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not influence the value of the incommensurate vector Q because the fluctuations are slow. The point is that the semiclassical analysis is based on relatively short distance and time, with corresponding typical momenta and en- ergies Q < q$_{semi}$ < p$_{F}$ ∝ √$_{x}$$_{,}$ ω$_{semi}$ ∼ ϵ$_{F}$ ∝ x . Quantum fluctuations come from smaller momentum/energy scales q$_{quant}$ < Q , ω$_{quant}$ ∝ x 3 / $^{2}$, as discussed in Ref. 29. In the double layer case, one has to include the in- 

terlayer hopping in the effective action (2). As a re- sult, holon wave function ψ attains bonding/antibonding index with respect to the interlayer hybridization [30]. The holon energy at each of the two nodal points reads ϵ ( p ) = β p $^{2}$/ 2 ± gQ ± Δ / 2, where ± gQ is the pseudospin splitting and Δ / 2 is the antibonding(a)/bonding(b) splitting. Effectively there are four different bands per nodal direction ( b − , b +, a − , a +), altogether eight bands. The filling configuration of these bands is determined by minimizing the semiclassical energy, which yields three different doping regimes [30] 

where λ = 2 g $^{2}$/ ( πβρ$_{s}$ ). The points x$_{0}$ and x$_{1}$ are Lifshitz points. Filling configuration of these three regimes are shown in Fig.1. Areas of the filled holon pockets are 

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Note that the total occupied area is 2 x because there are two layers. From (4) we conclude that there is one MQO frequency in the first regime, three frequencies in the sec- ond regime, and two frequencies in the third regime. Let us consider two scenarios for the doping dependence of the hybridization splitting: (A) a constant gap Δ = Δ$_{0}$, (B) Δ = Δ$_{0}$(1 + νx ), since the splitting Δ is due to tunneling via interlayer oxygen chains [31], a linear de- pendence on doping is possible. 

Values of the incommensurate wave vector Q deter- mined by neutron scattering [19–21] are presented in Fig.2. Comparing Fig.2 with Eqs.(3) one finds x$_{0}$ ≈ 0 . 06, x$_{1}$ ≈ 0 . 13. Using Eqs.(3), we can determine parameters of the model. We fix g = J and ρ$_{s}$ = 0 . 175 J , as they are predicted by the extended t - J model. Within the scenario (A) the fit gives, λ = 1 . 23, β = 2 . 95 J ( m ∗ = 1 . 35 m$_{e}$ ), Δ$_{0}$ = 0 . 556 J . Within the scenario (B), the data from Fig.2 are not sufficient to determine the additional parameter ν . However, we can assume that value of λ in 

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YBCO and LSCO is the same, and use the value λ = 1 . 31 obtained in Ref. [29] from fitting the data for LSCO. This gives β = 2 . 78 J ( m ∗ = 1 . 43 m$_{e}$ ), Δ$_{0}$ = 0 . 37 J , ν = 6 . 9 within the scenario (B). Note that values of β in both fits are close to that predicted by the extended t - J model, as demonstrated in the paragraph after Eq.(2), and the cor- responding effective masses are close to that measured in MQO away from the QCP [11, 12], m ∗ = 1 . 6 ± 0 . 1 m$_{e}$ . 

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Frequencies of MQO are determined by these param- eters via Eqs.(4) and (1). The results are presented in Fig.3. Difference between the scenario (A) (solid lines) and the scenario (B) (dashed lines) is fairly small. There is quite a reasonable agreement with experimental data, especially having in mind that the theory has no fitting parameters related to MQO. Even a high frequency point ≈ 1690 T is reproduced. Notice that MQO always con- tain higher harmonics. Most likely the blue circle at ≈ 1100 T is such a harmonic [11]. Filling of each pocket is only a fraction of the total doping, this explains why the filling extracted from one single frequency seems to violate the Luttinger sum rule. The main frequency at x ≈ 0 . 09 comes from the b + pocket, which has Fermi 

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