In connection with recent studies of extremely long-living cyclotron spin-flip excitations [1-3] (CSFE) - actually magneto-excitons in a quantum Hall electron gas, the contribution to light absorption related to such a magneto-excitonic ensemble is discussed. The CSFE relaxation found experimentally in the unpolarized quantum Hall system created in a real GaAs/AlGaAs heterostructure reaches 100 $\mu$s [4] at finite temperature $T\!\simeq\!0.5\,$K,
that seems to be a record value for a delocalized state excited in the conduction band of mesoscopic systems. Such a slow relaxation suggests that ensemble of the weakly interacting excitations, obeying the Bose-Einstein statistics, can experience at sufficiently high concentration a transition to a coherent state - Bose-Einstein condensate,  where all momenta of CSFEs have the same value. In the work a comparative analysis of both incoherent and coherent cases is done.
Role of randomness of the electrostatic field is discussed. In the incoherent phase the distribution of CSFE momenta is determined by a smooth random potential. Due to cool-down processes, diffusion and drift, which are fast compared to the CSFE lifetime, the magnetoexciton gets ``stuck'' in the smooth random electrostatic potential with minimum total energy, i.e. with zero group velocity.
 Appearance of the coherent phase is associated with the interaction of magnetoexcitons. The intensity of optical absorption in the coherent phase under some conditions is found to be an order of magnitude higher than that in the incoherent phase. Conditions for a phase transition from the incoherent state to the coherent one are discussed. The considered problem is related to optical probing of the 2D electron system in the experimental
study of spin-cyclotron excitations in the quantum-Hall system. The obtained results are of interest for future experimental studies of CSFEs in a spin-unpolarized quantum-Hall system.

1.  C. Kallin and B.I. Halperin, Phys. Rev. B 30, 5655 (1984).
2.  S. Dickmann and I.V. Kukushkin, Phys. Rev. B 71, 241310(R) (2005).
3.  S. Dickmann, Phys. Rev. Lett. 110, 166801 (2013).
4.  L.V. Kulik , A.V. Gorbunov, A.S. Zhuravlev, V.B. Timofeev, S. Dickmann, I.V. Kukushkin, Nature Sci. Rep. 5, 10354 (2015).

S. Dickmann

JETP Letters 109, issue 1 (2019)