Optical Lineshape Models and the Generalized Einstein Relation between Absorption and Stimulated Emission

TL;DR

This paper investigates optical lineshape models using the generalized Einstein relation, finding that quantum Brownian oscillator models satisfy detailed balance and are compatible with thermodynamic principles.

Contribution

The study demonstrates that quantum Brownian oscillator lineshapes obey the generalized Einstein relation, unlike other classical or semi-classical models, highlighting their suitability for optical spectroscopy.

Findings

Quantum Brownian oscillator lineshapes obey the generalized Einstein relation.

Classical and semi-classical models fail to satisfy detailed balance.

The model's lineshapes are consistent across damping regimes and energy scales.

Abstract

Recently, Ryu et al. 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 energy were each varied from much less to greater than the vibrational quantum of energy. All quantum Brownian oscillator lineshapes obey the generalized Einstein relation within the numerical precision of the calculation (14 to 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.