The Mesoscopic Nature of Quantum Dots in Photon Emission

Semiconductor quantum dots share many properties with atoms such as discrete spectrum, which implies the ability to emit high purity single photons. However, they have unique features as well that are unknown to other emitters: they embody tens of thousands of atoms attaining large mesoscopic sizes, and lack the common atomic symmetries. Here we discuss two effects that are mediated by the mesoscopic nature and render quantum dots fundamentally different than atoms. The mesoscopic size and lack of parity symmetry causes the electric-dipole approximation to not be applicable to In(Ga)As quantum dots. As a consequence, the latter do not fulfil the atomic selection rules and thus interact with the electric and magnetic components of light on the same electronic transition. The multi-atomic nature also causes a collective mesoscopic effect in monolayer-fluctuation GaAs quantum dots, namely single-photon superradiance, giving rise to a giant light-matter coupling strength. Semiconductor quantum dots (QDs) provide the essential link between light and matter and can be integrated monolithically into photonic devices. These nanometersized purposefully engineered impurities combine the atomic-like discrete spectra and excellent single-photon purity with the large light-matter interaction strength inherent to solid-state systems [1]. The ability to tailor the photonic environment around QDs has resulted in tremendous progress in manipulating single QD excitations. Strong coupling between a QD and a cavity [2–4] and near-unity coupling to a photonic waveguide [5–10] are a few out of many exciting realizations [1]. The atomic-like properties of QDs are supplemented by a range of new effects owing to their solid-state nature. For instance, vibrations of the underlying crystal lattice, known as phonons, may decohere the light-matter interaction [11–13] or couple non-resonant QD excitations to an optical cavity [14–18]. Similarly, the mesoscopic P. Tighineanu (B) · A.S. Sørensen · S. Stobbe · P. Lodahl The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark e-mail: [email protected] P. Lodahl e-mail: [email protected] © Springer International Publishing AG 2017 P. Michler (ed.), Quantum Dots for Quantum Information Technologies, Nano-Optics and Nanophotonics, DOI 10.1007/978-3-319-56378-7_5 165 166 P. Tighineanu et al. ensemble of the nuclei composing the QD can be used to tailor the hyperfine interaction with the electron in spin-based quantum-information science [19]. Recently it was found [20] that QDs may break the dipole approximation, which is often assumed to be valid also in solid-state quantum optics. These realizations unveil the complex nature of QDs, which embody tens to hundred thousand atoms attaining “mesoscopic” sizes that interact relentlessly with the surrounding solid-state environment. In this chapter we present a unified description of the mesoscopic nature of QDs [20–23]. In particular, we discuss twomesoscopic effects that exist solely due to the large physical size of QDs: the breakdown of the dipole theory of In(Ga)As QDs and collective enhancement of light-matter interaction with monolayer-fluctuation GaAs QDs. The small size L of most quantum emitters compared to the wavelength of light λ has ensured the success of the dipole theory, which states that emitters interact with light as dimensionless entities (point dipoles). Since QDs attain mesoscopic sizes of 10–30nm [24], the dipole approximation does not necessarily hold because the figure of merit 2πnL/λ0 ≈ 0.5 is not negligible. Here, typical values for the wavelength in vacuum λ0 = 900 nm, refractive index n = 3.42 and L = 20nm have been used. This figure of merit may be further enhanced in the vicinity of metal nanostructures, where additional propagating modes (surface plasmons) beyond the light cone arise. It has been observed that the spontaneous-emission dynamics from QDs placed near a metal interface show pronounced deviations from the dipole theory [20]. A theory of light-matter interaction beyond the dipole theory can explain these experimental findings by introducing a single mesoscopic moment to be considered along with the dipole moment in light-matter interactions [21]. Notably, this theory is more general than previously developed models [25–32] because it considers the symmetry of the full quantum-mechanicalwavefunction and not only the slowly varying envelope.We show that the discrete atomistic symmetry explains themicroscopic origin of the large mesoscopic moment observed experimentally. In particular, the developed theory pinpoints that large structural inhomogeneities at the crystal-lattice level lead to a violation of parity symmetry in In(Ga)AsQDs [22].Quantumdots therefore break the atomic selection rules and probe electric and magnetic fields on the same electronic transition [21]. Moreover, the mesoscopic size of QDs may ease the observation of dipole-forbidden transitions in photonic nanostructures [33]. It has been shown that, in the opposite limit of highest possible (spherical) symmetry present in, e.g., colloidal QDs, a shell theorem is valid, which states that the Purcell enhancement in an arbitrary photonic environment is protected by symmetry and does not depend on the QD size [32]. The second part of this chapter is devoted to presenting another mesoscopic property of QDs, namely collective enhancement of light-matter interaction leading to single-photon superradiance. Quantum dots benefit from their multi-body nature with an enhanced coupling to light compared to atoms, which renders them promising candidates for improving the efficiency of single-photons sources, solar cells and nano-lasers, to name a few important practical applications. Commonly employed QDs have, however, an upper limit for the interaction strength with light, regardless of their size and shape. It has therefore been a long-sought goal in quantum photonics 5 The Mesoscopic Nature of Quantum Dots in Photon Emission 167 to develop solid-state emitters beyond this upper limit [34–37]. We demonstrate that the fundamental excitation of a monolayer-fluctuation QD [38] is analogous to the phenomenon of single-photon superradiance defined by Dicke for a non-interacting ensemble [39]. This effect leads to an enhanced coupling to light far beyond that of conventional QDs, which may be of interest for fundamental science and technology alike. In particular, such rapid radiative decays will likely exceed relevant dephasing mechanisms resulting in highly coherent flying quantum bits. Furthermore, new and so far largely unexplored solid-state quantum-electrodynamics regimes involving energy non-conserving virtual processes, such as the ultra-strong coupling between light and matter, may become within reach at optical frequencies [23]. 5.1 Fundamentals of Light-Matter Interaction with Quantum Dots In this section we lay the fundamental as well as the experimentally relevant aspects describing the interaction between QDs and light. 5.1.1 Effective-Mass Theory The commonly employed bandstructuremethod for QDs is the effective-mass theory. It assumes that the bands, which are exact solutions in the bulk semiconductor, are weakly perturbed by the nanostructure. Formally, a quantized eigenstate within an electronic band can be written as a product of a periodic Bloch function, u(r), which captures the properties on the length scale of the crystal unit cell, and a slowly varying envelope, ψ(r), that inherits the size and symmetry of the mesoscopic QD potential j (r) = ψ j (r)u j (r), (5.1) where j = {e, hh, lh} labels either of the three relevant bands in zincblende semiconductors: electron, heavy hole, and light hole, respectively. It can be shown [40] that ψ j is subject to a Schrödinger-type equation