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AIP Advances

Derivation of second-order nonlinear optical conductivity by the projection-diagram method

Nam Lyong Kang, Sang Don Choi

Published Feb 29, 2012• 6 citations
Publisher

Abstract

A projection-diagram method is introduced for optical conductivity with lineshape functions, which takes into account the population criterion that the electron and phonon distribution functions are multiplicatively combined along with the energy conservation factors for proper interpretation of emission and absorption of phonons and photons in all the processes of electron transitions. It is further shown that the second order nonlinear optical conductivity of the system of electrons interacting with phonons, obtained using this method, is identical with that derived by using the state dependent projectors and the KC reduction identities [J. Phys. A: Math. Theor. 43, 165203 (2010)]. We expect that this method can reduce the amount of many-body calculation and can be of help in providing physical intuition into solid state quantum dynamics and representing perturbation expressions for such systems.

DOI
10.1063/1.3693406

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The Journal of Chemical Physics•Jul 1, 1999•61 citations
A projection-diagram method is introduced for optical conductivity with lineshape functions, which takes into account the population criterion that the electron and phonon distribution functions are multiplicatively combined along with the energy conservation factors for proper interpretation of emission and absorption of phonons and photons in all the processes of electron transitions. It is further shown that the second order nonlinear optical conductivity of the system of electrons interacting with phonons, obtained using this method, is identical with that derived by using the state dependent projectors and the KC reduction identities [J. Phys. A: Math. Theor. 43, 165203 (2010)]. We expect that this method can reduce the amount of many-body calculation and can be of help in providing physical intuition into solid state quantum dynamics and representing perturbation expressions for such systems.
A revised version of the combined projection technique introduced earlier by the present authors [J. Korean Phys. Soc. 59 , 3372 (2011)] is introduecd and is applied to deriving the linear optical conductivity containing the linewidth function for optical transitions of electrons interacting with phonons in solids. The result is shown to satisfy the population criterion which claims that the Fermi distribution functions for electrons and the Planck distribution functions for phonons should be combined in multiplicative forms in the many-body formalism. The main part of the result is further shown to be indentical with that obtained by using the projection-reduction method [Progr. Theor. Phys. 96 , 307 (1996)].
Some many-body theories of optical transitions in solids were examined from projection and population criteria. The results showed that state-independent projection methods cannot be applied to electron systems with non-uniform energy spectra. Moreover, neglecting some important terms in a second-order approximation leads to invalidation of the population criterion. In addition, a valid theory satisfying these two criteria can be obtained using a proper state-dependent projection operator and Kang-Choi reduction identity, and the result can be interpreted using a diagram, which can model the quantum dynamics of electron in solids.
Electron-phonon interactions underpin a variety of phenomena, ranging from transport and superconductivity to polarons and ultrafast carrier dynamics. Despite being one of the most intensely studied subjects in condensed matter physics, research on electron-phonon physics mostly focused on linear, first-order couplings. Second- and higher-order nonlinear couplings are commonly ignored because their calculations are too demanding and we lack computational frameworks that can provide both diagonal and off-diagonal coupling matrix elements. In this work, we report a theory and computational method for computing nonlinear electron-phonon interactions of any order in real materials. Our approach combines the advantages of unit-cell calculations of electron wavefunctions and supercell calculations of phonon perturbations, is systematically improvable, can be used with either semilocal or nonlocal exchange-correlation functionals, and is amenable to Wannier-Fourier interpolation. As a first proof of concept, we illustrate this method by computing second-order electron-phonon coupling matrix elements in diamond, lithium fluoride, and graphite as representative nonpolar semiconductors, polar semiconductors, and metals, respectively. Furthermore, we generalize the ab initio polaron equations to second-order electron-phonon couplings, and we show that second-order couplings are essential to achieve quantitative accuracy in polaron formation energies and hopping barriers. The present methodology will find application in the study of all properties and phenomena that are currently being investigated within the linear electron-phonon coupling approximation, from phonon-mediated superconductivity to excited-states dynamics, both in harmonic and anharmonic systems.
The optical absorption of a many (continuum) polaron gas is derived in the framework of a variational approach at zero temperature and weak electron-phonon coupling strength. We derive a compact formula for the optical conductivity of the many-polaron system, taking into account many-body effects in the electron or hole system. Within the method presented here, these effects are contained completely in the dynamical structure factor of the electron or hole system. This allows one to build on well-established studies of the interacting electron gas. Based on this approach a feature in the absorption spectrum of the many-polaron gas, related to the emission of a plasmon together with a phonon, is identified. As an application and illustration of the technique, we compare the theoretical many-polaron optical absorption spectrum as derived in the present work with the ‘‘d``d‘‘d-band'' absorption feature in ${\mathrm{Nd}}_{2}{\mathrm{CuO}}_{4\ensuremath{-}\ensuremath{\delta}} (\ensuremath{\delta}<0.004).$ Similarities are shown between the theoretically and the experimentally derived first frequency moment of the optical absorption of a family of differently doped ${\mathrm{Nd}}_{2\ensuremath{-}x}{\mathrm{Ce}}_{x}{\mathrm{CuO}}_{4}$ materials.
We have formulated expressions providing the number of terms (Y-G diagrams)1contributing to an n-th order susceptibility of a non-linear optical process. Obtaining an accurate count of these terms is important for diverse reasons: a) To accurately specify the number of terms needed to evaluate the nonlinear optical susceptibility either analytically or numerically, b) To help develop sequences of diagrammatic contributions which can be summed to provide analytical results for the strong interaction limit. An example is the two-level system for semiclassical and quantized fields, c) To treat two or more processes (terms) which could give rise to interference, resulting, for example, in extra resonances due to a lack of complete coherence, d) To generalize results from quantum electrodynamics to situations for which multiple photon interactions and zero point fluctuations associated with non-vacuum states need to be considered. We shall discuss the number of possible diagrams in terms of three factors, which we call a permutation factor relating to the number of possible frequencies that can be assigned to each interaction vertex, a commutation factor relating to interaction vertex assignment on the two sides of the diagram, and a state factor relating to the number of diagrams generated from an enumeration of specific material state assignments between interaction vertices. We will discuss the application of these results to several situations of interest including second harmonic generation, degenerate four-wave mixing, and resonant CARS (coherent anti-Stokes Raman scattering) in a four-level system. The relevance of the present results to experimental results which have been obtained will be discussed along with future directions which need to be addressed including the summability of diagrams under specific approximations, the calculation of expectation values, and the nonperturbative representation of processes.
The third-order nonlinear optical response of energy transfer systems is theoretically investigated. A system composed of two chromophores having the same electronic transition energies is considered. The dynamics of energy transfer between the two chromophores is assumed to occur via a hopping (incoherent) mechanism. We introduce new types of pathways incorporating the hopping processes occurring while the system is in population states and reconstruct a third-order response function which is computationally viable. The nuclear propagators in the electronic population states are written as convolution integrals between those of the nonreactive two-state system weighted by some factors for the energy transfer. The response function is given by multitime correlation functions and these are analyzed by the cumulant expansion method. Based on this approach, the three-pulse photon echo peak shift for several models of energy transfer systems is discussed. It is shown that the rephasing capability of the induced signal is reduced by the memory loss due to resonant energy transfer. A previous model which incorporates resonant energy transfers in an intuitive way is reviewed and modified to supplement the loss of dynamic correlation of nuclear motion within the framework of the theory. The response function obtained by our new approach gives a more accurate description than the existing theory and a comparative discussion is given. The effect of inhomogeneity in rate constants on the third-order signal is discussed and the temperature dependence of the echo signal is examined.