The spectral densities of Einstein coefficients for absorption and stimulated emission in a two-level quantum system are not equal to each other beyond the absorption (emission) line if the homogeneous broadening caused by interaction with a thermostat is much larger than the natural width. In this study, a relationship between these coefficients is derived on the basis of general thermodynamic requirements. This relationship is associated with new effects such as population inversion upon the nonresonant absorption of continuous radiation and radiation amplification without population inversion.
The spectral densities of Einstein coefficients for absorption and stimulated emission in a two-level quantum system are not equal to each other beyond the absorption (emission) line if the homogeneous broadening caused by interaction with a thermostat is much larger than the natural width. In this study, a relationship between these coefficients is derived on the basis of general thermodynamic requirements. This relationship is associated with new effects such as population inversion upon the nonresonant absorption of continuous radiation and radiation amplification without population inversion.
When the gas particles, which can be considered as a two level quantum system, are affected by frequent collisions with buffer gas particles (thermal reservoir), the spectral densities of the Einstein coefficients for absorption and induced emission are unequal to each other far from the absorption (emission) line. The relationship between the spectral densities at the specified condition reveals new physical phenomena. The mostly convincing example is the fact that population inversion is established in a two level system under nonresonance optical excitation. This effect is experimentally observed as a lasing on sodium resonance transitions.
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.
Kinetic quantum equations are derived for a density matrix with collision integrals describing nonlinear effects in spectra line wings. These equations take into account the earlier established inequality of the spectral densities of Einstein coefficients for absorption and stimulated radiation emission by a two-level quantum system in the far wing of a spectral line in the case of frequent collisions. The relationship of the absorption and stimulated emission probabilities with the characteristics of radiation and an elementary scattering event is found.
Amplification of probe radiation by an inversionless two-level system in the 'red' wing of its spectral line is studied theoretically during the resonance absorption of intense cw laser radiation. The effect is caused by the inequality of the spectral densities of the Einstein coefficients for absorption and stimulated emission outside the absorption line under conditions when homogeneous broadening, due to the interaction of particles with the buffer gas, significantly exceeds the natural broadening (at high pressures of the buffer gas). It is found that the larger the inversionless gain, the higher the buffer gas pressure and the pump radiation intensity. It is established that at high enough pump intensity, the probe field is amplified along the entire region of the 'red' wing of the line (at any negative detunings of the probe field frequency).
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.
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.