We present a quantum description of the interaction between a Bose-Einstein condensate and a single-mode quantized radiation field in the presence of a strong far-off-resonant pump laser. In the linear regime, the atomic medium is described approximately by two momentum states coupled to the radiation mode. We calculate the evolution of the operators in the Heisenberg picture and their expectation values, such as average and variance of the occupation numbers, atom-atom and atom-field correlations, and two-mode squeezing parameters. Then, we disentangle the evolution operator and obtain the exact evolution of the state vector in the linear regime. This allows us to demostrate that the system can be atom-atom or atom-field thermally entangled. We define the quasiclassical and the quantum recoil limits, for which explicit expressions of the average population numbers are obtained.

Continuous variable remote state preparation and teleportation are analyzed using Wigner functions in phase space. We suggest a remote squeezed state preparation scheme between two parties sharing an entangled twin beam, where homodyne detection on one beam is used as a conditional source of squeezing for the other beam. The scheme works also with noisy measurements, and provide squeezing if the homodyne quantum efficiency is larger than 50%. Phase space approach is shown to provide a convenient framework to describe teleportation as a generalized conditional measurement, and to evaluate relevant degrading effects, such the finite amount of entanglement, the losses along the line, and the nonunit quantum efficiency at the sender location.

Free Electron Laser (FEL) and Collective Atomic Recoil Laser (CARL) are described by the same model of classical equations for properly defined scaled variables. These equations are extended to the quantum domain describing the particle's motion by a Schrödinger equation coupled to a selfconsistent radiation field. The model depends on a single collective parameterρ which represents the maximum number of photons emitted per particle. We demonstrate that the classical model is recovered in the limitρ ≫ 1, in which the Wigner function associated to the Schrödinger equation obeys to the classical Vlasov equation. On the contrary, forρ ≤ 1, a new quantum regime is obtained in which both FELs and CARLs behave as a two-state system coupled to the self-consistent radiation field and described by Maxwell-Bloch equations.

We present the results of an experiment on light scattering from an elongated Bose-Einstein condensate ͑BEC͒ interacting with a far-off-resonant pump laser. By collective atomic recoil lasing ͑CARL͒ a coherent superposition of two atomic wave packets with different momenta is created. Varying the intensity of a weak counterpropagating laser beam we observe the transition from the pure superradiant regime to the Bragg scattering regime, where Rabi oscillations in a two-level system are observed. The process is limited by the decoherence between the two atomic wave packets. In the superradiant regime the experiment gives evidence of a contribution to decoherence which depends on the initial velocity of the condensate. The system is described by the CARL-BEC model, which is a generalization of the Gross-Pitaevskii model to include the self-consistent evolution of the scattered field and a phase-diffusion decoherence process, which accounts for the observed damping.

We investigate the interaction of a moving BEC with a far detuned laser beam. Superradiant Rayleigh scattering arises from the spontaneous formation of a matter-wave grating due to the interference of two wavepackets with different momenta. The system is described by the CARL-BEC model which is a generalization of the Gross-Pitaevskii model to include the self-consistent evolution of the scattered field. The experiment gives evidence of a damping of the matter-wave grating which depends on the initial velocity of the condensate. We describe this damping in terms of a phase-diffusion decoherence process, in good agreement with the experimental results.The experimental realization of Bose-Einstein condensates (BECs) with alkali trapped atoms has opened the possibility of investigating several fundamental aspects of quantum mechanics in macroscopic, i.e. many particle systems [1]. In superradiant Rayleigh scattering the coherent nature of the condensate leads to strong correlations between successive scattering events, as shown in the pioneering work of Ketterle and coworkers [2]. This process was then the basis for the first demonstration of phase-coherent matter wave amplification [3]. The effect studied by Ketterle is an example of a spontaneous formation of a regular density grating in an atomic system, arising from a collective instability as in the Collective Atomic Recoil Laser (CARL) [4]. In the absence of thermal broadening (as it happens in a BEC), CARL appears as a promising source of macroscopically entangled or number-squeezed atom-atom and/or atomphoton systems [5,6,7]. However, in a real BEC several effects due, for instance, to spontaneous emission, inhomogeneous broadening and collisions, may seriously inhibit the CARL process and destroy the coherence in the matter wave field [8]. The control of decoherence in the photon-BEC interaction would be a significant step toward the achievement of macroscopic entanglement of coherent matter waves.In this paper we investigate both theoretically and experimentally the influence of the initial velocity of the condensate on superradiant Rayleigh scattering. In the experiment we produce an elongated BEC of rubidium atoms and expose it to a single off-resonant laser pulse directed along the condensate symmetry axis. The laser is far detuned from any atomic resonance and the only scattering mechanism present is Rayleigh scattering [2]. In an elongated condensate a preferential direction for the scattered photons emerges, causing superradiant Rayleigh scattering. In this regime the atoms, initially scattered randomly, interfere with the atoms in the original momentum state creating a matter-wave grating with the right periodicity to further scatter the laser photons in the same mode. Both the matter-wave grating and the scattered light are then coherently amplified. In our geometry photons are back-scattered with k s ≈ − k, where k is the wave-vector of the laser photon, and the atoms move away from the original condensate with a relative momentum 2hk in the dire...