Large gain quantum-limited qubit state measurement using a two mode nonlinear cavity

A single nonlinear cavity dispersively coupled to a qubit functions as a large gain detector near a bifurcation, but also has an unavoidable large backaction that prevents QND measurement at weak couplings~[1]. We show theoretically that a modified setup involving two cavities (one linear, one nonlinear) and a dispersively coupled qubit allows for a far more optimal measurement. In particular, operating near a point of bifurcation, one is able to both achieve a large gain as well as a near quantum-limited backaction. We present analytic results for the gain and noise of this detector and a heuristic understanding of the physics, thus presenting a complete description of this new way of performing weak qubit state measurements. The setup we describe can easily be realised in experiments with superconducting circuits involving Josephson junctions~[2,3].\\[4pt] [1] C. LaFlamme, A.A. Clerk, Phys. Rev. A \textbf{83}, 033803 (2011)\\[0pt] [2] F.R. Ong \emph{et al.}, Phys. Rev. Lett. \textbf{106}, 167002 (2011)\\[0pt] [3] M. Hatridge \emph{et al.}, Phys. Rev. B \textbf{83}, 134501 (2011)