7 This state is reached independently of the initial state of the cavity modes, provided that each step is implemented for a sufficiently long time T . An analogous dynamics can be implemented in the strong-coupling regime, i.e., when τ Ω$_{b}$ ≳ 1. Here, one aims at creating a trapping state in the Bogoliubov ba- sis for each Bogoliubov oscillator in a two-step procedure, where the initial state of the atomic beam and the atomic parameter are changed as in the procedure outlined for the weak-coupling case. In this case, the creation of trap- ping states in the Bogoliubov basis, with numbers n$_{1}$ and n$_{2}$ for each Bogoliubov mode, corresponds to entan- gled states of the cavity modes of the form ˆ S ( ξ ) | n$_{1}$, n$_{2}$ 〉 . Choosing a broad velocity distribution, one finds that the steady state is a vacuum two-mode squeezed state, ˆ S ( ξ ) | 0 , 0 〉 . The proposal we discussed so far is based on a stochas- tic dynamics, where the average action of each atom leads the cavity modes to a stationary, EPR-entangled state. The scheme does not require atomic detection, nor con- trol of the number of atoms, nor of the interaction times (atomic velocities). A possible experimental setup of this scheme is sketched in Fig. 1(a). Prior to the interaction region, the atoms are prepared in a coherent superposi- tion of two Rydberg states | g 〉 and | e 〉 connected by a dipole transition. Inside the resonator a classical field saturates the dipole transition, thereby pumping on res- onance the two nondegenerate modes of the resonator, as shown in Fig. 2. prepared in state |−〉 and the detuning of the classical field is set to the value Δ = − Δ$_{0}$. Inside the resonator each atom undergoes the dynamics given by Eq. (28b), such that at the end of the interaction, on average, exci- tations have been removed from mode ˆ b$_{2}$ . In the weak-coupling regime, the equation of motion of the cavity field modes is given by the master equation for the density matrix ˆ ρ$_{t}$ , which during each step j ( j = 1 , 2) reads where τ is the interaction time, and r$_{at}$ is the atomic arrival rate. Correspondingly, during each step the average num- ber of Bogoliubov excitations is exponentially damped according to the equation 〈 ˆ b † $_{j}$ˆ b$_{j}$ 〉 t = 〈 ˆ b † $_{j}$ˆ b$_{j}$ 〉 $_{0}$exp( − γt ), and vanishes at times t ≫ 1 /γ (see also Sec. II C). In terms of the original field modes, this procedure implies that the atoms pump in phase only the two-mode squeezed state. Asymptotically, after the implementation of the two steps for a sufficiently long time (such that the cavity field is still stable over this time) the field state approaches the state While the interaction time between cavity and each atom needs not to be controlled, on the other hand the dynamics between atom and cavity is here assumed to be Hamiltonian, and characterized by a two-level transition which can be tuned on resonance with the cavity modes by means of an external field. Let us now discuss these as- sumptions individually. Selecting a two-level transition, here denoted by the electronic states | g 〉 → | e 〉 imposes constrains on the field polarizations, such that they all couple with the dipole transition, while coupling to other states is avoided. In an open-cavity geometry [22], this can be achieved by means of an electric potential between the two mirrors, which removes through Stark shifting the degeneracy of circular Rydberg states, and using cir- cular polarizations for both the cavity modes and the pump field. The two-step procedure needs a change in the transition frequency of the two-level atom, which can be achieved by an external static field. The assumption of Hamiltonian dynamics between in- dividual atoms and resonator relies on the fact that the atom must not decay during the interaction with the cav- ity modes, and dissipation of the cavity field should be negligible during the experiment. Experiments with mi- crowave resonators [22, 23] are characterized by inter- action times of the order of tens of µ s, which warrant which is a two-mode squeezed state, and whose degree of squeezing r$_{µ}$ is solely determined by the ratio | Δ / Ω | . ∂ ˆ ρ$_{t}$ ∂ t ∣ ∣ step j = − γ 2 $^{(}$ˆ b † $_{j}$ˆ b$_{j}$ ˆ ρ$_{t}$ − 2 ˆ b$_{j}$ ˆ ρ$_{t}$ ˆ b † j + ˆ ρ$_{t}$ ˆ b † $_{j}$ˆ b$_{j}$ $^{)}$, (30) γ = r$_{at}$ Ω$_{b}$ $^{2}$τ 2 , (31) ˆ ρ$_{∞}$ = | 0 , 0 〉 b 〈 0 , 0 | = ˆ S $^{†}$( r$_{µ}$ ) | 0 , 0 〉 a 〈 0 , 0 | ˆ S ( r$_{µ}$ ) , (32) FIG. 3: Schematic representation of the interaction processes needed, in order to prepare the cavity modes in a EPR state. In step 1, the atoms are prepared in the state | + 〉 before crossing the resonator. Inside the resonator they are driven by a classical field, and the atomic frequency is shifted with respect to ω$_{L}$ according to the energy level scheme displayed at the bottom of the figure. In step 2, the initial state is |−〉 and the atomic transition frequency is shifted, such that the detuning with the field has opposite sign. C. Experimental parameters Step 1: e Step 2: e ω$_{0,2}$ ω$_{0,1}$