Engineering atomic quantum reservoirs for photons Susanne Pielawa 1 , $^{2}$, Luiz Davidovich$^{3}$, David Vitali$^{4}$, and Giovanna Morigi1 , 5 1 Departament de F´ısica, Universitat Aut` onoma de Barcelona, 08193 Bellaterra, Spain 2 Department of Physics, Harvard University, Cambridge MA 02138, U.S.A. 3 Instituto de Fisica, Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, Brazil 4 Dipartimento di Fisica, Universit` a di Camerino, 62032 Camerino, Italy and 5 Theoretische Physik, Universit¨ at des Saarlandes, D-66041 Saarbr¨ ucken, Germany (Dated: November 17, 2018) We present protocols for creating entangled states of two modes of the electromagnetic field, by using a beam of atoms crossing microwave resonators. The atoms are driven by a transverse, classical field and pump correlated photons into (i) two modes of a cavity and (ii) the modes of two distant cavities. The protocols are based on a stochastic dynamics, characterized by random arrival times of the atoms and by random interaction times between atoms and cavity modes. The resulting effective model yields a master equation, whose steady state is an entangled state of the cavity modes. In this respect, the atoms act like a quantum reservoir, pulling the cavity modes into an entangled, Einstein-Podolski-Rosen (EPR) state, whose degree of entanglement is controlled by the intensity and the frequency of the transverse field. This scheme is robust against stochastic fluctuations in the atomic beam, and it does not require atomic detection nor velocity selection. The quest for quantum control of mesoscopic systems imposes the development of novel strategies, which go beyond the implementation of fully coherent Hamilto- nian dynamics. The main objective is to combat the effects of noise, which give rise to dissipation and de- coherence. Protocols based on quantum error correc- tion [1, 2], quantum feedback [3, 4], dynamical decou- pling [5, 6], and decoherence-free subspaces [7–9] aim at minimizing the detrimental effects of coupling to an ex- ternal environment. In this context, quantum reservoir engineering makes use of noise statistics as a resource for implementing robust quantum dynamics. The basic idea is to implement a stochastic dynamics whose stationary state is a nonclassical state. This is achieved by manip- ulating the coupling to the reservoir, whose properties are known only through the statistical averages [10]. A prominent example of quantum reservoir engineering is laser cooling, which achieves low temperatures of single atoms or ions by tailoring the scattering cross section, such that in average the scattered photon carries away mechanical energy of the atomic center of mass [11, 12]. More recently, this concept has been generalized in var- ious directions, for the generation of nontrivial many- body states and nonequilibrium quantum phases [13], and for the implementation of dissipation-driven quan- tum computation [14]. The idea of using an atomic beam as a reservoir for a field in a cavity actually goes back to quantum laser theory, see for instance Ref. [15], where a thermal atomic beam crossing the laser cavity acts as a thermal bath for the laser field. Actual implementations of this idea have been made in experiments on cavity quantum electrody- namics, where a beam of atoms in the lower state of a two-level space is used to generate a vacuum state of the field in a microwave cavity, by absorbing thermal photons initially present in the cavity [16–19]. Recently, we proposed a method for preparing quan- tum states of the electromagnetic field based on quantum reservoir engineering [20]. This method is implemented in a typical setup of microwave Cavity Quantum Electro- dynamics (CQED) as in Fig. 1(a), where the resonator is pumped by a beam of atoms with random arrival times, and needs neither atomic detection, nor detailed control of the sequence of atoms. We showed that, by suit- ably preparing the initial state of the incoming atoms, two-mode squeezing, i.e., Einstein-Podolski-Rosen corre- arXiv:1001.3281v2 [quant-ph] 6 Apr 2010 I. INTRODUCTION FIG. 1: Setup of the system for creation of EPR state of a) two modes of the same cavity and b) two spatially sep- arated cavities. In both cases atoms from a beam are first prepared in a coherent superposition of two Rydberg states | g 〉 and | e 〉 by a combination of laser and microwave fields. The atoms have random arrival times, and a low pumping rate warrants that at most one atom is inside the resonator at a time [22, 23]. While in the cavities, the dipole transi- tion | g 〉 → | e 〉 is saturated by a transverse microwave field, thereby pumping on resonance two nondegenerate modes of either one or two cavites, which are led asymptotically to a two-mode squeezed state. High-Q cavity a) Microwave Generator Cavity 1 Cavity 2 Excitation Excitation b) Atoms Oven Driving Microwave Field Microwave Generator Driving Microwave Field Excitation