Single quantum dot nanolaser - Wiley Online Library

Laser Photonics Rev. 5, No. 5, 607–633 (2011) / DOI 10.1002/lpor.201000039

LASER & PHOTONICS REVIEWS

Abstract Recent theoretical and experimental progress on nanolasers is reviewed with a focus on the emission properties of devices operating with a few or even an individual semiconductor quantum dot as a gain medium. Concepts underlying the design and operation of these devices, microscopic models describing light-matter interaction and semiconductor effects, as well as recent experimental results and lasing signatures are discussed. In particular, a critical review of the “self-tuned gain” mechanism which gives rise to quantum-dot mode coupling in the off-resonant case is provided. Furthermore recent advances in the modeling of single quantum dot lasers beyond the artificial atom model are presented with a focus on the exploration of similarities and differences between the atomic and semiconductor systems.

Single quantum dot nanolaser Stefan Strauf 1,* and Frank Jahnke 2 1. Introduction The invention of the laser, which celebrates its 50 anniversary, has revolutionized science and engineering and the way we live. While early laser systems had resonator dimensions of centimeters to meters, the continuous miniaturization of solid-state lasers has led to optical microcavities [1, 2] with dimensions down to the diffraction limit of the laser light, corresponding to a size reduction of about 7 orders of magnitude. If light is confined into such a small mode volume, i.e. a nanocavity is formed, the light-matter interaction becomes increasingly dominated by cavity-quantum electrodynamic (cavity-QED) effects. Various types of micro- and nanocavity lasers have been developed in the past. In vertical-cavity surface-emitting lasers (VCSEL) two distributed Bragg reflectors are used to confine the light in one-dimension on the scale of the light wavelength [3, 4]. This design allows for a much more efficient coupling of the laser output to optical fibers in comparison to edge emitters, which has led to a huge technological success [5]. When the VCSEL design is combined with an additional microstructuring, e.g. in the form of pillar fabrication [6], an efficient three-dimensional photon confinement can be realized in order to exploit cavity-QED effects. In parallel other designs of three-dimensional microresonators have been utilized, which include microdisks [7], microspheres [8] as well as two-dimensional photonic crystals (PCs) [9]. 1 2 *

PCs refer to a periodic dielectric nanostructure that introduces photonic band gaps. The term was established in connection with the seminal work of Yablonovitch [10] and John [11] in 1987. In its general meaning, distributed Bragg mirrors and VCSELs represent one-dimensional PCs and the roots of this research date back to the work of Lord Rayleigh. For applications like nanolasers, two-dimensional PCs have been introduced that consist of quasi-periodic two-dimensional arrangements of air holes in a semiconductor waveguide with quantum wells (QWs) [12, 13] or quantum dots (QDs) [14–17] as active material. These twodimensional PC cavities have a large potential in terms of high cavity finesse and low mode volume, and represent strong competitors for the VCSEL pillar design. Three-dimensional woodpile PC nanocavities containing self-assembled QDs as active medium have been fabricated using micromanipulation techniques [18, 19].These devices can be made to couple with a gain medium polarized in any direction because of the omnidirectional reflectivity property of the complete photonic band gap [10]. Very recently, lasing signatures of a woodpile PC nanocavity have been reported, which is of interest for novel applications such as three-dimensional integrated photonic circuits [20, 21]. For laser action, the strongly reduced gain length of these micro- and nanocavity devices needs to be compensated by high cavity reflectivities and low cavity losses, achieved by Bragg reflection and/or total internal reflection of the light. Among the original motivations for the

Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken NJ 07030, USA Institute for Theoretical Physics, University of Bremen, 28334 Bremen, Germany Corresponding author: e-mail: [email protected]

© 2011 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

LASER & PHOTONICS REVIEWS 608

fabrication of micro- and nanocavities was the strongly increased free spectral range of cavity modes allowing for single-mode laser operation. With the availability of photonic bandgap structures cavity-QED effects like altered spontaneous emission (SE) properties facilitate a reduction of the laser threshold, as discussed in Sect. 2. A prerequisite for laser emission is the optical gain of the active material. Already in conventional edge-emitting semiconductor lasers, the use of QW heterostructures had many beneficial properties based on the carrier confinement and the modified carrier density of states. New active materials utilizing semiconductor QDs with efficient three-dimensional carrier confinement are under current intense investigation [22]. While QDs are often considered as the active material of the next generation of conventional semiconductor lasers, at present the device performance is limited by size and compositional fluctuations of QDs that are inevitably related to current growth processes. Traditionally, semiconductor lasers utilize either several layers of QWs or high-density QDs as active medium in order to provide enough gain to overcome the losses in the device. With the advent of ultra-small mode volume nanocavities, the development of cavity-QED lasers with few or even a single QD emitter became possible. The ultimate limit is reached when a single QD is strongly coupled to a highly confined single cavity mode of the electromagnetic field. The system is the analog to the one-atom maser [23] or the single-atom laser [24] in atomic quantum optics. In recent years nanolasers build on a PC platform with few semiconductor QDs as active medium have gained tremendous interest in the research community [25]. This interest is largely driven by device applications such as optical communication, on-chip interconnects and beam steering, biochemical sensing, and quantum information processing. PC nanolasers provide wavelengths tunable coherent light sources with small footprints, ultra-low lasing thresholds [16, 17], low-power consumption, fast signal modulation in excess of 100 GHz [26, 27], and they can be directly integrated with other optical elements such as waveguides, beam splitters, modulators and detectors, to create photonic circuits on a chip. Besides device applications, the ultrasmall mode volume of PC nanocavities provides an ideal testbed to study light-matter interaction and cavity-QED effects of individual QDs in the weak and strong coupling regime, as well as Coulomb correlations and other few-body effects in these artificial few electron systems. On the theoretical side, the discrete level structure of QDs and the resulting similarity to atomic systems has been widely used throughout the literature by invoking atomic models to describe QD-based systems. In an atomic system one usually assumes optical transition between two electronic configurations, i.e. the “laser levels” that are resonantly coupled to a high-Q cavity mode. This coupling is described by the Jaynes-Cummings Hamiltonian [28] and the corresponding theoretical description of the single-atom laser was introduced in [29]. However, unlike atoms, the energy spectra of QDs are more complex and the question arises if atomistic models are adequate for QDs, which are


S. Strauf and F. Jahnke: Single quantum dot nanolaser

typically governed by multiple excited carriers and their interactions as well as the spatial and energetic proximity of a continuum of delocalized states [30]. The goal of this paper is to review the recent progress in fabrication, characterization and theoretical description of PC nanolasers with few QD emitters. Note that this review focuses on dielectric nanocavities and excludes plasmonic nanocavities in which surface plasmon oscillations dominate [31]. The paper is organized as follows. Cavity-QED effects are summarized in Sect. 2, which addresses modified spontaneous emission properties as well as the laser threshold reduction. Section 3 introduces the design aspect of high-Q and low mode volume PC nanocavities displaying desired mode profiles. Section 4 describes recent experiments on the characterization of the light-matter interaction in PC-QD nanolasers. After a brief review of cavity-QED effects of individual QDs we focus on the experimental signatures of lasing in these systems. In particular, Sect. 4.3 reviews various QD-mode detuning experiments which highlight the deviation of the QD energy spectrum from the ideal two-level system of the often applied artificial atom picture. Section 5 introduces recent advances in the modeling of single QD nanolasers beyond the artificial atom model. There we explore similarities and differences between the atomic and semiconductor systems.

2. Cavity-QED effects 2.1. Weak and strong coupling Cavity QED focuses on understanding the interactions between electromagnetic cavity modes and matter at the quantum level. In the weak coupling regime, the SE rate of an emitter can be influenced by the Purcell effect [32, 33], which is based on modifications of the density of states of the electromagnetic field in a resonator [34]. Weak coupling corresponds to a perturbative regime and the rate of SE is determined according to Fermi’s golden rule by the electromagnetic field density of states at the position of the emitter. Hence, by placing the emitter at a cavity field anti-node (node) or by tuning the cavity into (out of) resonance with the emitter frequency, the spontaneous decay rate can be enhanced (inhibited). The altered SE due to modifications of the photonic density of states and the electric-field strength at the position of the emitter can be quantified by the Purcell factor 0 =τ c , which describes the SE lifetime changes of FP = τsp sp c ) in a two-level emitter in the presence of the cavity (τsp 0 ). For a singlecomparison to the vacuum SE lifetime (τsp frequency emitter tuned in resonance with the cavity mode and positioned at the cavity-field anti-node, one finds FP =

3 4π 2

λ 3 Q m

n

V

(1)

with the cavity quality factor Q, the mode volume V , the

www.lpr-journal.org

REVIEW ARTICLE Laser Photonics Rev. 5, No. 5 (2011)

light wavelength λm and the refractive index of the cavity material n. Note that for a QD embedded in a host material with refractive index n, its radiative rate is n times what it would be in vacuum [35]. Furthermore note that this relationship is valid for dielectric cavities while plasmonic nanocavities require a modified treatment [36]. Control of SE has many applications. For example, LEDs, which are poised to replace fluorescent and incandescent lighting, rely on SE to generate light, but a large amount of the SE is confined inside the material and cannot be extracted from the device, severely limiting performance. Similarly, in lasers, which are coherent light sources, SE that does not couple to the lasing mode constitutes an unwanted loss of excitation in the gain material that raises the lasing threshold. As a result, there is a strong motivation to achieve control over SE and inhibit it when it is not desired or, alternatively, concentrate it into useful forms. In fact, controlling SE can be considered one of the most important objectives of current photonics research with the potential for important and dramatic advances in device performance. If the interaction strength of the coupled cavity and emitter system overcomes the dissipative losses, then it is in the so-called strong coupling regime where the light-matter interaction becomes reversible (non-perturbative) and the strongly coupled emitter modifies the cavity spectrum itself. If the discrete transition of a single emitter is coupled to a single mode of the electromagnetic field (selected by the cavity), then the Jaynes-Cummings model describes the appearance of new dressed states belonging to the coupled system. The periodic energy exchange between an excited emitter and the empty cavity is known as vacuum-field Rabi oscillations and manifest themselves as vaccum-field Rabi splitting in the anti-crossing of the cavity and emitter resonances. Much of the initial work in this field was accomplished in the 1980s with Rydberg atoms displaying SE enhancement [37] and inhibition [38]. Strong coupling has been demonstrated via normal-mode splitting with many atoms [39] and, after systematically increasing the cavity quality and lowering the mode volume, vacuum field Rabi splitting [40] and vacuum field Rabi oscillations [41] have been achieved with a single atom. For semiconductor systems in the perturbative cavityQED regime, enhanced spontaneous emission from GaAs QWs [42] and InGaAs QDs [43] in monolithic VCSEL microcavities has been demonstrated. In the non-perturbative cavity-QED regime, normal-mode coupling of QW excitons in planar VCSEL microcavities has been realized [44] and the transition from strong to weak coupling in connection with excitonic nonlinearities has been studied [45]. Vacuum field Rabi splitting of a single QD emitter in a threedimensional nanocavity recently became possible with pillar VCSELs [46] and PC resonators [47]. The current excitement in the field comes from possible applications in solid-state quantum information science. For example, strong coupling can be utilized as a bidirectional coherent interface between flying quantum bits (photons) and localized matter quantum bits (charge or spin of electrons or Cooper pairs).

www.lpr-journal.org

609

2.2. Nanolaser: Definition of lasing Conventional lasers operate in the weak-coupling regime. In lasers with cavity dimensions strongly exceeding the light wavelength, like edge-emitting laser diodes, fiber lasers, gas lasers etc., the free spectral range of cavity modes is small and a large number of modes interacts with the active material. As a result, a large fraction of the SE from the active material is coupled into nonlasing modes. This is quantified with the β factor, which is defined as the ratio of the SE rate into the laser mode over the total SE rate. In conventional lasers, this ratio is typically of the order of 10 6 . The input-output curves of conventional lasers reveal a sharp onset of stimulated emission at the laser threshold. According to a rate equation analysis [48, 49], the intensity jump at the threshold scales with 1/β , as illustrated in Fig. 1. In the limit of β = 1 the intensity increases linearly, which has led to the term of a “thresholdless laser”. This limit can be realized in a microcavity with high cavity Q and low mode volume by utilizing the Purcell effect: the SE into the laser mode is enhanced while SE into nonlasing modes is inhibited. In [50] the phase transition analog for a conventional laser has been discussed. Similar to a phase transition that is no longer well defined in a system with a small number of degrees of freedom, the laser threshold is no longer well defined if the number of photon modes available for SE becomes small. With semiconductor nanolasers, it has been demonstrated that for three-dimensional photon confinement with cavity quality factors Q > 104 and a mode volume approaching the cubic wavelength of light, the laser threshold in the input-output curve largely disappears [16,52,53]. This raises the question about the verification of the onset of lasing, or

Figure 1 (online color at: www.lpr-journal.org) Input-output curve of a laser for various β factors obtained from a numerical solutions of the rate equations. A cavity lifetime of 17ps (corresponding to Q 34,000 for an emission energy of 1.32 eV) and a spontaneous emission lifetime (enhanced by the Purcell effect) of 50ps have been used. Photon number and pump rate are scaled with β for a better comparison of the results. The figure is taken from [51].




in more general terms, a quantum mechanical analysis of the emission properties of high-β lasers. Within his quantum theory of coherence, Glauber introduced a set of photon correlation functions, some of which have been successfully used in the past to characterize the microcavity laser emission. The normalized first-order correlation function is given by g(1) (t ; τ ) =

hb† (t )b(t + τ )i hb† (t )b(t )i

(2)

where b† and b are photon creation and annihilation operators for the laser mode. g(1) describes amplitude correlations and is related to the coherence properties of the radiation as measured in a Michelson interferometer. In a stationary situation, where g(1) does not depend on t, the coherence time of the radiation can be determined according to R τcoh = ∞∞ jg(1) (τ )j2 dτ and the emission spectrum follows from a Fourier transform in τ. In conventional lasers the coherence time increases strongly at the laser threshold and the laser line width in the emission spectrum narrows according to the SchawlowTownes limit. For microcavity lasers a substantial reduction of the coherence time with increasing β has been demonstrated [54], see Fig. 2. Similarly, for high-β microdisk lasers, the emission linewidth remained well above the Shawlow-Townes limit [55]. In both cases, the strongly

increased SE into the laser mode represents no longer a small perturbation. Generally, in strong photon-confinement microcavity lasers, a small number of photons in the laser mode is subject to increased fluctuations due to SE. The second-order or intensity autocorrelation function depends on the average number of photon pairs, consisting of photons appearing at the times t and t + τ, g(2) (t ; τ ) =

hb† (t )b† (t + τ )b(t + τ )b(t )i : hb† (t )b(t )i hb† (t + τ )b(t + τ )i

(3)

For the special case τ = 0, the simultaneous appearance of two photons is analyzed. In a stationary situation, g(2) does not depend on t and the τ = 0 result reflects characteristic properties of the photon statistics. For thermal light with photon bunching in the count statistics, the normalized stationary autocorrelation function leads to g(2) (0) = 2 while for coherent radiation obeying Poisson statistics one finds g(2) (0) = 1. For a conventional laser, g(2) (0) exhibits a sharp transition from 2 to 1 at the laser threshold, which represents the transition from thermal to coherent emission. With increasing β the transition is more gradually and for β = 1 a broad transition area emerges, as can be seen in Fig. 2. Furthermore, for weak pumping the emission is no longer fully thermal.

3. Design of PC nanocavities

Figure 2 (online color at: www.lpr-journal.org) Theoretical results for nanolasers with QDs as active materials and various β factors. A cavity Q 44,500 and a spontaneous emission lifetime of 80ps have been used on the basis of experiments in [54]. The QD number has been increased from 5 to 50 and 500 for lowering β from 1 to 0.1 and 0.01. The semiconductor model used in these calculations is described in [51, 54].

=


Light-matter interaction strength in nanocavities is characterized by two key figures, the optical mode volume V and the Q factor, which is defined as Q = ω0U =P, where U is the time averaged energy stored in the cavity, P is the energy loss per cycle, i.e. the far field radiation intensity, and ω0 is the frequency of the confined mode. Q is a measure for how long the cavity stores light, i.e. the photon hold time τcav = Q=ω0 . Since the square of the electrical field strength E is inversely proportional to the mode volume, high Q-factors and ultra-small mode volumes are required to maximize the interaction strength between the optical field and, for example, the transition dipole moment of excitons inside a QD. Tremendous progress was recently made on the fabrication of various types of optical microcavities such as microdiscs, micropillars, microtoroids, microspheres, and PC nanocavities [1, 2]. Among them, PC nanocavities display the smallest mode volumes with V 0:5 (λ =n)3 . Photonic crystals are dielectric materials engineered with periodicity on a length scale comparable to the wavelength of light [10, 11], which are often achieved by etching a regular array of air holes into a planar slab of dielectric materials. Nanocavities are formed by breaking that periodicity in a controlled way, for example by leaving out one or several air holes. In these 2D slab cavities light is confined in the in-plane direction by distributed Bragg reflection and out of plane by total internal reflection (TIR), thereby creating a quasi 3D optical confinement. This is in contrast to pillar

www.lpr-journal.org


VCSEL resonators, in which Bragg reflection is used in one dimension and TIR in the two other directions, or the whispering gallery modes in microdiscs with light confinement in all three dimensions by TIR. Designing an ideal PC nanocavity for a nanolaser supporting only a single mode with high Q and low V at a particular mode wavelength λm with a desired mode profile is highly nontrivial [56–62]. A widespread technique is to start with an initial geometry and calculate the parameters and optical properties using finite-difference time domain (FDTD) simulations, a technique which timesteps Maxwell’s equation over a spatially discretized structure. Further optimization can be achieved by varying the geometry in a parametric search. Using FDTD one can also predict dynamical effects such as the Purcell effect [63, 64], coupling to input or output channels [65,66], or the β -factor of a cavity mode [67]. The inverse approach starts by choosing a desired mode defined in a Bloch-wave distribution and seeks for the matching cavity geometry supporting that mode [61]. For a recent review on other techniques to model PC nanocavities, such as the mesoscopic confinement description, which is based on approximating the distributed Bragg reflection field profile with a Fabry-Perot model, see Lalanne et al. [57].

3.1. Comparison of S1 and L3 type cavities A first requirement is that the resulting cavity mode wavelength λm must be close to the QD emission wavelength. For a given semiconducting material with refractive index n and slab thickness d one can choose a lattice type, for example a square lattice with a lattice constant a and air hole radius r and only one missing air hole in the center. This geometry is known as the S1-type cavity and a corresponding scanning electron microscope (SEM) image is shown in Fig. 3a, which was fabricated by electron beam lithography (EBL) and inductively coupled plasma etching [16, 68, 69]. The resulting λm can be calculated in FDTD simulations by placing broadband magnetic dipole sources selectively exciting TE modes of the slab. Comparison to experiments are achieved by mapping the resulting mode wavelength using micro photoluminescene (μ-PL) spectroscopy [70]. Typical parameters for an S1 cavity fabricated into a GaAs slab designed to match λm with InGaAs QDs emitting around 930 nm are n = 3:4, d = 180 nm, a = 300 nm, and r=a = 0:38. With an effective mode volume of only 0:1 (λ =n)3 the S1 cavity has one of the smallest mode volumes and is on a first glance attractive for making a nanolaser [71]. The corresponding 3D mode profile can be calculated with FDTD simulations by exciting a single mode at its location of maximum field intensity with a spectrally narrow source. As can be seen in Fig. 3c, the highest field intensity of the quadrupole mode in the S1 cavity is unfortunately predominantly localized in the air holes, where the QDs can not be positioned, or close to the air/semiconductor interface. Since QDs in close proximity (40 nm or smaller) to these interfaces have detrimental optical properties [72], the field

www.lpr-journal.org

611

Figure 3 (online color at: www.lpr-journal.org) Nanocavity design. SEM images of a GaAs based square lattice S1-type cavity (a) and a hexagonal lattice L3-type cavity (b). The corresponding FDTD calculations of the electrical field intensity profiles for the S1 and L3 geometry are shown in (c) and in (d), respectively. The figures are taken from [68].

profile of a regular S1 cavity is thus not very well suited for creating a single QD nanolaser, in particular since QDs near the interface can introduce absorption losses rather than gain. However, lasing with InGaAsP QWs operating at 1.5 μm can be achieved with S1 cavities, for both the dipole mode and the quadrupole mode, due to the low nonradiative surface recombination rate in this material [73, 74]. The first optically pumped PC laser was demonstrated in 1999 by Painter et al. utilizing a hexagonal lattice with one air hole missing (H1 or L1 cavity) and QWs as active material [12]. An electrically pumped PC laser based on a similar H1 cavity was demonstrated 2004 by Park et al. [75]. In contrast to the S1 cavity, the H1 cavity localizes the light better within the dielectric material but the Q factor in these lasers was limited to Q=250 (Q=2500) for the optically (electrically) pumped H1 laser. Akahane et al. realized that the abrupt interface between the defect region and the DBR mirror region causes large losses reducing the Q factor, which becomes more serious with decreasing cavity size [76]. In order to reach higher Q=V values they suggested that one has to confine the light in a more gentle way. One way is to create a nanocavity by taking out 3 holes in a row, a geometry known as L3-type cavity [77]. As an example, Fig. 3b shows an SEM image of an L3-type cavity and Fig. 3d the corresponding mode profile. In contrast to the S1 cavity the mode profile of the L3 cavity avoids the air/semiconductor interface and is thus better suited to realize a single QD nanolaser as it provides higher Q=V values and avoids coupling to lossy QDs near the interface. Optimizing nanocavities to achieve high Q-factors can be done by modeling the cavity field by a set of plane waves with corresponding wave vectors ~k using Fourier analysis of the mode profile [59, 60]. Momentum components of



the mode profile with k values smaller than kTIR = 2π =λm are responsible for losses since they don’t satisfy the TIR conditions in the vertical direction and escape the GaAs slab as radiation into the far field. These losses can dramatically reduce the Q factor of the cavity and special care must be taken to eliminate those k components. To this end one can shift and shrink adjacent holes as illustrated in the title figure by the white arrows. Theoretical Q factors approaching 100,000 can be achieved by shifting holes for about 15–18% of a and shrinking about 15–25% of r [58,76]. Improved outcoupling into the vertical direction while maintaining a high Q can be achieved by superposing a grating-like structure into the arrangement of several hole rows adjacent to the cavity area [78–80]. This skilful fine tuning of the resonator geometry to maximize Q is somewhat analog to the fine tuning of a music instrument utilizing standing sound waves such as a violin. Experimentally Q factors of 45,000 (62,000) have been observed for Si-based L3-type (L7-type) cavities probed around 1.5 μm, which are close to the theoretical values [76, 80]. In contrast, GaAs based L3-type cavities are limited to experimental Q values around 20,000 in the 930 nm wavelength region where the InAs QDs emit [16,17], while the designed Q is about 100,000 [58]. The discrepancy between the fabricated device and the intended design can be caused by fabrication errors such as hole size or lattice constant variations and rough side walls [81,82], but also by absorption losses at etched interfaces or by intrinsic absorption effects in the GaAs slab. In particular, it was found that GaAs has at 980 nm about six-fold higher absorption losses for TE modes as compared to 1460 nm, which was assigned predominantly to bulk (defect related) absorption and partly to the Urbach tail and surface state absorption [83]. This effect generally limits the achievable Q=V values for the case of single QD nanolasers made from self-assembled InAs QDs.

3.2. Ultra-high Q cavities Another principle to design nanocavities is based on a photonic double heterostructure confining light within a 1D line defect acting as a waveguide and resulting in experimental Q factors up to 6 105 measured in Si at a mode volume of 1:2 (λ =n)3 [84] and 8 105 for slightly larger V and local width modulation [85]. Similar high Q-factors up to 7 105 have also been achieved in the GaAs material system [86, 87]. Here light is confined due to the mode-gap effect along the waveguide rather than the photonic bandgap effect. The mode gap is created at the interface where two photonic crystal lattices with different lattice constant are combined, somewhat analog to the lattice mismatch in heteroepitaxy when two crystalline materials with different lattice constant grow on top of each other, which can result in electronic confinement [88]. Further improvement can be achieved by tapering the hole radius slowly over the length of the waveguide which yields theoretical Q values up to 8 107 [89]. These heterostructure nanocavities allow to trap and delay photons in Si for up to one ns [90] and can



perform all-optical switching [91–93], but they have not yet been applied to the GaAs material system for the purpose of creating a single QD nanolaser. Quality factors in excess of 108 can be implemented on a chip by using silica microtoroids which display relatively large mode volumes [94]. At these extremely high Q factors light is trapped for more than 40 ns which enabled the new field of cavity optomechanics reporting exciting new phenomena such as on-chip frequency comb generation [95] and cooling of mechanical resonator modes by dynamic backaction [96, 97]. Although generally very promising, integration with epitaxially grown QDs has not yet been demonstrated.

4. Experiments on PC nanolasers 4.1. Cavity-QED experiments with individual QDs In principle, semiconductor QDs are governed by the same cavity-QED physics (described by the Jaynes-Cummings Hamiltonian) as atomic systems. However, the technical implementations are quite different which has major consequences. i) QDs are permanently embedded in a solidstate and are a priori scalable. However, two self-assembled QDs are not alike, since they nucleate at random positions over the wafer and they vary in their transition frequencies due to size, strain, and composition fluctuations [98]. This constitutes severe technological challenges for the realization of single QD nanolaser and other deterministic and scalable cavity QED devices. ii) QDs interact with their environment via carrier-phonon interaction and Coulomb interaction. This leads to particular dephasing properties, like the pure dephasing due to LA phonons [99] with a characteristic signature in the lineshape of optical QD transitions and a specific temperature dependence [100, 101]. Furthermore, the QD dephasing is excitation dependent and linked to energy renormalizations that additionally influence the optical properties [102], as further discussed in Sects. 4.3 and 5.2. iii) While control of the mode wavelength λm in the relatively large cavities used for experiments with individial atoms can be carried out for example by actuating one of the mirrors, tuning the mode frequency of PC nanocavities is not so straightforward. Thus research in QD cavity-QED has to deal with techniques to control the spectral resonance condition between λm and the exciton emission wavelength λe as well as the spatial position of the emitter with respect to the electric field maximum at the antinode ~Emax . Deviations from the ideal resonance condition and emitter position lead to modifications of Eq. (1), which can be described as

FP =

λm2 3λm3 Q 4π 2 n3 V λm2 + 4Q2 (λe

2 E (~r) ~

;

λm )2 ~E 2 max

(4)

where ~E (~r) is the electric field at the QD location [103].

www.lpr-journal.org


4.1.1. Frequency tuning of randomly positioned QDs

A first generation of QD cavity-QED experiments was carried out with self-assembled QDs which nucleate randomly out of their wetting layer [98]. After fabrication of a cavity the relative position between field maxima and QDs is fixed and not necessarily optimized. Weak coupling signatures such as inhibition and enhancement of SE from a single QD have been observed by achieving the spectral resonance condition via temperature tuning [104–110]. While λm is relatively unaffected by temperature, λe is pinned to the semiconductor bandgap, which becomes smaller with increasing temperature, such that relative energy shifts of about 3-5 meV can be realized. To achieve strong coupling the QD must be located at the antinode of the cavity field with a precision usually better than 50 nm. Thus the corresponding generation of strong coupling experiments relied on random chance and, depending on QD density, often required the measurement of hundreds of devices before finding signatures of strong coupling in temperature tuning experiments [46, 47, 110–112]. A drawback of the temperature tuning technique in QD cavity-QED is that the exciton dephasing strongly increases with temperature. For an InGaAs QD, a reduction of the dephasing time from 630 ps at 7K down to 11 ps at 75K has been demonstrated in the weak excitation regime [99]. The correspondingly reduced coherence length in the photon emission is a severe problem for quantum information processing schemes relying on efficient two photon interference at a beam splitter [113–115]. It was found that λm of PC nanocavities and microdiscs is strongly affected by residual gas deposition (adsorption) at cryogenic temperatures [69]. This effect can be utilized to achieve the QD mode energy resonance condition at 4K, thereby avoiding strong QD dephasing at higher temperatures. Figure 4 shows a tuning experiment carried out at 4K using residual gas deposition. On resonance the data exhibit a clear intensity enhancement but anticrossing between QD and mode was not observed, which is indicative of a QD in

Figure 4 (online color at: www.lpr-journal.org) a) Energy tuning experiment based on residual gas deposition to achieve the spectral resonance condition. The insets shows the corresponding μ-PL spectra. b) Photon antibunching signature of a single QD in resonance with an L3-type PC mode [S. Strauf, unpublished].

www.lpr-journal.org

613

the weak coupling regime. Similar experiments with controlled Xe gas deposition have been performed in L3 cavities which reached into the strong coupling regime [116]. In order to confirm that light originates from a single QD one typically records the photon antibunching signature. For light emission from an individual two-level system the second order correlation function, as defined in Eq. (3), drops down to zero for zero delay time, indicating that simultaneous emission of two photons does not occur. Antibunching was first observed by Kimble et al. for single atoms [117]. Later antibunching was demonstrated for other kinds of quantum emitters, such as individual molecules [118], semiconductor QDs [119], nitrogen vacancy centers [120], acceptor-bound excitons [121], or singlewalled carbon nanotubes [122]. Figure 4b shows the corresponding antibunching signature for a QD-exciton tuned into resonance with an L3-type cavity mode under 82 MHz pulsed laser excitation. The normalized zero delay time peak area of 0.35 is indicative of single photon emission from a single QD in the PC nanolaser. Figure 5a compares the pump power dependence of exciton emission from a single QD located in an unprocessed area of the GaAs slab (bulk) with the intensity of a single QD near the cavity center. For the S1 or L3-type PC nanocavities, which are not optimized for far-field emission [78–80], one typically observes a 10 to 12-fold intensity enhancement for single QDs tuned into spectral resonance, as demonstrated in Fig. 5a. The QD on spectral resonance with the cavity appears also to saturate at a higher pump power [123]. Interestingly, the enhanced emission on resonance is not directly connected to the Purcell effect, i.e. a faster lifetime of the exciton emission. The corresponding measurements of the single QD lifetime are shown in Fig. 5b. This lack of QD-cavity mode coupling is caused by the spatial misalignment between the QD and the areas of high field intensity of the mode, which is particularly pronounced at the very low InGaAs QD areal densities of about 1–5 108 cm 2

Figure 5 (online color at: www.lpr-journal.org) a) Single QD intensity vs. pump power for a single QD located in an unprocessed area of the GaAs slab (A) and a QD near the cavity center in an L3-type device (B). b) Lifetime measurements for single QDs in three different dielectric environments recorded by time-correlated photon counting with an avalanche photodiode. The system resolution was optimized to 200 ps. Data are recorded at 4K. [S. Strauf, unpublished]



typically used in these devices [16, 69]. The measurements in Fig. 5b demonstrate that the photonic band gap effect can lead to three-fold inhibition of the exciton transition rate in an S1 cavity and up to one order of magnitude inhibition in an L3-type cavity, while the emission efficiency into the far-field can be simultaneously enhanced by an order of magnitude. Concurrent SE inhibition and enhanced light extraction was also demonstrated in PC nanocavities with QWs as active medium [124, 125]. Thus one can not directly interpret the measured single QD intensity enhancement as being caused by SE enhancement and one has to carefully distinguish contributions from SE enhancement and light extraction when analyzing time integrated data [126, 127]. In contrast, the SE lifetime is a direct measure for the Purcell effect, and SE enhancement has been often observed in the time domain in nanocavities with higher QD densities between 109 cm 2 and 1010 cm 2 , where the chances of spatial alignment are significantly increased [107, 128, 129]. For a recent review on SE control using PC nanocavities see Noda et al. [125]. Another ex-situ tuning technique is digital etching, which is based on gradually reducing the surface oxide layer of the GaAs slab in citric acid, resulting in removal of less than a nm of material per etch step [68]. With this technique λm can be tuned up to 80 nm in steps of 2 nm. Additionally, removal of the surface oxide increases the Q factor by about 30%. Regrowth of the surface oxide can be avoided by removing the PC chip out of the liquid, drying with nitrogen gas, and immediate capping with a glass slide and loading into the cryostat. This technique was used to achieve systematically higher Q factors for the lasing experiments discussed in Sect. 4.2 and 4.3.

4.1.2. Actively positioned QDs

To drastically improve the chances of finding pronounced light-matter interaction, mode maxima and QD position


must be actively controlled. Recently, a sophisticated controlled coupling method was introduced based on strain mediated stacking [130] of several QDs combined with atomic force microscopy tracing to locate the strain sites of a buried seed QD, followed by subsequent fabrication of PC cavities with respect to these markers [126, 131]. Figure 6 shows the resulting anticrossing signature of a single QD in strong coupling with a single L3 nanocavity mode. In this particular experiment a third peak arises near zero detuning. The origin of this third peak requires further investigations. Recently it has been pointed out that a combination of emission from higher rungs of the Jaynes-Cummings ladder (leading to peaks in between the vacuum doublet) and pure dephasing (which broadens these peaks) can lead to a triplet structure [132]. While signatures of weak and strong coupling can now be observed in a deterministic way there are also drawbacks of the discussed technique. The tracer QDs contribute partly to the optical emission and the quality factor of the L3 mode is significantly degraded since the topmost tracer QD creates a hill at the center of the L3 mode with lateral dimension of 100-300 nm, as shown in the inset of Fig. 6. One way to overcome these limitations is based on locating the position of the QD emission at 4K with a confocal microscope and tracing this position with respect to markers on the surface, thereby avoiding complications caused by a tracer stack of QDs. At a first glance this approach seems limited as a typical laser spot size is about 1 micron and carrier diffusion can further blur the QD location. However, the position of an object can be determined with much higher precision (experimentally 10 nm accuracy was demonstrated [133]) than the lateral separation of two nearby objects, which is diffraction limited. Using this far-field optical lithography technique signatures of single QD strong-coupling have been demonstrated for micropillar [134, 135] and for PC nanocavities [133]. In all of these approaches QDs still nucleate at random positions in the epitaxial growth such that no lateral or-

Figure 6 (online color at: www.lprjournal.org) Left: Spectral polariton anticrossing signature for a single QD in strong coupling with a single L3 nanocavity mode. Mode tuning was achieved utilizing the residual-gas deposition technique. Right: Corresponding optical spectra normalized to the exciton peak. The inset shows an atomic force microscope image of an L3 cavity featuring the topmost tracer QD of a seed QD located 63 nm below the surface. The figure is taken from [131].


www.lpr-journal.org


615

der exists between QDs. Therefore, spatial arrangement of several QDs with respect to a cavity mode is still an open issue and formation of quantum networks [136] seems quite challenging. To overcome these limitations direct control of the QD nucleation sites was carried out with patterned templates in the epitaxial growth [137–139] and first signatures of single QD weak coupling have been recently reported for micropillars [140]. For a recent review see [141]. While these demonstrations of deterministic spatial QD-mode coupling are very promising, a single QD nanolaser based on active positioning was not yet demonstrated and more efforts are necessary to create robust, deterministic, and scalable QD-CQED systems with several actively positioned QDs.

4.2. Experimental signatures of lasing Following the discussion in Sect. 2.2, the transition from SE into the lasing regime for conventional low-β lasers, which are not dominated by cavity-QED effects, can be experimentally characterized by several key signatures: – Observation of a sharp onset in the input/output curve – Schawlow-Townes linewidth narrowing of the cavity mode – Coherence time build up according to changes of g(1) (τ ) – Modification of the probability of two-photon coincidences described by a transition of g(2) (τ = 0) from 2 (SE) to 1 (coherent state) – Mode competition Note that according to the quantum theory of coherence applied to lasers, these are not independent criteria. Changes in emission intensity, coherence properties, as well as two photon-coincidences evolve from the underlying photon statistics and, hence, are not altered independently. In particular, linewith and coherence time likewise represent coherence properties. The purpose of the above criteria is to identify characteristic signatures in experiments. Figure 7 shows three typical signatures of lasing for a commercial VCSEL diode operating at room temperature with a multilayer QW gain medium. A sharp onset in the input/output curve in Fig. 7a occurs at a threshold current of 36 mA. Since the β -factor of planar VCSELs is about 0.001 many transverse modes compete for the gain, such that the output spectrum exhibits a transition from multimode to single mode emission above threshold, until ultimately the mode with the lowest loss dominates the spectrum, as shown in the insets of Fig. 7b. Another signature is the transition in the intensity autocorrelation function g(2) (τ = 0) from 2 (representing thermal light) to 1 (characteristic for coherent light) at the laser threshold. When compared to the theory curve shown in Fig. 2 for β = 0:01, the experimental data in Fig. 7b deviate from the expected value of thermal light, i.e g(2) (τ = 0) = 2 for pump powers below threshold. This is a measurement artefact in these Hanbury-Brown and Twiss type experiments carried out with single-photon counting avalanche photodiodes (APDs). Due to the typical APD timing jitter of about 600ps (optimized 150ps) g(2) (τ ) measurements

www.lpr-journal.org

Figure 7 (online color at: www.lpr-journal.org) Three signatures of the phase transition into the lasing regime for a VCSEL diode (β 0:001) recorded at room temperature. a) Sharp onset in the input/output curve. b) g(2) (τ = 0) measured as the bunching height, spectrally filtered at the lasing mode, and plotted as a function of pump current normalized to the threshold current Ith = 36 mA. The inset shows the corresponding mode spectra taken at a drive current of 32.5 mA (left), 34.8 mA (middle), and at Ith (right), with the lasing mode highlighted by the shading. Figure taken from [53].

=

of short coherence times (τcoh < 100ps) emitted by a laser driven at or below threshold are hampered. The origin is that due to the finite time resolution also pair events for τ > 0 are detected and that g(2) (τ ) decays to unity on a scale of the coherence time. A detailed explanation is found in [52]. A reduction in τcoh from 1 ns to less than 50 ps in the low excitation limit was directly measured for a micropillar laser with 3 μm diameter and β = 0:12 by recording the first order autocorrelation g(1) (τ ), as defined in Eq. (2), with a Michelson interferometer [52]. If the experimental τcoh data are convolved into the g(2) (τ ) function, the resulting g(2) (τ ) = 1 + [g(2) (0) 1] exp ( 2τ =τcoh ), when additionally corrected for the APD timing jitter, can be used to reproduce the measured peak in the second order autocorrelation around threshold. An alternative method based on a streak camera in single-photon counting mode has recently been introduced [142] which provides a track record of the individual photon emission events as well as a strongly increased time resolution of 2ps. In these experiments, the decay of g(2) from 2 to 1 for smaller β , the broadening of the transition region for microlasers with larger β [142], as well as the time evolution of the g(2) (t ; τ ) function was directly measured [143]. In contrast to the results shown in Fig. 7, for high-β nanocavity lasers, which are strongly influenced by cavityQED effects, the signatures of lasing are less pronounced, making their detection in the experiment more challenging. The laser transition in a system with a small number of optical modes available for emission processes is analogous to a phase transition in the limit of a small system size [50], i.e. a




Figure 8 (online color at: www.lpr-journal.org) (a) Linewidth narrowing and (b) input/output curve for an L3-type PC nanolaser with a very low QD density of 8 QDs per μm2 . The solid lines are fits to the rate equation based on atomic models. (c) and (d) Same for a non-lasing device located on the same wafer showing monotonic behavior without a kink. (e) g(2) (τ = 0) traces similar to Fig. 7b) but recorded for PC nanocavities with 11, 7, and 3 missing air holes. The solid lines are calculated using the standard photon number probability distribution. The lasing transition region is marked yellow and broadens the higher the corresponding β factor is. Figures (a,b) taken from [16] and (e) from [53].

sharp and clearly defined transition does not exist anymore. For example, an L3-type PC cavity with a β approaching unity confines only one (polarization non-degenerate) mode in the entire QD emission spectrum. Thus the typical signature of mode selection/competition shown in Fig. 7b for the microcavity laser does not apply anymore to a singlemode nanolaser. Figure 8 shows three signatures of lasing for an L3-type nanocavity with a very low density of QDs (8 QDs per μm2 ) as active medium. The s-shaped input/output curve in Fig. 8(b) displays a rather soft turn-on behavior and when fitted to the atomistic rate equation model (solid lines, compare also Fig. 1) yields β = 0:85, which is near unity where a vanishing lasing threshold behavior is expected [34]. For comparison, FDTD simulations for a nanocavity with a mode profile similar to the L3-type cavity yield β = 0:87, assuming a SE bandwidth of 25 nm, which is reduced from unity due to the presence of leaky modes in the thin slab PC design [67]. A soft threshold behavior has been also seen, e.g., in [15, 52, 144]. Care must be taken when using the kink in the input/output curves as the only signature for QD lasing, since several QD configurations (exciton, biexciton, etc.) can contribute to the total photon production rate. For example, a kink can also occur in the input/output trace when the biexciton state contribution takes over at higher pump powers and the exciton contribution quenches [30], as further discussed in Sect. 5.3. Thus in order to demonstrate lasing for high-β nanolasers it is essential to consider the interplay of several characteristic signatures for lasing. Another signature is the linewidth of the lasing mode which varies for a conventional laser inversely with the output power according to the Schawlow-Townes limit, but remains well above this limit for small mode volume semiconductor lasers, as discussed in Sect. 2. In particular, it was predicted that VCSEL with quantum wells as a gain medium display a pronounced plateau in the linewidth data around the threshold regime [145]. The underlying cause of the plateau formation is the coupling between intensity


and phase noise (gain refractive-index coupling), which is of particular importance in semiconductor lasers with quantum well gain. To quantify this linewidth broadening Henry introduced the linewidth enhancement factor α, which increases the linewidth at and above threshold according to (1 + α 2 ) [146]. The experimental data for an L3-type QD nanolaser display such a plateau in the linewidth narrowing trace, although at a reduced magnitude, as shown in Fig. 8(a). Note that to compare the linewidth narrowing with the intensity data in Fig. 8(b) the linewidth is plotted here as a function of pump power (instead of output power). A similar plateau was recently observed in threedimensional PC QD nanolaser [20]. In contrast, the L3-type cavity shown in Fig. 8(c,d) displays only a monotonous linewidth narrowing and a linear input/output curve until saturation, indicating that this device does not reach into the lasing regime. Such behavior corresponds to operation in the light-emitting-diode regime where there is a constant cavity loss but an increase in gain [147]. To some extent it is surprising that QD lasers show such a pronounced linewidth re-broadening around threshold since simple models predict a small if not zero α factor due to their discrete density of states. However, several experiments have shown that α factors of QD lasers can have a comparable magnitude to those of quantum wells with a continuous density of states [148]. One explanation put forward is that the finite carrier capture time and the plasma effect in QDs might be responsible for this behavior [148]. Another explanation is that the presence of excited QD states results in an alpha-factor above one [149]. Recent calculations of the α factor on the basis of a microscopic semiconductor theory [150] also predicted large values for a homogeneously broadened QD system, which are strongly reduced by inhomogeneous broadening. While the physical origin seems still under debate, it appears that a signature for lasing in PC QD nanolaser is not just monotonous linewidth narrowing but rather the observation of a characteristic plateau in the linewidth narrowing

www.lpr-journal.org


trace, as shown in Fig. 8(a), which is caused by the nonlinear interplay between carrier density and photon density. A direct signature for lasing action in high-β nanolasers is the transition from SE into the coherent light state which is shown in Fig. 8e for three different devices with 11, 7, and 3 missing holes in the Γ K direction forming L11, L7, and L3-type nanocavities, respectively. The larger the line defect the more modes can fit into the nanocavity. In particular, three modes into the L11, two modes into the L7, and one mode into the L3 device, corresponding to ideal β factors of 0.33, 0.5, and 1 from a simple mode counting argument when leaky modes are not taken into consideration. The solid line fits are calculated using the standard photon number probability distribution leading also to the stated β -factors [53]. With this set of devices the phase transition into the lasing regime was approached systematically. As the nominal number of modes in the laser decreases the transition to the coherent state broadens and occurs over a much wider range of pump powers (marked yellow in Fig. 8(e)). Similarly, the maximum value and the slopes of the g(2) (t ; τ ) data decrease suggesting reduced photon number fluctuations in high-β nanolasers [52]. In contrast, the lasing transition of the VCSEL diode shown in Fig. 7 occurs in a normalized pump region which is orders of magnitude narrower than those of the QD PC nanolasers, making the transition easily detectable in the experiment. The lasing threshold values extracted from the high-β device shown in Fig. 8 are of the order of 100 nW and, when corrected for the actual absorbed pump power, are as low as 4 nW (49 mW=cm2 ) [16]. These values are remarkably low, in fact orders of magnitude lower than observed in any macroscopic laser, VCSEL, microdisk laser [144, 151, 152], micropillar laser [52, 142, 153, 154], or PC nanocavities [12, 15, 26, 155–161], and comparable to recent work on PC nanolasers [162].

4.3. Detuning experiments - the quest for the gain mechanism Assuming ideal two-level like QDs that exhibit spectrally sharp exciton resonances and a statistical distribution of the QD emission over about 50 nm due to unavoidable QD size, strain, and composition fluctuations, it is rather surprising that simultaneous spectral and spatial coupling as described in Eq. (4) seems to occur in the L3-type nanocavities of Fig. 8 with only about 8 2 QDs per μm2 [16]. As the chances that even one of these QDs is both spectrally and spatially coupled to the lasing mode are below 1%, the question arises what does actually lase in these QD nanolasers? To address this important question we will discuss several related experiments in this section. The results will also challenge the analogy between the single atom laser and the single QD nanolaser. Unlike atoms, the energy spectra of QDs are more complex. A striking difference between cavity-QED experiments with atoms and with QDs is that in the latter case efficient light-matter coupling is also observed if the sharp exciton transitions of the s-shell are spectrally detuned from the

www.lpr-journal.org

617

Figure 9 (online color at: www.lpr-journal.org) Mode spectra of an L3-type nanolaser recorded at pump powers below the lasing threshold (a) and around the lasing threshold (b). Several individual QD s-shell exciton recombination lines are visible in the vicinity of the cavity mode which saturate close to the lasing threshold ( 100 nW). Data are recorded at 4K. [S. Strauf, unpublished] c) Spectra of randomly positioned QDs in another L3-type cavity shown on a log-scale (right y-axis) with the corresponding lifetimes (left y-axis) recorded at several spectral positions. While individual QD show pronounced inhibition of SE, the region of the broad background which is spectrally on resonance with the cavity mode is strongly Purcell enhanced (145ps), under various spectral detuning conditions. Figure taken from [16].

cavity mode wavelength. This was first recognized in experiments on L3-type nanocavities with an ultra-low areal QD density [16], for which the lasing signatures have been discussed in the previous chapter 4.2. Figure 9 shows the corresponding mode spectra, where a few sharp exciton recombination lines are visible at pump powers well below the lasing threshold (1 nW). When spectrally filtered, these sharp lines displayed pronounced photon antibunching signatures, both for on and off-resonance conditions, confirming that they originate from a single quantum emitter (see Fig. 4b). Temperature tuning experiments revealed that typically none of these sharp QD transitions was by chance in spectral resonance with the cavity mode at 4K. In addition, it was often observed (> 100 devices) that s-shell excitons display pronounced inhibition of SE up to an order of magnitude [16] as shown in Figs. 5b and 9c, indicating that these QDs are spatially off resonance, although spectrally in the vicinity of the mode emission. In contrast, the broad background visible on the log scale in Fig. 9c was found to be strongly Purcell enhanced, if spectrally on resonance with the cavity mode, with lifetimes down to 145 ps. The vanishing probability for simultaneous spatial and spectral coupling of the sharp s-shell exciton lines and the cavity mode is also supported by FDTD calculations as shown in Fig. 3d. For the L3 mode, areas with a field intensity I equal or larger than one tenth of its maximum Imax cover about 0.1 μm2 . Hence less than one QD within the mode volume has efficient coupling due to a favorable spatial position. Even for the cavity field intensity reduced



to one hundreds of its maximum value (areal coverage of 0.9 μm2 ) only 6-8 QDs are expected within this region. These QDs are not necessarily spectrally near the cavity mode emission. What can be concluded from these experiments is that lasing in PC cavities with ultra-low QD densities occurs regardless of the spectral resonance condition between the sharp s-shell exciton emission lines and the cavity mode spectrum, while individual QD lines show pronounced photon antibunching, and the broad background is Purcell enhanced and thus clearly coupled, which is in strong contrast to atomic systems [16,163]. On the other hand, the localized QD states are required to excite the cavity mode emission since devices fabricated in a region without QDs (but with the InAs wetting layer) show no cavity mode emission at all, i.e. lasing is not sustained by the energetic tail from the wetting layer alone. It was furthermore found that lasers operating at shorter wavelengths near the QD p-shell centered at 920 nm, despite having similar Q factors, exhibit substantially higher lasing threshold powers, as shown in Fig. 10. To tune the cavity mode over the entire emission range of the QD gain medium 12 PC nanolasers with different lattice constants were fabricated. The threshold power under optical pumping, as estimated from the kink in the input/output curve (red squares) or the onset of linewidth narrowing (blue circles), increased by over an order of magnitude when the mode is sustained by emission from the QD p-shell. This behavior can be understood by the fact that the QD p-shell becomes only significantly occupied, after the s-shell states have been saturated. Xie et al. [144] used a microdisk with a larger mode volume and a QD density of 50 QDs/μm2 , where the coupling probability is significantly increased. A pronounced reduction in lasing thresholds by a factor of three demonstrates that the s-shell exciton of a single QD, when tuned into reso-

Figure 10 (online color at: www.lpr-journal.org) Lasing threshold versus operating wavelength as estimated from the kink in the input/output curve (red squares) and the onset of linewidth narrowing (blue circles). The p-shell devices lase with at least one order of magnitude higher thresholds. Data are recorded at 4 K. [S. Strauf, unpublished]



nance with the cavity mode, can indeed influence the lasing characteristics. However, in that work it was also found that lasing signatures of the cavity mode are present under offresonant conditions similar to earlier findings [16]. Almost identical results, i.e. a three-fold reduced lasing threshold and sustained lasing under off-resonance conditions was also reported by other groups using micropillars with about 200 QDs per cavity [153] and L3-type PC cavities with an ultra-low density of less than one randomly positioned QD per cavity [162]. In addition it was shown that the sharp s-shell emission displayed pronounced photon antibunching, under off-resonance [16, 144] and when tuned into spectral resonance with the mode [153, 162]. While observation of antibunching from a sharp s-shell emission is an unambiguous proof that the light emission stems from an individual quantum emitter with an anharmonic energy spectrum, it is not a priori clear that lasing itself is sustained by s-shell excitons. Since lasing occurs already under off-resonance conditions, tuning of the s-shell exciton into resonance with the cavity mode creates a higher photon number inside the mode, thereby reducing the lasing threshold. Furthermore, the cavity mode was shown to display photon antibunching even when the s-shell exciton was spectrally detuned from the cavity mode [164, 165]. Thus in order to reveal the gain mechanism in present single QD nanolaser experiments, it is essential to understand the cavity mode feeding under off-resonance conditions. A second generation of experiments pioneered by Imamoglu et al. utilized actively positioned QDs (as described in Sect. 4.1.2) to achieve stronger cavity-QED effects as only one QD is present inside the device, and that QD is near or at the field maximum of the cavity mode [126,131,133,134]. Applying the gas-deposition technique described in Sect. 4.1.1 it was found that the cavity mode is visible for spectral detuning from the sharp s-shell exciton resonance in the energy range from -20 nm to +4 nm, a remarkably broad energy range [131]. Furthermore, it was demonstrated with second-order photon cross-correlation measurements that the photon emission from the cavity mode and the s-shell excitons is anti-correlated at the level of single quanta, proving that the mode is driven solely by the QD, despite the large energy mismatch up to 20 nm [131]. However, the mode emission itself did not display photon antibunching under these very large detunings. Similar signatures in photon cross correlation experiments between s-shell excitons and the cavity mode were also found in micropillars with a larger density of randomly positioned QDs. In this case, resonant excitation of the p-shell of a particular QD was used to suppress the contribution from other QD emitters. As a result the cavity mode revealed photon antibunching when detuned by 0.7 meV from the s-shell exciton emission [164]. Recently Englund et al. demonstrated similar signatures for a single QD which was 1.2 nm detuned from an L3-type PC cavity mode, and also showed Rabi splitting when tuned into resonance [165]. This cavity emission for arbitrary detunings clearly challenges the standard picture that QDs may be fully described as artificial atoms with discrete energy levels. On the flip side it also offers new possibilities for resonant single QD

www.lpr-journal.org


spectroscopy and background-free photon statistics as recently demonstrated by several groups [165–167]. It was also proposed as a resource for advanced solid-state singlephoton sources [109, 168] since the common spectral diffusion of a quantum emitter [169] is effectively canceled out by the “self-tuned” coupling effect [170]. The question remains what microscopic mechanism could be responsible for the non-resonant QD-cavity coupling. Most likely more than one mechanism contributes, depending on the magnitude of the detuning. Strauf et al. originally proposed both, a contribution from phonon mediated coupling, which is visible in the broadened spectral lineshape of s-shell excitons, and a contribution from a spectrally broad and Purcell enhanced quasicontinuum created by multiexcitons inside the s-shell and p-shell which are coupled to extended wetting layer states, which was called for short a “self-tuned gain mechanism”, however, without further quantifying it [16]. Calculations of the carrier-phonon interaction show that the exciton lineshape broadens in the base up to 2 meV at 7 K due to quadratic coupling of acoustic phonons [101], in agreement with experiments [100]. With respect to QDs inside cavities, it was predicted that dephasing for nonzero detuning leads to a qualitative change in the cavity spectrum as dephasing shifts the emission intensity towards the cavity frequency. This intensity shifting effect was predicted for detunings up to 1 meV [170, 171]. Further theoretical work on phonon-assisted cavity feeding predicted QD-mode coupling up to 2 meV at liquid helium temperatures, based on phonon scattering rates calculated by the independent Boson Hamiltonian [172] or the Schrieffer-Wolff perturbation approach [173]. In time resolved experiments on deterministically positioned QDs it was found for detuning energies up to three mode linewidth (about 2-3 meV) that the emission within the cavity mode arises from the spectrally closest s-shell exciton state, since the mode follows the exact SE lifetime of the detuned exciton state, where either the neutral, charged, or the biexciton states were probed, strongly suggesting phonon induced dephasing as the coupling mechanism [135]. The phonon mediated dephasing process was recently further confirmed in temperature dependent experiments under resonant s-shell excitation in micropillar cavities [166] and in L3-type PC cavities [165]. In light of the theoretical predictions and experimental observations it seems thus plausible that strong dephasing causes exciton mode coupling for detuning energies up to about 3 meV. On the other hand, spectral detuning and cross correlation experiments demonstrate that single photon feeding into the cavity mode occurs for detuning energies up to 20 nm (27 meV) which are about an order of magnitude larger [131], as illustrated in the experiment in Fig. 11. These observations can thus not be explained by pure dephasing or by the phonon assisted mechanisms alone. Kaniber et al. carried out similar experiments on L3-type cavities with randomly positioned QDs and found crosscorrelation signatures up to detunings of 19 meV. It was furthermore shown that shifting the mode away from the s-shell exciton transition enhances the purity of the photon antibunching signature, while the broad background coupled

www.lpr-journal.org

619

Figure 11 Spectra of an actively positioned QD with respect to an L3-type cavity mode labeled M (shaded grey) for various detuning energies achieved by digital etching of the surface oxide in citric acid. The number of etch steps are indicated. Figure is taken from [68].

to the cavity mode does not display photon antibunching, i.e., there is multi photon emission. As a possible explanation it has been suggested that charged excitons decay into a continuum of final states mediated by the photoinduced hybridization effect with the wetting layer, i.e. by photon-induced shake-up processes which are enhanced by the dielectric environment of the cavity, although no further evidence is provided [174]. Recently Winger et al. revealed a more detailed picture of the microscopic origin of the broad single-QD background emission and regarded it as an intrinsic feature in QD cavity QED systems with multiexciton states [175]. Excitation (or capture) of carriers into the p and d-shell leads to a series of manifolds. Using a configuration interaction approach, multiexciton eigenstates up to four electron-hole pairs were calculated. Since the higher orbital angular momentum states are subject to strong hybridization with the wetting layer, as was found earlier by Karrai et al. [176], these multiexciton transitions merge into an excited-state quasi-continuum distributed over an about 15 meV broad energy range. Possible optical transitions between the various broad band exciton manifolds, separated roughly by the band gap energy, leads to the omnipresent single-QD background emission. As this background also appears in the energy range where the sharp s-shell excitons emit, the model provides a microscopic picture of the above-discussed “selftuning” mechanism. In support of this nonresonant feeding mechanism, Laucht et al. have recently demonstrated that temporal correlation between the multiexciton background and the cavity mode exists, whereas emission from the sharp s-shell excitons occurs a few ns delayed and uncorrelated from the mode emission [177]. This is also consistent with second order photon cross-correlation signatures which



show that mode and s-shell exciton emission do not occur simultaneously [131, 175]. Interestingly, the intensity of the cavity mode itself is slightly nonlinear in the input/output curve of Winger et al. [175], resembling the essential features of high β lasing as presented for example in Fig. 8. However, the authors interpret the behavior of the cavity to first follow the s-shell exciton at low pump powers, which is linear, then the biexciton at intermediate pump powers, which is superlinear, and finally the mode follows the p-shell, which is linear up to saturation. Moreover, the work by Winger et al. challenges the view that the transition of the cavity mode emission from pronounced photon bunching at lower pump powers to Poissonian emission at higher pump powers is solely caused by QD lasing. Using a Monte Carlo random walk of excitation and photon-emission events within the multiexciton configurations of a single QD, they calculate that the cavity mode displays pronounced photon bunching at lower pump powers and a Poissonian emission at higher pump powers, which matches the observation in their experiments. Ritter et al. [30] have recently introduced a microscopic description of a single QD nanolaser going beyond the atomistic models, which also takes into account multiexciton configurations together with excitation-induced dephasing and energy renormalization, as further detailed in Sect. 5.3. This model also reveals a nonlinear input/output curve and attributes the change in slope to the photon contribution from s-shell exciton, biexciton, and p-shell. The experimentally observed transition from photon anti-bunching to bunching with increasing pumping is traced back to the appearance of additional emission channels while the occurrence of Poissonian emission for large pump rates is traced back to stimulated photon emission. What can be concluded from the theoretical approaches in [30, 175] is that it is very difficult to extract a reliable β factor from fits to the input/output curve of a single QD nanolaser. The application of atomistic rate equations is questionable, since they are derived for a multi-emitter system. A four-level model, as used for atomic systems, reflects static configurations and predicts strong saturation effects not seen in experiments. The description of the active material needs to include the cavity-emitter detuning and emission-backround effects for a proper analysis of the single QD emitter experiments. The conclusion from recent experiments [177] is that a sharp s-shell exciton line and a broad emission background feeding a detuned cavity resonance do not coexist. For strong excitation, a large number of various multiexciton states, possibly hybridizing also with wetting layer states [176], provide a broad emission background that can efficiently feed the cavity mode, while in the same singleQD nanolaser system for weak excitation, discrete s- and p-shell exciton resonances appear. Note that in the strong excitation regime the s-states likely remain occupied with excitons. However, they recombine at different energies due to their Coulomb interaction with other excited carriers. Since the experimental spectra collect data from many repeated excitation cycles, the results represent statistical



averages over many realisations of various multi-exciton states. These results challenge the interpretation that the s-shell exciton alone lases and provides the gain in present high-β nanolasers. On the other hand, it can be argued that if there is spatially only one QD within the cavity mode, then 100% of the gain can be provided by that single QD through the “self-tuned gain mechanism” [16, 175] involving both, the contributions from the single QD excitations and their coupling to the delocalized wetting-layer states. As the single QD background effect is not solely from the QD alone, one should understand the single QD gain medium as a hybrid system between QD and quantum well, in stark contrast to atomic systems.

4.4. Lasing under strong coupling Strong coupling between excitons and a microcavity mode was first demonstrated in 1992 by Weisbuch et al. who used a quantum well inside a VCSEL resonator [44]. Such a system is classical in the sense that a large ensemble of exciton oscillators contributes to the Rabi splitting and that the addition or loss of a single photon has negligible impact. In contrast, with a single QD strongly coupled to a singlemicrocavity mode [46, 47] vaccum-field Rabi splitting is realized, as is evident in experiments detecting quantum correlations in the emitted light field [131, 132] (see also Sect. 4.1.2). Nomura et al. have recently shown that lasing and strong coupling can coexist for a single QD coupled to an L3-type PC cavity mode [17]. For the s-shell exciton tuned into resonance with the cavity mode, a Rabi splitting with an exciton-mode coupling strength of g = 68 μeV was obtained, similar to previous reports [46, 47, 111, 131]. An interesting observation is that the cavity emission displayed also characteristic signatures for lasing, such as a nonlinear kink in the light input-output curve, photon bunching, and linewidth narrowing of the mode, in agreement with earlier results presented in Sect. 4.2. The results reported by Nomura et al. are remarkable in the sense that lasing and strong coupling can coexist in the intermediate pump regime. In contrast, earlier studies observed that with increasing pump power the Rabi splitting collapses and the system gradually evolves into the weak coupling regime, without any signatures for lasing [178]. This behavior was attributed to incoherent feeding of other background emitters into the cavity mode, known as cavity pumping [179], which randomizes the arrival time of the excitation and thereby averages out Rabi oscillations at higher pump powers. Nomura et al. used a four-level atomistic model including the effect of cavity pumping to analyze their experiment. However, in light of the detuning experiment presented in Sect. 4.3, the multiexciton physics and dephasing effects of the single QD gain medium, and recent findings that the s-shell exciton emission and the cavity mode emission can be temporarily delayed with respect to each other, it is questionable if atomistic few-level models are adequate to fit input/output curves and extract parameters such as the

www.lpr-journal.org


repumping rate in support of s-shell exciton lasing. In the limit of a single-emitter laser, the question what differences to expect between atomic and QD systems becomes even more prominent than for the many-emitter case where inhomogeneous broadening due to QD size and composition fluctuations masks experimental observations.

5. Theory of a single QD laser As discussed in previous chapters, there are fundamental differences between single-atom lasers and the current realization of single-QD nanolasers. While individual atoms can be isolated from the environment to a high degree, selforganized QDs are embedded in a semiconductor material which leads to various interaction effects. Furthermore, as discussed below, in QDs interaction processes can simultaneously act upon several configurations and the system configurations can change dynamically while atoms have static configurations. In this chapter, we present a theoretical analysis how multi-exciton effects, coupling of carriers between QD and WL states, and excitation-induced effects influence the emission properties of single-QD lasers.

621

to solve the von-Neumann equation for the density matrix of the system. This allows to determine arbitrary single time expectation values like the mean-photon number, occupation probabilites of the levels, photon correlation functions like g(1) (τ ) leading to the emission spectrum and g(2) (τ = 0) describing photon bunching and antibunching. Using the quantum regression theorem, also two-time expectation values are accessible. The theoretical treatment of the single-atom laser has been introduced in [29]. The von-Neumann equation for the statistical operator ρ of the system has the form ∂ρ ∂t

In atomic laser systems one considers the resonant coupling of the transition between two atomic levels to the laser cavity mode. In general, the interaction between the quantized light field and two electronic levels is described by the JaynesCummings Hamiltonian. Additional atomic levels are used for the pump process and for the depletion of the lower laser level in order to increase the inversion of the laser transition. An example for electronic transitions in a four-level laser system is shown in Fig. 12. For a single atomic few-level emitter interacting with one mode of the quantized optical field it is directly possible

=

κ 2bρb† 2

b† bρ

!

ρb† b

!

!

(6)

;

=

γi j 2σi j ρσi+j 2 (i; j )

∑

σi+j σi j ρ

!

ρσi+j σi j

:

(7)

j ih j

Here the atomic transition operators σi+j = j i and σi j = i j have been used. Finally the pump process can be represented either by coupling of the levels 1 and 4 to a coherent optical field or as incoherent pumping via a reverse Lindblad term [29]. This description of single-atom emitters coupled resonantly to a single high-Q cavity has been successfully used to describe the corresponding experimental results in [181]. Several properties of this system are important to note. Photon anti-bunching is observed for weak pumping, provided that the cavity decay rate is larger than the spontaneous emission rate into the laser mode. The four-level system shows saturation in the regime of strong pumping when the pump rate γ14 exceeds the atomic rates γ43 and γ21 . Using other level schemes, in which the incoherent

j ih j

www.lpr-journal.org

(5)

where b† and b are the creation and annihilation operators for photons in the laser mode. In the atomic system, dissipation is caused by the various electronic transitions represented by the blue and magenta arrows in the example of Fig. 12. Spontaneous emission from the laser transition into nonlasing modes 3 2 is of central importance for the jump in the I-O-curve. A similar transition 4 1 reduces the efficiency of the pump process. Another group of transitions involves the feeding of the upper laser level 4 3 and the depletion of the lower laser level 2 1. In atomic systems these are usually facilitated by spontaneous photon emission processes. The atomic transitions due to spontaneous emission into an empty continuum of photon modes leads in Born-Markov approximation to Lindblad terms Lsp ρ

Figure 12 (online color at: www.lpr-journal.org) Four-level laser scheme. The cavity mode interacts with the atomic levels 2 and 3, the pump process excites the system from state 1 to 4, relaxation processes couple the levels 4 and 3 as well as 2 and 1. The picture follows the model of [29].

1 [HJC ; ρ ] + L ρ ; i¯h

where HJC is the Jaynes-Cummings Hamiltonian for the coupling between the laser transition of the atomic system and the laser mode of the resonator. Dissipative processes are included via Lindblad terms collected in L ρ. For the photons in the laser mode, dissipation originates from cavity losses with the rate κ. This can be described by coupling the laser mode to a bath of other modes. Treating this interaction in Born-Markov approximation leads to [180] Lc ρ

5.1. Description of the single-atom laser

=



pump process directly removes electrons from the laser levels, pump-induced dephasing leads to output quenching for elevated pump rates. Examples for such a situation are twolevel models in which incoherent pumping is described by a Lindblad term that directly promotes the excitation from lower to upper laser levels [182] (output quenching can be seen in Fig. 4 of this reference) or three-level laser scheme where the excitation process moves the system out of the lower laser level to an upper pump level, see Fig. 5 of [29]. Another scheme suggests a two-level system coupled to two reservoires for independent pumping of electrons and holes [183].

5.2. Beyond the artificial atom model In the past, it has been proposed to adopt the single-atom laser models to semiconductor QD systems. In this context, it is important to note that the levels in the atomic model represent system states due to different electronic configurations in a multi-electron atom. By adopting e.g. an atomic four-level system, like the one in Fig. 12, one implies that other configurations have a large energy separation and/or negligible dipole coupling efficiency to the considered levels. Hence a scheme like in Fig. 12 in general describes the dynamics of the system represented as a point moving in a discrete configuration space rather than that of a single electron between four possible orbitals. Self-organized QDs are frequently grown on a wetting layer and/or embedded in a quantum well. The confinement structure is surrounded by semiconductor bulk material with a larger band gap. This leads to an energy level structure, as sketched in Fig. 13, where the discrete conductionband states due to three-dimensional confinement are energetically below a quasi-continuum of delocalized states. The latter consists for lower energies of states with twodimensional carrier motion in the wetting layer or quantum well. At higher energies the barrier states of the bulk material with three-dimensional carrier motion contribute.


In a self-organized growth process one often finds lensshaped QDs with cylindrical symmetry around the growth direction. This allows to classify electronic states in terms of two-dimensional angular-momentum quantum numbers. Then the lowest confined s-state is only spin-degenerate, the first excited p-state has an additional two-fold angularmomentum degeneracy and can accomodate a total of four electrons. In the ground-state of an undoped semiconductor, all valence-band (conduction-band) states are populated (empty). By exciting electrons and holes into the conductionand valence-band states, respectively, various system configurations emerge. Their energies are influenced by the Coulomb interaction among excited carriers. The situation is similar to a multi-electron atom and this is what has earned QDs the name of artificial atoms. Quantitatively this analogy has its limitations. QDs are much larger than atoms, which proportionally reduces the Coulomb interaction between excited electrons and holes. This considerably weakens the strength of the configuration interaction and leads to important differences between individual atoms and QDs. i) In QDs many configurations can lead to optical transition energies separated by only a few meV. Also the quasi-continuum of delocalized states remains only some ten meV away from the localized states. In atoms the configuration interaction is much larger which makes it easier to find a small number of configurations in the energy window of interest. ii) The close proximity of energy levels in QDs allows for efficient carrier scattering processes between the localized states and between localized and delocalized states due to carrier-phonon and carrier-carrier Coulomb interaction, see Sect. 6. iii) Since the elementary interaction processes take place between single-particle states, at the many-particle level the same interaction processes can simultaneously act upon different configurations. Hamiltonians for the carrier-photon interaction, for the carrier-phonon interaction, and for the Coulomb interaction of carriers are initially formulated with carrier field operators defined on a single-particle basis. For example, among the elementary

Figure 13 Left column: TEM images providing a top view of an ensemble of selfassembled QDs and the cross section of a single QD. From [184]. Right: Energy levels for a self-assembled QD. The threedimensional carrier confinement leads to localized states for electrons (holes) with discrete energies below (above) a quasicontinuum of delocalized states due to carrier motion in wetting-layer states and/or bulk states.


www.lpr-journal.org


interaction processes in QDs are the interband transistion of s-shell electrons under photon emission or absorption. These elementary processes can take place in the presence of other carriers, and the Coulomb configuration interaction then changes the corresponding transition energies. Specifically, s-shell electron-hole-pair recombination without other excited carriers leads to photon emission at the s-exciton energy, in the presence of another s-exciton with opposite-spin carriers the emitted photon has the energy of the (ground-state) biexciton, and when accompanied by a p-exciton emission occures at the s-p-biexciton energy. Only when the configuration interaction is large, just one of these transitions is resonantly coupled to a high-Q cavity mode, like the transition between the s-exciton and the ground state. The reduced configuration interaction in self-assembled QD allows for a transition to contribute via multiple configurations. iv) The influence of interaction-induced dephasing and screening effects strongly depends on the excitation conditions. In the highly excited system, carriers are present in the delocalized states. Their interaction with the QD carriers leads to screening of the configuration interaction and to energy renormalization of the QD states (which is associated with carrier scattering processes). This results in changing configuration energies when moving from weak to strong excitation while in single atoms the configurations are fixed and scattering states are typically not important. A typical scheme of carrier scattering processes in a QD laser is shown in Fig. 14. The pump process excites electron-

623

hole-pairs in the delocalized states, either due to carrier injection in a p-n-junction or via optical pumping. Carrier capture into the QD states and carrier relaxation between the QD states rapidly populates the lowest confined states for electrons and holes. The corresponding carrier scattering processes are discussed in Sect. 6. The cavity mode is usually coupled to the lowest QD interband transition. In edge-emitting QD lasers a high-density of QDs is used in the active region to overcome the large cavity losses of these low-Q resonators. At the same time, the device efficiency is reduced by the interplay between inhomogeneous broadening due to variations of the laser transition energies for the individual QDs and the off-resonant QD-cavity mode coupling discussed above. For a single-QD laser the situation is even more demanding since the photon production has to be realized with a single emitter. This has two implications: i) A nanocavity with very low cavity losses and strongly reduced spontaneous emission into non-lasing modes is required. Then the weakly pumped system is usually in the strong-coupling regime. ii) Strong pumping is required to ensure a high photon emission rate from the QD that matches the cavity loss rate. Exciting a large carrier density in the delocalized states facilitates efficient carrier scattering into the laser levels and the Purcell effect is used to increase the photon emission rate. By stepwise approaching these conditions with continuously improved samples, stimulated emission which dominantly results from a single QD has been realized.

5.3. Multi-configuration QD model For a general semiconductor model, one starts from the single-particle states which can be obtained in effectivemass approximation or from atomistic models. Configuration interaction determines the multi-exciton states for the confined carriers. For a non-perturbative treatment of the light-matter-interaction acting on the multi-exciton configurations, the von-Neumann equation is formulated with the Jaynes-Cummings Hamiltonian HJC and the Coulomb Hamiltonian HCoul , ∂ρ ∂t

Figure 14 (online color at: www.lpr-journal.org) Quantum dot laser scheme. The cavity mode interacts with the lowest QD interband transition between levels 2 and 3. The pump process excites electrons and holes into the delocalized state. These carriers can be captured into the QD states. Depending on the level spacing, direct capture into the laser transitions (not shown) can additionally contribute. The same type of intraband processes also lead to a redistribution of carriers between the QD levels. In QDs with shallow confinement potential, excited QD states can be absent. The picture follows the model of [30].

www.lpr-journal.org

= i¯1h

[

HJC + HCoul ; ρ ] + L sc ρ :

(8)

This equation can be evaluated in terms of basis states resulting from the placement of electrons and holes in various QD states. To describe photons in the laser mode, the Fock-state basis is used. Considering a given number of confined QD states and assuming a sufficiently large but finite maximal number of possible photon excitations, the Hilbert space of the problem is finite and closed equations for the density matrix elements can be formulated. Of central importance in the semiconductor model are the dissipative processes summarized in Eq. (8) as the term L sc ρ. As in the single-atom laser model, cavity losses and spontaneous recombination of various levels due to coupling to nonlasing modes are included. New are excitationdependent scattering and dephasing processes as additional




Figure 15 System configurations corresponding to two electrons in the four confined QD states of Fig. 14. The top row shows the state occupation for electrons in the conduction and valence-band picture while the bottom row corresponds to electrons (filled circles) and holes (open circles) in the electron-hole picture. The ground states (0X ) is given by two valence electrons in the states j1 i , j2 i . Single excitons with electrons and holes in the lowest QD state (s-state) and excited QD state (p-state) are represented by 1Xs and 1X p , respectively. The sp-biexciton is labeled as 2Xsp . Selection rules typically identify 0Xs and 0X p as optically dark configurations. The picture is taken from [30].

contributions to L sc ρ. Their microscopic origin is discussed in the next Sect. 6. The simplest possible way to include their effect in the von-Neumann equation (8) is to assume Lindblad terms which couple the QD carriers to a bosonic bath of phonons as well as to a fermionic bath of carriers in the delocalized states. On the one hand, the macroscopic size of the host material that embeds the QD permits the bath approximation. On the other hand, the convenient and frequently used form of the Lindblad terms follows only in Born-Markov approximation. As a compromise for using this form, the scattering rates are fitted to independent calculations along the lines of Sect. 6. These calculations are performed beyond the Born and Markov approximation. A more consistent treatment remains a challenge for future investigations. The excitation of carriers into delocalized states also leads to screening of the Coulomb interaction. This is accounted for by using screened Coulomb-matrix elements for the formulation of HCoul . The results of the discussed semiconductor model clearly depend on the number of confined QD states, their confinement energies (which enter in the calculation of optical interband transition energies and intraband carrier scattering processes), and the confinement wave functions (which influence via dipole, Coulomb and phonon interaction matrix elements the strength of the interaction processes.) These input entities can be determined on various levels of refinement. Atomistic tight-binding calculations have been used to determine confinement energies, wave functions and interaction matrix elements for QD systems [185, 186]. As an alternative to this computationally expensive approach, that represents a research field on its own, frequently continuum methods based on effective mass models [187] or kp-Hamiltonians [188–190] have been applied. To illustrate the semiconductor model for a single QD laser, we assume a confinement situation with two localized states for electrons as well as holes which are only spindegenerate. These states are represented by the levels 1 to 4 in Fig. 14. The pump process generates electron-hole-pairs in the delocalized states. The capture of these carriers in the


QD states is described by an incoherent Lindblad scattering process. Furthermore, the scattering of carriers into the lowest QD levels is included by excitation-dependent Lindblad rates. The scattering rates are determined from independent microscopic calculations described in Chapter 6. For additional details see also [30]. Finally, to reduce the number of possible carrier configurations, optical pumping with circularly polarized light into the delocalized states has been considered. Since spin-flip processes can be neglected on the time-scale of (cavity-enhanced) recombination processes, then only one spin-subsystem is excited. The resulting six possible QD carrier configurations are shown in Fig. 15. In this case, s-exciton, p-exciton and s-p-biexciton are optically active. For the solution of the von-Neumann equation (8) we assume that the cavity is resonant with the s-exciton transition. At weak excitation the s-p-biexciton is detuned from the cavity mode due to the corresponding biexciton binding energy. The input/output curve for the system is displayed in Fig. 16a for an extended range of pump rates. For weak pumping, among the excited states only the s-exciton has an appreciable probability and, hence, provides the only contribution to the cavity emission. With increasing pump rate, the exciton contribution grows linearly while the biexciton contribution grows quadratically, as can be seen by the dashed and dotted lines in Fig. 16a. For large pump rates the contribution of the s-exciton is reduced since it becomes more likely to find more than one electron-hole-pair in the QD. Furthermore, output quenching for the s-exciton emission is observed due to dephasing since the pump-induced excitation of QD carriers can drive the system out of the s-exciton state into the next higher excitation states (here the s-p-biexciton). While for atomic systems this situation would result in output quenching, in the semiconductor system new emission channels can open for strong excitation. In the discussed example with a small number of possible configurations, it is the s-p-biexciton that takes over the emission process. As long as the exciton emission dominates the cavity output, the second-order photon correlation function exhibits photon-antibunching, as expected for a single emitter

www.lpr-journal.org


625

Figure 16 (online color at: www.lpr-journal.org) (a) Steady-state mean photon number hni vs. pump rate P (solid line) together with its contributions from the exciton (dashed line) and biexciton (dotted line) transitions according to Fig. 15. (b) Second-order photon correlation function g(2) (τ = 0) versus pump rate P. (c) Excitation-induced dephasing (dashed line) and reduction of the Coulomb X due to WL carrier screening (solid line). (d) Photon emission spectrum for selected pump rates. Lines are exchange energy Esp vertically displaced. For all curves the light-matter coupling constant g = 0:9 ps 1 , cavity losses at rate κ = 0:1 ps 1 and spontaneous emission rates γ23 = γ14 = 0:01 ps 1 are used. The picture is taken from [30].

system with non-harmonic optical transition energies, see Fig. 16b. In the regime of intermediate pumping, alternative photon emission channels (here the s-p-biexciton in addition to the s-exciton recombination) can independently contribute. This naturally explains the transition into pronounced photon bunching in Fig. 16b, which is also seen in the single QD experiments of [162]. The detuning of the biexciton to the cavity resonance is reduced for large excitation due to screening of the Coulomb-interaction. Furthermore, excitation-induced dephasing broadens the biexciton emission line, thus further increasing its coupling to the cavity mode. These effects cleary rely on carriers in the delocalized wetting-layer states. Note that the regime of stimulated emission is only reached when the photon production rate is able to compensate the cavity loss rate. This implies strong pumping (with an appreaciable wetting layer carrier density) and ultrafast QD carrier relaxation (providing the optical dephasing). The reduction of the Coulomb exchange energy due to screening

www.lpr-journal.org

of wetting-layer carriers and the increasing dephasing rate are shown in Fig. 16c for the above-discussed example. The small cavity losses necessary to realize stimulated emission with a single QD emitter together with the small dephasing of the single emitter for weak pumping lead to the vacuum Rabi splitting of the degenerate exciton and cavity lines for small pump rates in Fig. 16d. Interaction-induced dephasing drives the transition into the weak-coupling regime. This transition does not necessarily coincide with the transition into stimulated emission, as demonstrated in the experimental observation of lasing in the strong-coupling regime [17]. For the small number of carrier configurations used in the above example and a cavity loss rate according to current experiments, stimulated emission is only reached with a light-matter coupling strength of the QD interband transitions g, that leads to a vacuum Rabi splitting exceeding the experimental observations by a factor of 9. Increasing the number of configurations by considering more shells for electrons and holes allows to



reduce g to reach stimulated emission. In the end, the true number of configurations to be considered is determined by the experimental QD realization.

5.4. Estimation of β -factor in nanocavities As described in the previous section, the emission properties of a single QD laser do not only depend on contributions from the localized QD states. Semiconductor specific effects like the dephasing and screening due to excited carriers can influence and even dominate the emission properties. Different slopes in the input-output curve can originate from the interplay of excitonic emission (for weak pumping), its saturation, and the increasing contributions of multi-exciton states for strong pumping. Hence, fitting the input/output curve with atomic (multi-emitter) rate equations (like the range of pump rates from 10 1 to 102 /ps in Fig. 16a) leaves out important aspects of the underlying physics. In this example β is close to unity. The estimation of the β -factor for nanolasers with several QD emitters has also its limitations as demonstrated in [51]. A possible deviation from full inversion of the laser transition and a modified source term of spontaneous emission in comparison to atomic systems [191] leads to deviations in the magnitude of the jump of the output intensity in the threshold region, in comparison to rate equation predictions. Clearly it is more reliable to estimate β from mode calculations in accordance with its original introduction as a property of the laser resonator.

6. Carrier scattering processes in QDs Carrier scattering processes play an important role in the physics underlying QD emitters. They are responsible for the capture of carriers from the delocalized into the localized QD states as well as for the carrier relaxation between the localized QD states, as shown schematically in Fig. 14. Fast supply of carriers in the laser levels is an important condition for the device efficiency. On the other hand, these scattering processes lead to dephasing in the optical processes. In this section, we review a microscopic description of carrier scattering processes, which has been used for the calculation of excitation dependent scattering and dephasing rates in the previous Sect. 5.3. The two important interactions contributing are the coupling of the QD states to lattice vibrations via the carrierphonon interaction and the carrier-carrier Coulomb interaction. On the one hand, these interaction effects trace back to the same origin as in any semiconductor structure, represented e.g. by second-quantization Hamiltonians with electron and hole operators. On the other hand, the discrete nature of the QD states influences the appearance of these interaction effects and requires modifications in their theoretical description. For example, the interaction of QD carriers with LO-phonons requires a non-perturbative treatment [192–196] while for the interaction with carriers in delocalized states a perturbative treatment usually suffices.



Also the pure dephasing due to LA-phonons results in a particular lineshape of the QD transition that reflects higherorder multi-phonon processes [101, 197]. Regarding the Coulomb interaction, it becomes possible to account for the full configuration interaction for a finite number of discrete QD states. This can be accomplished by numerical diagonalization of the free-carrier and Coulomb Hamiltonian in a basis of the QD carrier configurations [198–202] or by including the Coulomb Hamiltonian in the direct solution of the von-Neumann equation [30]. Such a treatment represents a situation in which the excited electrons and holes are restricted to the localized QD states, e.g., for weak excitation and at low temperatures. QD lasers driven into stimulated emission represent a different situation. In the high-excitation regime a substantial density of excited carriers remains in the delocalized states. These carriers can act as scattering partners for carrier relaxation inside the QD and for the capture of carriers from the delocalized into the QD states. Extensive theoretical studies have documented the efficiency of these processes [203–207]. The scattering processes are naturally linked to energy renormalizations as well as to dephasing of optical processes coupled to the respective states. Furthermore, excited carriers in delocalized states can efficiently screen the Coulomb interaction. This weakens the configuration interaction of QD carriers and leads to excitation-dependent system configurations as discussed in Sect. 5. In the following Sect. 6.1 we review the description of carrier-carrier Coulomb scattering under the premise that at least one of the involved carriers has its initial and/or final state in the delocalized spectrum. This extends the part of the Coulomb interaction solely between the QD carriers, as treated in configuration interaction calculations, and accounts for carrier capture, relaxation and dephasing processes. In Sect. 6.2 carrier scattering due to non-perturbative interaction with LO-phonons is outlined.

6.1. Carrier-carrier interaction Kinetic equations can be employed to analyze the redistribution of the carrier occupation probability fν (t ) for the states ν. In the simplest form, two-particle collisions are described with Boltzmann scattering rates. Then the temporal change of the carrier population fν (t ) due to scattering processes into and out of the states ν follow from ∂ fν ∂t

= (1

fν ) Sνin

fν Sνout :

(9)

with scattering rates Sνin,out . To obtain this form, a Markov approximation has been used, where one assumes that the population changes at a given time t depend explicitly only on the population functions fν (t ) at the same time. The scattering probabilities are proportional to the population f of the initial states and to the non-occupation 1 f of the final states. For the Coulomb interaction, where two carriers are scattered from the states ν1 and ν3 into ν and ν2 and vice

www.lpr-journal.org


627

versa, the scattering integral Sνin is given by Sνin =

2π h¯

fν

1(

∑

ν1 ;ν2 ;ν3

1

Wνν2 ν3 ν1 [Wνν 2 ν3 ν1

fν2 ) fν3 δ (e εν

Wνν ] 2 ν1 ν3

eεν1 + eεν2

eεν3 ):

(10)

A similar expression for the out-scattering contribution Sνout is obtained by replacing f ! 1 f . The population factors account for the availability of scattering partners as discussed above. The sum involves all available initial states for in-scattering processes as well as all possible scattering partners, that can be provided in QD or delocalized states. In Markov approximation, the delta function accounts for strict energy conservation of the scattering processes. This approximation is valid in the long-time limit, where scattering events are considered for times that are much larger than the inverse scattering rates. Even in this case, it is necessary to account for renormalizations of the singleparticle energies e εν by the Coulomb interaction instead of free-carrier energies εν as done in the perturbation theory. The scattering cross-section is also determined by the matrix elements of the screened Coulomb interaction Wνν2 ν3 ν1 . The first and second term in the square brackets correspond to direct and exchange Coulomb scattering, respectively. A treatment beyond the Markov approximation is possible within a quantum-kinetic description [208]. Then the delta-function is replaced by the spectral functions of the involved carriers, and time-integrals over the past of the system account for memory effects. The most simple form of a quantum-kinetic equation requires two further approximations. i) Within the generalized Kadanoff-Baym ansatz (GKBA) closed equations for single-time population functions are obtained; see [208] for detailed discussions. ii) The quasi-particle properties of the involved carriers are described by renormalized energies and damping (finite lifetime of the quasi-particle state) that can be summarized in complex energies b εν . The latter approximation is related to the Fermi-liquid theory and valid for large carrier densities, where excitonic effects can be neglected. The quantumkinetic equation with the discussed approximations has the form ∂ fν (t ) ∂t

Zt =

dt 0

∞

n [

1

fν (t 0 )] Sνin (t ; t 0 )

fν (t 0 ) Sνout (t ; t 0 )

;

(11)

where the in-scattering rate, that now depends on two time arguments, is given by Sνin (t ; t 0 ) =

1 Re ∑ h¯ 2 ν1 ;ν2 ;ν3

Wνν ν ν t fν t 0 1

0 (t ) Wνν 2 ν3 ν1

2 3 1( ) [

1 ( )[

fν2 (t 0 )] fν3 (t 0 )e

(12) 0 Wνν (t )] 2 ν1 ν3 i b h¯ (εν

bεν1 +bεν2 bεν3 )(t t 0 )

:

The out-scattering rate is again obtained from f ! 1 f . It turns out, that especially for the calculation of dephasing

www.lpr-journal.org

and its influence on optical spectra, the quantum-kinetic description leads to more realistic results. As discussed in [102], for the description of dephasing in optical spectra, equations similar to (11) and (12) are used for the interbandtransition dynamics. The determination of the Coulomb interaction matrix elements, that enter in equations (10) and (12), involves two steps. The bare (unscreened) interaction matrix elements can be calculated from the single-particle wave-functions of the corresponding electronic states. A second step involves the inclusion of screening effects due to excited carriers. From the nature of screening as well as from the excitation conditions at elevated temperatures (considered for optoelectronic applications) it appears natural to account only for screening due to the carriers in the delocalized WL states. A consistent theory of screening for the QD-WL system is developed in [207]. The renormalized single-particle energies, that enter in the scattering rates (10) and (12), can be obtained from the free-carrier energies by adding Hartree, exchange, and correlation selfenergy contributions. The Hartree selfenergy involves the electrostatic interaction with all other excited electrons and holes. Only in a system with local charge neutrality, the Hartree terms cancel. In QDs, local charging appears as a result of different electron and hole populations, different envelope wave functions for electrons and holes, as well as due to the charge separation of electrons and holes in a built-in electrostatic field [207].

6.2. Carrier-phonon interaction A prerequisite for efficient carrier-carrier scattering is the availability of scattering partners. Furthermore, Coulomb scattering is unable to dissipate kinetic energy from the carrier system. Note that the carrier-carrier scattering leads to an evolution of the carrier system towards a quasiequilibrium state for electrons and holes. However, it conserves the average kinetic energy of the carriers. Pumping of carriers with above-average kinetic energy and the preferred recombination of carriers in the highly populated states with below-average kinetic energy effectively heats up the carrier system [209]. This excess kinetic energy can only be dissipated to the crystal lattice via the emission of phonons. A thermalization of the carriers with respect to the crystal lattice is essential for efficient laser operation, since the increasing carrier temperature diminishes the optical gain via a reduction of the population inversion. The early work on carrier-phonon interaction in semiconductor QDs has been based on the application of timedependent perturbation theory. This method has previously been applied to bulk semiconductors or quantum wells. In these cases, its success is based on the application to carriers in continuum states. For the discussion of carrier-phonon interaction involving localized QD states with discrete energies, one has to consider the fact that the transition matrix elements provide only efficient coupling to phonons with small momenta [192, 210]. The strongest contribution to the



interaction of carriers with lattice vibrations is typically provided by LO-phonons due to the Coulomb-like coupling of carriers to lattice displacements in polar crystals. The nearly constant LO-phonon energy, at least for the considered small phonon momenta, and the energy-conservation requirement from Fermi’s golden rule led to the prediction of inefficient carrier transitions between the QD states, except for the unlikely coincidence of transition energies between the QD states with the LO-phonon energy. Deformation-potential coupling with LA-phonons determines the low-density and low-temperature QD emission lineshape via pure dephasing. The interaction with LA-phonons is much less important for the redistribution of carriers. For small phonon momenta, their energy is too small to facilitate efficient carrier scattering processes between the QD states. On these ground, particularly due to the application of perturbation theory, inefficient carrier-phonon scattering processes in QD systems have been predicted [211]. To prevent this so-called phonon bottleneck, only much less efficient higher-order processes, like those involving combinations of LA- and LO-phonons [212,213] or the contributions of carrier-carrier scattering processes at elevated carrier densities have been proposed. Early investigations of QD-polarons have been based on the “random phase approximation” (RPA) [192, 193] or on the direct diagonalization of a restricted state space [195], where only localized states are considered. For a single discrete electronic state, an analytic treatment of the interaction with LO-phonons is possible within the “independent boson model” [214]. The numerical extension to several discrete electronic states is discussed in [194]. However, in this work the influence of delocalized WL states with a quasi-continuum of energies, that plays a key role for carrier generation in optoelectronic devices, has been neglected. When investigations are restricted to localized states with discrete energies, RPA results deviate from a more accurate direct diagonalization of a restricted system. On the other hand, the inclusion of WL states provides a natural source of broadening of the electronic states, that justifies the application of the RPA. The next step in the analysis of carrier-phonon scattering in QD systems was the development of a theory that accounts for the scattering of renormalized quasi-particles (QD-polarons). This became possible within a quantumkinetic theory that has been applied to the problem of carrier capture and relaxation [196]. Such a quantum-kinetic description naturally incorporates both, non-Markovian effects on ultrafast timescales and a non-perturbative treatment of the carrier-phonon interaction in the polaron picture. Polaronic renormalization effects have been found to be the main origin for fast carrier capture and relaxation processes, as well as efficient carrier thermalization on a picosecond timescale. The results strongly deviate from a perturbative treatment using Fermi’s golden rule. In the following, the retarded Green’s function Grα (τ ) with the time τ and the state index α is used to describe the quasi-particle renormalizations of QD electrons and holes in the single-particle states α due to the non-perturbative interaction with LO-phonons. In the absence of interaction, this



function oscillates with the free-carrier energy εα according to i i (13) Grα (τ ) = Θ(τ ) e h¯ εα τ h¯ and the Fourier transform 1 (14) Grα (ω ) = h¯ ω εα + iη with the frequency ω has a pole at the free-carrier energy with η > 0, η 0. Quasi-particle renormalizations are obtained from the solution of the Dyson equation

!

∂ εα Grα (τ ) = δ (τ ) i¯h ∂τ Zτ + dτ 0 Grα (τ 0 ) ∑ Grβ (τ τ 0 ) D> αβ (τ 0

τ 0) ;

(15)

β

where the sum involves all available states β . The Dyson equation contains a self-energy for the many-body description of interaction processes. In Eq. (15) the so-called random-phase approximation has been used to formulate the interaction term with the phonon propagator

j

j

q) 2 D? αβ (τ ) = ∑ Mαβ (~

~q

n

LO

eiωLO τ + (1 + nLO ) eiωLO τ

(16)

:

Here nLO is a Bose-Einstein function for the population of the phonon modes (assumed to act as a bath in thermal equilibrium), ωLO is the LO-phonon frequency, and the sum over the three-dimensional phonon wave vector ~q involves all phonon modes. For an introduction into the many-body theory in general and Green’s functions in particular, we refer the reader to [214, 215]. The Dyson equation (15) takes into account all possible virtual transitions due to carrier-phonon interaction for a carrier in the state α. The single-particle energies for both localized and delocalized states are considered and the corresponding wave functions enter via the interaction matrix elements Mαβ . The Fourier transform of the retarded Green’s funcbα (ω ) = tion directly provides the spectral function G r 2 Im Gα (ω ) that reflects the density of states (DOS) for a carrier in the state α. Only for non-interacting carriers, the spectral function contains a delta-function at the freeparticle energies, expressing that each state is associated with a single energy. This picture changes due to the quasiparticle renormalizations. The dynamics of the carrier population due to interaction with LO-phonons can be described with a quantum-kinetic equation [196, 215] ∂ fα (t ) = 2Re ∑ ∂t β

n

Z

t ∞

fα (t 0 ) 1

1

dt 0 Grβ (t

t 0 ) Grα (t

0 fβ (t 0 ) D< αβ (t

t 0)

t)

0 fα (t 0 ) fβ (t 0 ) D> αβ (t

o t)

:

(17)

www.lpr-journal.org


In comparison to Boltzmann scattering rates, the deltafunctions are replaced by convolutions of the phonon propagator with retarded Green’s functions. The latter account for the polaronic renormalizations of the initial and final carrier states. Furthermore, memory effects are included by the time dependence of the population factors on the past evolution of the system. Boltzmann scattering integrals follow as a limiting case, when the Markov approximation is applied (population functions fα (t 0 ) are taken at the external time t) and when quasiparticle renormalizations are neglected with the use of freecarrier retarded Green’s functions (13). Examples for ultrafast QD carrier scattering processes in the polaron picture are given in [196] Conditions for reaching a thermal quasi-equilibrium in the carrier system are discussed in [216]. For a recent comparison with experimental results, see [217, 218].

7. Conclusions and future directions Recent pioneering experiments have shown that a single QD emitter can drive a nanocavity system into stimulated emission. However, as the in-depth analysis of the gain mechanism and new theoretical developments have shown, the system strongly deviates from its atomic counterpart. Self-organized QDs are far from being isolated systems. Interactions with lattice vibrations and carriers in the host material do not only provide scattering and dephasing channels. The availability of multiple electronic configurations for photon emission processes reveals a richer physics than an atom-like three or four level system. Further reduced cavity losses and controlled positioning of individual QDs will provide cleaner experimental results, e.g., to reach deeper into the strong coupling regime. Future research is also expected to aim on achieving more atomiclike systems, for example, by using QDs with stronger confinement or by finding ways to grow QDs without wetting layers, such as top-down fabricated dots which are known to emit brightly [219]. Another goal is to deterministically couple several QDs to the mode. The latter might be challenging with the GaAs platform, but feasible with the ultrahigh quality factors of silicon microcavities. Quantum dot cavity QED research remains both challenging and exciting and future progress will ultimately enable the processing of quantum information in new and more powerful ways. Acknowledgements. Frank Jahnke acknowledges financial support through the German Science Foundation (DFG) and a grant for CPU time at the Forschungszentrum Juelich. He thanks his collaborators Weng Chow, Manfred Bayer, Detlef Hommel, Peter Michler and his co-workers Paul Gartner, Christopher Gies, and Jan Wiersig for the joint work on many interesting projects. Stefan Strauf acknowledges partial financial support through the National Science Foundation (NSF). He thanks his collaborators Dirk Bouwmeester, Pierre Petroff, Matthew Rakher, Evelyn Hu, Larry Coldren, Kevin Hennessy, Yong-Seok Choi, Antonio Badolato, and Lucio Claudio Andreani, for the joint work on nanolasers and the many interesting and stimulating discussions. The authors

www.lpr-journal.org

629

thank Hyatt Gibbs, Peter Michler, and Glenn Solomon for a critical reading of the manuscript. Received: 2 December 2010, Revised: 19 January 2011, Accepted: 8 February 2011 Published online: 14 March 2011 Key words: Cavity-QED, photonic crystals, quantum dots, lasing, Jaynes-Cummings interaction, photon statistics, strong coupling.

Stefan Strauf received his Ph.D. degree from Bremen University (Germany) in 2001. During his postdoc at UC Santa Barbara from 2003-2006 he demonstrated single photon sources and photonic crystal quantum dot nanolasers with world record performance. In 2006 Dr. Strauf joined Stevens Institute of Technology as an Assistant Professor of Physics and Engineering Physics. His current research focuses on quantum photonics and optoelectronics with quantum dots, carbon nanotubes, and graphene. Frank Jahnke received his Ph.D. from the University of Rostock (Germany) in 1990. He was postdoctoral research associate at the Forschungszentrum (Germany) and at the Optical Julich ¨ Sciences Center, University of Arizona (USA). Frank Jahnke obtained his Habilitation at the University of Marburg (Germany) in 1997. In 2000 he joined the University of Bremen (Germany) as a full professor for Theoretical Physics. His research activities involve quantum optical effects and many-body interactions in semiconductor nanostructures.

References [1] K. Vahala, Nature 424, 839–845 (2003). [2] K. Vahala (ed.), Optical microcavities, Advanced Series in Applied Physics, Vol. 5 (World Scientific, New Jersey, 2005). [3] K. Iga, F. Koyanama, and S. Kinoshita, IEEE J. Quantum Electron. 24, 1845 (1988). [4] J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, and L. T. Florez, IEEE J. Quantum Electron. 27, 1332 (1991). [5] P. L. Gourley, Nature 371, 571 (1994). [6] T. Gutbrod, M. Bayer, A. Forchel, J. Reithmaier, T. Reinecke, S. Rudin, and P. Knipp, Phys. Rev. B 57(16), 9950– 9956 (1998). [7] S. McCall, A. Levi, R. Slusher, S. Pearton, and R. Logan, Appl. Phys. Lett. 60(3), 289–291 (1992). [8] V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, Phys. Rev. A 54(3), R1777– R1780 (1996). [9] T. F. Krauss, R. M. D. L. Rue, and S. Brand, Nature 383, 699–702 (1996).



[10] E. Yablonovitch, Phys. Rev. Lett. 58(20), 2059–2062 (1987). [11] S. John, Phys. Rev. Lett. 58(23), 2486–2489 (1987). [12] O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, Science 284(June), 1819 (1999). [13] S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, Science 293(5532), 1123–1125 (2001). [14] T. Yoshie, O. Shchekin, H. Chen, D. Deppe, and A. Scherer, Electron. Lett. 38(17), 967 – 968 (2002). [15] J. Hendrickson, B. C. Richards, J. Sweet, S. Mosor, C. Christenson, D. Lam, G. Khitrova, H. M. Gibbs, T. Yoshie, A. Scherer, O. B. Shchekin, and D. G. Deppe, Phys. Rev. B 72(19), 193303 (2005). [16] S. Strauf, K. Hennessy, M. T. Rakher, Y. S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, Phys. Rev. Lett. 96(12), 127404 (2006). [17] M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, Nature Phys. 6(4), 279–283 (2010). [18] K. Aoki, D. Guimard, M. Nishioka, M. Nomura, S. Iwamoto, and Y. Arakawa, Nature Photon. 2(11), 688–692 (2008). [19] A. Tandaechanurat, S. Ishida, K. Aoki, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, Appl. Phys. Lett. 94(17), 171115 (2009). [20] A. Tandaechanurat, S. Ishida, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, Nature Photon. 5, 91–94 (2011). [21] S. Strauf, Nature Photon. 5, 72–74 (2011). [22] D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (John Wiley and Sons, Chichester, 1999). [23] D. Meschede and H. Walter, G. Müller, Phys. Rev. Lett. 54, 551 (1985). [24] K. An, J. J. Childs, R. R. Dasari, and M. S. Feld, Phys. Rev. Lett. 73, 3375 (1994). [25] S. Noda, Science 314(5797), 260–261 (2006). [26] H. Altug, D. Englund, and J. Vuckovic, Nature Phys. 2(7), 484–488 (2006). [27] D. Englund, H. Altug, B. Ellis, and J. Vuckovic, Lase. and Photon. Rev. 2, 264–274 (2008). [28] P. Meystre and M. Sargent III, Elements of Quantum Optics, 2nd edition (Springer, New York, 1991). [29] Y. Mu and C. M. Savage, Phys. Rev. A 46, 5944 (1992). [30] S. Ritter, P. Gartner, C. Gies, and F. Jahnke, Opt. Express 18(10), 9909–9921 (2010). [31] M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, Nature 460(7259), 1110–U68 (2009). [32] E. Purcell, H. Torrey, and R. Pound, Phys. Rev. 69, 37 (1946). [33] S. Haroche and D. Kleppner, Phys. Today 42, 24 (1989). [34] H. Yokoyama, K. Nishi, T. Anan, Y. Nambu, S. D. Brorson, E. P. Ippen, and M. Suzuki, Opt. Quantum Electron. 24, S245–S272 (1992), 10.1007/BF00625827. [35] A. Thränhardt, C. Ell, G. Khitrova, and H. M. Gibbs, Phys. Rev. B 65(3), 035327 (2002). [36] A. Koenderink, Opt. Lett. 35, 4208–4210 (2010). [37] P. Goy, J. M. Raimond, M. Gross, and S. Haroche, Phys. Rev. Lett. 50, 1903–1906 (1983). [38] R. Hulet, E. Hilfer, and D. Kleppner, Phys. Rev. Lett. 55, 2137 (1985). [39] D. Meschede, H. Walther, and M. G., Phys. Rev. Lett. 54, 551–554 (1985).



[40] R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. Rev. Lett. 68, 1132 (1992). [41] M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 76(11), 1800–1803 (1996). [42] H. Yokoyama, K. Nishi, T. Anan, H. Yamada, S. D. Brorson, and E. P. Ippen, Appl. Phys. Lett. 57(26), 2814–2816 (1990). [43] L. A. Graham, D. L. Huffaker, and D. G. Deppe, Appl. Phys. Lett. 74(17), 2408–2410 (1999). [44] C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, Phys. Rev. Lett. 69, 3314 (1992). [45] G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999). [46] J. P. Reithmaier, G. Sek, A. Loffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, Nature 432(7014), 197–200 (2004). [47] T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, Nature 432, 200–203 (2004). [48] H. Yokoyama and S. D. Brorson, J. Appl. Phys. 66, 4801 (1989). [49] G. Björk and Y. Yamamoto, IEEE J. Quantum Electron. 27, 2386–2396 (1991). [50] P. R. Rice and H. J. Carmichael, Phys. Rev. A 50, 4318 (1994). [51] C. Gies, J. Wiersig, M. Lorke, and F. Jahnke, Phys. Rev. A 75, 013803 (2007). [52] S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, and C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, Phys. Rev. Lett. 98, 043906 (2007). [53] Y. S. Choi, M. T. Rakher, K. Hennessy, S. Strauf, A. Badolato, P. M. Petroff, D. Bouwmeester, and E. L. Hu, Appl. Phys. Lett. 91(3), 031108 (2007). [54] S. Ates, C. Gies, S. M. Ulrich, J. Wiersig, and S. Reitzenstein, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, Phys. Rev. B 78, 155319 (2008). [55] U. Mohideen, R. Slusher, F. Jahnke, and S. Koch, Phys. Rev. Lett. 73, 1785 (1994). [56] O. Painter, J. Vuckovic, and A. Scherer, J. Opt. Soc. Am. B, Opt. Phys. 16(2), 275–285 (1999). [57] P. Lalanne, S. Sauvan, and J. Hugonin, Laser Photon. Rev. 6, 514–526 (2008). [58] L. Andreani, D. Gerace, and M. Agio, Photon. Nanostruct. Fund. Appl. 3, 120–128 (2005). [59] K. Srinivasan and O. Painter, Optics Express 10(15), 670– 684 (2002). [60] K. Srinivasan and O. Painter, Optics Express 11, 579 (2003). [61] D. Englund, I. Fushman, and J. Vuckovic, Optics Express 13(16), 5961–5975 (2005). [62] H. Benisty, Nature Phys. 1, 9 (2005). [63] Y. Xu, R. Lee, and A. Yariv, Phys. Rev. A 61, 033808 (2000). [64] H. Y. Ryu, J. K. Hwang, and Y. H. Lee, Ieee J. Quantum Electron. 39(2), 314–322 (2003). [65] G. Kim, Y. H. Lee, A. Shinya, and M. Notomi, Opt. Express 12, 6624–6631 (2004). [66] P. Yao, V. S. Manga Rao, and S. Hughes, Laser Photon. Rev. 4, 499–516 (2010). [67] J. Vuckovic, O. Painter, Y. Xu, A. Yariv, and A. Scherer, IEEE J. Quantum Elect. 35, 1168–1175 (1999).

www.lpr-journal.org


[68] K. Hennessy, A. Badolato, A. Tamboli, P. M. Petroff, E. Hu, M. Atature, J. Dreiser, and A. Imamoglu, Appl. Phys. Lett. 87(2), 021108 (2005). [69] S. Strauf, M. T. Rakher, I. Carmeli, K. Hennessy, C. Meier, A. Badolato, M. J. A. DeDood, P. M. Petroff, E. L. Hu, E. G. Gwinn, and D. Bouwmeester, Appl. Phys. Lett. 88(4), 043116 (2006). [70] K. Hennessy, C. Reese, A. Badolato, C. F. Wang, A. Imamoglu, P. M. Petroff, E. Hu, G. Jin, S. Shi, and D. W. Prather, Appl. Phys. Lett. 83(18), 3650–3652 (2003). [71] R. Han-Youl, H. Jeong-Ki, and Y. H. Lee, IEEE J. Quantum Elect. 39, 314–22 (2003). [72] C. F. Wang, A. Badolato, I. Wilson-Rae, P. M. Petroff, E. Hu, J. Urayama, and A. Imamoglu, Appl. Phys. Lett. 85(16), 3423–3425 (2004). [73] H. Y. Ryu, S. H. Kim, H. G. Park, J. K. Hwang, Y. H. Lee, and J. S. Kim, Appl. Phys. Lett. 80(21), 3883–3885 (2002). [74] H. Y. Ryu, S. H. Kim, H. G. Park, and Y. H. Lee, J. Appl. Phys. 93(2), 831–837 (2003). [75] H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, Science 305(5689), 1444–1447 (2004). [76] Y. Akahane, T. Asano, B. S. Song, and S. Noda, Nature 425(6961), 944–947 (2003). [77] Y. Akahane, T. Asano, B. S. Song, and S. Noda, Appl. Phys. Lett. 83, 1512–1514 (2003). [78] M. Toishi, D. Englund, A. Faraon, and J. Vuckovic, Opt. Express 17, 14618–14626 (2009). [79] N. V. Q. Tran, S. Combrie, and A. De Rossi, Phys. Rev. B 79, 041101(R) (2009). [80] S. L. Portalupi, M. Galli, C. Reardon, T. F. Krauss, L. O’Faolain, L. C. Andreani, and D. Gerace, Optics Express 18(15), 16064–16073 (2010). [81] T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, Appl. Phys. Lett. 79(26), 4289–4291 (2001). [82] M. Borselli, T. J. Johnson, and O. Painter, Optics Express 13(5), 1515–1530 (2005). [83] C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, Appl. Phys. Lett. 90(5), 051108 (2007). [84] B. S. Song, S. Noda, T. Asano, and Y. Akahane, Nature Materials 4(3), 207–210 (2005). [85] E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, Appl. Phys. Lett. 88(4), 041112 (2006). [86] E. Weidner, S. Combrie, N. V. Q. Tran, A. De Rossi, J. Nagle, S. Cassette, A. Talneau, and H. Benisty, Appl. Phys. Lett. 89, 221104 (2006). [87] S. Combrie, A. De Rossi, Q. V. Tran, and H. Benisty, Opt. Lett. 33, 1908 (2008). [88] H. Kroemer, Rev. Mod. Phys. 73, 783–793 (2001). [89] S. H. Kwon, T. Snner, M. Kamp, and A. Forchel, Optics Express 16, 4605 (2008). [90] T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, Nature Photon. 1(1), 49–52 (2007). [91] T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, Appl. Phys. Lett. 87(15), 151112 (2005). [92] T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, Appl. Phys. Lett. 90(3), 031115 (2007). [93] K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, Nature Photon. 4(7), 477–483 (2010).

www.lpr-journal.org

631

[94] D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Nature 421(6926), 925–928 (2003). [95] P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450(7173), 1214–1217 (2007). [96] T. J. Kippenberg and K. J. Vahala, Science 321(5893), 1172– 1176 (2008). [97] A. Schliesser and T. J. Kippenberg, Advances In Atomic, Molecular, and Optical Phys., Vol 58 58, 207–323 (2010). [98] P. Petroff, A. Lorke, and A. Imamoglu, Phys. Today 54, 46 (2001). [99] P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, Phys. Rev. Lett. 87(15), 157401 (2001). [100] L. Besombes, K. Kheng, L. Marsal, and H. Mariette, Phys. Rev. B 63(15), 155307 (2001). [101] E. A. Muljarov and R. Zimmermann, Phys. Rev. Lett. 93(23), 237401 (2004). [102] M. Lorke, T. R. Nielsen, J. Seebeck, P. Gartner, and F. Jahnke, Phys. Rev. B 73, 085324 (2006). [103] B. Gayral and J. Gerard, Phys. Rev. Lett. 90, 229701 (2003). [104] P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu, and A. Imamoglu, Science 290(5500), 2282–+ (2000). [105] C. Santori, M. Pelton, G. Solomon, Y. Dale, and E. Yamamoto, Phys. Rev. Lett. 86(8), 1502–1505 (2001). [106] M. Pelton, C. Santori, J. Vuckovic, B. Y. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, Phys. Rev. Lett. 89(23), 233602 (2002). [107] D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, Phys. Rev. Lett. 95(1), 013904 (2005). [108] D. Gevaux, A. J. Bennett, R. M. Stevenson, A. Shields, P. Atkinson, J. Griffiths, G. A. C. Anderson, D. Jones, and D. A. Ritchie, Appl. Phys. Lett. 88, 131101 (2006). [109] S. Strauf, Nature Photon. 4(3), 132–134 (2010). [110] P. Michler (ed.), Single Semiconductor Quantum Dots, Nanoscience and Nanotechnology (Springer, Berlin, Heidelberg, 2009). [111] E. Peter, P. Snellart, D. Martrou, A. Lemaitre, J. Hours, J. Gerard, and J. Bloch, Phys. Rev. Lett. 95, 067401 (2005). [112] G. Khitrova, H. Gibbs, M. Kira, S. Koch, and A. Scherer, Nat. Phys. 2, 81–90 (2006). [113] C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044—-2046 (1987). [114] E. Knill, R. Laflamme, and G. J. Milburn, Nature 409, 46– 52 (2001). [115] C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, Nature 419(6907), 594–597 (2002). [116] S. Mosor, J. Hendrickson, B. C. Richards, J. Sweet, G. Khitrova, H. M. Gibbs, T. Yoshie, A. Scherer, O. B. Shchekin, and D. G. Deppe, Appl. Phys. Lett. 87, 141105 (2005). [117] H. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39(11), 691–695 (1977). [118] T. Basché, W. E. Moerner, M. Orrit, and H. Talon, Phys. Rev. Lett. 69, 1516–1519 (1992). [119] P. Michler, A. Imamolu, M. Mason, P. J. Carson, G. F. Strouse, and S. K. Buratto, Nature 406, 968–970 (2000). [120] C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter, Phys. Rev. Lett. 85(2), 290–293 (2000).



[121] S. Strauf, P. Michler, M. Klude, D. Hommel, G. Bacher, and A. Forchel, Phys. Rev. Lett. 89(17), 177403 (2002). [122] A. Hogele, C. Galland, M. Winger, and A. Imamoglu, Phys. Rev. Lett. 100(21), 217401 (2008). [123] T. D. Happ, I. I. Tartakovskii, V. D. Kulakovskii, J. P. Reithmaier, M. Kamp, and A. Forchel, Phys. Rev. B 66(4), 041303 (2002). [124] M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, Science 308(5726), 1296–1298 (2005). [125] S. Noda, M. Fujita, and T. Asano, Nature Photon. 1(8), 449–458 (2007). [126] A. Badolato, K. Hennessy, M. Atature, J. Dreiser, E. Hu, P. M. Petroff, and A. Imamoglu, Science 308(5725), 1158– 1161 (2005). [127] Y. Ota, M. Nomura, N. Kumagai, K. Watanabe, S. Ishida, S. Iwamoto, and Y. Arakawa, Appl. Phys. Lett. 93(18), 183114 (2008). [128] T. Kuroda, N. Ikeda, T. Mano, Y. Sugimoto, T. Ochiai, K. Kuroda, S. Ohkouchi, N. Koguchi, K. Sakoda, and K. Asakawa, Appl. Phys. Lett. 93(11), 111103 (2008). [129] A. Laucht, F. Hofbauer, N. Hauke, J. Angele, S. Stobbe, M. Kaniber, G. Bohm, P. Lodahl, M. C. Amann, and J. J. Finley, New J. Phys. 11(February), 023034 (2009). [130] Q. H. Xie, A. Madhukar, P. Chen, and N. P. Kobayashi, Phys. Rev. Lett. 75, 2542–2545 (1995). [131] K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature, S. Gulde, S. Falt, E. L. Hu, and A. Imamoglu, Nature 445(7130), 896–899 (2007). [132] A. Gonzalez-Tudela, E. del Valle, E. Cancellieri, C. Tejedor, D. Sanvitto, and F. P. Laussy, Optics Express 18(7), 7002– 7009 (2010). [133] S. Thon, M. Rakher, H. Kim, J. Gudat, W. Irvine, P. Petroff, and D. Bouwmeester, Appl. Phys. Lett. 94, 111115 (2009). [134] A. Dousse, L. Lanco, J. Suffczynski, E. Semenova, A. Miard, A. Lemaitre, I. Sagnes, C. Roblin, J. Bloch, and P. Senellar, Phys. Rev. Lett. 1010, 267404 (2008). [135] J. Suffczynski, A. Dousse, K. Gauthron, A. Lemaitre, I. Sagnes, L. Lanco, J. Bloch, P. Voisin, and P. Senellart, Phys. Rev. Lett. 103(2), 027401 (2009). [136] A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, Phys. Rev. Lett. 83(20), 4204–4207 (1999). [137] T. Mano, R. Notzel, G. J. Hamhuis, T. J. Eijkemans, and J. H. Wolter, Appl. Phys. Lett. 81, 1705–1707 (2002). [138] B. D. Gerardot, G. Subramanian, S. Minvielle, H. Lee, J. A. Johnson, W. V. Schoenfeld, D. Pine, J. S. Speck, and P. M. Petroff, J. Crystal Growth 236(4), 647–654 (2002). Mehta, D. Reuter, A. D. Wieck, [139] M. S. Michaelis de Vasconcellos, A. Zrenner, and C. Meier, Appl. Phys. Lett. 97, 143101 (2010). [140] C. Schneider, A. Huggenberger, T. Sunner, T. Heindel, M. Strauss, S. Gopfert, P. Weinmann, S. Reitzenstein, L. Worschech, M. Kamp, S. Hofling, and A. Forchel, Nanotechnology 20(43), 434012 (2009). [141] D. Dalacu, M. Reimer, S. Frderick, D. Kim, J. Lapointe, P. Poole, G. Aers, R. Williams, W. Ross McKinnon, M. Korkusinski, and P. Hawrylak, Laser Photon. Rev. 4, 283–299 (2010). [142] J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Bestermann, M. Bayer, C. Kistner, S. Reitzenstein, S. H. A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).



[143] M. Assmann, F. Veit, M. Bayer, C. Gies, F. Jahnke, S. Reitzenstein, S. Hofling, L. Worschech, and A. Forchel, Phys. Rev. B 81(16), 165314 (2010). [144] Z. G. Xie, S. Gotzinger, W. Fang, H. Cao, and G. S. Solomon, Phys. Rev. Lett. 98(11), 117401 (2007). [145] G. Bjoerk, A. Karlsson, and Y. Yamamoto, Appl. Phys. Lett. 60, 304 (1992). [146] C. H. Henry, IEEE J. Quantum Electron. 18, 259 (1982). [147] L. A. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995). [148] S. Melnik, G. Huyet, and A. Uskov, Optics Express 14, 2950–2955 (2006). [149] J. Vazquez, H. Nilsson, J. Z. Zhang, and I. Galbraith, IEEE J. Quantum Electron. 42, 986 (2006). [150] M. Lorke, F. Jahnke, and W. Chow, Appl. Phys. Lett. 90, 051112 (2007). [151] P. Michler, A. Kiraz, L. Zhang, C. Becher, and A. Imamoglu, Appl. Phys. Lett. 77, 184 (2000). [152] K. Srinivasan, M. Borselli, K. Painter, A. Stintz, and S. Krishna, Optics Express 14, 1094–1105 (2006). [153] S. Reitzenstein, C. Bockler, A. Bazhenov, A. Gorbunov, A. Loffler, M. Kamp, V. D. Kulakovskii, and A. Forchel, Optics Express 16(7), 4848–4857 (2008). [154] D. Bajoni, P. Senellart, E. Wertz, I. Sagnes, A. Miard, A. Lemaitre, and J. Bloch, Phys. Rev. Lett. 100(4), 047401 (2008). [155] T. Yoshie, O. Shchekin, H. Chen, D. Deppe, and A. Scherer, Electr. Lett. 38, 967–968 (2002). [156] H. Y. Ryu, S. H. Kwon, Y. J. Lee, Y. H. Lee, and J. S. Kim, Appl. Phys. Lett. 80(19), 3476–3478 (2002). [157] M. Loncar, T. Yoshie, A. Scherer, P. Gogna, and Y. Qiu, Appl. Phys. Lett. 81, 2680–2682 (2002). [158] H. Y. Ryu, M. Notomi, E. Kuramoti, and T. Segawa, Appl. Phys. Lett. 84(7), 1067–1069 (2004). [159] M. Nomura, S. Iwamoto, K. Watanabe, N. Kumagai, Y. Nakata, S. Ishida, and Y. Arakawa, Optics Express 14(13), 6308–6315 (2006). [160] D. Englund, H. Altug, and J. Vuckovic, Appl. Phys. Lett. 91(7), 071124 (2007). [161] L. J. Martinez, B. Alen, I. Prieto, J. F. Galisteo-Lopez, M. Galli, L. C. Andreani, C. Seassal, P. Viktorovitch, and P. A. Postigo, Optics Express 17(17), 15043–15051 (2009). [162] M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, Optics Express 17(18), 15975–15982 (2009). [163] S. Strauf, Nature Phys. 6(4), 244–245 (2010). [164] D. Press, S. Gotzinger, S. Reitzenstein, C. Hofmann, A. Loffler, M. Kamp, A. Forchel, and Y. Yamamoto, Phys. Rev. Lett. 98(11), 117402 (2007). [165] D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vuckovic, Phys. Rev. Lett. 104(7), 073904 (2010). [166] S. Ates, S. M. Ulrich, A. Ulhaq, S. Reitzenstein, A. Loffler, S. Hofling, A. Forchel, and P. Michler, Nature Photon. 3(12), 724–728 (2009). [167] A. Ulhaq, S. Ates, S. Weiler, S. M. Ulrich, S. Reitzenstein, A. Loffler, S. Hofling, L. Worschech, A. Forchel, and P. Michler, Phys. Rev. B 82(4), 045307 (2010). [168] S. Strauf, N. G. Stoltz, M. T. Rakher, L. A. Coldren, P. M. Petroff, and D. Bouwmeester, Nature Photon. 1(12), 704–708 (2007). [169] P. Frantsuzov, M. Kuno, B. Janko, and R. Marcus, Nature Phys. 4, 519–522 (2008).

www.lpr-journal.org


[170] A. Auffeves, J. M. Gerard, and J. P. Poizat, Phys. Rev. A 79(5), 053838 (2009). [171] A. Naesby, T. Suhr, P. T. Kristensen, and J. Mork, Phys. Rev. A 78(4), 045802 (2008). [172] U. Hohenester, A. Laucht, M. Kaniber, N. Hauke, A. Neumann, A. Mohtashami, M. Seliger, M. Bichler, and J. J. Finley, Phys. Rev. B 80(20), 201311 (2009). [173] U. Hohenester, Phys. Rev. B 81(15), 155303 (2010). [174] M. Kaniber, A. Laucht, A. Neumann, J. M. Villas-Boas, M. Bichler, M. C. Amann, and J. J. Finley, Phys. Rev. B 77(16), 161303 (2008). [175] M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, and A. Imamoglu, Phys. Rev. Lett. 103(20), 207403 (2009). [176] K. Karrai, R. J. Warburton, C. Schulhauser, A. Hogele, B. Urbaszek, E. J. McGhee, A. O. Govorov, J. M. Garcia, B. D. Gerardot, and P. M. Petroff, Nature 427(6970), 135– 138 (2004). [177] A. Laucht, M. Kaniber, A. Mohtashami, N. Hauke, M. Bichler, and J. J. Finley, Phys. Rev. B 81(24), 241302 (2010). [178] S. Munch, S. Reitzenstein, P. Franeck, A. Loffler, T. Heindel, S. Hofling, L. Worschech, and A. Forchel, Optics Express 17(15), 12821–12828 (2009). [179] F. P. Laussy, E. del Valle, and C. Tejedor, Phys. Rev. Lett. 101(8), 083601 (2008). [180] H. J. Carmichael, Statistical Methods in Quantum Optics 1 (Springer, Berlin, 1998). [181] J. McKeever, A. Boca, A. D. Boozer, J. R. Buck, and H. J. Kimble, Nature 425, 268 (2003). [182] E. del Valle, F. P. Laussy, and C. Tejedor, Phys. Rev. B 79(23), 235326 (2009). [183] O. Benson and Y. Yamamoto, Phys. Rev. A 59(6), 4756– 4763 (1999). [184] S. Anders, C. S. Kim, B. Klein, M. W. Keller, R. P. Mirin, and A. G. Norman, Phys. Rev. B 66(12), 125309 (2002). [185] N. Baer, S. Schulz, S. Schumacher, P. Gartner, G. Czycholl, and F. Jahnke, Appl. Phys. Lett. 87, 231114 (2005). [186] N. Baer, S. Schulz, P. Gartner, S. Schumacher, G. Czycholl, and F. Jahnke, Phys. Rev. B 76, 075310 (2007). [187] A. Wojs, P. Hawrylak, S. Fafard, and L. Jacak, Phys. Rev. B 54, 5604 (1996). [188] O. Stier, M. Grundmann, and D. Bimberg, Phys. Rev. B 59(8), 5688–5701 (1999). [189] R. Santoprete, B. Koiller, R. B. Capaz, P. Kratzer, Q. K. K. Liu, and M. Scheffler, Phys. Rev. B 68(23), 235311 (2003). [190] M. Winkelnkemper, A. Schliwa, and D. Bimberg, Phys. Rev. B 74, 155322 (2006). [191] T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, and A. Wieck, Phys. Rev. B 76, 165318 (2007). [192] T. Inoshita and H. Sakaki, Phys. Rev. B 56, 4355 (1997). [193] K. Kral and Z. Khas, Phys. Rev. B 57, 2061 (1998). [194] T. Stauber, R. Zimmermann, and H. Castella, Phys. Rev. B 62, 7336 (2000).

www.lpr-journal.org

633

[195] O. Verzelen, R. Ferreira, G. Bastard, T. Inoshita, and H. Sakaki, phys. stat. sol. (a) 190, 213 (2002). [196] J. Seebeck, T. R. Nielsen, P. Gartner, and F. Jahnke, Phys. Rev. B 71, 125327 (2005). [197] B. Krummheuer, V. M. Axt, and T. Kuhn, Phys. Rev. B 65(19), 195313 (2002). [198] M. Braskén, M. Lindberg, D. Sundholm, and J. Olsen, Phys. Rev. B 61(11), 7652–7655 (2000). [199] P. Hawrylak, G. A. Narvaez, M. Bayer, and A. Forchel, Phys. Rev. Lett. 85(2), 389–392 (2000). [200] A. Hartmann, Y. Ducommun, E. Kapon, U. Hohenester, and E. Molinari, Phys. Rev. Lett. 84(24), 5648–5651 (2000). [201] N. Baer, P. Gartner, and F. Jahnke, Eur. Phys. J. B 42, 231 (2004). [202] L. He, G. Bester, and A. Zunger, Phys. Rev. Lett. 95(24), 246804 (2005). [203] U. Bockelmann and T. Egeler, Phys. Rev. B 46, 15574 (1992). [204] A. V. Uskov, F. Adler, H. Schweizer, and M. H. Pilkuhn, J. Appl. Phys. 81, 7895 (1997). [205] M. Braskén, M. Lindberg, M. Sopanen, H. Lipsanen, and J. Tulkki, Phys. Rev. B 58, 15993 (1998). [206] T. R. Nielsen, P. Gartner, and F. Jahnke, Phys. Rev. B 69, 235314 (2004). [207] T. Nielsen, P. Gartner, M. Lorke, J. Seebeck, and F. Jahnke, Phys. Rev. B 72, 235311 (2005). [208] H. Haug and A. P. Jauho, Quantum Kinetics in Transport & Optics of Semiconductors, 1. ed. (Springer, Berlin, 1996). [209] F. Jahnke and S. W. Koch, Optics Lett. 18, 1438 (1993). [210] U. Bockelmann and G. Bastard, Phys. Rev. B 42, 8947 (1990). [211] H. Benisty, C. M. Sotomayor-Torres, and C. Weisbuch, Phys. Rev. B 44, 10945 (1991). [212] T. Inoshita and H. Sakaki, Phys. Rev. B 46, 7260 (1992). [213] H. Jiang and J. Singh, IEEE J. Quantum Electron. 34, 1188 (1998). [214] G. D. Mahan, Many-Particle Physics, 2. ed. (Plenum, New York, 1990). [215] W. Schäfer, M. Wegener, Semiconductor Optics and Transport Phenomena, 1. ed. (Springer-Verlag, Berlin, 2002). [216] P. Gartner, J. Seebeck, and F. Jahnke, Phys. Rev. B 73, 115307 (2006). [217] G. J. Beirne, M. Reischle, R. Robach, W. M. Schulz, M. Jetter, J. Seebeck, P. Gartner, C. Gies, F. Jahnke, and P. Michler, Phys. Rev. B 75, 195302 (2007). [218] H. Kurtze, J. Seebeck, P. Gartner, D. R. Yakovlev, D. Reuter, A. D. Wieck, M. Bayer, and F. Jahnke, Phys. Rev. B 80(23), 235319 (2009). [219] S. Kalliakos, C. P. Garcia, V. Pellegrini, M. Zamfirescu, L. Cavigli, M. Gurioli, A. Vinattieri, A. Pinczuk, B. S. Dennis, L. N. Pfeiffer, and K. W. West, Appl. Phys. Lett. 90(18), 181902 (2007).
