Abstract
Localisation microscopy of multiple weak, incoherent point sources with possibly different intensities in one spatial dimension is equivalent to estimating the amplitudes of a classical mixture of coherent states of a simple harmonic oscillator. This enables us to bound the multi-parameter covariance matrix for an unbiased estimator for the locations in terms of the quantum Fisher information matrix, which we obtained analytically. In the regime of arbitrarily small separations we find it to be no more than rank two—implying that no more than two independent parameters can be estimated irrespective of the number of point sources. We use the eigenvalues of the classical and quantum Fisher information matrices to compare the performance of spatial-mode demultiplexing and direct imaging in localisation microscopy with respect to the quantum limits.
Highlights
Locating multiple single emitters is a key challenge in fluorescence microscopy
We provide an analytical lower bound on an unbiased estimators covariance matrix for localisation microscopy—simultaneously estimating the locations of N incoherent, weak point sources of unequal but known intensities in one spatial dimension
As the inverse of the quantum Fisher information matrix (QFIM) lower bounds the covariance matrix, our result implies that no more than two independent parameters can be estimated in localisation microscopy in the limit of arbitrarily small separations
Summary
Locating multiple single emitters is a key challenge in fluorescence microscopy. Using methods from classical and quantum estimation theory, it was shown theoretically that two arbitrarily close incoherent point sources may be resolved, and that this may be achieved in practice using a spatial-mode demultiplexing (SPADE) measurement. As the inverse of the QFIM lower bounds the covariance matrix, our result implies that no more than two independent parameters can be estimated in localisation microscopy in the limit of arbitrarily small separations. In this limit, we provide a mathematical explanation for our observation in terms of an approximation of the light field involving only the first two Hermite–Gauss modes.
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