Time crystals are nonequilibrium many-body phases in which the state of the system dynamically approaches a limit cycle. While these phases have recently been the focus of intensive research, it is still far from clear whether they can host quantum correlations and whether the latter can be exploited for parameter estimation. Here, we consider the nonequilibrium behaviour of an open quantum light-matter system, as realised in current trapped ion experiments. While light and matter are uncorrelated in the stationary phase, the time-crystal phase features bipartite correlations of both quantum and classical nature. Although an open quantum system seems to be detrimental for parameter estimation, the continuous observation of the emission field leads to a Heisenberg (linear) scaling of the sensitivity in time throughout the time-crystal phase. Thus, the classical correlations built-up in the emission field rather than the previously described quantum correlations within the system itself are a useful resource for sensing applications. This suggests that a simple measurement protocol such as a maximum likelihood measure may be sufficient to achieve the Heisenberg scaling in time.
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