## Abstract

The Dicke phase transition has recently come to prominence due to the
experimental observation (at the ETH and elsewhere) of a
self-organization transition in a BEC inside an optical cavity. In this
talk I will discuss two other examples. When a BEC is placed in a double
well potential each atom can in general be in a superposition of both
wells. If a single impurity atom is now added we find that there is a
critical interaction strength above which the system spontaneously
breaks its symmetry so that the impurity atom becomes localized in one
well and the majority of bosons in the other. I will explain how this
system can be mapped onto the Dicke model and back this up with the
results of a calculation of the critical exponents of the fidelity
susceptibility which indicate that this transition falls into the same
universality class as the Dicke model (and the Lipkin-Meshkov-Glick
model). We take the view that this system is a toy model for quantum
measurement with the BEC acting as a meter that measures the position of
the impurity atom thereby describing wave function collapse as a phase
transition. Time permitting, I will mention a second example which
concerns the membrane-in-the-middle version of optomechanics realized in
experiments at Yale. I point out that there is a symmetry breaking phase
transition of the Dicke-type in this
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driven
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model where the membrane
suddenly displaces to the left or the right.