Ultrahigh Purcell factors and Lamb shifts using slow-light metamaterial waveguides

Employing a medium-dependent quantum optics formalism and a Green function solution of Maxwell's equations, we study the enhanced spontaneous emission factors (Purcell factors) and Lamb shifts from a quantum dot or atom near the surface of a %embedded in a slow-light metamaterial waveguide. Purcell factors of approximately 250 and 100 are found at optical frequencies for $p-$polarized and $s-$polarized dipoles respectively placed 28\thinspace nm (0.02\thinspace $\lambda_{0}$) above the slab surface, including a realistic metamaterial loss factor of $\gamma /2\pi =2 \mathrm{THz}$. For smaller loss values, we demonstrate that the slow-light regime of odd metamaterial waveguide propagation modes can be observed and related to distinct resonances in the Purcell factors. Correspondingly, we predict unusually large and rich Lamb shifts of approximately -1 GHz to -6 GHz for a dipole moment of 50 Debye. We also make a direct calculation of the far field emission spectrum, which contains direct measurable access to these enhanced Purcell factors and Lamb shifts.