corresponding mode ( j = 1 , 2). Our goal is to generate a dynamics described by a master equation whose steady state is a two-mode squeezed state, ˆ ρ$_{St}$ = | ψ 〉〈 ψ | , with Here, | 0 , 0 〉 is the vacuum state of both cavity modes and the subset {| 4 n $^{2}$〉} are fixed points of the dynamics, i.e., trapping states. The initial state of the cavity determines in which of these trapping states the cavity will be found. In general the final state is a statistical mixture of these states. The other limiting case is found for large σ , i.e. a broad distribution for the atom’s velocity. For σ ≳ π , each of the coefficients samples approximately equally over the period of the sinusoidal function in Eq. (11), avoiding trapping states, and we can therefore set B$_{n}$ = 1 $_{2}$. For the fidelity in this approximation we find is the two-mode squeezing operator, with ξ a complex pa- rameter. It can be verified that state | ψ 〉 is the vacuum state of harmonic oscillators, whose annihilation opera- tors ˆ b$_{1}$ and ˆ b$_{2}$ are related to the cavity modes operators by the relation and analogously for the creation operators. We refer to modes ˆ b$_{1}$ and ˆ b$_{2}$ as Bogoliubov modes, in analogy with the Bogoliubov transformation used in solid-state physics [46]. State ˆ ρ$_{St}$ can be hence the dark state of an inter- action Hamiltonian ˆ H$_{1}$ , which has, say, the form of the Jaynes-Cummings Hamiltonian ˆ H$_{JC}$ in Eq. (1) but with the operator ˆ a (ˆ a $^{†}$) replaced by the operator ˆ b$_{1}$ ( ˆ b † $_{1}$). We note that it is also the dark state of the interaction Hamil- tonian ˆ H$_{2}$ , which has the same form as ˆ H$_{1}$ but now with ˆ b$_{2}$ and ˆ b † 2 in place of ˆ b$_{1}$ and ˆ b † $_{1}$. In particular, ˆ ρ$_{St}$ is si- multaneously dark state of both interactions, and it is unique. Hence, a dynamics can be constructed, which has as unique steady state ˆ ρ$_{St}$ , by implementing sequen- tially two interactions which effectively damp oscillators ˆ b$_{1}$ and ˆ b$_{2}$ , respectively. In the following we will show how to engineer such dynamics. and one sees that in this limit the fidelity of the ground state is determined only by the initial occupation number ¯ n (parametrized by µ ) and by the number of atoms rt which have crossed the cavity. where we used the relation ∫ t 0 τ $^{n}$e − $^{Dτ}$d τ = n ! D n +1 $^{[}$1 − e − Dt ∑ n m =0 ( Dt ) m m ! $^{]}$. For an initial ther- mal state with c$_{n}$ (0) = (1 − µ $^{2}$) µ 2 $^{n}$, where µ is given by the initial average number of thermal photons via $^{〈}$ˆ a $^{†}$ˆ a 〉 = µ 2 1 − µ 2 , this simplifies to In the following we discuss in detail and extend a pro- posal for generating EPR-entangled states of two cavity modes by resonant interaction with a beam of atoms, which was first presented in Ref. [20]. The setup is sketched in Fig. 1(a), and is composed by a resonator, with two modes in which a single atom can emit photons. The atoms are part of a beam crossing the resonator, and are driven by a classical microwave field while they inter- act with the cavity modes. The interaction is shaped in such a way that the atoms emit correlated photons into the cavity modes. The underlying mechanism is four- wave mixing, where emission into the cavity modes is enhanced by resonant coupling with the Rabi sidebands induced by a classical field that saturates the atomic transition [45], with the creation of EPR-correlations be- ing enforced by the initial quantum state of the injected atoms. Contrary to typical setups based on optical para- metric amplifiers, the atoms pump the resonator through resonant single-photon processes. The Hamiltonian of driving field, atom, and cavity modes, in the reference frame rotating at the frequency ω$_{L}$ of the classical field, has the form where g$_{j}$ are the coupling constants between the two-level atom and each cavity mode, detunings δ$_{j}$ = ω$_{L}$ − ω$_{j}$ , Δ = ω$_{L}$ − ω$_{0}$ , and describes the coupling between dipole and classical field, with strength Ω, see Fig. 2. Let the coupling to the classical field be much stronger than the coupling to the cavity modes, | Ω | ≫ | g$_{λ}$ | , it is then convenient to express Hamiltonian (19) in the basis of eigenstates |±〉 of ˆ H $_{0}$, with Before discussing details of the mechanism, let us make some general remarks on the main ideas regarding the generation of EPR states through reservoir engineering. We first label by 1 and 2 the two cavity modes, such that ω$_{j}$ is the frequency and ˆ a$_{j}$ , ˆ a † j are the annihilation and creation operators of an energy quantum ℏ ω$_{j}$ of the ˆ H RF = ˆ H 0 − ∑ j =1 , 2 ℏ δ$_{j}$ ˆ a † $_{j}$ˆ a$_{j}$ + ∑ j =1 , 2 ℏ g$_{j}$ $^{(}$ˆ σ $^{†}$ˆ a$_{j}$ + ˆ σ ˆ a † j $^{)}$, (19) ˆ H 0 = − ℏ Δˆ σ $^{†}$ˆ σ + ℏ Ω $^{(}$ˆ σ † + ˆ σ ) (20) ˆ H 0 |±〉 = − ℏ (Δ ∓ d ) / 2 |±〉 , (21) F $_{sc}$( t ) = 1 − µ $^{2}$e − rt (1 − µ $^{2}$) / $^{2}$, (15) ˆ b$_{1}$ = ˆ S $^{†}$( ξ )ˆ a$_{1}$ ˆ S ( ξ ) , (18a) ˆ b$_{2}$ = ˆ S $^{†}$( ξ )ˆ a$_{2}$ ˆ S ( ξ ) , (18b) F $_{sc}$( t ) = 1 − e − r$_{at}$t/ 2 ∞ ∑ n $^{=}$0 c$_{n}$ (0) n − 1 ∑ m =0 ( rt/ 2) m m ! , (14) ˆ S ( ξ ) = exp $^{(}$ξ $^{∗}$ˆ a$_{1}$ ˆ a$_{2}$ − ξ ˆ a † $_{1}$ˆ a † 2 ) (17) | ψ 〉 = ˆ S $^{†}$( ξ ) | 0 , 0 〉 . (16) 5 III. GENERATING EPR ENTANGLED RADIATION WITH AN ATOMIC RESERVOIR A. Engineering the coupling to the reservoir