# Optical Response of an Interacting Polaron Gas in Strongly Polar Crystals
## 1 Introduction  
Polaron manifestations in the optical response of polar crystals, such as complex oxides and high-\(T_{c}\) superconductors are the subject of intense investigations [1; 2; 3; 4; 5; 6; 7] (for a review, see also [8; 9]). Several features in the infrared optical-absorption spectra of complex oxides have been associated with large polarons [10] or with a mixture of large and small polarons [11]. The analysis in those papers was performed using a single-polaron model, so that the concentration dependence of the optical-absorption spectra could not properly take into account many-polaron effects.  
The many-body theory of the optical absorption of a gas of interacting polarons [12; 13] allows one to study the density dependence of the optical-absorption spectra. The calculation [12] was performed in the single-branch approximation for optical phonons. In Ref. [13], the optical conductivity of \(n\)-doped SrTiO\({}_{3}\) was calculated accounting for the electron-phonon interaction with multiple longitudinal optical (LO) and transverse optical (TO) phonon branches. The calculations of the optical conductivity of a weak-coupling polaron gas [12; 13] compare fairly well with the experimental data [1; 2] and therefore confirm the contribution of large polarons in the optical response. Strontium titanate represents an especially interesting case due to its unique features, particularly a high static dielectric constant at low temperatures and essentially non-parabolic shape of the conduction band which consists of three subbands. This requires a treatment of the optical conductivity beyond the frequently used lowest-order perturbation approximation.  
The first-principle methods are powerful for the theoretical study of both equilibrium and response properties of polarons. At present, ab initio calculations of the polaron band energies are developed,for example, in Refs. [14; 15; 16; 17; 18]. Also, the polaron mobility has been calculated from the first principles (see, for review, [19]). In the present work, we consider a complementary semianalytic approach, which has its own advantages. First, it is much less time- and memory-consuming for computation. Second, more important, it allows sometimes a clear physical interpretation of features of obtained spectra, as they follow from a used model.  
The present work is focused on the many-polaron optical response in strongly polar crystals like SrTiO\({}_{3}\). The strong polarity means that the ratio of the static and high-frequency dielectric constant is large: \(\varepsilon_{0}/\varepsilon_{\infty}\gg 1\). This does not necessarily lead to a high electron-phonon coupling constant: in strontium titanate the effective coupling constant \(\alpha\approx 2\) as determined in [13]. Even in this moderate-coupling case, electron collective excitations (attributed to plasmons only in the long-wavelength limit) are substantially mixed with LO phonons [20] and therefore can result in a non-trivial spectrum of the optical conductivity. Therefore, in this paper, we determine the optical conductivity of a many-polaron gas taking into account mixing of phonons and electron collective excitations in the total dielectric function of the electron-phonon system, and taking also into account the multi-subband structure of the conductivity band. The method is applied to \(n\)-doped strontium titanate.