Spectrum analysis with quantum dynamical systems

Measuring the power spectral density of a stochastic process, such as a stochastic force or magnetic field, is a fundamental task in many sensing applications. Quantum noise is becoming a major limiting factor to such a task in future technology, especially in optomechanics for temperature, stochastic gravitational wave, and decoherence measurements. Motivated by this concern, here we prove a measurement-independent quantum limit to the accuracy of estimating the spectrum parameters of a classical stochastic process coupled to a quantum dynamical system. We demonstrate our results by analyzing the data from a continuous-optical-phase-estimation experiment and showing that the experimental performance with homodyne detection is close to the quantum limit. We further propose a spectral photon-counting method that can attain quantum-optimal performance for weak modulation and a coherent-state input, with an error scaling superior to that of homodyne detection at low signal-to-noise ratios.

Figure 1

Some examples of the hidden stochastic process $X\left(t\right)$ and generator $\widehat{Q}$. (a) $X\left(t\right)$ is the classical force and $\widehat{Q}$ is the mechanical position. (b) $X\left(t\right)$ is the c-number forced displacement and $\widehat{Q}$ is proportional to the photon-number operator. (c),(d) $X\left(t\right)$ is the phase modulation and $\widehat{Q}$ is proportional to the photon-flux operator.

Figure 2

(a) Adaptive homodyne detection. (b) Spectral photon counting with a diffraction grating and a lens. (c) Spectral photon counting with an optical-resonator array.

Figure 3

Log-log plots of the quantum limit ${\tilde{\mathcal{J}}}^{-1}$ [inverse of Eqs. (3.19), black solid line] and homodyne limit ${\tilde{j}}^{-1}$ [inverse of Eqs. (3.21), blue dashed line] on the mean-square errors versus an SNR quantity $C\equiv 8{\theta }_1{S}_I/{\theta }_2$. (Top plot) Limits on ${\mathrm{\Sigma }}_{11}$ [normalized in a unit of ${\theta }_1^2/\left({\theta }_{2}T\right)]$; (bottom plot) limits on ${\mathrm{\Sigma }}_{22}$ (normalized in a unit of ${\theta }_2/T$). No measurement can achieve an error below the quantum limit (gray “forbidden” region), while the homodyne performance (blue “homodyne” region) cannot go below the homodyne limit. For $C\gg 1$, the limits approach constants, while for $C\ll 1$ the homodyne limit has a significantly worse error scaling.

Figure 4

Log-log plots of the quantum limit ${\tilde{\mathcal{J}}}^{-1}$ [inverse of Eqs. (3.19), black solid line], the homodyne limit ${\tilde{j}}^{-1}$ [inverse of Eqs. (3.21), blue dash line], and the experimental mean-square estimation errors $\mathrm{\Sigma }$ versus the SNR quantity $C\equiv 8{\theta }_1{S}_I/{\theta }_2$. (Top plot) Experimental ${\mathrm{\Sigma }}_{11}=\{4.0\pm 1.2,2.0\pm 0.6,2.0\pm 0.6,4.4\pm 1.1\}$ [in a unit of ${\theta }_1^2/\left({\theta }_{2}T\right)]$ versus $C=\{23.5,64.8,113,254\}$, compared with the homodyne limit and the quantum limit. (Bottom plot) Experimental ${\mathrm{\Sigma }}_{22}=\{8.7\pm 3.2,4.4\pm 1.6,5.2\pm 1.7,6.4\pm 1.4\}$ (in a unit of ${\theta }_2/T$) versus the same $C$ values, compared with the homodyne limit and the quantum limit.