We present a theoretical model for a transversally driven Bose-Einstein condensate coupled to an optical cavity. We focus on the interplay between different coherent couplings, which can trigger a structural phase transition, known as the superradiant phase transition. Our approach, based on band-structure theory and a mean-field description, enables a comprehensive analysis of the nature of the system's excited modes, precursing the phase transitions. By incorporating dissipative couplings, intrinsic to these systems, we find non-Hermitian phenomena such as the coalescence of crossing precursor modes and the emergence of exceptional points. The general formulation of our model allows us to explain the role of an angle between the transverse pump and the cavity deviating from ${90}^{\circ }$. This offers us a unified perspective on the plethora of different implementations of such systems.

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• Atoms, ions, & molecules in cavities
• Bose-Einstein condensates
• Cavity quantum electrodynamics
• Cold atoms & matter waves
• Collective effects in atomic physics
• Critical phenomena
• Crystallization
• Dynamical phase transitions
• Hybrid quantum systems
• Light-matter interaction
• Mechanical effects of light on material media
• Non-reciprocal propagation
• Open quantum systems & decoherence
• Optical lattices & traps
• Pattern formation
• Quantum description of light-matter interaction
• Quantum optics
• Quantum phase transitions
• Quantum simulation
• Quasiparticles & collective excitations
• Superfluidity
• Synchronization