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 $z\left(t\right)$ 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 multitime moments $\langle z\left(t_n\right)\cdots z\left(t_1\right)\rangle$ 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.

Figure 1

Schematics of continuous spin noise spectroscopy. The fluctuating electron spin orientation in a quantum system induces a fluctuating polarization of the probe beam via the Faraday effect. Polarization noise and photon shot noise contribute to the noise signal $z\left(t\right)$. The noise spectrum $S\left(\omega \right)$ exhibits a peak at the spin precession frequency and a constant offset due to shot noise. The system $\rho$ may include coupling of the electron spin (e) with a nuclear spin (n).

Figure 2

The quantities $a$ and $\gamma$ can be determined from a measured noise spectrum and allow for an estimate of the measurements strength $\beta$ via ${\beta }^2=\gamma a$.

Figure 3

Calculated spin noise spectra of the coupled spin system in ZnO:In for high spin temperatures and increasing magnetic fields.

Figure 4

Fourth-order correlation spectra $S_z^{\left(4\right)}({\omega }_1,-{\omega }_1,{\omega }_2,-{\omega }_2)$ of the spin-spin system in ZnO:In at $B_x=0,10$, and 100 mT. Negative correlations appear in blue. The function asinh is used for color scaling.

Figure 5

Comparison of spin noise spectra of the coupled spin system in ZnO:In at $B_x=100\mathrm{mT}$ for isotropic and anisotropic hyperfine interaction along the $x$ axis. (a) Second-order spectra $S_{\mathrm{q}}\left(\omega \right)$. (b) Fourth-order correlation spectra $S_z^{\left(4\right)}({\omega }_1,-{\omega }_1,{\omega }_2,-{\omega }_2)$.

Figure 6

Bispectrum $S_z^{\left(3\right)}({\omega }_1,{\omega }_2,-{\omega }_1-{\omega }_2)$ (upper row) and correlation spectrum $S_z^{\left(4\right)}({\omega }_1,-{\omega }_1,{\omega }_2,-{\omega }_2)$ (lower row) for a 10 mT magnetic field in in-plane ($0^{\circ }$) and out-of-plane directions (${30}^{\circ },{60}^{\circ },{90}^{\circ }$).

Figure 7

Power spectra $S_{\mathrm{q}}\left(\omega \right)$ for a 10 mT magnetic field in in-plane ($0^{\circ }$) and out-of-plane directions (${30}^{\circ },{60}^{\circ },{90}^{\circ }$).