

Interfacing single photons and single quantum dots with photonic nanostructures

Peter Lodahl    Sahand Mahmoodian    Søren Stobbe Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
May 30, 2020

Photonic nanostructures provide means of tailoring the interaction between light and matter and the past decade has witnessed a tremendous experimental and theoretical progress in this subject. In particular, the combination with semiconductor quantum dots has proven successful. This manuscript reviews quantum optics with excitons in single quantum dots embedded in photonic nanostructures. The ability to engineer the light-matter interaction strength in integrated photonic nanostructures enables a range of fundamental quantum-electrodynamics experiments on, e.g., spontaneous-emission control, modified Lamb shifts, and enhanced dipole-dipole interaction. Furthermore, highly efficient single-photon sources and giant photon nonlinearities may be implemented with immediate applications for photonic quantum-information processing. The review summarizes the general theoretical framework of photon emission including the role of dephasing processes, and applies it to photonic nanostructures of current interest, such as photonic-crystal cavities and waveguides, dielectric nanowires, and plasmonic waveguides. The introduced concepts are generally applicable in quantum nanophotonics and apply to a large extent also to other quantum emitters, such as molecules, nitrogen vacancy ceters, or atoms. Finally, the progress and future prospects of applications in quantum-information processing are considered.


I Introduction

Quantum electrodynamics (QED) studies the interaction between light and matter at the most fundamental level where single quanta of light (photons) and single entities of matter (quantum emitters) are controllably coupled. Founded by Paul Dirac in the 1920s, the theory of QED encompasses intriguing quantum phenomena such as quantum superposition states and entanglement, and has proven to be remarkably precise. The progress on QED experiments with atoms and photons have spun out of research on atomic spectroscopy. A remarkable experimental frontrunner was the demonstration of the Lamb shift W. E Lamb and Retherford (1947) proving that the anticipated degeneracy of the and states of atomic hydrogen was lifted. The Lamb shift can be interpreted as being due to the interaction of the emitter with the quantum vacuum and its experimental demonstration stimulated further developments of QED, notably leading to renormalization theory. The Casimir effect constitutes another landmark in the history of QED; it describes how two mirrors experience an attractive force due to the radiation pressure of vacuum Casimir (1948). The development of tools for experimenting with single photons and single atoms started in the 1970s following the invention of the laser. The first experimental demonstration that an excited atom emits a single photon at a time was reported by Kimble et al. (1977), which marked the birth of experimental quantum optics. Since then a range of exciting experiments on fundamental aspects of light-matter interaction at the single-photon level have appeared using atoms where, e.g., cavities can be exploited for increasing the interaction strength. For a recent review of atom-based cavity QED including a historical account of the field, see Haroche (2013).

In parallel with the development of atomic QED, major research efforts have been focused on solid-state alternatives. Solid-state systems have the obvious experimental asset that the elaborate experimental techniques needed for trapping and cooling single atoms are not required. Both the emitter and the optical environment can be engineered to enhance the photon-emitter coupling, and consequently two different research disciplines have merged into solid-state QED. The first is material-science research, which has developed methods to synthesize solid-state single-photon emitters with excellent optical properties. In particular, the discovery of photoluminescence from single self-assembled quantum dots Marzin et al. (1994) embedded in GaAs where atomically smooth heterostructures can be grown with molecular beam epitaxy was the first in series of major breakthroughs. Since then growth methods have developed tremendously and today quantum dots can be tailored to have excellent optical properties. The second is nanophotonics research where the optical environment of the emitter is engineered by nanofabrication methods, which builds on the original insight of Purcell that radiative processes are not immutable properties of the emitter, but can be controlled by the environment Purcell (1946). Pioneering solid-state QED experiments were carried out by Drexhage (1970), who showed that the radiative lifetime of europium ions is influenced by the presence of a nearby dielectric interface. This experiment constitutes the first example that the light-matter coupling efficiency can be tailored by structuring the environment of the emitter, which is the essence of modern research in solid-state QED. Today experimental techniques have matured very significantly, and experiments are routinely performed on single solid-state quantum emitters in highly complex photonic nanostructures; the present manuscript reviews this progress. We are here concerned with quantum emitters at optical frequencies and the majority of the considered examples pertain to quantum dots. Nonetheless most of the concepts are of much broader scope and apply equally well to other solid-state emitters and atoms, or to two-level emitters implemented in superconducting circuits.

The research field of solid-state QED has widespread and far-ranging implications and applications. For instance, QED systems are widely proposed for quantum-information processing and optimized photon-matter interfaces are at the heart of photonic devices such as lasers and solar cells. An outstanding challenge in quantum physics today is to construct scalable quantum networks that exploit quantum parallelism for encoding and processing information. To this end, a variety of different physical systems exploiting either photon or matter degrees of freedom have been considered Nielsen and Chuang (2000), and each system has pros and cons: photons are robust carriers of quantum information over long distances but tend to interact weakly, which makes quantum-computing protocols experimentally demanding Kok et al. (2007). In contrast, e.g., electrons confined in quantum dots may interact very strongly but this also makes the system vulnerable to decoherence processes from the environment and the interaction has limited spatial range Hanson et al. (2007). An efficient quantum interface between light and matter implemented in a scalable quantum architecture is therefore expected to have wide applications in quantum-information processing Kimble (2008) since it allows encoding quantum information in both light and matter variables, thus potentially benefiting from the advantages of each system.

The main challenge in order to realize efficient light-matter interfaces is that the interaction between a single photon and a single emitter tends to be very weak. A natural photon-emitter process is that of spontaneous emission: a single two-level emitter with an electron prepared in the excited state undergoes a transition to the ground state by emitting a single photon. Spontaneous emission constitutes an example of a general class of quantum-mechanics problems where a simple quantum system is coupled to a large reservoir since the emitter comprises only two states while the photon can be emitted to any of a continuum of optical states each characterized by different wave vectors. Consequently, the challenge for efficient photon-emitter interfacing is to strongly enhance the coupling to one preferred mode and/or suppress the coupling to all unwanted modes. High-finesse optical cavities have been a popular approach for accomplishing this and very significant experimental progress has been obtained within the last few decades with single atoms Haroche (2013). In recent years, solid-state alternatives have emerged with the benefit that the systems can be engineered by modern nanofabrication techniques. This leads to a whole range of new design strategies based on dielectric and metallic nanostructures including waveguides and nanocavities. Two different classes of solid-state single-photon emitters have been considered, using either superconducting circuits emitting in the microwave regime Wallraff et al. (2004) or two-level quantum emitters with optical transitions, e.g., semiconductor quantum dots Michler et al. (2000), nitrogen vacancy centers in diamond Kurtsiefer et al. (2000), or single molecules Lounis and Moerner (2000). A benefit of optical methods is that many functionalities can be highly integrated on a photonic chip O’Brien et al. (2009) since the wavelength of the electromagnetic radiation determines the typical length scale of the building blocks. As a consequence, highly integrated and engineered photonic circuits could potentially be constructed, which is a quest in the burgeoning research field of quantum nanophotonics.

A variety of different optical emitters have been studied in nanophotonic experiments including laser-cooled and trapped atoms, rare-earth ions, single molecules, impurity centers (e.g., nitrogen vacancy centers), colloidal quantum dots, and self-assembled quantum dots. In the present Review, most of the discussed examples concern epitaxially grown III-V semiconductor quantum dots (primarily InGaAs or GaAs) that typically emit in the near infrared. These emitters constitute many-particle mesoscopic systems that contain rich and exciting quantum physics. The present knowledge of and control over InGaAs quantum dots, regarding level structure, emission dynamics, stability, and coherence, is now sufficiently mature that a serious exploration of their potential in quantum optics and quantum-information processing seems viable. It should be stressed that such experiments are carried out at cryogenic conditions, typically at a temperature of , which is necessary to prevent thermal depopulation and achieve sufficiently good coherence properties as required in most quantum-optics experiments. In contrast, room-temperature experiments with solid-state emitters typically suffer from severe decoherence due to Coulomb and phonon-induced scattering that makes coherent quantum-optics experiments infeasible. The ability to tailor light-matter interfaces at room temperature may nonetheless find a range of other important applications, e.g., in the context of light harvesting or efficient light sources. We emphasize that while the present review in focused on the quantum aspects of light-matter interaction, many of the discussed concepts are generally applicable and could equally well be exploited in room-temperature applications.

Ii Semiconductor quantum dots

Characteristic size of a quantum dot relative to a single atom and a photonic crystal. A single atom (a) measures a few Ångström while self-assembled InGaAs quantum dots (b) typically have dimensions of tens of nanometer and consist of approximately
Figure 1: Characteristic size of a quantum dot relative to a single atom and a photonic crystal. A single atom (a) measures a few Ångström while self-assembled InGaAs quantum dots (b) typically have dimensions of tens of nanometer and consist of approximately atoms. The micrograph in (b) shows an uncapped quantum dot obtained by scanning tunneling microcopy. Single quantum dots can be embedded in photonic nanostructures for quantum-optics experiments, an example of which is (c) that shows a scanning electron micrograph image of a photonic-crystal waveguide, where the photonic lattice constant typically is around . (b) Reprinted with permission from Márquez et al. (2001).

The existence of a discrete and anharmonic electronic spectrum is the prerequisite for many quantum-optics experiments since it enables generating single photons when an electron undergoes a transition between two levels. An obvious choice is a single atom, which represents a clean quantum system with discrete electronic states. The ability to create discrete electronic states in a solid-state system enables a range of new opportunities for integrated quantum-optics experiments, and this can be achieved in a quantum dot. A quantum dot is a semiconductor “artifical atom” that although consisting of tens of thousands of atoms has optical properties similar to single atoms due to the quantum confinement of electrons to a nanometer length scale. Since quantum dots are solid-state emitters they can readily be implemented in photonic nanostructures such as nanowires, plasmonic nanoantennas, and photonic crystals, as discussed in the following sections. Figure 1 indicates the size of a quantum dot relative to a single atom and a typical photonic nanostructure. In the present section we briefly review the structural and optical properties of quantum dots. Unless explicitly indicated, we are solely considering quantum dots in the family of InAs/GaAs/AlAs III-V semiconductors, which are grown by epitaxial methods.

ii.1 Growth and structural properties

Structural properties of quantum dots. (a)-(c) Atomic-force micrographs (AFMs) of uncapped quantum dots for (a) Stranski-Krastanov quantum dots of InAs, (b) interface fluctuations of GaAs, and (c) droplet epitaxy quantum dots of GaAs displaying the surface topography with bright (dark) colors indicating high (low) features. (d), (f), and (h) illustrate the confinement potentials, where dark, neutral, and bright gray indicate AlGaAs, GaAs, and InAs, respectively for the three types of quantum dots shown above. The electron (hole) wave function is shown in blue (red). For interface-fluctuation quantum dots (f) the in-plane motion of electrons and holes may become correlated (purple exciton wave function) and could lead to a giant oscillator strength. (e), (g), and (i) illustrate the wave functions along the growth axis
Figure 2: Structural properties of quantum dots. (a)-(c) Atomic-force micrographs (AFMs) of uncapped quantum dots for (a) Stranski-Krastanov quantum dots of InAs, (b) interface fluctuations of GaAs, and (c) droplet epitaxy quantum dots of GaAs displaying the surface topography with bright (dark) colors indicating high (low) features. (d), (f), and (h) illustrate the confinement potentials, where dark, neutral, and bright gray indicate AlGaAs, GaAs, and InAs, respectively for the three types of quantum dots shown above. The electron (hole) wave function is shown in blue (red). For interface-fluctuation quantum dots (f) the in-plane motion of electrons and holes may become correlated (purple exciton wave function) and could lead to a giant oscillator strength. (e), (g), and (i) illustrate the wave functions along the growth axis for the respective type of quantum dots. For Stranski-Krastanov quantum dots, an asymmetric confinement potential leads to a significant offset between the electron and hole. (b) Reprinted with permission from Peter et al. (2005). (c) Reprinted with permission from Mano et al. (2009).

One of the important experimental advantages of quantum dots is that they are made from semiconductor materials for which a wealth of growth and processing technology has been developed over the past decades. Sophisticated crystal-growth procedures combined with semiconductor processing methods such as electron-beam lithography, etching, and deposition constitute the generic nanofabrication platform on which the significant experimental progress within quantum nanophotonics during the past decades is built. In this section we discuss the most common methods for growing quantum dots and the impact of the growth method on their optical properties.

Since semiconductors are very sensitive to impurities and defects, quantum dots are fabricated by epitaxial methods such as molecular-beam epitaxy where heterostructures are grown with monolayer precision under ultra-high-vacuum conditions Shchukin and Bimberg (1999); Stangl et al. (2004); Biasiol and Heun (2011). The most common approach for InGaAs quantum dots is the Stranski-Krastanov method that relies on the self-assembly of InAs or InGaAs quantum dots on a GaAs surface due to the 7% larger lattice constant of InAs compared to that of GaAs. As a consequence, only a thin wetting layer of InAs can be deposited on GaAs before the strain is relaxed by the nucleation of quantum dots in the form of randomly positioned islands as shown in Fig. 2(a). In order to protect the quantum dots from oxidation and to prevent interaction with surface states, a GaAs capping layer is grown atop the quantum dots. While Stranski-Krastanov quantum dots have a pyramidal shape before capping, cf. Fig. 1(b), they develop the shape of a truncated pyramid after capping Eisele et al. (2008) due to a significant material intermixing as shown in Fig. 2(d). This in turn leads to an inhomogeneous indium distribution and a strain that varies throughout the quantum dot. Typically, quantum dots are grown with heights in the range of 1- and in-plane sizes in the range of 10-. Controlling the size and therefore the quantum confinement as well as the material composition enable tailoring the emission wavelength. Size variations between different quantum dots within a single growth run are inevitable, i.e., a quantum¨-dot ensemble will be inhomogeneously broadened, implying that individual tuning of single quantum dots would generally be required in order to couple them mutually. It has been found that the indium is concentrated in an inverted pyramid inside the quantum dot Liu et al. (2000), which leads to variations in the confinement potential along the growth axis as indicated in Fig. 2(e) where also schematic electron and hole envelope wave functions are shown. Notably, the hole resides above the electron and this leads to a significant static electric dipole along the growth axis Fry et al. (2000).

Quantum dots may also form by growing a thin quantum well but terminating the growth while the lower and upper monolayers of the quantum well are formed Gammon et al. (1996) leading to monolayer fluctuations as shown in Fig. 2(b) and (f) with a confinement potential as indicated in Fig. 2(g). This allows growing practically unstrained GaAs quantum dots embedded in AlGaAs and these interface-fluctuation quantum dots can be larger in the plane perpendicular to the growth direction than typical Stranski-Krastanov quantum dots. Interface-fluctuation quantum dots benefit from a relatively narrow inhomogeneous broadening and predictable wave functions along the growth direction as compared to Stranski-Krastanov quantum dots since intermixing is absent. However, the integration in optical nanostructures is more challenging since they are embedded in AlGaAs, whose surface is prone to oxidation. They are particularly promising for obtaining quantum dots with an enhanced oscillator strength as discussed in Sec. II.5.2.

Droplet epitaxy is an emerging growth technique, where droplets of gallium are saturated with arsenic, resulting in relatively large and low-density GaAs quantum dots in AlGaAs as shown in Fig. 2(c). Since droplet epitaxy quantum dots are embedded in AlGaAs surroundings, the above mentioned challenges pertain to their integration in photonic nanostructures as well. Since the AlGaAs capping layer is often grown at low temperatures, a high-temperature post-growth annealing is required to make them optically active and the demonstrated quantum efficiency cannot yet compete with Stranski-Krastanov quantum dots Tighineanu et al. (2013) although narrow linewidths have been reported Mano et al. (2009); Sallen et al. (2011). The post-growth annealing results in intermixing and the resulting confinement is therefore smaller than the apparent size as shown in Fig. 2(h) and (i).

In quantum-optics experiments it is often required to isolate a single quantum dot, which may be achieved by various methods, such as etching away the material surrounding a single quantum dot or evaporating an opaque metallic aperture masking other quantum dots. Such methods may modify the optical properties in adverse ways and are therefore not generally compatible with nanophotonic devices, such as waveguides or cavities. The most versatile experimental approach at present is to use samples with a low quantum-dot density combined with confocal microscopy. A main drawback of the standard growth methods is that the lateral positions of the quantum dots are not controlled and it has proven a successful strategy to deterministically align nanophotonic structures to a single quantum dot that is first located by microscopy techniques Badolato et al. (2005); Hennessy et al. (2007); Dousse et al. (2008); Thon et al. (2009); Kojima et al. (2013).

It would generally be ideal to grow periodic arrays of quantum dots for deterministic and scalable integration with photonic nanostructures and many approaches are presently studied. This is currently an active field of crystal-growth research Kiravittaya et al. (2009) and relies on growing quantum dots on patterned substrates obtained, e.g., by electron-beam lithography, and subsequently etching small recesses creating nucleation sites for quantum dots. This processing leads to challenges because impurities and defects are introduced and growing high-quality material immediately atop the regrowth interface is difficult. Therefore a common approach involves growing a seed layer of quantum dots that is optically inactive due to defects at the regrowth interface but introduces a suitable amount of strain so that another quantum-dot layer can be grown atop. The optical quality of positioned quantum dots tends to be lower than that of Stranski-Krastanov quantum dots giving broader linewidths and lower quantum efficiency Albert et al. (2010) and patterned regrowth of quantum dots is not yet widely applied in quantum-nanophotonics experiments. Recent promising progress includes the demonstration of emission of indistinguishable Jöns et al. (2013) and polarization-entangled Juska et al. (2013) photons although the yield, i.e., the fraction of sites with an optically active high-quality quantum dot is so far below unity. It is nonetheless anticipated that future progress eventually will render positioned quantum dots indispensable for more controlled experiments Gallo et al. (2008) and for scaling up quantum architectures. It should also be mentioned that even if the inherent optical properties of positioned quantum dots do not reach the quality of Stranski-Krastanov material, the Purcell effect, see Sec. IV.3, or resonant excitation, see Sec. V.3, can be employed to enhance the effective quantum efficiency or reduce the effects of linewidth broadening, respectively.

The material composition and size determine the energy range of the emission spectrum of quantum dots. In particular, the lowest-energy transition of InAs quantum dots typically falls in the range of 850 to while GaAs quantum dots typically emit at around 670 to . The choice of material affects also photonic properties, in particular through the real part of the index of refraction. Also bulk and surface absorption are strongly material dependent and play an important role for the performance of nanophotonic devices Michael et al. (2007). In general, longer wavelengths imply that fabrication of nanostructures is easier and that the relative importance of fabrication imperfections is reduced. Furthermore, the material absorption is smaller at longer wavelengths, which therefore allow for, e.g., higher -factors of optical nanocavities. On the other hand, silicon-based photodetectors such as avalanche photodiodes are much more efficient at shorter wavelengths and therefore lead to higher photon-count rates, although this obstacle may be overcome by the use of superconducting detectors. A tradeoff between these factors determine the most ideal wavelength for a given experimental setting.

ii.2 Excitons in quantum dots

The fundamental optical excitation in a quantum dot consists of an electron in the conduction band and a hole in the valence band. In bulk semiconductors and quantum wells there is an important distinction between an uncorrelated electron-hole pair and an exciton, which is an electron-hole pair bound by direct and exchange Coulomb interaction. In quantum dots this distinction is often not required. In small quantum dots, the motion of electrons and holes is dominated by quantum confinement, which implies that they are mutually independent, but exciton effects are required to explain the fine structure. The optically active states in quantum dots are therefore always excitonic. A further discussion of the effects of Coulomb interaction and confinement is given in Sec. II.5.2.

The unfilled orbital shells of atomic Al, Ga, In, and As are , , , and , respectively, so predominantly covalent bonds are formed in GaAs and in the other relevant binary and ternary alloys. This leads to tetrahedral bonds and a zincblende crystal structure and in the absence of spin-orbit effects there would be three degenerate valence bands. Including spin orbit, the split-off valence band is shifted to lower energy but the light- and heavy-hole bands remain degenerate Yu and Cardona (2010). The heterojunctions of InGaAs in GaAs and GaAs in AlGaAs have type-I energy-band alignment, which ensures carrier confinement at both the conduction- and the valence-band edge. This is essential for the efficient interaction with light and allows for quantized states for both electrons and holes. The aspect ratio of quantum dots is larger than unity and the dominant quantization axis is the growth direction. This lifts the heavy-/light-hole degeneracy so that the transitions from the conduction band to the heavy-hole band have the lowest energy. Strain plays a major role in InGaAs quantum dots and lifts the degeneracy further. While neglecting the light-hole band is often a good approximation, a substantial band mixing can occur in quantum dots with a pronounced structural asymmetry Belhadj et al. (2010). Sophisticated numerical models have been developed in order to encompass these effects, such as the method Stier et al. (1999), which is a continuum theory and the empirical pseudo-potential theory Bester (2009), which is an atomistic approach. Despite these significant advances, the comparison to experiments is limited by the lack of knowledge about the exact atomic configuration of quantum dots. Such information could be extracted using, e.g., high-resolution transmission electron microscopy or scanning-tunneling microscopy but these techniques are time consuming and often destructive and therefore challenging to combine with optical spectroscopy. Fortunately, under the conditions relevant for quantum-optics experiments, i.e., quantum dots with a large aspect ratio at low temperatures and small carrier populations, many features of quantum dots can be described remarkably well by using a simple two-band effective-mass model where only the heavy-hole valence band and the conduction band are included. We restrict the discussion in this review to such a model. In the effective-mass approximation, the electronic energy bands are taken into account by the band-edge effective mass of the carriers. We are mainly concerned with quantum dots of sizes where the energy-level spacing is large compared to the Coulomb energy. In this case, Coulomb effects may be included perturbatively and the motion of the carriers in the conduction and valence band may be considered independent. This is known as the strong-confinement regime and throughout the review we restrict the discussion to this regime unless explicitly noted. It should be noted that in quantum dots the energy-level spacing may actually be comparable to the Coulomb energy leading to a complex interplay between confinement and Coulomb interaction, which is denoted the intermediate-confinement regime. However, the strong-confinement model has so far been able to describe experiments successfully. As already mentioned and discussed further below, even in the strong-confinement regime, Coulomb exchange determines the fine structure.

ii.3 The transition matrix element

The key property of a quantum emitter determining the strength of its interaction with light is the transition matrix element between the ground and excited states. This may also be expressed as a transition dipole moment or an oscillator strength. For atoms this is an immutable intrinsic property but for quantum dots it may be controlled by modifying the exciton wave function and changing the selection rules of optical transitions. Here we discuss the physics underlying the transition matrix element of quantum dots.

The quantum state of an electron in the conduction band, c, or heavy-hole valence band, v, consists of three parts,


where , , and is the envelope wave function, the electronic Bloch function evaluated at the -point of the bandstructure, and the spin state, respectively. The envelope wave function is obtained from the effective-mass Schrödinger equation , where is the electron rest mass and is the anisotropic effective mass, which in general is a tensor but here taken to be scalar for simplicity. is the confinement potential, is the electron eigenenergy, and is the band-edge energy.

Optical transitions are induced by the minimal-coupling Hamiltonian, which in the generalized Coulomb gauge and the dipole approximation can be written as , where is the elementary charge, is the momentum operator, and is the vector potential of the electromagnetic field that is evaluated at the position of the quantum dot, . The relevant quantity for spontaneous emission is therefore the momentum matrix element that follows from Eq. (1),


leading to three selection rules for optical transitions: i) the envelope wave functions must have the same parity, ii) the Bloch functions must have opposite parity, and iii) the electron spin must remain unchanged. The Bloch matrix element is , where is the volume of a unit cell, and depends only on bulk material properties as is quantified by the Kane energy to be introduced below. Formally the selection rules can be relaxed by band mixing, but experimentally transitions obeying them have been observed to dominate Johansen et al. (2008). The first selection rule is only approximately valid since parity often is not a good quantum number for quantum-dot envelope functions.

The electronic states are commonly described in the equivalent electron-hole picture in which the valence-band states are transformed to the hole picture according to , , and . In this picture the decay of an electron from the conduction band to the valence band is viewed as the recombination of an electron and a hole and we define the electron and hole pseudo-spin states, which describe the total angular momentum of the Bloch functions and the spin, i.e., , , , and . Here the arrows with subscripts denote the projected spin so that, e.g., is an electron with , where is along the growth direction. A quadruplet of exciton states are formed enabling dipole-allowed transitions for the states and , and dipole-forbidden transitions for and . It is common practice to suppress the envelope wave function in the notation because they only contribute to the matrix element with a pre-factor, i.e., the wave-function overlap. The evaluation of the corresponding dipole matrix elements is obtained by converting back to the electron picture and using Eq. (2). To summarize the implications of the notation with an example, the matrix element for transitions between the state and the state void of excitations, i.e., the quantum-dot ground state, , corresponds to .

The Bloch functions inherit the symmetry of the atomic orbitals: the conduction-band Bloch functions have s-symmetry and the valence band has p-symmetry. Hence we define , , , and , where , , and denote functions with even parity, odd parity along , and odd parity along , respectively. With these definitions we obtain, e.g., for that , where () is a unit vector along (). Evidently, the polarization of the dipole moment is given by the term , which describes circular polarization. The magnitude of the dipole moment depends on the wave-function overlap , which can be controlled by, e.g., external static electric fields, and the Bloch matrix element . The latter is an entirely bulk semiconductor quantity given by the Kane energy Vurgaftman et al. (2001).

Quantum dots are most often grown on (001) substrates where the symmetry leads to in-plane anisotropic confinement potentials even for rotationally symmetric quantum dots Bester et al. (2003) and typically an in-plane structural asymmetry is also present. The preferential elongation is along , which is also apparent from Fig. 2(b). The exchange interaction splits the four excitonic states into two doublets, which are separated by the dark-bright energy splitting Bayer et al. (2002): the dark-exciton doublet, and , and the bright-exciton doublet, and . Furthermore, the exchange interaction splits the bright doublet due to the reduced in-plane symmetry with a fine-structure splitting typically on the order of Seguin et al. (2005). The splitting between the two dark excitons is of the order of Poem et al. (2010). The resulting excitonic states are shown in Fig. 3 along with the radiative transitions as well as the spin-flip and non-radiative transitions that are discussed further below. Using the procedure outlined above, it is straightforward to show that and have linearly polarized dipole transitions to the ground state with orientations along and , respectively.

ii.4 Multiexcitonic states

The lowest-energy confined states in quantum dots and the transitions between them. The full blue (empty red) circles indicate the electron (hole) configuration in the conduction (valence) band
Figure 3: The lowest-energy confined states in quantum dots and the transitions between them. The full blue (empty red) circles indicate the electron (hole) configuration in the conduction (valence) band -shells of the quantum dot. The pseudo-spin states are discussed in the text. The biexciton may decay to one of the two bright exciton states by emission of a horizontally (H) or vertically (V) polarized photon, respectively. The negative (positive) trion decays to a single electron (hole) by emission of circularly polarized light with the helicity depending on the additional carrier. Furthermore, spin-flip processes (gray arrows) couple bright and dark excitons. Non-radiative processes (black arrows) are generally present and can for some transitions be dominant; here only the non-radiative decay of the bright excitons are indicated explicitly. Note that the ordering of the states in the figure follows the occupancy, while the emission energies of the radiative excitonic complexes depend not only on the occupancy but also on confinement and correlation effects.

Quantum dots can contain multiple electrons and/or holes leading to additional transitions with different optical properties than single neutral excitons. The simplest examples of multiexcitonic states are trions and biexcitons that are useful in quantum-optics experiments requiring multi-level schemes.

Multiple bound states may exist as solutions to the effective-mass equation for both the conduction and valence bands and these eigenstates are commonly denoted -, -, and -shells, etc., in analogy with the convention in atomic physics and not to be confused with the symmetries of the Bloch functions discussed above. We note that the approximate selection rule for the envelope functions imply that only transitions between the same shell in the conduction and the valence band need to be considered. Excitons in higher-lying states are often not relevant for optical emission because they decay to the -shells on a few-picosecond time scale by emission of phonons but depending on the type of excitation in a given experiment they may affect the decay cascade towards the -shells. In the following we consider only the -shells. Here only four types of excitonic quasiparticles exist due to the Pauli exclusion principle: the neutral exciton, the negative trion (two electrons and one hole), the positive trion (one electron and two holes), and the biexciton (two electrons and two holes). The charge configuration, pseudo-spin state, and dipole-allowed transitions of these states are shown in Fig. 3 and are discussed further below. We note that non-radiative processes have not been studied systematically for trions and biexcitons.

The exchange interaction vanishes for the trions due to Kramer’s theorem according to which a system with an odd number of fermions is at least two-fold degenerate if it is governed by a Hamiltonian that is symmetric under time reversal Messiah (1999), which is the case for a trion in the absence of external magnetic fields. The state of the two identical carriers must be antisymmetrized because trions are fermions. The negative trion is given by , where is the hole-spin state. For the positive trion, , where is the electron-spin state. The transition matrix elements for the decay of a trion to a single electron or hole are circularly polarized with the helicity depending on the spin of the additional carrier. As opposed to excitons they have no corresponding dark states. The additional carrier in trions may be prepared in a spin eigenstate and trions are therefore particularly relevant for spin physics. Quantum-dot spin physics was recently reviewed by Warburton (2013) and Urbaszek et al. (2013).

Biexcitons have the pseudo-spin configuration and they can decay radiatively to either of the bright excitons. Depending on which exciton it decays to, a cascade of either two horizontally polarized or two vertically polarized photons is triggered, i.e., the emitted photonic state is or , cf. Fig. 4.(c). Since biexcitons contain two excitons they are observed at higher excitation densities and decay approximately twice as fast as excitons since they have twice the number of radiative decay channels. If the fine-structure splitting is much smaller than the natural linewidth, biexcitons are a source of polarization-entangled photons because the emission cascade leads to the entangled photonic state , where and denote left- and right-hand circular polarization, respectively Benson et al. (2000). The fine-structure splitting can be reduced by growing quantum dots on the higher-symmetry (111) substrates Juska et al. (2013); Kuroda et al. (2013) or by applying various tuning schemes, such as electric fields Bennett et al. (2010).

ii.5 Optical properties of quantum dots

 Examples of excitonic level schemes in quantum dots of relevance for quantum-optics experiments. (a) The most basic three-level optical transition scheme of a single bright exciton,
Figure 4: Examples of excitonic level schemes in quantum dots of relevance for quantum-optics experiments. (a) The most basic three-level optical transition scheme of a single bright exciton, that can emit a photon by decaying radiatively to the ground state, . Also non-radiative decay processes and coupling to the dark exciton state, , through spin-flip processes may occur. The dark state can also recombine non-radiatively. This level scheme leads to a bi-exponential decay of the emitted intensity and holds also for the other bright exciton, (not shown). (b) Optical -scheme formed by the two bright exciton states that decay to the ground state. The two orthogonally polarized bright excitons are split by the fine-structure splitting, . The non-radiative processes and dark states indicated in (a) are excluded for simplicity. (c) Four-level scheme formed by the biexciton level, the two bright exciton levels, and the ground state. The biexciton decays through a cascaded process of either emitting two horizontally or two vertically polarized photons, respectively.

Quantum dots are reliable sources of photons for quantum optics and various level schemes can be realized by different excitation and detection strategies. Figure 4 summarizes the most relevant levels and decay processes. In many applications the three-level scheme of Fig. 4(a) is applied, which suffices for, e.g., describing single-photon emission. We consider this level scheme throughout the review unless otherwise noted. In this case we define the excited state . In the following we discuss the key figures of merit of the three-level system and elaborate on the single-photon-emission properties and coherence in Sec. II.6.

ii.5.1 Quantum-dot decay dynamics

A detailed understanding of the dynamics of quantum dots is essential in order to exploit them as reliable photon sources in quantum-photonics applications. The various exciton states can be coupled by spin-flip processes where, e.g., an exchange-mediated process between electron and hole Roszak et al. (2007) or spin-orbit coupling Liao et al. (2011) flips the spin of the exciton while longitudinal acoustic (LA) phonons provide or remove the energy difference between the two states. Spin-flip processes are generally much slower than the radiative decay processes so it is a good approximation to include only spin flips between and as well as between and . Transitions between and require changing the spin of both the electron and the hole and are therefore negligible. This implies that and are decoupled and leads to two identical three-level schemes of the form shown in Fig. 4(a). This constitutes the basic level scheme relevant for many quantum-photonics experiments employing a single (neutral) exciton in a quantum dot. For the case of non-resonant excitation where both bright and dark exciton states are populated, the population of the bright exciton follows a bi-exponential decay Smith et al. (2005); Wang et al. (2011),


with the fast and slow decay rates and , and corresponding amplitudes, and . Here denotes the radiative decay rate for the bright exciton, () denotes the non-radiative decay rate of the bright (dark) exciton, and is the bright-dark spin-flip rate. The initial population of bright and dark excitons can be assumed identical, i.e., for weak non-resonant excitation although corrections to this assumption may be imposed by the decay cascade involving higher-excited states in the quantum dot Poem et al. (2010). It has been found experimentally that the recombination of dark excitons is dominated by non-radiative processes and that Johansen et al. (2010), i.e., the intrinsic non-radiative decay rates of bright and dark exciton states are approximately equal, which reflects that the binding energy of bright and dark excitons are very similar. This approximation has been used to obtain Eq. (3). We note that with externally appplied strain, light- and heavy-hole mixing Huo et al. (2014) may lead to a radiative contribution to the recombination of dark excitons. Since most experiments are carried out at temperatures where the thermal energy exceeds the energy splitting between dark and bright states , the probability of emission and absorption of phonons is approximately equal and it is a good approximation to assume , which has also been assumed to reach the simplified dynamics of Eq. (3).

By fitting experimental decay curves with the bi-exponential model of Eq. (3), the radiative, non-radiative, and spin-flip rates can be extracted for a single quantum dot. With this method, single quantum dots can be employed for mapping the local light-matter interaction strength, as discussed in Sec. IV.3. Considerable variations in the spin-flip and non-radiative rates are generally found across a quantum-dot ensemble and between different growth runs, and the described quantitative method of determining these processes is thus required. We note that trions and biexcitons do not have a fine structure implying that their population dynamics follows a single-exponential decay with no direct access to the non-radiative rates.

ii.5.2 The oscillator strength

The intrinsic capability of a dipole emitter to interact with light is characterized by the magnitude of the transition momentum matrix element, , as introduced above. In the dipole and rotating-wave approximations this is equivalent to retaining only the dipole term in the multipolar gauge Craig and Thirunamachandran (1998). In this approximation, the interaction Hamiltonian can be expressed in terms of the transition dipole moment operator, , and the electric field amplitude, , evaluated at the position of the emitter, , as . This description is common in atomic physics where the dipole approximation is an excellent approximation but verifying its validity for quantum dots is non-trivial as discussed in Sec. IV.5. The dipole matrix element is related to the momentum matrix element via , where is the optical transition energy. The optical transition strength is commonly quantified by the oscillator strength, which is a dimensionless parameter defined as the decay rate in a homogeneous medium divided by the decay rate of a classical oscillator. It can be expressed as


A large oscillator strength is desirable since it increases the light-matter interaction strength and therefore increases the generation rate of single photons. A large oscillator strength also reduces the relative impact of other interactions that may give rise to undesired effects, such as phonon-dephasing or non-radiative decay. Evidently the squared wave-function overlap, , must be maximized to enhance the oscillator strength cf. the discussion of Fig. 5 below.

While the wave-function overlap cannot exceed unity and thus sets the upper limit for the oscillator strength, the strong-confinement model underlying Eq. (4) is only valid when the Coulomb interaction can be neglected or included perturbatively. For large quantum dots with a radius larger than the exciton Bohr radius, this assumption breaks down. The exciton Bohr radius is , where denotes the vacuum permittivity, is the relative static permittivity of the material, and the reduced mass is defined as with the electron (hole) effective mass, (). The exciton Bohr radius in nanostructures is modified by confinement and in quantum dots where the quantization axis is dominant the Bohr radius is reduced by a factor of Bastard et al. (1982). When the quantum-dot radius is much larger than the exciton Bohr radius the exciton is weakly confined leading to a hydrogen-like state. In the weak-confinement regime, the oscillator strength turns out to be proportional to the exciton volume Hanamura (1988); Andreani et al. (1999); Stobbe et al. (2012) or, for a disc-shaped quantum dot, to the area, i.e.,


This remarkable phenomenon is known as the giant-oscillator-strength effect and may be considered an analogue of superradiance. It implies that the oscillator strength can be significantly enhanced by increasing the lateral extent of the quantum dot. This prediction has lead to significant experimental efforts to observe the giant-oscillator-strength effect, which is challenging since non-radiative processes could mask the effect. The indirect extraction of the oscillator strength in large InGaAs quantum dots obtained from the founding work on strong coupling in cavity QED Reithmaier et al. (2004) has turned out to be inconsistent with direct measurements of the oscillator strength from time-resolved spectroscopy, since non-radiative recombination was found to be relevant Stobbe et al. (2010). This discrepancy is likely a result of the influence of additional excitons feeding the cavity, as discussed in further details in Sec. VII. Indications of a large oscillator strength have been reported for GaAs interface-fluctuation quantum dots from time-resolved experiments Peter et al. (2005); Hours et al. (2005).

ii.5.3 The quantum efficiency

Emission-energy dependence of fundamental quantum-optical properties of quantum dots. (a) Emission spectrum of an InGaAs quantum-dot ensemble. (b) Total decay rate (red), which is the sum of the radiative (black) and non-radiative (blue) decay rates. The solid blue line is the predictions from a scaling model for non-radiative recombination at the quantum-dot surface. The solid black line is a fit using a numerical solution to the effective-mass equation with the quantum-dot heights shown in the upper scale. (c) Quantum efficiency, (d) oscillator strength, and (e) wave-function overlap where the data points (solid lines) are directly extracted from the data (theory) in (b). The wave-function overlap governs the radiative properties of excitons. Data reproduced from

Exciton-recombination processes in quantum dots may suffer from non-radiative contributions, which add undesirable losses to the radiative decay. The relative strength of the radiative and the non-radiative decays is quantified by the fraction of recombination events leading to photon emission, i.e., the quantum efficiency,


where is the radiative decay rate in a homogeneous reference medium, e.g., GaAs in the case of Stranski-Krastanov quantum dots. In optics applications it is desirable to have

Measuring the quantum efficiency can be challenging since it is non-trivial to separate radiative and non-radiative processes. For example, if , which is typically the case, the fast decay rate probed in a time-resolved photoluminescence experiment is . Figure 5 shows experimental data on the key optical properties of quantum dots where a systematic variation in the projected local density of optical states (LDOS; discussed at length in Sec. IV) was used to modify the radiative rate while leaving the non-radiative decay unaffected Johansen et al. (2008) and thus enabling their separation. The energy dependence of the total decay rate across the emission spectrum of the quantum-dot ensemble (Fig. 5(a)) is shown in Fig. 5(b) and turns out to be influenced by the variation in as reflected in the drop in quantum efficiency for high energies shown in Fig. 5(c). The radiative rate and thus also the oscillator strength, cf. Fig. 5(d), were successfully modeled by Eq. (4) that predicts that they are proportional to the overlap of the envelope wavefunctions as shown in Fig. 5(e). The observed energy dependence is due to the different effective masses of electrons and holes such that the wavefunction confinement inside the quantum dot differs with energy. While this detailed information was extracted for an ensemble of quantum dots by applying a controlled modification of the LDOS, the radiative and non-radiative rates can also be extracted at the single quantum-dot level by recording the bi-exponential decay dynamics.

Little is known about the physical mechanism of non-radiative processes in quantum dots but the few existing experiments Stobbe et al. (2009) indicate that the non-radiative decay rate scales with the surface-to-volume ratio of the quantum dot. This shows that charge trapping at lattice defects at the interface between the quantum dot and the surroundings is likely a contribution to non-radiative recombination in Stranski-Krastanov quantum dots; this finding is consistent with the large variations in the non-radiative decay rate between different quantum dots that is typically found. Recently also mesoscopic light-matter interaction effects were found to contribute to this additional decay process Tighineanu et al. (2014b); for a further discussion of these effects see Sec. IV.5. Since the LDOS can be strongly altered in photonic nanostructures it is sometimes convenient to define an effective quantum efficiency, , which is the quantum efficiency of the emitter at a particular position, wavelength, and orientation in a photonic structure. Since may be significantly enhanced by enhancing the LDOS, the influence of non-radiative effects may be partially suppressed.

ii.5.4 Excitation schemes and tuning

One of the simplest yet very powerful experimental techniques for assessing the optical properties of quantum dots is photoluminescence spectroscopy, where the quantum dot is optically excited and the spontaneous emission recorded with a spectrometer equipped with a charged-coupled-device camera. The excitation energy can be either 1) above-band, i.e., above the band gap of the barrier material, 2) quasi-resonant, i.e., matching an excited excitonic state or continuum of states in the wetting layer, or 3) resonant, i.e., exactly matching the energy of the exciton transition under investigation. Detailed spectroscopic insight is obtained by monitoring the emission from the exciton while scanning the laser wavelength, which is known as photoluminescence excitation spectroscopy. Resonant excitation is particularly challenging and has been demonstrated only rather recently because the strong excitation laser must be efficiently filtered. Resonant excitation is discussed in detail in Sec. V. An alternative approach is to embed the quantum dots in a diode and mask the wafer surface with opaque metal except a small aperture atop the quantum dot, which allows probing resonant-excitation properties in the photocurrent. The diode may also be used in forward bias to create a single-photon light-emitting diode Yuan et al. (2002); Salter et al. (2010), which is appealing for certain applications but the drawback of this electroluminescence is that the energy and polarization of the injected carriers is more difficult to control as compared to controlling an excitation laser although there is currently significant progress Conterio et al. (2013). Nonetheless the ability to electrically excite quantum dot distinguish them from other optical emitters and may become of major practical relevance.

For many applications in quantum nanophotonics, precise spectral tuning of quantum dots is required. For instance, in the case of a cavity with a -factor of , the cavity linewidth is approximately and the observation of cavity-QED effects requires tuning the quantum dot into resonance with the mode to this precision. A wide tuning range is generally required since a quantum-dot ensemble is usually strongly inhomogenously broadened (can be up to 50-) and fabrication imperfections also lead to a significant spread in the actual cavity resonance frequencies. One possible mechanism exploits temperature tuning where both the cavity-resonance and exciton-transition energies shift in the same direction (red-shift with increasing temperature) but at a different rate so that they may be tuned into mutual resonance Englund et al. (2007). This has proven to be a very useful method that is easy to implement but may compromise coherence properties since increasing the temperature also increases the phonon population leading to dephasing. Another option is tuning with magnetic fields Stevenson et al. (2006), which depending on the orientation of the field relative to the quantum dot also changes the fine structure and the optical selection rules Bayer et al. (2002). However, this method has rather limited tuning range and it appears unfeasible to apply large local magnetic fields to individual quantum dots. Another option is adsorption of inert gasses, which changes the dielectric environment that in turn spectrally shift the optical response of the nanostructures while leaving the quantum dots unaffected. This works well for photonic crystals due to the relatively large field amplitude in the holes where the gas is adsorbed. However, for more advanced applications involving multiple quantum dots to be independently tuned, this method is not useful. Applying external strain enables not only tuning of the energy levels but also allows modifying the underlying electronic band structure to create light-hole excitons Huo et al. (2014). However, it appears challenging to incorporate integrate such strain tuning with high-quality photonic nanostructures. The most appealing tuning mechanism is to apply a static electric field to tune the quantum dots via the quantum-confined Stark effect. Electric fields may be readily applied between doped layers below and above the quantum dots forming a vertical p-i-n diode, which under reverse bias can be used to apply large fields. With a proper device design involving the tunneling barriers, a tuning range of corresponding to Bennett et al. (2010) has been demonstrated. The doping levels can be so small that absorptive losses do not limit the -factor of cavities or the transmission of waveguides. Independent tuning of multiple quantum dots could be achieved by etching isolation trenches Winger et al. (2011) or by local dopant implantation Ellis et al. (2011).

ii.6 Coherent single-photon emission from quantum dots

Quantum dots are mesoscopic emitters embedded in a solid-state environment and a number of processes not encountered in atomic physics play an important role for the single-photon emission from quantum dots. In particular, phonons, charge fluctuations associated with lattice defects or impurities, and spin fluctuations in the ensemble of nuclei give rise to dephasing, spin-flip, and non-radiative decay processes. Such processes can influence the quality of the single-photon emission from quantum dots and in particular determine the coherence properties. Highly coherent single photons are required for quantum-information processing applications and in the present section we briefly outline the main decoherence processes and governing parameters for quantum dots.

In order to assess the quality of a single-photon source it is essential to develop precise experimental methods. To this end, the photon statistics of a quantum state of light is determined by recording the second-order correlation function defined as


where and are annihilation and creation operators for the optical mode probed in the experiment and is a time delay introduced in between two subsequent measurements of the number of photons at time and , respectively. Experimentally, can be recorded by dividing the light beam on a beam splitter and recording with two different photodetectors the number of photons with an electronically controlled delay in each path. The second-order correlation function is independent of in a stationary experiment. This is referred to as the Hanbury-Brown-Twiss setup. The value at is particularly important because it directly determines the quality of the single-photon source. For Fock states, , where is the number of photons and thus for a true single-photon source. For a coherent (thermal) state, (). Consequently, can be employed as a measure of the single-photon purity of a light source. In the case of quantum dots, the recombination of a single exciton generates a single photon, but the recorded signal may be polluted by photons emitted through other recombination processes on different transitions. In experiments, much effort is devoted to the selective excitation of a single exciton in a quantum dot and the subsequent spectral filtering of the emitted light in order to suppress any multi-photon contributions from other emission processes.

Information about the coherence of the emitter can be obtained from the spontaneous-emission spectrum. A number of different dephasing mechanisms will in general influence a quantum dot that can be included at various levels of sophistication when modelling experimental data. In the simplest approximation, the dephasing reservoir is assumed to be Markovian and described by a single rate . In this case the emission spectrum exhibits the well-known Lorentzian form with a linewidth determined by the spontaneous-emission lifetime and dephasing times according to


where is the total coherence time (i.e., the inverse linewidth), is the inverse total decay rate of the emitter, and is the pure-dephasing time. We note that due to dephasing, the coherence of the emitter usually decays faster than the population. In the absence of dephasing, , the two-fold slowdown of coherence relative to population reflects that the former depends linearly on the complex expansion coefficient of the excited state of the emitter while the latter depends quadratically. The theory of spontaneous-emission dynamics and coherence is treated in detail in Secs. IV.1 and IV.2. The pure-dephasing model captures the basic feature of a broadening of the emission spectrum and may be used to describe, e.g., the broadening of the zero-phonon line of the emission spectrum Muljarov and Zimmermann (2004). Other dephasing processes in quantum dots include charge-fluctuation noise and spin noise from the coupling of the exciton to the spins of the nuclei Kuhlmann et al. (2013a). The charge noise arises from the presence of a fluctuating distribution of charges in the vicinity of the quantum dot and leads to slow (longer than milliseconds) spectral diffusion of the quantum-dot resonance. To a large extent, this can be overcome by applying resonant excitation on electrically contacted quantum dots whereby near-transform-limited optical transitions have been observed Kuhlmann et al. (2013b). Residual spectral diffusion can be overcome by locking the quantum dot to a stable reference Prechtel et al. (2013). It should be emphasized that an adequate description of these dephasing processes would include how the functional form of the emission spectrum is modified by the dephasing, which is not included in the pure-dephasing model where the emission spectrum remains Lorentzian. Furthermore, phonon broadening is generally much faster (picosecond time scales) than the emitter lifetime and therefore also not captured by the pure-dephasing model. The phonon sidebands of the emission spectra are discussed below.

Another experimental test of single-photon coherence exploits a Hong-Ou-Mandel interferometer. In such an experiment, two single photons are interfered on a 50/50 beam splitter and the two outputs are recorded with single-photon detectors as shown in Fig. 6(a). This constitutes a generalization of the Hanbury-Brown-Twiss setup to a case where two input beams are incident on the beam splitter as opposed to just one. The degree of indistinguishability for a photon source that can be extracted with a Hong-Ou-Mandel interferometer is defined as Bylander et al. (2003); Kiraz et al. (2004); Kaer et al. (2013)


where the expressions in the second line hold for the case of pure (i.e., Markovian) dephasing. The lack of any coincidence detection events in a Hong-Ou-Mandel measurement signifies that the two photons are fully indistinguishable, which is only the case if they are fully coherent. Such single-photon quantum interference is a direct ingredient in many quantum-information protocols and therefore an important test of a single-photon source. With quantum dots, Hong-Ou-Mandel interferometry has been performed by interfering two consecutively emitted photons from the same quantum dots, see Fig. 6(b). The photons were found to be partly indistinguishable, which is a consequence of dephasing of the quantum-dot levels. The degree of indistinguishability is determined from the peak area of the coincidence events detected when two consecutively emitted photons meet on the beam splitter. As opposed to the emission spectrum, which is sensitive to the slow spectral-diffusion processes, indistinguishability measurements on two consecutively emitted photons from the same source are only sensitive to fast dephasing occurring on a time scale shorter than or similar to the time delay between the two photons, such as phonon processes. As a consequence, a quantum-dot transition suffering from slow spectral diffusion, which is often the case for non-resonant excitation schemes, may still be capable of emitting highly indistinguishable photons.

a) Sketch of a Hong-Ou-Mandel (HOM) interferometer capable of measuring the degree of indistinguishability of two consecutively emitted photons from a quantum dot. The setup employs an asymmetric Michelson-Morley interferometer for compensating the path-length difference between the two photons enabling them to meet in one quarter of all incidences at the second passage through the beam splitter and thus to interfere. The quantum interference is gauged by recording the correlation function
Figure 6: a) Sketch of a Hong-Ou-Mandel (HOM) interferometer capable of measuring the degree of indistinguishability of two consecutively emitted photons from a quantum dot. The setup employs an asymmetric Michelson-Morley interferometer for compensating the path-length difference between the two photons enabling them to meet in one quarter of all incidences at the second passage through the beam splitter and thus to interfere. The quantum interference is gauged by recording the correlation function . (b) Example of HOM measurements for a quantum dot in a micropillar cavity. For distinguishable pairs of photons a five-peak spectrum with peak area ratios of 1:2:2:2:1 from the two encounters of the beam splitter is expected. A strong suppression of the central peak (labelled 3) relative to the neighboring peaks 1-2 and 4-5 is the experimental evidence for quantum interference of the two photons that is possible only for (partly) indistinguishable photons. Reprinted with permission from Santori et al. (2002).

In the following we will discuss further dephasing induced by phonons that is unavoidable for quantum dots since they are embedded in a solid-state material. It should be emphasized that at elevated temperatures additional effects play a role, such as Coulomb-induced scattering to wetting layer states Steinhoff et al. (2012), but such effects are outside the scope of the present account and for further details, see Kira and Koch (2011). By cooling down the quantum dots to close to zero absolute temperature, the influence of phonons can be strongly suppressed, although spontaneous emission of phonons prevails. Longitudinal acoustic (LA) phonons are a major broadband source of dephasing for quantum dots. A quantum dot can absorb or emit phonons through inelastic processes that generally require a description of coupling to a non-Markovian reservoir. LA phonons induce an asymmetric broadening of the emission spectrum reflecting that the probability of absorbing (emitting) a phonon of frequency is proportional to (), where is the phonon occupation at the relevant wave vector . A typical LA-phonon energy that leads to phonon sidebands of a quantum dot is about Madsen et al. (2013) where the relevant phonon energy decreases with the size of the exciton. At a temperature of , we estimate . The coupling between an exciton in a quantum dot and a phonon reservoir can be described by the interaction Hamiltonian Krummheuer et al. (2002); Mahan (2000); Kaer et al. (2012)


where and are bosonic annihilation and creation operators for the phononic mode and the raising and lowering operators for the quantum dot are introduced in Sec. IV.1. The quantum dot-phonon interaction strength is defined as


with and where a linear LA-phonon dispersion relation, , is assumed for the relevant frequency range with the angular-averaged speed of sound. Furthermore, is the mass density, the quantization volume of the phonon modes, and and the deformation potential and electronic wave function for the state , respectively. The quantum-dot-phonon interaction strength depends sensitively on the quantum-dot size since it is proportional to the spatial Fourier transform of the wavefunction for the electron in either the ground or excited state. As a consequence, large excitons are more robust to dephasing compared to smaller excitons. Figure 7 shows examples of calculated spontaneous-emission spectra for a single quantum dot at various temperatures clearly demonstrating the asymmetric broadening due to LA phonons.

Calculated normalized emission spectrum (logarithmic scale) for a quantum dot coupled to an LA-phonon reservoir at different temperatures. The phonon sidebands are highly asymmetric at low temperatures. The parameters used are experimental values reported in
Figure 7: Calculated normalized emission spectrum (logarithmic scale) for a quantum dot coupled to an LA-phonon reservoir at different temperatures. The phonon sidebands are highly asymmetric at low temperatures. The parameters used are experimental values reported in Madsen et al. (2013). Note that the peak of each spectrum is normalized to unity. The inset shows the fraction of the intensity in the sidebands as a function of temperature.

Iii Photonic nanostructures

The essential challenge for quantum-optics experiments at the single-photon level is to strongly enhance and control the interaction between light and matter such that an emitted single photon preferentially couples to one well-defined optical mode. To this end, photonic nanostructures are very well suited due to their ability to tailor the electromagnetic field on a length scale that can be made a fraction of the optical wavelength. Photonic nanostructures with excellent optical properties can be fabricated with modern nanofabrication methods such as molecular-beam epitaxy, electron-beam lithography, and etching. In the present section we introduce the various types of photonic nanostructures that have been employed in QED experiments and compare their relevant figures of merit.

iii.1 Photonic crystals

Photonic crystals are inhomogeneous dielectric materials where the refractive index is modulated periodically on a length scale determined by the optical wavelength. In such a structure, light propagation is controlled by optical Bragg scattering of light, which is the optical analogue of electron Bragg diffraction employed in crystallographic experiments on solids. The strength of Bragg scattering increases with the refractive-index contrast of the materials composing the photonic crystal. Furthermore, any optical loss such as absorption is detrimental to the functionality of a photonic crystal since it suppresses light interference and therefore Bragg scattering.

A photonic crystal can be described by a spatially periodic dielectric permittivity that in general has both a real and an imaginary part where the latter is linked to the absorption. The most effective photonic crystals are typically made from semiconductor materials and constructed for a frequency range where absorption is sufficiently small such that it can be ignored to a good approximation, i.e., . Furthermore, for the narrow band frequency applications that are usually considered in the context of quantum optics, any frequency dependence of the dielectric permittivity can be ignored. Here we will mainly consider semiconductor materials such as gallium arsenide or silicon with the large refractive index of where large-contrast photonic crystals can be obtained, e.g., by etching air voids.

A multitude of photonic crystals with different crystal symmetries or material compositions and based on different fabrication strategies have been studied over the years since photonic crystals were first proposed Bykov (1975); Yablonovitch (1987); John (1987). Here we are mainly concerned with photonic-crystal membranes made of GaAs as illustrated schematically in Fig. 8(a). These structures have proven very well suited for quantum-optics experiments since they can be fabricated with high precision and InGaAs quantum dots with good optical properties can be naturally incorporated. For a thorough review of photonic crystals including details on fabrication, optical experiments, and numerical modeling, see Busch et al. (2007) and Joannopoulos et al. (2008).

(a) Band diagram of a triangular-lattice photonic-crystal membrane (shown in the left inset) for TE-like modes with membrane refractive index
Figure 8: (a) Band diagram of a triangular-lattice photonic-crystal membrane (shown in the left inset) for TE-like modes with membrane refractive index (corresponding to GaAs), hole radius to lattice constant ratio of , and membrane thickness . The high-symmetry points of the Brillouin zone are shown in the bottom inset. The light blue shaded area indicates the continuum of unbounded radiation modes inside the light cone, and the yellow shading shows the 2D photonic band-gap area. (b)-(c) Spatial map in the - () plane of the inhibition factor at a scaled frequency of for (b) an -dipole and (c) a -dipole where is the density of state for a homogeneous medium of GaAs. (d) Frequency dependence of the LDOS plotted on a logarithmic scale for an -dipole (red curve) and a -dipole (blue curve) positioned in a photonic-crystal membrane at the crosses shown in (b) and (c).

The optical modes of a photonic crystal obey Bloch’s theorem, i.e., the electric field satisfies Joannopoulos et al. (2008)


where is a Bloch wave vector and is any vector in the Bravais lattice spanning the periodic photonic crystal. The Bloch modes are generally strongly dispersive, as can be seen in the dispersion diagram in Fig. 8(a) that plots the frequency versus for various high-symmetry directions in the reciprocal lattice. This strong structural dispersion can be employed for tailoring light propagation, and, e.g., slow light can be obtained, as considered in further detail in Sec. VI. A photonic band gap opens if Bragg scattering is so pronounced that no modes exist for a range of frequencies. A complete photonic band gap inhibiting all modes for any propagation direction and polarization can only be obtained in photonic crystals with periodicity in all three dimensions, and even then only for very high refractive-index contrast and certain crystal lattices. Importantly, even in lower-dimensional photonic-crystal structures, pronounced pseudo-gaps exist that modify major parts of the optical modes. Consequently, the density of optical states can be strongly modified, which is the basis for all quantum-optics applications of photonic crystals. Photonic-crystal membranes, cf. Fig. 8, have proven to be particularly well suited. In these structures the propagation of light in the plane of the membrane is suppressed due to the 2D photonic band gap, while leakage out of the structure is restricted to a narrow cone of wave vectors in the case of a high-refractive-index-contrast semiconductor-air interface. In quantum-optics experiments the relevant leakage rate is that experienced by a quantum emitter in the nanostructure that is generally strongly dependent on emitter position and frequency. The photonic-crystal membranes have the obvious experimental advantage compared to full 3D photonic crystals that embedded quantum emitters can conveniently be probed by laser excitation from the top of the membrane, which enables spectroscopy on a single emitter. Furthermore, very mature planar fabrication methods can be readily employed to obtain high-quality photonic crystals.

(a) Micropillar cavity (b) Photonic-crystal cavity (c) Nanobeam cavity (d) Microdisk cavity
Reithmaier et al. (2004) Ohta et al. (2011) Srinivasan and Painter (2007)
Loo et al. (2010) Hennessy et al. (2007) Ohta et al. (2011) Srinivasan and Painter (2007)
Table 1: Overview of nanophotonic cavities. Each panel displays a scanning electron micrograph of a real device along with a sketch illustrating the operational principle for a quantum emitter coupling to the structure. Furthermore, state-of-the-art experimental results are listed, to be discussed in Sec. VII. (a) Micropillar cavity (micrograph reprinted with permission from Reithmaier et al. (2004)). The Bragg stack above and below the center of the pillar confines light to the central region as shown in the inset. (b) Modified photonic-crystal L3 cavity implemented in a membrane. The photonic band gap localizes light in the defect region and the schematic shows how a quantum dot preferentially emits into the cavity mode. (c) A nanobeam cavity (micrograph reprinted with permission from Ohta et al. (2011)). The cavity mode is confined by 1D Bragg diffraction in the high-refractive-index material of the nanorod. (d) Microdisk cavity (micrograph reprinted with permission from Srinivasan and Painter (2007)). The emitter couples to optical modes that travel circularly around the microdisk. The asterisk () indicates that in these cases was extracted from spectral rather than time-resolved data, i.e., it will be largened by dephasing processes.
(a) Photonic-crystal waveguide (b) Photonic nanowire (c) Plasmonic nanowire
Sapienza et al. (2010) Claudon et al. (2010) Akimov et al. (2007)
Arcari et al. (2014) Rao and Hughes (2007b) Claudon et al. (2010) Bleuse et al. (2011) Akimov et al. (2007) Chang et al. (2006)
Table 2: Overview of nanophotonic waveguides. Each panel displays a scanning electron micrograph of a real device with a sketch illustrating the operational principle for a quantum emitter coupling to the structure, as well as experimentally measured (theoretically calculated) -factors and Purcell factors, and ( and ), as discussed in Sec. VI. The bottom panel shows the references for the Purcell and -factors. (a) Photonic-crystal waveguide membrane containing a single layer of quantum dots in the center of the membrane (indicated as yellow triangles) (micrograph reprinted with permission from Sapienza et al. (2010)). (b) Photonic nanowire made of GaAs containing an InAs quantum dot (illustrated as a triangle). The upper part of the nanowire is tapered in order to maximize the outcoupling efficiency and a gold mirror is embedded in the structure below the nanowire (micrograph reprinted with permission from Claudon et al. (2010)). The stated value of the experimental -factor represents a lower bound determined from estimates of the total out-coupling efficiency. (c) Plasmonic nanowire made of silver that is coupled to a single quantum dot (placed within the red circle) (micrograph reprinted with permission from Akimov et al. (2007)).

In quantum-optics experiments with dipole emitters, the LDOS is the relevant quantity that determines spontaneous emission and more generally the local light-matter interaction strength. It specifies the number of optical states at the frequency per frequency bandwidth and volume as experienced by the emitter, and is defined as


where is the position of the emitter and a unit vector specifying the orientation of the transition dipole moment. The normalized mode functions, , in which the electric field is expanded, are described in Sec. IV.1. The LDOS is obtained by summing over all the mode functions with eigenfrequencies that enter through a Dirac delta function . The LDOS was first introduced by Sprik et al. (1996) and subsequently evaluated for the case of photonic crystals by Busch and John (1998).

The LDOS in a photonic crystal is strongly dependent on position and orientation. Figure 8(d) shows an example of a calculation of the frequency dependence of the LDOS for a GaAs photonic-crystal membrane. A frequency region with a strongly suppressed LDOS is revealed, which corresponds to the 2D photonic band gap for the transverse-electric-like (TE-like) fields in which case are the dominant field that are even functions with respect to . In the band gap, a dipole oriented in the plane of the membrane will therefore only radiate weakly via the small residual coupling to the non-guided radiation modes that are found above the light line in the dispersion diagram in Fig. 8(a). The high refractive index of GaAs ensures that the coupling to radiation modes is strongly suppressed. Indeed Fig. 8(b) shows that the LDOS can be inhibited by up to a factor of relative to that of a homogenous medium of GaAs. Furthermore, the periodic spatial dependence of the inhibition factor in the photonic-crystal lattice is displayed for the two orthogonal in-plane dipole orientations that are relevant for quantum dots. While both dipoles are found to be suppressed at all positions compared to the case of a homogeneous medium, a strong anti-correlation is generally observed, i.e., two orthogonally polarized dipoles would observe a very different LDOS. The ability to suppress the coupling to unwanted leaky modes by more than two orders of magnitude is the main asset of photonic-crystal membranes in quantum-optics experiments, which was first realized by Koenderink et al. (2006). As discussed in Secs. VI and VII, by introducing waveguides and cavities in the photonic crystals, the coupling to a preferred optical mode can be enhanced significantly. As a consequence, photonic crystals offer the possibility to tailor the local light-matter interaction strength by 3-4 orders of magnitude by combining suppression of unwanted leaky modes with enhancement of a single mode. With this approach a nearly perfect photon-emitter interface may be obtained; the physics of which is the main objective of the present review.

iii.2 Photonic cavities

Resonating light in a cavity provides a way of enhancing light-matter interaction since the coupling to one localized mode can be strongly enhanced compared to all other modes. In photonic nanostructures the cavities can have very small mode volumes, which enhances the interaction strength. The two decisive cavity-QED figures of merit are the quality factor and the effective mode volume of the localized quasi-mode. In the following, various nanophotonic approaches to high- and small- cavities are reviewed.

Micropillar cavities have been widely exploited as high- cavities. They can be fabricated by epitaxial growth of alternating layers of refractive indices and (e.g., GaAs and AlGaAs) each of thickness , and by subsequently etching a micropillar with a typical diameter of a few microns and a height of around . The alternating layers form a Bragg mirror with a reflectivity that is controlled by the number of layers. An extended spacer layer of length, e.g., in between two such Bragg mirrors forms a highly localized cavity mode where quantum dots can be positioned, see Table 1(a). The two Bragg mirrors are often grown with different numbers of layers in order to make a one-sided cavity consisting of a highly-reflecting mirror and an out-coupling mirror with an optimized transmission. The diameter of the micropillar is chosen to restrict the lateral extension of the guided mode confined in the high-index material, which leads to a small mode volume. Choosing the diameter below 1-, however, is found to significantly reduce due to the sensitivity to sidewall roughness Gazzano et al. (2013b). Typical mode volumes accessible in micropillar cavities are at the level of and tapered cavities have recently been proposed as a way of localizing light even better Lermer et al. (2012).

Highly localized cavity modes can be obtained by introducing defects in photonic-crystal membranes and a multitude of different design possibilities have been explored in the literature. Indeed this flexibility of being able to tailor a cavity mode by controlling the geometry of the photonic crystal and the defect area provides a very important asset of the photonic-crystal platform. One of the most successful and important designs so far is the L3 cavity that is obtained from a 2D triangular lattice in a photonic-crystal membrane by leaving out three holes, cf. Table 1(b), leading to a mode volume of less than . It was realized that the -factor can be very significantly boosted by more than an order of magnitude by displacing the holes at each end of the cavity by just a fraction of a lattice constant, whereby leakage to radiation modes can be strongly suppressed Akahane et al. (2003). This remarkable sensitivity to the detailed design reflects the large potential of photonic-crystal cavities. A cavity -factor of has been observed experimentally in modified L3 cavities in silicon at a wavelength of Lai et al. (2014). Lower -factors are generally observed for GaAs cavities since they are matched to the shorter wavelength of InGaAs quantum dots of around and therefore more sensitive to fabrication imperfections, residual scattering and absorption due to the embedded quantum dots, as well as bulk and surface absorption, which is more pronounced at shorter wavelengths Michael et al. (2007). A cavity -factor of has been reported in a GaAs L3 cavity containing a single quantum dot Hennessy et al. (2007).

A number of different photonic-crystal cavity structures have been considered that potentially have higher -factors including waveguide heterostructures Song et al. (2005) for which a -factor of in a GaAs cavity with quantum dots has been reported Ota et al. (2011), nanobeam cavities Ohta et al. (2011), or even random Anderson-localized modes in photonic-crystal waveguides due to naturally occurring fabrication imperfections Topolancik et al. (2007); Smolka et al. (2011). The nanobeam cavity displayed in Table 1(c) is a 1D photonic crystal consisting of etched holes in a narrow and thin membrane with a central defect area defining the cavity region. Finally microdisk cavities, see Table 1(d), constitute another class of resonators where the trapping of light is due to total internal reflection rather than Bragg scattering. In a microdisk cavity the light is confined to the rim of the disk in a whispering-gallery mode that can have a very high , generally, however, at the expense of larger mode volume compared to cavities employing Bragg scattering. Table 1 summarizes the various cavity configurations including figures of merit in relation to their usage for cavity-QED experiments. Such cavity-QED experiments employing single quantum dots are reviewed in Sec. VII.

iii.3 Nanophotonic waveguides

Nanophotonic waveguides enable routing photons between different locations on an optical chip, and could therefore be applied in integrated quantum networks for connecting stationary qubits (e.g., encoded in quantum dots) with flying qubits (photons). Nanophotonic waveguides can be highly dispersive, which may be employed for enhancing light-matter interaction as an alternative to the cavity case. In a waveguide, the quantum emitter can efficiently and over a wide bandwidth couple single photons directly to a propagating optical mode for immediate applications without the necessity of coupling out of a localized mode, which would be the case for high- cavities. In reality low- extended cavities may constitute the best compromise in order to obtain both a large Purcell enhancement and highly efficient and broadband coupling.

Waveguides can readily be implemented in photonic-crystal membranes. The most simple design consists of leaving out a row of holes along the - direction of a triangular lattice, which is referred to as a W1 waveguide, see Table 2(a). Figure 16(a) in Sec. VI.1 displays the projected dispersion diagram for a photonic-crystal waveguide membrane. It is found that bands arise in the 2D band gap region corresponding to propagating modes that are spatially confined to the waveguide. These guided modes are highly dispersive; i.e., the group velocity of light varies with frequency. The group velocity can be strongly reduced in a photonic-crystal waveguide and in general be tailored by controlling the structural parameters of the photonic lattice. Furthermore, since the waveguide modes appear below the light line in the dispersion diagram (cf. Fig. 16), an ideal photonic-crystal waveguide features lossless propagation. In reality, unavoidable fabrication imperfections (cf. Sec. III.4) induce a finite leak rate of coupling vertically out of the waveguide. Two main features of the photonic-crystal waveguides are important for their quantum-optics applications: the ability to suppress the coupling to radiation modes due to the band gap and the simultaneous enhancement of coupling to the photonic-crystal waveguide mode that is enhanced by slow light. This is considered in detail in Sec. VI. A number of other potential applications of photonic-crystal waveguides for integrated photonics have been envisioned that exploit photonic band-gap guiding of light. For a review of this topic see Joannopoulos et al. (2008).

Photonic nanowires constitute another class of waveguide-type photonic nanostructure. They are extended cylindrical rods of a high-refractive-index dielectric material, see Table 2(b). Photonic nanowires can either be fabricated by etching or be epitaxially grown, e.g., in the form of GaAs nanowires with embedded quantum dots. For diameters in the range of 150- the nanowire supports a single well-confined guided mode at Friedler et al. (2009), which is a typical wavelength for InGaAs quantum dots. The bandwidth of this mode is up to , which makes nanowires insensitive to the wavelength of the employed emitter. As opposed to photonic-crystal waveguides, the mode of the nanowire is weakly dispersive, which means that slow-light enhancement of light-matter interaction cannot be obtained. Therefore applications of nanowires in quantum optics are primarily based on the ability to suppress leaky modes rather than enhancing a single mode, which makes them less sensitive to fabrication imperfections at the expense of a less efficient light-matter coupling strength compared to photonic-crystal waveguides.

Finally, metallic waveguide structures can guide light even for a wire diameter that is just a tiny fraction of the optical wavelength in free space. Such sub-wavelength guidance can potentially lead to a very large electromagnetic field strength per photon, as required in quantum-optics experiments. In metallic waveguides, the light is guided in the form of surface-plasmon polaritons that are surface waves confined to a metal-dielectric interface that propagate along the interface while being exponentially damped in the direction perpendicular to the interface. An example of a plasmonic nanowire is shown in Table 2(c). Just like dielectric nanowires, the response of plasmonic nanowires is broadband, with the added benefit that the strong field confinement and dispersion of plasmons imply that the light-matter interaction strength can be strongly enhanced. Unfortunately, plasmonic nanowaveguides are inherently lossy due to absorption in the metal, so while they may be useful for locally enhancing light-matter interaction, the subsequent guiding of photons is preferably carried out in dielectric structures. To accommodate this point, plasmon-based quantum circuits have been proposed where the plasmon mode is adiabatically coupled to a dielectric waveguide Chang et al. (2006). Another possibility is to use a slot waveguide created by placing two plasmon waveguides next to each other where a plasmon mode can form in the gap region between the waveguides with enhanced propagation length. Plasmon nanowires can be fabricated by, e.g., chemical synthesis of crystalline structures, metal evaporation in trenches written by electron-beam lithography, or etching of thin metal films. While the theoretical potential of plasmonic nanostructures seems very promising Chang et al. (2006), the fabrication of high-quality plasmonic waveguides still remains a challenge given the design, material, and fabrication restrictions presently valid Chen et al. (2010). Consequently, at present dielectric waveguides are advantageous for quantum-optics applications.

iii.4 The role of fabrication imperfections

Photonic nanostructures can be fabricated by a range of different techniques using either top-down approaches such as electron-beam lithography and etching or bottom-up approaches based on self-assembly of nanoscale objects. Almost all the examples of nanostructures discussed here are fabricated by patterning semiconductor substrates by electron-beam lithography followed by reactive-ion and/or wet-chemical etching. These methods are compatible with self-assembled quantum dots and are also most commonly employed for fabricating integrated photonic circuits. A detailed account of the various fabrication methods is outside the scope of the present review, but for further details we refer to Busch et al. (2007) and references therein. Common for all methods is that statistical imperfections are unavoidably introduced during the fabrication process, which potentially can reduce the functionality of the device by, e.g., inducing unwanted optical loss. The fabrication imperfections can be of various sorts including roughness, polydispersity, displacements, vacancies, etc. Such random imperfections are treated by means of statistical physics and eventually can lead to pronounced random multiple scattering of light even for sub-wavelength features because high-refractive-index-contrast composites induce large scattering cross sections. In state-of-the-art dielectric photonic nanostructures the amount of fabrication imperfections can be reduced to a standard deviation at the level of a few nanometers Garcia et al. (2013). The relevance of such an amount of imperfections is fully dependent on the actual device under consideration, and the role of fabrication imperfections should be addressed individually for each application.

Extensive theoretical and experimental work has been carried out on the role of fabrication imperfections in photonic-crystal waveguides Hughes et al. (2005); Mazoyer et al. (2009); Garcia et al. (2013). Imperfections lead to two effects: the backscattering of the propagating mode into the counter-propagating direction in the waveguide and out-of-plane scattering due to coupling to leaky modes that are above the light line. The former (latter) is characterized by an average length () leading to a total extinction length that determines how the average intensity decays along the waveguide. It should be emphasized that while a single realization of imperfections leads to a complex intensity speckle pattern, the extinction length predicts how light decays on average and is obtained after an ensemble average over configurations of disorder. The back-scattering length is strongly dispersive in the photonic-crystal waveguide and in a perturbative description was predicted to scale as Hughes et al. (2005). In the context of 1D Anderson localization the back-scattering length is referred to as the localization length and has been measured to be below in the slow-light regime of state-of-the-art photonic-crystal waveguides Smolka et al. (2011), which has been the basis for experiments on cavity QED with Anderson-localized modes Sapienza et al. (2010). The resulting average loss length associated with out-of-plane leakage was found to be around . By operating outside the slow-light regime, the propagation distance can be significantly increased. Alternatively, waveguides shorter than the back-scattering length can be applied, such that the strong light-matter interaction achievable in photonic waveguides can be exploited without suffering from Anderson-localization effects. This will be the setting of the description of waveguide quantum optics of Sec. VI.

Iv Spontaneous emission of single photons from solid-state emitters in photonic nanostructures

Illustration of spontaneous emission from a two-level emitter in a photonic medium. Single photons (red wave packets) are emitted one at a time from a quantum dot (QD) placed in a nanophotonic structure. For a full description of spontaneous emission in nanophotonics, the potential influence of the back action (indicated by red arrows) of the vacuum electric field on the quantum emitter must be accounted for, which can lead to intricate non-exponential dynamics and strongly modified emission spectra.

Much of the physics discussed in the present review concerns a single quantum emitter that is emitting (or absorbing) a single quantum of light, a photon, into (or from) an engineered photonic environment. In this section we describe a general theoretical framework for this setting, which is illustrated in Fig. 9, and encompasses photon emission in any inhomogeneous photonic nanostructure. The quantum emitter is described as a two-level dipole emitter but the formalism can be readily generalized to describe more complex level schemes, cf. the discussion of quantum-dot level schemes in Sec. II.5. The coupling to the photonic nanostructure is described by the projected LDOS, which quantifies the magnitude of vacuum fluctuations responsible for spontaneous emission of photons. The LDOS is linked to the electric-field Green’s tensor that contains all details about the spatio-temporal properties of the electromagnetic field.

In the following, the basics of the LDOS formalism is discussed in some detail, since it is a powerful and generally applicable framework for QED in any nanophotonic structure and appears not to be textbook material. Key quantities addressed include the spontaneous-emission spectrum and dynamics beyond the Markov approximation. The Markov approximation holds when radiation back action from the reservoir is negligible, but may break down in, e.g., photonic crystals and nanocavities due to the large local electromagnetic field strength and the strongly dispersive behavior. Photonic nanostructures can be employed for tailoring the quantum vacuum and thereby to control fundamental QED-processes and interactions like spontaneous emission, the Lamb shift, Casimir forces, or dipole-dipole interactions Milonni (1993). In particular we explain in the following how the Lamb shift of electronic transitions can be strongly modified in photonic nanostructures. Another benefit of the LDOS formalism is that the spontaneous-emission rates are rigorously introduced. This contrasts more specific QED models, such as the commonly used dissipative Jaynes-Cummings model of cavity-QED discussed in Sec. VII, where spontaneous-emission rates are included as phenomenological parameters.

iv.1 Equations of motion for spontaneous emission in the local-density-of-states description

We consider a single two-level dipole emitter with the excited state and ground state separated by the transition frequency and coupled to a continuum of optical modes at frequencies . The total Hamiltonian in the rotating-wave approximation is


where , are annihilation and creation operators, respectively, for photons in the mode and , are the emitter-population inversion and raising operators, respectively. The coupling rate to each optical mode is , which contains the transition dipole moment of the emitter and the local electric field at the position of the emitter, , for the mode specified by and there are two polarization components for each wave vector. h.c. denotes the Hermitian conjugate of the preceding term. As mentioned in Sec. II.3, the interaction Hamiltonian is often used in solid-state quantum optics and leads also to Eq. (14). Dephasing processes are omitted in the following but Markovian dephasing can be readily implemented through a non-Hermitian term in the Hamiltonian, which is valid in a Monte-Carlo wave-function description of spontaneous emission Meystre and Sargent (2007). Furthermore, a thorough discussion of field quantization in a dielectric medium is outside the scope of the present account, but further information can be found in Wubs et al. (2003).

The equation of motion for the excited state of the emitter can be formulated as


with and where the combined quantum state of light and matter has been expanded as . Here is the collective vacuum state of all modes in the radiation reservoir and corresponds to one photon in the mode with wave vector . By the restriction of having only a single excitation in the system, the formalism is suitable for describing spontaneous emission or single-photon absorption. In photonic nanostructures the interaction strength and thus the dynamics depends on the position of the emitter, , but this argument has been omitted in and for simplicity of notation. The last term in Eq. (15) vanishes in the case of spontaneous emission , but it can readily be included for describing single-photon absorption, as was detailed in Chen et al. (2011) for the case of a 1D waveguide geometry.

The quantized electromagnetic field is expanded according to


where defines the field amplitude entering in the local coupling rate . The mode functions constitute a normalized set of basis functions used to expand the field and obey the wave equation Yao et al. (2010); Novotny and Hecht (2012)


with a generalized normalization condition given by In photonic crystals a convenient choice of basis functions is Bloch modes. It is useful to express the electric field in terms of Green’s tensor , which solve Maxwell’s wave equation in the photonic medium for a delta-function source term Yao et al. (2010); Novotny and Hecht (2012), i.e.,


where is the unit tensor. Here the case of no free charges is considered where the generalized transversal condition applies. Furthermore, longitudinal components in Green’s tensor only contribute in absorptive media leading to non-radiative decay, which generally should be minimized in quantum-photonics applications. Green’s tensor can be expanded in the mode functions according to


where denotes the outer product. The electric field radiated by a dipole source at can be obtained in the Green-tensor formalism from


where .

The equation of motion for the emitter, Eq. (15), can now be expressed in terms of Green’s tensor. It follows from Eq. (19) that Barnett et al. (1996)


which links the summation over the radiation reservoir in Eq. (15) to the LDOS, which was defined in Eq. (13). It follows consequently that


Equation (22) provides a complete description of spontaneous emission in any inhomogeneous photonic environment and accounts fully for the back-action of the vacuum electric field of the environment, which enters through the LDOS. The LDOS is the mode density that is “seen” by a dipole emitter. It is a classical quantity obtained by solving Maxwell’s equations but it also determines the mode density of the vacuum electromagnetic field that is required to describe spontaneous emission. Thus, the variance of the projected vacuum electric field (the ‘vacuum fluctuations’) is given by


which is indeed proportional to the LDOS. It should be emphasized that while the applied wavefunction approach fully accounts for non-Markovian coupling to the photonic reservoir, any non-Markovian dephasing processes cannot be captured by such a formalism. Non-Markovian dephasing is most appropriately treated by density operator theory, cf. Sec. VII.2.2.

It is instructive to express Eq. (22) as


with the introduction of the memory kernel Vats et al. (2002)


The kernel expresses the memory of the radiation reservoir, i.e., to what extent the state of the reservoir at previous times influences A special and simple case is that of Wigner-Weisskopf theory that holds when varies insignificantly over the linewidth of the emitter such that the memory kernel can be approximated as


In this case the memory kernel is singular, i.e., the radiation reservoir is memoryless. This is also referred to as the Markov approximation where back-action from the radiation reservoir is negligible and the population of the excited state of the emitter decays exponentially in time with a radiative rate of


The corresponding emission spectrum is a Lorentzian of width , as explained in Sec. IV.2. The power emitted by the dipole is also proportional to the LDOS because for continuous-wave excitation the radiated power is . Equation (28) shows that spontaneous emission can be controlled by modifying the LDOS, which is essential in quantum nanophotonics for engineering the light-matter interaction strength. This will be elaborated in further detail in Sec. IV.3.

iv.2 The spontaneous-emission spectrum and the Lamb shift

The spontaneous-emission spectrum is another important measurable quantity modified by interaction with the radiation reservoir. Following the approach by Vats et al. (2002), the Fourier transform of Eq. (25) is


where is assumed, i.e., the population decays to the radiation reservoir at very long times. The Fourier transform of the memory kernel is