Quantum physics has intrigued scientists and philosophers alike, because it challenges our notions of reality and locality-concepts that we have grown to rely on in our macroscopic world. It is an intriguing open question whether the linearity of quantum mechanics extends into the macroscopic domain. Scientific progress over the last decades inspires hope that this debate may be decided by table-top experiments. IntroductionThe last three decades have witnessed what has been termed 1 the second quantum revolution: A renaissance of research on the quantum foundations, hand in hand with growing experimental capabilities, 2 revived the idea of exploiting quantum superpositions for technological applications, from information science [3][4][5] to precision metrology. [6][7][8] Quantum mechanics has passed all precision tests with flying colors, but it still seems to be in conflict with our common sense. Since quantum theory knows no boundaries everything should fall under the sway of the superposition principle, including macroscopic objects. This is at the bottom of Schrödinger's thought experiment transforming a cat into a state which strikes us as classically impossible. And yet, 'Schrödinger kittens' of entangled photons 9 and ions 10 have been realized in the lab.So why are the objects around us never found in superpositions of states that would be excluded in a classical description? One may emphasize the smallness of Planck's constant, or point to decoherence theory, which describes how a system will effectively lose its quantum features when coupled to a quantum environment of sufficient size. 11,12 The formalism of decoherence, however, is based on the framework of unitary quantum mechanics, implying that some interpretational exercise is required not to become entangled in a multitude of parallel worlds. 13 More radically, one may ask whether quantum mechanics breaks down beyond a certain mass or complexity scale. As will be discussed below, such ideas can be motivated by the apparent incompatibility of quantum theory and general relativity. It is safe to state, in any case, that quantum superpositions of truly massive, complex objects are terra incognita. This makes them an attractive challenge for a growing number of sophisticated experiments.We start by reviewing several prototypical tests of the superposition principle, focusing on the quantum states of motion displayed by material objects. Particle position and momentum variables have a well-defined classical analogue, and they are therefore particularly suited to probe the macroscopic domain. We note that aspects of macroscopicity can also be addressed in experiments with photons, [14][15][16] Figure 1A): The single-valuedness of the wave function entails that the magnetic flux encircled by a closed-loop supercurrent must be quantized. In particular, one can define a symmetric and an antisymmetric linear combination of two supercurrents, which circulate simultaneously in opposing directions. Billions of electrons may contribute coherently to the wave...

Emergent quantum technologies have led to increasing interest in decoherence -the processes that limit the appearance of quantum effects and turn them into classical phenomena. One important cause of decoherence is the interaction of a quantum system with its environment, which 'entangles' the two and distributes the quantum coherence over so many degrees of freedom as to render it unobservable. Decoherence theory [1][2][3][4] has been complemented by experiments using matter waves coupled to external photons [5][6][7] or molecules [8], and by investigations using coherent photon states [9], trapped ions [10] and electron interferometers [11,12]. Large molecules are particularly suitable for the investigation of the quantumclassical transition because they can store much energy in numerous internal degrees of freedom; the internal energy can be converted into thermal radiation and thus induce decoherence. Here we report matter wave interferometer experiments in which C 70 molecules lose their quantum behaviour by thermal emission of radiation. We find good quantitative agreement between our experimental observations and microscopic decoherence theory. Decoherence by emission of thermal radiation is a general mechanism that should be relevant to all macroscopic bodies.In this Letter we investigate the decoherence of molecular matter waves. We change the internal temperature of the molecules in a controlled way before they enter a near-field interferometer, and observe the corresponding reduction of the interference contrast. The idea behind this effort is to demonstrate a most fundamental decoherence mechanism that we encounter in the macroscopic world: All large objects, but also molecules of sufficient complexity, are able to store energy and to interact with their environment via thermal emission of photons. It is generally believed that warm macroscopic bodies emit far too many photons to allow the observation of de Broglie interferences, whereas individual atoms or molecules can be sufficiently well isolated to exhibit their quantum nature. However, there must be a transition region between these two limiting cases. Interestingly, as we show in this study, C 70 fullerene molecules have just the right amount of complexity to exhibit perfect quantum interference in our experiments [13] at temperatures below 1000 K, and to gradually lose all their quantum behaviour when the internal temperature is increased up to 3000 K. We can thus trace the quantum-to-classical transition in a controlled and quantitative way. The complexity of large molecules adds a novel quality with respect to previously performed experiments with atoms [5][6][7]: the energy in molecules may be equilibrated in many internal degrees of freedom during the free flight, and a fraction of the vibrational energy will eventually be reconverted into emitted photons. Therefore the internal dynamics of the molecule is also relevant for the quantum behaviour of the centre-of-mass state. In contrast to resonance fluorescence, which was investigated wit...

We review recent progress and future prospects of matter wave interferometry with complex organic molecules and inorganic clusters. Three variants of a near-field interference effect, based on diffraction by material nanostructures, at optical phase gratings, and at ionizing laser fields are considered. We discuss the theoretical concepts underlying these experiments and the experimental challenges. This includes optimizing interferometer designs as well as understanding the role of decoherence. The high sensitivity of matter wave interference experiments to external perturbations is demonstrated to be useful for accurately measuring internal properties of delocalized nanoparticles. We conclude by investigating the prospects for probing the quantum superposition principle in the limit of high particle mass and complexity.

We study the loss of spatial coherence in the extended wave function of fullerenes due to collisions with background gases. From the gradual suppression of quantum interference with increasing gas pressure we are able to support quantitatively both the predictions of decoherence theory and our picture of the interaction process. We thus explore the practical limits of matter wave interferometry at finite gas pressures and estimate the required experimental vacuum conditions for interferometry with even larger objects.PACS numbers: 03.65.Yz,39.20.+q Matter wave interferometers are based on quantum superpositions of spatially separated states of a single particle. However, as is well known, the concept of wave-particle duality does not apply to a classical object which by definition never occupies macroscopically distinct states simultaneously. By performing interference experiments with particles of increasing complexity one can therefore probe the borderline between these incompatible descriptions.It is still a matter of debate how to explain the quantum-to-classical transition in a unified framework. Some theories contain an element beyond the unitary evolution of quantum mechanics [1, 2] -which includes the 'collapse' of the wave function as taught in many standard textbooks. Decoherence theory, on the other hand, remains within the framework of the quantum theory [3,4,5]. It explains the decay of quantum coherences as being caused by the interaction of the quantum object with its environment.So far, several decoherence experiments in atom interferometry focused on the loss of coherence due to scattering of a single [6,7] or a few [8] laser photons by an atom. Other authors proposed or realized schemes to encode which-path information in internal atomic degrees of freedom, thereby reducing the interference contrast as well, in spite of a negligible change in the atomic centerof-mass state [9,10]. These studies are complemented by experiments which quantitatively followed the decoherence of a coherent photon state in a high-finesse microwave cavity [11] or of the motional state of a trapped ion [12]. However, all these experiments worked with few-level systems and engineered environments.In the present letter we quantitatively investigate a mechanism which seems to be among the most natural and most effective sources of decoherence in our macroscopic world, namely collisions with gas particles. From the controlled suppression of quantum interference as a function of the gas pressure we are able to test both the predictions of decoherence theory and our picture of the collisional interaction.We note that the effect of atomic collisions in an atom interferometer was already investigated in [13]. How- ever, decoherence effects were not observed in these experiments, since the detected atoms did not change the state of the colliding gas sufficiently to leave behind the required path information for decoherence. In contrast to that, our experiment uses massive C 70 -fullerene molecules, and is based on a Talbot-L...

Research on matter waves is a thriving field of quantum physics and has recently stimulated many investigations with electrons 1 , neutrons 2 , atoms 3 , Bose-condensed ensembles 4 , cold clusters 5 and hot molecules 6 . Coherence experiments with complex objects are of interest for exploring the transition to classical physics 7-9 , for measuring molecular properties 10 , and they have even been proposed for testing new models of space-time 11 . For matter-wave experiments with complex molecules, the strongly dispersive effect of the interaction between the diffracted molecule and the grating wall is a major challenge because it imposes enormous constraints on the velocity selection of the molecular beam 12 . Here, we describe the first experimental realization of a new set-up that solves this problem by combining the advantages of a so-called Talbot-Lau interferometer 13 with the benefits of an optical phase grating.Several methods have been developed in the past for the coherent manipulation of matter waves with de Broglie wavelengths in the nanometre and picometre range. For instance, free-standing material gratings were used in the diffraction of electrons 14 , atoms 15,16 and molecules 5,6,17 . In addition, coherent beam splitting at non-resonant standing light waves, often designated the KapitzaDirac effect, has been observed for all of these species [18][19][20] . Recent implementations of near-field interferometry 13,[21][22][23] underlined the particular advantages of the Talbot-Lau concept for experiments with massive objects: the required grating period scales only weakly (d ∼ √ l) with the de Broglie wavelength, and the design accepts beams of low spatial coherence, which makes high signals possible even for weak sources.A symmetric Talbot-Lau interferometer (TLI) consists of three identical gratings. The first one prepares the transverse coherence of the weakly collimated beam. Quantum near-field diffraction at the second nanostructure generates a periodic molecular density distribution at the position of the third mask, which represents a self-image of the second grating, if the grating separation equals a multiple of the Talbot length L T = d 2 /l. The mask can be laterally shifted to transform the molecular interference pattern into a modulation of the molecular beam intensity that is recorded behind the interferometer.In the established TLI design with three nanofabricated gratings 23 , the molecule-wall interaction with the grating bars imprints a further phase shift ϕ on the matter wave, which depends on the molecular polarizability α, the velocity v z and the distance r to the wall within the grating slit. Because of its strongly nonlinear r-dependence, this interaction restricts the interference contrast to very narrow bands of de Broglie wavelengths, as we show in Fig. 1a for the example of the fullerene C 70 . In this simulation, we use the full Casimir-Polder potential 24 , even though the long-distance (retarded) approximation, decaying as α/r 4 , closely reproduces the results. The sha...

The wave nature of matter is a key ingredient of quantum physics and yet it defies our classical intuition. First proposed by Louis de Broglie a century ago, it has since been confirmed with a variety of particles from electrons up to molecules. Here we demonstrate new high-contrast quantum experiments with large and massive tailor-made organic molecules in a near-field interferometer. Our experiments prove the quantum wave nature and delocalization of compounds composed of up to 430 atoms, with a maximal size of up to 60 Å, masses up to m=6,910 AMU and de Broglie wavelengths down to λdB=h/mv≃1 pm. We show that even complex systems, with more than 1,000 internal degrees of freedom, can be prepared in quantum states that are sufficiently well isolated from their environment to avoid decoherence and to show almost perfect coherence.

We propose an experimentally accessible, objective measure for the macroscopicity of superposition states in mechanical quantum systems. Based on the observable consequences of a minimal, macrorealist extension of quantum mechanics, it allows one to quantify the degree of macroscopicity achieved in different experiments.

Matter-wave interferometry performed with massive objects elucidates their wave nature and thus tests the quantum superposition principle at large scales. Whereas standard quantum theory places no limit on particle size, alternative, yet untested theories-conceived to explain the apparent quantum to classical transition-forbid macroscopic superpositions. Here we propose an interferometer with a levitated, optically cooled and then free-falling silicon nanoparticle in the mass range of one million atomic mass units, delocalized over 4150 nm. The scheme employs the near-field Talbot effect with a single standing-wave laser pulse as a phase grating. Our analysis, which accounts for all relevant sources of decoherence, indicates that this is a viable route towards macroscopic high-mass superpositions using available technology.