Ultracold bosonic gases coupled to an optical cavity mode

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Ultracold atoms coupled to a photonic mode of an optical cavity show fascinating phenomena such asthe Dicke phase transition. This is a self-organized phase transition in which the atoms spontaneouslyorder into a checkerboard density pattern and the photonic mode becomes occupied due to thefeedback mechanism between the atoms and the photonic mode. In this project we will investigatebosonic atoms which are coupled to photonic modes and are additionally confined to optical latticestructures. We will mainly focus on a novel coupling dominantly to the tunneling process of the atomswith or without spatially dependent phase imprint which has been proposed by the PI. We expectinteresting phases to occur such as the self-organization of complex Mott-insulating and superfluidphases or in the presence of a phase imprint, Meissner and vortex phases.The full solution of the dynamics of the combined atom-cavity system is very demanding mainlydue to three reasons: (i) the strongly correlated nature of the interacting bosonic atoms in optical lat-tices, (ii) the coupling of the spatially extended cavity mode to the atoms, which can induce effective,long-range interactions between the atoms, and (iii) the dissipative nature of the cavity mode by theleaking of the photons out of the cavity due to imperfect mirrors. We describe the combined and dis-sipative system by a Markovian master equation with a Lindblad dissipator. In order to determine theresulting dynamics we will employ different approaches. In a first approach, we will reduce the modelby adiabatically eliminating the cavity mode to an effective interacting bosonic model - with long-rangeprocesses - subjected to a self-consistency condition. For the solution of this self-consistent modelwe will develop a self-consistent implementation of the matrix product state (MPS) algorithm andthe quantum Monte-Carlo method in collaboration with project P6. The second approach aims at afull numerical simulation in quasi-one-dimensional systems. The implementation of a time-dependentMPS taking into account the dissipative and extended cavity mode with a large local Hilbert space isone of the main challenges of the project.