  We introduce the Euclidean-time correlator of axial charge and electric
current as an observable that can be used to study the out-of-equilibrium
Chiral Magnetic Effect (CME) in first-principle lattice QCD simulations with
background magnetic field. This observable directly reflects the fact that in
the background magnetic field, a state with nonzero axial charge features
nonzero electric current. For free fermions, the axial-vector correlator only
receives contributions from the Lowest Landau Level, and features a linear
dependence on both magnetic field and temperature with a universal coefficient.
With an appropriate regularization, non-vanishing axial-vector correlator is
compatible with the vanishing of the CME current in thermal equilibrium state
with nonzero chiral chemical potential $\mu_5$. We demonstrate that the
real-time counterpart of the Euclidean-time axial-vector correlator is
intimately related to the real-time form of the axial anomaly equation, which
strongly limits possible corrections in full QCD. We present numerical results
for the Euclidean-time axial-vector correlator in $SU(2)$ lattice gauge theory
with $N_f = 2$ light quark flavours, demonstrating perfect agreement with free
fermion result on both sides of the chiral crossover. The proposed methodology
should help to answer the question whether the QCD corrections might be
responsible for non-observation of CME in RHIC isobar run.
