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Roger Penrose proposed that a spatial quantum superposition collapses as a back-reaction from spacetime, which is curved in different ways by each branch of the superposition. In this sense, one speaks of gravity-related wave function collapse. He also provided a heuristic formula to compute the decay time of the superposition—similar to that suggested earlier by Lajos Diósi, hence the name Diósi–Penrose model. The collapse depends on the effective size of the mass density of particles in the superposition, and is random: this randomness shows up as a diffusion of the particles’ motion, resulting, if charged, in the emission of radiation. Here, we compute the radiation emission rate, which is faint but detectable. We then report the results of a dedicated experiment at the Gran Sasso underground laboratory to measure this radiation emission rate. Our result sets a lower bound on the effective size of the mass density of nuclei, which is about three orders of magnitude larger than previous bounds. This rules out the natural parameter-free version of the Diósi–Penrose model.

We propose in this White Paper a concept for a space experiment using cold atoms to search for ultra-light dark matter, and to detect gravitational waves in the frequency range between the most sensitive ranges of LISA and the terrestrial LIGO/Virgo/KAGRA/INDIGO experiments. This interdisciplinary experiment, called Atomic Experiment for Dark Matter and Gravity Exploration (AEDGE), will also complement other planned searches for dark matter, and exploit synergies with other gravitational wave detectors. We give examples of the extended range of sensitivity to ultra-light dark matter offered by AEDGE, and how its gravitational-wave measurements could explore the assembly of super-massive black holes, first-order phase transitions in the early universe and cosmic strings. AEDGE will be based upon technologies now being developed for terrestrial experiments using cold atoms, and will benefit from the space experience obtained with, e.g., LISA and cold atom experiments in microgravity. KCL-PH-TH/2019-65, CERN-TH-2019-126.

In the context of spontaneous wave function collapse models, we investigate the properties of the continuous spontaneous localization (CSL) collapse rate for rigid bodies in a superposition of two states located at different places. By exploiting the Euler-Maclaurin formula, we show that for standard matter the rate for a continuous mass distribution accurately reproduces the exact rate (i.e., the one for a discrete distribution). We compare the exact rate with previous estimates in the literature and we asses their validity. We find that the reduction rate displays a peculiar mass density difference effect, which we investigate and describe in detail. We show that the recently proposed layering effect is a consequence of the mass density difference effect.

Despite the unquestionable empirical success of quantum theory, witnessed by the recent uprising of quantum technologies, the debate on how to reconcile the theory with the macroscopic classical world is still open. Spontaneous collapse models are one of the few testable solutions so far proposed. In particular, the continuous spontaneous localization (CSL) model has become subject of intense experimental research. Experiments looking for the universal force noise predicted by CSL in ultrasensitive mechanical resonators have recently set the strongest unambiguous bounds on CSL. Further improving these experiments by direct reduction of mechanical noise is technically challenging. Here, we implement a recently proposed alternative strategy that aims at enhancing the CSL noise by exploiting a multilayer test mass attached on a high quality factor microcantilever. The test mass is specifically designed to enhance the effect of CSL noise at the characteristic length rc=10-7 m. The measurements are in good agreement with pure thermal motion for temperatures down to 100 mK. From the absence of excess noise, we infer a new bound on the collapse rate at the characteristic length rc=10-7 m, which improves over previous mechanical experiments by more than 1 order of magnitude. Our results explicitly challenge a well-motivated region of the CSL parameter space proposed by Adler.

The CSL model predicts a progressive breakdown of the quantum superposition principle, with a noise randomly driving the state of the system towards a localized one, thus accounting for the emergence of a classical world within a quantum framework. In the original model the noise is supposed to be white, but since white noises do not exist in nature, it becomes relevant to identify some of its spectral properties. Experimental data set an upper bound on its frequencies, while in this paper we bound it from below. We do so in two ways: by considering a 'minimal' measurement setup, requiring that the collapse is completed within the measurement time; and in a measurement modeling-independent way, by requiring that the fluctuations average to zero before the measurement time.

We discuss a scheme for macrorealistic theories of the Leggett-Garg form (Leggett and Garg 1985 Phys. Rev. Lett. 54 857). Our scheme is based on a hybrid optomechanical system. It seems reasonable to test these inequalities with an optomechanical system, since in an optomechanical cavity it is possible to create non-classical states of the mirror through a projective measurement on the cavity field. We will present the protocol to generate such non-classicality for a general optomechanical cavity and after we will carry out a theoretical test for one of the possible formulations of these inequalities using a hybrid optomechanical system. Specifically, the inequality will be investigated for an harmonic oscillator coupled to a two-level system, which replaces the light field of the cavity. The aim is to reproduce, with this system, the evolution of a single spin-1/2 for which the inequality is violated; this is achievable through the conditioning of the two-level system which will be used as an ancilla.

The continuous spontaneous localization (CSL) model predicts a tiny break of energy conservation via a weak stochastic force acting on physical systems, which triggers the collapse of the wave function. Mechanical oscillators are a natural way to test such a force; in particular, a levitated micromechanical oscillator has been recently proposed to be an ideal system. We report a proof-of-principle experiment with a micro-oscillator generated by a microsphere diamagnetically levitated in a magnetogravitational trap under high vacuum. Due to the ultralow mechanical dissipation, the oscillator provides a new upper bound on the CSL collapse rate, which gives an improvement of two orders of magnitude over the previous bounds in the same frequency range, and partially reaches the enhanced collapse rate suggested by Adler. Although being performed at room temperature, our experiment has already exhibited advantages over those operating at low temperatures. Our results experimentally show the potential for a magnetogravitational levitated mechanical oscillator as a promising method for testing the collapse model. Further improvements in cryogenic experiments are discussed.

We present a technique for recovering the spectrum of a non-Markovian bosonic bath and/or non-Markovian noises coupled to a harmonic oscillator. The treatment is valid under the conditions that the environment is large and hot compared to the oscillator, and that its temporal autocorrelation functions are symmetric with respect to time translation and reflection - criteria which we consider fairly minimal. We model a demonstration of the technique as deployed in the experimental scenario of a nanosphere levitated in a Paul trap, and show that it would effectively probe the spectrum of an electric field noise source from 102 to 106 Hz with a resolution inversely proportional to the measurement time. This technique may be deployed in quantum sensing, metrology, computing, and in experimental probes of foundational questions.

Abstract: The proposed mission “Space Atomic Gravity Explorer” (SAGE) has the scientific objective to investigate gravitational waves, dark matter, and other fundamental aspects of gravity as well as the connection between gravitational physics and quantum physics using new quantum sensors, namely, optical atomic clocks and atom interferometers based on ultracold strontium atoms. Graphical abstract: [Figure not available: see fulltext.].

What gravitational field is generated by a massive quantum system in a spatial superposition? Despite decades of intensive theoretical and experimental research, we still do not know the answer. On the experimental side, the difficulty lies in the fact that gravity is weak and requires large masses to be detectable. However, it becomes increasingly difficult to generate spatial quantum superpositions for increasingly large masses, in light of the stronger environmental effects on such systems. Clearly, a delicate balance between the need for strong gravitational effects and weak decoherence should be found. We show that such a trade off could be achieved in an optomechanics scenario that allows to witness whether the gravitational field generated by a quantum system in a spatial superposition is in a coherent superposition or not. We estimate the magnitude of the effect and show that it offers perspectives for observability.

Collapse models describe phenomenologically the quantum-to-classical transition by adding suitable nonlinear and stochastic terms to the Schrödinger equation, thus (slightly) modifying the dynamics of quantum systems. Experimental bounds on the collapse parameters have been derived from various experiments involving a plethora of different systems, from single atoms to gravitational wave detectors. Here, we give a comprehensive treatment of the continuous spontaneous localization (CSL) model, the most studied among collapse models, for Fermi liquids. We consider both the white and non-white noise case. Application to various astrophysical sources is presented.

In this work we study the unitary time-evolutions of quantum systems defined on infinite-dimensional separable time-dependent Hilbert spaces. Two possible cases are considered: a quantum system defined on a stochastic interval and another one defined on a Hilbert space with stochastic integration measure (stochastic time-dependent scalar product). The formulations of the two problems and a comparison with the general theory of open quantum systems are discussed. Possible physical applications of the situations considered are analyzed.

Gravitational decoherence and gravitational wave function collapse are presented as two related but conceptually distinct ideas. Gravitational decoherence measures the effect of gravitational perturbations on the evolution of quantum systems, in particular their progressive lack of coherence. Gravitational wave function collapse starts with the assumption that the Schrodinger equation is not entirely right, and must be supplemented with extra terms, which cause the (random) collapse of the wave function; the collapse is then linked to gravity. Some of the most popular models are reviewed, with an emphasis on their conceptual status, stage of development, and open questions.

Wave function collapse models assume that the Schrödinger equation is not entirely right, and must be supplemented with extra terms, which cause the (random) collapse of the wave function. In some of these models, the collapse is linked to gravity. We review the most popular gravitational wave function collapse models, with an emphasis on their conceptual status, stage of development, comparison with experiments, and open questions.

Collapse models implement a progressive loss of quantum coherence when the mass and the complexity of quantum systems increase. We will review such models and the current attempts to test their predicted loss of quantum coherence.

The continuous spontaneous localization (CSL) model strives to describe the quantum-to-classical transition from the viewpoint of collapse models. However, its original formulation suffers from a fundamental inconsistency in that it is explicitly energy nonconserving. Fortunately, a dissipative extension to CSL has been recently formulated that solves such an energy-divergence problem. We compare the predictions of the dissipative and nondissipative CSL models when various optomechanical settings are used and contrast such predictions with available experimental data, thus building the corresponding exclusion plots.

Abstract: Models of spontaneous wave function collapse describe the quantum-to-classical transition by assuming a progressive breakdown of the superposition principle when the mass of the system increases, providing a well-defined phenomenology in terms of a non-linearly and stochastically modified Schrödinger equation, which can be tested experimentally. The most popular of such models is the continuous spontaneous localization (CSL) model: in its original version, the collapse is driven by a white noise, and more recently, generalizations in terms of colored noises, which are more realistic, have been formulated. We will analyze how current non-interferometric tests bound the model, depending on the spectrum of the noise. We will find that low frequency purely mechanical experiments provide the most stable and strongest bounds. Graphical abstract: [Figure not available: see fulltext.].

Recently, nonthermal excess noise, compatible with the theoretical prediction provided by collapse models, was measured in a millikelvin nanomechanical cantilever experiment [A. Vinante, Phys. Rev. Lett. 119, 110401 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.110401]. We propose a feasible implementation of the cantilever experiment able to probe such noise. The proposed modification, completely within the grasp of current technology and readily implementable also in other types of mechanical noninterferometric experiments, consists in replacing the homogeneous test mass with one composed of different layers of different materials. This will enhance the action of a possible collapse noise above that given by standard noise sources.

The continuous spontaneous localization (CSL) model is the best known and studied among collapse models, which modify quantum mechanics and identify the fundamental reasons behind the unobservability of quantum superpositions at the macroscopic scale. Albeit several tests were performed during the last decade, up to date the CSL parameter space still exhibits a vast unexplored region. Here, we study and propose an unattempted non-interferometric test aimed to fill this gap. We show that the angular momentum diffusion predicted by CSL heavily constrains the parametric values of the model when applied to a macroscopic object.