Jisu Ryu

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.

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

Femtosecond two-dimensional Fourier transform spectroscopy is used to determine the static bandgap inhomogeneity of a colloidal quantum dot ensemble. The excited states of quantum dots absorb light, so their absorptive two-dimensional (2D) spectra will typically have positive and negative peaks. It is shown that the absorption bandgap inhomogeneity is robustly determined by the slope of the nodal line separating positive and negative peaks in the 2D spectrum around the bandgap transition; this nodal line slope is independent of excited state parameters not known from the absorption and emission spectra. The absorption bandgap inhomogeneity is compared to a size and shape distribution determined by electron microscopy. The electron microscopy images are analyzed using new 2D histograms that correlate major and minor image projections to reveal elongated nanocrystals, a conclusion supported by grazing incidence small-angle X-ray scattering and high-resolution transmission electron microscopy. The absorption bandgap inhomogeneity quantitatively agrees with the bandgap variations calculated from the size and shape distribution, placing upper bounds on any surface contributions.

In experiments with high photon flux, it is necessary to rapidly remove the sample from the beam and to delay re-excitation until the sample has returned to equilibrium. Rapid and complete sample exchange has been a challenge for air-sensitive samples and for vibration-sensitive experiments. Here, a compact spinning sample cell for air and moisture sensitive liquid and thin film samples is described. The principal parts of the cell are a copper gasket sealed enclosure, a 2.5 in. hard disk drive motor, and a reusable, chemically inert glass sandwich cell. The enclosure provides an oxygen and water free environment at the 1 ppm level, as demonstrated by multi-day tests with sodium benzophenone ketyl radical. Inside the enclosure, the glass sandwich cell spins at ≈70 Hz to generate tangential speeds of 7-12 m/s that enable complete sample exchange at 100 kHz repetition rates. The spinning cell is acoustically silent and compatible with a ±1 nm rms displacement stability interferometer. In order to enable the use of the spinning cell, we discuss centrifugation and how to prevent it, introduce the cycle-averaged resampling rate to characterize repetitive excitation, and develop a figure of merit for a long-lived photoproduct buildup.

Pump-probe polarization anisotropy measurements with 15 fs pulses are employed to investigate the electronic structure of PbS quantum dots. The initial anisotropy at the bandgap is anomalously low (<0.1) and suggests large electronic couplings.