arXiv subjects

Enrico Serra

Publications and source records attributed to Enrico Serra.

At least 37 records · Page 2Linked to original sources

arXiv2018

Variational and stability properties of constant solutions to the NLS equation on compact metric graphs

We consider the nonlinear Schrödinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the graph is compact, for every value of the mass there is a constant solution. Our scope is to analyze (in dependence of the mass) the variational properties of this solution, as a critical point of the energy functional: local and global minimality, and (orbital) stability. We consider both the subcritical regime and the critical one, in which the features of the graph become relevant. We describe how the above properties change according to the topology and the metric properties of the graph.

math.AP↗

arXiv2018

Silicon Nitride MOMS Oscillator for Room Temperature Quantum Optomechanics

Optomechanical SiN nano-oscillators in high-finesse Fabry-Perot cavities can be used to investigate the interaction between mechanical and optical degree of freedom for ultra-sensitive metrology and fundamental quantum mechanical studies. In this work we present a nano-oscillator made of a high-stress round-shaped SiN membrane with an integrated on-chip 3D seismic filter properly designed to reduce mechanical losses. This oscillator works in the 200 kHz - 5 MHz range and features a mechanical quality factor of $Q\simeq10^7$ and a Q-frequency product in excess of $6.2 \times 10^{12}$ Hz at room temperature, fulfilling the minimum requirement for quantum ground-state cooling of the oscillator in an optomechanical cavity. The device is obtained by MEMS DRIE bulk micromachining with a two-side silicon processing on a Silicon-On-Insulator (SOI) wafer. The microfabrication process is quite flexible and additional layers could be deposited over the SiN membrane before the DRIE steps, if required for a sensing application. Therefore, such oscillator is a promising candidate for quantum sensing applications in the context of the emerging field of Quantum Technologies.

physics.app-ph↗

arXiv2018

Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs

We investigate the existence of ground states for the focusing nonlinear Schroedinger equation on a prototypical doubly periodic metric graph. When the nonlinearity power is below 4, ground states exist for every value of the mass, while, for every nonlinearity power between 4 (included) and 6 (excluded), a mark of $L^2$-criticality arises, as ground states exist if and only if the mass exceeds a threshold value that depends on the power. This phenomenon can be interpreted as a continuous transition from a two-dimensional regime, for which the only critical power is 4, to a one-dimensional behavior, in which criticality corresponds to the power 6. We show that such a dimensional crossover is rooted in the coexistence of one-dimensional and two-dimensional Sobolev inequalities, leading to a new family of Gagliardo-Nirenberg inequalities that account for this continuum of critical exponents.

math.AP↗

arXiv2018

Normal--mode splitting in a weakly coupled optomechanical system

Normal--mode splitting is the most evident signature of strong coupling between two interacting subsystems. It occurs when two subsystems exchange energy between themselves faster than they dissipate it to the environment. Here we experimentally show that a weakly coupled optomechanical system at room temperature can manifest normal--mode splitting when the pump field fluctuations are anti-squashed by a phase-sensitive feedback loop operating close to its instability threshold. Under these conditions the optical cavity exhibits an effectively reduced decay rate, so that the system is effectively promoted to the strong coupling regime.

quant-ph↗

arXiv2017

Enhancing sideband cooling by feedback--controlled light

We realise a phase-sensitive closed-loop control scheme to engineer the fluctuations of the pump field which drives an optomechanical system, and show that the corresponding cooling dynamics can be significantly improved. In particular, operating in the counter-intuitive "anti-squashing" regime of positive feedback and increased field fluctuations, sideband cooling of a nanomechanical membrane within an optical cavity can be improved by 7.5~dB with respect to the case without feedback. Close to the quantum regime of reduced thermal noise, such feedback-controlled light would allow going well below the quantum backaction cooling limit.

quant-ph↗

arXiv2017

Multiple positive bound states for the subcritical NLS equation on metric graphs

We consider the Schroedinger equation with a subcritical focusing power nonlinearity on a noncompact metric graph, and prove that for every finite edge there exists a threshold value of the mass, beyond which there exists a positive bound state achieving its maximum on that edge only. This bound state is characterized as a minimizer of the energy functional associated to the NLS equation, with an additional constraint (besides the mass prescription): this requires particular care in proving that the minimizer satisfies the Euler--Lagrange equation. As a consequence, for a sufficiently large mass every finite edge of the graph hosts at least one positive bound state that, owing to its minimality property, is orbitally stable.

math.AP↗

arXiv2017

Enhancement of three-mode optomechanical interaction by feedback-controlled light

We realise a feedback-controlled optical Fabry-Perot cavity in which the transmitted cavity output is used to modulate the input amplitude fluctuations. The resulting phase-dependent fluctuations of the in-loop optical field, which may be either sub-shot- or super-shot-noise, can be engineered to favorably affect the optomechanical interaction with a nanomechanical membrane placed within the cavity. Here we show that in the super-shot-noise regime ("anti-squashed light") the in-loop field has a strongly reduced effective cavity linewidth, corresponding to an increased optomechanical cooperativity. In this regime feedback improves the simultaneous resolved sideband cooling of two nearly degenerate membrane mechanical modes by one order of magnitude.

quant-ph↗

arXiv2017

Nonlinear dynamics on branched structures and networks

Nonlinear dynamics on graphs has rapidly become a topical issue with many physical applications, ranging from nonlinear optics to Bose-Einstein condensation. Whenever in a physical experiment a ramified structure is involved, it can prove useful to approximate such a structure by a metric graph, or network. For the Schroedinger equation it turns out that the sixth power in the nonlinear term of the energy is critical in the sense that below that power the constrained energy is lower bounded irrespectively of the value of the mass (subcritical case). On the other hand, if the nonlinearity power equals six, then the lower boundedness depends on the value of the mass: below a critical mass, the constrained energy is lower bounded, beyond it, it is not. For powers larger than six the constrained energy functional is never lower bounded, so that it is meaningless to speak about ground states (supercritical case). These results are the same as in the case of the nonlinear Schrodinger equation on the real line. In fact, as regards the existence of ground states, the results for systems on graphs differ, in general, from the ones for systems on the line even in the subcritical case: in the latter case, whenever the constrained energy is lower bounded there always exist ground states (the solitons, whose shape is explicitly known), whereas for graphs the existence of a ground state is not guaranteed. For the critical case, our results show a phenomenology much richer than the analogous on the line.

math.AP↗

arXiv2016

On the lack of bound states for certain NLS equations on metric graphs

The purpose of this paper is to prove some results on the absence of bound states for certain nonlinear Schrödinger equations on noncompact metric graphs with localized nonlinearity. In particular, we show how the topological and metric properties of graphs affect the existence/nonexistence of bound states. This work completes the discussion initiated in [17, 18].

math.AP↗

arXiv2016

Negative energy ground states for the $L^2$-critical NLSE on metric graphs

We investigate the existence of ground states with prescribed mass for the focusing nonlinear Schrödinger equation with $L^2$-critical power nonlinearity on noncompact quantum graphs. We prove that, unlike the case of the real line, for certain classes of graphs there exist ground states with negative energy for a whole interval of masses. A key role is played by a thorough analysis of Gagliardo-Nirenberg inequalities and on estimates of the optimal constants. Most of the techniques are new and suited to the investigation of variational problems on metric graphs.

math-ph↗

arXiv2015

Bound states of the NLS equation on metric graphs with localized nonlinearities

We investigate the existence of multiple bound states of prescribed mass for the nonlinear Schrödinger equation on a noncompact metric graph. The main feature is that the nonlinearity is localized only in a compact part of the graph. Our main result states that for every integer k, the equation possesses at least k solutions of prescribed mass, provided that the mass is large enough. These solutions arise as constrained critical points of the NLS energy functional. Estimates for the energy of the solutions are also established.

math.AP↗

arXiv2015

Probing deformed commutators with macroscopic harmonic oscillators

A minimal observable length is a common feature of theories that aim to merge quantum physics and gravity. Quantum mechanically, this concept is associated to a nonzero minimal uncertainty in position measurements, which is encoded in deformed commutation relations. In spite of increasing theoretical interest, the subject suffers from the complete lack of dedicated experiments and bounds to the deformation parameters are roughly extrapolated from indirect measurements. As recently proposed, low-energy mechanical oscillators could allow to reveal the effect of a modified commutator. Here we analyze the free evolution of high quality factor micro- and nano-oscillators, spanning a wide range of masses around the Planck mass $m_{\mathrm{P}}$ (${\approx 22\,μ\mathrm{g}}$), and compare it with a model of deformed dynamics. Previous limits to the parameters quantifying the commutator deformation are substantially lowered.

gr-qc↗

arXiv2015

Threshold phenomena and existence results for NLS ground states on graphs

We investigate the existence of ground states of prescribed mass, for the nonlinear Schroedinger energy on a noncompact metric graph G. While in some cases the topology of G may rule out or, on the contrary, guarantee the existence of ground states of any given mass, in general also some metric properties of G, and their quantitative relation with the actual value of the prescribed mass, are relevant as concerns the existence of ground states. This may give rise to interesting phase transitions from nonexistence to existence of ground states, when a certain quantity reaches a critical threshold.

math.AP↗

arXiv2014

NLS ground states on graphs

We investigate the existence of ground states for the subcritical NLS energy on metric graphs. In particular, we find out a topological assumption that guarantees the nonexistence of ground states, and give an example in which the assumption is not fulfilled and ground states actually exist. In order to obtain the result, we introduce a new rearrangement technique, adapted to the graph where it applies. Owing to such a technique, the energy level of the rearranged function is improved by conveniently mixing the symmetric and monotone rearrangement procedures.

math.AP↗

arXiv2014

Lack of ground state for NLS on bridge-type graphs

We prove the nonexistence of ground states for NLS on bridge-like graphs, i.e. graphs with two halflines and four vertices, of which two at infinity, with Kirchhoff matching conditions. By ground state we mean any minimizer of the energy functional among all functions with the same mass.

math.AP↗

arXiv2013

A minimization approach to hyperbolic Cauchy problems

Developing an original idea of De Giorgi, we introduce a new and purely variational approach to the Cauchy Problem for a wide class of defocusing hyperbolic equations. The main novel feature is that the solutions are obtained as limits of functions that minimize suitable functionals in space-time (where the initial data of the Cauchy Problem serve as prescribed boundary conditions). This opens up the way to new connections between the hyperbolic world and that of the Calculus of Variations. Also dissipative equations can be treated. Finally, we discuss several examples of equations that fit in this framework, including nonlocal equations, in particular equations with the fractional Laplacian.

math.AP↗

arXiv2012

Phonon momentum and damping of mechanical resonators

The concept of physical momentum associated to phonons in a crystal, complemented with some fundamental reasoning, implies measurable effects in crystals even at a macroscopic scale. We show that, in close analogy with the transfer of momentum in the kinetic theory of gases, physical momentum carried by of phonons couples the thermal and the velocity field in a vibrating crystal. Therefore an heat flow applied to a vibrating crystal can sustain or damp the oscillation, depending on the interplay between the temperature and the velocity gradient. We derive the general equations of this effect and show that its experimental confirmation is within reach of current technology.

cond-mat.mes-hall↗

arXiv2007

A note on the radial solutions for the supercritical Henon equation

We prove the existence of a positive radial solution for the Hénon equation with arbitrary growth. The solution is found by means of a shooting method and turns out to be an increasing function of the radial variable. Some numerical experiments suggest the existence of many positive oscillating solutions.

math.AP↗