to be initially in the ground state ${\left|3\right\rangle}$. From this ground state, it could be excited by either the single-photon *probe* field, with central frequency $\omega_p$ and coupling to transition ${\left|3\right\rangle} \leftrightarrow {\left|2\right\rangle}$, or by the single-photon *trigger* field, with central frequency $\omega_t$ and coupling to transition ${\left|3\right\rangle}
\leftrightarrow {\left|4\right\rangle}$. We assume that the five levels are Zeeman sub-levels of an alkali atom, and that both pulses have a sufficiently narrow bandwidth. In this way, the Zeeman splittings can be chosen so that the atomic medium is coupled only to a given circular polarization of either the probe or trigger field, while it is transparent for the orthogonally polarized mode, which crosses the gas undisturbed [@Ottaviani03]. As a consequence, the logical basis for each qubit practically coincides with the two lowest Fock states of the mode with the “right” polarization, $|0_j\rangle$ and $|1_j\rangle$ ($j=p,t$), while the “wrong” polarization modes will be neglected from now on.

![Energy levels of the *M*-scheme. $\Omega_j$ are the Rabi frequencies of classical fields, while $g_{p,t}$ denote couplings of the quantized probe and trigger fields to their respective transitions. $\delta_j$ are the detuning of the fields from resonance.[]{data-label="fig:5levelscheme"}](levelscheme2.eps)

A classical pump field, with frequency $\omega_1$ and 