abstract
- We use a master equation to study the dynamics of two coupled macroscopic quantum systems (e.g.\ a Josephson junction made of two Bose-Einstein condensates or two spin states of an ensemble of trapped ions) subject to a weak continuous measurement. If the coupling between the two systems is suddenly switched on the resulting dynamics leads to caustics (fold and cusp catastrophes) in the number-difference probability distribution, and at the same time the measurement gradually induces a quantum-to-classical transition. Decoherence is often invoked to help resolve paradoxes associated with macroscopic quantum mechanics, but here, on the contrary, caustics are well-behaved in the quantum (many-particle) theory and divergent in the classical (mean-field) theory. Caustics thus represent a breakdown of the classical theory towards which decoherence seems to inevitably lead. We find that measurement backaction plays a crucial role in softening the resulting singularities and calculate the modification to the Arnol'd index which governs the scaling of the caustic's amplitude with the number of atoms. The Arnol'd index acts as a critical exponent for the formation of singularities during quantum dynamics and its modification by the open nature of the system is analogous to the modification of the critical exponents of phase transitions occurring in open systems.