Higher order moments, cumulants, and spectra of continuous quantum noise measurements

We present general quantum mechanical expressions for higher order moments, cumulants, and spectra of continuously measured quantum systems with applications in spin noise spectroscopy, quantum transport, and measurement theory in general. Starting from the so-called stochastic master equation of continuous measurement theory, we find that the leading orders of the fluctuating detector output $z(t)$ with respect to the measurement strength $\beta$ are a white shot noise background, a constant measurement offset, and the leading order quantum noise of the measured operator $A$. Starting from quantum expressions for the multi-time moments $\langle z(t_n)\cdots z(t_1) \rangle$ we derive three- and four-time cumulants that are valid in all orders of $\beta$ covering the full regime between the weak and strong measurement limit (Zeno-limit). Intriguingly, quantum expressions for the cumulants were found that exhibit the same simple structure as those for the moments after introduction of only a slightly modified system propagator. Very compact expressions for the cumulant-based third and fourth order spectra (bispectrum and trispectrum) follow naturally. We illustrate the usefulness of higher order spectra by treating a real world two-spin system with strong hyperfine interaction.Moreover, spin noise spectroscopy is shown to have the potential for investigating the transition from weak measurements to the famous quantum Zeno regime for realistic probe laser intensities.