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Quantum Control for Advanced Quantum Metrology

Periodic Reporting for period 1 - ConAQuMe (Quantum Control for Advanced Quantum Metrology)

Reporting period: 2016-05-01 to 2018-04-30

In the last years the novel, potentially revolutionary, possibilities offered by quantum mechanical systems have been deeply explored giving rise to a whole spectrum of Quantum Technologies, which are expected to have a big impact on the everyday life of future generations. Quantum metrology represents one of the most promising of these second generation quantum technologies: its aim is to exploit quantum mechanics to perform ultra-precise measurement not achievable by means of classical approaches.
Yet, a major obstacle still stands in the way of their full exploitation: preserving quantum coherence, the necessary ingredient for quantum protocols to outperform their classical counterparts, is a very difficult task. As it happens in ``classical technologies'', where the effect of noise is neutralized by monitoring and controlling the signal and its environment, quantum control is going to play a fundamental role for the development of noise-resilient protocols based on quantum mechanics.
The main goal of this fellowship has been to develop a more general framework for noisy quantum metrology, encompassing quantum control techniques based on time-continuous measurements and real-time feedback, and considering more general noisy evolutions and the case of multi-parameter estimation.
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 ( ) and on the institutional repository of University of Milan ( ).
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 ).
In conclusion, thanks to this project we have obtained two main results beyond the state of the art:
-we have been able to explore the usefulness of time-continuous monitoring for quantum metrology. In particular we have shown how the desired quantum enhancement can be restored, also in the presence of noise, when these particular strategies are employed. As a byproduct we have developed an efficient and reliable algorithm to derive the ultimate limits on generic estimation strategies based on time-continuous measurements and we have made available the code to the research community.
- we have addressed the problem of multi-parameter estimation, that will be fundamental for the progress of future quantum-enhanced sensors, proving the usefulness of entangling measurements, and in general studying the role of correlations, with experimental proof-of-principle verification in photonics setups.

Our effort will help in shortening the gap between the theoretical design of quantum metrology protocols and their actual implementation and usage in our everyday life.