Luciano Di Fiore

FRINGE-COUNTING TECHNIQUE USED TO LOCK A SUSPENDED INTERFEROMETER

We implement a digital fringe-counting technique to measure in real time the relative mirror displacement of a suspended Michelson interferometer with modulated optical path length for oscillations much larger than the laser wavelength (λ). This provides the proper error signal for a servo mechanism that reduces the relative displacement within λ/2. The implemented technique does not require extra optics or polarizers and thus can be used for interferometric gravitational wave detectors as a starting procedure to get the system locked.

Real-time digital control of optical interferometers by the mechanical-modulation technique

We discuss the application of digital systems to the automatic control of dual-wave optical interferometers. We show that, if the mechanical-modulation technique is used for error-signal extraction, digital techniques can be used both for error-signal extraction and for control-signal generation. Therefore, apart from two front/end amplifiers that are necessary to match the dynamics of the detectors and actuators to the dynamics of the analog-to-digital converters and digital-to-analog converters, no other analog devices are required. In particular, the mechanical-modulation technique requires the synchronous demodulation of the photodiode output signal. Hence we need to implement a digital lock-in amplifier whose algorithm is described here. Finally, we describe one of the possible applications of this digital control procedure, such as the control of a classic Mach-Zehnder interferometer in air.

Digitally controlled interferometer prototype for gravitational wave detection

In this article, we describe the architecture of the 3 m suspended Michelson interferometer prototype for gravitational wave detection which is operational in Napoli. The characteristic which makes this interferometer different from the existing ones is the digital implementation of the control system, the monitoring system, the data acquisition system, and the archiving system. This architecture makes this interferometer a good test bench for the study, the development, and the test of general techniques for the automatic control of interferometers for gravitational wave detection. In particular, it is now being used for the development and the test of some subsystems of the very long baseline interferometric VIRGO antenna for gravitational wave detection. [The Virgo Project, Final Design of the Italian–French large base interferometric antenna Virgo for gravitational wave detection of which the authors are proponents and in whose construction the Authors are collaborating (INFN, Italy, and CNRS, France,...

Earth-Moon Lagrangian points as a testbed for general relativity and effective field theories of gravity

We first analyse the restricted four-body problem consisting of the Earth, the Moon and the Sun as the primaries and a spacecraft as the planetoid. This scheme allows us to take into account the solar perturbation in the description of the motion of a spacecraft in the vicinity of the stable Earth-Moon libration points L4 and L5 both in the classical regime and in the context of effective field theories of gravity. A vehicle initially placed at L4 or L5 will not remain near the respective points. In particular, in the classical case the vehicle moves on a trajectory about the libration points for at least 700 days before escaping away. We show that this is true also if the modified long-distance Newtonian potential of effective gravity is employed. We also evaluate the impulse required to cancel out the perturbing force due to the Sun in order to force the spacecraft to stay precisely at L4 or L5. It turns out that this value is slightly modified with respect to the corresponding Newtonian one. In the second part of the paper, we first evaluate the location of all Lagrangian points in the Earth-Moon system within the framework of general relativity. For the points L4 and L5, the corrections of coordinates are of order a few millimeters and describe a tiny departure from the equilateral triangle. After that, we set up a scheme where the theory which is quantum corrected has as its classical counterpart the Einstein theory, instead of the Newtonian one. In other words, we deal with a theory involving quantum corrections to Einstein gravity, rather than to Newtonian gravity. By virtue of the effective-gravity correction to the long distance form of the potential among two point masses, all terms involving the ratio between the gravitational radius of the primary and its separation from the planetoid get modified. Within this framework, for the Lagrangian points of stable equilibrium, we find quantum corrections of order two millimeters, whereas for Lagrangian points of unstable equilibrium we find quantum corrections below a millimeter. In the latter case, for the point L1, general relativity corrects Newtonian theory by 7.61 meters, comparable, as an order of magnitude, with the lunar geodesic precession of about 3 meters per orbit. The latter is a cumulative effect accurately measured at the centimeter level through the lunar laser ranging positioning technique. Thus, it is possible to study a new laser ranging test of general relativity to measure the 7.61-meter correction to the L1 Lagrangian point, an observable never used before in the Sun-Earth-Moon system. Performing such an experiment requires controlling the propulsion to precisely reach L1, an instrumental accuracy comparable to the measurement of the lunar geodesic precession, understanding systematic effects resulting from thermal radiation and multi-body gravitational perturbations. This will then be the basis to consider a second-generation experiment to study deviations of effective field theories of gravity from general relativity in the Sun-Earth-Moon system.

On the photon Green functions in curved spacetime

Quantization of electrodynamics in curved spacetime in the Lorentz gauge and with arbitrary gauge parameter makes it necessary to study Green functions of non-minimal operators with variable coefficients. Starting from the integral representation of photon Green functions, we link them to the evaluation of integrals involving Γ-functions. Eventually, the full asymptotic expansion of the Feynman photon Green function at small values of the world function, as well as its explicit dependence on the gauge parameter, are obtained without adding by hand a mass term to the Faddeev–Popov Lagrangian. Coincidence limits of the second covariant derivatives of the associated Hadamard function are also evaluated, as a first step towards the energy–momentum tensor in the non-minimal case.

High accuracy digital temperature control for a laser diode

A digital servo‐loop was implemented to control the temperature of a laser diode. By using a conditionally stable loop we obtained a temperature stability of about ±20 μK over periods of hours.

“Quasi-complete” mechanical model for a double torsion pendulum

We present a dynamical model for the double torsion pendulum nicknamed PETER, where one torsion pendulum hangs in cascade, but off-axis, from the other. The dynamics of interest in these devices lies around the torsional resonance, that is at very low frequencies (mHz). However, we find that, in order to properly describe the forced motion of the pendulums, also other modes must be considered, namely swinging and bouncing oscillations of the two suspended masses, that resonate at higher frequencies (Hz). Although the system has obviously 6+6 Degrees of Freedom, we find that 8 are sufficient for an accurate description of the observed motion. This model produces reliable estimates of the response to generic external disturbances and actuating forces or torques. In particular, we compute the effect of seismic floor motion (tilt noise) on the low frequency part of the signal spectra and show that it properly accounts for most of the measured low frequency noise.

Feasibility of a magnetic suspension for second generation gravitational wave interferometers

Abstract This paper deals with the use of a magnetic levitation system as a part of a multi-stage seismic attenuator for gravitational wave interferometric antennas. The proposed configuration uses permanent magnets in attraction to balance the suspended weight, plus a closed loop position control to obtain a stable levitation. The system is analyzed using a MATLAB simulation code to compute the forces exerted by extended magnets. The validity of this model has been tested by a comparison with the experimental data from a levitated suspension prototype.

On solar system dynamics in general relativity

Recent work in the literature has advocated using the Earth–Moon–planetoid Lagrangian points as observables, in order to test general relativity and effective field theories of gravity in the solar system. However, since the three-body problem of classical celestial mechanics is just an approximation of a much more complicated setting, where all celestial bodies in the solar system are subject to their mutual gravitational interactions, while solar radiation pressure and other sources of nongravitational perturbations also affect the dynamics, it is conceptually desirable to improve the current understanding of solar system dynamics in general relativity, as a first step towards a more accurate theoretical study of orbital motion in the weak-gravity regime. For this purpose, starting from the Einstein equations in the de Donder–Lanczos gauge, this paper arrives first at the Levi-Civita Lagrangian for the geodesic motion of planets, showing in detail under which conditions the effects of internal structure and finite extension get canceled in general relativity to first post-Newtonian order. The resulting nonlinear ordinary differential equations for the motion of planets and satellites are solved for the Earth’s orbit about the Sun, written down in detail for the Sun–Earth–Moon system, and investigated for the case of planar motion of a body immersed in the gravitational field produced by the other bodies (e.g. planets with their satellites). At this stage, we prove an exact property, according to which the fourth-order time derivative of the original system leads to a linear system of ordinary differential equations. This opens an interesting perspective on forthcoming research on planetary motions in general relativity within the solar system, although the resulting equations remain a challenge for numerical and qualitative studies. Last, the evaluation of quantum corrections to location of collinear and noncollinear Lagrangian points for the planar restricted three-body problem is revisited, and a new set of theoretical values of such corrections for the Earth–Moon–planetoid system is displayed and discussed. On the side of classical values, the general relativity corrections to Newtonian values for collinear and noncollinear Lagrangian points of the Sun–Earth–planetoid system are also obtained. A direction for future research will be the analysis of planetary motions within the relativistic celestial mechanics set up by Blanchet, Damour, Soffel and Xu.

Quantum time delay in the gravitational field of a rotating mass

We examine quantum corrections of time delay arising in the gravitational field of a spinning oblate source. Low-energy quantum effects occurring in Kerr geometry are derived within a framework where general relativity is fully seen as an effective field theory. By employing such a pattern, gravitational radiative modifications of Kerr metric are derived from the energy-momentum tensor of the source, which at lowest order in the fields is modelled as a point mass. Therefore, in order to describe a quantum corrected version of time delay in the case in which the source body has a finite extension, we introduce a hybrid scheme where quantum fluctuations affect only the monopole term occurring in the multipole expansion of the Newtonian potential. The predicted quantum deviation from the corresponding classical value turns out to be too small to be detected in the next future, showing that new models should be examined in order to test low-energy quantum gravity within the solar system.