Photon Counting Statistics for Blinking CdSe−ZnS Quantum Dots: A Lévy Walk Process

Blinking-Based Multiplexing: A New Approach for Differentiating Spectrally Overlapped Emitters

Multicolor single-molecule imaging is widely applied to answer questions in biology and materials science. However, most studies rely on spectrally distinct fluorescent probes or time-intensive sequential imaging strategies to multiplex. Here, we introduce blinking-based multiplexing (BBM), a simple approach to differentiate spectrally overlapped emitters based solely on their intrinsic blinking dynamics. The blinking dynamics of hundreds of rhodamine 6G and CdSe/ZnS quantum dots on glass are obtained using the same acquisition settings and analyzed with a change point detection algorithm. Although substantial blinking heterogeneity is observed, the analysis yields a blinking metric with 93.5% classification accuracy. We further show that BBM with up to 96.6% accuracy is achieved by using a deep learning algorithm for classification. This proof-of-concept study demonstrates that a single emitter can be accurately classified based on its intrinsic blinking dynamics and without the need to probe its spectral color.

Stochastic Kinetics of Nanocatalytic Systems

Catalytic reaction events occurring on the surface of a nanoparticle constitute a complex stochastic process. Although advances in modern single-molecule experiments enable direct measurements of individual catalytic turnover events occurring on a segment of a single nanoparticle, we do not yet know how to measure the number of catalytic sites in each segment or how the catalytic turnover counting statistics and the catalytic turnover time distribution are related to the microscopic dynamics of catalytic reactions. Here, we address these issues by presenting a stochastic kinetics for nanoparticle catalytic systems. We propose a new experimental measure of the number of catalytic sites in terms of the mean and variance of the catalytic event count. By considering three types of nanocatalytic systems, we investigate how the mean, the variance, and the distribution of the catalytic turnover time depend on the catalytic reaction dynamics, the heterogeneity of catalytic activity, and communication among catalytic sites. This work enables accurate quantitative analyses of single-molecule experiments for nanocatalytic systems and enzymes with multiple catalytic sites.

Insights on heterogeneity in blinking mechanisms and non-ergodicity using sub-ensemble statistical analysis of single quantum-dots

Photo-luminescence (P-L) intermittency (or blinking) in semiconductor nanocrystals (NCs), a phenomenon ubiquitous to single-emitters, is generally considered to be temporally random intensity fluctuations between "bright" ("On") and "dark" ("Off") states. However, individual quantum-dots (QDs) rarely exhibit such telegraphic signals, and yet, a vast majority of single-NC blinking data are analyzed using a single fixed threshold which generates binary trajectories. Furthermore, while blinking dynamics can vary dramatically over NCs in the ensemble, the extent of diversity in the exponents (m_On/Off) of single-particle On-/Off-time distributions (P(t_On/Off)), often used to validate mechanistic models of blinking, remains unclear due to a lack of statistically relevant data sets. Here, we subclassify an ensemble of QDs based on the emissivity of each emitter and subsequently compare the (sub)ensembles' behaviors. To achieve this, we analyzed a large number (>1000) of blinking trajectories for a model system, Mn⁺² doped ZnCdS QDs, which exhibits diverse blinking dynamics. An intensity histogram dependent thresholding method allowed us to construct distributions of relevant blinking parameters (such as m_On/Off). Interestingly, we find that single QD P(t_On/Off)s follow either truncated power law or power law, and their relative proportion varies over subpopulations. Our results reveal a remarkable variation in m_On/Off amongst as well as within subensembles, which implies multiple blinking mechanisms being operational amongst various QDs. We further show that the m_On/Off obtained via cumulative single-particle P(t_On/Off) is distinct from the weighted mean value of all single-particle m_On/Off, evidence for the lack of ergodicity. Thus, investigation and analyses of a large number of QDs, albeit for a limited time span of a few decades, are crucial to characterize the spatial heterogeneity in possible blinking mechanisms.

Biexciton and trion energy transfer from CdSe/CdS giant nanocrystals to Si substrates

Observation of energy transfer (ET) from multiexcitonic (MX) complexes in nanocrystal quantum dots (NQDs) has been severely restricted due to efficient nonradiative Auger recombination leading to very low MX emission quantum yields. Here we employed "giant" CdSe/CdS NQDs with suppressed Auger recombination to study ET of biexcitons (BX) and charged excitons (trions) into Si substrate. Photoluminescence (PL) measurements of (sub)monolayers of gNQDs controllably assembled on various interacting surfaces and augmented by single gNQD's imaging reveal appearance of BX spectral signatures and progressive acceleration of PL lifetimes of all excitonic species on Si substrates. From statistical analysis of a large number of PL lifetime traces, representative exciton, trion and BX ET efficiencies are measured as ∼75%, 55% and 45% respectively. Detailed analysis of the MX's radiative rates demonstrate the crucial role of the radiative (waveguide) ET in maintaining high overall transfer efficiency despite the prevalent Auger recombination. Our observations point towards practical utilization of MX-bearing nanocrystals in future optoelectronics architectures.

Evidence in Support of Exciton to Ligand Vibrational Coupling in Colloidal Quantum Dots

The Perspective focuses on the investigation of an unresolved conflict in semiconductor colloidal quantum dots (CQDs) research, concerning the influence of the immediate surrounding on the optical properties of the materials. Today's advanced synthetic colloidal procedures offer formation of a high-quality inorganic crystallite, capped with various organic/inorganic molecular ligands. The Perspective aims to clarify whether exciton recombination processes in CQDs are influenced by the type of crystallite-ligand bonding and, moreover, whether these excitonic processes experience direct coupling to the ligands' vibrational modes. Most ligands used have redox characteristics whose functional groups are added on to the CQDs' surface via coordination, covalent or ionic bonding. The surface-ligand bonding introduces electronic states either above or below the intraband/interband energy gap, resulting in electronic passivation or in creation of trapping states that affect intraband and interband relaxation processes. Furthermore, crystalline electronic states may have a direct coupling to molecular vibrational states via direct overlap of electronic wave functions or through a long-range energy-transfer process. Also, photoejected carriers resulting from an Auger process or ionization processes may diffuse temporarily onto a ligand site. These scenarios are discussed in the current publication with supporting theoretical and experimental observations.

Room-temperature single-photon generation from solitary dopants of carbon nanotubes

On-demand single-photon sources capable of operating at room temperature and the telecom wavelength range of 1,300-1,500 nm hold the key to the realization of novel technologies that span from sub-diffraction imaging to quantum key distribution and photonic quantum information processing. Here, we show that incorporation of undoped (6,5) single-walled carbon nanotubes into a SiO2 matrix can lead to the creation of solitary oxygen dopant states capable of fluctuation-free, room-temperature single-photon emission in the 1,100-1,300 nm wavelength range. We investigated the effects of temperature on photoluminescence emission efficiencies, fluctuations and decay dynamics of the dopant states and determined the conditions most suitable for the observation of single-photon emission. This emission can in principle be extended to 1,500 nm by doping of smaller-bandgap single-walled carbon nanotubes. This easy tunability presents a distinct advantage over existing defect centre single-photon emitters (for example, diamond defect centres). Our SiO2-encapsulated sample also presents exciting opportunities to apply Si/SiO2-based micro/nano-device fabrication techniques in the development of electrically driven single-photon sources and integration of these sources into quantum photonic devices and networks.

Two-state theory of binned photon statistics for a large class of waiting time distributions and its application to quantum dot blinking

A theoretical method is proposed for the calculation of the photon counting probability distribution during a bin time. Two-state fluorescence and steady excitation are assumed. A key feature is a kinetic scheme that allows for an extensive class of stochastic waiting time distribution functions, including power laws, expanded as a sum of weighted decaying exponentials. The solution is analytic in certain conditions, and an exact and simple expression is found for the integral contribution of "bright" and "dark" states. As an application for power law kinetics, theoretical results are compared with experimental intensity histograms from a number of blinking CdSe/ZnS quantum dots. The histograms are consistent with distributions of intensity states around a "bright" and a "dark" maximum. A gap of states is also revealed in the more-or-less flat inter-peak region. The slope and to some extent the flatness of the inter-peak feature are found to be sensitive to the power-law exponents. Possible models consistent with these findings are discussed, such as the combination of multiple charging and fluctuating non-radiative channels or the multiple recombination center model. A fitting of the latter to experiment provides constraints on the interaction parameter between the recombination centers. Further extensions and applications of the photon counting theory are also discussed.

Asymptotic densities of ballistic Lévy walks

We propose an analytical method to determine the shape of density profiles in the asymptotic long-time limit for a broad class of coupled continuous-time random walks which operate in the ballistic regime. In particular, we show that different scenarios of performing a random-walk step, via making an instantaneous jump penalized by a proper waiting time or via moving with a constant speed, dramatically effect the corresponding propagators, despite the fact that the end points of the steps are identical. Furthermore, if the speed during each step of the random walk is itself a random variable, its distribution gets clearly reflected in the asymptotic density of random walkers. These features are in contrast with more standard nonballistic random walks.

Correlation of atomic structure and photoluminescence of the same quantum dot: pinpointing surface and internal defects that inhibit photoluminescence

In a size regime where every atom counts, rational design and synthesis of optimal nanostructures demands direct interrogation of the effects of structural divergence of individuals on the ensemble-averaged property. To this end, we have explored the structure-function relationship of single quantum dots (QDs) via precise observation of the impact of atomic arrangement on QD fluorescence. Utilizing wide-field fluorescence microscopy and atomic number contrast scanning transmission electron microscopy (Z-STEM), we have achieved correlation of photoluminescence (PL) data and atomic-level structural information from individual colloidal QDs. This investigation of CdSe/CdS core/shell QDs has enabled exploration of the fine structural factors necessary to control QD PL. Additionally, we have identified specific morphological and structural anomalies, in the form of internal and surface defects, that consistently vitiate QD PL.

Infinite densities for Lévy walks

Motion of particles in many systems exhibits a mixture between periods of random diffusive-like events and ballistic-like motion. In many cases, such systems exhibit strong anomalous diffusion, where low-order moments 〈|x(t)|(q)〉 with q below a critical value q(c) exhibit diffusive scaling while for q>q(c) a ballistic scaling emerges. The mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion, e.g., the known diffusion equations and central limit theorems fail to describe both aspects. In this paper we resolve this problem by resorting to the concept of infinite density. Using the widely applicable Lévy walk model, we find a general expression for the corresponding non-normalized density which is fully determined by the particles velocity distribution, the anomalous diffusion exponent α, and the diffusion coefficient K(α). We explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data.

Competition between auger recombination and hot-carrier trapping in PL intensity fluctuations of type II nanocrystals

Performing time-tagged, time-correlated, single-photon-counting studies on individual colloidal nanocrystal quantum dots (NQDs), the evolution of photoluminescence (PL) intensity-fluctuation behaviors in near-infrared (NIR) emitting type II, InP/CdS core-shell NQDs is investigated as a function of shell thickness. It is observed that Auger recombination and hot-carrier trapping compete in defining the PL intensity-fluctuation behavior for NQDs with thin shells, whereas the role of hot-carrier trapping dominates for NQDs with thick shells. These studies further reveal the distinct ramifications of altering either the excitation fluence or repetition rate. Specifically, an increase in laser pump fluence results in the creation of additional hot-carrier traps. Alternately, higher repetition rates cause a saturation in hot-carrier traps, thus activating Auger-related PL fluctuations. Furthermore, it is shown that Auger recombination of negatively charged excitons is suppressed more strongly than that of positively charged excitons because of the asymmetry in the electron-hole confinement in type II NQDs. Thus, this study provides new understanding of how both NQD structure (shell thickness and carrier-separation characteristics) and excitation conditions can be used to tune the PL stability, with important implications for room-temperature single-photon generation. Specifically, the first non-blinking NQD capable of single-photon emission in the near-infrared spectral regime is described.

Influence of the core size on biexciton quantum yield of giant CdSe/CdS nanocrystals

We present a systematic study of photoluminescence (PL) emission intensity and biexciton (BX) quantum yields (QYBX) in individual "giant" CdSe/CdS nanocrystals (g-NCs) as a function of g-NC core size and shell thickness. We show that g-NC core size significantly affects QYBX and can be utilized as an effective tuning parameter towards higher QYBX while keeping the total volume of the g-NC constant. Specifically, we observe that small-core (2.2 nm diameter) CdSe/CdS NCs with a volume of ∼200 nm(3) (shell comprises 4 CdS monolayers) show very low average and maximum QYBX's of ∼3 and 7%, respectively. In contrast, same-volume medium-core (3 nm diameter) NCs afford higher average values of ∼10%, while QYBX's of ∼30% are achieved for same-volume large-core (5.5 nm diameter) CdSe/CdS NCs, with some approaching ∼80%. These observations underline the influence of the g-NC core size on the evolution of PL emissive states in multi-shell NCs. Moreover, our study also reveals that the use of long anneal times in the growth of CdS shells plays a critical role in achieving high QYBX.

Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem

Strong anomalous diffusion, where ⟨|x(t)|(q)⟩ ∼ tqν(q) with a nonlinear spectrum ν(q) ≠ const, is wide spread and has been found in various nonlinear dynamical systems and experiments on active transport in living cells. Using a stochastic approach we show how this phenomenon is related to infinite covariant densities; i.e., the asymptotic states of these systems are described by non-normalizable distribution functions. Our work shows that the concept of infinite covariant densities plays an important role in the statistical description of open systems exhibiting multifractal anomalous diffusion, as it is complementary to the central limit theorem.

Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking

This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.

Fast current blinking in individual PbS and CdSe quantum dots

Fast current intermittency of the tunneling current through single semiconductor quantum dots was observed through time-resolved intermittent contact conductive atomic force microscopy in the dark and under illumination at room temperature. The current through a single dot switches on and off at time scales ranging from microseconds to seconds with power-law distributions for both the on and off times. On states are attributed to the resonant tunneling of charges from the electrically conductive AFM tip to the quantum dot, followed by transfer to the substrate, whereas off states are attributed to a Coulomb blockade effect in the quantum dots that shifts the energy levels out of resonance conditions due to the presence of the trapped charge, while at the same bias. The observation of current intermittency due to Coulomb blockade effects has important implications for the understanding of carrier transport through arrays of quantum dots.

Theory of fractional Lévy kinetics for cold atoms diffusing in optical lattices

Recently, anomalous superdiffusion of ultracold 87Rb atoms in an optical lattice has been observed along with a fat-tailed, Lévy type, spatial distribution. The anomalous exponents were found to depend on the depth of the optical potential. We find, within the framework of the semiclassical theory of Sisyphus cooling, three distinct phases of the dynamics as the optical potential depth is lowered: normal diffusion; Lévy diffusion; and x∼t(3/2) scaling, the latter related to Obukhov's model (1959) of turbulence. The process can be formulated as a Lévy walk, with strong correlations between the length and duration of the excursions. We derive a fractional diffusion equation describing the atomic cloud, and the corresponding anomalous diffusion coefficient.

Lévy walks with velocity fluctuations

The standard Lévy walk is performed by a particle that moves ballistically between randomly occurring collisions when the intercollision time is a random variable governed by a power-law distribution. During instantaneous collision events, the particle randomly changes the direction of motion but maintains the same constant speed. We generalize the standard model to incorporate velocity fluctuations into the process. Two types of models are considered, namely (i) with a walker changing the direction and absolute value of its velocity during collisions only, and (ii) with a walker whose velocity continuously fluctuates. We present a full analytic evaluation of both models and emphasize the importance of initial conditions. We show that, in the limit of weak velocity fluctuations, the integral diffusion characteristics and the bulk of diffusion profiles are identical to those for the standard Lévy walk. However, the type of underlying velocity fluctuations can be identified by looking at the ballistic regions of the diffusion profiles. Our analytical results are corroborated by numerical simulations.

Pump-intensity- and shell-thickness-dependent evolution of photoluminescence blinking in individual core/shell CdSe/CdS nanocrystals

We report a systematic study of photoluminescence (PL) intensity and lifetime fluctuations in individual CdSe/CdS core/shell nanocrystal quantum dots (NQDs) as a function of shell thickness. We show that while at low pump intensities PL blinking in thin-shell (4-7 monolayers, MLs) NQDs can be described by random switching between two states of high (ON) and low (OFF) emissivities, it changes to the regime with a continuous distribution of ON intensity levels at high pump powers. A similar behavior is observed in samples with a medium shell thickness (10-12 MLs) without, however, the PL intensity ever switching to a complete "OFF" state and maintaining ca. 30% emissivity ("gray" state). Further, our data indicate that highly stable, blinking-free PL of thick-shell (15-19 MLs) NQDs ("giant" or g-NQDs) is characterized by nearly perfect Poisson statistics, corresponding to a narrow, shot-noise limited PL intensity distribution. Interestingly, in this case the PL lifetime shortens with increasing pump power and the PL decay may deviate from monoexponential. However, the PL intensity distribution remains shot-noise limited, indicating the absence of significant quantum yield fluctuations at a given pump power intensity during the experimental time window.

Strong field line shapes and photon statistics from a single molecule under anomalous noise

We revisit the line-shape theory of a single molecule with anomalous stochastic spectral diffusion. Waiting time profiles for bath induced spectral jumps in the ground and excited states become different when a molecule, probed by continuous-wave laser field, reaches the steady state. This effect is studied for the stationary dichotomic continuous-time-random-walk spectral diffusion of a single two-level chromophore with power-law distributions of waiting times. Correlated waiting time distributions, line shapes, two-point fluorescence correlation function, and Mandel Q parameter are calculated for arbitrary magnitude of laser field. We extended previous weak field results and examined the breakdown of the central limit theorem in photon statistics, indicated by asymptotic power-law growth of Mandel Q parameter. Frequency profile of the Mandel Q parameter identifies the peaks of spectrum, which are related to anomalous spectral diffusion dynamics.

Complexity and synchronization

We study a fully connected network (cluster) of interacting two-state units as a model of cooperative decision making. Each unit in isolation generates a Poisson process with rate g . We show that when the number of nodes is finite, the decision-making process becomes intermittent. The decision-time distribution density is characterized by inverse power-law behavior with index mu=1.5 and is exponentially truncated. We find that the condition of perfect consensus is recovered by means of a fat tail that becomes more and more extended with increasing number of nodes N . The intermittent dynamics of the global variable are described by the motion of a particle in a double well potential. The particle spends a portion of the total time tau(S) at the top of the potential barrier. Using theoretical and numerical arguments it is proved that tau(S) is proportional to (1/g)ln(const x N) . The second portion of its time, tau(K), is spent by the particle at the bottom of the potential well and it is given by tau(K)=(1/g)exp(const x N) . We show that the time tau(K) is responsible for the Kramers fat tail. This generates a stronger ergodicity breakdown than that generated by the inverse power law without truncation. We establish that the condition of partial consensus can be transmitted from one cluster to another provided that both networks are in a cooperative condition. No significant information transmission is possible if one of the two networks is not yet self-organized. We find that partitioning a large network into a set of smaller interacting clusters has the effect of converting the fat Kramers tail into an inverse power law with mu=1.5 .

The effects of anomalous diffusion on power-law blinking statistics of CdSe nanorods

In this study of fluorescence blinking statistics for nanorods, we present a diffusion-controlled reaction model that leads to a more general formula: t(-m) exp[-(Gammat)(n)]. This formula describes a short-time power law with a crossover to a stretched exponential decay at later times. Based on quantum Brownian motion for a coupled central harmonic oscillator coupled to heat bath oscillators, we show that the position distribution follows anomalous diffusion with time-dependent diffusion coefficient and drift coefficient. The first and the second moments of the energy fluctuations are shown to be related to the exponent m and n for the blinking statistics.

Observation of correlated emission intensity and polarization fluctuations in single CdSe/ZnS quantum dots

Time-resolved single-nanoparticle spectroscopy has been carried out to examine the luminescence characteristics of individual CdSe/ZnS core/shell quantum dots. In particular, the possible correlations between emission intensity, lifetime, spectrum, and polarization fluctuations have been investigated. The emission polarization was found to be correlated with the luminescence intensity in a nonlinear way. The low-emissive states were found to correlate with red-shifted spectrum, increased nonradiative decay, and low degree of emission polarization. The observations are consistent with the model that charged quantum dots can be emissive.

Towards non-blinking colloidal quantum dots

At a single-molecule level, fluorophore emission intensity fluctuates between bright and dark states. These fluctuations, known as blinking, limit the use of fluorophores in single-molecule experiments. The dark-state duration shows a universal heavy-tailed power-law distribution characterized by the occurrence of long non-emissive periods. Here we have synthesized novel CdSe-CdS core-shell quantum dots with thick crystalline shells, 68% of which do not blink when observed individually at 33 Hz for 5 min. We have established a direct correlation between shell thickness and blinking occurrences. Importantly, the statistics of dark periods that appear at high acquisition rates (1 kHz) are not heavy tailed, in striking contrast with previous observations. Blinking statistics are thus not as universal as thought so far. We anticipate that our results will help to better understand the physico-chemistry of single-fluorophore emission and rationalize the design of other fluorophores that do not blink.

Blinking suppression and intensity recurrences in single CdSe-oligo(phenylene vinylene) nanostructures: experiment and kinetic model

We report time-resolved single molecule fluorescence imaging of individual CdSe quantum dots that are functionalized with oligomeric conjugated organic ligands. The fluorescence intensity trajectories from these composite nanostructures display both a strong degree of blinking suppression and intensity fluctuations with characteristic recurrence times on the order of 10-60 s. In addition, fluorescence decay rate measurements of individual hybrid nanostructures indicate significantly modified non-radiative quantum dot decay rates relative to conventional ZnS-capped CdSe quantum dots. We show that a modified diffusive reaction coordinate model with slow fluctuations in quantum dot electron energies (1S(e), 1P(e)) can reproduce the experimentally observed behaviour.

Evidence for a diffusion-controlled mechanism for fluorescence blinking of colloidal quantum dots

Fluorescence blinking in nanocrystal quantum dots is known to exhibit power-law dynamics, and several different mechanisms have been proposed to explain this behavior. We have extended the measurement of quantum-dot blinking by characterizing fluctuations in the fluorescence of single dots over time scales from microseconds to seconds. The power spectral density of these fluctuations indicates a change in the power-law statistics that occurs at a time scale of several milliseconds, providing an important constraint on possible mechanisms for the blinking. In particular, the observations are consistent with the predictions of models wherein blinking is controlled by diffusion of the energies of electron or hole trap states.

Occupation time statistics in the quenched trap model

We investigate the distribution of the occupation time of a particle undergoing a random walk among random energy traps and in the presence of a deterministic potential field. When the distribution of energy traps is exponential with a width T(g), we find in thermal equilibrium a transition between Boltzmann statistics when T>T(g) to Lamperti statistics when T < T(g). We explain why our main results are valid for other models of quenched disorder, and discuss briefly implications on single particle experiments.