We show here that the dephasing between the ionizing laser pulse and the proton bunch is small when compared to the plasma wavelength.

In the AWAKE experiment a short (100fs) and intense (10¹⁴W/cm²) laser pulse created the plasma by field ionization of a rubidium vapor. The laser pulse also seeds the self-modulation instability (SMI) and set the zero of the wakefields phase. The SMI creates a periodic modulation of the proton bunch that eventually propagates at the speed of the proton bunch. The protons have a (initial) kinetic energy of E₀=400GeV, which means that their velocity and that of the bunch is v_b=(1-1/γ²)^1/2c, where γ=E₀/m_pc², m_p the proton mass.
The laser pulse propagates at the group velocity of the laser pulse central (angular) frequency ω₀ in the plasma with plasma electron frequency ω_pe: v_l=v_g=(1-ω_pe²/ω₀²)^1/2c. The plasma (angular) electron frequency is given by ω_pe=(n_ee²/ε₀m_e)^1/2, where n_e is the plasma density.
The proton bunch takes a time t_p to travel a plasma length L given by t_p=L/v_b.
During that time, the laser pulse travels a length L+ΔL given by L+ΔL=v_lt_p=v_l/v_bL.
Therefore the two dephase by ΔL_dephase=(L+ΔL)-L=(v_l/v_b-1)L. The relative dephasing can be written as: ΔL_dephase/L=(1-^ω_pe2⁄_ω0²)^1⁄₂/(1-¹⁄_γ²)^1⁄₂-1.

This expression can be evaluated with AWAKE parameters.
Since the bunch γ factor is large one can Taylor expand (1-1/γ²)^1/2≈(1-1/2γ²) (when 1/γ²<<1).
Similarly, when the laser frequency is much larger than the plasma frequency (ω₀²>>ω_pe²): (1-ω_pe²/ω₀²)^1/2≈(1-ω_pe²/2ω₀²). Therefore and further:

ΔL_dephase/L≈(1-ω_pe²/2ω₀²)/(1-1/2γ²)-1≈(1-ω_pe²/2ω₀²)(1+1/2γ²)-1

and

ΔL_dephase/L≈(1-ω_pe²/2ω₀²)(1+1/2γ²)-1≈1+1/2γ²-ω_pe²/2ω₀²-1.

Thus:

ΔL_dephase/L≈1/2γ²-ω_pe²/2ω₀².

One can define the critical electron plasma density such that (n_crite²/ε₀m_e)^1/2=ω₀, i.e. for which the plasma frequency is equal to the laser frequency. With this definition:

ΔL_dephase/L≈1/2γ²-n_e/2n_crit.

For the AWAKE case, λ₀=780nm so that ω₀=2.43x10¹⁵rad/s. At the baseline plasma density of 7x10¹⁴cm^-3, ω_pe=1.49x10¹²rad/s. Therefore ω_pe²/ω₀²=n_e/n_crit≈3.76x10^-7<<1 is well satisfied.
In addition, for γ=400, 1/γ²=6.25x10⁶<<1.
We therefore have (at this particular plasma density) ΔL_dephase/L≈2.94x10^-6.
Note also that ΔL_dephase is positive so that the laser pulse moves forward with respect to the proton bunch.
In the AWAKE case, the plasma length is L=10m and ΔL_dephase≈2.94x10^-5m.
For this dephasing to be negligible it needs to be small when compared to the plasma wavelength λ_pe≈1.26mm. The condition ΔL_dephase<<λ_pe is well satisfied (ΔL_dephase/λ_pe≈0.023) and for practical purposes there is no dephasing. This is true even at the lowest and highest densities envisaged, n_e=10¹⁴cm^-3 and 10¹⁵cm^-3, respectively.
In addition, the ionization occurs within the laser pulse and we have L_laser=30μm (100fs), so that even if there was evolution of the ionization location within the laser pulse, this evolution would also have no practical effect on the wakefield phase.
We finally note that this is true for the zero of the wakefields. The wakefields phase itself, behind the ionizing laser evolves (for example) during the growth of the SMI [A. Pukhov et al., Phys. Rev. Lett. 107, 145003 (2011)].