QW model fit to the experimental data was made assuming QD inhomogeneous broadening of D inh ¼ 20 meV [see Eq. (28)]. Figure 14 shows the steady-state electron population distribution calculated for three different excitations. The black curve at pump power far below lasing threshold may be described by a single Fermi-Dirac function, indicating equilibrium condition within QD and QW populations, as well as between QD and QW populations. At 7 mW pump power, which is within the transition from spontaneous emission to lasing, quasi equilibrium remains within QD and QW populations. However, the discontinuity in the blue curve indicates that the QD and QW populations are not in thermal equilibrium. Here, a dynamical population bottleneck exists where carriers in the QW are not efficiently channeled to the QD states. At the even higher power of 20mW, which is after the lasing threshold, a population hole exists in the QD population (dip in red curve). Within the population hole are QDs contributing resonantly and non-resonantly to the laser field. The gain suppression with the population bottleneck prior to lasing and the enhancement of gain from the off-resonant QDs after lasing accentuates the "S" shape input-output curve. Consequently, neglecting these contributions results in underestimation of b.
Similar influence of off-resonant QDs is also reported in a single-photon source experiment. Figure 15 shows measured excitation dependences of intensity spectrum and g (2) (0) for a device consisting one InAs QD that is resonant with a mode from a 1.8 lm diameter pillar cavity with Q ¼ 15 000. At low excitation, the spectrum double peak with a vacuum Rabi splitting %60 leV occurring with g ð2Þ ð0Þ % 0, indicate a single-photon source operating under strong light-matter coupling condition. This draws much attention when first observed because of the promise for higher single-photon source performance and more potential applications within quantum computing. [40][41][42] However, like the AMO counterpart, there is relatively little activity in implementing strong coupling physics in optoelectronic devices at present. Therefore, we will not dwell on it, but instead return to the b extraction question.
As the laser pump increases pass 2 lW, the double peak in the lPL spectrum vanishes indicating a transition from strong to weak coupling. Eventually, g 2 ð Þ 0 ð Þ reaches unity and the device lases. Judging from the pronounced "S" shape in the measured input/output curve in Fig. 16(a), one would infer a relatively low b for the device. On the other hand, the cavity dimensions and the presence of strong coupling point to a b closer to unity. The resolution comes from modeling the experiment in detail. The stimulation revealed that the FIG. 14. Steady-state electron population per electronic state versus state energy, from the calculation giving the solid curve in Fig. 13. The pump powers are 4 mW, 7 mW, and 20 mW (black, blue, and red curves, respectively). The energy is relative to the bulk InAs conduction band edge. Note that the maximum population is unity from the Exclusion Principle. laser excitation at 532 nm also excited nearby, non-resonant QDs. By including the contributions from the six background QDs estimated for the sample, the fitting to experiment produces b % 0:22. As shown in Fig. 16(b), the measured "S" shape comprises two input/output curves: one from the resonant QD inside the cavity and the other from non-resonant QDs in the vicinity. Simulation shows that the resonant QD provides an average photon population of n p % 0:2 at saturation (dashed trace), which is insufficient to reach lasing threshold according to the discussion in Sec. IV B on laser physics. At high excitation the plot shows the onset of nonresonant QD contribution (dotted trace). The addition of resonant and non-resonant contributions then allows the device to reach lasing threshold, as in the case of the inhomogeneously broadened laser discussed in Figs. 13 and 14. There, system starts with thermal photon statistics at low excitation and with a relatively large number of QDs contributing to the emission. Here, one starts in a photon-number (Fock) state and single QD instead. However, in both cases, neglecting off-resonant QDs arising from inhomogeneous broadening results in underestimating b.

Inter-QD correlations
There are also cases of underestimation of b when inter-QD correlation is present. The correlation depends sensitively on the QD number. With a small number of QDs in the system, collisions are reduced to the extent necessary for the correlations to exist. In this situation, the electron-hole polarization receives contributions such as hb † i c † j c j b i i, where i 6 ¼ j label the QDs. A consequence is the possibility of giant photon bunching as well as sub-and super-radiance. 44 We will discuss how an error can occur in the extraction of b from experiment. Figure 17(a) is a plot of g (2) (0) versus pump power for a device similar to Device D in Table I, except for having 40 instead of 60 QDs in the active region. The points are from experiment and the curve is calculated using the QD-QW model with inter-QD correlation taken into account. An important feature is g (2) (0) > 2 at low excitation, which indicates super-thermal bunching. Figure 17(b) shows the corresponding effect on the input-output curve. The solid curve labeled QD-QD is from the same calculation as in Fig. 17(a). The fit is obtained using b ¼ 0:72. At first glance, the "S"  Fig.  15(a). The red horizontal line marking the onset of lasing is positioned according to when g 2 ð Þ 0 ð Þ ! 1 in Fig.  15(b). Plotted in (b) and (c) are the calculated pump rate dependences of average intracavity photon number and g 2 ð Þ 0 ð Þ. The dashed and dotted curves in (b) are for contributions from resonant QD and from non-resonant QDs, respectively. They combine to give the "S" shape in the solid curve. The device is the same as in Fig.15. looks very pronounced for a device with such a high b. Repeating the calculations without the inter-QD correlation gives the dashed curve, which has the flatter appearance expect with high b. The difference in the curves indicates the inter-QD correlation contributions. There is a region, before lasing threshold where the solid curve falls below the dashed one. Here, radiation is suppressed (subradiance). Then, at higher excitation, the inter-QD correlation switch role and enhances radiation (superradiance). With subradiance and superradiance occurring side-by-side, a pronounced "S" shape results. The dotted curve is the fit to experimental points without the inter-QD correlation. It gives b ¼ 0:23, which we know is incorrect because of the unmistakable g (2) (0) > 2 in Fig. 17(a), indicating sub-and superradiant effects at low excitation power.

Quantum-confined Stark effect
As many of us are aware, LEDs are rapidly replacing incandescent and fluorescent lights because of advantages in durability, adaptability and environmental impact. 45 The LED's efficiency in converting electricity to light allows it to be solar powered. The social impact cannot be overstated, for solar-powered LEDs enable the introduction of lighting or the replacement of health-wise dangerous kerosene lanterns in less developed and remote regions in the world.
Studies to date suggest that even higher efficiency is possible, as well as imperative for the wellbeing of the LED industry. A serious problem involves light output saturation at a very low current density of tens of A/cm 2 . This saturation translates to <10 mW in light output from a single LED before efficiency degrades. This concern is often referred to as the efficiency droop. 46 When investigating a laser solution to the droop problem, we discovered another active medium property that may lead to an underestimation of b. It involves the presence of quantum-confined Stark effect (QCSE) arising from mechanical strain in certain crystal structures. For example, compared to zinc-blende crystals such as GaAs and InP, a wurtzite III-N crystal with quantum confinement along the c-axis can have a strong strain-induced electric field. 47 This piezoelectric field changes the envelope functions for different QW subbands under low and high carrier densities as depicted in Fig. 18. Band structure effects in the presence of screening of QCSE is determined by solving the coupled Poisson and k Á p equations. Notice especially the change in the overlap of the ground-state electron and hole envelope functions (red curves), which significantly influences optical emission.
The appreciable change in envelope function overlap leads to significant dependence of the dipole matrix element on excitation. The dipole matrix element determining emission strength has the form 13 where i and j are electron and hole levels involved with the optical transition, l bulk is the bulk material dipole matrix element, the coefficients C n;i and A m;j account for state mixing, and the summation involving r is over heavy-, light-, and crystal-split-hole states. Important to the carrier-density dependence of l i;j is the envelope function overlap, which is described by the spatial integral over electron and hole envelope functions u e;n and u r;m , respectively, and governed by the combined effects of quantum confinement and piezoelectric field. The influence in the quantum-confined Stark effect is illustrated in Fig. 18(c), which is a plot of g ¼ jl TE 1;1 =l bulk j as a function of carrier density for a 3 nm In 0.2 Ga 0.8 N QW between GaN barriers.
A quick way to see the effects of Fig. 18(c) on inputoutput behavior is to use the 2-rate equation model. Changing A g and B to g 2 A g and g 2 B in Eqs. (21) and (23), we computed the curves in Fig. 19(a). For each b, there is a significantly larger intensity jump, and interestingly, an intensity jump remains at b ¼ 1. The larger jump in photon number at threshold with QCSE comes from the suppression of intracavity photon population prior to onset of QCSE screening. This is different from the subradiance in Sec. IV C 2 in that the emission suppression is purely a singleparticle effect.
In an actual device, the QCSE effect is combined with contributions from carrier capture and escape, band dispersions and nonradiative losses. We use the QD-QW model to include these contributions. Figure 19(b) summarizes the results by plotting the photon number jump at threshold n versus the spontaneous emission factor b, with and without QCSE. Comparing Figs. 19(a) and 19(b) indicates a reduction in the jump with the added contributions. The reason is the increase in lasing threshold from the added contributions, which in turn led to partial screening of QCSE prior to the onset of stimulated emission. While Fig. 19 was computed for a QD active medium, we found similar behavior, i.e., increase in photon number jump at threshold with QCSE, from calculations for InGaN QW lasers.

A. Scope
The goal we had set for ourselves was to write, as close as possible, a stand-alone introduction of quantum optics in optoelectronic engineering. Basic concepts are explained at the level of graduate level physics and electrical engineering. To accomplish that, we limited discussion to two specific examples we feel are excellent for illustrating how scientists and engineers are implementing CQED in the engineering of optoelectronic components. We complement our focused approach by citing contributions chosen from a vast literature to provide examples of the broader as well as historical perspective.

B. Progression from physics to devices
The progression from physics idea to commercial realization involves several steps. In between, one needs to arrive at a device concept with perhaps some targeted application advantages. Then, comes the engineering development to confirm those advantages. Only afterwards will there be a possibility of sufficient interest to proceed with production engineering. In terms of the progression towards devices, the examples we used involving quantum-dot micro-and nanocavity emitters, are passed the device conceptualization stage. While many experiments remained optically pumped, optoelectronic devices are being demonstrated.
A good illustration of the progression is provided by the VCSEL. The proposal and proof of concept by Iga and his co-workers on what is today the VCSEL came in the late 1970s. 48 A motivation then and remains today is fully monolithic fabrication. Perhaps, one may associate the underlying physics with experiments using semiconductor etalons and superlattices to study optical nonlinearities or strong lightmatter interaction. 49 Roughly a decade later, device research intensified, after important contributions resulting in, e.g., room-temperature cw operation, and milliampere threshold current. 50,51 It took until the late 1990s, after demonstration of competitiveness with in-plane lasers, e.g., in efficiency, 52 that engineering for production and commercialization began. Today, companies are manufacturing millions of VCSELs per year for appliances used in our daily lives.

C. To the physicist
A physics contribution that usually comes to mind is new ideas for better devices and new applications. The single-photon source described in Sec. III is a good example. With nonclassical photon statistics, one reduces noise to the point that a semiconductor QD can be used in quantum information processing.
However, when introducing new ideas to optoelectronics, it is useful to keep in mind that there already exists a huge infrastructure. Important are compatibility with processing techniques and equipment that have been developed with considerable cost. Also important is integration into the rest of the system, including the power source and readout. Given that one has to deal with many factors beyond just coming up with a great physics idea, why would anyone be interested in device development? The answer is the lure of problems that are in urgent need for solutions. Figure 20 shows the example of expected growth in the number devices connected to the internet. Many of these devices contain optoelectronic components, which must continue to improve, e.g., in bandwidth and energy saving, to support the projected use increase.

D. To the engineer
Advances in semiconductor device fabrication have enabled optoelectronic components to perform close to fundamental limitations. In some cases, understanding of the operation of these components takes the device physics into the realm of quantum optics and traditional concepts of optical gain per unit light, optical confinement, and laser operation itself are may no longer be meaningful in their design and evaluation. We hope that this review provides a timely account of the relevant quantum optics for the "traditional" device physicist and engineer who are working to adapt his or her thinking to new concepts, so as to help accelerate evolution of phenomena such as high-b lasing and nonclassical light generation into manufacturable products.
There is another role for quantum optics besides introducing new functionalities. Not as obvious is work on device characterization and optimization. A model derived from firstprinciples, with a minimum of free parameters, can extract device parameters that are difficult to measure directly. Section II discusses how photon statistics in the form of intensity correlation g 2 ð Þ ð0Þ can be used to distinguish between lasing and nonlasing devices in marginal cases. In Sec. IV, we describe an example where a CQED model is used to extract the spontaneous emission factor b from measured input-output curves.
The need for characterization also extends to the active medium. For example, with QD devices grown on silicon, threading dislocations from lattice mismatch and QD inhomogeneity from Stranski-Krastanov growth remain obstacles to be overcome. 53 Useful for developing growth processes is information on the inhomogeneous broadening and non-radiative carrier loss rate. 54 To extract these quantities from experiment requires models that treat scattering effects more rigorously than in the CQED models discussed in Sec. IV. For QWs, this means scattering contributions calculated using second-order perturbation theory, such as that resulting in the Lindblad equations. To treat QD scattering accurately, it is necessary to go beyond perturbation theory. There, quantum kinetic treatments including memory effects have been developed. For overviews on the Lindblad and quantum kinetic approaches, see Refs. 55 and 56, respectively. The ideal device model would be one requiring only the epitaxy-growth sheet and cavity design as input.
Finally, this review is far from providing a complete picture of the impact quantum optics is or could be making in optoelectronics. Perhaps not as ready for engineering development as our chosen examples, but nevertheless intriguing for future devices are the quantum optical phenomena involving strong coupling physics, entangled-photon creation, polariton lasing, quantum coherence, and many more. Textbook discussions of these phenomena may be found in Refs. 1 and 20.
For brevity, we left out some details involving algebra in Sec. IV B. For the interested reader, we fill in some of the gaps in this Appendix. First, the reduction to the two-rateequation model is possible by invoking the rate-equation approximation. The assumption is basically that the polarization changes sufficiently fast to follow any time variation in the photon and carrier populations. Then, one may write Eq. (19) as FIG. 20. Devices (in units of 10 9 ) including optoelectronic ones involved with the internet versus year. The largest growth is in the "internet of things" which is a collective term for any hardware, software, data, or service that involve data transfer via the internet.
where a † a s ð Þ