of freedom $\xi_\mathbf{m}$ associated with a single atom. The sites are numbered by the indices $\mathbf{n}$ or $\mathbf{m}$ for electrons and ions respectively. Operators $c^{\dagger}_{\mathbf{n}\sigma}$ creates an electron on site ${\bf n}$ with spin $\sigma$. Coulomb repulsion $V(\mathbf{n}-\mathbf{n}')$ has been screened up to the first nearest neighbors, with on site repulsion $U$ and nearest-neighbor repulsion $V_c$. In contrast, the Fröhlich interaction is assumed to be long-range, due to unscreened interaction with c-axis high-frequency phonons [@ale5]. The form of the interaction with c-axis polarized phonons has been specified via the force function[@alekor], $f_{\mathbf{m}}(\mathbf{n})=\kappa\left[(\mathbf{m}-\mathbf{n})^2+1\right]^{-3/2}$, where $\kappa$ is a constant. The dimensionless electron-phonon coupling constant $\lambda$ is defined as $\lambda=\sum_{\mathbf{m}}f^{2}_{\mathbf{m}}(0)/2M\omega^2 zT(a)$ which is the ratio of the polaron binding energy to the kinetic energy of the free electron $zT(a)$, and the lattice constant is taken as $a=1$.

![Polaron to bipolaron mass ratio for a range of $\bar{\omega}=\omega_0/T(a)$ and $\lambda$ on the staggered ladder. Mobile small bipolarons are seen even in the adiabatic regime $\bar{\omega}=0.5$ for couplings $\lambda$ up to 2.5 (Reproduced from J. P. Hague et al., Phys. Rev. Lett [**98**]{}, 037002 (2007), (c) American Physical Society, 2007. [@jim2]" ). ](fig4.ps){height="105mm"}

In the limit of high phonon frequency $\omega\gg T(a)$ and large on-site 