Gravitationally Induced Entanglement between Two Massive Particles is Sufficient Evidence of Quantum Effects in Gravity

Violation of a Leggett-Garg Inequality Using Ideal Negative Measurements in Neutron Interferometry

Leggett-Garg inequalities (LGIs) have been proposed in order to assess how far the predictions of quantum mechanics defy "macroscopic realism." With LGIs, correlations of measurements performed on a single system at different times are described. We report on an experiment that demonstrates the violation of an LGI with neutrons. The final measured value of the Leggett-Garg correlator K=1.120±0.007(stat)±0.019(sys), obtained in a neutron interferometric experiment, is clearly above the limit K=1 predicted by macrorealistic theories. The experimental results are analyzed within the framework of dynamical theory of neutron diffraction, evidently reproducing the obtained values.

Quantum control and Berry phase of electron spins in rotating levitated diamonds in high vacuum

Levitated diamond particles in high vacuum with internal spin qubits have been proposed for exploring macroscopic quantum mechanics, quantum gravity, and precision measurements. The coupling between spins and particle rotation can be utilized to study quantum geometric phase, create gyroscopes and rotational matter-wave interferometers. However, previous efforts in levitated diamonds struggled with vacuum level or spin state readouts. To address these gaps, we fabricate an integrated surface ion trap with multiple stabilization electrodes. This facilitates on-chip levitation and, for the first time, optically detected magnetic resonance measurements of a nanodiamond levitated in high vacuum. The internal temperature of our levitated nanodiamond remains moderate at pressures below 10-5 Torr. We have driven a nanodiamond to rotate up to 20 MHz (1.2 × 109 rpm), surpassing typical nitrogen-vacancy (NV) center electron spin dephasing rates. Using these NV spins, we observe the effect of the Berry phase arising from particle rotation. In addition, we demonstrate quantum control of spins in a rotating nanodiamond. These results mark an important development in interfacing mechanical rotation with spin qubits, expanding our capacity to study quantum phenomena.

Testing the Conjecture That Quantum Processes Create Conscious Experience

The question of what generates conscious experience has mesmerized thinkers since the dawn of humanity, yet its origins remain a mystery. The topic of consciousness has gained traction in recent years, thanks to the development of large language models that now arguably pass the Turing test, an operational test for intelligence. However, intelligence and consciousness are not related in obvious ways, as anyone who suffers from a bad toothache can attest-pain generates intense feelings and absorbs all our conscious awareness, yet nothing particularly intelligent is going on. In the hard sciences, this topic is frequently met with skepticism because, to date, no protocol to measure the content or intensity of conscious experiences in an observer-independent manner has been agreed upon. Here, we present a novel proposal: Conscious experience arises whenever a quantum mechanical superposition forms. Our proposal has several implications: First, it suggests that the structure of the superposition determines the qualia of the experience. Second, quantum entanglement naturally solves the binding problem, ensuring the unity of phenomenal experience. Finally, a moment of agency may coincide with the formation of a superposition state. We outline a research program to experimentally test our conjecture via a sequence of quantum biology experiments. Applying these ideas opens up the possibility of expanding human conscious experience through brain-quantum computer interfaces.

Quo Vadis Particula Physica?

In this brief paper, I give a very personal account on the state of particle physics on the occasion of Paul Frampton's 80th birthday.

Magnomechanically induced transparency and tunable slow-fast light via a levitated micromagnet

In this paper, we theoretically investigate the magnomechanically induced transparency (MIT) phenomenon and slow-fast light propagation in a microwave cavity-magnomechanical system which includes a levitated ferromagnetic sphere. Magnetic dipole interaction determines the interaction between the photon, magnon, and center of mass motion of the cavity-magnomechanical system. As a result, we find that apart from coupling strength, which has an important role in MIT, the levitated ferromagnetic sphere's position provides us a parameter to manipulate the width of the transparency window. In addition, the control field's frequency has crucial influences on the MIT. Also this hybrid magnonic system allows us to demonstrate MIT in both the strong coupling and intermediate coupling regimes. More interestingly, we demonstrate tunable slow and fast light in this hybrid magnonic system. In other words, we show that the group delay can be adjusted by varying the control field's frequency, the sphere position, and the magnon-photon coupling strength. These parameters have an influence on the transformation from slow to fast light propagation and vice versa. Based on the recent experimental advancements, our results provide the possibility to engineer hybrid magnonic systems with levitated particles for the light propagation, and the quantum measurements and sensing of physical quantities.

Steady motional entanglement between two distant levitated nanoparticles

Quantum entanglement in macroscopic systems is not only essential for practical quantum information processing, but also valuable for the study of the boundary between quantum and the classical world. However, it is very challenging to achieve the steady remote entanglement between distant macroscopic systems. We consider two distant nanoparticles, both of which are optically trapped in two cavities. Based on the coherent scattering mechanism, we find that the ultrastrong optomechanical coupling between the cavity modes and the motion of the levitated nanoparticles could be achieved. The large and steady entanglement between the filtered output cavity modes and the motion of nanoparticles can be generated if the trapping laser is under the red sideband. Then through entanglement swapping, the steady motional entanglement between the distant nanoparticles can be realized. We numerically simulate and find that the two nanoparticles with 10 km distance can be entangled for the experimentally feasible parameters, even in room temperature environments. The generated continuous variable multipartite entanglement is the key to realizing the quantum enhanced sensor network and the sensitivity beyond the standard quantum limit.

Measuring gravity with milligram levitated masses

Gravity differs from all other known fundamental forces because it is best described as a curvature of space-time. For that reason, it remains resistant to unifications with quantum theory. Gravitational interaction is fundamentally weak and becomes prominent only at macroscopic scales. This means, we do not know what happens to gravity in the microscopic regime where quantum effects dominate and whether quantum coherent effects of gravity become apparent. Levitated mechanical systems of mesoscopic size offer a probe of gravity, while still allowing quantum control over their motional state. This regime opens the possibility of table-top testing of quantum superposition and entanglement in gravitating systems. Here, we show gravitational coupling between a levitated submillimeter-scale magnetic particle inside a type I superconducting trap and kilogram source masses, placed approximately half a meter away. Our results extend gravity measurements to low gravitational forces of attonewton and underline the importance of levitated mechanical sensors.

Mass-Independent Scheme to Test the Quantumness of a Massive Object

The search for empirical schemes to evidence the nonclassicality of large masses is a central quest of current research. However, practical schemes to witness the irreducible quantumness of an arbitrarily large mass are still lacking. To this end, we incorporate crucial modifications to the standard tools for probing the quantum violation of the pivotal classical notion of macrorealism (MR): while usual tests use the same measurement arrangement at successive times, here we use two different measurement arrangements. This yields a striking result: a mass-independent violation of MR is possible for harmonic oscillator systems. In fact, our adaptation enables probing quantum violations for literally any mass, momentum, and frequency. Moreover, coarse-grained position measurements at an accuracy much worse than the standard quantum limit, as well as knowing the relevant parameters only to this precision, without requiring them to be tuned, suffice for our proposal. These should drastically simplify the experimental effort in testing the nonclassicality of massive objects ranging from atomic ions to macroscopic mirrors in LIGO.

Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity

We consider two interacting systems when one is treated classically while the other system remains quantum. Consistent dynamics of this coupling has been shown to exist, and explored in the context of treating space-time classically. Here, we prove that any such hybrid dynamics necessarily results in decoherence of the quantum system, and a breakdown in predictability in the classical phase space. We further prove that a trade-off between the rate of this decoherence and the degree of diffusion induced in the classical system is a general feature of all classical quantum dynamics; long coherence times require strong diffusion in phase-space relative to the strength of the coupling. Applying the trade-off relation to gravity, we find a relationship between the strength of gravitationally-induced decoherence versus diffusion of the metric and its conjugate momenta. This provides an experimental signature of theories in which gravity is fundamentally classical. Bounds on decoherence rates arising from current interferometry experiments, combined with precision measurements of mass, place significant restrictions on theories where Einstein's classical theory of gravity interacts with quantum matter. We find that part of the parameter space of such theories are already squeezed out, and provide figures of merit which can be used in future mass measurements and interference experiments.

Quantum reference frames for an indefinite metric

The current theories of quantum physics and general relativity on their own do not allow us to study situations in which the gravitational source is quantum. Here, we propose a strategy to determine the dynamics of objects in the presence of mass configurations in superposition, and hence an indefinite spacetime metric, using quantum reference frame (QRF) transformations. Specifically, we show that, as long as the mass configurations in the different branches are related via relative-distance-preserving transformations, one can use an extension of the current framework of QRFs to change to a frame in which the mass configuration becomes definite. Assuming covariance of dynamical laws under quantum coordinate transformations, this allows to use known physics to determine the dynamics. We apply this procedure to find the motion of a probe particle and the behavior of clocks near the mass configuration, and thus find the time dilation caused by a gravitating object in superposition. Comparison with other models shows that semi-classical gravity and gravitational collapse models do not obey the covariance of dynamical laws under quantum coordinate transformations.

High-Q Magnetic Levitation and Control of Superconducting Microspheres at Millikelvin Temperatures

We report the levitation of a superconducting lead-tin sphere with 100  μm diameter (corresponding to a mass of 5.6  μg) in a static magnetic trap formed by two coils in an anti-Helmholtz configuration, with adjustable resonance frequencies up to 240 Hz. The center-of-mass motion of the sphere is monitored magnetically using a dc superconducting quantum interference device as well as optically and exhibits quality factors of up to 2.6×10^{7}. We also demonstrate 3D magnetic feedback control of the motion of the sphere. The setup is housed in a dilution refrigerator operating at 15 mK. By implementing a cryogenic vibration isolation system, we can attenuate environmental vibrations at 200 Hz by approximately 7 orders of magnitude. The combination of low temperature, large mass, and high quality factor provides a promising platform for testing quantum physics in previously unexplored regimes with high mass and long coherence times.

Signatures of Quantum Gravity in the Gravitational Self-Interaction of Photons

We propose relativistic tests of quantum gravity using the gravitational self-interaction of photons in a cavity. We demonstrate that this interaction results in a number of quantum gravitational signatures in the quantum state of the light that cannot be reproduced by any classical theory of gravity. We rigorously assess these effects using quantum parameter estimation theory and discuss simple measurement schemes that optimally extract their signatures. Crucially, the proposed tests are free of QED photon-photon scattering, are sensitive to the spin of the mediating gravitons, and can probe the locality of the gravitational interaction. These protocols provide a new avenue for studying the quantum nature of gravity in a relativistic setting.

Layered materials as a platform for quantum technologies

Layered materials are taking centre stage in the ever-increasing research effort to develop material platforms for quantum technologies. We are at the dawn of the era of layered quantum materials. Their optical, electronic, magnetic, thermal and mechanical properties make them attractive for most aspects of this global pursuit. Layered materials have already shown potential as scalable components, including quantum light sources, photon detectors and nanoscale sensors, and have enabled research of new phases of matter within the broader field of quantum simulations. In this Review we discuss opportunities and challenges faced by layered materials within the landscape of material platforms for quantum technologies. In particular, we focus on applications that rely on light-matter interfaces.

Quantum Uncertainty Limit for Stern-Gerlach Interferometry with Massive Objects

We analyze the fundamental coherence limit of a nano-object with an embedded spin in a Stern-Gerlach interferometer. This limit stems from the which-path information provided by the object's rotational degrees of freedom due to the evolution of their quantum uncertainty. We show that such interferometry is straightforward in a weak magnetic field and short duration. Large wave packet separation is made possible with proper fine-tuning over long durations. This opens the door to fundamental tests of quantum theory and quantum gravity. The results and conclusions are extendable to any type of interferometry with complex objects.

Locally Mediated Entanglement in Linearized Quantum Gravity

The current interest in laboratory detection of entanglement mediated by gravity was sparked by an information-theoretic argument: entanglement mediated by a local field certifies that the field is not classical. Previous derivations of the effect modeled gravity as instantaneous; here we derive it from linearized quantum general relativity while keeping Lorentz invariance explicit, using the path-integral formalism. In this framework, entanglement is clearly mediated by a quantum feature of the field. We also point out the possibility of observing "retarded" entanglement, which cannot be explained by an instantaneous interaction. This is a difficult experiment for gravity, but is plausible for the analogous electromagnetic case.

Entanglement Witness for the Weak Equivalence Principle

The Einstein equivalence principle is based on the equality of gravitational and inertial mass, which has led to the universality of a free-fall concept. The principle has been extremely well tested so far and has been tested with a great precision. However, all these tests and the corresponding arguments are based on a classical setup where the notion of position and velocity of the mass is associated with a classical value as opposed to the quantum entities.Here, we provide a simple quantum protocol based on creating large spatial superposition states in a laboratory to test the quantum regime of the equivalence principle where both matter and gravity are treated at par as a quantum entity. The two gravitational masses of the two spatial superpositions source the gravitational potential for each other. We argue that such a quantum protocol is unique with regard to testing especially the generalisation of the weak equivalence principle by constraining the equality of gravitational and inertial mass via witnessing quantum entanglement.

Generation of Entanglement from Mechanical Rotation

Many phenomena and fundamental predictions, ranging from Hawking radiation to the early evolution of the Universe rely on the interplay between quantum mechanics and gravity or more generally, quantum mechanics in curved spacetimes. However, our understanding is hindered by the lack of experiments that actually allow us to probe quantum mechanics in curved spacetime in a repeatable and accessible way. Here we propose an experimental scheme for a photon that is prepared in a path superposition state across two rotating Sagnac interferometers that have different diameters and thus represent a superposition of two different spacetimes. We predict the generation of genuine entanglement even at low rotation frequencies and show how these effects could be observed even due to the Earth's rotation. These predictions provide an accessible platform in which to study the role of the underlying spacetime in the generation of entanglement.

Optomechanical Quantum Entanglement Mediated by Acoustic Phonon Fields

We present exact solutions for the quantum time evolution of two spatially separated, local inductor-capacitor (LC) oscillators that are coupled optomechanically to a long elastic strip that functions as a quantum thermal acoustic field environment. We show that the optomechanical coupling to the acoustic environment gives rise to causal entanglement dynamics between the two LC oscillators in the absence of resonant photon exchange between them, and that significant entanglement develops regardless of the environment temperature. Such a process establishes that distributed entanglement may be generated between superconducting qubits via a connected phonon bus bar, without the need for resonant phonon release and capture.

Force-Gradient Sensing and Entanglement via Feedback Cooling of Interacting Nanoparticles

We show theoretically that feedback cooling of two levitated, interacting nanoparticles enables differential sensing of forces and the observation of stationary entanglement. The feedback drives the two particles into a stationary, nonthermal state which is susceptible to inhomogeneous force fields and which exhibits entanglement for sufficiently strong interparticle couplings. We predict that force-gradient sensing at the zepto-Newton per micron range is feasible and that entanglement due to the Coulomb interaction between charged particles can be realistically observed in state-of-the-art setups.

A way forward for fundamental physics in space

Space-based research can provide a major leap forward in the study of key open questions in the fundamental physics domain. They include the validity of Einstein’s Equivalence principle, the origin and the nature of dark matter and dark energy, decoherence and collapse models in quantum mechanics, and the physics of quantum many-body systems. Cold-atom sensors and quantum technologies have drastically changed the approach to precision measurements. Atomic clocks and atom interferometers as well as classical and quantum links can be used to measure tiny variations of the space-time metric, elusive accelerations, and faint forces to test our knowledge of the physical laws ruling the Universe. In space, such instruments can benefit from unique conditions that allow improving both their precision and the signal to be measured. In this paper, we discuss the scientific priorities of a space-based research program in fundamental physics.

Diamagnetic Composites for High-Q Levitating Resonators

Levitation offers extreme isolation of mechanical systems from their environment, while enabling unconstrained high-precision translation and rotation of objects. Diamagnetic levitation is one of the most attractive levitation schemes because it allows stable levitation at room temperature without the need for a continuous power supply. However, dissipation by eddy currents in conventional diamagnetic materials significantly limits the application potential of diamagnetically levitating systems. Here, a route toward high-Q macroscopic levitating resonators by substantially reducing eddy current damping using graphite particle based diamagnetic composites is presented. Resonators that feature quality factors Q above 450 000 and vibration lifetimes beyond one hour are demonstrated, while levitating above permanent magnets in high vacuum at room temperature. The composite resonators have a Q that is >400 times higher than that of diamagnetic graphite plates. By tuning the composite particle size and density, the dissipation reduction mechanism is investigated, and the Q of the levitating resonators is enhanced. Since their estimated acceleration noise is as low as some of the best superconducting levitating accelerometers at cryogenic temperatures, the high Q and large mass of the presented composite resonators positions them as one of the most promising technologies for next generation ultra-sensitive room temperature accelerometers.

Quantum Signatures of Black Hole Mass Superpositions

We present a new operational framework for studying "superpositions of spacetimes," which are of fundamental interest in the development of a theory of quantum gravity. Our approach capitalizes on nonlocal correlations in curved spacetime quantum field theory, allowing us to formulate a metric for spacetime superpositions as well as characterizing the coupling of particle detectors to a quantum field. We apply our approach to analyze the dynamics of a detector (using the Unruh-deWitt model) in a spacetime generated by a Banados-Teitelboim-Zanelli black hole in a superposition of masses. We find that the detector exhibits signatures of quantum-gravitational effects corroborating and extending Bekenstein's seminal conjecture concerning the quantized mass spectrum of black holes in quantum gravity. Crucially, this result follows directly from our approach, without any additional assumptions about the black hole mass properties.

Quantum correlations beyond entanglement in a classical-channel model of gravity

A direct quantization of the Newtonian interaction between two masses is known to establish entanglement, which if detected would witness the quantum nature of the gravitational field. Gravitational interaction is yet compatible also with gravitational decoherence models relying on classical channels, hence unable to create entanglement. Here, we show in paradigmatic cases that, despite the absence of entanglement, a classical-channel model of gravity can still establish quantum correlations in the form of quantum discord between two masses. This is demonstrated for the Kafri-Taylor-Milburn (KTM) model and a recently proposed dissipative extension of this. In both cases, starting from an uncorrelated state, a significant amount of discord is generally created. This eventually decays in the KTM model, while it converges to a small stationary value in its dissipative extension. We also find that initial local squeezing on the state of the masses can significanlty enhance the generated discord.

Dynamics and Entanglement in Quantum and Quantum-Classical Systems: Lessons for Gravity

Motivated by quantum gravity, semiclassical theory, and quantum theory on curved spacetimes, we study the system of an oscillator coupled to two spin-1/2 particles. This model provides a prototype for comparing three types of dynamics: the full quantum theory, the classical oscillator with spin backreaction, and spins propagating on a fixed oscillator background. From calculations of oscillator and entanglement entropy evolution, we find the three systems give equivalent dynamics for sufficiently weak oscillator-spin couplings but deviate significantly for intermediate couplings. These results suggest that semiclassical dynamics with backreaction does not provide a suitable intermediate regime between quantum gravity and quantum theory on curved spacetime.

Spin-Controlled Quantum Interference of Levitated Nanorotors

We describe how to prepare an electrically levitated nanodiamond in a superposition of orientations via microwave driving of a single embedded nitrogen-vacancy (NV) center. Suitably aligning the magnetic field with the NV center can serve to reach the regime of ultrastrong coupling between the NV and the diamond rotation, enabling single-spin control of the particle's three-dimensional orientation. We derive the effective spin-oscillator Hamiltonian for small amplitude rotation about the equilibrium configuration and develop a protocol to create and observe quantum superpositions of the particle orientation. We discuss the impact of decoherence and argue that our proposal can be realistically implemented with near-future technology.

Stochastic Variational Method for Viscous Hydrodynamics

In this short review, we focus on some of the subjects, related to J. Cleymans’ pioneering contribution of statistical approaches to the particle production process in heavy-ion collisions. We discuss these perspectives from the effects of stochastic processes in collective variables of hydrodynamic description, which is described by a stochastic variational method. In this connection, we stress also the necessity of the inclusion of surface and quantum effects in the study of relativistic heavy-ion reactions.

Unruh Effect for Mixed Neutrinos and the KMS Condition

The quantization of mixed (neutrino) fields in an accelerated background reveals a non-thermal nature for Unruh radiation, which can be fitted by a Tsallis-like distribution function. However, for relativistic flavor neutrinos, which are represented by the standard Pontecorvo states, such a correction turns out to be negligible and thermality is restored. We show that the usage of Pontecorvo states for the calculation of the decay rate of an accelerated proton in the laboratory and comoving frames leads to consistent results and correctly implements the KMS thermal condition. Thus, the employment of these states in the above framework is not at odds with the principle of general covariance, in contrast to recent claims in the literature.

Gravity, Quantum Fields and Quantum Information: Problems with Classical Channel and Stochastic Theories

In recent years an increasing number of papers have attempted to mimic or supplant quantum field theory in discussions of issues related to gravity by the tools and through the perspective of quantum information theory, often in the context of alternative quantum theories. In this article, we point out three common problems in such treatments. First, we show that the notion of interactions mediated by an information channel is not, in general, equivalent to the treatment of interactions by quantum field theory. When used to describe gravity, this notion may lead to inconsistencies with general relativity. Second, we point out that in general one cannot replace a quantum field by a classical stochastic field, or mock up the effects of quantum fluctuations by that of classical stochastic sources (noises), because in so doing important quantum features such as coherence and entanglement will be left out. Third, we explain how under specific conditions semi-classical and stochastic theories indeed can be formulated from their quantum origins and play a role at certain regimes of interest.

Enhancing Gravitational Interaction between Quantum Systems by a Massive Mediator

In 1957 Feynman suggested that the quantum or classical character of gravity may be assessed by testing the gravitational interaction due to source masses in superposition. However, in all proposed experimental realizations using matter-wave interferometry, the extreme weakness of this interaction requires pure initial states with extreme squeezing to achieve measurable effects of nonclassical interaction for reasonable experiment durations. In practice, the systems that can be prepared in such nonclassical states are limited to small masses, which in turn limits the strength of their interaction. Here we address this key challenge-the weakness of gravitational interaction-by using a massive body as an amplifying mediator of gravitational interaction between two test systems. Our analysis shows that this results in an effective interaction between the two test systems that grows with the mass of the mediator, is independent of its initial state and, therefore, its temperature. This greatly reduces the requirement on the mass and degree of delocalization of the test systems and, while still highly challenging, brings experiments on gravitational source masses a step closer to reality.

Quantum Information in Relativity: The Challenge of QFT Measurements

Proposed quantum experiments in deep space will be able to explore quantum information issues in regimes where relativistic effects are important. In this essay, we argue that a proper extension of quantum information theory into the relativistic domain requires the expression of all informational notions in terms of quantum field theoretic (QFT) concepts. This task requires a working and practicable theory of QFT measurements. We present the foundational problems in constructing such a theory, especially in relation to longstanding causality and locality issues in the foundations of QFT. Finally, we present the ongoing Quantum Temporal Probabilities program for constructing a measurement theory that (i) works, in principle, for any QFT, (ii) allows for a first- principles investigation of all relevant issues of causality and locality, and (iii) it can be directly applied to experiments of current interest.

Non-Gaussian Mechanical Motion via Single and Multiphonon Subtraction from a Thermal State

Quantum optical measurement techniques offer a rich avenue for quantum control of mechanical oscillators via cavity optomechanics. In particular, a powerful yet little explored combination utilizes optical measurements to perform heralded non-Gaussian mechanical state preparation followed by tomography to determine the mechanical phase-space distribution. Here, we experimentally perform heralded single-phonon and multiphonon subtraction via photon counting to a laser-cooled mechanical thermal state with a Brillouin optomechanical system at room temperature and use optical heterodyne detection to measure the s-parametrized Wigner distribution of the non-Gaussian mechanical states generated. The techniques developed here advance the state of the art for optics-based tomography of mechanical states and will be useful for a broad range of applied and fundamental studies that utilize mechanical quantum-state engineering and tomography.

Can We Detect the Quantum Nature of Weak Gravitational Fields?

A theoretical framework for the quantization of gravity has been an elusive Holy Grail since the birth of quantum theory and general relativity. While generations of scientists have attempted to find solutions to this deep riddle, an alternative path built upon the idea that experimental evidence could determine whether gravity is quantized has been decades in the making. The possibility of an experimental answer to the question of the quantization of gravity is of renewed interest in the era of gravitational wave detectors. We review and investigate an important subset of phenomenological quantum gravity, detecting quantum signatures of weak gravitational fields in table-top experiments and interferometers.

Levitodynamics: Levitation and control of microscopic objects in vacuum

[Figure: see text].

Ultra-coherent nanomechanical resonators based on inverse design

Engineered micro- and nanomechanical resonators with ultra-low dissipation constitute a promising platform for various quantum technologies and foundational research. Traditionally, the improvement of the resonator's performance through nanomechanical structural engineering has been driven by human intuition and insight. Such an approach is inefficient and leaves aside a plethora of unexplored mechanical designs that potentially achieve better performance. Here, we use a computer-aided inverse design approach known as topology optimization to structurally design mechanical resonators with optimized performance of the fundamental mechanical mode. Using the outcomes of this approach, we fabricate and characterize ultra-coherent nanomechanical resonators with, to the best of our knowledge, record-high Q ⋅ f products for their fundamental mode (where Q is the quality factor and f is the frequency). The proposed approach - which can also be used to improve phononic crystals and coupled-mode resonators - opens up a new paradigm for designing ultra-coherent micro- and nanomechanical resonators, enabling e.g. novel experiments in fundamental physics and extreme sensing.

Quantum gravitational decoherence from fluctuating minimal length and deformation parameter at the Planck scale

Schemes of gravitationally induced decoherence are being actively investigated as possible mechanisms for the quantum-to-classical transition. Here, we introduce a decoherence process due to quantum gravity effects. We assume a foamy quantum spacetime with a fluctuating minimal length coinciding on average with the Planck scale. Considering deformed canonical commutation relations with a fluctuating deformation parameter, we derive a Lindblad master equation that yields localization in energy space and decoherence times consistent with the currently available observational evidence. Compared to other schemes of gravitational decoherence, we find that the decoherence rate predicted by our model is extremal, being minimal in the deep quantum regime below the Planck scale and maximal in the mesoscopic regime beyond it. We discuss possible experimental tests of our model based on cavity optomechanics setups with ultracold massive molecular oscillators and we provide preliminary estimates on the values of the physical parameters needed for actual laboratory implementations.

Large Quantum Delocalization of a Levitated Nanoparticle Using Optimal Control: Applications for Force Sensing and Entangling via Weak Forces

We propose to optimally control the harmonic potential of a levitated nanoparticle to quantum delocalize its center-of-mass motional state to a length scale orders of magnitude larger than the quantum zero-point motion. Using a bang-bang control of the harmonic potential, including the possibility of inverting it, the initial ground-state-cooled levitated nanoparticle coherently expands to large scales and then contracts to the initial state in a time-optimal way. We show that this fast loop protocol can be used to enhance force sensing as well as to dramatically boost the entangling rate of two weakly interacting nanoparticles. We parameterize the performance of the protocol, and therefore the macroscopic quantum regime that could be explored, as a function of displacement and frequency noise in the nanoparticle's center-of-mass motion. This noise analysis accounts for the sources of decoherence relevant to current experiments.

Kapitza-Dirac Blockade: A Universal Tool for the Deterministic Preparation of Non-Gaussian Oscillator States

Harmonic oscillators count among the most fundamental quantum systems with important applications in molecular physics, nanoparticle trapping, and quantum information processing. Their equidistant energy level spacing is often a desired feature, but at the same time a challenge if the goal is to deterministically populate specific eigenstates. Here, we show how interference in the transition amplitudes in a bichromatic laser field can suppress the sequential climbing of harmonic oscillator states (Kapitza-Dirac blockade) and achieve selective excitation of energy eigenstates, cat states, and other non-Gaussian states. This technique can transform the harmonic oscillator into a coherent two-level system or be used to build a large-momentum-transfer beam splitter for matter waves. To illustrate the universality of the concept, we discuss feasible experiments that cover many orders of magnitude in mass, from single electrons over large molecules to dielectric nanoparticles.

Realization of a complete Stern-Gerlach interferometer: Toward a test of quantum gravity

The Stern-Gerlach effect, found a century ago, has become a paradigm of quantum mechanics. Unexpectedly, until recently, there has been little evidence that the original scheme with freely propagating atoms exposed to gradients from macroscopic magnets is a fully coherent quantum process. Several theoretical studies have explained why a Stern-Gerlach interferometer is a formidable challenge. Here, we provide a detailed account of the realization of a full-loop Stern-Gerlach interferometer for single atoms and use the acquired understanding to show how this setup may be used to realize an interferometer for macroscopic objects doped with a single spin. Such a realization would open the door to a new era of fundamental probes, including the realization of previously inaccessible tests at the interface of quantum mechanics and gravity.

Measurement of gravitational coupling between millimetre-sized masses

Gravity is the weakest of all known fundamental forces and poses some of the most important open questions to modern physics: it remains resistant to unification within the standard model of physics and its underlying concepts appear to be fundamentally disconnected from quantum theory^1-4. Testing gravity at all scales is therefore an important experimental endeavour^5-7. So far, these tests have mainly involved macroscopic masses at the kilogram scale and beyond⁸. Here we show gravitational coupling between two gold spheres of 1 millimetre radius, thereby entering the regime of sub-100-milligram sources of gravity. Periodic modulation of the position of the source mass allows us to perform a spatial mapping of the gravitational force. Both linear and quadratic coupling are observed as a consequence of the nonlinearity of the gravitational potential. Our results extend the parameter space of gravity measurements to small, single source masses and low gravitational field strengths. Further improvements to our methodology will enable the isolation of gravity as a coupling force for objects below the Planck mass. This work opens the way to the unexplored frontier of microscopic source masses, which will enable studies of fundamental interactions^9-11 and provide a path towards exploring the quantum nature of gravity^12-15.

Some Notes on Quantum Information in Spacetime

The results obtained since the 70s with the study of Hawking radiation and the Unruh effect have highlighted a new domain of authority of relativistic principles [...].

Aharonov-Bohm Phase is Locally Generated Like All Other Quantum Phases

In the Aharonov-Bohm (AB) effect, a superposed charge acquires a detectable phase by enclosing an infinite solenoid, in a region where the solenoid's electric and magnetic fields are zero. Its generation seems therefore explainable only by the local action of gauge-dependent potentials, not of gauge-independent fields. This was recently challenged by Vaidman, who explained the phase by the solenoid's current interacting with the electron's field (at the solenoid). Still, his model has a residual nonlocality: it does not explain how the phase, generated at the solenoid, is detectable on the charge. In this Letter, we solve this nonlocality explicitly by quantizing the field. We show that the AB phase is mediated locally by the entanglement between the charge and the photons, like all electromagnetic phases. We also predict a gauge-invariant value for the phase difference at each point along the charge's path. We propose a realistic experiment to measure this phase difference locally, by partial quantum state tomography on the charge, without closing the interference loop.

Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems

The standard formulation of quantum theory relies on a fixed space-time metric determining the localisation and causal order of events. In general relativity, the metric is influenced by matter, and is expected to become indefinite when matter behaves quantum mechanically. Here, we develop a framework to operationally define events and their localisation with respect to a quantum clock reference frame, also in the presence of gravitating quantum systems. We find that, when clocks interact gravitationally, the time localisability of events becomes relative, depending on the reference frame. This relativity ia a signature of an indefinite metric, where events can occur in an indefinite causal order. Even if the metric is indefinite, for any event we can find a reference frame where local quantum operations take their standard unitary dilation form. This form is preserved when changing clock reference frames, yielding physics covariant with respect to quantum reference frame transformations.

Optomechanics with levitated particles

Optomechanics is concerned with the use of light to control mechanical objects. As a field, it has been hugely successful in the production of precise and novel sensors, the development of low-dissipation nanomechanical devices, and the manipulation of quantum signals. Micro- and nano-particles levitated in optical fields act as nanoscale oscillators, making them excellent low-dissipation optomechanical objects, with minimal thermal contact to the environment when operating in vacuum. Levitated optomechanics is seen as the most promising route for studying high-mass quantum physics, with the promise of creating macroscopically separated superposition states at masses of 10⁶ amu and above. Optical feedback, both using active monitoring or the passive interaction with an optical cavity, can be used to cool the centre-of-mass of levitated nanoparticles well below 1 mK, paving the way to operation in the quantum regime. In addition, trapped mesoscopic particles are the paradigmatic system for studying nanoscale stochastic processes, and have already demonstrated their utility in state-of-the-art force sensing.

Cooling of a levitated nanoparticle to the motional quantum ground state

Quantum control of complex objects in the regime of large size and mass provides opportunities for sensing applications and tests of fundamental physics. The realization of such extreme quantum states of matter remains a major challenge. We demonstrate a quantum interface that combines optical trapping of solids with cavity-mediated light-matter interaction. Precise control over the frequency and position of the trap laser with respect to the optical cavity allowed us to laser-cool an optically trapped nanoparticle into its quantum ground state of motion from room temperature. The particle comprises 108 atoms, similar to current Bose-Einstein condensates, with the density of a solid object. Our cooling technique, in combination with optical trap manipulation, may enable otherwise unachievable superposition states involving large masses.