Engineering entanglement for metrology using rotating matter waves

We present the first exact diagonalisation study of a rotating Bose-Einstein condensate stirred by a quadratic anisotropic potential near the threshold of the first vortex nucleation beyond the lowest Landau level (LLL) approximation. Although the nucleation of the first vortex in this system has been the focus of extensive experimental and theoretical research in both mean-field studies, as well as in exact diagonalisation ones, they have all relied on the assumption of the interaction strength being weak enough so that the system is well described by the LLL approximation. The LLL approximation accurately predicts the rotation frequency at which the first vortex penetrates the gas, and correctly describes the appearance of a quantum phase transition at a critical rotation frequency c, which leads to quantum correlations between the atoms and entanglement in the ground state at c. However, our research shows that the LLL approximation fails at describing the details of the entangled state and the quantum Fisher information, which bounds the accuracy of phase measurements in metrology schemes, even for weak interactions. Our results reveal a rich system that allows for the engineering of different promising entangled states for quantum metrology, which are also amenable to experimental investigation. Finally, we propose an inteferometric scheme that makes use of these entangled states, which is shown to have the potential of delivering nearly Heisenberg-limited precision for the measurement of small rotations. The scheme is conceptually very simple, and it is also within reach of current experimental technologies.