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Quantum localization (single-body or many-body) comes with the emergence oflocal conserved quantities -- whose conservation is precisely at the heart ofthe absence of transport through the system. In the case of fermionic systemsand $S=1/2$ spin models, such conserved quantities take the form of effectivetwo-level systems, called $l$-bits. While their existence is the definingfeature of localized phases, their direct experimental observation remainselusive. Here we show that strongly localized $l$-bits bear a dramaticuniversal signature, accessible to state-of-the-art quantum simulators, in theform of periodic cusp singularities in the Loschmidt echo following a quantumquench from a N\'eel/charge-density-wave state. Such singularities areperfectly captured by a simple model of Rabi oscillations of an ensemble ofindependent two-level systems, which also reproduces the short-time behavior ofthe entanglement entropy and the imbalance dynamics. In the case of interactinglocalized phases, the dynamics at longer times shows a sharp crossover to afaster decay of the Loschmidt echo singularities, offering an experimentallyaccessible signature of the interactions between $l$-bits.

Loschmidt echo singularities as dynamical signatures of strongly localized phases