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The diagonal elements of the time correlation matrix are used to probe closedquantum systems that are measured at random times. This enables us to extracttwo distinct parts of the quantum evolution, a recurrent part and anexponentially decaying part. This separation is strongly affected when spectraldegeneracies occur, for instance, in the presence of spontaneous symmetrybreaking. Moreover, the slowest decay rate is determined by the smallest energylevel spacing, and this decay rate diverges at the spectral degeneracies.Probing the quantum evolution with the diagonal elements of the timecorrelation matrix is discussed as a general concept and tested in the case ofa bosonic Josephson junction. It reveals for the latter characteristicproperties at the transition to Hilbert-space localization.

Probing many-body systems near spectral degeneracies