Abstract
Quantum technologies are opening novel avenues for applied and fundamental science at an impressive pace. In this perspective article, we focus on the promises coming from the combination of quantum technologies and space science to test the very foundations of quantum physics and, possibly, new physics. In particular, we survey the field of mesoscopic superpositions of nanoparticles and the potential of interferometric and noninterferometric experiments in space for the investigation of the superposition principle of quantum mechanics and the quantumtoclassical transition. We delve into the possibilities offered by the stateoftheart of nanoparticle physics projected in the space environment and discuss the numerous challenges, and the corresponding potential advancements, that the space environment presents. In doing this, we also offer an abinitio estimate of the potential of spacebased interferometry with some of the largest systems ever considered and show that there is room for tests of quantum mechanics at an unprecedented level of detail.
Quantum mechanics is one of the most successful physical theories humankind has ever formulated. Nonetheless, its interpretation and range of validity elude our full grasping. One of the basic features of quantum physics is the superposition principle which, when applied to the macroscopic world, leads to counterintuitive states akin to the celebrated Schödinger’s cat. While models beyond quantum mechanics, challenging some of its interpretational issues, have been formulated in their early days, testing the predictions of the theory when applied to the macroscopic world has proven to be a tall order. The main reason for this is the intrinsic difficulty in isolating large systems from their environment.
Space offers a potentially attractive arena for such an endeavor, promising the possibility to create and verify the quantum properties of macroscopic superpositions far beyond current Earthbased capabilities^{1,2,3,4}. In this work, we focus on the efforts to test the boundaries of quantum physics in space employing nanoparticles, which are one of the bestsuited candidates for quantum superpositions of highmass objects. It should be noticed that, while we will focus on testing quantum physics, large spatial superpositions of massive systems are bound to be sensitive probes for many other physical phenomena, from dark matter and dark energy searches^{5,6,7,8,9,10,11,12,13} to gravimetry and Earth observation applications^{14,15}.
In this perspective article, we delve into the possibilities offered by the stateoftheart nanoparticle physics projected in the space environment. In doing so, we offer an abinitio estimate of the potential of spacebased interferometry with some of the largest systems ever considered and show that there is room for testing quantum mechanics at an unprecedented level of detail.
In particular, after a brief introduction to the problem at hand and its relevance in fundamental physics, we discuss the advantages potentially offered by a space environment for quantum experiments based on large quantum superpositions of nanoparticles. We also give a selfcontained overview of the current stateoftheart for spacemission proposals and distinguish two classes of experiments that can be performed in space: noninterferometric and interferometric ones. The former does not require the creation of macroscopic superpositions and exploit the freeevolution spread of the position of a quantum particle. The latter, in contrast, require the creation and verification of large superpositions but also offer the benefit of a direct test of both the superposition principle of quantum mechanics and of competing theories. Both classes of experiments take advantage of the long freefall times in space and can be used to cast stringent constraints on theoretical predictions. To showcase this last aspect, we present an abinitio estimate of the constraints that can be expected from spacebased interferometry with large nanoparticles.
Superposition of macroscopic systems: the case for space
The predictions of quantum physics have been confirmed with a high degree of precision in a multitude of experiments, from the subatomic scale up to matterwave interferometry with tests masses of nearly 10^{5} atomic mass units (amu)^{16}. The basis for observing matterwave interference is the quantum superposition principle, one of the pillars of quantum physics. While quantum physics does not pose any fundamental limitation to the size of quantum superposition states, the Gedankenexperiment of Schrödinger’s cat^{17} illustrates the controversies entailed by the superposition principle when extended to the macroscopic world. Many proposals have been formulated in an attempt to establish a mechanism that would lead to the emergence of a classical world at macroscopic scales. Among them, we find Bohmian mechanics^{18,19}, decoherence histories^{20}, the manyworld interpretation^{21}, and collapse models^{22,23} to name a few. The latter differs from the other proposals in the fact that they predict a phenomenology that deviates from one of standard quantum mechanics, albeit in a delicate fashion. In this sense, collapse models represent an alternative construction to standard quantum theory, more than an alternative interpretation recovering all the predictions of the latter. In light of the central role that they play in the experimental investigation of quantum macroscopicity^{24,25}, in the following, we will focus on such models as benchmarks for precision tests of quantum mechanics.
In 2010, a proposal for experimentally creating and verifying a state akin to the one of Schrödinger’s cat based on the use of massive mechanical resonators was put forward within the context of the MAQRO proposal^{1}. The latter put forward the vision of harnessing the unique environment provided by space to test quantum physics in a dedicated, mediumsized space mission to be conducted within the framework of the “Cosmic Vision programme” run by the European Space Agency (ESA). The scope of the endeavor was to create a macroscopic superposition of motional states of a massive particle and probe its quantum coherence by allowing the wave functions of the components of such superposition to interfere, as in a doubleslit experiment. The spacebased environment would guarantee unprecedented levels of protection from environmental noises, as well as favorable working conditions for the engineering of the catlike state^{1}.
Nearfield interferometry has later been identified as a viable route for the achievement of the original goals of MAQRO^{2}, holding the promises for testing the superposition principle with particles of mass up to 10^{11} amu. This would be at least six orders of magnitude larger than the current record^{16}. It would also far exceed the projected upper bound to the masses that could be used in similar groundbased experiments. Such terrestrial upperbounds are strongly limited by the achievable freefall times on Earth^{26} (cf. subsection Possible advantages of a space environment). The basic payload consists of optically trapped dielectric nanoparticles with a target mass range from 10^{7} to 10^{11} amu. The main scientific objectives are to perform both nearfield interferometric and noninterferometric experiments. In both cases, highvacuum and cryogenic temperatures are needed. The particles, after loading, are initially trapped in an optical cavity and their centerofmass degree of freedom cooled down by a 1064 nm laser entering the quantum regime. For this purpose, two TEM_{00} modes with orthogonal polarization are to be used for trapping and sideband cooling along the cavity axis. The transverse motion is instead cooled employing a TEM_{01} and a TEM_{10} mode. After the initial state preparation, the particle can be released from the optical trap and undergo different evolutions—free fall expansion, coherent manipulations, and quantum detection—depending on the experiments to be performed.
The feasibility of the avenue identified in MAQRO has recently been investigated in a Quantum Physics Payload platForm (QPPF) study at the ESA Concurrent Design Facility^{4}. Such study has identified (a) the core steps towards the realization of a spacebased platform for highprecision tests of quantum physics, and (b) the potential of such platforms to test quantum physics with increasing test mass with the scope to ascertain potential deviations from the predictions of quantum physics due, for instance, to gravity. The ultimate goal of these endeavors is to provide a reference mission design for quantum physics experiments in space.
The QPPF^{4} study culminated with the identification of a suitable combination of feasible freefall times, temperature and pressures [cf. Table 1], setting a target of 2034 for the launch of the mission. While several technical challenges remain to be addressed, the QPPF study has consolidated the intention to leverage on the expertize in nearfield interferometry and optomechanics where stateoftheart experiments with large molecule at near 10^{5} amu have been reported^{16}, groundstate cooling achieved^{27}, and proofofconcept proposals for groundbased interferometry with largemass experiments put forward^{26,28}. In this respect, it should be mentioned that theoretical proposals based on other approaches, most noticeably magnetic levitation, have recently attracted the attention of part of the community^{29}. These proposals envision testing the superposition principle with groundbased experiments, overcoming some of the existing limitations through the low noise level, long coherentoperation times, and lack of need for light fields driving the dynamics promised by magnetic levitation. The use of masses of the order of 10^{13} amu has been forecast in this context^{29}. While extremely interesting, magnetic levitation technologies for quantum experiments are still at an early stage^{30,31,32,33} with, at present, no quantum superposition having been created which such techniques.
In search of the quantumtoclassical boundary
The extreme fragility of spatial quantum superpositions in the presence of environmental interactions (ubiquitous in any realistic setting) makes testing the superposition principle at macroscopic scales a tall order. Indeed, such interactions result in a suppression of quantum coherence in a position that can be described by the following master equation in position representation^{34}
where \({\hat{\rho }}_{t}\) is the statistical operator of the system at time t, \(\hat{H}\) is the system’s Hamiltonian, and
the last term of Eq. (1) describes the deviations from unitary dynamics occurring at a rate Γ(x), which quantifies the decoherence effect. The typical behavior of the latter, with a quadratic dependence for small spatial separations and saturating for large ones, is shown in Fig. 1. Such deviations from unitarity can be due to environmental noises or nonstandard modifications of quantum mechanics^{23,35,36}. The environmental influence is always present, and it inevitably disturbs the experiment compromising the possibility to detect superposition states. The typical noise effects on the experimental setups addressed in this paper are due to collisions with residual gas particles, blackbody radiation, vibrations, and in general any noise propagating through the experimental setup. Quantitatively, for a sphere made of fused silica with a radius of 60 nm and an internal temperature of 40 K, placed in a vacuum in an environment at 20 K and a pressure of 10^{−11} Pa [cf. Table 1], one has that for spatial superpositions larger than a nanometer but smaller than a millimeter, which is the range of interest for the interferometric test we will consider here, the gas collisions give a constant contribution^{36} \({{\Gamma }}(xx^{\prime}) \sim 1.1\) s^{−1}, while the contribution from blackbody radiation depends explicitly on the superposition size as \({{\Gamma }}(xx^{\prime}) \sim 4.9\times 1{0}^{12}\times  xx^{\prime} { }^{2}\) m^{−2}s^{−1}.
As mentioned, Eq. (1) is conducive to an investigation on potential deviations from standard quantum theory due, for instance, to collapse models. Due to its interesting phenomenology, theoretical interest, and current strong experimental effort in testing it^{23,37,38,39,40,41}, in the following, we will focus on the Continuous Spontaneous Localization (CSL) model. The CSL model describes, through a stochastic and nonlinear modification of the Schrödinger equation, the collapse of the wave function as a spontaneous process whose strength increases with the mass of the system^{42}. Its action is characterized by two phenomenological parameters: λ_{CSL} and r_{c}. These are, respectively, the collapse rate, which quantifies the strength of the collapse noise and setting the spatial resolution of the collapse. Theoretical considerations lead to different proposed values for such parameters: λ_{CSL} = 10^{−16} s^{−1} and r_{c} = 10^{−7} m for Ghirardi, Rimini, and Weber^{43}; λ_{CSL} = 10^{−8±2} s^{−1} for r_{c} = 10^{−7} m, and λ_{CSL} = 10^{−6±2} s^{−1} for r_{c} = 10^{−6} m by Adler^{44}. Consequently, one can describe the evolution of the density matrix of a system with Eq. (1) where, in addition to the decoherence effects ascribed to the environment, a term accounting for spontaneous collapse appears. The form of such a term and its effects are discussed in detail in the Noninterferometric tests section. This reveals the importance of careful characterization of environmental sources of decoherence in view of probing new physics, which is the aim of the space experiments with large nanoparticles reviewed here. It should be noted indeed that, the experimental setups considered here are relevant also for testing other models predicting nonstandard decoherence mechanisms^{45,46,47} or models like the DiósiPenrose (DP) one^{48,49,50} in which the wave function collapse is related to gravity.
Possible advantages of a space environment
The main advantage offered by space for quantum experiments with large particles is undoubtedly a long freefall time. While freelyfalling systems are not necessary for some noninterferometric experiments, they are the golden standard for the interferometric ones. For the latter, long freefall times are of crucial importance to achieve better sensitivity and to increase the mass of the particles in quantum superposition as the rate of the wavefunction spreading is set by 1/m. In stateoftheart interferometric experiments, and for masses of up to 10^{6} amu, the necessary freefall times are far below 1 s and can be readily achieved in laboratory experiments^{16,26,28}. However, going to significantly higher test masses requires correspondingly longer freefall times^{1,2,26} such as to eventually rendering it inevitable to perform such experiments in space (see Fig. 2). Long freefall times help also in noninterferometric settings. The latter do not require the creation and verification of quantum superpositions but are based on the modified dynamics predicted by alternative models to quantum mechanics—as for example the heating induced by the CSL noise on massive particles. In this context, letting the particle fall freely allows reducing the effects of all the sources of noise that affect the center of mass motion. Among them certainly is acceleration noise typically originating from mechanical vibrations. However, one should also include other forces acting on the particle’s motion and which might be present in the experiment thus maximizing the effects induced by modifications of quantum mechanics. In what follows, we provide a brief yet rigorous account of the most relevant of such forces.
An equally important challenge is the isolation from vibrations, which contribute to the overall decoherence mechanisms acting on the system. Especially in the lowfrequency regime, space experiments can provide strong advantages compared to those performed on the ground. For example, ensuring that an interference pattern with a period of d = 1 μm formed during an evolution time of T = 100 s is not washed out requires a maximum acceleration noise of \({S}_{aa}^{\max } \sim 3{d}^{2}/8\pi {T}^{3}\) corresponding to \(\sqrt{{S}_{aa}} \sim 3.5\times 1{0}^{10}\ {\rm{m}}\ {{\rm{s}}}^{2}/\sqrt{{\rm{Hz}}}\). Such low noise can be achieved in space. The most impressive achievement so far has been LISA Pathfinder with an acceleration noise as small as \(\sqrt{{S}_{aa}} \sim 1{0}^{15}\ {\rm{m}}\ {{\rm{s}}}^{2}/\sqrt{{\rm{Hz}}}\) in the mHz regime^{51}. This value has to be compared to those of the stateofart groundbased experiments. For example, the Bremen droptower allows for up to around 9 s of free fall^{52} in an environment characterized by an acceleration noise of \(\sim\!\!1{0}^{5}\ {\rm{m}}\ {{\rm{s}}}^{2}/\sqrt{{\rm{Hz}}}\). We also mention that the exceptionally low level of noise achieved in LISA Pathfinder has already allowed this experiment to provide bounds on the CSL parameters that are more than three orders of magnitude stronger than those provided by the groundbased gravitational waves detector LIGO^{53,54,55}, thus demonstrating the advantages—in terms of isolation from vibration—of spacebased experiments.
A potential additional advantage of a dedicated space mission is data statistics. In stateoftheart matterwave experiments^{16}, many test particles pass through the interferometer simultaneously. In proposals, considered in the QPPF, suggesting to prepare the initial state of the test particle by optomechanical means, the test particles pass through the interferometer individually^{1,28,56}. Thus, the time of the experiment is inevitably longer, and growing with the number of data points required. For example, in an experiment with 10^{4} data points, where a single shot takes 100 s (10 s), the data collection would last more than 11.5 days (27 h). This compares very favorably to the typical number of two or three runs per day that can be performed in a microgravity environment on the ground as at the Bremen droptower, which is limited by the necessity of setting and resetting the pressure in the entire tower between two consecutive drops. It should be mentioned that this limitation is not present in the Hannover Einstein Elevator platform where, for freefall times of ≤4 s, 300 runs per day are possible^{57}.
Stateoftheart technological platforms
The space environment promises, in principle, to provide a unique combination of low temperature, extremely high vacuum, and very long freefall times. In particular, the temperature in space is naturally limited by the temperature of the microwave background radiation of about 3 K while the vacuum is instead limited by the presence of cosmic and solar radiation^{58}. However, in actual spacebased experiments, additional shielding may be required. For example, if the spacecraft is in an orbit about one astronomical unit from the Sun, the payload will have to be shielded from direct solar radiation. The spacecraft will require stabilization using micro thrusters, which will introduce force noise, and there will need to be stationkeeping maneuvers. These measures, as well as the gravitational field of the spacecraft, will reduce the achievable freefall time. In addition, the equipment necessary to operate the payload and the spacecraft typically will be in an enclosure kept at a stable temperature of about 300 K. Inside the spacecraft, it is therefore not trivial to achieve cryogenic temperatures and extremely high vacuum levels. Achieving the vacuum levels and temperatures necessary for macroscopic tests of quantum physics, therefore, requires careful considerations. In the context of the MAQRO mission concept, it has been suggested to use purely passive radiative cooling and direct outgassing to space in order to achieve these requirements^{1,2,59,60}. During the QPPF study, this concept was adapted to protect the scientific instrument with a cover and to enhance the cooling performance by additional active cooling using a hydrogen sorption cooler^{4}.
The protective cover limits outgassing to space, and the QPPF study concluded that the achievable pressure would at best be 10^{−11} Pa instead of the aimed pressure of 10^{−13} Pa. As a result, the experiments were constrained to a test particle mass up to 2 × 10^{9} amu and freefall time up to 40 s. Because this has a significant impact on the science objectives, improving the achievable vacuum in spacebased experiments will be a critical issue to be solved before a space mission of this type can be launched. Two other critical issues were identified in the QPPF study^{4}. Firstly, the mechanism needed for loading the test particles into an optical trap in extremely high vacuum conditions needs further scrutiny and several viable alternatives are under consideration. The QPPF study suggested desorbing particles from microelectromechanical systems (MEMS) and guide them to the experiment using linear Paul traps^{4}, a mechanism that is currently under further investigation by ESA. An alternative suggestion makes use of a combination of linear Paul traps and hollowcore photoniccrystal fibers^{2} or the desorption of particles from a piezoelectric substrate using surface acoustic waves. The challenge of such desorptionbased approaches is to make sure that the desorbed submicron particles do not carry a net charge^{61}, and that their centerofmass motion is sufficiently cold to allow for optical trapping. At the same time, a sufficiently low internal temperature of the particles is required to avoid decoherence due to the emission of blackbody radiation^{1,56}. Secondly, the optical gratings used for preparing nonclassical states have grating apertures comparable to the size of the nanoparticles to be employed. This can decohere the quantum states via photon scattering. A recent study^{62} investigated the latter issue extending the formalism of nearfield interferometry beyond the pointparticle approximation and offering the basis for the analysis reported in the Interferometric tests section.
Noninterferometric tests
In this section, we focus on noninterferometric tests of quantum mechanics. Differently from the interferometric ones, this class of tests does not rely on the availability of quantum superpositions but is based on sideeffects of modifications of quantum mechanics. Consequently, they can be performed also in presence of strong decoherence, although the latter will influence the effectiveness of the test. For this reason, they currently provide the most stringent tests of collapse models on the ground.
A plethora of different experiments belong to this class and exploit different physical systems. Among them, precision measurements of the internal energy of a solid, expected to vary due to the collapse noise, have been exploited^{63,64,65}. The modifications to the free evolution dynamics of BoseEinstein condensate due to the presence of the collapse mechanism have been investigated^{66,67}. And Xray measurements—which exploit the fact that the collapse mechanism makes charged particles emit radiation^{68,69,70}—have already provided strong limits on the DiósiPenrose model^{71}. In this context, also optomechanical experiments are of particular relevance^{41,53,54,72,73,74,75,76}. They are typically used to characterize noise^{77,78,79}, and thus possibly discriminate between standard and nonstandard noise sources^{41}.
One of the most promising noninterferometric tests in space is based on monitoring the expansion of the centerofmass position spread of a freelyfalling nanoparticle^{80}. The main reason, as it is shown in Eq. (2) below, is that the position variance grows as the cube of time, making evident the advantage of the long freefall time that can be achieved in space. It could be argued that long times can also be achieved in ground experiments by suspending the particles using an harmonic trap. However, the use of such a trap would certainly introduce additional noises and, more importantly, it would imply a position variance growth that scales only linearly with time^{67,81}.
Given the evolution in Eq. (1), it is easy to show that its nonunitary part does not affect the average position 〈x_{t}〉 of the particle, but changes its variance \({\sigma }^{2}=\langle {x}_{t}^{2}\rangle {\langle {x}_{t}\rangle }^{2}\) by a factor 〈Δσ^{2}〉 that, for a free system and in the x ≪ a regime [cf. Fig. 1], reads
The diffusion rate Λ is the sum of different contributions stemming from residual gas collisions, blackbody radiation, and nonstandard sources, such as the CSL or the DiósiPenrose model. For the CSL model and a homogeneous sphere of radius R and mass M, one has^{75,82}
while for the DP model one obtains
We have used the dimensionless parameters η_{CSL} = R/r_{c} and η_{DP} = R/R_{0} with R_{0} a free parameter that is characteristic of the DP model^{83}. These expressions can be then used to set bounds on, respectively, CSL and DP parameters with spacebased experiments, as we discuss next.
Long freefall times: opportunities and challenges for spacebased experiments
A possible spacebased experiment, as envisioned in the MAQRO proposal and QPPF, is as follows. A nanosphere is initially trapped by an harmonic optical potential and its centerofmass motion is optically cooled. The trapping is then removed and the nanosphere remains in freefall for a time t after which its position is measured. Achieving a high position resolution is possible by, for example, combining a coarsegrained standard optical detection on a CMOS chip with a highresolution backscattering detection scheme^{84}, which could eventually provide a position accuracy on the order of ε = 10^{−12} m at a typical bandwidth of 100 kHz, by controlling the measurement backaction^{85}. By repeating such a procedure N times, one can reconstruct the position spread σ^{2} and thus quantify the effects of the nonunitary dynamics through Eq. (2). To detect effects as those predicted by the CSL or the DP model, one needs to minimize the competing standard decoherence effects (from collisions and blackbody radiation), which contribute to the total Λ in Eq. (2).
We are now in a position to estimate the bounds on the CSL parameters. To do this, we employ the values in Table 1. We consider silica nanospheres with a 120 nm diameter as test particles and an internal temperature fixed at 40 K. Moreover, we also assume levels of vibrational noise similar to those obtained in LISA Pathfinder^{51}. With these assumptions, the strongest competing effect to the CSL noise is the collisional decoherence, which limits the bounds on the CSL parameters. Such a bound is indeed obtained by setting Λ_{CSL} equal to the collisional contribution to the diffusive constant^{36} Λ. We show the corresponding bound as the solid red line in Fig. 3, where such bound is compared to groundbased ones achieved by stateoftheart experiments on the CSL model.
For what concerns the DP model, the stateoftheart experimental bounds indicate that the free parameter R_{0} is limited to^{71} \({R}_{0}\ge {R}_{0}^{* }=0.5\times 1{0}^{10}\) m. Because the DPinduced collapse becomes stronger for smaller R_{0}, the maximum effect is obtained for \({R}_{0}^{* }\). Such a value of R_{0} leads to a position spread in the aforementioned setup of \(\sqrt{\langle {{\Delta }}{\sigma }^{2}\rangle } \sim 3\times 1{0}^{26}\) m for t = 100 s, well beyond the stateofart position measurement sensitivity ε.
An important aspect to consider for experiments performed in space is their limited lifetime. Especially when one considers long freefall times, this will have an impact on the statistical accuracy with which one can determine the variance of the measured data points^{2,4}. The long freefall time t required to see potential deviations from the quantum predictions has to compete with a finite time T available to take the complete data set. At best, the number of data points can be N = T/t. This limit on the number of data points implies a statistical uncertainty in determining the position spread. To quantify it, we assume that the initial quantum state of the test particle is the ground state of an harmonic oscillator with a mechanical frequency ω. Consequently, the measured position will be normally distributed and the corresponding fractional uncertainty of the variance of the measured position will be^{86} \(\sqrt{2/(N1)}\approx \sqrt{2t/T}\). Assuming the deviations from the quantum predictions to be small, the statistical uncertainty of the variance is \({{\Delta }}{x}_{{\rm{f}}}^{2}\approx \sqrt{2t/T}{x}_{{\rm{s}}}^{2}\), where \({x}_{{\rm{s}}}^{2}\approx {t}^{2}\omega \hslash /2m\) is the variance of the wavepacket predicted by quantum physics for times much longer than 1/ω. By taking ω = 10^{5} Hz for the trap frequency, a free evolution time t = 100 s and a total time T of 30 days, we have that Δx_{f} ~ 3 × 10^{−5} m which has to be compared to the sensitivity ε ~ 10^{−12} m. For these parameters, the statistical uncertainty will dominate over the position sensitivity ε already after about? 0.1 ms. Such statistical uncertainty becomes a fundamental limitation for the experiment. The corresponding upper bound on the CSL parameters are represented by the dashed, red line in Fig. 3. To reach such a limit, the pressure would need to be reduced by more than two orders of magnitude, down to P = 3 × 10^{−14} Pa, with respect to the conditions set by the continuous red bound.
Figure 3 and the analysis above suggest that noninterferometric experiments performed with the parameters in Table 1 can enhance only partially the exploration of the CSL parameter space. A more substantial improvement would require to solve technical challenges, such as a significant pressure reduction. Alternatively, one can pursue the path of interferometric experiments, which is discussed in the next section.
Interferometric tests
Here, we will provide an overview of the current stateoftheart for proposals of interferometric experiments testing the superposition principle of quantum mechanics for higher masses than the current experimental record on the ground by using a space environment. We will discuss the challenges faced by such experiments, and we will provide novel simulation results estimating the interference visibility expected in spacebased experimental tests of the superposition principle of quantum mechanics.
Nearfield interferometry
After Clauser envisioning its use for “small rocks and live viruses” experiments^{87} and its initial demonstration for C^{70} molecules interferometry^{88}, for almost two decades the most successful technique harnessed for interferometric tests of quantum physics has been nearfield interferometry^{89,90}. With this technique, and employing three optical gratings^{91}, in 2019 the Arndt’s group in Vienna was able to successfully build and demonstrate spatial the quantum superposition of big molecules with masses beyond^{16} 10^{4} amu.
Recently, the possibility to consistently describe the effects of an optical grating on large dielectric particles with radii comparable to the optical wavelengths^{62,92} has opened the possibility to use optical grating to study quantum interference on even larger particles. At the same time, concrete proposals to go beyond the current mass record, employing individually addressed dielectric particles and single optical grating^{2,4,28} have shown the experimental viability of nearfield interferometry to actually perform such larger mass superposition experiments.
We thus focus specifically on these implementations to give an overview of how a nearfield interferometric scheme works.
We refer to Fig. 4 for a schematic representation of a singlegrating nearfield setup. Contrary to the case of lighter systems, where molecular beams are engineered, each nanoparticle in the experiment is individually addressed. We thus have, at each run of the experiment, four main stages:

(a)
The nanoparticle is trapped and cooled down in an optical cavity for a time t_{c} after which the centerofmass degree of freedom is in a very lowtemperature thermal state characterized by the momentum and position variances σ_{p}, σ_{z}. No cooling down to the ground state is required.

(b)
The particle is released and free fall for a time t_{1}. During this time, residual gas collisions and thermal radiation are the main sources of decoherence. The free evolution of the postcooling state needs to guarantee that the coherence length is sufficient to cover at least two adjacent “slits” of the optical grating.

(c)
A retroreflected pulsed laser provides a purephase grating^{92} for the dielectric nanoparticle. Scattering and absorption of grating photons constitute the main decoherence channels in the short interaction time with the grating.

(d)
Second period of free evolution for a time t_{2} during which the same sources of decoherence as in point (b) act. This stage has to last enough time for the interference pattern to form.

(e)
The position of the particle is measured via optical detection^{2,4}.
By repeating this protocol (a–e) many times, an interference pattern can form in the measured position distribution. This pattern can be mathematically described by a probability distribution function P(z) which can be analytically derived from a phasespace treatment of the interferometric experiment^{28,92}:
where \(\delta =m/(\sqrt{2\pi }{\sigma }_{p}({t}_{1}+{t}_{2}))\), t_{T} = md^{2}/h is the Talbot time and D = d(t_{1} + t_{2})/t_{1} is a geometric magnification factor. In this last expression, the B_{n}’s are known as the generalized Talbot coefficients^{92,93} and account for the coherent and incoherent effects of the optical grating, while the kernels R_{n} account for environmental decoherence, due to absorption, emission, and scattering of thermal radiation and collisions with residual gas, during the freefalling times t_{1}, t_{2}. This expression remains formally unchanged when classical particles following ballistic trajectories are considered, but the explicit expressions for the decoherence kernels and Talbot coefficients will change. Expression (5), with the proper coefficients, can thus be used to describe the classical shadow pattern arising from a completely classical description of the system (see Fig. 5). Finally, nonlinear modifications of quantum mechanics—but also other sources of positional decoherence like e.g., stochastic gravitational waves background^{94,95}—can easily be included in Eq. (5) by introducing their respective noise kernels R_{n}. We refer the interested reader to literature^{26,62} and the Supplementary Note 1 for a detailed derivation and explicit expressions of the functions entering Eq. (5).
In order to go beyond the current nearfield interferometry mass record, large particles need to be used. Here, “large” refers to spherical particles with a radius R comparable to or greater than the grating period d such that kR ≳ 1, where k = 2π/λ is the wavevector of the optical grating. In the following, we will use the formalism based on Mie scattering theory^{62} to account for a large particle traversing an optical grating. For what concerns the purephase character of the grating—i.e., its coherent effect on the particle’s state—it can be shown that the unitary evolution of the particle’s state \(\hat{\rho }\) (reduced along the longitudinal direction z) when traversing the grating assumes, in the eikonal approximation, the form \(\left\langle z\right\rho \leftz^{\prime} \right\rangle \to \exp \left[i{\phi }_{0}({\cos }^{2}kz{\cos }^{2}kz^{\prime})\right]\left\langle z\right\rho \leftz^{\prime} \right\rangle ,\) where ϕ_{0} is the eikonal phase factor characterizing the coherent evolution. This is the same as in the case of a pointlike particle and the only difference introduced by the use of Mie scattering theory^{96,97} is found in the structure of the eikonal phase ϕ_{0} which can be expressed as
in terms of the laser and particle parameters. Here, cϵ_{0}∣E_{0}∣^{2}/2 is the intensity parameter of the incident light, E_{L} and a_{L} are the grating laser energy and spot area, respectively, and F_{0} is obtained from Mie theory upon the evaluation at z = −λ/8 of the longitudinal conservative force acting on the particle^{62,92}. Equation (6) reduces to the wellknown result \({\phi }_{0}=2{\mathcal{R}}(\chi){E}_{{\rm{L}}}/(\hslash c{\epsilon }_{0}{a}_{{\rm{L}}})\) with χ the polarizability for a pointlike particle. For what concerns the incoherent effects of the grating, the finite size of the particles leads to modify the Talbot coefficients with respect to the pointlike case. We refer the reader to Supplementary Note 1 for further details.
Finally, large particle nearfield interferometric experiments present several technical challenges^{26}. Common to both ground and spacebased experiments is the challenge of diminishing as far as possible any environmental noise which would suppress the interference pattern. This can be achieved by a combination of ultrahigh vacuum and cryogenic conditions. Moreover, for experiments aiming at using single particles in several (~10^{4}) runs, a fast reloading/recycling technique must be developed^{4,98,99,100}. On top of these challenges, the key limitation for groundbased experiments is the short freefall time. This is due to the Earth’s gravitational field and limits such experiments to a few seconds of free evolution. While this challenge can be overcome in principle, it will require a substantial modification of the scheme to go beyond masses of the order of 10^{7} amu^{26,29}. This is not the case for spacebased experiments, where current estimates show the promise to reach masses of the order of 10^{9}–10^{11} amu and freefalling times of the order of hundreds of seconds^{2,4}. In the following section, we substantiate these claims by presenting an optimized analysis of spacebased nearfield interferometry showing the actual possibilities offered by a space environment.
Optimization for large particles: the current frontiers
We present in this section the results of a numerical investigation of the possibilities offered by spacebased experiments in conjunction with nearfield interferometry as discussed in the previous section. We employ the formalism based on Mie scattering theory^{26} to account for the finite size of the particles with respect to the grating period, and we use the experimental parameters, as summarized in Supplementary Note 2, which have been extracted from the QPPF study about the MAQRO mission^{4}. We are able to include in our analysis all the major known sources of environmental decoherence which can affect the interference pattern. In particular, we account for scattering and absorption of grating photons at stage (c) of the protocol, residual gas collisions, and blackbody thermal radiation decoherence during the free evolution stage (b–d). We then include the effect of modifications of quantum mechanics in the form of the CSL model with whitenoise^{23}. What we present here is the first fully consistent analysis of such a setup and its potential for fundamental physics studies, which does not rely on the Rayleigh approximation, which cannot be consistently used unless for order of magnitude estimates.
One complication of nearfield interferometry—in contrast with the textbook case of farfield interferometry—that needs to be taken into account when performing an experiment is that also perfectly classical particles following ballistic trajectories through the optical grating would form a typical interferencelike figure known as Moiré shadow pattern^{89} (see Fig. 5). It is thus of crucial importance to discriminate between this pattern and a quantum mechanical one^{88}. This is a prerequisite for both claiming to be able to test the superposition principle and for any analysis of modifications of standard quantum mechanics. Thus, we introduce the first figure of merit (ℵ_{QC}) to estimate the "distance” between the quantum interference patter’s probability distribution (pdf) and the pdf of the shadow pattern which would result from classical mechanics
where L = 10^{−7} m is the spatial window in which the position measurement is performed and P_{Clas} (P_{Q}) is the pdf predicted by classical (quantum) mechanics. A similar quantity can then be obtained to discriminate between a quantum interference pattern and the pattern deriving from modifications of quantum mechanics. We will focus here on the CSL model with white noise. Thus the second figure of merit that we will employ is ℵ_{QCSL}, which is given by Eq. (7) with P_{Clas} → P_{CSL}.
In the following, we assume to be able to discriminate values of ℵ ≥ 0.05 (i.e., difference bigger than 5%) which appears to be an experimentally justifiable choice^{101}. Moreover, we optimize over the parameters t_{1}, t_{2}, E_{L}/a_{L} of the setup, which can be easily controlled, to maximize the figure of merits. As we will see, the optimization leads to values of the figure of merits well above the 5% threshold. Before illustrating the results of the analysis, let us comment on the choice of parameters. On the one hand, the freefalling times t_{1}, t_{2} are extremely important in the formation of the interference figure, whether it is t_{1} which guarantees a sufficient spreading of the initial state to a coherence length covering more than two “slits” or t_{2} which allows the interference to happen. These two times can also be easily adjusted in a spacebased experiment by simply changing the activation times of the grating and measurement lasers. On the other hand, the parameter E_{L}/a_{L} enters directly in Eq. (6) and thus determines the purephase coherent effect of the grating. This parameter can also be easily tuned, being a property of the way the grating laser is operated. We keep instead fix all the other parameters entering our analysis (see Supplementary Table 1). These are: the wavelength of the grating laser, which is dictated by current technological possibilities; the material(s) parameters of the nanosphere, we considered silica (SiO2) particles which are widely employed in optomechanical experiments for their optical properties; environmental parameters, which have been extracted from the QPPF study^{4} and represent the current stateoftheart for spacebased setups. Furthermore, always referring to the QPPF study on the stability of a possible mission’s spacecraft, we constrain the total freefall time to t_{1} + t_{2} ≤ 100 s. Note that, the QPPF study concludes that, due to vacuum restriction, the interference pattern for the proposed MAQRO mission would be visible for freefalling times of up to 40 s. However, the 100 s benchmark is among the scientific objectives of the community, as reported in the QPPF. We thus chose to present our results with this constraint on the times. Nonetheless, our analysis shows that a freefall time of 100 s would be achievable within the parameters of the QPPF without spoiling the interference pattern.
As outlined above, the first step in the analysis is to consider when ℵ_{QC} is large enough to guarantee the possibility to certify a quantum mechanical interference pattern and then consider the corresponding ℵ_{QCSL}. Figure 6 shows the results of our numerical investigation in this respect. The panels in the first row show the values of ℵ_{QC}, i.e., the distance between the classical shadow pattern and the quantum interference one, for particles masses {10^{7}, 10^{8}, 10^{9}, 10^{10}, 10^{11}} amu as a function of t_{1}, t_{2} and for the values of E_{L}/a_{L} which maximize the distinguishability. The latter is reported, as a function of t_{1}, t_{2}, in the Supplementary Note 2 (see Supplementary Fig. 2 therein). From the first row of Fig. 6, we see that ℵ_{QC} takes values definitely larger than the experimentally justifiable threshold of 5% for freefall times t_{1} + t_{2} ≤ 100 s, opening the way to direct tests of the quantum superposition principle with mesoscopic quantum systems in large spatial superpositions. The panels in the second row in Fig. 6 show instead of the comparison between the quantum interference pattern and the one which would arise if the CSL noise—with parameters chosen at λ_{CSL} = 10^{−8} s^{−1} and r_{c} = 10^{−7} m as proposed by Adler^{102}—was present. The panels on the second row are obtained by evaluating the cost function ℵ_{QCSL} at the same values of E_{L}/a_{L} used for the upper row, i.e., the values that, at fixed {t_{1}, t_{2}}, maximize the quantumtoclassical distinguishability. It should be noted that, for the comparison between CSL and quantum mechanics, we do not necessarily need to restrict our attention to only the values of the parameter E_{L}/a_{L} that maximize the classicalquantum distinguishability. Indeed, by direct inspection of the interference figures it can be deduced that, in general, the classical and CSL patterns are quite different as far as they are not both flattened out by the effects of the noises (environmental or fundamental). This means that we can look for other parameter values which increase the distance between the quantum and CSL patterns. We show this on the third row of Fig. 6 where we report the values of ℵ_{QCSL} at the values of E_{L}/a_{L} which maximize it. As it can be seen, the difference with the panels of the second row is not large apart for very light masses, meaning that the combined maximum distinguishability is nearly achievable.
Finally, Fig. 7 extends the previous analysis to the whole parameter space of the CSL model. This exclusion plot is obtained for values of the parameters t_{1}, t_{2}, E_{L}/a_{L} which maximize the distinguishability between the quantum and CSL predictions, i.e., ℵ_{QCSL}, as shown in the third row of Fig. 6. The solid lines in Fig. 7 show the upper bounds that could be achieved with spacebased nearfield interferometry experiments with particle masses up to 10^{11} amu. As it can be seen, already the use of 10^{9} amu particles (green solid line) has the potential to rule out collapse models even beyond the values GRW originally proposed for the parameters, a feat that is outside the reach of current experiments. This is one of the main results of this work. It shows that nearfield spacebased experiments hold the promise to push tests of quantum mechanics—and of collapse models—way beyond what is possible with groundbased experiments and have the ability to directly access a large and unexplored area of parameter space {λ_{CSL}, r_{c}} of the considered modifications of quantum mechanics.
In conclusion, we should cite that, while the analysis presented in this section makes use of the formalism developed to account for the finite size of the particles^{62}, and we have included all major sources of decoherence following the technical details laid down in the ESA’s QPPF report^{4}, the description of the system suffers from an unavoidable level of idealization. Without entering in the discussion of technical challenges like the load and reuse of the nanoparticles in several runs of the experiment, we can still point out some of the idealizations made that enter directly into the simulations of the interferometric setup. In particular, throughout this work we have assumed: the particles to be perfectly spherical, thus neglecting rotational degrees of freedom; the particles to be homogeneous, which has allowed us to use the formalism^{62} derived from Miescattering theory; finally, we have employed the sphere’s bulk material refraction index which is tabulated in the literature. This last point is discussed in some detail in recent works^{62}, where it is shown how the coherence properties of the grating interaction strongly depend on the refraction index. It is thus a crucial step for any realization of interferometric spacebased experiments with large nanoparticles to conduct preliminary experiments to determine the physical properties of the nanoparticles, with particular reference to their refractive index which could deviate from the bulk material one.
Conclusion and outlook
In this perspective article, we have discussed the unique possibilities offered by the space environment for investigating the quantum superposition principle by dedicated interfrometric and noninterferometric experiments and to test quantum mechanics in the parameter regime of largemass particles, impossible to reach on the ground by today’s technology. In particular, we have focused our attention on the generation and certification of spatial quantum superpositions of particles with sizes of the order of hundreds of nanometers and the possibilities that this offers for fundamental tests of quantum theory and alternatives thereof^{103}.
After arguing for the advantages offered by space, being the long freefall times and the availability of lownoise conditions, we considered two main experimental strategies for fundamental studies in space. The first one is the indirect approach of noninterferometric experiments, which does not require the creation of spatial superposition. This strategy has been proven key in recent work on the ground to test collapse models in otherwise unreachable parameter regimes. The second strategy is the more direct one based on interferometric experiments. Here, nearfield interferometry with large dielectric nanospheres is the current powerhouse, proven experimental technology, and shows its potential when combined with the advantages of the space environment. We have reported a detailed forecast of the potential offered by these techniques based on stateoftheart parameter values and showed how spacebased experiments offer the possibility to both certify the creation of macroscopic superpositions and essentially rule out an entire family of alternative models to standard quantum mechanics. Most importantly, we have not found a fundamental showstopper for performing both interferometric and noninterferometric experiments in space.
Needless to mention, large spatial superpositions of highmass systems will provide a fine probe for further tests of fundamental physics. This includes: the domain of highenergy particle physics beyond the standard model, when it comes to testing candidates of Dark Matter^{5,6,7,8,9,10} and possible effects in particle interactions related to Dark Energy^{11,12,13}; the lowenergy regime of the interplay between quantum mechanics and gravity^{46,104,105,106,107}; precision tests of gravity^{14,15,83,94,108,109,110,111}; the test of the equivalence principle and of general relativity’s predictions, such as gravitational waves, in a parameter range complementary to existing experiments such as LIGO, VIRGO, GEO600, and the planned LISA space antenna^{112,113}, and framedragging effects^{114}. Furthermore, largemass experiments in space will unavoidably provide a formidable platform for applications in Earth and planet observation^{115,116}, where largemass mechanical systems have already shown a superb capability as force and acceleration sensors^{78,117,118,119,120,121,122,123,124,125}, including in rotational mechanical modes^{55,126}.
It is clear that the realization of largemass, fundamental physics experiments in space is an immensely challenging project. Therefore, the most important next step is to form a community of scientists, industry, and space agencies for defining a concrete roadmap for the accomplishment of a successful space mission by working on finetuned theoretical analysis of conditions for the experiment, coming up with new proposals to test further new physics in the largemass regime and, last but not least, to push the development of technology readiness for space. Such a roadmap should include performing proofofprinciple largemass experiments in microgravity environments— such a sounding rockets, drop towers and Einstein elevators, space stations, CubeSats, and potentially on the Moon and Mars—in alignment with international and national space agencies plans for future fundamental physics experiments in space. We hope that the results of this work will stimulate the physics community to further investigate the possibilities offered by spacebased experiments of this kind.
Data availability
Data are available upon reasonable requests to the corresponding authors.
Code availability
Code is available upon reasonable requests to the corresponding authors.
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Acknowledgements
A. Belenchia acknowledges support from the MSCA IF project pERFEcTO (Grant No. 795782) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project number BR 5221/41. A. Bassi acknowledges financial support from the INFN, the University of Trieste and the support by grant number (FQXiRFPCPW2002) from the Foundational Questions Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley Community Foundation. S. Donadi acknowledges financial support from INFN and the Fetzer Franklin Fund. G.G. acknowledges support from the Spanish Agencia Estatal de Investigación, project PID2019107609GBI00, from the QuantERA grant C’MONQSENS!, by Spanish MICINN PCI20191118692, and from COST Action CA15220. R.K. acknowledges support by the Austrian Research Promotion Agency (projects 854036, 865996) and by the Slovenian Research Agency (research projects N10180, J22514, J19145 and P10125). M.P. is supported by the DfESFI Investigator Programme (grant 15/IA/2864), the Royal Society Wolfson Research Fellowship (RSWF\R3\183013), the Leverhulme Trust Research Project Grant (grant no. RGP2018266), and the UK EPSRC (grant no. EP/T028106/1). H.U. acknowledges support from The Leverhulme Trust (RPG2016046) and the UK EPSRC (EP/V000624/1). A. Bassi, M.C., M.P., and H.U. are supported by the H2020 FET Project TEQ (Grant No. 766900). All the authors acknowledge partial support from COST Action QTSpace (CA15220) and thank Alexander Franzen for the creation of one of the figures in the manuscript.
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G.G. and A.Belenchia. led the development of the project with strong input from M.C. and S.D. and with significant contributions from A.Bassi, R.K., M.P., and H.U. All authors contributed to the preparation of the manuscript and its finalization.
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Gasbarri, G., Belenchia, A., Carlesso, M. et al. Testing the foundation of quantum physics in space via Interferometric and noninterferometric experiments with mesoscopic nanoparticles. Commun Phys 4, 155 (2021). https://doi.org/10.1038/s42005021006567
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DOI: https://doi.org/10.1038/s42005021006567
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