The Laser Interferometer Space Antenna (LISA) senses gravitational waves by measuring distance fluctuations between three spacecraft (SCs). These measurements rely on precise tracking of a beat-note phase that is formed on a quadrant photodiode (QPD) at each SC by interference of a local laser with a laser sent from a distant SC. The crucial prerequisite of the phase tracking is a successful acquisition of the beat-note frequency. This paper aims to optimize the carrier-to-noise density ratio (CNR) during this process and to evaluate the resulting probability of detection (PD). The CNR is generally lowest during the beat-note acquisition process, since the pointing accuracy relies on coarse acquisition techniques. Based on analytical models, we examine which combinations of QPD segments for the signal readout yield the highest CNR, i.e., are least susceptible to pointing errors. We find from simulations that the highest CNR is ensured by taking the maximum of a combination of two segments in the vertical and horizontal directions. For pointing errors (3 sigma) of 3.9 mu rad and 4.3 mu rad, this yields an improvement of around 3.7 dB and 5.6 dB in the CNR, respectively, in comparison to a combination of all four segments. In addition, the PD for various configurations of the baselined Fourier-peak detection is analyzed. Here, we find that the PD is most sensitive to the CNR compared to the design parameters of the acquisition scheme, in particular the FFT length. Moreover, it is shown that the aforementioned improvements in the CNR can lead to a significant enhancement of the PD.

The LISA Pathfinder (LPF) mission successfully demonstrated the feasibility of the technology needed for the future space borne gravitational wave observatory LISA. A key subsystem under study was the laser interferometer, which measured the changes in relative distance in between two test masses (TMs). It achieved a sensitivity of 32.0−1.7+2.4 fm/Hz, which was significantly better than the prelaunch tests. This improved performance allowed direct observation of the influence of laser frequency noise in the readout. The differences in optical path lengths between the measurement and reference beams in the individual interferometers of our setup determined the level of this undesired readout noise. Here, we discuss the dedicated experiments performed on LPF to measure these differences with high precision. We reached differences in path length difference between (368±5) μm and (329.6±0.9) μm which are significantly below the required level of 1 mm or 1000 μm. These results are an important contribution to our understanding of the overall sensor performance. Moreover, we observed varying levels of laser frequency noise over the course of the mission. We provide evidence that these do not originate from the laser frequency stabilization scheme which worked as expected. Therefore, this frequency stabilization would be applicable to other missions with similar laser frequency stability requirements.2 MoreReceived 2 October 2023Accepted 9 January 2024DOI:https://doi.org/10.1103/PhysRevD.109.042003Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasGravitational wave detectionMetrologyNoiseOptics & lasersPhysical SystemsLaser applicationsTechniquesData analysisGravitational wave detectorsOptical interferometrySatellite data analysisGravitation, Cosmology & AstrophysicsAtomic, Molecular & Optical

Future space observatories achieve detection of gravitational waves by interferometric measurements of a carrier phase, allowing to determine relative distance changes, in combination with an absolute distance measurement based on the transmission of pseudo-random noise chip sequences. In addition, usage of direct-sequence spread spectrum modulation enables data transmission. Hereafter, we report on the findings of a novel performance evaluation of planned receiver architectures, performing phase and distance readout sequentially, addressing the interplay between both measurements. An analytical model is presented identifying the power spectral density of the chip modulation at frequencies within the measurement bandwidth as the main driver for phase noise. This model, verified by numerical simulations, excludes binary phase-shift keying modulations for missions requiring pico-meter noise levels at the phase readout, while binary offset carrier modulation, where most of the power has been shifted outside the measurement bandwidth, exhibits superior phase measurement performance. Ranging analyses of the delay-locked loop reveal strong distortion of the pulse shape due to the preceding phase tracking introducing ranging bias variations. Numerical simulations show that these variations, however, which originate from data transitions, are compensated by the delay tracking loop, enabling sub-meter ranging accuracy, irrespective of the modulation type.

Systematic errors affecting center-of-gravity (CoG) measurements may occur from coarse sampling of the point-spread-function (PSF) or from signal truncation at the boundaries of the region-of-interest (ROI). For small ROI and PSF widths, these effects are shown to become dominant, but this can be mitigated by introducing novel unbiased estimators that are largely free of systematic error and perform particularly well for low photon numbers. Analytical expressions for the estimator variances, comprising contributions from photon shot noise, random pixel noise, and residual systematic error, are derived and verified by Monte Carlo simulations. The accuracy and computational speed of the unbiased estimators are compared to those of other common estimators, including iteratively weighted CoG, thresholded CoG, iterative least squares fitting, and two-dimensional Gaussian regression. Each estimator is optimized with respect to ROI size and PSF radius and its error compared to the theoretical limit defined by the Cramer Rao lower bound (CRLB). The unbiased estimator with full systematic error correction operating on a small ROI [3×3] emerges as one of the most accurate estimators while requiring significantly less computing effort than alternative algorithms.

The laser interferometer space antenna (LISA) senses gravitational waves by measuring distance fluctuations between three spacecraft (SC). These measurements rely on precise tracking of a beat note phase that is formed on a quadrant-photo-diode (QPD) at each SC by interference of a local laser with a laser sent from a distant SC. The crucial prerequisite of the phase tracking is a successful acquisition of the beat note frequency. This article aims to optimize the carrier-to-noise density ratio (CNR) during this process, and to evaluate the resulting probability of detection (PD). CNR is generally lowest during the beat note acquisition process since pointing accuracy relies on coarse acquisition techniques. Based on analytical models, we examine which combinations of QPD segments for the signal read-out yield the highest CNR, i.e., they are least susceptible to pointing errors. We find from simulations that the highest CNR is ensured by taking the maximum of a combination of two segments in vertical and horizontal direction. For pointing errors (3σ) of 3.9 μrad and 4.3 μrad this yields an improvement of around 3.7 dB and 5.6 dB in CNR, respectively, in comparison to a combination of all four segments. In addition, the PD for various configurations of the baselined Fourier peak detection is analyzed. Here we find that the PD is most sensitive to the CNR compared to the design parameters of the acquisition scheme, in particular the FFT length. Moreover, it is shown that aforementioned improvements in CNR can lead to a significant enhancement of the PD.

In the context of the Laser Interferometer Space Antenna (LISA), the laser subsystems exhibit frequency fluctuations that introduce significant levels of noise into the measurements, surpassing the gravitational wave signal by several orders of magnitude. Mitigation is achieved by means of time-shifting individual measurements in a data processing step known as time-delay interferometry (TDI). The suppression performance of TDI relies on accurate knowledge and consideration of the delays experienced by the interfering lasers. While considerable efforts have been dedicated to the accurate determination of interspacecraft ranging delays, the sources for delays onboard the spacecraft have been either neglected during TDI processing or assumed to be known. Contrary to these assumptions, analog delays of the phasemeter front end and the laser modulation are not only large but also prone to change with temperature and heterodyne frequency. This motivates our proposal for a novel method enabling a calibration of these delays on-ground and in-space, based on minimal functional additions to the receiver architecture. Specifically, we establish a set of calibration measurements and elucidate how these measurements are utilized in data processing, leading to the mitigation of the delays in the TDI Michelson variables. Following a performance analysis of the calibration measurements, the proposed calibration scheme is assessed through numerical simulations. We find that in the absence of the calibration scheme, the assumed drifts of the analog delays increase residual laser noise at high frequencies of the LISA measurement band. A single, on-ground calibration of the analog delays leads to an improvement by roughly one order of magnitude, while recalibration in space may improve performance by yet another order of magnitude. Towards lower frequencies, ranging error is always found to be the limiting factor for which countermeasures are discussed.

Time-delay interferometry (TDI) is a data processing technique for space-based gravitational-wave detectors to create laser-noise-free equal-optical-path-length interferometers virtually on the ground. It relies on the interspacecraft signal propagation delays, which are delivered by intersatellite ranging monitors. Also delays due to onboard signal propagation and processing have a nonnegligible impact on the TDI combinations. However, these onboard delays were only partially considered in previous TDI-related research; onboard optical path lengths have been neglected so far. In this paper, we study onboard optical path lengths in TDI. We derive analytical models for their coupling to the second-generation TDI Michelson combinations and verify these models numerically. Furthermore, we derive a compensation scheme for onboard optical path lengths in TDI and validate its performance via numerical simulations.

The detection of gravitational waves requires a strain sensitivity at unprecedented precision. The planned space observatory LISA overcomes this extreme challenge by heterodyne laser interferometry at picometer-precision based on the exploitation of carrier phase measurements between spacecraft separated by millions of kilometers. In addition, data transmission and absolute ranging, necessary to mitigate effects of laser frequency fluctuations in post-processing, are achieved with direct-sequence spread spectrum signals. The foreseen receivers shall typically operate in a sequential phase-locked loop and delay-locked loop configuration for consecutive phase and distance measurement. Recent analysis observed code tracking delay variations, identified as ranging bias variations, as a result of this sequential arrangement. Hereafter, we present an analytical analysis of these ranging bias variations. Comparisons to numerical simulations reveal the compelling influence of the cross-correlation of the chip sequences on the ranging bias variations for a fixed modulation scheme and thus affirm the necessity of numerical analysis. In addition, a generic model for the quantisation error of a digital delay-locked loop is introduced that may be used for analysis and design of digital code tracking loops in various applications. Finally, comparison to a numerical simulation reveals that at small ranging bias variations, the code tracking error is fully described by the quantisation error, while at high ranging bias variations, this effect is negligible and the code tracking error is dominated by ranging bias variations.

The Laser Interferometer Space Antenna (LISA) mission is a space-borne observatory designed to detect and characterize gravitational wave sources inaccessible to ground-based detectors. The mission relies on laser interferometry to measure changes in space-time. In this context, non-avoidable noise sources within the LISA system, including tilt-to-length (TTL) coupling, reduce the detector’s resolution and complicate achieving the mission’s goals unless appropriate mitigation strategies are implemented. This paper applies time-delay interferometry infinity (TDI- ∞ ) for TTL noise estimation in LISA and assesses its suitability from the perspective of system identification and calibration for the first time. The recently published TDI- ∞ concept indicates a different frequency response within the LISA measurement band compared to the standard methodology of TDI second-generation. Our main contribution is demonstrating the advantages of the TDI- ∞ algorithm for TTL noise calibration in space-borne interferometer constellations. Specifically, we show that this algorithm improves the estimation performance of TTL noise due to its frequency behavior while requiring less computation time. The reduction in computational effort afforded by the TDI- ∞ algorithm could accelerate the availability of calibrated TDI data for astrophysical analysis if required by the LISA science community. The improvement in estimation performance underscores the concept’s potential to enhance the detector’s sensitivity further.

We present a comprehensive simulation of the spatial acquisition of optical links for the LISA mission in the in-field pointing architecture, where a fast pointing mirror is used to move the field-of-view of the optical transceiver, which was studied as an alternative scheme to the baselined telescope pointing architecture. The simulation includes a representative model of the far-field intensity distribution and the beam detection process using a realistic detector model, and a model of the expected platform jitter for two alternative control modes with different associated jitter spectra. For optimally adjusted detector settings and accounting for the actual far-field beam profile, we investigate the dependency of acquisition performance on the jitter spectrum and the track-width of the search spiral, while scan speed and detector integration time are varied over several orders of magnitude. Results show a strong dependency of the probability for acquisition failure on the width of the auto-correlation function of the jitter spectrum, which we compare to predictions of analytical models. Depending on the choice of scan speed, three different regimes may be entered which differ in failure probability by several orders of magnitude. We then use these results to optimize the acquisition architecture for the given jitter spectra with respect to failure rate and overall duration, concluding that the full constellation could be acquired on average in less than one minute. Our method and findings can be applied to any other space mission using a fine-steering mirror for link acquisition.

The objective of the proposed macroscopic quantum resonators (MAQRO) mission is to harness space for achieving long free-fall times, extreme vacuum, nano-gravity, and cryogenic temperatures to test the foundations of physics in macroscopic quantum experiments at the interface with gravity. Developing the necessary technologies, achieving the required sensitivities and providing the necessary isolation of macroscopic quantum systems from their environment will lay the path for developing novel quantum sensors. Earlier studies showed that the proposal is feasible but that several critical challenges remain, and key technologies need to be developed. Recent scientific and technological developments since the original proposal of MAQRO promise the potential for achieving additional science objectives. The proposed research campaign aims to advance the state of the art and to perform the first macroscopic quantum experiments in space. Experiments on the ground, in micro-gravity, and in space will drive the proposed research campaign during the current decade to enable the implementation of MAQRO within the subsequent decade.

An analytical model is derived for the probability of failure (P-fail) to spatially acquire an optical link with a jittering search beam. The analytical model accounts for an arbitrary jitter spectrum and considers the associated correlations between jitter excursions on adjacent tracks of the search spiral. An expression of P-fail in terms of basic transcendental functions is found by linearizing the exact analytical model with respect to the correlation strength. Predictions from the models indicate a strong decrease of P-fail with increasing correlation strength, which is found to be in excellent agreement with results from Monte Carlo simulations. The dependency of P-fail on the track width and scan speed is investigated, confirming previous assumptions about the impact of correlations. Expressions and applicable constraints are derived for the limits of full and no correlations, and the optimal track width to minimize the acquisition time is computed for a range of scan speeds. The model is applicable to optical terminals equipped with a fast beam steering mirror, as often found for optical communications missions in space.

The Laser Interferometer Space Antenna (LISA) mission features a three-spacecraft long-arm constellation intended to detect gravitational wave sources in the low-frequency band up to 1 Hz via laser interferometry. The paper presents an open-loop control strategy for point-ahead angle (PAA) correction required to maintain the optical links of the moving constellation. The control strategy maximizes periods between adjustments at the constellation level and is shown to be optimal from the perspective of estimating and correcting tilt-to-length (TTL) coupling. TTL is a noise source that couples angular spacecraft jitter and jitter of optical subassemblies with longitudinal interferometer measurements. Without precise TTL noise estimation and correction, TTL coupling fundamentally limits the detector's sensitivity.

The objective of the proposed MAQRO mission is to harness space for achieving long free-fall times, extreme vacuum, nano-gravity, and cryogenic temperatures to test the foundations of physics in macroscopic quantum experiments. This will result in the development of novel quantum sensors and a means to probe the foundations of quantum physics at the interface with gravity. Earlier studies showed that the proposal is feasible but that several critical challenges remain, and key technologies need to be developed. These new technologies will open up the potential for achieving additional science objectives. The proposed research campaign aims to advance the state of the art and to perform the first macroscopic quantum experiments in space. Experiments on the ground, in micro-gravity, and in space will drive the proposed research campaign during the current decade to enable the implementation of MAQRO within the subsequent decade.

We investigate how the probability of acquiring an optical link between a scanning and a target spacecraft depends on the spectral shape, power, and dimensionality of the beam jitter, as well as on the choice of detector integration time, beam detection radius, and scan speed. For slow scans and long integration times, the probability of failure (Pfail) is determined by the integrated jitter power up to a critical frequency, which we verify by comparing the results of an analytical model to those of Monte Carlo simulations. Jitter above the critical frequency leads to a loss of correlation between integration windows and decreases Pfail for both 1D (radial) and 2D (radial and tangential) jitter, as long as the RMS jitter amplitude does not exceed the beam diameter. In the opposite limit of fast scans and short integration times, emergent correlations between jitter fluctuations on two adjacent scanning tracks also decrease Pfail. The analytical model is additionally used to assess the effect of multiple overlapping tracks and the impact of target drifts in the uncertainty plane.

The Laser Interferometer Space Antenna (LISA) mission aims to observe gravitational waves featuring a three-spacecraft interferometer constellation with an arm length of 2.5 Mio km. In support of maximizing LISA's scientific return, the paper presents the design of coordinated spacecraft constellation maneuvers tailored for high-precision estimation of tilt-to-length (TTL) coupling noise under the influence of gravitational wave events. Tilt-to-length coupling couples the angular jitter of the three LISA spacecraft and their optomcchanical assemblies to longitudinal measurement noise of the interferometers. The effect is a fundamental limitation of LISA's sensitivity and spaceborne long-arm interferometers, in general. The limitation can be avoided by estimating tilt-to-length coupling noise and correcting the scientific measurements. Former research allows limited conclusions to be drawn about the feasibility of accurate tilt-to-length coupling estimation because only the effect of system-internal noise sources has been addressed. Since tilt-to-length coupling noise in LISA is estimated and corrected based on science instrument measurements, the calibration shall be robust against the disturbing impact of gravitational wave events. The paper quantifies the deterioration of tilt-to-length parameter observabilities in the view of stellar and galactic events without maneuver execution. Then the developed maneuver design method is shown to provide accurate estimation results in the more realistic scenario. Although the approach temporarily reduces the number of independent scientific signals, the scientific operation mode of the spacecraft constellation can be maintained.

This erratum corrects a typing error in Appl. Opt.60, 3936 (2021)APOPAI0003-693510.1364/AO.419594. The correction does not affect the results and conclusions of the original paper.

This paper aims to compare various aspects of systems engineering between space and ground-based astronomy projects. After a brief roundup of the development of systems engineering practices in space, we discuss the rapidly progressing adoption of similar methods in complex ground projects. Special attention is given to the analysis of increasing system complexity on ground which leads to a commensurate increase in project effort and cost. The importance of development of enabling technologies and improvement of engineering methodologies are discussed by specific examples.

Do the laws of quantum physics still hold for macroscopic objects - this is at the heart of Schrödinger’s cat paradox - or do gravitation or yet unknown effects set a limit for massive particles? What is the fundamental relation between quantum physics and gravity? Ground-based experiments addressing these questions may soon face limitations due to limited free-fall times and the quality of vacuum and microgravity. The proposed mission Macroscopic Quantum Resonators (MAQRO) may overcome these limitations and allow addressing such fundamental questions. MAQRO harnesses recent developments in quantum optomechanics, high-mass matter-wave interferometry as well as state-of-the-art space technology to push macroscopic quantum experiments towards their ultimate performance limits and to open new horizons for applying quantum technology in space. The main scientific goal is to probe the vastly unexplored ‘quantum-classical’ transition for increasingly massive objects, testing the predictions of quantum theory for objects in a size and mass regime unachievable in ground-based experiments. The hardware will largely be based on available space technology. Here, we present the MAQRO proposal submitted in response to the 4th Cosmic Vision call for a medium-sized mission (M4) in 2014 of the European Space Agency (ESA) with a possible launch in 2025, and we review the progress with respect to the original MAQRO proposal for the 3rd Cosmic Vision call for a medium-sized mission (M3) in 2010. In particular, the updated proposal overcomes several critical issues of the original proposal by relying on established experimental techniques from high-mass matter-wave interferometry and by introducing novel ideas for particle loading and manipulation. Moreover, the mission design was improved to better fulfill the stringent environmental requirements for macroscopic quantum experiments.

Passive cooling of scientific instruments via thermal radiation to deep space offers many advantages over active cooling in terms of mission cost, lifetime and the achievable quality of vacuum and microgravity. Motivated by the mission proposal MAQRO to test, the foundations of quantum physics harnessing a deep-space environment, we investigate the performance of a radiatively cooled instrument, where the environment of a test particle in a quantum superposition has to be cooled to less than 20 K. We perform a heat-transfer analysis between the instrument components and a transfer-function analysis on thermal oscillations induced by the spacecraft interior and dissipative sources. The thermal behavior of the instrument is discussed for an orbit around a Lagrangian point and for a highly elliptical Earth orbit. Finally, we investigate possible design improvements. These include a mirror-based design of the imaging system on the optical bench (OB) and an extension of the heat shields. (C) 2016 The Authors. Published by Elsevier Ltd.