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toward the low-energy part of Raman spectrum with the photon energy shift ω < γ$_{1}$/ 2, which is determined by the excitation of the electron-hole pairs in the low-energy (de- generate) bands with vp ≪ γ$_{1}$ . At such low energies, the band structure as well as Landau level structure can be described by the effective two-band Hamiltonian written in the basis of orbitals on the sites A 1 and B 2, 1 

To describe the excitation of the low-energy modes cor- responding to the transitions between low-energy band states described by ˆ H$_{eff}$ , we take only the part of R which acts in that two-dimensional Hilbert space, keep terms in the lowest relevant order in vp/γ$_{1}$ ≪ 1 and γ$_{1}$/ Ω ≪ 1, and write down an effective amplitude R $_{eff}$, 

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We point out that the above matrix cannot be obtained within a theory constrained by the two-band approxi- mation, Eq. (5), from the very beginning. Seemingly, one may try to define a contact-interaction-like term due to the terms quadratic in the electron momentum p in Eq. (5), which carries a prefactor e $^{2}$ℏ$^{2}$v2 ϵ$_{0}$γ$_{1}$ Ω , which may suggest a greater magnitude of scattering than prefac- tor e $^{2}$ℏ$^{2}$v2 ϵ$_{0}$ Ω 2 above. However, the scattering amplitude ob- tained within this model can only be applied to photons with Ω < γ$_{1}$ , which is hardly relevant for Raman spec- troscopy since the latter is usually performed with laser beams using Ω ∼ 1 . 3 − 2 . 8eV. 15–23 The angle-resolved probability of the Raman scatter- 

ing, w (˜ q ≈ 0 ), determined using Fermi’s golden rule and with the help of Eq. (6), is 

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where f$_{i}$ and f$_{f}$ are filling factors of the initial and fi- nal electronic state, respectively, and the spin and valley degeneracies have already been taken into account. This gives 33 

Above, the first term with polarization factor Ξ$_{s}$ de- scribes the contribution of photons scattered with the same circular polarization as the incoming beam. The second term, with polarization factor Ξ$_{o}$, represents the scattered photons with circular polarization opposite to the incoming beam. 

In turn, the angle-integrated spectral density of Raman 

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scattering g ( ω ) is 

<img src='content_image/414065.jpg'>

Here, the constant spectral density g as a function of ω reflects the parabolicity of the low-energy bands and thus, energy-independent density of states in the bilayer. This is different in monolayer graphene, where g ( ω ) ∝ ω , reflecting the energy-dependent density of states of electron-hole pairs. 29 The characteristic of monolayer graphene crossed polarisation of in/out photons is re- tained in the case of the bilayer system. Experimentally, constant spectral density g in undoped bilayer graphene is impossible to distinguish from a homogeneous back- ground. However, if the chemical potential µ is not at the neutrality point, then transitions with ω < 2 µ are essen- tially blocked. Although new processes, resulting in the creation of the intraband electron-hole pair excitations and very small ω , are possible for µ ̸ = 0, their contribu- tion carries additional prefactor v/c ∼ 1 $_{300}$. 26 Explicit cal- culation performed for the monolayer graphene showed that the quantum efficiency of the intraband transitions was of the order of 10 − $^{15}$.29 In contrast, for chemical po- tential µ ∼ 50meV (corresponding to additional carrier density n$_{0}$ 1 . 5 × 10 $^{12}$cm − $^{2}$), the lost quantum efficiency due to the blocked interband transitions is, according to Eq. (8), Δ I ∼ 10 − $^{12}$. 

## III. INTER-LANDAU-LEVEL TRANSITIONS IN BILAYER GRAPHENE RAMAN 

The quantization of electron states into Landau levels gives the Raman spectrum due to the electronic excita- tions a pronounced structure which can be used to detect their contribution experimentally. We only consider here low-energy Landau levels, as at high energies the Landau level broadening due to, for example, electron-phonon in- teraction, will smear out the LL spectrum. In strong magnetic fields, low-energy Landau levels are sufficiently described 34 by 

<img src='content_image/414060.jpg'>

where λ$_{B}$ = √ ℏ /eB is the magnetic length, n is the Landau level index and α = + denotes the conduction and α = − the valence band. Also, ψ$_{n}$ is the normalised n -th Landau level wavefunction. In a neutral bilayer, all LLs have additional fourfold degeneracy (two due to the electron spin and two due to the valley). Moreover, levels n = 0 and n = 1 are degenerate at ϵ = 0 giving rise to an 8-fold degenerate LL. We can project our effective 

