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To enhance the consistency between the quantum descriptions of waves and particles, we quantise mechanical point particles in this paper in the same physically-motivated way as we previously quantised light in quantum electrodynamics (Bennett et al 2016 Eur. J. Phys. 37 014001). To identify the relevant Hilbert space, we notice that mechanical particles can occupy any position x while moving at any velocity v . Afterwards, we promote the classical states ( x ,  v ) to pairwise orthogonal quantum states | x , v ⟩ and demand that these evolve according to Newton’s equations of motion. The resulting quantum theory is mass-independent, when Newton’s equations of motion are mass-independent, as one would expect. The basic formulation of quantum mechanics emerges from quantum mechanics in configuration space as a semi-classical approximation when a fixed mass is imposed and several other adjustments are made.

Wave-particle duality as a fundamental tenet of quantum mechanics is crucial for advancing comprehension of quantum theories and developing quantum technologies with practical applications. However, taking into account experimental impact factors to develop a feasible measurement for wave-like and particle-like properties of light fields is an ongoing challenge, and the non-classicality extraction and determination remains to be explored. In this work, feasibly measurable second-order photon correlations based on Hanbury Brown−Twiss and Hong−Ou−Mandel interferences are employed to analyze the evolution of wave−particle duality for various input states. The wave-particle dualities of chaotic, coherent and mixed classical states as functions of time delay and coherence time are investigated. The realistic impacts of background noise, detection efficiency, intensity ratio and phase differences on the wave−particle duality of non-classical (Fock and squeezed coherent) states are unveiled. In noisy backgrounds with low detection efficiencies, efficient enhancement and extraction of non-classicality and a continuous transition from classical to non-classical region are achieved in single photon state mixed with coherent state by adjusting the phase difference from 0 to π / 2 . The non-classicality of squeezed coherent state can be induced by the classical wave-like and particle-like properties. The research provides a practical precision measurement of wave−particle duality that is helpful for the improvement of high-resolution quantum imaging and sensing.

Within the framework of quantum mechanics, the wave function squared describes the probability density of particles. In this article, another description of the wave function is given which embeds quantum mechanics into the traditional fields of physics, thus making new interpretations dispensable. The new concept is based on the idea that each microscopic particle with non-vanishing rest mass is accompanied by a matter wave, which is formed by adjusting the phases of the vacuum fluctuations in the vicinity of the vibrating particle. The vibrations of the particle and wave are phase-coupled. Particles move on continuous approximately classical trajectories. By the phase coupling mechanism, the particle transfers the information on its kinematics and thus also on the external potential to the wave. The space dependence of the escorting wave turns out to be equal to the wave function. The new concept fundamentally differs from the pilot wave concept of Bohmian mechanics.

Wave–particle duality as the defining characteristic of quantum objects is a typical example of the principle of complementarity. The wave–particle–entanglement (WPE) complementarity, initially developed for two-qubit systems, is an extended form of complementarity that combines wave–particle duality with a previously missing ingredient, quantum entanglement. For two-qubit systems in mixed states, the WPE complementarity was further completed by adding yet another piece that characterizes ignorance, forming the wave–particle–entanglement–ignorance (WPEI) complementarity. A general formulation of the WPEI complementarity can not only shed new light on fundamental problems in quantum mechanics, but can also have a wide range of experimental and practical applications in quantum-mechanical settings. The purpose of this study is to establish the WPEI complementarity for general multi-dimensional bipartite systems in pure or mixed states, and extend its range of applications to incorporate hierarchical and infinite-dimensional bipartite systems. The general formulation is facilitated by well-motivated generalizations of the relevant quantities. When faced with different directions of extensions to take, our guiding principle is that the formulated complementarity should be as simple and powerful as possible. We find that the generalized form of the WPEI complementarity contains unequal-weight averages reflecting the difference in the subsystem dimensions, and that the tangle, instead of the squared concurrence, serves as a more suitable entanglement measure in the general scenario. Two examples, a finite-dimensional bipartite system in mixed states and an infinite-dimensional bipartite system in pure states, are studied in detail to illustrate the general formalism. We also discuss our results in connection with some previous work. The WPEI complementarity for general finite-dimensional bipartite systems may be tested in multi-beam interference experiments, while the second example we studied may facilitate future experimental investigations on complementarity in infinite-dimensional bipartite systems.

Tunnelling is one of the most characteristic phenomena of quantum physics, underlying processes such as photosynthesis and nuclear fusion, as well as devices ranging from superconducting quantum interference device (SQUID) magnetometers to superconducting qubits for quantum computers. The question of how long a particle takes to tunnel through a barrier, however, has remained contentious since the first attempts to calculate it 1 . It is now well understood that the group delay 2 —the arrival time of the peak of the transmitted wavepacket at the far side of the barrier—can be smaller than the barrier thickness divided by the speed of light, without violating causality. This has been confirmed by many experiments 3 – 6 , and a recent work even claims that tunnelling may take no time at all 7 . There have also been efforts to identify a different timescale that would better describe how long a given particle spends in the barrier region 8 – 10 . Here we directly measure such a time by studying Bose-condensed 87 Rb atoms tunnelling through a 1.3-micrometre-thick optical barrier. By localizing a pseudo-magnetic field inside the barrier, we use the spin precession of the atoms as a clock to measure the time that they require to cross the classically forbidden region. We study the dependence of the traversal time on the incident energy, finding a value of 0.61(7) milliseconds at the lowest energy for which tunnelling is observable. This experiment lays the groundwork for addressing fundamental questions about history in quantum mechanics: for instance, what we can learn about where a particle was at earlier times by observing where it is now 11 – 13 . Using the spin precession of Bose-condensed 87 Rb atoms as a clock, direct measurements are made of the time required for Rb atoms to quantum tunnel through a classically impenetrable barrier.

Optics--a field of physics focusing on the study of light--is also central to many areas of biology, including vision, ecology, botany, animal behavior, neurobiology, and molecular biology. The Optics of Life introduces the fundamentals of optics to biologists and nonphysicists, giving them the tools they need to successfully incorporate optical measurements and principles into their research. Sönke Johnsen starts with the basics, describing the properties of light and the units and geometry of measurement. He then explores how light is created and propagates and how it interacts with matter, covering topics such as absorption, scattering, fluorescence, and polarization. Johnsen also provides a tutorial on how to measure light as well as an informative discussion of quantum mechanics.

Two-particle interference is a fundamental feature of quantum mechanics, and is even less intuitive than wave–particle duality for a single particle. In this duality, classical concepts—wave or particle—are still referred to, and interference happens in ordinary space-time. On the other hand, two-particle interference takes place in a mathematical space that has no classical counterpart. Entanglement lies at the heart of this interference, as it does in the fundamental tests of quantum mechanics involving the violation of Bell's inequalities 1 , 2 , 3 , 4 . The Hong, Ou and Mandel experiment 5 is a conceptually simpler situation, in which the interference between two-photon amplitudes also leads to behaviour impossible to describe using a simple classical model. Here we report the realization of the Hong, Ou and Mandel experiment using atoms instead of photons. We create a source that emits pairs of atoms, and cause one atom of each pair to enter one of the two input channels of a beam-splitter, and the other atom to enter the other input channel. When the atoms are spatially overlapped so that the two inputs are indistinguishable, the atoms always emerge together in one of the output channels. This result opens the way to testing Bell's inequalities involving mechanical observables of massive particles, such as momentum, using methods inspired by quantum optics 6 , 7 , and to testing theories of the quantum-to-classical transition 8 , 9 , 10 , 11 . Our work also demonstrates a new way to benchmark non-classical atom sources 12 , 13 that may be of interest for quantum information processing 14 and quantum simulation 15 . The Hong–Ou–Mandel effect—in which two indistinguishable photons that enter a 50:50 beam-splitter are found only as a pair at one of the two outputs, leading to a dip in the coincidence rate of the detectors—is now realized with 4 He atoms instead of photons; this opens the way to performing basic quantum-physics experiments with mechanical observables of massive particles. 4 He in the Hong-Ou-Mandel experiment The Hong-Ou-Mandel effect, in which two indistinguishable photons entering a 50:50 beam splitter lead to a dip in the coincidence rate of the detectors, demonstrates basic features of the theory of quantum mechanics and has no classical analogue. It is the basis of many other experiments in quantum information and quantum optics. Here Marc Cheneau and colleagues realize the Hong-Ou-Mandel experiments with helium-4 atoms instead of photons, building on previous developments on atom pair production. Since atoms, as opposed to photons, are massive particles, this experiment opens possibilities to perform basic quantum physics experiments, like Bell tests, with observables of massive particles. In the long run, this could contribute to research on the effect of mass and gravity in quantum mechanics, especially in relation to the quantum-to-classical transition.

Based on the concept of geometric coherence, we propose a wave-particle duality relation that establishes an equation between distinguishability and visibility for any pure state in the composite system of a general n-path interferometer and a which-path detector. This relation has the desired property that any increase of distinguishability will inevitably cause a decrease in visibility, thereby quantitatively capturing the essence of complementarity. Additionally, we reveal a connection between the fidelity function and discrimination of pure states. Specifically, we generalize the previous result in Xiong and Wu (2018 J. Phys. A 51 414005) by showing that for any quantum state ρ, the geometric coherence can be exactly expressed as the optimal guessing probability in the discrimination of the corresponding ensemble of pure states.

Anderson localization of matter waves in a Bose-Einstein condensate Anderson localization of waves in disordered media was originally predicted fifty years ago, in the context of transport of electrons in crystals. The phenomenon is much more general and has been observed in a variety of systems, but never directly for matter waves. The authors use a non-interacting Bose–Einstein condensate of ultracold atoms to study Anderson localization. The effect is clearly demonstrated through investigations of the transport properties and spatial and momentum distributions. The highly controllable nature of the system may render it useful for investigations of the interplay between disorder and interaction, and to uncover exotic quantum phases. Anderson localization of waves in disordered media was originally predicted 1 fifty years ago, in the context of transport of electrons in crystals 2 . The phenomenon is much more general 3 and has been observed in a variety of systems, including light waves 4 , 5 . However, Anderson localization has not been observed directly for matter waves. Owing to the high degree of control over most of the system parameters (in particular the interaction strength), ultracold atoms offer opportunities for the study of disorder-induced localization 6 . Here we use a non-interacting Bose–Einstein condensate to study Anderson localization. The experiment is performed with a one-dimensional quasi-periodic lattice—a system that features a crossover between extended and exponentially localized states, as in the case of purely random disorder in higher dimensions. Localization is clearly demonstrated through investigations of the transport properties and spatial and momentum distributions. We characterize the crossover, finding that the critical disorder strength scales with the tunnelling energy of the atoms in the lattice. This controllable system may be used to investigate the interplay of disorder and interaction (ref. 7 and references therein), and to explore exotic quantum phases 8 , 9 .

A droplet bouncing on a liquid bath can self-propel due to its interaction with the waves it generates. The resulting \"walker\" is a dynamical association where, at a macroscopic scale, a particle (the droplet) is driven by a pilot-wave field. A specificity of this system is that the wave field itself results from the superposition of the waves generated at the points of space recently visited by the particle. It thus contains a memory of the past trajectory of the particle. Here, we investigate the response of this object to forces orthogonal to its motion. We find that the resulting closed orbits present a spontaneous quantization. This is observed only when the memory of the system is long enough for the particle to interact with the wave sources distributed along the whole orbit. An additional force then limits the possible orbits to a discrete set. The wave-sustained path memory is thus demonstrated to generate a quantization of angular momentum. Because a quantum-like uncertainty was also observed recently in these systems, the non-locality generated by path memory opens new perspectives.