transition amplitude R eff onto the eigenstates Ψ$_{n}$ α to find the electronic Raman spectrum in the presence of a strong external magnetic field. This leads to the following selection rules for allowed electronic transitions from the initial level n $^{−}$: Among those, i ) is the dominant transition. These selec- tion rules, represented schematically in Fig. 2(a), show that using Raman spectroscopy, one can probe different electronic excitations than in optical spectroscopy, where the selection rules are Δ n = ± 1. 30,35 For a neutral bi- layer, the angle-integrated spectral density g ( ω ) of Ra- man scattering in the magnetic field is equal to: per incoming photon, which at the field B ∼ 10T gives I$_{n}$ − → n + ∼ 10 − 12 for Ω ∼ 1eV photons, comparable to similar transitions in monolayer graphene. 29 A weaker feature in Fig. 2(b) is the first and the only visible ( n ∓ 1) $^{−}$→ ( n ± 1) + peak due to both 2 − → 0 and 0 → 2 + transitions, positioned to the left of the 2 − → 2 + peak. The quantum efficiencies of the ( n ± 1) − → ( n ∓ 1) + transitions are smaller by the factor ( γ$_{1}$ Ω ) 2 in comparison to the n − → n + transitions. This is different from the monolayer graphene case, where the corresponding ratio between quantum efficiencies of ( n ± 1) − → ( n ∓ 1) + and n − → n + transitions is ( ω Ω ) $^{2}$, much smaller than for the bilayer. The term δg ( ω ) can be further emphasized by changing the energy of incoming photons Ω. Shown in Figs. 3(a) and 3(b), is a comparison of the total spectral density g ( ω ) and contributions due to each mode sep- arately, for two different energies of incoming photons, Ω = 2eV and Ω = 1eV. The intensity scale is the same on both figures and in each case, the total spectral den- sity g ( ω ), the contributions due to the n − → n + and ( n ± 1) − → ( n ∓ 1) + modes are shown in the solid, dot- dashed and dashed line, respectively. The dominant con- tribution, resulting from the Raman scattering accom- panied by the n − → n + electronic transitions, is pro- Here, we use Lorentzian γ ( x ) = π − $^{1}$Γ / ( x 2 + Γ $^{2}$) with a width specified by Γ to model the broadening of Landau levels. The term δg ( ω ) describes the spectral density of the ( n ∓ 1) − → ( n ± 1) + transitions, which is a correction to the dominant contribution due to the n − → n + tran- sitions given by the first term on the right hand side of Eq. (11). An example of the low-energy electronic contribution to the Raman spectrum in the neutral bilayer in strong magnetic field is shown with a solid line in Fig. 2(b). The dominant features are peaks due to the n − → n + transi- tions with the first being the 2 − → 2 + transition. Note that within the LL indexing scheme applied here, indices 0 and 1 are only used to denote one valley-degenerate level each (no α index is needed). Lifting the valley de- generacy by introducing charge asymmetry between lay- ers will not allow any additional n − → n + transition be- cause valley-split levels for n = 0 , 1 belong to different valleys and excitation between them would require a huge momentum transfer. The quantum efficiency of a single n − → n + peak in Fig. 2(b) is approximately I$_{n}$ − → n + ≈ ( v 2 c 2 e $^{2}$/λ$_{B}$ ϵ$_{0}$π Ω ) 2 = v $^{4}$e$^{5}$B π $^{2}$c$^{4}$ϵ 2 0 ℏ Ω 2 (12) g ( ω ) ≈ 16 Ξ$_{s}$ ( e 2 4 πϵ$_{0}$ ℏ c v c ) $^{2}$( ℏ v λ$_{B}$ Ω ) 2 ∑ n ≥ 2 γ ( ω − 2 ϵ$_{n}$ + ) + δg ( ω ); (11) δg ( ω ) = 8 Ξ$_{o}$ ( γ$_{1}$ Ω ) 2 ( e 2 4 πϵ$_{0}$ ℏ c v c ) $^{2}$( ℏ v λ$_{B}$ Ω ) 2   ∑ n =1 , 2 2 γ ( ω − ϵ$_{(}$$_{n}$$_{+1)}$ + )+ ∑ n ≥ 3 γ ( ω − ϵ$_{(}$$_{n}$$_{+1)}$ + − ϵ$_{(}$$_{n}$$_{−}$$_{1)}$ + )   . i ) n − → n $^{+}$; ii ) ( n ∓ 1) − → ( n ± 1) $^{+}$. (10) 5 FIG. 2: (a) Schematic of allowed inter-LL transitions accom- panying the Raman scattering. The solid (dashed) line rep- resents the first dominant (weaker) transition 2 − ← 2 + (pair 2 − ← 0 and 0 ← 2 $^{+}$). (b) The low-energy electronic contribu- tion to the Raman spectrum in bilayer graphene. The solid (dashed) line represents the spectrum in the presence (ab- sence) of an external magnetic field and chemical potential µ = 0 ( µ = 0). For the spectrum in a magnetic field, cor- responding inter-LL transitions have been attributed to each peak. GLYPH<G65> GLYPH<G57> GLYPH<G57> GLYPH<G77> GLYPH<G77>