Quantum Metrology in Correlated Environments

We analytically obtain the precision bounds of frequency measurements in correlated Markovian and non-Markovian environments by using a variational approach. It is verified that in standard Ramsey spectroscopy setup, the metrological equivalence of product and maximally entangled states persists in maximally correlated Markovian and non-Markovian environments. We find that the optimal measurement can achieve a much higher resolution than standard Ramsey spectroscopy in the correlated environments. When the number of particles in the maximally entangled states is even, the precision bound decreases with interrogation time; and when the number is odd, the precision bound is independent of interrogation time, both in correlated Markovian and general non-Markovian environments. In addition, the opposite case can appear in some special non-Markovian environments.