Quantum probes for the spectral properties of a classical environment

We address the use of simple quantum probes for the spectral characterization of classical noisy environments. In our scheme a qubit interacts with a classical stochastic field describing environmental noise and is then measured after a given interaction time in order to estimate the characteristic parameters of the noise. In particular, we address estimation of the spectral parameters of two relevant kinds of non-Gaussian noise: random telegraph noise with a Lorentzian spectrum and colored noise with a 1/f \u3b1 spectrum. We analyze in detail the estimation precision achievable by quantum probes and prove that population measurement on the qubit is optimal for noise estimation in both cases. We also evaluate the optimal interaction times for the quantum probe, i.e., the values maximizing the quantum Fisher information (QFI) and the quantum signal-to-noise ratio. For random telegraph noise the QFI is inversely proportional to the square of the switching rate, meaning that the quantum signal-to-noise ratio is constant and thus the switching rate may be uniformly estimated with the same precision in its whole range of variation. For colored noise, the precision achievable in the estimation of \u201ccolor,\u201d i.e., of the exponent \u3b1, strongly depends on the structure of the environment, i.e., on the number of fluctuators describing the classical environment. For an environment modeled by a single random fluctuator estimation is more precise for pink noise, i.e., for \u3b1 = 1, whereas by increasing the number of fluctuators, the quantum signal-to-noise ratio has two local maxima, with the largest one drifting towards \u3b1 = 2, i.e., brown noise