David M. Jonas

Recently, Ryu et al. [Proc. Natl. Acad. Sci. U. S. A. 121, e2410280121 (2024)] generalized Einstein’s three coefficients for absorption, stimulated emission, and spontaneous emission between two quantum levels to a set of four spectra between two broadened bands. The spectra obey generalized Einstein relationships at thermal equilibrium; Einstein’s relations are obtained as an approximation for line spectra. Here, the generalized Einstein relation between absorption and stimulated emission dipole-strength spectra is applied to investigate optical lineshape models. Lineshapes for the Bloch model, the stochastic model, and the semi-classical Brownian oscillator model do not obey the generalized Einstein relation and, therefore, fail to satisfy detailed balance with Planck blackbody radiation. The quantum Brownian oscillator model treats a harmonic quantum vibration that is bi-linearly coupled to a thermal bath of quantum harmonic oscillators, which generate damping and a random force. The two-state quantum Brownian oscillator lineshape model provides lineshapes for transitions between two displaced, but otherwise identical, harmonic potential energy surfaces on which the same quantum vibration is coupled to the same thermal bath of quantum harmonic oscillators. The absorption and stimulated emission lineshapes were calculated using the quantum Brownian oscillator model in under-damped, critically damped, and over-damped cases. The thermal and reorganization energies were each varied from values much smaller than the vibrational quantum of energy to values greater than it. All quantum Brownian oscillator lineshapes obey the generalized Einstein relation within the numerical precision of the calculation (14–30 digits), suggesting this lineshape model is compatible with detailed balance. The formula giving the electric-dipole transition cross section in terms of these lineshapes is presented.

Recently, Ryu et al. showed that two broadened bands connected by a set of four Einstein-coefficient spectra for stimulated and spontaneous single-photon transitions will obey detailed balance at equilibrium if the spectra satisfy generalized Einstein relations. Here, quantum mechanical expressions for Einstein-coefficient spectra are obtained in the electric-dipole approximation using an intramolecular Boltzmann distribution and the quantized field operators in isotropic, dispersive media of Nienhuis and Alkemade [Physica B+C 81, 181–188 (1976)]. These expressions suggest relationships between Einstein-coefficient spectra and dipole-strength spectra. The electrodynamic relationship between the spectral density for electromagnetic energy and the spectral density for the square of the electric field is developed and used to define dipole-strength spectra in terms of conditional transition probabilities per unit time. These rigorously relate dipole-strength spectra to Einstein-coefficient spectra, thus establishing quantum formulas for dipole-strength spectra and new generalized Einstein relations between dipole-strength spectra. For transitions between two bands, the dipole-strength spectra depend on a single total dipole strength but replace Einstein’s degeneracy ratio and transition frequency with a change in standard chemical potential and a single underlying lineshape that is manifested differently in the four spectra. At equilibrium, the relations specify the Stokes’ shift between forward and reverse transitions. The relationships between dipole-strength spectra, spontaneous emission spectral densities, and stimulated transition cross sections depend on the refractive index, the dielectric constant, and the local field, but not on the derivative of the refractive index. The broadband relationships reduce to known relationships for narrow spectra inside materials and for line spectra in vacuum.

We present Einstein coefficient spectra and a detailed-balance derivation of generalized Einstein relations between them that is based on the connection between spontaneous and stimulated emission. If two broadened levels or bands overlap in energy, transitions between them need not be purely absorptive or emissive. Consequently, spontaneous emission can occur in both transition directions, and four Einstein coefficient spectra replace the three Einstein coefficients for a line. At equilibrium, the four different spectra obey five pairwise relationships and one lineshape generates all four. These relationships are independent of molecular quantum statistics and predict the Stokes’ shift between forward and reverse transitions required by equilibrium with blackbody radiation. For Boltzmann statistics, the relative strengths of forward and reverse transitions depend on the formal chemical potential difference between the initial and final bands, which becomes the standard chemical potential difference for ideal solutes. The formal chemical potential of a band replaces both the energy and degeneracy of a quantum level. Like the energies of quantum levels, the formal chemical potentials of bands obey the Rydberg-Ritz combination principle. Each stimulated Einstein coefficient spectrum gives a frequency-dependent transition cross-section. Transition cross-sections obey causality and a detailed-balance condition with spontaneous emission, but do not directly obey generalized Einstein relations. Even with an energetic width much less than the photon energy, a predominantly absorptive forward transition with an energetic width much greater than the thermal energy can have such an extreme Stokes’ shift that its reverse transition cross-section becomes predominantly absorptive rather than emissive.

Solution-processed semiconductors are in demand for present and next-generation optoelectronic technologies ranging from displays to quantum light sources because of their scalability and ease of integration into devices with diverse form factors. One of the central requirements for semiconductors used in these applications is a narrow photoluminescence (PL) line width. Narrow emission line widths are needed to ensure both color and single-photon purity, raising the question of what design rules are needed to obtain narrow emission from semiconductors made in solution. In this review, we first examine the requirements for colloidal emitters for a variety of applications including light-emitting diodes, photodetectors, lasers, and quantum information science. Next, we will delve into the sources of spectral broadening, including "homogeneous" broadening from dynamical broadening mechanisms in single-particle spectra, heterogeneous broadening from static structural differences in ensemble spectra, and spectral diffusion. Then, we compare the current state of the art in terms of emission line width for a variety of colloidal materials including II-VI quantum dots (QDs) and nanoplatelets, III-V QDs, alloyed QDs, metal-halide perovskites including nanocrystals and 2D structures, doped nanocrystals, and, finally, as a point of comparison, organic molecules. We end with some conclusions and connections, including an outline of promising paths forward.

The ensemble Stokes' shift gives the average single-molecule linewidth for quantum-confined nanomaterials.

Fast and efficient energy transfer in photosynthetic antennas supports all life on earth. Nonadiabatic energy transfer drives unusual vibrations through tight coupling with electronic motion. Polarization dependent vibrational motion drives polarization independent femtosecond energy transfer.

Oleylamine is widely used in nanomaterial and metal-halide perovskite synthesis. Impurities in commercially available oleylamine reagents are identified, purification procedures are explained, beneficial effects on lead chloride/oleylamine mixtures are documented, and impacts on materials synthesis are discussed. Suspensions of 0.12 mole fraction PbCl₂, PbBr₂, and PbI₂ in purified, dried, and filtered oleylamine were found to form clear solutions upon heating to 100 °C. The infrared, Raman, and NMR spectra of oleylamine and a common trans impurity present in literature reference IR spectra are presented.

The remarkable photonic and photochemical properties of colloidal quantum dots (QD) depend critically on the dynamics of carrier interactions and relaxation. Despite their importance, a quantitative experimental evaluation of these processes has proven elusive due to the inherent challenge of exactly separating single-exciton and multi-exciton dynamics, whose spectroscopic signatures overlap in time, spectrum, and excitation fluence. Here, we measure pump fluence-dependent absolute pump-probe transients of indium arsenide QDs, refreshing the sample using beam scanning to limit repetitive excitation. Focusing on the low fluence limit near the onset of bi-exciton formation, excitation conditions were precisely controlled and characterized by averaging Poisson-distributed excitation statistics over all three spatial dimensions of the pump and probe beam spatial profiles to determine the average excitation probability. A saturation model is developed to uniquely decompose the pump-probe signal into single-exciton and bi-exciton signals. This method harnesses the distinct pump-fluence scaling of absolute pump-probe signals from singly- and doubly-excited QDs without any assumptions regarding the relative timescales or amplitudes of single-exciton and bi-exciton signals. Probing in the bulk-like region of the QD absorption spectrum, the signal from bi-excitons is found to be 1.8 times the signal from single excitons at T = 0, consistent with the conventionally assumed factor of 2 within the 95% confidence intervals. The bi-exciton signal contains the same hot carrier relaxation dynamics as that from single excitons, but signal from a second exciton additionally exhibits a 26 ps exponential decay attributed to Auger recombination.