  The dispersive coupling between a qubit and a cavity mode is widely used for
performing non-destructive readout of the qubit state. In this approach, it is
typically required that the dispersive strong coupling regime is achieved. Here
we show that the use of an auxiliary cavity mode reduces by orders of magnitude
the required value of the dispersive coupling, for a given decay rate of the
cavity mode. The analysis is performed within the input-output formalism, in
terms of the photon scattering matrix elements and of the signal-to-noise
ratio. We derive simple analytical expressions for the optimal parameters and
recover the standard single-mode result as a limiting case. The present results
can also be applied to the qubit readout based on longitudinal cavity-qubit
interactions, and to any sensing scheme where the cavity frequency is used as a
probe to estimate some physical parameter of interest.

Quantum Physics;Mesoscale and Nanoscale Physics