Correlators Larger Than One in Continuous Measurement of a Superconducting Qubit

The interplay of the phase and informational backactions can lead to counter intuitive results. We consider the effect of the phase backaction on the correlator〈I(t) I(t+τ)〉for the output signal I(t) from a continuous measurement of a qubit. We demonstrate that in the presence of Rabi oscillations the correlator can become larger than 1, even though │〈I(t)〉│≤ 1 due to normalization. The bound of 1 for the correlator can be exceeded only when the phase backaction is present. The correlators can be calculated using the generalized "collapse recipe", which we prove using the quantum Bayesian formalism. The recipe can be further generalized to the case of multi-time correlators and arbitrary number of phase-sensitive detectors, measuring non-commuting qubit observables. The theory agrees well with experimental results for continuous cQED measurement of a transmon qubit. The experimental correlator exceeds the bound of 1 for a sufficiently large angle between the amplified and informational quadratures, causing the phase backaction. The demonstrated effect can be useful for calibration of the quadrature misalignment.