Cavity QED with Cold Atoms in an Optical Lattice

Ultracold atoms in an Optical Lattice (OL) opened new domains in both theoretical and experimental physics, and represents an important area of present day research. The quantum phase transition from the superfluid into the Mott insulator phase has been realized experimentally after its theoretical prediction by the Bose-Hubbard model. In the Mott insulator phase the system can be considered as an artificial crystal, the fact that makes it attractive for a wide range of condensed matter studies. The recent implementation of a Bose-Einstein Condensate (BEC) of ultracold trapped atoms in a Cavity Quantum-Electro Dynamic (CQED), brings near the experimental achievement of ultracold atoms trapped in an OL in the CQED limit. In our previous research we studied solid-state effects in the system of OL ultracold atoms within a cavity, where we introduced collective electronic excitations (excitons) and cavity polaritons. Here, the OL forms of classical laser field off-resonance with the atomic transitions, while the cavity field is quantized and resonantly coupled to the atomic transitions. Cavity polaritons prove as an observation tool of OL defects. The excitation and de- excitation of vibrational modes of higher Bloch bands through their coupling to electronic excitation is also investigated. In the first part of the proposal we aim to extend our research to new topics. The OL and CQED assisted-cooling of vibrational and rotational modes of ultracold molecules should be studied. Cavity photon polarization mixing induced of different anisotropic effects in OLs should be treated with applications for quantum information. Cavity polaritons as an observation tool should be extended to detect the superfluid phase in OLs. The coupling of cavity polaritons to vibrational modes of higher Bloch bands should be derived and used as a mechanism for polariton relaxation in OLs. Furthermore, we aim to study the strong magnetic coupling of a BEC to a planner microwave resonator integrated on an atom chip. The system allows quantum interconnect of solid-state and atomic qubits, which is useful for quantum memory and communication. Our main aim in the proposal is to study new quantum phases and electronic excitation interactions in an OL within a cavity. A system of ground and excited state kinds of ultracold atoms in an OL is well described by the two- components Bose-Hubbard model. Within a cavity, the atom-photon resonant coupling can change the kind of atom through emission or absorption of a cavity photon. The system is described now by the coupled cavity photons and two-components Bose-Hubbard model, which we aim to investigate in order to achieve new quantum phases. The life time of excited atoms and cavity photons are included, and a cavity external pump added. Hence we aim to study the new quantum phases as a function of the mean cavity photon number, which can be controlled via the external pump. Moreover, the effect of electronic excitation transfer among different atom sites on the quantum phases should be studied. In the last part we aim to study nonlinear interactions of electronic excitations in OLs, where two sources of interactions should be considered. The first is dynamic interactions induced of electrostatic interactions, and can form bound-states. The second is induced of the Fermionic properties of electronic excitations, as for two-level atoms a single excitation at most is allowed at each atom (saturation effect). For low excitation density in an OL we should apply a bosonization procedure which results in interacting bosons with kinematic interactions. Such interactions are a source of nonlinearity for nonlinear optics in OLs, where we concentrate in nonlinear processes of importance for studying OL properties, with emphasize on new effects typical for OLs. The kinematic interaction should be used to check the diluteness and the relaxation time of a Bose gas of polaritons, and the results should be used to examine the possibility of achieving a BEC of cavity polaritons, where the advantages of a BEC in OLs should be emphasized.

Ultracold atoms in an Optical Lattice (OL) opened new domains in both theoretical and experimental physics, and represents an important area of present day research. The quantum phase transition from the superfluid into the Mott insulator phase has been realized experimentally after its theoretical prediction by the Bose-Hubbard model. In the Mott insulator phase the system can be considered as an artificial crystal, the fact that makes it attractive for a wide range of condensed matter studies. The recent implementation of a Bose-Einstein Condensate (BEC) of ultracold trapped atoms in a Cavity Quantum-Electro Dynamic (CQED), brings near the experimental achievement of ultracold atoms trapped in an OL in the CQED limit. In our previous research we studied solid-state effects in the system of OL ultracold atoms within a cavity, where we introduced collective electronic excitations (excitons) and cavity polaritons. Here, the OL forms of classical laser field off-resonance with the atomic transitions, while the cavity field is quantized and resonantly coupled to the atomic transitions. Cavity polaritons prove as an observation tool of OL defects. The excitation and de- excitation of vibrational modes of higher Bloch bands through their coupling to electronic excitation is also investigated. In the first part of the proposal we aim to extend our research to new topics. The OL and CQED assisted-cooling of vibrational and rotational modes of ultracold molecules should be studied. Cavity photon polarization mixing induced of different anisotropic effects in OLs should be treated with applications for quantum information. Cavity polaritons as an observation tool should be extended to detect the superfluid phase in OLs. The coupling of cavity polaritons to vibrational modes of higher Bloch bands should be derived and used as a mechanism for polariton relaxation in OLs. Furthermore, we aim to study the strong magnetic coupling of a BEC to a planner microwave resonator integrated on an atom chip. The system allows quantum interconnect of solid-state and atomic qubits, which is useful for quantum memory and communication. Our main aim in the proposal is to study new quantum phases and electronic excitation interactions in an OL within a cavity. A system of ground and excited state kinds of ultracold atoms in an OL is well described by the two- components Bose-Hubbard model. Within a cavity, the atom-photon resonant coupling can change the kind of atom through emission or absorption of a cavity photon. The system is described now by the coupled cavity photons and two-components Bose-Hubbard model, which we aim to investigate in order to achieve new quantum phases. The life time of excited atoms and cavity photons are included, and a cavity external pump added. Hence we aim to study the new quantum phases as a function of the mean cavity photon number, which can be controlled via the external pump. Moreover, the effect of electronic excitation transfer among different atom sites on the quantum phases should be studied. In the last part we aim to study nonlinear interactions of electronic excitations in OLs, where two sources of interactions should be considered. The first is dynamic interactions induced of electrostatic interactions, and can form bound-states. The second is induced of the Fermionic properties of electronic excitations, as for two-level atoms a single excitation at most is allowed at each atom (saturation effect). For low excitation density in an OL we should apply a bosonization procedure which results in interacting bosons with kinematic interactions. Such interactions are a source of nonlinearity for nonlinear optics in OLs, where we concentrate in nonlinear processes of importance for studying OL properties, with emphasize on new effects typical for OLs. The kinematic interaction should be used to check the diluteness and the relaxation time of a Bose gas of polaritons, and the results should be used to examine the possibility of achieving a BEC of cavity polaritons, where the advantages of a BEC in OLs should be emphasized.