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Optical properties of semiconductor quantum dots: Few-particle states and coherent-carrier control Ulrich Hohenester† Istituto Nazionale per la Fisica della Materia (INFM) and Dipartimento di Fisica, Università di Modena e Reggio Emilia, Via Campi 213A, I-41100 Modena, Italy In quantum dots, often referred to as artificial atoms, electrons and holes are confined in all three space directions. This strong quantum confinement results in a discrete, atomic-like carrier density of states. In turn, the coupling to the solid-state environment (e.g., phonons) is strongly suppressed and Coulomb correlations among charge carriers are strongly enhanced. Indeed, in the optical spectra of single dots spectrally narrow emission peaks have been observed (indicating a small environment coupling), which undergo discrete energy shifts when more carriers are added to the dot (indicating energy renormalizations due to additional Coulomb interactions). In this paper we first give a short introduction into this rapidly growing field of research and review some of the recent experimental and theoretical work. We then provide details of a full configuration-interaction approach for the calculation of few-particle electron-hole states and optical properties of semiconductor quantum dots. Finally, a short survey over the recent work of the author is presented. PACS numbers: 85.30.Vw,71.35.-y,73.20.Dx,78.66.Fd CONTENTS I. Introduction II. Carrier states in semiconductor quantum dots A. Physical system B. Single-particle states C. Coulomb interactions D. Few-particle states III. Optical properties of quantum dots A. Light-semiconductor coupling B. Linear absorption C. Nonlinear absorption D. Luminescence E. Coherent-carrier control IV. Results A. Near-field spectra B. Multi-excitons C. Multi-charged excitons D. Coherent-carrier control V. Summary Acknowledgments A. Single-particle states and matrix elements 1. Single-particle states 2. Optical matrix elements 3. Coulomb matrix elements B. Few-particle states 1. Direct diagonalization 2. Optical matrix elements 3. Biexcitons References I. 1 2 2 2 4 5 6 6 6 7 8 8 9 9 9 10 10 12 12 13 13 13 13 13 13 14 14 15 INTRODUCTION In recent years much attention is being devoted to the properties of semiconductor quantum dots. In these sys- † Electronic address: [email protected] tems, carriers are subject to a confining potential in all spatial directions, giving rise to a discrete energy spectrum (“artificial atoms”) and to novel phenomena of interest for fundamental physics as well as for applications to electronic and optoelectronic devices (Bimberg et al. 1998, Jacak et al. 1998). The extension and the shape of the dot confining potential varies, depending on the nanostructure fabrication technique: The dots that are studied most extensively by optical methods are induced by quantum well (QW) thickness fluctuations (Brunner et al. 1994, Flack et al. 1996, Gammon et al. 1996, Hess et al. 1994, Zrenner et al. 1994), or obtained by spontaneous island formation in strained layer epitaxy (Grundmann et al. 1995, Leon et al. 1995, Marzin et al. 1994), self-organized growth on patterned substrates (Hartmann et al. 2000a), stressor-induced QW potential modulation (Obermüller et al. 1999, Rinaldi et al. 2000) cleaved-edge overgrowth (Schedelbeck et al. 1997, Wegscheider et al. 1997), as well as chemical selfaggregation techniques (Alivisatos 1996, Murray et al. 1995). The resulting confinement lengths fall in a wide range between 1µm and 10 nm. In spite of the continuing progress, all the available fabrication approaches still suffer from the effects of inhomogeneity and dispersion in dot size, which leads to large linewidths when optical experiments are performed on large dot ensembles. A major advancement in the field has come from different types of local optical experiments, that allow the investigation of individual quantum dots thus avoiding inhomogeneous broadening (see Zrenner 2000, for a review). With these advances it became possible to study in great detail a variety of novel physical phenomena, such as fine-structure splitting (Gammon et al. 1996), atomic-like excitation spectra (Brunner et al. 1994, Gammon et al. 1995, Hessman et al. 1996), 2 Coulomb renormalizations of few-particle states (Bayer et al. 2000a, Dekel et al. 1998, Hartmann et al. 2000a, Landin et al. 1998, Motohisa et al. 1998, Warburton et al. 2000), interdot couplings (Schedelbeck et al. 1997), Zeeman splittings and diamagnetic shifts (Bockelmann et al. 1997, Cingolani et al. 1999, Findeis et al. 2000b, Heller and Bockelmann 1997, Rinaldi et al. 1996, 1998). The theoretical analysis of such experiments requires a detailed understanding of carrier states in quantum dots (accounting for effects such as band offsets, strain, or piezoelectric fields) and of Coulomb-type renormalizations of few-particle carrier states. Single-particle electron and hole states have been computed for small quantum dots by use of ab-initio approaches (Rohlfing and Louie 1998), and for larger dots by approximate techniques such as pseudopotential (Wang et al. 1999), eightband k.p (Jiang and Singh 1997, Pryor 1998, Stier et al. 1999), or effective-mass studies (Braskén et al. 1997, Jacak et al. 1998). Coulomb renormalizations of fewparticle electron-hole states have been treated within restricted (Hawrylak 1999, Narvaez and Hawrylak 2000) or full (Braskén et al. 2000, Hartmann et al. 2000a, Hohenester and Molinari 2000a) configuration-interaction models, or by use of density-functional (Nair and Masumoto 2000) or density-matrix descriptions (Hohenester and Molinari 2000a, Hohenester et al. 1999b). In this paper we first review some of the recent work about carrier states and optical properties of semiconductor quantum dots. Details of our computational approach will be presented. We have broken down this paper into the following parts: In Sec. II we discuss different approaches for calculating single-particle and Coulombrenormalized few-particle states; Sec. III is devoted to a discussion of optical spectroscopy techniques and of the theoretical framework accounting for light-semiconductor coupling; Sec. IV briefly discusses recent papers of the author and presents introductory material; finally, in Sec. V we draw some conclusions. II. CARRIER STATES IN SEMICONDUCTOR QUANTUM DOTS In this section we set out to review the basic ideas underlying the theoretical description of few-particle states in semiconductor quantum dots. A. Physical system We shall consider the problem of carriers confined in a single quantum dot under the action of external electromagnetic fields. The carriers will experience their mutual Coulomb interactions as well as the interaction with phonons or other environmental degrees of freedom. It turns out to be convenient to describe the problem by the Hamiltonian: H = (Hc + Hcc ) + Hop + Henv , (1) where Hc accounts for non-interacting carriers (electrons and holes) in presence of the dot confinement potential and of possible time-independent external electric and magnetic fields; Hcc describes Coulomb interactions among charge carriers; Hop accounts for the coupling to light; finally, Henv describes the coupling to environmental degrees of freedom (e.g., phonons or carriers in the wetting layer). 1 B. Single-particle states The three-dimensional confinement of carrier states in semiconductor quantum dots is due to the use of materials with different conduction and valence band offsets and due to strain fields and related effects (e.g., piezoelectric fields; Bimberg et al. 1998). In the following we briefly discuss the most widely used theoretical approaches accounting for such effects, and develop the theoretical framework used in this work. 1. Ab-initio approaches True ab-initio calculations of carrier states in semiconductor quantum dots are inherently restricted to dots containing only a few tens to hundreds of atoms; in this respect, hydrogen-terminated silicon clusters represent the most thoroughly studied class of systems (see, e.g., Rohlfing and Louie 1998). The theoretical framework for the calculation of the electronic and optical properties is usually based on the density-functional theory within the local-density approximation (Dreizler and Gross 1990); in addition the GW method (Arayasetiawan and Gunnarson 1998) is employed to account for electron-hole Coulomb correlations (Albrecht et al. 1998, Rohlfing and Louie 2000). However, for the quantum dots of our present concern, which typically consist of several thousands of atoms, such first-principles techniques are not directly applicable. To overcome these limitations a number of different approaches have been proposed. Zunger and collaborators have developed an atomistic pseudopotential approach (see, e.g., Franceschetti et al. 1998, Wang et al. 1999, and references therein) with a suitably chosen parameterization of pseudopotentials; for details the reader is referred to the literature. 1 Note that our choice of accounting for static electric and magnetic fields in Hc and for light coupling in Hop is somewhat arbitrary, and is primarily thought to highlighten the important role of carrier-light coupling in our present study of the optical properties of semiconductor quantum dots. 3 2. Envelope-function approximation Another solution scheme, which we shall pursue in this work, is provided by the envelope-function approximation for the calculation of carrier states in quantum dots (see, e.g., Haug and Koch 1993, Yu and Cardona 1996). Here, the main assumption is that the dot confinement potential varies smoothly on a length scale of the lattice constant.2 The low-energy carrier wavefunctions then approximately consist of bulk-like atomic parts un,k∼ =0 (r) (with n the band index and k the wavevector) which are smoothly modulated by the envelope function φn (r) (for details see Yu and Cardona 1996). Within the spirit of the envelope-function approximation all details of the atomic part un,k∼ =0 (r) of the carrier wavefunctions are lumped into a few band-structure parameters (e.g., effective masses or optical bulk matrix elements, whose values are either fitted to experiment or obtained from firstprinciples calculations), and only the envelope function φn (r) is kept to describe the modulations of the atomic bulk wavefunctions. a. k.p approach Conveniently the envelope-function approach comes together with the k.p approximation (Kane 1966), which provides an efficient approach for calculating the material dispersion and wavefunctions. Details of this approach depend on the number of bands considered in the calculation of the total wavefunction ψ(r) = X un,k∼ =0 (r)φn (r). (2) n For III-V semiconductors an exhaustive description of the bandstructure around the Γ point is provided by an eightband model (Kane 1966, Trebin et al. 1979) containing: the s-like conduction-band states |s, ± 21 i (with spinup and spin-down orientation); the p-like valence-band states | 32 , ± 32 i, | 32 , ± 12 i (heavy-hole and light-hole bands in bulk semiconductors) and | 12 , ± 12 i (spin-orbit split-off band). Within this scheme carrier states in quantum dots are described by the effective one-particle Hamiltonian (Jiang and Singh 1997, Pryor 1998, Stier et al. 1999): h = t + hstr + v + δh, (3) with: t the kinetic term (in presence of possible external electric and magnetic fields), whose eigenvalues give the bulk bandstructure; hstr the strain contributions; v 2 Note that the envelope-function approximation even provides an excellent description for quantum-well-like confinement where this assumption is no longer valid. the confinement potential due to the conduction- and valence-band offsets; finally, δh accounts for higher-order contributions within the k.p approach which lead to a mixing of electron (hole) states with different spin orientations (Khaetskii and Nazarov 2000, Maialle et al. 1993). Eight-band k.p studies of carrier states in pyramidal InAs/GaAs self-assembled quantum dots have been carried out by Jiang and Singh (1997), Pryor (1998), and Stier et al. (1999). In the calculations for these dots the authors have demonstrated the importance of treating bandstructure effects at such high level of accuracy to account for the strong valence-band mixing due to the substantially different band offsets (gap energy Eg = 1.519 eV in GaAs vs. Eg = 0.418 eV in InAs) and due to the pronounced effects of strain fields (lattice mismatch of approximately seven percent). We note here in passing that, despite the accuracy of such 8×8 k.p studies, the details of carrier wavefunctions in InAs dots are still a matter of intense debate. This controversy is due to the open issues of the detailed shape of overgrown InAs/GaAs dots and of possible In diffusion into the surrounding GaAs (Liu et al. 2000). Using Starkeffect spectroscopy together with theoretical modeling, Fry et al. (2000) have shown that in capped InAs/GaAs dots holes can become localized above electrons—in contrast to the predictions of calculations assuming pyramidal dot structures (Stier et al. 1999). b. Effective-mass approximation In this work we shall make further simplifications as compared to the above envelope-function and k.p approach. First, we shall not treat strain-induced effects explicitly, but shall rather assume that an effective confinement potential —accounting for band offsets and strain effects— can be provided. Such a procedure is well justified for epitaxially grown quantum dots (see, e.g., Hartmann et al. 1997, 1998) where strain can be safely neglected; in addition, because of the uncertainties regarding the detailed shape and material composition of many dots, an appropriately parameterized confinement potential, with parameters possibly fitted to experiment, is known to provide a sound approach for semi-quantitative comparison with experiment (Bayer et al. 2000a, Hawrylak et al. 2000, Rinaldi et al. 2000). Second, we make use of the effective-mass approximation; for electrons we assume an isotropic mass me , and for heavy holes ↔ a diagonal effective-mass tensor mh = diag(mt , mt , ml ) with mt = mo /(γ1 + γ2 ) the transverse mass in (x, y)directions and ml = mo /(γ1 − 2γ2 ) the longitudinal mass in z-direction (Haug and Koch 1993, Liu and Bryant 1999), where mo is the free-electron mass and γ1 , γ2 are Luttinger’s parameters; contributions from the light-hole band are neglected because of the usually narrow confinement in z-direction and the resulting large inter-subband splitting. Inclusion of valence-band mixing is expected to introduce minor quantitative changes in the electronic 4 energies and envelope functions of the quantum dot system (see Braskén et al. 1997, for a quantitative estimate of valence-band mixing). The single-particle properties for electrons and holes in presence of an external magnetic field B = (0, 0, B) in z-direction are then described by (Haug and Koch 1993, Rossi 1998): he = te (πe ) + ve (r) − eΦ(r) + µB ge Sz B + δhe hh = th (πh ) + vh (r) + eΦ(r) − 31 µB gh Jz B + δhh , (4) with the kinetic terms te (p) = p2 /(2me ) and th (p) = ↔ −1 1 e p; the momenta πe,h = −i∇± 2c (B×r) within 2 p(mh ) symmetric gauge; the confinement potentials ve (r) and vh (r) for electrons and holes, respectively; the scalar potential Φ(r) accounting for an external electric field; the Zeeman interaction energies µB ge Sz B and 31 µB gh Jz B for electrons and holes, respectively (Blackwood et al. 1994, van Kesteren et al. 1990), with µB Bohr’s magneton, ge the electron g-factor, gh the hole g-factor, Sz = ± 12 the electron spin orientation, and Jz = ± 23 the hole angular momentum in z-direction; finally δhe and δhh are contributions which mix electron (hole) states with different spin orientations. The eigenenergies eµ and hν for electrons and holes, respectively, and the corresponding envelope wavefunctions φeµ (r) and φhν (r) are obtained from the solutions of Eq. (4), as outlined in the following section and in appendix A.1. been demonstrated to be a good approximation for selfassembled quantum dots formed by strained-layer epitaxy (Jacak et al. 1998). Here: v(r) = 1 d mt ωo2 (x2 + y 2 ) + Vo θ(|z| − ), 2 2 (6) where ωo is the level splitting due to the in-plane harmonic potential, Vo is the band offset, and d is the width of the quantum well. With this choice of the confinement potential the envelope-function equation factorizes into the in-plane and longitudinal parts, which both can be solved analytically. The solutions of the in-plane part provide the well-known Fock-Darwin states, as outlined in some detail by Jacak et al. (1998) or Chakraborty and Pietiläinen (1995). C. Coulomb interactions When the dot is populated by more than one electron or hole the properties of few-particle carrier states are not only governed by the single-particle energies and states but, in addition, by mutual Coulomb interactions among charge carriers. In nanoscale semiconductor quantum dots such Coulomb renormalizations are enhanced because of the strong three-dimensional quantum confinement. Depending on the approximations employed in the calculation of single-particle states Coulomb interactions have to be treated on different levels of sophistication. 3. Solution of the single-particle envelope-function equation 1. Ab-initio approaches In our computational approach we start from the single-particle envelope-function Hamiltonian (with magnetic field in z-direction) Ab-initio-type calculations, such as the densityfunctional–local-density-approximation or pseudopotential approach, provide wavefunctions ψ which show rapid oscillations on the length scale of the lattice constant (atomic part) with smooth modulations on the length scale of the semiconductor nanostructures (envelope part). When computing Coulomb matrix elements within this ψ-basis dielectric screening has to be treated with special care. Quite generally, dielectric screening in semiconductors is a dynamic process where one charge carrier (electron or hole) polarizes its surrounding medium; in turn, a second carrier experiences not only the bare Coulomb potential exerted by the first carrier but, in addition, also by the induced polarization cloud. From bulk semiconductors it is well known that such polarization-induced screening can reduce the bare Coulomb interaction by a factor of the order of ten.3 The dielectric screening function κ(r, r 0 ) is a nonlocal quantity. Within ab-initio-type approaches κ is h= p2x + p2y p2 B 2 (x2 + y 2 ) B lz + z + v(r) + − , (5) 2mt 2ml 2mt 2mt where mt and ml are the transversal and lateral mass, respectively, l = r×p is the angular-momentum operator and B = ±eB/c. Here, on the right-hand side: the first two terms account for the kinetic energy in the in-plane and longitudinal directions; the third term accounts for the diamagnetic shift in presence of an external magnetic field; finally, the last term describes the Zeeman splitting of states with non-zero angular momentum. For arbitrary confinement potentials v(r), Eq. (5) is solved by lattice discretization and direct diagonalization of the Hamiltonian matrix; details of our numerical approach are presented in appendix A.1. In this paper we shall alternatively assume a prototypical dot confinement which is parabolic in the (x, y)-plane and box-like along z; such confinement potentials have 3 We shall neglect the frequency-dependence of the dielectric screening function throughout and assume static screening. 5 treated at different levels of sophistication (Franceschetti et al. 1998, Ögüt et al. 1997, Rohlfing and Louie 1998, 2000), where the details of dielectric screening are still matter of intense debate (see, e.g., Delerue et al. 2000, Franceschetti et al. 1999, Ögüt et al. 1999, Reboredo et al. 1999). 2. Envelope-function approximation Dielectric screening within the envelope function approximation is usually accounted for by reducing the bare Coulomb matrix elements by the static dielectric constant κ. In this work we only keep monopole-monopole Coulomb terms which conserve the number of electrons and holes and concentrate on the two most important interaction channels: direct Coulomb interactions, which are responsible for the strong few-particle renormalizations; short-ranged electron-hole exchange interactions, which give rise to the bright-dark splitting of excitons. Other types of Coulomb interactions, such as Auger-type processes or terms which lead to the splitting of longitudinal and transversal excitons, are discussed in some detail by Axt and Mukamel (1998) and Maialle et al. (1993). where the bright-dark splitting az of excitons in quantum dots is enhanced with respect to its bulk value because of motional narrowing (Maialle et al. 1993), i.e., the confinement-induced increased overlap of electron and hole wavefunctions; for InGaAs/GaAs dots Bayer et al. (1999, 2000b) have reported a bright-dark exciton splitting of 200 µeV as compared to the 20 µeV bulk GaAs value, i.e., an enhancement by a factor of ten. If not noted differently, in the following we shall not consider the fine splitting due to the electron-hole exchange interaction. D. Few-particle states We next show how to compute few-particle states of quantum dots. In our calculations we account explicitly for the dot confinement and external fields (Hc ) and for mutual Coulomb interactions among electrons and holes (Hcc ). It turns out to be convenient to introduce the m † Fermionic field operators (ψµσ ) which create an electron (hole) in single-particle state µ with spin orientation σ = ±.4 With this notation the single-particle Hamiltonian Hc of Eq. (1) is of the form Hoe + Hoh , with: Hom = a. Direct Coulomb interaction X m † m m µσ (ψµσ ) ψµσ , (8) µσ With the envelope functions φm µ (r) (m = e, h) the direct Coulomb matrix elements are of the form (Haug and Koch 1993, Rossi 1998): and the Coulomb term Hcc reads: Hcc = ∗ n∗ 0 n 0 m φm µ0 (r)φν 0 (r )φν (r )φµ (r) , κ|r − r 0 | (7) where qm = ∓e for electrons and holes, respectively. Details of our computational approach for calculation of V are presented in appendix A.3. We note that in case of cylinder symmetry of the dot confinement the angular momentum in z-direction becomes a constant of motion and the Coulomb-matrix elements conserve total angular momentum (Jacak et al. 1998). Vµmn 0 µ,ν 0 ν = qm qn Z drdr 0 b. Electron-hole exchange interaction The electron-hole exchange interaction accounts for the fact that an electron which is promoted from the valence to the conduction band no longer experiences the exchange interactions with all valence-band electrons. In the electron-hole picture this process gives rise to a repulsive interaction between electrons and holes with opposite spin orientations. Following the work of van Kesteren et al. (1990) and Blackwood et al. (1994) the short-range electron-hole exchange interaction is described in lowest order by an effective Hamiltonian of the form az Jz Sz , 1 2 X X m † n † n m Vµmn 0 µ,ν 0 ν (ψµ0 σ ) (ψν 0 σ 0 ) ψνσ 0 ψµσ . mn,σ,σ 0 µµ0 ,νν 0 (9) (Note that in Eq. (9) we have not included the electronhole exchange interaction.) 1. Direct diagonalization In our computational approach the few-particle electron-hole states are computed within a full configuration-interaction model (McWeeny 1992). We first construct a set of basis functions |`i within the Hilbert space of electron-hole excitations; appropriate truncation of the Hilbert space to approximately ten single-particle states and 100–1000 few-particle states of lowest energy for electrons and holes, respectively, is necessary to keep the numerics tractable (for details see appendix B.1). We then obtain for a given number of electrons and holes the few-particle energies Eλ and states Ψλ` = h`|λi by direct diagonalization of the eigenvalue problem: 4 For notational simplicity in the following we shall write for holes σ = ± instead of Jz = ± 32 . 6 X (Hc + Hcc )``0 Ψλ`0 = Eλ Ψλ` , (10) `0 with H``0 = h`|H|`0 i. Such full configuration-interaction (CI) approaches appear to be the method of choice for the quantum dots of our present concern, since they allow to treat Coulomb renormalizations with as few approximations as possible (i.e., a suitable truncation of the Hilbert space) and since, within the strong-confinement regime, only a limited number of basis functions has to be retained (thus keeping the numerics tractable for a moderate number of electrons and holes). Note that the rather straightforward CI implementation of appendix B.1 without attempting major optimization of performance is due to the strong-confinement regime and the resulting perturbative character of Coulomb interactions; a more sophisticated CI approach for carrier states in dots within the weak-confinement regime is presented by Braskén et al. (2000), where, however, the larger basis set |`i prohibited the calculation of optical spectra. We finally note that other approaches for the calculation of few-particle carrier states, such as truncated CI techniques (Hawrylak and A. Wojs 1996, Jacak et al. 1998, Wojs and Hawrylak 1997), density-matrix descriptions (Hohenester et al. 1999b), or approaches based on density-functional theory have in general the drawback that they treat certain types of Coulomb interactions within a mean-field approximation, which can lead to spurious effects in the optical spectra. III. OPTICAL PROPERTIES OF QUANTUM DOTS The coupling of light with optical (i.e., electron-hole) excitations in quantum dots allows the study of fewparticle states by means of optical spectroscopy. This section is devoted to a short survey over different spectroscopy techniques and the theoretical framework accounting for light-matter coupling. A. Light-semiconductor coupling In lowest order the light-semiconductor coupling is described by a process where a photon is absorbed and an electron is promoted from the valence to the conduction band (i.e., creation of electron-hole pair), and by the reversed process where an electron-hole pair is destroyed and a photon is created; for simplicity, in this work we shall neglect all higher-order processes (e.g., two-photon absorption; Fedorov et al. 1996). The interaction of dot carriers with light is then accounted for by a Hamiltonian of the form −E.P (Haug and Koch 1993, Rossi 1998), where E is the electric field of the light and P is the material polarization. We next employ a number of approximations (Axt and Mukamel 1998, Haug and Koch 1993, Rossi 1998, Scully and Zubairy 1997): First, we neglect optical intraband transitions; second, we make use of the rotating-wave approximation; third, within our envelopefunction–one-band effective-mass approach we make use of the dipole approximation and neglect the directional dependence of the bulk dipole matrix element µo ; we furthermore assume that circularly polarized light creates electron-hole pairs with well defined spin orientations. Then (Haug and Koch 1993, Rossi 1998): Hcl = − 12 XZ dr [Eσ(+) (r, t)Pσ (r) + Eσ(−) (r, t)Pσ† (r)], σ (11) where Eσ(±) is the electric field profile of the light with polarization σ and with the superscript (±) accounting for clockwise or counterclockwise rotating components, e h P± (r) = µo ψ∓ (r)ψ± (r) is the interband-dipole operator with the field operators ψσm † (r), which create an electron or hole with spin orientation σ at position r.5 Note that in Eq. (11) we have explicitly assumed a real-space dependence of the electric field profile. Such a choice will turn out to be convenient for the discussion of near-field spectra; in case of far-field excitation the r-dependence of E can be safely neglected. In the following we discuss the cases where the electric field is treated either classically (see absorption) or quantum-mechanically (see luminescence). B. Linear absorption Let us assume that the dot structure is weakly perturbed by a light field, which we shall treat on a classical level. Within the linear-response regime we assume (without loss of generality) a monofrequent laser excitation at frequency ω with an (inhomogeneous) electric field profile Eσ (r, t) = Eσ (r) exp(iωt). This external light field induces an interband polarization Pσ (r, t) = hPσ (r)it (Rossi 1998), where the brackets denote a suitable ensemble average. The total absorbed power α(ω) at a given frequency ω is then proportional to the imaginary part of the Fourier-transformed interband polarization (Haug and Koch 1993, Mauritz et al. 1999, 2000): Z iωt α(ω) ∝ = drdt Eσ (r)e Pσ (r, t) . (12) The calculation of α(ω) thus requires the knowledge of Pσ (r, t) in presence of the driving laser field. Within 5 These m through: field operators are related to ψµσ X m m ψσ (r) = φm µσ (r)ψµσ . µ 7 linear response it is possible to obtain under very general conditions Kubo-like expressions for the induced interband polarization (Kubo et al. 1985); alternative approaches for calculating Pσ (r, t) for nonequilibrium semiconductor nanostructures are usually based on the density-matrix (Rossi 1998) or non-equilibrium Green’s function approaches (Haug and Jauho 1996). 2. Near-field vs. far-field absorption Using Eqs. (12) and (13) we then get within timedependent perturbation theory7 for the absorption spectra (Mauritz et al. 1999, 2000): α(ω) ∝ X Z dr ξ(r − R)Ψx (r, r)2 Dγ (ω − Ex ), (15) x 1. Excitons In this work we pursue a somewhat simplified approach based on a wavefunction description, and environment coupling is introduced by adding a small imaginary part iγ to the eigenenergies. Let us assume that before arrival of the driving laser the dot system is in its vacuum state |0i, i.e., no electrons and holes present. The laser then creates through the interband-polarization operator Pσ† (r) [Eq. (11)] an electron-hole pair, which propagates in presence of the dot confinement and of mutual Coulomb interactions. Thus, the linear optical properties are governed by the electron-hole (exciton) eigenergies Ex and wavefunctions Ψx ,6 which we obtain from Eq. (10): (eµ + hν )Ψxµν + X eh x x Vµµ̄,ν ν̄ Ψµ̄ν̄ = Ex Ψµν , (13) µ̄ν̄ with Ψxµν = h0|ψνh ψµe |xi (note that for notational simplicity we have dropped all spin indices). Within the field of semiconductor optics Eq. (13) is usually referred to as the excitonic eigenvalue problem (Axt and Mukamel 1998, Haug and Koch 1993, Rossi and Molinari 1996a,b). A particularly intriguing form of this equation is obtained for the real-space representation of the exciton wavefunction: Ψx (re , rh ) = X φeµ (re )Ψxµν φhν (rh ), (14) µν where the left-hand side of Eq. (13) gets the form [he (re ) + hh (rh ) − e2 /(κ|re − rh |)]Ψx (re , rh ), with the single-particle Hamiltonians he and hh of Eq. (4), which clearly emphasizes the Coulomb-type renormalization of exciton states. 6 Few-particle states with more than one electron or hole only contribute to the higher-order optical response (Axt and Mukamel 1998, Axt and Stahl 1995, Victor et al. 1995) and are thus neglected. where Dγ (ω) = 2γ/(ω 2 + γ 2 ) is a broadened deltafunction and, for convenience, we have chosen an electricfield distribution of the form E(r) = Eo ξ(r − R), with ξ(r) a normalized distribution. Two limiting cases can be identified. For a spatially homogeneous electromagnetic field (ξ ∝ 1, i.e., far-field), the oscillator strength is given R by the spatial average of the excitonic wavefunction ∝ | dr Ψx (r, r)|2 . For near-field excitation ξ(r − R) is centered around the tip position R of the near-field probe; in the hypothetical limit of an infinitely narrow probe, ξ(r − R) = δ(r − R), one is probing the local value of the exciton wavefunction; within an intermediate regime of a narrow but finite probe, Ψλ (r, r) is averaged over a region which is determined by the spatial extension of the light beam; therefore, excitonic transitions which are optically forbidden in the far-field may become visible in the near-field (Bryant 1998, Liu and Bryant 1999, Mauritz et al. 1999, 2000, Simserides et al. 2000). C. Nonlinear absorption We next turn to the discussion of nonlinear optical spectroscopy (Bonadeo et al. 1998b, 2000, Ikezawa et al. 1997). Quite generally, this technique requires two lasers at different frequencies: a strong pump laser with electric field Eσpp exp(iωp t) and a weak probe pulse with Eσt t exp(iωt t). Nonlinear absorption measures the absorption changes of the weak probe pulse ∝ χ(3) |Eσpp |2 Eσt t exp(iωt t), with the third-order susceptibility χ(3) , and thus provides information about the optical properties of the dot system in presence of the strong pump pulse. As compared to linear absorption, this technique has the advantage of giving a much better noise-signal ratio, which is of crucial importance for the measurement of single quantum dots (Bonadeo et al. 7 For the time-dependent state vector we find: Z X [Ψx (r, r)]∗ |Ψit ∝ |0i − 12 µo dr E(r)e−iωt |xi, ω − Ex + iγ x where we have assumed that before arrival of the laser the system is in |0i and we have performed the adiabatic approximation (note that the damping constant γ, accounting for environment coupling to, e.g., phonons, is essential to perform this limiting procedure). 8 1998b, 2000);8 In addition, nonlinear spectroscopy provides information about biexcitonic features in the optical spectra (Axt and Mukamel 1998). 1. Biexcitons To gain some insight into the latter point, in the following we briefly discuss the nature of biexcitonic states. Since the external light fields create electron-hole pairs with given spin orientations, it turns out to be convenient to define the exciton operators: D. Luminescence Luminescence (spontaneous emission) involves a process where one electron-hole pair is destroyed and one photon is created. Its description thus requires a quantum-mechanical treatment of light (Scully and Zubairy 1997). Introducing a suitable quantization of the light field according to Haug and Koch (1993) we then obtain from Eq. (11) the carrier-photon interaction: Hcl = − X 1 (2πωkσ )− 2 [ia†kσ Pσ (r) − iakσ Pσ† (r)], (19) kσ b†xσ = X e e Ψxµν (ψµσ )† (ψνσ e h µν )† σe =σ,σh =−σ . (16) Apparently, b†xσ |0i creates an optically active exciton state |xi. With these operators we then construct the biexciton states |λi according to: |λi = ησσ0 X Ψ̄λxx0 b†xσ b†x0 σ0 |0i, (17) where the Bosonic field operators a†kσ create a photon with polarization σ and wavevector k. Their free motion P is governed by kσ ωkσ a†kσ akσ with √ ωkσ = c|k|/n, where c is the speed of light and n ∼ = κ is the semiconductor refraction index. Assume that the dot is initially in the few-particle state |λi. The luminescence spectrum is then obtained from Fermi’s golden rule: xx0 0 1 4 with ησσ0 = for parallel exciton spins (σ = σ ) and one otherwise, and Ψ̄λxx0 the biexciton wavefunctions within the exciton basis (16).9 As outlined in appendix B.3, these wavefunctions are obtained, in analogy to the excitonic eigenvalue problem (13), from the four-particle Schrödinger equation: (Ex + Ex0 )Ψ̄λxx0 + X V̄xx0 ,x̄x̄0 Ψ̄λx̄x̄0 = Ēλ Ψ̄λxx0 , Iσ (ω) ∝ X |(Pσ )λλ0 |2 Dγ (ω − Eλ0 + Eλ ), (20) λ0 Rwith the total interband-polarization operator Pσ = dr Pσ (r). As Pσ removes one electron-hole pair, it connects states λ and λ0 with different numbers of electrons and holes; in appendix B.2 we discuss how to compute such matrix elements within our direct-diagonalization approach. (18) x̄x̄0 E. Coherent-carrier control with V̄ the exciton-exciton Coulomb matrix elements [Eqs. (B7,B8)] and Ēλ the biexciton energies. As can be inferred from Eq. (B8), the magnitude of these Coulomb elements, which are responsible for biexciton renormalizations, depends on the asymmetry of electron and hole wavefunctions of the exciton states contributing to Ψ̄λxx0 (Hohenester and Molinari 2000a, Hohenester et al. 1999a). 8 Another experimental technique circumventing this problem is photo-luminescence excitation spectroscopy (PLE); for details see, e.g., Bonadeo et al. (1998a, 2000), Hawrylak et al. (2000), Heller and Bockelmann (1997) or Toda et al. (2000). 9 Here we have exploited that the exciton wavefunctions Ψx µν of Eq. (13) provide a complete set of basis functions within the P x ∗ x subspace of electron-hole excitations. Thus, x (Ψµν ) Ψµ0 ν 0 = P 0 x ∗ x δµµ0 δνν 0 and µν (Ψµν ) Ψµν = δxx0 . Note that for parallel spins the states b†xσ b†x0 σ |0i provide an overcomplete set of basis functions (Axt and Mukamel 1998). Up to now we have discussed situations where the optical properties of semiconductor nanostructures are investigated under steady-state conditions. Much of the recent excitement about optical spectroscopy in semiconductors, on the other hand, is due to the possibility of manipulating electron-hole states by means of ultrafast coherent-carrier control (see, e.g., Heberle et al. 1995, Rossi 1998, Shah 1996). In its simplest form, coherent-carrier control requires two temporally delayed sub-picosecond laser pulses (Heberle et al. 1995): the first one creates electron-hole pairs which propagate in phase with the exciting laser pulse; when the second pulse arrives at later time, its action depends on the phase relation between the two pulses and on how much polarization induced by the first pulse is left in the system (i.e., dephasing). In absence of dephasing the second pulse can, depending on the phase relation between the two pulses, either remove all exciton population, or increase it by at most a factor of four. However, such ideal performance is usually spoiled by dephasing processes which are due to scatterings of excitons with environmental degrees of freedom (e.g., phonons) or due to exciton-exciton 9 IV. RESULTS In this section we turn to a brief discussion of the results obtained within our full configuration-interaction approach. We start by discussing in Sec. IV.A model calculations of optical near-field spectroscopy; Secs. IV.B and IV.C are devoted to the optical properties of multiexciton and multi-charged excitons, respectively; finally, we discuss in Sec. IV.D selected topics of coherent-carrier control in quantum dots. A. Near-field spectra The major advancement in the field of quantum dot spectroscopy is due to the improvement of different types of local spectroscopy, which allow the study of individual dots and thus overcome the limitations related to inhomogeneous broadening. Among local spectroscopies, the approaches based on scanning near-field optical microscopy (SNOM; Hecht et al. 2000, Paesler and Moyer 1996) are especially interesting as they bring the spatial resolution well below the diffraction limit of light: With the development of small-aperture optical fiber probes, sub-wavelength resolutions were achieved (λ/8 − λ/5 or λ/40; Betzig et al. 1991, Saiki and Matsuda 1999) and the first applications to nanostructures became possible (Zrenner 2000). As the resolution increases, local optical techniques in principle allow direct access to the space and energy distribution of quantum states within the dot. This opens, however, a number of questions regarding the interpretation of these experiments, that were often neglected in the past. First of all, for spatially inhomogeneous electromagnetic (EM) fields it is no longer possible to define and measure an absorption coefficient that locally relates the absorbed power density with the light intensity (since the susceptibility χ(r, r 0 ) cannot be approximated by a local tensor). In the linear regime, a local absorption coefficient can still be defined, which is however a complicated function that depends on the specific EM field distribution (Mauritz et al. 1999, 2000). The interpretation of near-field spectra therefore requires calculations based on a reasonable assumption for the profile of the EM field. 20 0.0 0.2 0.4 0.6 0.8 1.0 10 X (nm) scatterings. Within the last couple of years the study of such dephasing has attracted enormous interest (see, e.g., Rossi 1998, Shah 1996, and references therein) because of the most interesting fundamental questions related to the interplay of coherent and incoherent carrier dynamics, and of promising technological applications. Implementation of coherent-carrier control in single semiconductor quantum dots is an extremely demanding task, and only recently first experimental results have become available (Bonadeo et al. 1998a, Toda et al. 2000). There, the authors have reported that, because of severe phase-space restrictions and the resulting strong suppressions of scatterings, dephasing rates are strongly reduced as compared to semiconductors of higher dimensionality. 0 -10 -20 0 20 40 60 Photon Energy (meV) 80 100 FIG. 1. Contour plot of near-field spectrum (infinite resolution) of a single quantum dot (Hohenester and Molinari 2000a, Simserides et al. 2000). The solid line shows the optical far-field spectrum; because of symmetry only a small portion of the excitons couples in the far field to the light. Secondly, the quantum states that are actually probed are few-particle states of the interacting electrons and holes photoexcited in the dot. Even in the linear regime, excitonic effects are known to dominate the optical spectra of dots since Coulomb interactions are strongly enhanced by the three-dimensional confinement. Nearfield spectra probe exciton wavefunctions (see Fig. 1), and their spatial coherence and overlap with the EMfield profile will determine the local absorption (Mauritz et al. 1999, 2000) The theoretical framework presented in Secs. II and III is especially suited for a realistic description of the quantum states of the interacting electron and holes photoexcited in the linear regime. In this respect, our theoretical analysis presented in Simserides et al. (2000) improves drastically over previous approaches, which generally focused on a more detailed treatment of the EM field distributions (Bryant 1998, Chavez-Pirson and Chu 1999, Girard et al. 1998, Hanewinkel et al. 1997). B. Multi-excitons We next turn to the case of a dot populated by a larger number of electron-hole pairs. In a typical single-dot experiment (Zrenner 2000), a pump pulse creates electronhole pairs in continuum states (e.g., wetting layer) in the vicinity of the quantum dot, and some of the carriers become captured in the dot; from experiment it is known 10 that there is a fast subsequent carrier relaxation due to environment coupling to the few-particle states of lowest energy (Bayer et al. 2000a, Dekel et al. 1998, Landin et al. 1998, Zrenner 2000); finally, electrons and holes in the dot recombine by emitting a photon. By varying in a steady-state experiment the pump intensity and by monitoring luminescence from the dot states, one thus obtains information about the few-particle carrier states. Fig. 2 shows luminescence spectra computed within our direct-diagonalization approach [Eqs. (10,20)] for different numbers of electron-hole pairs (Hohenester and Molinari 2000b, Rinaldi et al. 2000). Indeed, whenever one additional electron-hole pair is added to the dot the optical spectrum changes because of the additional Coulomb interactions. A comparison of such calculated luminescence spectra with experiment is presented for multi-excitons in Rinaldi et al. (2000) (see Bayer et al. (2000a) for related work). and Pötz 1999, Rossi 1998). Such a controlled “wavefunction engineering” is an essential prerequisite for a successful implementation of ultrafast optical switching and quantum-information processing in a pure solid state system. In this respect, semiconductor quantum dots appear to be among the most promising candidates because of their discrete atomic-like density of states and the resulting suppressed coupling to the solid-state environment. It thus has been envisioned that optical excitations in quantum dots could be successfully exploited for quantum-information processing (Imamoglu et al. 1999, L. Quiroga 1999, Zanardi and Rossi 1998, 1999). In the following we briefly discuss selected topics of coherentcarrier control in semiconductor quantum dots. Coulomb interactions not only lead to renormaliztions of multi-exciton states but also of multi-charged excitons, i.e., complexes consisting of multiple electrons and a single hole. Experimental realization of such carrier complexes can be found, e.g., in Hartmann et al. (2000a), where GaAs/AlGaAs quantum dots are remote-doped with electrons from donors located in the vicinity of the dot. Employing the mechanism of photo-depletion of the quantum dot together with the slow hopping transport of impurity-bound electrons back to the dot (Hartmann et al. 2000a,b), it is possible to efficiently control the number of surplus electrons in the dot from one to approximately six. Other possibilities of remote doping are to place dots within a n-i field-effect structure (Warburton et al. 1997) and to apply an external gate voltage to control the number of surplus electrons (Findeis et al. 2000a, Warburton et al. 2000). Quite generally, with increasing doping the spectral changes are similar to those presented for multi-exciton states. More specifically, the main peaks red-shift because of exchange and correlation effects, and each few-particle state leads to a specific fingerprint in the optical response. This unique assignment of peaks or peak multiplets to given few-particle configurations allows to unambiguously determine the detailed few-particle configuration of carriers in quantum dots in optical experiments; this fact is used in Hartmann et al. (2000a,b) to study the impurity-dot transport. D. Coherent-carrier control Coherent-carrier control in semiconductor nanostructures has recently attracted an enormous interest because it allows the coherent manipulation of carrier wavefunctions on a timescale shorter than typical dephasing times (see, e.g., Bonadeo et al. 1998a, Heberle et al. 1995, Hu Luminescence (arb. units) C. Multi-charged excitons -20 6X (6e+6h) 5X (5e+5h) 4X (4e+4h) 3X (3e+3h) 2X (2e+2h) X (1e+1h) -10 0 10 20 Photon Energy (meV) 30 FIG. 2. Luminescence spectra for a quantum dot (for details see Hohenester and Molinari 2000a) and for different multi-exciton states (number of electron-hole pairs according to expression in parentheses). The insets sketch the respective electron-hole configurations before photon emission (spin orientation according to orientation of triangles). Photon energy zero is given by the groundstate exciton X0 . 1. Quantum-information processing Quantum-information processing (i.e., quantum computation, quantum cryptography, or quantum teleportation) represent exciting new arenas which exploit intrinsic quantum mechanical properties (see, e.g., Bennet and DiVincenzo 2000, DiVincenzo 2000, Steane 1997, and references therein). Basic elements to process quantum information are quantum bits (qubits), which, in analogy to classical bits, are defined as suitably chosen two-level sys- 11 Absorption (arb. units) tems. Much of the present excitement about quantuminformation processing originates from the seminal discoveries of Shor and others, who showed that —provided quantum-information processing can be successfully implemented for ∼100–1000 qubits— quantum algorithms can perform some hard computations much faster than classical algorithms, and can allow the reduction of exponentially complex problems to polynomial complexity (Bennet and DiVincenzo 2000). -10 0 10 20 30 Photon Energy (meV) 40 FIG. 3. Absorption spectra quantum dot (see Troiani et al. 2000a,b, for details), which is initially prepared in: (a) vacuum state; (b) exciton |X0 i state (exciton groundstate); (c) exciton |X1 i state; (d) biexciton |Bi state. Quite generally, only a few basic requirements are needed for a successful implementation of quantuminformation processing, which, according to DiVincenzo (2000), can be summarized in the following five points: (i) A scalable physical system with well characterized qubits; (ii) the ability to initialize the state of the qubits; (iii) long relevant decoherence times, much longer than typical qubit-manipulation times; (iv) a “universal” set of quantum gates; and (v) a qubit-specific measurement capability. Apparently, the main difficulty for a successful implementation of quantum-information processing in a “real physical system” concerns the unavoidable coupling of qubits to their environment, which leads to the process of decoherence where some qubit or qubits of the computation become entangled with the environment, thus in effect “collapsing” the state of the quantum computer. Recently, we have demonstrated (Troiani et al. 2000a,b) that exciton states in quantum dots can serve as qubits, where the requested quantum gates can be implemented by use of ultrafast spectroscopy and coherentcarrier control. The essence of our proposal is summarized in Fig. 3. Fig. 3(a) shows the absorption spectrum of an empty dot (i.e., no electrons and holes present): Two pronounced absorption peaks X0 and X1 can be identified whose energy splitting is of the order of the dot confinement (here ∼25 meV). As the central step within our proposal we ascribe the different exciton states to the computational degrees of freedom (qubits). Thus, for the specific case of Fig. 1 we have two qubits (X0 and X1 ), which have value one if exciton x is populated and zero otherwise (x = X0 , X1 ). Fig. 3 (b) reports the absorption spectrum for the case that the first qubit (exciton X0 ) is set equal to one (is populated): Due to state filling, the character of transition X0 changes from absorption to gain (i.e., negative absorption); in addition, the higher-energetic transition is shifted to lower energy, which is attributed to the formation of a biexcitonic state B whose energy is reduced by an amount of ∆ because of exchange interactions between the two electrons and holes, respectively (Hawrylak 1999, Hohenester and Molinari 2000a). These findings can be formalized as briefly discussed in the following. By use of the exciton operators (16) our computational space is given by: The vacuum state |0i; the single-exciton states (where only one qubit is equal to one) |xi = b†x |0i (we only consider one spin orientation); and |11i = b†X0 b†X1 |0i, the state where both qubits are equal to one (note that, quite generally, it is not obvious that the biexciton state B corresponds to the state |11i since Coulomb interactions can mix different single-exciton states x; however, as discussed in Troiani et al. (2000a,b) at the example of a prototypical dot confinement, in the strong confinement regime |Bi is almost parallel to |11i, thus providing the producttype Hilbert space required for quantum-information processing). Then, the Hamiltonian describing the exciton dynamics in a single dot can be written in analogy to nuclear-magnetic-resonance (NMR) based quantuminformation processing implementations (Bennet and DiVincenzo 2000, Gershenfeld and Chuang 1997): H= X Ex n̂x − ∆ n̂X0 n̂X1 , (21) x=X0 ,X1 with Ex the single-exciton energies and n̂x = b†x bx the exciton occupation operator. From related NMR work it is well known that a Hamiltonian of this form can account for the conditional and unconditional qubit operations required for QIP (see Troiani et al. 2000a,b, for details). 12 2. Adiabatic passage Quantum coherence among charge states is known to have strong impact on the optical properties of confined systems. A striking example of such coherence is provided by trapped states, where two intense laser fields drive an effective three-level system into a state which is completely stable against absorption and emission from the radiation fields (see, e.g., Scully and Zubairy 1997). This highly nonlinear coherent-population trapping is exploited in the process of Stimulated Raman Adiabatic Passage (STIRAP) (Bergmann et al. 1998) to produce complete population transfer between quantum states: Coherent control of optical excitations of atoms and molecules, as well as of chemical-reaction dynamics has been demonstrated in the last years (Bergmann et al. 1998). Successful implementations of such schemes in semiconductor nanostructures would have a strong impetus on the development of novel opto-electronic devices. However, in contrast to atomic systems carrier lifetimes in the solid state are much shorter because of the continuous density-of-states of charge excitations and stronger environment coupling. Although the possibility of trapped states and coherent population transfer in semiconductor quantum wells has been demonstrated theoretically (Binder and Lindberg 1998, Lindberg and Binder 1995, Pötz 1997), its efficiency is expected to be extremely poor as compared to atomic systems. It has, on the other hand, been noted that the situation could be much more favorable in semiconductor quantum dots (Pötz 1997) because of phase-space restrictions. In Hohenester et al. (2000) we propose a solid-state implementation of STIRAP in two coupled semiconductor quantum dots, which are remote-doped with a single surplus hole. The right-dot and left-dot hole states, respectively, are optically connected through the chargedexciton state. Proper combination of pulsed Stokes and pump laser fields then allows a coherent population transfer between the two dot states without suffering significant losses due to environment coupling of the charged exciton state (see Hohenester et al. 2000, for details). 2, that the fluorescence of the laser-driven system exhibits long dark periods, associated to the excitation of the extremely weak 0 ↔ 2 transition, which are followed by bright periods, where many photons are emitted from the decay of the short-lived state 1. In Hohenester (2000) we propose to use continuous laser excitation and continuous monitoring of luminescence for the observation of single spin-flip processes in semiconductor quantum dots. Contrary to the proposal of Dalibard et al., within this scheme the rare scattering events originate from the coupling of the dot carriers to the elementary excitations of their solid-state environment. Thus, the dot serves as a sensor of its environment, which makes possible the observation of the otherwise almost inaccessible rare scattering events. Besides the most challenging prospect of measuring single scatterings in solid state, the long-lived spin excitations in quantum dots have recently attracted strong interest (Gupta et al. 1999, Kamada et al. 1999, Khaetskii and Nazarov 2000) because of their potential utilization for quantum-information processing (Imamoglu et al. 1999, Loss and DiVincenzo 1998). V. SUMMARY In conclusion, we have discussed the theoretical framework for calculating carrier states in semiconductor quantum dots. We have developed a numerical approach for computing single-particle states within the envelopefunction and effective-mass approximations and fewparticle electron-hole states by use of a full configurationinteraction model. The results have been used to predict the optical spectra of quantum dots. More specifically, we have shown that because of the strong quantum confinement Coulomb interactions between electrons and holes are strongly enhanced, and that each electron-hole configuration leads to its characteristic signature in the optical spectra (absorption and luminescence). Our results have been found to be in good agreement with experiment. Finally, we have shown how multi-excitonic states can be exploited for a favourable implementation of novel coherent-control techiques and quantum information processing in semiconductor quantum dots. 3. Quantum measurement A final example which emphasizes the important role of the interplay between coherent and incoherent carrier dynamics in laser-excited single quantum dots is provided by continuously monitored luminescence. The question of how to theoretically describe environment coupling and scatterings in a single system arose almost a decade ago, when it became possible to store single ions in a Paul trap and to continuously monitor their resonance fluorescence (see Plenio and Knight 1998, for a review). In the seminal work of Dalibard et al. (1992) the authors showed for a V-scheme, where the groundstate 0 of an atom is coupled to a short-lived state 1 and to a metastable state ACKNOWLEDGMENTS I am indebted to Elisa Molinari for the opportunity to work in her group and for most generous support, both scientifically and personally, within the last couple of years. I gratefully acknowledge helpful discussions and support from my collegous Costas Simserides, Filippo Troiani, and Rosa Di Felice. I thank the members of the networks “Ultrafast” and “SQID” for lively and stimulating discussions. In addition, I acknowledge successful and pleasant collaborations with Arno Hartmann, 13 Yann Ducommun, Ross Rinaldi, Roberto Cingolani, Artur Zrenner, Giovanna Panzarini, and Fausto Rossi. Special thanks are due to Claudia Ambrosch-Draxl and to the members of the institute of theoretical physics of Graz university for their kind hospitality. Finally, this work would have been hardly possible without the love and support of Moritz, Zoe, and Olga. APPENDIX A: SINGLE-PARTICLE STATES AND MATRIX ELEMENTS Z dr φhν (r)φeµ (r), (A3) with µo the dipole matrix element of the bulk semiconP ductor. Thus, Mνµ = µo i φhν,i φeµ,i for the latticediscretized wavefunctions. 3. Coulomb matrix elements 1. Single-particle states In this appendix we show how to numerically solve the single-particle Schrödinger equation (5) in absence of magnetic fields. The confinement potential P v(r) is assumed to be given at the positions ri = k λki ∆k êk of a sufficiently dense real-space grid, with integers λki ∈ [ 21 Nk , 12 Nk − 1], and meshsize ∆k and unit vector êk in the k-th Cartesian directions. Because of our use of the envelope-function approximation, the low-energy singleparticle envelope functions φµ (r) have an only small rdependence. Introducing the abbreviations φµ,i = φµ (ri ) and vi = v(ri ), Eq. (5) can be approximately solved by direct diagonalization of the lattice-discretized eigenvalue problem: X 2 ∂i,k − + vi φµ,i = µ φµ,i , 2mk Mνµ = µo (A1) k In the calculation of the Coulomb matrix elements of Eq. (7) the limit r − r 0 → 0 has to be treated with some care. First, we note Rthat Eq. (7) is of the form R dr ρ(r)Φ(r), with Φ(r) = dr 0 ρ(r 0 )/κ|r−r 0 |. P The integrals canPbe approximately expressed through i ρi Φi and Φi = j V̄ij ρj , where: V̄ij = 3 Z Y k=1 1 2 ∆k − 12 ∆k dξk 1 κ|ri − rj − ξ| (A4) is the Coulomb potential averaged over a cube around position ri −rj (note that this integral remains finite even for ri − rj = 0), and we have used that, because of the weak r-dependence of the envelope functions φ(r), within each cube ρ can be approximated by ρi = ρ(ri ). In our computational approach we solve the convolution-type P summation Φi = V̄ ρ ij j by Fourier transformation; j only those Coulomb matrix elements which are larger than one percent of the largest element are kept. with APPENDIX B: FEW-PARTICLE STATES φµ (ri − ∆k êk ) − 2φµ (ri ) + φµ (ri + ∆k êk ) . = ∆2k (A2) Since the real-space representation (A1) of the Hamiltonian is a sparse matrix (i.e., the kinetic part only involves a few off-diagonal elements and the contributions proportional to vi are diagonal) and since we are only interested in the low-energy eigenstates, Eq. (A1) can be easily solved for typical grid dimensions of 64 × 64 × 64 (we assume periodic boundary conditions). 2 ∂i,k φµ,i 2. Optical matrix elements In the calculation of the electron-hole states and of their optical properties we need the optical and Coulomb matrix elements within the single-particle bases φeµ and φhν for electrons and holes, respectively. By use of the envelope-function and dipole approximations (Haug and Koch 1993, Rossi 1998) the optical matrix element for the transition between electron state µ and hole state ν is of the form: 1. Direct diagonalization In this Appendix we describe our direct-diagonalization approach for the calculation of the manyparticle states |λi for a given number of electrons and holes. Since the Coulomb elements of Eq. (7) conserve the spin orientations of electrons and holes, we construct our basis functions |`i within the spin-restricted Hilbert spaces: |`i = Y m,σ (ψµmN σ )† . . . (ψµm2 σ )† (ψµm1 σ )† N =Nσm |0i, (B1) µ ~ =~ µm `,σ where |0i denotes the vacuum state (i.e., no electrons and holes) and ` labels the set of quantum numbers µ ~ for electrons and holes with spin orientations σ = ±; for given σ and m, we require (without loss of generality) µN > · · · > µ2 > µ1 . In order to keep the numerics tractable a suitable truncation of the Hilbert space is required. This truncation is performed along two different lines: First, we only keep a 14 small number of single-particle states of lowest energy m µ for electrons and holes, respectively (typically ten states); second, we only keep those µ ~ ’s for which the sum of single P PNσm m particle energies σ i=1 (µi σ )µ~ =~µm < Eom (Eom cho`,σ sen such that typically 100–1000 different configurations for electrons and holes are kept). This separate truncation for electron and hole states turns out to be convenient because of the different magnitudes of level splittings e and h due to the substantially different electron and hole masses. A further reduction of the Hilbert space can be achieved in case of cylinder symmetry, where the angular momentum in z-direction becomes a constant of motion, and only subspaces with given total angular momentum Mz have to be considered (Jacak et al. (1998); m in case of cylinder symmetry these sub-Hilbert spaces remain in general small enough that no additional Eo truncation is required). Next, the many-particle eigenstates are obtained by direct diagonalization of the Hamiltonian matrix (Hc + For Hc , these contributions are diagonal Hcc )``0 . P PNσm m δ``0 m,σ i=1 (µi σ )µ~ =~µm . For the case of Coulomb in`,σ teractions we note that V can connect at most two of the four subspaces (e, ±) and (h, ±). If V connects states µ ~ =µ ~m ~0 = µ ~m `,σ and µ `0 ,σ within the same subspace (m, σ) n and if all µ ~ `,σ0 = µ ~ n`0 ,σ0 within the other subspaces, we obtain the contribution: m Nσ k1 −1 Nσ k3 −1 X X X X (−1)k1 +k2 +k3 +k4 (Vµmm 0 k µ 1 k1 =2 k2 =1 k3 =2 k4 =1 0 k3 ,µk2 µk4 0 − Vµmm 0 k µ 2 0 k3 ,µk1 µk4 ;k3 ,k4 )δµ~µ~ ;k , 1 ,k2 (B2) 0 Q where µ ~ ; k1 , k2 gives a vector of length Nσm − 2 with the entries at positions k1 and k2 removed, and δµ~µ~ = k δµk µ0k . Correspondingly, if V connects states µ ~ =µ ~m ~m ~ n`0 ,σ0 within two different subspaces ν =µ ~ n`,σ0 and µ ~0 = µ ν0 = µ `,σ , ~ `0 ,σ , ~ m̄ 0 m̄ ~ `0 ,σ̄ within the remaining subspaces, we obtain the contribution: (m, σ), (n, σ ) and if µ ~ `,σ̄ = µ m m n n Nσ Nσ Nσ0 Nσ0 X X X X 0 (−1)k1 +k2 +k3 +k4 Vµmn 0 k µ 1 k1 =1 k2 =1 k3 =1 k4 =1 where µ ~ ; k1 gives a vector of length Nσm −1 with the entry at position k1 removed. In our computational approach it turns out be convenient to first compute and store the Coulomb matrix elements for all different combinations of sub-Hilbert spaces (m, σ); these tabulated values are then used to construct the Hamiltonian matrix (Hc + Hcc )``0 . Because of the 0 k2 ,νk3 νk4 0 ;k2 ν ;k4 , δ δµµ ;k 1 ν ;k3 (B3) Kronecker δ’s, only a small fraction of all matrix elements is non-zero (with the above mentioned truncation of Coulomb elements only ∼1%), and the 100–1000 eigenstates of lowest energy Eλ are obtained, for a Hamiltonian matrix of typical dimension 104 , by direct diagonalization. 2. Optical matrix elements We finally show how to compute the optical matrix elements (see also appendix A.2): M± ``0 = h`| X h e Mνµ ψν∓ ψµ± |`0 i. (B4) µν Here we have assumed that photons with well-defined circular polarization σ = ± can create electrons and holes with given (opposite) spin orientations; the dipole operator in Eq. (B4) connects few-particle states |`i and |`0 i with e 0 e h 0 h e 0 e different numbers of electrons and holes, N and N 0 (more specifically, N∓ = N∓ , N± = N± , and N± = N± − 1, h h 0 N∓ = N∓ − 1 for σ = ±). Thus: 0 e 0 h N± N∓ M± ``0 = X X (−1)ke +kh M(~µh0 ke =1 kh =1 where (~ µ)k denotes the k-th component of vector µ ~. ` µ ~ e` ,± µ ~h µ ~ e` ,∓ µ ~h ` ,∓ ` ,± δ δ δ δ , e e e h ) (~ µ ) ~ 0 ;ke µ ~ 0 ~ 0 ;kh µ µ ~ h0 ,∓ kh `0 ,± ke µ ` ,± ` ,∓ ` ,∓ (B5) ` ,± 3. Biexcitons In this appendix we show how to derive Eq. (18); our analysis closely follows the work of Axt and Mukamel 15 (1998). First, we obtain from Eqs. (10) and (B2,B3) the two-electron–two-hole Schrödinger equation: Ēλ Ψλµν,µ0 ν 0 = (eµ + hν + eµ0 + hν0 )Ψλµν,µ0 ν 0 ee λ hh λ +Vµµ̄,µ 0 µ̄0 Ψµ̄ν,µ̄0 ν 0 + V ν ν̄,ν 0 ν̄ 0 Ψµν̄,µ0 ν̄ 0 eh λ eh λ +Vµµ̄,ν ν̄ Ψµ̄ν̄,µ0 ν 0 + Vµ0 µ̄0 ,ν 0 ν̄ 0 Ψµν,µ̄0 ν̄ 0 eh λ eh λ +Vµµ̄,ν 0 ν̄ 0 Ψµ̄ν,µ0 ν̄ 0 + V 0 0 µ µ̄ ,ν ν̄ Ψµν̄,µ̄0 ν 0 , (B6) h e with Ψλµν,µ0 ν 0 = h0|ψνh0 σ0 ψµe 0 σe0 ψνσ ψµσ |λi, and we ase h h sume an implicit summation over bared indices. Inserting ansatz (17) into Eq. 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