9 we will show, the atomic velocity can change without affecting the efficiency of the protocol. For the following treatment we assume that the mode waist is much smaller than the wavelength, as it is of- ten the case, so that the maser field has a well defined phase over the interaction region. In particular, we de- note by ψ$_{1}$ = kz$_{1}$ and ψ$_{2}$ = kz$_{2}$ the phases at each res- onator. The atom is continuously driven by the maser field, and the phase of the dressed states is here assumed to follow adiabatically the phase of the field as the atom moves. For simplicity, in what follows we assume the phases ψ$_{1}$ = ψ$_{2}$ = 0. Within these approximations, we write the effective dynamics of the individual atom in- teracting sequentially with the cavity modes in terms of semiclassical dressed states of the maser field, given in Eqs. (23). Setting the detuning between maser and cav- ity modes ω$_{L}$ − ω$_{1}$ = d and ω$_{L}$ − ω$_{2}$ = − d , with d the frequency splitting between the dressed states given in Eq. (22), we obtain the corresponding effective Hamilto- nian, which reads ˆ H$_{eff}$$_{,j}$ = ˆ H$_{0}$ + ˆ H$_{j}$ with FIG. 5: Logarithmic plot of time T required in each step to reach the EPR state as a function of average interaction time ¯ τ (in units of Ω − 1 b ). The EPR state here chosen is charac- terized by µ = . 95 (¯ n$_{0}$ = 9 . 3), and we require that the cor- responding fidelity at T$_{tot}$ = 2 T is F = 0 . 99. The blue lines represent analytical expressions for fidelities F$_{wc}$ , Eq. (9), and F$_{sc}$ , Eq. (15). The red line has been obtained from numeri- cal solution of Eq. (10), where coefficients B$_{n}$ were calculated taking σ = 0 . 05 φ$_{0}$ in the Gaussian distribution function of Rabi angles φ$_{0}$ = ¯ τ Ω$_{b}$. The most efficient regime, in terms of fastest protocols for a given fidelity, lies between weak and strong coupling. The plot shows slower convergence at Rabi angles corresponding to φ$_{0}$ = π/ √ j ( j positive integer) due to trapping states. 0.001 0.01 0.05 0.1 0.5 1 5 10 50 100 20 50 100 200 500 1000 2000 5000 10000 Τ g T g Ε weak coupling strong coupling “trapping states” τ (units of Ω ) b T (units of є Ω ) b -1 -1 -1 ˆ H$_{0}$ = ℏ d$^{[}$ ˆ π$_{z}$ 2 − ˆ a † $_{1}$ˆ a$_{1}$ + ˆ a † $_{2}$ˆ a$_{2}$$^{]}$ , (36a) ˆ H$_{1}$ = − ℏ g$_{1}$ sin 2 θf$_{1}$ ( z ) $^{[}$ˆ a † $_{1}$ˆ π + + ˆ a$_{1}$ ˆ π − $^{]}$, (36b) ˆ H$_{2}$ = ℏ g$_{2}$ cos 2 θf$_{2}$ ( z ) $^{[}$ˆ a † $_{2}$ˆ π − + ˆ a$_{2}$ ˆ π + $^{]}$, (36c) In addition, the atoms are driven by a maser which prop- agates almost parallel to the z -axis and which has wave vector k , frequency ω$_{L}$ and intensity Ω (we neglect any spatial gradient and assume that the maser intensity is uniform along z ). The Hamiltonian describing the coherent interaction of one individual atom of the beam with both cavities reads where the angle θ is given in Eq. (24). The space- dependence of the cavity spatial mode functions can be substituted by the mean value of the function over the interaction region, which is constant over an interval of time, such that the total pulse area is preserved [16]. Hamiltonian Eq. (36) has the same form as Eq. (27) in Sec. III, with the only difference that the atom interacts with each mode in well-separated time intervals, due to the distant location of the cavities. We will now show that the combined state of the two distant cavity modes can be pulled into a two-mode squeezed state as a result of the interaction with an atomic beam, of which the number of atoms and the individual interaction times are known only statistically. where g$_{λ}$ is the strength of the coupling between cav- ity mode and dipolar transition, while the spatial mode function takes the form The line of reasoning extends the protocol presented in Sec. III. In this case, however, one must consider that the interaction of the atom with each mode is sequen- tial. For this purpose we introduce the evolution opera- tor ˆ U$_{j}$ ( τ ) for the dynamics of the atom interacting with the resonator j over the interval of time τ (here taken in interaction picture with respect to Hamiltonian ˆ H$_{0}$ in Eq. (36a)) and we evaluate the total density matrix ˆ ρ → t of atom and resonator after the atom has interacted with both cavities in interaction picture, assuming that the atom interacts sequentially first with mode 1 and then with mode 2. Denoting by Δ t the interval of time in which this occurs, the density matrix after the interac- tion reads with w the mode waist. The atomic center of mass is located at the time-changing position z ( t ) = z (0) + vt , where v is the velocity of the atom. The assumption of classical center-of-mass motion is justified by the pa- rameters of the typical experimental situation, where the atoms exit an oven and the velocity selection brings to distributions still within the classical regime. In addi- tion, the assumption of uniform motion is not necessary, but convenient for the theoretical treatment. Indeed, as ˆ H ( t ) = ℏ ω$_{0}$ ˆ σ $^{†}$ˆ σ (35) + ℏ Ω $^{(}$e − i( ω$_{L}$t − kz ( t $^{))}$ˆ σ † + e i( ω$_{L}$t − kz ( t $^{))}$ˆ σ ) + ∑ [ ℏ ω$_{λ}$ ˆ a † $_{λ}$ˆ a$_{λ}$ + ℏ g$_{λ}$f$_{λ}$ ( z ( t ))(ˆ a$_{λ}$ ˆ σ † + ˆ a † $_{λ}$ˆ σ ) ] , f$_{λ}$ ( z ) = exp( − ( z − z$_{λ}$ ) $^{2}$/ 2 w $^{2}$) $_{/}$√$_{2}$$_{πw}$ $^{2}$, ˆ ρ $^{→}$( t + Δ t ) = ˆ U$_{2}$ ( τ ) ˆ U$_{1}$ ( τ )ˆ ρ ( t ) ˆ U † $_{1}$( τ ) ˆ U † $_{2}$( τ ) . (37)