## 1 Introduction  
Recently, there has been a great deal of interest in the subject of the population (and coherence) manipulation in quantum states, which provoked by the emergence of the quantum information paradigm [1; 2]. The exciting researches are currently progressed in different solid-state systems such as semiconductor quantum dots [3; 4; 5; 6] and nitrogen-vacancy centers in diamond [7; 8] based on the interplay of the solid-state nature of the material engineering and the atom-like character of the sharp discrete energy levels. In those solid-state materials and structures, the techniques and concepts demonstrated previously in quantum optics of atoms and molecules [9; 10; 11] can be put to practical use. Rare earth sesquioxides, R\({}_{2}\)O\({}_{3}\) (R is rare earth), are alternative attractive candidates to exploit the coherent quantum phenomena because the intra-4\(f\) transitions in rare-earth ions are weakly perturbed by the crystalline environments, and they exhibit a resonance with a very narrow inhomogeneous linewidth [12; 13; 14; 15]. In the various R\({}_{2}\)O\({}_{3}\), we focus on the single crystalline (Er\({}_{x}\)Sc\({}_{1-x}\))\({}_{2}\)O\({}_{3}\) including Er\({}_{2}\)O\({}_{3}\) since the trivalent Er (Er\({}^{3+}\)) can interact with a telecommunication-band photon (\(\sim\)1.5 \(\upmu\)m) and is a potential platform for the coherent population manipulation in the quantum information network using the well-developed infrastructure [16; 17; 18; 19; 20]. However, for this purpose, some obstacles have to be overcome: a relatively small oscillator strength of a single ion [21; 22], a relatively large upconversion(UC) rate [23; 24; 25; 26; 27; 28; 29; 30], and a large concentration quenching [20; 22]. These issues complicate the physics of the energy transfer (ET) in Er\({}^{3+}\)-doped solids as well as in other R\({}_{2}\)O\({}_{3}\) solids. Figure 1 shows the various kinds of ET from one of the Stark levels in the \({}^{4}I_{13/2}\) manifold of Er\({}^{3+}\)[30], which are observed in (Er\({}_{x}\)Sc\({}_{1-x}\))\({}_{2}\)O\({}_{3}\) epitaxial thin films in this study. (Er\({}_{x}\)Sc\({}_{1-x}\))\({}_{2}\)O\({}_{3}\) crystals have a cubic bixbyite structure (Figure 2a). A unit cell of the crystal contains 32 Er and/or Sc ions, 24 at sites with \(C_{2}\) (non-inversion) symmetry and 8 at sites with \(C_{3i}\) (inversion) symmetry (denoted by \(C_{2}\) and \(C_{3i}\) sites hereafter). In Er\({}^{3+}\) in C\({}_{2}\) site, the (forced) electric-dipole transitions between the Stark levels are allowed. Since \(C_{3i}\) is a higher symmetry than \(C_{2}\), the electric-dipole transitions are forbidden strictly and only magnetic-dipole transitions between \({}^{4}I_{13/2}-^{4}I_{15/2}\) are allowed. Therefore, the UC occurs only in C\({}_{2}\) sites and gives rise to a severe population dissipation from a target state in the case of high power excitation [20]. Also, we observe mutual ET between Er\({}^{3+}\) at \(C_{2}\) and \(C_{3i}\) sites and the ET between Er\({}^{3+}\) with the same site symmetry (\(C_{2}\) or \(C_{3i}\)). In this work, we will refer to the former ET as the inter-site ET and the latter ET as the intra-site ET as shown in Figure 1. Moreover, the non-radiative relaxation occurs due to the crystal imperfection and multiple phonon emission. The intra-site ET induces the energy migration to open the channels to the quenching centers generated by the crystal defects and Er-ion clustering. By using the single crystalline (Er\({}_{x}\)Sc\({}_{1-x}\))\({}_{2}\)O\({}_{3}\) with various Er concentrations, we can investigate how the inter-Er distance affects the energy flow in a system. Also, in order to investigate the energy flow in detail, the initial energy injection into well-resolved point such as one of the Stark levels in the multiplet is very important. Figure 1: Various energy transfers that are observed in the transition between the Stark levels of \({}^{4}I_{15/2}\) and \({}^{4}I_{13/2}\) in the (Er\({}_{x}\)Sc\({}_{1-x}\))\({}_{2}\)O\({}_{3}\) crystal.