The Bernstein problem in Heisenberg groups

In these notes, we collect the main and, to the best of our knowledge, most up-to-date achievements concerning the Bernstein problem in the Heisenberg group; that is, the problem of determining whether the only entire minimal graphs are hyperplanes. We analyze separately the problem for t-graphs and for intrinsic graphs: in the first case, the Bernstein Conjecture turns out to be false in any dimension, and a complete characterization of minimal graphs is available in H1 for the smooth case. A positive result is instead available for Lipschitz intrinsic graphs in H1; moreover, one can see that the conjecture is false in Hn with n at least 5, by adapting the Euclidean counterexample in high dimension; the problem is still open when n is 2, 3 or 4

Quantum Optics Experiments in Space

Space has always been a primary source of inspiration for the development of the scientific, technological, artistic, philosophical and religious thinking of the whole humankind. Space explorations marked the history of the XX century, bringing an incredible technological advancement and allowing to investigate the natural phenomena over scales and into details which are simply not available on Earth. Nowadays, the Space is the benchmark of the new quantum revolution, which promises to change the way we communicate, measure and calculate, thank to the exploitation and the control of what happens at the microscopic scale. Indeed, the quantum theory, born at the beginning of the XX century to describe the behavior of the elementary particles of Nature, has reached today an incredible reliability. As any scientific theory, Quantum Mechanics is valid within the limits in which it has been experimentally verified, and the Space is the main stage where to validate quantum predictions at large scales, in a domain that is completely different with respect to the microscopic one from which it moved. The technological advances in photonics, which allows the manipulation and the control of the single quanta of light, the photons, make today feasible fundamental tests of Quantum Mechanics in Space, experiments to investigate, for example, if entanglement is preserved along thousands of kilometers or if the wave-particle duality survives even after a Space trip. Furthermore, Space makes available relativistic regimes, in which the velocities and the distances could allow to experimentally investigate the unresolved puzzle of modern physics, that is, the interplay between Quantum Mechanics and gravitation. For these reasons, this thesis is dedicated to the Quantum Optics experiment in Space I have been involved during my PhD

Interference at the Single Photon Level Along Satellite-Ground Channels

Quantum interference arising from superposition of states is a striking evidence of the validity of Quantum Mechanics, confirmed in many experiments and also exploited in applications. However, as for any scientific theory, Quantum Mechanics is valid within the limits in which it has been experimentally verified. In order to extend such limits, it is necessary to observe quantum interference in unexplored conditions such as moving terminals at large distance in Space. Here we experimentally demonstrate single photon interference at a ground station due to the coherent superposition of two temporal modes reflected by a rapidly moving satellite thousand kilometers away. The relative speed of the satellite induces a varying modulation in the interference pattern. The measurement of the satellite distance in real time by laser ranging allowed us to precisely predict the instantaneous value of the interference phase. We then observed the interference patterns with visibility up to $67\%$ with three different satellites and with path length up to 5000 km. Our results attest the viability of photon temporal modes for fundamental tests of Physics and Quantum Communications in Space.Comment: Version accepted for publication in Phys. Rev. Let

Proposal for an Optical Test of the Einstein Equivalence Principle

The Einstein Equivalence Principle (EEP) underpins all metric theories of gravity. Its key element is the local position invariance of non-gravitational experiments, which entails the gravitational red-shift. Precision measurements of the gravitational red-shift tightly bound violations of the EEP only in the fermionic sector of the Standard Model, however recent developments of satellite optical technologies allow for its investigation in the electromagnetic sector. Proposals exploiting light interferometry traditionally suffer from the first-order Doppler effect, which dominates the weak gravitational signal necessary to test the EEP, making them unfeasible. Here, we propose a novel scheme to test the EEP, which is based on a double large-distance optical interferometric measurement. By manipulating the phase-shifts detected at two locations at different gravitational potentials it is possible to cancel-out the first-order Doppler effect and observe the gravitational red-shift implied by the EEP. We present the detailed analysis of the proposal within the post-Newtonian framework and the simulations of the expected signals obtained by using two realistic satellite orbits. Our proposal to overcome the first-order Doppler effect in optical EEP tests is feasible with current technology.Comment: manuscript improve

Large-scale optical interferometry in general spacetimes

We introduce a convenient formalism to evaluate the frequency-shift affecting a light signal propagating on a general curved background. Our formulation, which is based on the laws of geometric optics in a general relativistic setting, allows to obtain a transparent generalization of the Doppler frequency-shift without requiring to perform Local Lorentz transformations. It is easily applicable to stationary spacetimes, and in particular to the near-Earth experiments where geometry is described in the parametrized post-Newtonian approximation. We apply our recipe to evaluate the phase-shift arising in large-scale optical interferometric experiments, as the optical version of the Colella-Overhauser-Werner experiment.Comment: 8 pages, 3 figure

Comparative analysis of plant genomes allows the definition of the "Phytolongins": a novel non-SNARE longin domain protein family

<p>Abstract</p> <p>Background</p> <p>Subcellular trafficking is a hallmark of eukaryotic cells. Because of their pivotal role in the process, a great deal of attention has been paid to the SNARE proteins. Most R-SNAREs, or "longins", however, also possess a highly conserved, N-terminal fold. This "longin domain" is known to play multiple roles in regulating SNARE activity and targeting via interaction with other trafficking proteins. However, the diversity and complement of longins in eukaryotes is poorly understood.</p> <p>Results</p> <p>Our comparative genome survey identified a novel family of longin-related proteins, dubbed the "Phytolongins" because they are specific to land plants. Phytolongins share with longins the N-terminal longin domain and the C-terminal transmembrane domain; however, in the central region, the SNARE motif is replaced by a novel region. Phylogenetic analysis pinpoints the Phytolongins as a derivative of the plant specific VAMP72 longin sub-family and allows elucidation of Phytolongin evolution.</p> <p>Conclusion</p> <p>"Longins" have been defined as R-SNAREs composed of both a longin domain and a SNARE motif. However, expressed gene isoforms and splice variants of longins are examples of non-SNARE motif containing longins. The discovery of Phytolongins, a family of non-SNARE longin domain proteins, together with recent evidence on the conservation of the longin-like fold in proteins involved in both vesicle fusion (e.g. the Trs20 tether) and vesicle formation (e.g. σ and μ adaptin) highlight the importance of the longin-like domain in protein trafficking and suggest that it was one of the primordial building blocks of the eukaryotic membrane-trafficking machinery.</p

Towards Quantum Communication from Global Navigation Satellite System

Satellite-based quantum communication is an invaluable resource for the realization of a quantum network at the global scale. In this regard, the use of satellites well beyond the low Earth orbits gives the advantage of long communication time with a ground station. However, high-orbit satellites pose a great technological challenge due to the high diffraction losses of the optical channel, and the experimental investigation of such quantum channels is still lacking. Here, we report on the first experimental exchange of single photons from Global Navigation Satellite System at a slant distance of 20000 kilometers, by exploiting the retroreflector array mounted on GLONASS satellites. We also observed the predicted temporal spread of the reflected pulses due to the geometrical shape of array. Finally, we estimated the requirements needed for an active source on a satellite, aiming towards quantum communication from GNSS with state-of-the-art technology.Comment: Revte

Regression of Environmental Noise in LIGO Data

We address the problem of noise regression in the output of gravitational-wave (GW) interferometers, using data from the physical environmental monitors (PEM). The objective of the regression analysis is to predict environmental noise in the gravitational-wave channel from the PEM measurements. One of the most promising regression method is based on the construction of Wiener-Kolmogorov filters. Using this method, the seismic noise cancellation from the LIGO GW channel has already been performed. In the presented approach the Wiener-Kolmogorov method has been extended, incorporating banks of Wiener filters in the time-frequency domain, multi-channel analysis and regulation schemes, which greatly enhance the versatility of the regression analysis. Also we presents the first results on regression of the bi-coherent noise in the LIGO data

Extending Wheeler's delayed-choice experiment to Space

Gedankenexperiments have consistently played a major role in the development of quantum theory. A paradigmatic example is Wheeler's delayed-choice experiment, a wave-particle duality test that cannot be fully understood using only classical concepts. Here, we implement Wheeler's idea along a satellite-ground interferometer which extends for thousands of kilometers in Space. We exploit temporal and polarization degrees of freedom of photons reflected by a fast moving satellite equipped with retro-reflecting mirrors. We observed the complementary wave-like or particle-like behaviors at the ground station by choosing the measurement apparatus while the photons are propagating from the satellite to the ground. Our results confirm quantum mechanical predictions, demonstrating the need of the dual wave-particle interpretation, at this unprecedented scale. Our work paves the way for novel applications of quantum mechanics in Space links involving multiple photon degrees of freedom.Comment: 4 figure

Intermodal quantum key distribution field trial with active switching between fiber and free-space channels

Intermodal quantum key distribution enables the integration of fiber networks and free-space channels, which are both necessary elements for the development of a global quantum network. We present a field trial of an intermodal quantum key distribution system - comprised of two polarization-based transmitters and a single receiver - in which the active channel is alternately switched between a free-space link of 620 m and a 17km-long deployed fiber in the metropolitan area of Padova. The performance of the free-space channel is evaluated against the atmospheric turbulence strength. The field trial lasted for several hours in daylight conditions, attesting the intermodal functionality between fiber and free-space channels. Our switching system represents a cost-effective solution for a trusted quantum key distribution network, reducing the number of necessary devices in different network topologies