We study the counting statistics for electrons and photons being emitted from a driven two level quantum dot. Our technique allows us to calculate their mutual correlations as well. We study different transport configurations by tuning the chemical potential of one of the leads to find that the electronic and photonic fluctuations can be externally manipulated by tuning the AC and transport parameters. We also propose special configurations where electron-photon correlation is maximal meaning that spontaneous emission of photons with a well defined energy is regulated by single electron tunneling. Interesting features are also obtained for energy dependent tunneling.
We study the collective interaction of excitons in closely spaced artificial molecules and arrays of nearly identical quantum dots with the electromagnetic modes. We discuss how collective fluorescence builds up in the presence of a small mismatch of the transition energy. We show that a superradiant state of a single exciton in a molecule of two dots with realistic energy mismatch undergoes a two-rate decay. We analyze also the stability of subdecoherent states for non-identical systems.
We study a two-level quantum dot embedded in a phonon bath and irradiated by a time-dependent ac field and develope a method that allows us to extract simultaneously the full counting statistics of the electronic tunneling and relaxation (by phononic emission) events as well as their correlation. We find that the quantum noise of both the transmitted electrons and the emitted phonons can be controlled by the manipulation of external parameters such as the driving field intensity or the bias voltage.
We experimentally investigate the fluorescence photon emission characteristics for single q-dots by using optical nanofibers. We demonstrate that single q-dots can be deposited along an optical nanofiber systematically and reproducibly with a precision of 5 μm. For single q-dots on an optical nanofiber, we measure the fluorescence photon numbers coupled into the nanofiber and the normalized photon correlations, by varying the excitation laser intensity. We estimate the fluorescence photon coupling efficiency into the nanofiber guided modes to be higher than 9.4\pm3%.
Single photon sources, which are compatible with quantum memories are an important component of quantum networks. In this article, we show optical investigations on isolated GaAs/Al$_{0.25}$Ga$_{0.75}$As quantum dots grown via droplet epitaxy, which emit single photons on resonance with the Rb-87-D$_2$ line (780 nm). Under continuous wave resonant excitation conditions, we observe bright, clean and narrowband resonance fluorescence emission from such a droplet quantum dot. Furthermore, the second-order correlation measurement clearly demonstrates the single photon emission from this resonantly…
The combination of single photon emitters (quantum dots) and tailored metal nanoparticles with defined size and shape allows a detailed study of the interaction between light and matter. The enhanced optical near-field of the nanoparticles can strongly influence the absorption and emission of nearby fluorescent quantum dots. We show that a controlled spatial arrangement enables the analysis and understanding of polarization dependent coupling between a metal nanoparticle and few or single fluorescent quantum dots. Modifications in the fluorescence spectrum and lifetime are analyzed and compare…
We demonstrate systematic resonance fluorescence recovery with near-unity emission efficiency in single quantum dots embedded in a charge-tunable device in a wave-guiding geometry. The quantum dot charge state is controlled by a gate voltage, through carrier tunneling from a close-lying Fermi sea, stabilizing the resonantly photocreated electron-hole pair. The electric field cancels out the charging/discharging mechanisms from nearby traps toward the quantum dots, responsible for the usually observed inhibition of the resonant fluorescence. Fourier transform spectroscopy as a function of the a…
Optically active quantum dots, for instance self-assembled InGaAs quantum dots, are potentially excellent single photon sources. The fidelity of the single photons is much improved using resonant rather than non-resonant excitation. With resonant excitation, the challenge is to distinguish between resonance fluorescence and scattered laser light. We have met this challenge by creating a polarization-based dark-field microscope to measure the resonance fluorescence from a single quantum dot at low temperature. We achieve a suppression of the scattered laser exceeding a factor of 10^7 and backgr…
Surface graphene guides were interfaced with an array of individual semiconductor quantum dots, whose position was commensurate with the optical guide modes. The surface guide served as a channel for a Field Effect Transistor (FET) while the dots were placed within the capacitor formed between the graphene channel and the gate electrode. We report on negative differential photo-related conductance under light and a diminishing fluorescence effect as a function of bias. We suggest that the quenched fluorescence may be hindered, to some degree, by incorporating the QD in a resonator, which is tu…
We report on enhanced fluorescence of lead sulfide quantum dots interacting with leaky modes of slab-type silicon photonic crystals. The photonic crystal slabs were fabricated supporting leaky modes in the near infrared wavelength range. Lead sulfite quantum dots which are resonant the same spectral range were prepared in a thin layer above the slab. We selectively excited the leaky modes by tuning wavelength and angle of incidence of the laser source and measured distinct resonances of enhanced fluorescence. By an appropriate experiment design, we ruled out directional light extraction effect…
We present the theory of resonance fluorescence from an asymmetric quantum dot driven by a two-component electromagnetic field with two different frequencies, polarizations and amplitudes (bichromatic field) in the regime of strong light-matter coupling. It follows from the elaborated theory that the broken inversion symmetry of the driven quantum system and the bichromatic structure of the driving field result in unexpected features of the resonance fluorescence, including the infinite set of Mollow triplets, the quench of fluorescence peaks induced by the dressing field, and the oscillating …
Photonic quantum technologies call for scalable quantum light sources that can be integrated, while providing the end user with single and entangled photons on-demand. One promising candidate are strain free GaAs/AlGaAs quantum dots obtained by droplet etching. Such quantum dots exhibit ultra low multi-photon probability and an unprecedented degree of photon pair entanglement. However, different to commonly studied InGaAs/GaAs quantum dots obtained by the Stranski-Krastanow mode, photons with a near-unity indistinguishability from these quantum emitters have proven to be elusive so far. Here, …
We present a new physical model resolving a long-standing mystery of the power-law distributions of the blinking times in single colloidal quantum dot fluorescence. The model considers the non-radiative relaxation of the exciton through multiple recombination centers. Each center is allowed to switch between two quasi-stationary states. We point out that the conventional threshold analysis method used to extract the exponents of the distributions for the on-times and off-times has a serious flaw: The qualitative properties of the distributions strongly depend on the threshold value chosen for …
Using a polaron master equation approach we investigate the resonance fluorescence spectra from coherently driven quantum dots (QDs) coupled to an acoustic phonon bath and a photonic crystal waveguide with a rich local density of photon states (LDOS). Resonance fluorescence spectra from QDs in semiconductor crystals are known to show strong signatures of electron-phonon interactions, but when coupled to a structured photonic reservoir, the QD emission properties are also determined by the frequency dependence of the LDOS of the photon reservoir. Here, we investigate the simultaneous role of co…
The electronic energy levels and optical transitions of a semiconductor quantum dot are subject to dynamics within the solid-state environment. In particular, fluctuating electric fields due to nearby charge traps or other quantum dots shift the transition frequencies via the Stark effect. The environment dynamics are mapped directly onto the fluorescence under resonant excitation and diminish the prospects of quantum dots as sources of indistinguishable photons in optical quantum computing. Here, we present an analysis of resonance fluorescence fluctuations based on photon counting statistics…
Resonant optical excitation of few-level quantum systems enables coherent quantum control, resonance fluorescence, and direct characterization of dephasing mechanisms. Experimental demonstrations have been achieved in a variety of atomic and solid-state systems. An alternative but intriguing quantum photonic platform is based on single layer transition metal chalcogenide semiconductors, which exhibit a direct band-gap with optically addressable exciton valley-pseudospins in a uniquely two-dimensional form. Here we perform resonance and near-resonance excitation of three-dimensionally confined …
Time-resolved resonance fluorescence (RF) on a single self-assembled quantum dot (QD) is used to analyze the generation and capture of photoinduced free charge carriers. We directly observe the capture of electrons into the QD in an intensity reduction of the exciton transition. The exciton transition is quenched until the captured electron tunnels out of the dot again in the order of milliseconds. Our results demonstrate that even under resonant excitation, excited, free electrons are generated, which can negatively influence the optical properties of a QD. This detrimental effect has been ne…
We report a new way to slow down the spontaneous emission rate of excitons in the wetting layer (WL) through radiative field coupling between the exciton emissions and the dipole field of metal islands. As a result, a long-lifetime decay process is detected in the emission of InAs/GaAs single quantum dots (QDs). It is found that when the separation distance from WL layer (QD layer) to the metal islands is around 20 nm and the islands have an average size of approximately 50 nm, QD lifetime may change from approximately 1 to 160 ns. The corresponding second-order autocorrelation function g(2) (…
Site-controlled growth of semiconductor quantum dots (QDs) represents a major advancement to achieve scalable quantum technology platforms. One immediate benefit is the deterministic integration of quantum emitters into optical microcavities. However, site-controlled growth of QDs is usually achieved at the cost of reduced optical quality. Here, we show that the buried-stressor growth technique enables the realization of high-quality site-controlled QDs with attractive optical and quantum optical properties. This is evidenced by performing excitation power dependent resonance fluorescence expe…
We report on a Python-toolbox for unbiased statistical analysis of fluorescence intermittency properties of single emitters. Intermittency, i.e., step-wise temporal variations in the instantaneous emission intensity and fluorescence decay rate properties are common to organic fluorophores, II-VI quantum dots and perovskite quantum dots alike. Unbiased statistical analysis of intermittency switching time distributions, involved levels and lifetimes is important to avoid interpretation artefacts. This work provides an implementation of Bayesian changepoint analysis and level clustering applicabl…
We present an open-system master equation study of the coherent and incoherent resonance fluorescence spectrum from a two-level quantum system under coherent pulsed excitation. Several pronounced features which differ from the fluorescence under a constant drive are highlighted, including a multi-peak structure and a pronounced off-resonant spectral asymmetry, in stark contrast to the conventional symmetrical Mollow triplet. We also study semiconductor quantum dot systems using a polaron master equation, and show how the key features of dynamic resonance fluorescence change with electron--acou…
We have developed a theory of the one-phonon intraband resonance Raman scattering (IRRS) in anisotropic quantum dots subjected to an arbitrarily directed magnetic field. The differential Raman cross section is obtained. The resonance structure of the Raman cross section is studied. It is shown that the quantum dot subjected in a magnetic field can be used as the detector of phonon modes. The interesting multiplet structure of the resonance peaks is studied.
The dynamic response of InAs/GaAs self-assembled quantum dots (QDs) to strain is studied experimentally by periodically modulating the QDs with a surface acoustic wave and measuring the QD fluorescence with photoluminescence and resonant spectroscopy. When the acoustic frequency is larger than the QD linewidth, we resolve phonon sidebands in the QD fluorescence spectrum. Using a resonant pump laser, we have demonstrated optical frequency conversion via the dynamically modulated QD, which is the physical mechanism underlying laser sideband cooling a nanomechanical resonator by means of an embed…
A set of theoretical results on infrared absorption and inelastic (Raman) scattering of light in semiconductor quantum dots is reviewed. When it is possible, qualitative comparisons with recent experimental results are presented.
Fluctuations in the fluorescence from macroscopic ensembles of colloidal semiconductor quantum dots have the spectral form of 1/f noise. The measured power spectral density reflects the fluorescence intermittency of individual dots with power-law distributions of "on" and "off" times, and can thus serve as a simple method for characterizing such blinking behavior.
We investigate resonant tunneling through molecular states of coupled double quantum dots embedded in an Aharonov-Bohm (AB) interferometer. The conductance through the system consists of two resonances associated with the bonding and the antibonding quantum states. We predict that the two resonances are composed of a Breit-Wigner resonance and a Fano resonance, those widths and Fano factor depending on the AB phase very sensitively. Further, we point out that the bonding properties, such as the covalent and the ionic bonding, can be identified by the AB oscillations.
We have studied the absorption of electromagnetic radiation of an anisotropic quantum dot taking into account the spin-flip processes that is associated with the interaction of the electrons with optical phonons. It is shown that these processes lead to the resonance absorption. Explicit formula is derived for the absorption coefficient. The positions of the resonances peaks are found.
We report two-photon resonance fluorescence of an individual semiconductor artificial atom. By non-linearly driving a single quantum dot via a two-photon transition, we probe the linewidth of the two-photon processes and show that, similar to their single-photon counterparts, they are close to being Fourier limited at low temperatures. The evolution of the population of excitonic states with the Rabi frequency exhibits a clear s-shaped behavior, indicative of the non-linear response via the two-photon excitation process. We model the non-linear response using a 4-level atomic system representi…
We derive a general formula for the non-Markovian fluorescence spectrum of a multi-level system interacting with a bosonic environment. To this end, we apply linear-response theory to describe the dynamics of a detector monitoring the emission spectrum of a general multi-level system. The resultant emission lineshape function is directly related to the two-time correlation of system observables, which we derive using Nakajima-Zwanzig Generalized Master Equation without assuming a Markov approximation.
The fluorescence spectrum of a strongly pumped two level system is characterised by the Mollow triplet that has been observed in a variety of systems, ranging from atoms to quantum dots and superconducting qubits. We theoretically studied the fluorescence of a strongly pumped intersubband transition in a quantum well. Our results show that the many-electron nature of such a system leads to a modification of the usual Mollow theory. In particular, the intensity of the central peak in the fluorescence spectrum becomes a function of the electron coherence, allowing access to the coherence time of…
We report the first experimental demonstration of interference-induced spectral line elimination predicted by Zhu and Scully [Phys. Rev. Lett. 76, 388 (1996)] and Ficek and Rudolph [Phys. Rev. A 60, 4245 (1999)]. We drive an exciton transition of a self-assembled quantum dot in order to realize a two-level system exposed to bichromatic laser field and observe nearly complete elimination of the resonance fluorescence spectral line at the driving laser frequency. This is caused by quantum interference between coupled transitions among the doubly dressed excitonic states, without population trapp…
Based on calculations of the electronic structure of coupled multiple quantum dots, we study systemically the transport properties of the system driven by an ac electric field. We find qualitative difference between transport properties of double coupled quantum dots (DQDs) and triple quantum dots. For both symmetrical and asymmetrical configurations of coupled DQDs, the field can induce the photon-assisted Fano resonances in current-AC frequency curve in parallel DQDs, and a symmetric resonance in serial DQDs. For serially coupled triple quantum dots(STQDs), it is found that the $Λ$-type ener…
We show that the numerical method [M.E. Torio, K. Hallberg, A.H. Ceccatto, and C.R. Proetto, Phys. Rev. B 65, 085302 (2002), cond-mat/0108167v2] does not reproduce correctly the Kondo physics in quantum dot systems with interaction.
We investigate the charging dynamics in epitaxially grown InAs quantum dots under resonant excitation with and without additional low-power above-band excitation. Time-resolved resonance fluorescence from a charged exciton (trion) transition is recorded as the above-band excitation is modulated on and off. The fluorescence intensity varies as the QD changes from charged to neutral and back due to the influence of the above-band excitation. We fit the transients of the time-resolved resonance fluorescence with models that represent the charging and neutralization processes. The time dependence …
We investigate the spectrum of the resonance fluorescence from the lateral triple quantum dots controlled by voltage and obtain some interesting features such as controllable triple narrow peaks. In our system we use tunneling instead of coupling lasers, and the positions, widths and heights of the resonance fluorescence peaks can be controlled by tuning the tunneling couplings. We explain the observed spectrum with the transition properties of the dressed states generated by the coupling of the two tunneling and the laser field. These features can also be viewed as the effects of Spontaneousl…
We propose a controllable way of tuning Fano resonances in open quantum dots in the absence of magnetic field. A quantum dot can be modified by changing the gate voltages that define the dot itself. An extra degree of freedom can be introduced by means of a controllable repulsive perturbation, such as the one induced by scanning a Atomic Force Microscope tip on a real sample. We numerically investigate the coupling between localized states in the dot and the continuum in the leads. The advantage of such position controllable perturbation is the selective manipulation of the quantum dot states.…
We use a laser-driven single (In,Ga)As quantum dot (QD) in the dressed state regime of resonance fluorescence ($T = 4$ K) to observe the four $D_1$-transition lines of alkali atomic cesium ($Cs$) vapor at room temperature. We tune the frequency of the dressing continuous-wave laser in the vicinity of the bare QD resonance $\sim 335.116$ THz ($\sim 894.592$ nm) at constant excitation power and thereby controllably tune the center and side channel frequencies of the probe light, i.e. the Mollow triplet. Resonances between individual QD Mollow triplet lines and the atomic hyperfine-split transiti…
The absorption of electromagnetic radiation of an anisotropic quantum dot is theoretically investigated taking into account the processes associated with simultaneous scattering from ionized impurities. It is shown that the scattering of electrons by impurities leads to the resonance absorption even if we have only one impurity in the quantum dot. Explicit formula is derived for the absorption coefficient. The positions of the resonances peaks are found. The effects of external magnetic field on the resonance absorption are studied.
The nature of fluorescence intermittency for semiconductor quantum dots (QD) and single molecules (SM) is proposed as a manifestation of Anderson localization. The power law like distribution for the \emph{on} time is explained as due to the interaction between QD/SM with a random environment. In particular, we find that the \emph{on}-time probability distribution behaves differently in localized and delocalized regimes. They, when properly scaled, are \emph{universal} for different QD/SM systems. The \emph{on}-time probability distribution function in the delocalized QD/SM regime can be appro…
We study the second-order polaronic resonance between 2-LO-phonon states and p-shell electron states in a quantum dot. We show that the spectrum in the resonance area can be quantitatively reproduced by a theoretical model using only standard material constants, without any free parameters. We propose also a perturbative approach to the problem based on a quasi-degenerate perturbation theory. This method not only considerably reduces the numerical complexity without considerable loss of accuracy but also gives some insight into the structure and origin of the resonance spectrum.
Resonance fluorescence from a quantum emitter is an ideal source to extract indistinguishable photons. By using the cross polarization to suppress the laser scattering, we observed resonance fluorescence from GeV color centers in diamond at cryogenic temperature. The Fourier-transform-limited linewidth emission with $T_2/2T_1\sim0.86$ allows for two-photon interference based on single GeV color center. Under pulsed excitation, the 24 ns separated photons exhibit a Hong-Ou-Mandel visibility of $0.604\pm0.022$, while the continuous-wave excitation leads to a coalescence time window of 1.05 radia…
The Fano resonance in quantum dots embedded in Aharonov-Bohm rings is examined theoretically, using two models of non-interacting electrons. The first model yields an analytical expression for the conductance G. G is written in an extended Fano form with a complex parameter. The shape of the resonance can be asymmetric or symmetric, depending on the magnetic flux enclosed in the ring. The "phase" of the resonance is changed continuously with increasing the flux in two-terminal situations. These are in accordance with recent experimental results. In the second model, we consider the dephasing e…
Scalable quantum photonic networks require coherent excitation of quantum emitters. However, many solid-state systems can undergo a transition to a dark shelving state that inhibits the fluorescence. Here we demonstrate that a controlled gating using a weak non-resonant laser, the resonant excitation can be recovered and amplified for single germanium vacancies (GeVs). Employing the gated resonance excitation, we achieve optically stable resonance fluorescence of GeV centers. Our results are pivotal for the deployment of diamond color centers as reliable building blocks for scalable solid stat…
Photoluminescence (PL) intermittency is a ubiquitous phenomenon detrimentally reducing the temporal emission intensity stability of single colloidal quantum dots (CQDs) and the emission quantum yield of their ensembles. Despite efforts for blinking reduction via chemical engineering of the QD architecture and its environment, blinking still poses barriers to the application of QDs, particularly in single-particle tracking in biology or in single-photon sources. Here, we demonstrate the first deterministic all-optical suppression of quantum dot blinking using a compound technique of visible and…
We report on resonance fluorescence from a single quantum dot emitting at telecom wavelengths. We perform high-resolution spectroscopy and observe the Mollow triplet in the Rabi regime--a hallmark of resonance fluorescence. The measured resonance-fluorescence spectra allow us to rule out pure dephasing as a significant decoherence mechanism in these quantum dots. Combined with numerical simulations, the experimental results provide robust characterisation of charge noise in the environment of the quantum dot. Resonant control of the quantum dot opens up new possibilities for on-demand generati…
Intensity squeezing, i.e., photon number fluctuations below the shot noise limit, is a fundamental aspect of quantum optics and has wide applications in quantum metrology. It was predicted in 1979 that the intensity squeezing could be observed in resonance fluorescence from a two-level quantum system. Yet, its experimental observation in solid states was hindered by inefficiencies in generating, collecting and detecting resonance fluorescence. Here, we report the intensity squeezing in a single-mode fibre-coupled resonance fluorescence single-photon source based on a quantum dot-micropillar sy…
The implementation and engineering of bright and coherent solid state quantum light sources is key for the realization of both on chip and remote quantum networks. Despite tremendous efforts for more than 15 years, the combination of these two key prerequisites in a single, potentially scalable device is a major challenge. Here, we report on the observation of bright and coherent single photon emission generated via pulsed, resonance fluorescence conditions from a single quantum dot (QD) deterministically centered in a micropillar cavity device via cryogenic optical lithography. The brightness…
We report on the observation of bright emission of single photons under pulsed resonance fluorescence conditions from a single quantum dot (QD) in a micropillar cavity. The brightness of the QD fluorescence is greatly enhanced via the coupling to the fundamental mode of a micropillar, allowing us to determine a single photon extraction efficiency of $(20.7\pm0.8)~\%$ per linear polarization basis. This yields an overall extraction efficiency of $(41.4\pm1.5)~\%$ in our device. We observe the first Rabi-oscillation in a weakly coupled quantum dot-micropillar system under coherent pulsed optical…
The mesoscopic characteristics of a quantum dot (QD), which make the dipole approximation (DA) break down, provide a new dimension to manipulate light-matter interaction [M. L. Andersen et al., Nat. Phys. 7, 215 (2011)]. Here we investigate the power spectrum and the second-order correlation property of the fluorescence from a resonantly driven QD placed on a planar metal. It is revealed that due to the pronounced QD spatial extension and the dramatic variation of the triggered surface plasmon near the metal, the fluorescence has a notable contribution from the quadrupole moment. The π-rotatio…
Spectral filtering of resonance fluorescence is widely employed to improve single photon purity and indistinguishability by removing unwanted backgrounds. For filter bandwidths approaching the emitter linewidth, complex behaviour is predicted due to preferential transmission of components with differing photon statistics. We probe this regime using a Purcell-enhanced quantum dot in both weak and strong excitation limits, finding excellent agreement with an extended sensor theory model. By changing only the filter width, the photon statistics can be transformed between antibunched, bunched, or …
Pioneering methods in recent optical lattice experiments allow to focus laser beams down to a spot size that is comparable to the lattice constant. Inspired by this achievement, we examine the resonance fluorescence spectra of two-level atoms positioned in adjacent lattice sites and compare the case where the laser hits only one atom (single site addressing) with cases where several atoms are illuminated. In contrast to the case where the laser hits several atoms, the spectrum for single site addressing is no longer symmetric around the laser frequency. The shape of the spectrum of fluorescent…
Multi-photon emitters are a sought-after resource in quantum photonics. Nonlinear interactions between a multi-level atomic system and a coherent drive can lead to resonant two-photon emission, but harvesting light from this process has remained a challenge due to the small oscillator strengths involved. Here we present a study of two-photon resonance fluorescence at microwave frequencies, using a superconducting, ladder-type artificial atom, a transmon, strongly coupled to a waveguide. We drive the two-photon transition between the ground and second-excited state at increasingly high powers a…
Signatures of inter-sideband coherence in the resonance fluorescence spectrum of an acoustically-modulated quantum dot
Institute of Theoretical Physics, Wrocław University of Science and Technology（弗罗茨瓦夫科技大学理论物理研究所）；Institute of Physics, University of Münster（明斯特大学物理研究所）
In the present work, the effect of resonant pumping schemes in improving the photon coherence is investigated on InAs/InGaAs/GaAs quantum dots emitting in the telecom C-band. The linewidths of transitions of multiple exemplary quantum dots are determined under above-band pumping and resonance fluorescence via Fourier-transform spectroscopy and resonance scans, respectively. The average linewidth is reduced from $9.74\,\mathrm{GHz}$ in above-band excitation to $3.50\,\mathrm{GHz}$ in resonance fluorescence underlining its superior coherence properties. Furthermore, the feasibility of coherent s…
We temporally resolve the resonance fluorescence from an electron spin confined to a single self-assembled quantum dot to measure directly the spin's optical initialization and natural relaxation timescales. Our measurements demonstrate that spin initialization occurs on the order of microseconds in the Faraday configuration when a laser resonantly drives the quantum dot transition. We show that the mechanism mediating the optically induced spin-flip changes from electron-nuclei interaction to hole-mixing interaction at 0.6 Tesla external magnetic field. Spin relaxation measurements result in …
We show how to calculate the first- and second-order statistics of the scattered fields for an arbitrary intensity coherent state light field interacting with a two-level system in a waveguide geometry. Specifically, we calculate the resonance fluorescence from the qubit, using input-output formalism. We derive the transmission and reflection coefficients, and illustrate the bunching and anti-bunching of light that is scattered in the forward and backward directions, respectively. Our results agree with previous calculations on one- and two-photon scattering as well as those that are based on …
