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tion to the Raman spectra and obtain an analytic descrip- tion for the selection rules and intensity of such Raman scattering in a bilayer placed in a strong (quantizing) magnetic field. 

## II. THEORY OF THE INELASTIC LIGHT SCATTERING IN BILAYER GRAPHENE 

Bilayer graphene consists of two coupled sheets of graphene with AB (Bernal) stacking characteristic of bulk graphite, 32 Fig. 1(a). The unit cell contains four in- equivalent atoms A 1, B 1, A 2 and B 2 where letters A and B denote two sublattices in the same layer and 1 / 2 stands for the bottom/top layer. The Fermi level in graphene lies in the vicinities of the corners of the hexagonal Bril- louin zone (also called valleys) known as K$_{+}$ and K$_{−}$ (Fig. 1(b)). The conventional tight-binding Hamiltonian based on π -orbitals of carbon atoms (one per atom, four in the unit cell) and expanded in momentum around the valleys reads 

Here, σ = ( σ$_{x}$, σ$_{y}$ ) and σ$_{x}$ , σ$_{y}$ , σ$_{z}$ are the Pauli matrices, electron momentum p is measured from the center of the valley, v ∼ 10 $^{6}$m/s (Refs. 11,12,18,23) is a parameter related to the nearest neighbour intralayer coupling γ$_{0}$ ∼ 3eV, $^{11,18,23}$, γ$_{1}$ ∼ 0 . 4eV (Refs. 5,9–12,18,19,21,23) is the direct interlayer coupling, v$_{3}$ is related to the weak direct A 1 ↔ B 2 interlayer hops [ v$_{3}$/v ∼ 0 . 1 (Refs. 1,18,23)] and ξ = ± is the valley index. The basis is constructed using components corresponding to atomic sites A 1 , B 2 , A 2 , B 1 in the valley K$_{+}$ and B 2 , A 1 , B 1 , A 2 in K$_{−}$ . One can also take into account terms quadratic in the electron momentum p : 

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where µ = − v 2 6 $_{γ$_{0}$}$. However, the influence of the δ ˆ H term on the results of the Raman spectra analysis is negligibly small, as shown in the Appendix. 

The part of the resulting electronic dispersion relevant for the Raman scattering of photons with energies Ω < 2 . 5eV, is illustrated in Fig. 1(c) for the valley K$_{+}$ . Two bands, later referred to as low-energy ones, touch each other at the neutrality point - the position of the Fermi energy in the neutral structure. Two other, referred to as high-energy bands, are split by the interlayer coupling, γ$_{1}$ , from the neutrality point. The v$_{3}$ parameter leads to the trigonal warping of the electronic dispersion. Its influence is most important for very low energies, ϵ < 5meV. 

To describe the process of inelastic scattering of light on electrons, we consider an experimental setup in which incoming laser light of energy Ω ≫ γ$_{1}$ , in-plane momen- tum q (out-of-plane component of momentum equal to 

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q$_{z}$ = √ Ω $^{2}$/c2 − q $^{2}$) and polarisation l is shined onto to the sample. Scattered photon has polarisation ˜ l , in-plane momentum ˜ q and energy ˜ Ω = Ω − ω , where ω is the Raman shift. We also assume the temperature T to be smaller than the Raman shift, k$_{B}$T < ω ( k$_{B}$ is the Boltz- mann’s constant). In our case, the inelastic light scatter- ing may occur via a one-step process (so called contact interaction) or a two-step process involving an interme- diate state. The two-step process, such as shown in Fig. 1(c), involves: the absorption (or emission) of a photon with energy Ω ( ˜ Ω) transferring an electron with momen- tum p from an occupied state in the valence band into a virtual intermediate state (energy is not conserved at this stage), followed by another electron emission (or ab- sorption) of the second photon with energy ˜ Ω (Ω). The one-step process is the usual inelastic scattering of an incoming photon on an electron with transfer of energy to the latter. As a result of both one and two-step pro- cesses, an electron-hole pair in the low-energy bands is created with the electron and the hole having almost the same momentum ( p + q − ˜ q and p , respectively), since q , ˜ q ≪ p and the momentum transfer from light is neg- ligible ( v/c ∼ 3 · 10 − $^{3}$). Therefore, p + q − ˜ q ≈ p and due to the approximately electron-hole symmetric band 

