# Birotons and \"Dark\" Quantum Hall Hierarchies
## 4 Discussion
Another problem of FQHE states is the spectrum of experimentally detectable neutral excitations. Direct measurement of neutral excitations of the Laughlin-Jain hierarchy using resonant Raman scattering reproduces the pattern observed in the activation experiments [6]. The energy of neutral excitations for the FQHE state 1/3 shows excellent agreement with data calculated for a biproton (magneto-graviton) with zero momentum. In contrast, for the following fractional states of the Laughlin-Jain hierarchy, 2/5 and 3/7, the energies of the excitations decrease drastically, showing no resemblance to the calculation results. One possible way of explaining the observed effect is to take into account \"dark\" FQHE states, namely the optical transitions between the ground states of different FQHE states, as the local redistribution of the electron density induced by the electromagnetic field of a light wave. The coexistence of two different optical transitions: (i) between different fractional states and (ii) within a single fractional state were already discussed in [25]. The authors of the paper observed analogous abnormal optical transitions in the spectrum of spin density neutral excitations for the main Laughlin state 1/3, measured by the resonant reflection for several heterostructures with GaAs/AlGaAs quantum wells [5; 26; 27]. The spectrum indicates the direct optical transitions that excite spin birotons (spin magneto-gravitons), as well as the transitions that give rise to excitations with the energy lying in  
Figure 3: **(a)** Energies of birotons with zero momentum, at a magnetic field of 10 T, calculated for different FQHE states in the range of filling factors [1/4; 3/4], given a \(\delta\)-function GaAs/AlGaAs quantum well. Fractional states of the Laughlin–Jain hierarchy (black circles) and “dark” hierarchies (red circles) are expressed in Coulomb energy units. The solid lines are included for clarity. The length of green arrows signifies the measured charge density excitation energies from [6], corrected for the finite quantum well width and magnetic field used in the experiment. The diamonds denote the even electron filling factors, 1/4, 3/10, 3/8, 1/2, 5/8, 7/10 (left to right). (b) Energies of neutral excitations with spin 1 (S = 1) for the Laughlin state 1/3 calculated for spin birotons and spin excitons with zero momentum, at a magnetic field of 10 T (black dots). The green dots show the measured excitation energies from [5; 6; 25]. Red ovals indicate the corresponding pairs of experimental and calculated neutral excitations, SE(0)-spin exciton [6] and SBMR(0)-spin biproton (magneto-graviton) [5]. Squares mark the experimentally observable excitations that have no theoretical counterpart. the bandgap of the fractional Hall dielectric \(\
u=1/3\) (Figure 3). Unlike birotons, which exhibit bosonic properties predicted for neutral excitations [5], the anomalous excitations do not show signs of Bose statistics [26; 27], suggesting possible transitions between the ground states of different fractional states rather than neutral excitations in the fractional QHE state \(1/3\).