Quantum noise spectroscopy as an incoherent imaging problem

I point out the mathematical correspondence between an incoherent imaging model proposed by my group in the study of quantum-inspired superresolution [M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016)] and a noise spectroscopy model also proposed by us [M. Tsang and R. Nair, Phys. Rev. A 86, 042115 (2012); S. Ng et al., Phys. Rev. A 93, 042121 (2016)]. Both can be regarded as random displacement models, where the probability measure for the random displacement depends on unknown parameters. The spatial-mode demultiplexing (SPADE) method proposed for imaging is analogous to the spectral photon counting method proposed by Ng et al. for optical phase noise spectroscopy: Both methods are discrete-variable measurements that are superior to direct displacement measurements (direct imaging or homodyne detection) and can achieve the respective quantum limits. Inspired by SPADE, I propose a modification of spectral photon counting when the input field is squeezed: The output field is simply unsqueezed before spectral photon counting. I show that this method is quantum optimal and far superior to homodyne detection for both parameter estimation and detection, thus solving the open problems in the work of Tsang and Nair and Ng et al.

Figure 1

(a) Schematic of an incoherent imaging system, where the quantum state $\rho$ of each image-plane photon can be modeled as a randomly displaced object, with the distribution of the incoherent sources determining the probability measure $P$ for the displacement and the point-spread function of the imaging system determining the initial state $|\psi \rangle$ of the photon. (b) Schematic of an optomechanical sensor, where the quantum state $\rho$ of the optical fields before the measurement can also be modeled as a randomly displaced object, with a probability measure $P$ governing the displacement and the initial state of the optical fields determining $|\psi \rangle$. In both cases, a measurement is modeled by a positive-operator-valued measure $\mathcal{E}$, and an estimator $\check{\beta }\left(\lambda \right)$ of a parameter of $P$ can be constructed from the measurement outcome $\lambda$.

Figure 2

Michelson implementation of the model given by Eqs. (11, 12, 13, 14, 15, 16, 17, 18, 19).

Figure 3

Comparison of USPC and homodyne detection in terms of their Fisher information given by Eqs. (42) as a function of $\theta$ in log-log scale. The Fisher information has been normalized with respect to the time-bandwidth product $BT$. The $\theta$ has been normalized so that ${\theta }^2$ is the ratio of the displacement spectrum $S_X$ to the quantum-limited phase-quadrature spectrum $S_{\eta }=1/4S_k$. Both axes are dimensionless.

Figure 4

Comparison of USPC and homodyne detection in terms of their Chernoff exponents given by Eqs. (49) and (50), assuming the flat spectra given by Eqs. (39) and (40). The horizontal axis $\varphi$ is the square root of the spectral SNR defined by Eq. (54). The Chernoff exponent has been normalized with respect to the time-bandwidth product $BT$. The plot is in log-log scale and both axes are dimensionless.