content_image/422968.jpg content_image/422977.jpg content_image/422972.jpg content_image/422973.jpg content_image/422970.jpg content_image/422969.jpg content_image/422964.jpg content_image/422965.jpg content_image/422974.jpg content_image/422971.jpg content_image/422966.jpg content_image/422975.jpg content_image/422967.jpg
and 

The states |±〉 are the semiclassical dressed states, and read 

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 

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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]. 

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 

<img src='content_image/422977.jpg'>

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 

Here, 

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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. 

## B. Effective dynamics: reaching the EPR state 

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 

<img src='content_image/422967.jpg'>

