6 The states |±〉 are the semiclassical dressed states, and read and with The corresponding energy levels are shown in Fig. 2. We introduce the raising and lowering operators for the dressed states basis, with ˆ π$_{z}$ = | + 〉 〈 + | − |−〉 〈−| . Using the semiclassi- cal dressed-state basis and operator notation we rewrite Eq. (19) as ˆ H RF = ˆ H$_{0}$ + ˆ H$_{int}$ , with and If | g$_{λ}$ | ≪ d we can choose which processes are resonant, and thus relevant for the dynamics, by changing the values of δ$_{1}$, δ$_{2}$ , and d . One is thus able to generate a diversity of dynamical processes, some of which are discussed in Refs. [20, 47]. with r$_{µ}$ = arctanh µ , while the value of µ is determined by the classical field parameters, We now can see that the interaction of the cavity modes with a beam of atoms, each initially prepared in the state | + 〉 ( |−〉 ) and undergoing the dynamics governed by Hamiltonian ˆ H$_{1}$ = ˆ H$_{0}$ + ˆ H 1 int ( ˆ H$_{2}$ = ˆ H$_{0}$ + ˆ H 2 $_{int}$), will give rise to an effective dynamics, whose steady state is the pure state | ψ 〉 of the cavity modes. The Hamiltonian governing the dynamics will be ˆ H$_{1}$ or ˆ H$_{2}$ depending on the sign of the detuning Δ, which can be controlled by appropriately shifting the atomic transition frequency. Let us set δ$_{1}$ = d and δ$_{2}$ = − d , as shown in Fig. 2. If | g$_{λ}$ | ≪ d we obtain from Eq. (26) the effective Hamilto- nian ˆ H RF ≈ ˆ H$_{eff}$ = ˆ H$_{0}$ + ˆ H$_{int}$ , with and ˆ H$_{0}$ = ℏ d (ˆ π$_{z}$/ 2 − ˆ a † $_{1}$ˆ a$_{1}$ + ˆ a † $_{2}$ˆ a$_{2}$ ). We have assumed g := g$_{1}$ = g$_{2}$ . The processes described by Eq. (27) are indicated by the arrows in Fig. 2. In the basis of the b operators, we have ˆ H$_{0}$ = ℏ d $^{(}$ˆ π$_{z}$/ 2 − ˆ b † $_{1}$ˆ b$_{1}$ + ˆ b † $_{2}$ˆ b$_{2}$ ) and We discuss now how the cavity modes can be prepared in the two-mode squeezed state asymptotically. This is achieved by an effective “ dissipation ” process in the b - basis, implemented in a two-step procedure sketched in Fig. 3. The first step consists in letting the atomic beam interact with the resonator with each atom prepared in the state | + 〉 and the detuning of the classical field set to the value Δ = Δ$_{0}$ > 0. Inside the cavity each atom un- dergoes the dynamics of Eq. (28a), such that at the end of the interaction, on average, excitations have been re- moved from mode ˆ b$_{1}$ . In the second step, the atoms are Here, B. Effective dynamics: reaching the EPR state ˆ H 1 int = − ℏ Ω$_{b}$ $^{(}$ˆ b$_{1}$ ˆ π − + ˆ b † $_{1}$ˆ π + $^{)}$, if Δ > 0 (28a) ˆ H 2 int = ℏ Ω$_{b}$ $^{(}$ˆ b † $_{2}$ˆ π − + ˆ b$_{2}$ ˆ π + $^{)}$, if Δ < 0 . (28b) Ω$_{b}$ = g $^{√}$(1 − µ ) / (1 + µ ) µ = tan 2 θ if | tan θ | < 1 (29) µ = (tan θ ) − 2 if | tan θ | > 1 . ˆ H$_{int}$ = ∑ λ ℏ g$_{λ}$ $^{[}$ˆ π$_{z}$ $^{(}$ˆ a$_{λ}$ + ˆ a † λ $^{)}$cos θ sin θ (26) + $^{(}$ˆ π $^{+}$ˆ a$_{λ}$ + ˆ a † $_{λ}$ˆ π − $^{)}$cos 2 θ − $^{(}$ˆ π $^{−}$ˆ a$_{λ}$ + ˆ π $^{+}$ˆ a † λ $^{)}$sin 2 θ $^{]}$. ˆ H$_{0}$ = ℏ d ˆ π$_{z}$/ 2 − ℏ ∑ λ δ$_{λ}$ ˆ a † $_{λ}$ˆ a$_{λ}$, ˆ π + = | + 〉 〈−| , ˆ π − = |−〉 〈 + | , (25) tan θ = 2 | Ω | d − Δ . (24) | + 〉 = sin θ | g 〉 + cos θ | e 〉 , (23a) |−〉 = cos θ g 〉 − sin θ e , (23b) d = $^{√}$Δ 2 + 4Ω 2 . (22) Δ | + | + | + ω$_{L}$ ω$_{L}$ d ω$_{2}$ ω$_{1}$ |g, N-1> |g, N> |g, N+1> |e, N> |e, N-1> |e, N-2> FIG. 2: Left ladder: energy of photon states | N 〉 (with N ≫ 1) of the semiclassical field at frequency ω$_{L}$ . Mid- dle ladder: corresponding energies of the doublets of states {| g, N 〉 , | e, N − 1 〉} , where Δ = ω$_{L}$ − ω$_{0}$ is the corresponding splitting in energy. Right ladder: energy of the semiclassical dressed states |±〉 , Eq. (23), with energy splitting d , Eq. (22). A transition | + 〉 → |−〉 is accompanied by absorption (emis- sion) of a photon of frequency ω$_{1}$ ( ω$_{2}$ ) from (into) the corre- sponding cavity mode. ˆ H$_{int}$ = ℏ g $^{(}$ˆ a † $_{2}$cos 2 θ − ˆ a$_{1}$ sin 2 θ $^{)}$ˆ π − + H . c . , (27)