10 In order to obtain the two-mode squeezing correlations, we need also processes in which the temporal sequence of photon absorption and emission is reversed, namely, processes of the sort and which require the presence of a second current of atoms propagating in the opposite direction, such that each atom first interacts with resonator 2 and then with resonator 1, see Fig. 6. After imposing this condition, one derives an effective dissipative dynamics for the Bo- goliubov modes ˆ b$_{1}$ and ˆ b$_{2}$ , defined as in Eq. (18), and which is valid over a time step Δ t such that there is at most one atom inside the resonator at a time. Provided that the atoms are initially prepared in the state | + 〉 ( |−〉 ) and the detuning is such that Δ = Δ$_{0}$ (Δ = − Δ$_{0}$), the master equation for the field density matrix in the coarse-grained time-scale is interact with two vertically arranged cavities above the fountain [52], or a optical conveyor belt [53], where the atoms are transported back and forth between the res- onators. These mechanisms have been employed so far in the optical regime, where the lifetime of the resonator modes is limited to tens of microseconds, in which case the implementation of this protocol is not straightfor- ward. where Δˆ ρ → t = ˆ ρ $^{→}$( t + Δ t ) − ˆ ρ ( t ) (same for $^{←}$), and r$_{at}$ is the atom pump rate, which is assumed to be the same in both directions. Hence, mode ˆ b$_{1}$ ( ˆ b$_{2}$ ) is exponentially damped according to an equation of the form given in Eq. (30). Preparation of the two resonators in a two- mode squeezed state then is achieved, provided that both dynamics take place, by changing detuning and state preparation after the first step has been implemented af- ter a sufficiently long time. As in the protocol for entangling two modes of the same resonator, this scheme does not require detection of final atomic states, nor control of the atomic velocities. More- over, the velocity of the atoms can change during prop- agation, without affecting the efficiency of the protocol. This result has been derived considering Hamiltonian dy- namics between cavity and individual atoms. Let us now discuss the limitations to this proposal. First, only one atom at a time must be present inside the cavities. Moreover, the atoms must not decay before they have interacted with both resonators. Atomic lifetime of the order of tens of milliseconds and atomic velocities of about 400 m/s require that the distance between the cavities is no larger than a few meters. Moreover, the resonators must be stable over the whole run of the ex- periment. Experimental implementation of this proposal is quite challenging, since one must have a geometry with two counterpropagating atomic beams, thereby avoiding col- lisions between the atoms. As an alternative, one could think of a ring, as realized for instance in ion-storage se- tups [51] in which the atomic beam is confined and which crosses two resonators placed at two different points of the ring. Another possibility is to implement an atomic fountain, where atoms traveling upwards and downwards In this paper we have extensively characterized the properties of an atomic beam as a reservoir for the modes of the electromagnetic field inside a resonator, for the purpose of creating entangled states of the cavity modes. The atoms can mediate the interaction between the modes of the same resonator or of distant cavities, es- tablishing a dynamics whose steady state is an Einstein- Poldosky-Rosen entangled state. As opposed to previous proposals, see for instance [36, 37], the atoms do not need to be initially correlated nor their number has to be controlled. Control on atomic velocity (interaction time) and atomic detection are not required. The de- gree of entanglement is controlled by an external maser field, which drives the atoms and tailors their interaction with the cavity modes. In this respect, the proposals discussed in this article are instances of quantum reser- voir engineering. Statistical properties of the cavity field can be evaluated by measuring the internal states of the emerging atoms [49]. Its state can also be determined FIG. 6: Top: sketch of the setup for entangling the modes of two distant resonators. Two atomic beams propagate in both directions and cross the resonators, interacting sequentially with each of them. Here one of the two required steps of the procedure is shown, where the atoms are prepared in the state | + 〉 . Bottom: energy levels which are coupled inside each resonator. The “ball” represents the atomic occupation. An atom traveling from right to left first emits a photon ω$_{2}$ into the second resonator, then emits a photon ω$_{1}$ into the first one, only if it has previously emitted into resonator 2, otherwise it may absorb a photon ω$_{1}$ . An atoms traveling from left to right absorbs/emits photons in the reversed sequence. V. CONCLUSIONS AND OUTLOOK ∂ ˆ ρ$_{t}$ ∂ t = r$_{at}$ Δˆ ρ $^{→}$( t ) + r$_{at}$ Δˆ ρ $^{←}$( t ) = − γ $^{[}$ˆ b † $_{j}$ˆ b$_{j}$ ˆ ρ − 2 ˆ b$_{j}$ ˆ ρ b † j + ˆ ρ ˆ b † $_{j}$ˆ b$_{j}$ $^{]}$, (39) ˆ ρ $^{←}$( t + Δ t ) = ˆ U$_{1}$ ( τ ) ˆ U$_{2}$ ( τ )ˆ ρ ( t ) ˆ U † $_{2}$( τ ) ˆ U † $_{1}$( τ ) (38) cavity 1 ω$_{1}$ ω$_{1}$ ω$_{2}$ cavity 2 ω$_{2}$