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negligible spontaneous decay, typically of the order of tens of ms. The requirement that the cavity does not decay over the duration of the experiment T$_{tot}$ = 2 T is instead more delicate, as it requires that the resonator is stable over the total interaction time with the atomic beam. Moreover, the time T must be sufficiently smaller than the coherence time of the driving fields, so that the amplitude Ω remains constant. 

Being the dynamics stochastic, the desired EPR-state is reached asymptotically. For this purpose, we estimate the time needed for reaching a given fidelity in the prepa- ration of the desired state. We first focus on the weak- coupling limit. Using Eq. (9) we find that the fidelity of the protocol has the form 

where we have assumed that each step is performed on the time interval T , such that the total duration of the protocol T$_{tot}$ = 2 T . Clearly, the initial state affects the time scale required for reaching the desired fidelity. In particular, when the cavity modes are initially in the vac- uum state one has 〈 ˆ b † $_{j}$ˆ b$_{j}$ 〉 0 = µ 2 1 − µ 2 =: ¯ n$_{0}$ . When they are in a thermal state with n$_{th}$ thermal photons in each mode, instead, thus 〈 ˆ b † $_{j}$ˆ b$_{j}$ 〉 0 = n th 0 = ¯ n$_{0}$ + n$_{th}$ + 2 n$_{th}$ ¯ n$_{0}$ . Figure 4 displays the estimated total experimental times and corresponding average number of photons per mode at steady state as a function of µ , where ¯ n$_{0}$ = µ $^{2}$/ (1 − µ $^{2}$) when the cavity modes are in the vacuum state at t = 0. For the degree of squeezing r$_{µ}$ ≈ 2 . 1 ( µ = 0 . 97), lead- ing to an average number of 16 photons per mode at steady state, and ¯ n$_{∞}$ = 0 . 01, corresponding to a fidelity F ≈ 0 . 98, then one has T$_{tot}$ ∼ 36 ms in case of an ini- tially empty cavity ( T$_{tot}$ ∼ 43 ms for 0.7 thermal pho- tons). Resonators stable over this time are available in present experiments [28]. From these results we also see that fluctuations in the coupling with the driving field, δ Ω, are negligible with current microwave sources. 

We now consider the implementation of the protocol in the strong-coupling limit. In this case the fidelity for preparing the cavity modes in the two-mode squeezed state takes the form 

where we have used Eq. (15), assuming a broad distribu- tion of atomic velocity and an initially empty cavity. 

In order to compare the efficiency of the two proce- dures, we now evaluate the time required to reach a desired fidelity in the weak and in the strong-coupling limit. We first observe that the system accesses the strong-coupling regime by increasing the Rabi angle φ . In the case here considered, φ can be changed by chang- ing the interaction time while keeping g , and thus Ω$_{b}$ fixed. Note that increasing the interaction time implies that the atomic arrival rate r$_{at}$ needs to be adjusted to avoid simultaneous presence of more than one atom in the 

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cavity. Let us therefore take the average number of atoms in the interval of time τ so that ϵ = r$_{at}$ ¯ τ ≪ 1. In the weak-coupling limit the fidelity in Eq. (33), shows that it is favorable to increase ¯ τ even at the expense of a slower rate r$_{at}$ (see Eq. (31)). In the strong-coupling limit, from Eq. (34) we find that increasing ¯ τ further gives slower convergence to the desired fidelity, hence slower proto- cols. Figure 5 displays the time required for reaching a desired fidelity F = 0 . 99 as a function of ¯ τ . One finds an optimal ¯ τ , for which the protocol is fastest, in between the two regimes. 

## IV. ENTANGLING TWO DISTANT CAVITIES WITH AN ATOMIC RESERVOIR 

Thus far we have considered two modes of the same mi- crowave cavity. However, as EPR entangled radiation is a possible resource for quantum communication with con- tinuous variables, e.g. for quantum teleportation [48], it would be desirable to create two entangled modes belong- ing to two different, spatially-separated cavities. In this section we extend the concepts discussed in Section III and present a scheme for entangling the modes of two spatially separated resonators using an atomic beam. 

We assume two open resonators, which are crossed by a beam of atoms propagating along the z axis and whose relevant modes are at frequency ω$_{1}$ and ω$_{2}$ , with anni- hilation and creation operators a$_{1}$, a † 1 and a$_{2}$, a † $_{2}$, respec- tively. The resonators spatial mode functions along the z axis are Gaussians centered at z$_{1}$ and z$_{2}$ , respectively, such that the distance | z$_{1}$ − z$_{2}$ | = D is much larger than the modes waist w , namely, D ≫ w and the fields mode functions have no spatial overlap. The atomic transition is quasi-resonant with a mode at frequency ω$_{1}$ in the first resonator, and at frequency ω$_{2}$ with the second resonator. 

