Quantum Metrology in Non-Markovian Environments

We analyze precision bounds for a local phase estimation in the presence of general, non-Markovian phase noise. We demonstrate that the metrological equivalence of product and maximally entangled states that holds under strictly Markovian dephasing fails in the non-Markovian case. Using an exactly solvable model of a physically realistic finite bandwidth dephasing environment, we demonstrate that the ensuing non-Markovian dynamics enables quantum correlated states to outperform metrological strategies based on uncorrelated states using otherwise identical resources. We show that this conclusion is a direct result of the coherent dynamics of the global state of the system and environment and therefore the obtained scaling with the number of particles, which surpasses the standard quantum limit but does not achieve Heisenberg resolution, possesses general validity that goes beyond specific models. This is in marked contrast with the situation encountered under general Markovian noise, where an arbitrarily small amount of noise is enough to restore the scaling dictated by the standard quantum limit.

Figure 1

Ratio $r$ between the optimal resolution achievable with uncorrelated and maximally entangled inputs as a function of the number of particles $n$. The dashed line shows the expected behavior in the absence of noise where $r=\sqrt{n}$ (Heisenberg limit), while $r$ becomes equal to 1 (pink line) when the noise is fully Markovian. In the presence of non-Markovian phase decoherence, product states and maximally entangled initial preparations are no longer metrologically equivalent. In the case of a zero temperature bath with an Ohmic spectral density ($s=1$), maximally entangled states allow for a higher resolution for any value of $n$ and $r$ displays a typical $n^{1/4}$ dependence as shown by the solid line in the figure.