Waveguide QED using Decoherence Free Subspaces
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Introduction
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Recent theoretical and experimental work has shown that an attractive configuration to engineer strong collective dissipation is given by one-dimensional (1d) photonic-like systems such as photonic crystal waveguides [@joannopoulos_book95a; @laucht12a; @goban13a; @yu14a; @tiecke14a; @sollner14a; @arcari14a; @goban15a; @young15a], optical fibers [@vetsch10a; @goban12a; @mitsch14a; @petersen14a; @beguin14a], metal [@chang07a; @dzsotjan10a; @gonzaleztudela11a; @bermudez15a] and graphene plasmonic [@koppens11a; @huidobro12a; @christensen11a; @martinmoreno15a] waveguides or superconducting circuits [@mlynek14a]. Their interaction with quantum emitters, usually referred to as waveguide QED, offers interesting characteristics: *i)* the density of modes of the waveguide is inversely proportional to the group velocity $1/v_g(\omega_{{{\rm a}}})$, and therefore is strongly enhanced when the atomic frequency is in a region of slow light, e.g., in photonic crystal waveguides close to a band edge. This enhancement implies achieving regions of a large decay rate into the waveguide, $\Gamma_\oned$, compared to other decay channels, denoted by $\Gamma^*$, characterized through the *Purcell Factor*, $P_\oned=\Gamma_\oned/\Gamma^*$; *ii)* the 1d guided modes retain a small modal area $\lesssim \lambda_{{\rm a}}^2$, for propagation lengths $L_{\mathrm{prop}} \gg \lambda_{{\rm a}}$ (the wavelength of the 1d mode associated to the atomic frequency considered); *iii)* the interaction is *strongly* long-range, favoring individual adressing, and it can even 