We wrongly claimed in our derivation of the four-time cumulant C 4 (z(t 1), z(t 2), z(t 3), z(t 4)) that the terms α 13 and α 31 cancel in the original paper. A correct treatment starts with (we use the short notation of Appendix F) α 13 + α 31 = T (AG 43 Aρ 0)T (AG 21 Aρ 0)θ 32 − T (AG 43 Aρ 0)T (AG 21 Aρ 0)/2. (1) The contribution of the sum Sum(α 13 + α 31) to C 4 contains all 24 permutations of the indices 1 to 4. Assuming a preliminary time-order t 4 > t 3 > t 2 > t 1 , one finds that most terms vanish [e.g., −T (AG 23 Aρ 0)T (AG 41 Aρ 0)/2 is zero since G 23 = 0 due to t 3 > t 2 ]. The surviving terms give rise to Sum(α 13 + α 31) = −T (AG 42 Aρ 0)T (AG 31 Aρ 0) − T (AG 41 Aρ 0)T (AG 32 Aρ 0)
Random-time sampling of a quantum system is introduced as a new approach to continuous quantum measurements with prospects for ultra-low-perturbation measurements. Random sampling is, e.g., naturally realized in an optical spin noise experiment when weak probe-laser light exhibits random single-photon events in the detector. We show that a direct evaluation of these detector timetraces yields power spectra that are equivalent to those of the usual continuous measurement regime. Surprisingly, this holds true even for average sampling rates much lower than the typical frequency range of the measured quantum dynamics. The third-order quantum polyspectrum (bispectrum) also contains the same information as its continuous counterpart. Many applications of randomtime sampling are envisioned for high-resolution spectroscopy, circuit quantum electrodynamics, quantum sensing, and quantum measurements in general.
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