  The mixing mechanism of axial-vectors $D_{s1}(2460)$ and $D_{s1}(2536)$ is
studied via intermediate hadron loops, e.g. $D^* K$, to which both states have
strong couplings. By constructing the two-state mixing propagator matrix that
respects the unitarity constraint and calculating the vertex coupling form
factors in a chiral quark model, we can extract the masses, widths and mixing
angles of the physical states. Two poles can be identified in the propagator
matrix. One is at $\sqrt{s}=2454.5 \ \textrm{MeV}$ corresponding to
$D_{s1}(2460)$ and the other at $\sqrt{s}=(2544.9-1.0i) \ \textrm{MeV}$
corresponding to $D_{s1}(2536)$. For $D_{s1}(2460)$, a large mixing angle
$\theta=47.5^\circ$ between ${}^3P_1$ and ${}^1P_1$ is obtained. It is driven
by the real part of the mixing matrix element and corresponds to
$\theta'=12.3^\circ$ between the $j=1/2$ and $j=3/2$ state mixing in the heavy
quark limit. For $D_{s1}(2536)$, a mixing angle $\theta=39.7^\circ$ which
corresponds to $\theta'=4.4^\circ$ in the heavy quark limit is found. An
additional phase angle $\phi=-6.9^\circ \sim 6.9^\circ$ is needed at the pole
mass of $D_{s1}(2536)$ since the mixing matrix elements are complex numbers.
Both the real and imaginary part are found important for the large mixing
angle. We show that the new experimental data from BaBar provide a strong
constraint on the mixing angle at the mass of $D_{s1}(2536)$, from which two
values can be extracted, i.e. $\theta_1=32.1^\circ$ or $\theta_2=38.4^\circ$.
Our study agrees well with the latter one. Detailed analysis of the mass shift
procedure due to the coupled channel effects is also presented.
