  Entangled states of rotating, trapped ultracold bosons form a very promising
scenario for quantum metrology. In order to employ such states for metrology,
it is vital to understand their detailed form and the enhanced accuracy with
which they could measure phase, in this case generated through rotation. In
this work we study the rotation of ultracold bosons in an asymmetric trapping
potential beyond the lowest Landau level (LLL) approximation. We demonstrate
that whilst the LLL can identify reasonably the critical frequency for a
quantum phase transition and entangled state generation, it is vital to go
beyond the LLL to identify the details of the state and quantify the quantum
Fisher information (which bounds the accuracy of the phase measurement). We
thus identify a new parameter regime for useful entangled state generation,
amenable to experimental investigation.
