Imaging using interferometer arrays based on the Van Cittert--Zernike theorem has been widely used in astronomical observation. Recently it was shown that superresolution can be achieved in this system for imaging two weak thermal point sources. Using quantum estimation theory, we consider the fundamental quantum limit of resolving the transverse separation of two strong thermal point sources using interferometer arrays, and show that the resolution is not limited by the longest baseline. We propose measurement techniques using linear beam splitters and photon-number-resolving detection to achieve our bound. Our results demonstrate that superresolution for resolving two thermal point sources of any strength can be achieved in interferometer arrays.
Imaging using interferometer arrays based on the Van Cittert--Zernike theorem has been widely used in astronomical observation. Recently it was shown that superresolution can be achieved in this system for imaging two weak thermal point sources. Using quantum estimation theory, we consider the fundamental quantum limit of resolving the transverse separation of two strong thermal point sources using interferometer arrays, and show that the resolution is not limited by the longest baseline. We propose measurement techniques using linear beam splitters and photon-number-resolving detection to achieve our bound. Our results demonstrate that superresolution for resolving two thermal point sources of any strength can be achieved in interferometer arrays.
It has been shown through quantum estimation theory that Rayleigh’s limit can be avoided for single-lens imaging, referred to as superresolution. The quantum estimation approach has recently been used to show superresolution is also possible for imaging based on interferometer arrays in the weak source limit. Following this line of discussion, we consider the resolution limit of estimating the separation between two point sources of arbitrary strength using interferometer arrays. By carefully designing the measurement, we find it is possible to overcome the well-known resolution limit of interferometer arrays as determined by the longest baseline. We construct an optimal measurement to achieve superresolution using linear beam-splitters and photon-number resolved detection.
We obtain the ultimate quantum limit for estimating the transverse separation of two thermal point sources using a given imaging system with limited spatial bandwidth. We show via the quantum Cramér-Rao bound that, contrary to the Rayleigh limit in conventional direct imaging, quantum mechanics does not mandate any loss of precision in estimating even deep sub-Rayleigh separations. We propose two coherent measurement techniques, easily implementable using current linear-optics technology, that approach the quantum limit over an arbitrarily large range of separations. Our bound is valid for arbitrary source strengths, all regions of the electromagnetic spectrum, and for any imaging system with an inversion-symmetric point-spread function. The measurement schemes can be applied to microscopy, optical sensing, and astrometry at all wavelengths.
We determine the ultimate potential of quantum imaging for boosting the resolution of a far-field, diffraction-limited, linear imaging device within the paraxial approximation. First, we show that the problem of estimating the separation between two pointlike sources is equivalent to the estimation of the loss parameters of two lossy bosonic channels, i.e., the transmissivities of two beam splitters. Using this representation, we establish the ultimate precision bound for resolving two pointlike sources in an arbitrary quantum state, with a simple formula for the specific case of two thermal sources. We find that the precision bound scales with the number of collected photons according to the standard quantum limit. Then, we determine the sources whose separation can be estimated optimally, finding that quantum-correlated sources (entangled or discordant) can be superresolved at the sub-Rayleigh scale. Our results apply to a variety of imaging setups, from astronomical observation to microscopy, exploiting quantum detection as well as source engineering.
Resolving high-contrast targets is a fundamental, yet highly challenging, task in astronomy. Using quantum estimation theory, we demonstrate that the ultimate limit for estimating the separation between two unequal-brightness thermal sources via interferometry remains constant, enabling the potential for superresolution. We give a comparative analysis of two primary stellar interferometric schemes: amplitude interferometry and intensity interferometry. Notably, the nulling strategy employed in amplitude interferometry, a configuration specifically proposed for exoplanet detection by leveraging destructive interference to suppress the brighter source, is quantum optimal for separation estimation. While intensity interferometry is less effective than amplitude interferometry in lossless scenarios and fails to achieve superresolution, it becomes competitive when optical loss in large-scale interferometry is considered. By applying these methodologies to modern stellar interferometry, we highlight the promise of large-scale interferometry for advancing high-resolution astronomical observation.
We present a quantum optical scheme for imaging transversely displaced thermal sources of arbitrary intensities by employing multiphoton interference with a reference single-photon Fock state at a beamsplitter. Obtaining an analytical form for transverse momenta-resolved L-photon probabilities in either output, we show via Fisher information analysis that separation estimators built using interference sampling of multiphoton events exhibit significantly enhanced precision vis-à-vis existing imaging schemes over a wide range of separations and brightness. Even-photon-number coincidences exhibit constant precision in the sub-Rayleigh regime, demonstrating quantum superresolution of our scheme beyond the diffraction limit. For sources emitting on average Ns∼1 photon per frame (such as in IR emission of thermal sources), precision bounds for our scheme scale linearly in Ns, exemplifying an enhanced precision of estimators in relation to weak sources Ns≪1, and matching the ultimate quantum scaling. Finally, transverse momenta resolution in the Fourier plane produces finite imaging precisions for intermediate and large source separations using coarse pixel sizes of order δy∼100μm for exemplary image spot sizes σx∼0.1μm, in contrast with existing schemes of diffraction-limited direct imaging and superresolved inversion interferometric imaging that are severely degraded by coarse pixel sizes and have limited use. Combining the relatively straightforward sensing operation of Hong-Ou-Mandel interferometers with multiphoton coincidence detection of arbitrarily bright thermal sources and inner variable resolution of transverse photonic momenta, our scheme offers a robust alternative to non-invasive single-particle tracking and imaging of bright sources in nanoscopic chemical and biological systems.
We propose two schemes for estimating the separation of two thermal sources via double homodyne and double array homodyne detection considering the joint measurement of conjugate quadratures of the image plane field.By using the Cramér–Rao bound, we demonstrate that the two schemes can estimate the separation well below the Rayleigh limit and have an advantage over direct imaging when the average photon number per source exceeds five.For arbitrary source strengths, double homodyne detection is superior to homodyne detection when the separation is above 2 5/4 σ / N s , σ is the beam width, N s is the average photon number per source.A larger separation can be estimated better via double array homodyne detection with the superiority of flexible operation compared with other schemes. High-speed and high-efficiency detection enables the measurement schemes with potential practical applications in fluorescence microscopy, astronomy and quantum imaging.