2 erly tailored dissipative dynamics, where the atoms act as reservoir of the photonic field. As mentioned before, a simple example is the preparation of the cavity mode in the vacuum state [16–19]. Creation of other Fock states of the cavity field has been achieved in milestones ex- periments made in the strong-coupling regime, when the resonator field saturates the atomic transition [32, 33]. In this case, by accurately selecting the atomic velocity so that the interaction time of each atom is fixed, the nonlinear dynamics of atom and cavity field possesses several fixed points, so-called trapping states [34], which approach Fock states of the cavity field in the limit of negligible dissipation. lations [21], is created between the cavity modes at steady state. In this paper we provide a detailed discussion of the proposal and analyze its robustness. We show, more- over, that further nonclassical states of the electromag- netic field can be realized by tuning different parameters of the setup in Fig. 1(a), and which are stationary states of the interaction with the atomic beam. We also propose a scheme for entangling two distant resonators in a setup like the one sketched in Fig. 1(b). Our procedure extends to the preparation of nonlocal EPR states previous ideas regarding the production of macroscopically separated fields [24–27]. This article is organized as follows. In Sec. II we re- view the basic concepts at the basis of quantum reservoir engineering in microwave cavity QED. In Sec. III we dis- cuss the specific scheme for creating EPR states of the modes of a resonator. In Sec. IV a method for entan- gling the modes of two distant cavities is proposed. The conclusions are presented in Sec. V. Several theoretical proposals have been discussed in the literature, which provide schemes for the preparation of an arbitrary single-mode quantum state of the electro- magnetic field in a resonator, involving resonant inter- action with a well-controlled sequence of atoms, without the need of atomic detection [35–37]. This latter require- ment is indeed important, since present experiments lack high-efficiency detectors. In this section we briefly review the basic properties of atom-photon interactions in microwave CQED from the point of view of quantum reservoir engineering, hence set- ting the ground for the proposals for establishing EPR- correlations presented in Sec. III and Sec. IV of this ar- ticle. For a comprehensive review the reader is referred to Refs. [16, 22, 23, 38] A typical setup of microwave cavity QED is sketched in Fig. 1(a). Preparation and monitoring of the cavity field is achieved by interaction with atomic beams, whose in- ternal state is prepared in a circular Rydberg state with large principal number, and possessing a dipolar tran- sition which couples resonantly with the cavity mode. While the internal state of the atoms and their velocity can be prepared with high precision, the atomic arrival time is a stochastic variable which is known only prob- abilistically, according to a Poissonian distribution. The interaction of the cavity field with individual atoms is hence warranted by setting very low rates, such that the probability that two atoms are found inside the resonator is negligible [22, 23]. The interaction between atoms and cavity mode is coherent to a large extent: The atomic circular Rydberg states are typically stable over the in- teraction time, and high-finesse resonators in microwave cavity QED can reach very long photon-storage times, which in state-of-the-art experiments can reach the or- der of fractions of seconds [23, 28]. These properties al- lowed for the realization of milestone experiments. Some paradigmatic ones are the preparation and measurement of nonclassical states of the microwave field [22, 23], the experimental characterization of loss of coherence of the quantum field [29] and of the transition from quantum to classical dynamics [30], and the quantum nondemoli- tion measurement of the number of photons of the cavity field [18, 31]. Most of these formidable results were obtained imple- menting Hamiltonian dynamics, where the interaction between atoms and cavity photons is essentially disper- sive. On the contrary, the realization of nonclassical states of the cavity field in the dissipative regime, where atoms and photons exchange energy, is based on prop- with g the coupling strength. Here, ˆ a , ˆ a † are the an- nihilation, creation operator of a cavity photon, while ˆ σ = | g 〉〈 e | , ˆ σ † = | e 〉〈 g | describe the dipole lowering and rising operators, with g 〉 and | e 〉 the ground and excited states, respectively, of the dipolar transition. For ω$_{0}$ = ω$_{c}$ , then [ ˆ H$_{0}$, ˆ H$_{JC}$ ] = 0 and the evolution operator can be written as The stability of the atomic states and of the resonator mode during the interaction time justifies the use of Hamiltonian dynamics for the interaction between a sin- gle atom and the cavity mode. The Hamiltonian govern- ing the dynamics of a single dipole and the cavity mode is well described by the Jaynes-Cummings Hamiltonian, ˆ H = ˆ H$_{0}$ + ˆ H$_{JC}$ . Here ˆ H$_{0}$ = ℏ ω$_{0}$ ˆ σ $^{†}$ˆ σ + ℏ ω$_{c}$ ˆ a $^{†}$ˆ a gives the unperturbed evolution of a dipole at frequency ω$_{0}$ and a harmonic oscillator, the cavity mode, at frequency ω$_{c}$ , while the interaction between the dipole and the electric field of the cavity reads A. Jaynes-Cummings Hamiltonian ˆ H$_{JC}$ = ℏ g ˆ a $^{†}$ˆ σ + H . c ., (1) ˆ U ( t ) = exp $^{(}$− i ˆ Ht/ ℏ $^{)}$= exp$^{(}$− i ˆ H$_{0}$t/ ℏ $^{)}$exp$^{(}$− i ˆ H$_{JC}$t/ ℏ ) (2) II. AN ATOMIC RESERVOIR FOR MICROWAVE PHOTONS