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Deciphering the vibronic lasing performances in an electron-phonon-photon coupling system

Rulin Miao, Yu Fu, Dazhi Lu, Fei Liang, Haohai Yu, Huaijin Zhang, Yicheng Wu

Published Feb 22, 2023• 6 citations

Abstract

Coupling between electronic motions and the lattice vibrations, phonons could broaden the spectral bandwidth of the fluorescence spectroscopy by the energy transferring, which was recognized from the beginning of last century and successfully applied in many vibronic lasers. However, the laser performances under electron-phonon coupling were mainly prejudged by the experimental spectroscopy. The multiphonon participated lasing mechanism is still elusive and should be in-depth investigated. Here, a direct quantitative relationship between the laser performance and phonon participating dynamic process was derived in theory. With a transition metal doped alexandrite (Cr 3+ :BeAl 2 O 4 ) crystal, the multiphonon coupled laser performance was manifested in experiments. Associated with the Huang-Rhys factor calculations and hypothesis, the multiphonon participated lasing mechanism with phonon numbers from 2 to 5 was discovered and identified. This work provides not only a credible model for understanding the multiphonon participated lasing, but should also boost the study of laser physics in the electron-phonon-photon coupled systems.

DOI: 10.1364/oe.482391

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Coupling between electronic motions and the lattice vibrations, phonons could broaden the spectral bandwidth of the fluorescence spectroscopy by the energy transferring, which was recognized from the beginning of last century and successfully applied in many vibronic lasers. However, the laser performances under electron-phonon coupling were mainly prejudged by the experimental spectroscopy. The multiphonon participated lasing mechanism is still elusive and should be in-depth investigated. Here, a direct quantitative relationship between the laser performance and phonon participating dynamic process was derived in theory. With a transition metal doped alexandrite (Cr 3+ :BeAl 2 O 4 ) crystal, the multiphonon coupled laser performance was manifested in experiments. Associated with the Huang-Rhys factor calculations and hypothesis, the multiphonon participated lasing mechanism with phonon numbers from 2 to 5 was discovered and identified. This work provides not only a credible model for understanding the multiphonon participated lasing, but should also boost the study of laser physics in the electron-phonon-photon coupled systems.

Herewith we present the quantum theory of the vibronic laser, which is based on the crucial role played by the host crystal phonons in the laser dynamics. In the solid state transition-metal ion laser the operation takes place between the vibronically broadened electronic levels of the gain center, thus the laser action is accompanied by the creation or annihilation of the host lattice phonons. The nonequilibrium coherent phonons, produced in the process of the nonradiative energy transition from the photoexcited impurity ion, pump the energy level, from whivh the laser action takes place. One can say that the laser photons are produced at the cost of lattice phonons. This is the reason why the coherent phonons strongly influence the character of the laser action and govern the complicated laser dynamics. In the laser equations, derived by us, the number of coherent phonons is treated as one of dynamical variables coupled with the number of photons. In solution we observe the eect of energy pulling between photons and phonons. Phonons follow all forms of the photon dynamics including self-pulsations and chaotic dynamics.

We study static and dynamic properties of an electron coupled to dispersive quantum optical phonons in the framework of the Holstein model defined on a one-dimensional lattice. Calculations are performed using the Lanczos algorithm based on a highly efficient construction of the variational Hilbert space. Even small phonon dispersion has a profound effect on the low-energy optical response. While the upward phonon dispersion broadens the optical spectra due to single-phonon excitations, the downward dispersion has the opposite effect. With increasing dispersion, a multiphonon excitation (MPE) state becomes the lowest excited state of the system at zero momentum and determines the low-frequency response of the optical conductivity where the threshold for optical absorption moves below the single-phonon frequency. Multiphonon states form a well-defined bandlike feature just above the polaron band as clearly seen in the electron spectral function. Low-energy MPEs should be observable in systems with strong optical phonon dispersion in optical as well as angle-resolved photoemission experiments.

A quantum-mechanical description of a vibronic multimode standing-wave transition-metal ion laser is presented. The impurity ion levels are coupled both by the radiation field and by the phonons of the host lattice. A dynamic Heisenberg–Langevin set of equations for the material system and the photon and phonon operators, including radiative and nonradiative damping terms and quantum-stochastic forces, has been derived. The numerical solutions of those equations show the crucial role of the phonons in the laser dynamics. The competition between the photons and the phonons leads to regular stable self-pulsation for certain sets of parameters.

The material metrics for optimal laser cooling of ion-doped solids are derived using atomic and molecular dynamics properties of the constituents. The anti-Stokes process is modeled as an optical phonon coupling of the bound electron, followed by a photon absorption. The transition dipole moment is estimated using a simplified charge-displacement model and the Judd--Ofelt theory of rare-earth ions both suggesting that transitions with high-energy gaps and similar angular momentum states should be used. The electron-phonon coupling is interpreted as a derivative of electronic energy with respect to displacement of the nearest neighboring ligands whose stretching mode frequency is approximated using the molecular data. The Debye-Gaussian model is used for the phonon density of states of diatomic crystal. Then, the Fermi golden rule is used for photon-induced, phonon-assisted electronic transition probability and applied to the cooling rate equation by defining a phonon-assisted transition dipole moment. Based on the material metrics, an example blend is investigated for its cooling performance and a general guide is proposed for selection of better performing laser cooling hosts. Furthermore, the cooling rate limits are discussed and three distinct characteristic times are identified with the photon-induced, phonon-assisted transition time controlling the rate. The metrics guide the selection of host materials for optimal cooling, and predict a noticeable increase in the absorption rate when using a blend of cation atoms.

Phonon sidebands, electron-phonon coupling and vibronic lasing in Tm

^{3+}

:Lu

_2

O

_3

sesquioxide

A vibronic Tm:Lu

_2

O

_3

laser generated 654 mW at 2238-2309 nm with a slope efficiency of 68.9% exceeding the Stokes limit. Tm:Lu2O3 features a strong electron-phonon interaction resulting in a profound long-wave phonon sideband for the

^3

F

_4

→

^3

H

_6

electronic transition extending up to at least 2.4 μm supporting vibronic laser operation.