Quantum metrology with multimode Gaussian states of multiple point sources

A unified theoretical framework is developed for measuring multimode Gaussian states of multiple point sources. The spatial and temporal mode functions are found analytically by aid of the circular symmetry of the object plane and aperture, as well as by the rectangular spectral density function of the background. They relate the modal operators and statistical moments to the locations of the point sources through the prolate spheroidal functions. An experimentally straightforward implementation is proposed, which consists of a nonabsorptive polarization- and phase-modulating spatial light modulator (SLM), three lenses, and a polarization beam splitter. Interestingly, it is found that the spatial modal annihilation operators are related to the modal field operators in the image plane by a simple complex-valued scale factor. The quantum Fisher information matrix and quantum Cramér-Rao bound (QCRB) are also discussed. The numerical calculation of QCRB is completed for the case where the spatially multimode aperture fields are prepared by the spatial propagation from temporally single-mode thermal states, coherent states, or quadrature squeezed states of the point sources in the object plane. The results show that the Rayleigh limit can always be surpassed by means of increasing the mean photon rate or the squeeze parameter.

Figure 1

(a) Quantum diffraction from a circular object plane $O$ of radius $b$ to a circular aperture $A$ of radius $a$. The background is uniform, while the $J$ point sources are independent of each other. (b) Mode channel model with beam splitters to account for modal losses. Each mode covers the entire circular areas of the object plane and the aperture, and the loss in each channel is described by a beam splitter along the propagation path.

Figure 2

(a) Implementation of the spatial mode decomposition. The SLM and PBS multiply the aperture field by the function ${\psi }_p(\mathbf{r}/a)/a\sqrt{z_p}$, and the lens L1 with focal length $f=R$ multiplies the aperture field by $exp(-i\pi {\mathbf{r}}^2/{\lambda }_{0}R)$. The other two lenses L2 and L3 with focal length $f=R$ are used to compensate for the phase shifts proportional to ${\mathbf{r}}^2$. $A$ and $I$ denote the aperture and image plane, respectively. (b) Part of the spatial mode functions ${\psi }_p(\mathbf{r}/a)/a\sqrt{z_p}$ sequentially loaded onto the SLM. The odd and even columns are the amplitudes and phases, respectively, and the two-element vector indices $p=(N,n)$ are labeled at the upper left corners.

Figure 3

QCRBs $Q_j$ as functions of the location of the point sources in the object plane. (a), (c), and (e) are for $\mathrm{\Delta }{r}_j$, and (b), (d), and (f) are for $\mathrm{\Delta }{\theta }_j$. The maximum spatial mode indices are ${(N,n)}_{\text{max}}=(9,9)$ for (a) and (b), $(9,18)$ for (c) and (d), and $(18,18)$ for (e) and (f). The units of $Q_j\left(r_j\right)$ are millimeters, and that of $Q_j\left({\theta }_j\right)$ are rad. The point sources are in coherent states, and the mean photon rates ${\mathcal{N}}_j=1.0\times {10}^4$. The boundaries of the object plane are marked with yellow circles. To enhance the details, the natural logarithms of $Q_j$ are used.

Figure 4

QCRBs $Q_j$ of $\mathrm{\Delta }{r}_j$ and $\mathrm{\Delta }{\theta }_j$ as functions of (a) the mean photon rate ${\mathcal{N}}_j$ and (b) the squeeze parameter $\varrho$, with $\vartheta =\pi /4$. The units of $Q_j\left(r_j\right)$ and the Rayleigh limit are millimeters, and the units of $Q_j\left({\theta }_j\right)$ are rad. The point source is at the location $r_j=1/2$ and ${\theta }_j=0$ in the object plane. The maximum spatial mode indices are ${(N,n)}_{\text{max}}=(9,18)$. For coherent states, $\varrho =0$.