Biexcitonic Fine Structure of CdSe Nanocrystals Probed by Polarization-Dependent Two-Dimensional Photon Echo Spectroscopy

Constructing Dynamical Symmetries for Quantum Computing: Applications to Coherent Dynamics in Coupled Quantum Dots

Dynamical symmetries, time-dependent operators that almost commute with the Hamiltonian, extend the role of ordinary symmetries. Motivated by progress in quantum technologies, we illustrate a practical algebraic approach to computing such time-dependent operators. Explicitly we expand them as a linear combination of time-independent operators with time-dependent coefficients. There are possible applications to the dynamics of systems of coupled coherent two-state systems, such as qubits, pumped by optical excitation and other addressing inputs. Thereby, the interaction of the system with the excitation is bilinear in the coherence between the two states and in the strength of the time-dependent excitation. The total Hamiltonian is a sum of such bilinear terms and of terms linear in the populations. The terms in the Hamiltonian form a basis for Lie algebra, which can be represented as coupled individual two-state systems, each using the population and the coherence between two states. Using the factorization approach of Wei and Norman, we construct a unitary quantum mechanical evolution operator that is a factored contribution of individual two-state systems. By that one can accurately propagate both the wave function and the density matrix with special relevance to quantum computing based on qubit architecture. Explicit examples are derived for the electronic dynamics in coupled semi-conducting nanoparticles that can be used as hardware for quantum technologies.

Exciton and biexciton transient absorption spectra of CdSe quantum dots with varying diameters

Transient absorption (TA) spectroscopy of semiconductor nanocrystals (NCs) is often used for excited state population analysis, but recent results suggest that TA bleach signals associated with multiexcitons in NCs do not scale linearly with exciton multiplicity. In this manuscript, we probe the factors that determine the intensities and spectral positions of exciton and biexciton components in the TA spectra of CdSe quantum dots (QDs) of five diameters. We find that, in all cases, the peak intensity of the biexciton TA spectrum is less than 1.5 times that of the single exciton TA spectrum, in stark contrast to a commonly made assumption that this ratio is 2. The relative intensities of the biexciton and exciton TA signals at each wavelength are determined by at least two factors: the TA spectral intensity and the spectral offset between the two signals. We do not observe correlations between either of these factors and the particle diameter, but we find that both are strongly impacted by replacing the native organic surface-capping ligands with a hole-trapping ligand. These results suggest that surface trapping plays an important role in determining the absolute intensities of TA features for CdSe QDs and not just their decay kinetics. Our work highlights the role of spectral offsets and the importance of surface trapping in governing absolute TA intensities. It also conclusively demonstrates that the biexciton TA spectra of CdSe QDs at the band gap energy are less than twice as intense as those of the exciton.

Origin of Vibronic Coherences During Carrier Cooling in Colloidal Quantum Dots

Recent two-dimensional electronic spectroscopy experiments [Tilluck et al. J. Phys. Chem. Lett. 2021, 12 (39), 9677-9683] indicate the creation of coherent vibronic wavepackets in the first femtoseconds of hot carrier cooling in hexadecylamine-passivated CdSe quantum dots. Here we present a quantum chemical study of the origin of these coherences in a CdSe nanocrystal. We find that coherent wavepacket motions along vibrational coordinates with alkylamine character promote nonradiative relaxation through conical intersections between the exciton states of the inorganic core. Electronic excitations in the core are found to pass energy to the vibrations of the ligands via two distinct mechanisms: excitation of core phonon modes that are coupled to the ligand vibrations and direct excitation of ligand vibrations by delocalization of the exciton onto the ligands, both of which naturally arise within a photochemical framework based on many-electron potential energy surfaces. If these findings are demonstrated to be general, vibronic coherences may be leveraged to control photophysical outcomes in colloidal quantum dots.

Electronic Structure and Excited State Dynamics of Cadmium Chalcogenide Nanorods

The cylindrical quasi-one-dimensional shape of colloidal semiconductor nanorods (NRs) gives them unique electronic structure and optical properties. In addition to the band gap tunability common to nanocrystals, NRs have polarized light absorption and emission and high molar absorptivities. NR-shaped heterostructures feature control of electron and hole locations as well as light emission energy and efficiency. We comprehensively review the electronic structure and optical properties of Cd-chalcogenide NRs and NR heterostructures (e.g., CdSe/CdS dot-in-rods, CdSe/ZnS rod-in-rods), which have been widely investigated over the last two decades due in part to promising optoelectronic applications. We start by describing methods for synthesizing these colloidal NRs. We then detail the electronic structure of single-component and heterostructure NRs and follow with a discussion of light absorption and emission in these materials. Next, we describe the excited state dynamics of these NRs, including carrier cooling, carrier and exciton migration, radiative and nonradiative recombination, multiexciton generation and dynamics, and processes that involve trapped carriers. Finally, we describe charge transfer from photoexcited NRs and connect the dynamics of these processes with light-driven chemistry. We end with an outlook that highlights some of the outstanding questions about the excited state properties of Cd-chalcogenide NRs.

Coherent Two-Dimensional and Broadband Electronic Spectroscopies

Over the past few decades, coherent broadband spectroscopy has been widely used to improve our understanding of ultrafast processes (e.g., photoinduced electron transfer, proton transfer, and proton-coupled electron transfer reactions) at femtosecond resolution. The advances in femtosecond laser technology along with the development of nonlinear multidimensional spectroscopy enabled further insights into ultrafast energy transfer and carrier relaxation processes in complex biological and material systems. New discoveries and interpretations have led to improved design principles for optimizing the photophysical properties of various artificial systems. In this review, we first provide a detailed theoretical framework of both coherent broadband and two-dimensional electronic spectroscopy (2DES). We then discuss a selection of experimental approaches and considerations of 2DES along with best practices for data processing and analysis. Finally, we review several examples where coherent broadband and 2DES were employed to reveal mechanisms of photoinitiated ultrafast processes in molecular, biological, and material systems. We end the review with a brief perspective on the future of the experimental techniques themselves and their potential to answer an even greater range of scientific questions.

Anisotropy in fifth-order exciton-exciton-interaction two-dimensional spectroscopy

Exciton-exciton-interaction two-dimensional (EEI2D) spectroscopy is a fifth-order variant of 2D electronic spectroscopy. It can be used to probe biexciton dynamics in molecular systems and to observe exciton diffusion in extended systems such as polymers or light-harvesting complexes. The exciton transport strongly depends on the geometrical and energetic landscape and its perturbations. These can be of both local character, such as molecular orientation and energetic disorder, and long-range character, such as polymer kinks and structural domains. In the present theoretical work, we investigate the anisotropy in EEI2D spectroscopy. We introduce a general approach for how to calculate the anisotropy by using the response-function formalism in an efficient way. In numerical simulations, using a Frenkel exciton model with Redfield-theory dynamics, we demonstrate how the measurement of anisotropy in EEI2D spectroscopy can be used to identify various geometrical effects on exciton transport in dimers and polymers. Investigating a molecular heterodimer as an example, we demonstrate the utility of anisotropy in EEI2D spectroscopy for disentangling dynamic localization and annihilation. We further calculate the annihilation in extended systems such as conjugated polymers. In a polymer, a change in the anisotropy provides a unique signature for exciton transport between differently oriented sections. We analyze three types of geometry variations in polymers: a kink, varying geometric and energetic disorder, and different geometric domains. Our findings underline that employing anisotropy in EEI2D spectroscopy provides a way to distinguish between different geometries and can be used to obtain a better understanding of long-range exciton transport.

Observing Multiexciton Correlations in Colloidal Semiconductor Quantum Dots via Multiple-Quantum Two-Dimensional Fluorescence Spectroscopy

Correlations between excitons, that is, electron-hole pairs, have a great impact on the optoelectronic properties of semiconductor quantum dots and thus are relevant for applications such as lasers and photovoltaics. Upon multiphoton excitation, these correlations lead to the formation of multiexciton states. It is challenging to observe these states spectroscopically, especially higher multiexciton states, because of their short lifetimes and nonradiative decay. Moreover, solvent contributions in experiments with coherent signal detection may complicate the analysis. Here we employ multiple-quantum two-dimensional (2D) fluorescence spectroscopy on colloidal CdSe_1-xS_x/ZnS alloyed core/shell quantum dots. We selectively map the electronic structure of multiexcitons and their correlations by using two- and three-quantum 2D spectroscopy, conducted in a simultaneous measurement. Our experiments reveal the characteristics of biexcitons and triexcitons such as transition dipole moments, binding energies, and correlated transition energy fluctuations. We determine the binding energies of the first six biexciton states by simulating the two-quantum 2D spectrum. By analyzing the line shape of the three-quantum 2D spectrum, we find strong correlations between biexciton and triexciton states. Our method contributes to a more comprehensive understanding of multiexcitonic species in quantum dots and other semiconductor nanostructures.

Visualizing Ultrafast Electron Transfer Processes in Semiconductor-Metal Hybrid Nanoparticles: Toward Excitonic-Plasmonic Light Harvesting

Recently, it was demonstrated that charge separation in hybrid metal-semiconductor nanoparticles (HNPs) can be obtained following photoexcitation of either the semiconductor or of the localized surface plasmon resonance (LSPR) of the metal. This suggests the intriguing possibility of photocatalytic systems benefiting from both plasmon and exciton excitation, the main challenge being to outcompete other ultrafast relaxation processes. Here we study CdSe-Au HNPs using ultrafast spectroscopy with high temporal resolution. We describe the complete pathways of electron transfer for both semiconductor and LSPR excitation. In the former, we distinguish hot and band gap electron transfer processes in the first few hundred fs. Excitation of the LSPR reveals an ultrafast (<30 fs) electron transfer to CdSe, followed by back-transfer from the semiconductor to the metal within 210 fs. This study establishes the requirements for utilization of the combined excitonic-plasmonic contribution in HNPs for diverse photocatalytic applications.

Massively parallel classical logic via coherent dynamics of an ensemble of quantum systems with dispersion in size

Quantum parallelism can be implemented on a classical ensemble of discrete level quantum systems. The nanosystems are not quite identical, and the ensemble represents their individual variability. An underlying Lie algebraic theory is developed using the closure of the algebra to demonstrate the parallel information processing at the level of the ensemble. The ensemble is addressed by a sequence of laser pulses. In the Heisenberg picture of quantum dynamics the coherence between the N levels of a given quantum system can be handled as an observable. Thereby there are N ² logic variables per N level system. This is how massive parallelism is achieved in that there are N ² potential outputs for a quantum system of N levels. The use of an ensemble allows simultaneous reading of such outputs. Due to size dispersion the expectation values of the observables can differ somewhat from system to system. We show that for a moderate variability of the systems one can average the N ² expectation values over the ensemble while retaining closure and parallelism. This allows directly propagating in time the ensemble averaged values of the observables. Results of simulations of electronic excitonic dynamics in an ensemble of quantum dot (QD) dimers are presented. The QD size and interdot distance in the dimer are used to parametrize the Hamiltonian. The dimer N levels include local and charge transfer excitons within each dimer. The well-studied physics of semiconducting QDs suggests that the dimer coherences can be probed at room temperature.

Coherent Exciton Dynamics in Ensembles of Size-Dispersed CdSe Quantum Dot Dimers Probed via Ultrafast Spectroscopy: A Quantum Computational Study

Interdot coherent excitonic dynamics in nanometric colloidal CdSe quantum dots (QD) dimers lead to interdot charge migration and energy transfer. We show by electronic quantum dynamical simulations that the interdot coherent response to ultrashort fs laser pulses can be characterized by pump-probe transient absorption spectroscopy in spite of the inevitable inherent size dispersion of colloidal QDs. The latter, leading to a broadening of the excitonic bands, induce accidental resonances that actually increase the efficiency of the interdot coupling. The optical electronic response is computed by solving the time-dependent Schrodinger equation including the interaction with the oscillating electric field of the pulses for an ensemble of dimers that differ by their size. The excitonic Hamiltonian of each dimer is parameterized by the QD size and interdot distance, using an effective mass approximation. Local and charge transfer excitons are included in the dimer basis set. By tailoring the QD size, the excitonic bands can be tuned to overlap and thus favor interdot coupling. Computed pump-probe transient absorption maps averaged over the ensemble show that the coherence of excitons in QD dimers that lead to interdot charge migration can survive size disorder and could be observed in fs pump-probe, four-wave mixing, or covariance spectroscopy.

Non-Markovian Exciton-Phonon Interactions in Core-Shell Colloidal Quantum Dots at Femtosecond Timescales

We perform two-dimensional coherent spectroscopy on CdSe/CdZnS core-shell colloidal quantum dots at cryogenic temperatures. In the two-dimensional spectra, sidebands due to electronic coupling with CdSe lattice LO-phonon modes are observed to have evolutions deviating from the exponential dephasing expected from Markovian spectral diffusion, which is instantaneous and memoryless. Comparison to simulations provides evidence that LO-phonon coupling induces energy-gap fluctuations on the finite timescales of nuclear motion. The femtosecond resolution of our technique probes exciton dynamics directly on the timescales of phonon coupling in nanocrystals.

Deciphering hot- and multi-exciton dynamics in core-shell QDs by 2D electronic spectroscopies

2D electronic spectroscopy maps acquired in different configurations unveil intraband hot carrier cooling and interband multi-exciton recombination dynamics.

Although the harnessing of multiple and hot excitons is a prerequisite for many of the groundbreaking applications of semiconductor quantum dots (QDs), the characterization of their dynamics through conventional spectroscopic techniques is cumbersome. Here, we show how a careful analysis of 2DES maps acquired in different configurations (BOXCARS and pump–probe geometry) allows the tracking and visualization of intraband Auger relaxation mechanisms, driving the hot carrier cooling, and interband bi- and tri-exciton recombination dynamics. The results obtained on archetypal core–shell CdSe/ZnS QDs suggest that, given the global analysis of the resulting datasets, 2D electronic spectroscopy techniques can successfully and efficiently dispel the intertwined dynamics of fast and ultrafast recombination processes in nanomaterials. Hence, we propose this analysis scheme to be used in future research on novel quantum confined systems.

Seeing Multiexcitons through Sample Inhomogeneity: Band-Edge Biexciton Structure in CdSe Nanocrystals Revealed by Two-Dimensional Electronic Spectroscopy

The electronic structure of multiexcitons significantly impacts the performance of nanostructures in lasing and light-emitting applications. However, these multiexcitons remain poorly understood due to their complexity arising from many-body physics. Standard transient-absorption and photoluminescence spectroscopies are unable to unambiguously distinguish effects of sample inhomogeneity from exciton-biexciton interactions. Here, we exploit the energy and time resolution of two-dimensional electronic spectroscopy to access the electronic structure of the band-edge biexciton in colloidal CdSe quantum dots. By removing effects of inhomogeneities, we show that the band-edge biexciton structure must consist of a discrete manifold of electronic states. Furthermore, the biexciton states within the manifold feature distinctive binding energies. Our findings have direct implications for optical gain thresholds and efficiency droop in light-emitting devices and provide experimental measures of many-body physics in nanostructures.

Fluorescence-Detected Two-Quantum and One-Quantum-Two-Quantum 2D Electronic Spectroscopy

We demonstrate two-quantum (2Q) coherent two-dimensional (2D) electronic spectroscopy using a shot-to-shot-modulated pulse shaper and fluorescence detection. Broadband collinear excitation is realized with the supercontinuum output of an argon-filled hollow-core fiber, enabling us to excite multiple transitions simultaneously in the visible range. The 2Q contribution is extracted via a three-pulse sequence with 16-fold phase cycling and simulated employing cresyl violet as a model system. Furthermore, we report the first experimental realization of one-quantum-two-quantum (1Q-2Q) 2D spectroscopy, offering less congested spectra as compared with the 2Q implementation. We avoid scattering artifacts and nonresonant solvent contributions by using fluorescence as the observable. This allows us to extract quantitative information about doubly excited states that agree with literature expectations. The high sensitivity and background-free nature of fluorescence detection allow for a general applicability of this method to many other systems.

Coherent two-dimensional fluorescence micro-spectroscopy

We have developed coherent two-dimensional (2D) fluorescence micro-spectroscopy which probes the nonlinear optical response at surfaces via fluorescence detection with sub-micron spatial resolution. This enables the investigation of microscopic variations in heterogeneous systems. An LCD-based pulse shaper in 4f geometry is used to create collinear trains of 12-fs visible/NIR laser pulses in the focus of an NA = 1.4 immersion-oil microscope objective. We demonstrate the capabilities of the new method by presenting 2D spectra, analyzed via phase cycling, as a function of position of selected sub-micron regions from a laterally nanostructured polycrystalline thin film of fluorinated zinc phthalocyanine (F₁₆ZnPc).

Multi-heterodyne two dimensional coherent spectroscopy using frequency combs

Optical multi-dimensional coherent spectroscopy is a powerful technique for studying the structure, properties and ultrafast dynamics of atoms, molecules, semiconductor materials and complex systems. Current implementations of multi-dimensional coherent spectroscopy have long acquisition times and/or limited spectral resolution. In addition, most of the techniques utilize complex geometries or phase cycling schemes to isolate non-linear signals. We demonstrate a novel approach of using frequency combs to perform rapid, high resolution and background free multi-dimensional coherent spectroscopy of semiconductor materials. Our approach is inspired by dual-comb spectroscopy, which has been proven to be a versatile tool for obtaining one dimensional absorption spectra with high resolution in a short acquisition time. We demonstrate the method using a GaAs multi-quantum well sample.

Frequency combs enable rapid and high-resolution multidimensional coherent spectroscopy

Dual laser frequency combs can rapidly measure high-resolution linear absorption spectra. However, one-dimensional linear techniques cannot distinguish the sources of resonances in a mixture of different analytes, nor can they separate inhomogeneous and homogeneous broadening. Here, we overcame these limitations by acquiring high-resolution multidimensional nonlinear coherent spectra with frequency combs. We experimentally differentiated and assigned the Doppler-broadened features of two naturally occurring isotopes of rubidium atoms (87Rb and 85Rb) according to the placement of their hyperfine energy states in a two-dimensional spectrum.

Bacteriophytochrome Photoisomerization Proceeds Homogeneously Despite Heterogeneity in Ground State

Phytochromes are red/far-red photoreceptors that are widely distributed in plants and prokaryotes. Ultrafast photoisomerization of a double bond in a biliverdin cofactor or other linear tetrapyrrole drives their photoactivity, but their photodynamics are only partially understood. Multiexponential dynamics were observed in previous ultrafast spectroscopic studies and were attributed to heterogeneous populations of the pigment-protein complex. In this work, two-dimensional photon echo spectroscopy was applied to study dynamics of the bacteriophytochromes RpBphP2 and PaBphP. Two-dimensional photon echo spectroscopy can simultaneously resolve inhomogeneity in ensembles and fast dynamics by correlating pump wavelength with the emitted signal wavelength. The distribution of absorption and emission energies within the same state indicates an ensemble of heterogeneous protein environments that are spectroscopically distinct. However, the lifetimes of the dynamics are uniform across the ensemble, suggesting a homogeneous model involving sequential intermediates for the initial photodynamics of isomerization.

Two-Dimensional Visible Spectroscopy For Studying Colloidal Semiconductor Nanocrystals

Possibilities offered by 2D visible spectroscopy for the investigation of the properties of excitons in colloidal semiconductor nanocrystals are overviewed, with a particular focus on their ultrafast dynamics. The technique of 2D electronic spectroscopy is illustrated with several examples showing its advantages compared to 1D ultrafast spectroscopic techniques (transient absorption and time-resolved photoluminescence).

Mapping multidimensional electronic structure and ultrafast dynamics with single-element detection and compressive sensing

Compressive sensing allows signals to be efficiently captured by exploiting their inherent sparsity. Here we implement sparse sampling to capture the electronic structure and ultrafast dynamics of molecular systems using phase-resolved 2D coherent spectroscopy. Until now, 2D spectroscopy has been hampered by its reliance on array detectors that operate in limited spectral regions. Combining spatial encoding of the nonlinear optical response and rapid signal modulation allows retrieval of state-resolved correlation maps in a photosynthetic protein and carbocyanine dye. We report complete Hadamard reconstruction of the signals and compression factors as high as 10, in good agreement with array-detected spectra. Single-point array reconstruction by spatial encoding (SPARSE) Spectroscopy reduces acquisition times by about an order of magnitude, with further speed improvements enabled by fast scanning of a digital micromirror device. We envision unprecedented applications for coherent spectroscopy using frequency combs and super-continua in diverse spectral regions.

Observation of an Excitonic Quantum Coherence in CdSe Nanocrystals

Recent observations of excitonic coherences within photosynthetic complexes suggest that quantum coherences could enhance biological light harvesting efficiencies. Here, we employ optical pump-probe spectroscopy with few-femtosecond pulses to observe an excitonic quantum coherence in CdSe nanocrystals, a prototypical artificial light harvesting system. This coherence, which encodes the high-speed migration of charge over nanometer length scales, is also found to markedly alter the displacement amplitudes of phonons, signaling dynamics in the non-Born-Oppenheimer regime.

Mapping the Vibronic Structure of a Molecule by Few-Cycle Continuum Two-Dimensional Spectroscopy in a Single Pulse

Accurate mapping of the electronic and vibrational structure of a molecular system is a basic goal of chemistry as it underpins reactivity and function. Experimentally, the challenge is to uncover the intramolecular interactions and ensuing dynamics that define this structure. Multidimensional coherent spectroscopy can map such interactions analogous to the way in which nuclear magnetic resonance provides access to the nuclear spin structure. Here we present two-dimensional coherent spectra measured using few-cycle continuum light. Critically, our approach instantaneously maps the energy landscape of a complex molecular system in a single laser pulse across 350 nm of bandwidth, thereby making it suitable for rapid molecular fingerprinting. We envision few-cycle supercontinuum spectroscopy based on the nonlinear optical response as a powerful tool to examine molecules in the condensed phase at the extremes of time, space, and energy.

Dispersion-free continuum two-dimensional electronic spectrometer

Electronic dynamics span broad energy scales with ultrafast time constants in the condensed phase. Two-dimensional (2D) electronic spectroscopy permits the study of these dynamics with simultaneous resolution in both frequency and time. In practice, this technique is sensitive to changes in nonlinear dispersion in the laser pulses as time delays are varied during the experiment. We have developed a 2D spectrometer that uses broadband continuum generated in argon as the light source. Using this visible light in phase-sensitive optical experiments presents new challenges in implementation. We demonstrate all-reflective interferometric delays using angled stages. Upon selecting an ~180  nm window of the available bandwidth at ~10  fs compression, we probe the nonlinear response of broadly absorbing CdSe quantum dots and electronic transitions of Chlorophyll a.

Multiple population-period transient spectroscopy (MUPPETS) of CdSe/ZnS nanoparticles. I. Exciton and biexciton dynamics

The nonradiative relaxation of both the exciton and biexciton in CdSe/ZnS core-shell nanoparticles have complicated, nonexponential kinetics. This paper presents data on this system from multiple population-period transient spectroscopy (MUPPETS), a method for two-dimensional kinetics. An initial report of a dispersed (nonexponential) biexciton decay [J. Am. Chem. Soc. 2013, 135, 1002] is confirmed in a more rigorous analysis. Additional transient-grating data allow a quantitative treatment of the full, complex MUPPETS data set. The MUPPETS signal has a strong fluence dependence. With extrapolation to the low fluence limit, the ratio of cross sections for ground-to-exciton and exciton-to-biexciton absorption is found to be close to the predictions of the uncorrelated-electron model. The full two-dimensional MUPPETS data set is reported for the first time and is analyzed to detect heterogeneity in the exciton decay. The exciton has a substantial (>40%) nonradiative decay, but it is not due to a subset of defective particles. A surface relaxation in response to formation of the exciton is suggested. This data set is the first capable of detecting correlations between the biexciton and exciton decay mechanism. None is found.

Exciton superposition states in CdSe nanocrystals measured using broadband two-dimensional electronic spectroscopy

Coherent superpositions among eigenstates are of interest in fields as diverse as photosynthesis and quantum computation. In this report, we used two-dimensional electronic spectroscopy (2D ES) to measure the decoherence time of a superposition of the two lowest-energy excitons in colloidal CdSe nanocrystals (cubic phase) in solution at room temperature. In the electron-hole representation, the quantum coherence is, remarkably, a twelve-particle correlation. By comparing the measured 2D ES to simulations, we also explored the effects of inhomogeneous broadening and examined the spectroscopic signatures of biexcitons.