The results of the project can be divided in two main parts.
The first part regards the study of the performances of quantum metrological protocols based on time-continuous measurements.
- We have applied quantum estimation theory to collapse models, where the Schrodinger equation is modified accommodating a stochastic term able to describe a “fundamental” process of decoherence. We have shown how the estimation precision of the value of the “collapse-induced” diffusion on an opto-mechanical system is incredibly enhanced if the environment is time-continuously monitored via homodyne detection.
- We have considered the estimation of parameters via time-continuous measurements for bosonic systems described by a Gaussian dynamics. We have derived a simple and reliable method to calculate the classical Fisher information corresponding to these kind of strategies. This will allow to derive the corresponding ultimate limits on precision, that were not calculable efficiently via the methods already existing in the literature
- This last result has been crucial for the following project. We have in fact derived the ultimate limits for quantum magnetometry via time-continuous measurements via an ensemble of N atoms. In the limit of large N, the whole dynamics can be treated as a single (Gaussian) bosonic field, and we have been able to derive analytical formulas for the corresponding effective quantum Fisher information. We have shown that, even by starting with a (classical) coherent spin state, a quantum-enhanced (Heisenberg) precision can be achieved thanks to the information obtained via the continuous measurements and thanks to the spin-squeezing generated via the measurement during the dynamics.
- We have finally addressed one of the main objectives of the project. We have studied quantum frequency estimation for N qubits subjected to independent Markovian noise, via strategies based on time-continuous monitoring of the environment. For parallel noise, i.e. dephasing, we showed that perfectly efficient time-continuous photo-detection allows to recover the unitary (noiseless) QFI, and thus to obtain a Heisenberg scaling of the precision for every value of the monitoring time. For finite detection efficiency, one falls back to the noisy standard quantum limit scaling, but with a constant enhancement due to an effective reduced dephasing. Also in the transverse noise case we obtained that the Heisenberg scaling is recovered for perfectly efficient detectors, and we find that both homodyne and photo-detection based strategies are optimal. For finite detectors efficiency, our numerical simulations show that, as expected, an enhancement can be observed, but we cannot give any conclusive statement regarding the scaling.
The second part of results regard multi-parameter estimation with experimental proof-of-principle verification in photonics setup, in collaboration with the quantum optics group of Università Roma Tre, led by Prof. Marco Barbieri. We have mainly focused on the joint estimation of phase and phase-diffusion.
- In the first work we have studied the usefulness of entangled measurements for the joint estimation of the two parameters, given multiple copies of the probe states. While entangled measurements have been shown to be not useful for single-parameter estimation, we showed how in this framework they can actually give a non-trivial enhancement and are actually necessary in order to attain the ultimate quantum limit on the estimation precision.
- In the second, related, work, we have explored the role of frequency correlations within a photon pair generated via parametric down-conversion, when used as a probe for a dispersive medium (characterised by phase and phase-diffusion).
Apart from these two main parts of the project, other results have been derived during the fellowship. We have studied how single-side measurement on entangled Gaussian states can be exploited to generate quantum coherence, and to detect Gaussian entanglement via studying the extractable work. In another collaboration we have studied the relationship between quantum thermometry and quantum speed limit.
All these results have been described in more detail in several academic papers that have been published in international peer-reviewed journals and have been made also available on the open-access arXiv ( http://arXiv.org ) and on the institutional repository of University of Milan ( air.unimi.it ).
Moreover they have been disseminated in several international conferences and academic visits during the two years of fellowships (e.g. PBQ16 in Olomouc, IQIS16 in Rome, APS March Meeting 2017 in New Orleans, IQIS17 in Florence, Quantum’17 in Torino ).
