We study the interplay of chaos and tunneling between two weakly coupled Bose-Josephson junctions. The classical phase space of the composite system has a mixed structure including quasi-integrable self-trapping islands for particles and excitations, separated by a chaotic sea. We show that the many-body dynamical tunneling gap between macroscopic Schrödinger cat states supported by these islands is chaos-enhanced. The many-body tunneling rate fluctuates over several orders of magnitude with small variations of the system parameters or the particle number.

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• BEC tunneling
• Bose gases
• Chaos
• Control & applications of chaos
• Dynamical tunneling
• Quantum chaos
• Semiclassical physics
• Tunneling & traversal time
• Josephson junctions
• Chaos & nonlinear dynamics