This Tutorial will introduce the mathematical framework for describing systems of identical particles, and explain the notion of indistinguishability. We will then focus our attention on dynamical systems of free particles and formally introduce the concept of many-particle interference. Its impact on many-particle transition probabilities is computationally challenging to evaluate, and it becomes rapidly intractable for systems with large numbers of identical particles. Hence, this Tutorial will build up towards alternative, more efficient methods for observing signatures of many-particle interference. A first type of signatures relies on the detection of a highly sensitive -but also highly fragile-processes of total destructive interference that occurs in interferometers with a high degree of symmetry. A second class of signatures is based on the statistical features that arise when we study the typical behaviour of correlations between a small number of the interferometer's output ports. We will ultimately show how these statistical signatures of many-particle interference lead us to a statistical version of the Hong-Ou-Mandel effect.
We study a general theory on the interference of two-photon wavepacket in a beam splitter. We find that the symmetry of two-photon spectrum plays an important role in the manners of interference. We distinguish the coalescence and anti-coalescence interferences, and prove that the anti-coalescence interference is the signature of photon entanglement.
We propose a simple numerical experiment of two slits interference of particles. It disproves the popular belief that such an interference is incompatible with a knowledge which slit each particle came through or, more generally, ``quantum particles could not have trajectories''. Our model is an illustration to the contextual interpretation of quantum probabilities.
Multi-particle interference is an essential ingredient for fundamental quantum mechanics phenomena and for quantum information processing to provide a computational advantage, as recently emphasized by Boson Sampling experiments. Hence, developing a reliable and efficient technique to witness its presence is pivotal towards the practical implementation of quantum technologies. Here we experimentally identify genuine many-body quantum interference via a recent efficient protocol, which exploits statistical signatures at the output of a multimode quantum device. We successfully apply the test to…
Matter wave revivals depend on a delicate interplay of constructive many-body quantum interferences in the developing dynamics of an ultracold bosonic system in an optical lattice. It is shown that the interplay between weak intersite tunneling and strong onsite interactions can lead to the quantum dynamics of a density wave displaying several features not found in the mean-field limit: occupancy oscillations, resurgent revivals, and a (anti-) synchronization of revival peaks and occupancy oscillation peaks. This implies cooperative interference effects that alternate between constructive and …
We predict the existence of high frequency modes in the interference pattern of two condensates made of fermionic-atom dimers. These modes, which result from fermion exchanges between condensates, constitute a striking signature of the dimer composite nature. From the 2-coboson spatial correlation function, that we derive analytically, and the Shiva diagrams that visualize many-body effects specific to composite bosons, we identify the physical origin of these high frequency modes and determine the conditions to see them experimentally by using bound fermionic-atom pairs trapped on optical lat…
Two-photon quantum interference at a beam splitter, commonly known as Hong-Ou-Mandel interference, was recently demonstrated with \emph{microwave-frequency} photons by Lang \emph{et al.}\,\cite{lang:microwaveHOM}. This experiment employed circuit QED systems as sources of microwave photons, and was based on the measurement of second-order cross-correlation and auto-correlation functions of the microwave fields at the outputs of the beam splitter. Here we present the calculation of these correlation functions for the cases of inputs corresponding to: (i) trains of \emph{pulsed} Gaussian or Lore…
We predict a generic manifestation of quantum interference in many-body bosonic systems resulting in a coherent enhancement of the average return probability in Fock space. This enhancement is both robust with respect to variations of external parameters and genuinely quantum insofar as it cannot be described within mean-field approaches. As a direct manifestation of the superposition principle in Fock space, it arises when many-body equilibration due to interactions sets in. Using a semiclassical approach based on interfering paths in Fock space, we calculate the magnitude of the backscatteri…
We study degenerate three photon down conversion as a potential scheme for generating nonclassical states of light which exhibit clear signatures of phase space interference. The Wigner function representing these states contains an interference pattern manifesting quantum coherence between distinct phase space components, and has substantial areas of negativity. We develop an analytical description of the interference pattern, which demonstrates how the oscillations of the Wigner function are built up by the superposition principle. We analyze the impact of dissipation and pump fluctuations o…
Motivated by recent proposals of Majorana qubits and the read-out of their quantum state we investigate a qubit setup formed by two parallel topological wires shunted by a superconducting bridge. The wires are further coupled to two quantum dots, which are also linked directly, thus creating an interference loop. The transport current through this system shows an interference pattern which distinguishes two basis states of the qubit in a QND measurement. We analyze various properties of the interference current and the read-out process, including the resulting dephasing and relaxation. We also…
Quantum theory permits interference between indistinguishable paths but, at the same time, restricts its order. Single-particle interference, for instance, is limited to the second order, that is, to pairs of single-particle paths. To date, all experimental efforts to search for higher-order interferences beyond those compatible with quantum mechanics have been based on such single-particle schemes. However, quantum physics is not bounded to single-particle interference. We here experimentally study many-particle higher-order interference using a two-photon-five-slit setup. We observe nonzero …
We propose entropic measures for the strength of single-particle and two-particle interference in interferometric experiments where each particle of a pair traverses a multi-path interferometer. Optimal single-particle interference excludes any two-particle interference, and vice versa. We report an inequality that states the compromises allowed by quantum mechanics in intermediate situations, and identify a class of two-particle states for which the upper bound is reached. Our approach is applicable to symmetric two-partite systems of any finite dimension.
Bohr's Complementarity Principle is quantitatively formulated in terms of the distinguishability of various paths a quanton can take, and the measure of the interference it produces. This phenomenon results from the interference of single-quanton amplitudes for various paths. The distinguishability of paths puts a bound on the sharpness of the interference the quanton can produce. However there exist other kinds of quantum phenomena where interference of two-particle amplitudes results in a two-particle interference, if the particles are indistinguishable. The Hong-Ou-Mandel (HOM) effect and t…
We analyze the double slit interference of a mesoscopic particle. We calculate the visibility of the interference pattern, introduce a characteristic temperature that defines the onset to decoherence and scrutinize the conditions that must be satisfied for an interference experiment to be possible.
Quantum interference of two independent particles in pure quantum states is fully described by the particles' distinguishability: the closer the particles are to being identical, the higher the degree of quantum interference. When more than two particles are involved, the situation becomes more complex and interference capability extends beyond pairwise distinguishability, taking on a surprisingly rich character. Here, we study many-particle interference using three photons. We show that the distinguishability between pairs of photons is not sufficient to fully describe the photons' behaviour …
Wave-particle duality in multi-path interference is fraught with issues despite substantial progress in recent years. It was experimentally shown that in certain specific conditions, getting path information in a multi-path experiment can actually increase the visibility of interference. As a result, it was argued that in multi-path interference experiments, visibility of interference and 'which-path' information are not always complementary observables. In the present work, a new wave-particle duality relation is presented, based on a sum of visibilities of interference from individual pairs …
Claims have been made that, in two-particle interference experiments involving bosons, Bohmian trajectories may entail observable consequences incompatible with standard quantum mechanics. By general arguments and by an examination of specific instances, we show that this is not the case.
Born's rule, one of the cornerstones of quantum mechanics, relates detection probabilities to the modulus square of the wave function. Single-particle interference is accordingly limited to pairs of quantum paths and higher-order interferences are prohibited. Deviations from Born's law have been quantified via the Sorkin parameter which is proportional to the third-order term. We here extend this formalism to many-particle interferences and find that they exhibit a much richer structure. We demonstrate, in particular, that all interference terms of order $(2M+1)$ and greater vanish for $M$ par…
In this note we demonstrate that a quantum-like interference picture could appear as a statistical effect of interference of deterministic particles, i.e. particles that have trajectories and obey deterministic equations, if one introduces a discrete time. The nature of the resulting interference picture does not follow from the geometry of force field, but is strongly attached to the time discreetness parameter. As a demonstration of this concept we consider a scattering of charged particles on the charged screen with a single slit. The resulting interference picture has a nontrivial minimum-…
The double-slit experiment is the most direct demonstration of interference between individual quantum objects. Since similar experiments with single particles and more slits produce interference fringes reducible to a combination of double-slit patterns it is usually argued that quantum interference occurs between pairs of trajectories, compactly denoted as second-order interference. Here we show that quantum mechanics in fact allows for interference of arbitrarily high order. This occurs naturally when one considers multiple quantum objects interacting in the presence of a nonlinearity, both…
All-order momentum correlation functions associated with the time-of-flight spectroscopy of three spinless ultracold bosonic interacting neutral atoms confined in a linear three-well optical trap are presented. The underlying Hamiltonian employed for the interacting atoms is an augmented three-site Hubbard model. Our investigations target matter-wave interference of massive particles, aiming at the establishment of experimental protocols for characterizing the quantum states of trapped attractively or repulsively interacting ultracold particles, with variable interaction strength. The manifest…
Entanglement can modify the interference patterns of multi-particle systems. We analyse, using the path integral formalism, a novel example of multi-particle interference and some unexplored aspects of this phenomenon by considering the two-slit arrangement with two distinguishable particles in a superposition state. The two components of entanglement, multiplicity of multi-particle terms and their coherence, can be studied separately by comparing the patterns of product, mixed and superposition states. This suggests a scheme to detect multi-particle superposition in some favourable cases. Mor…
Identical particles exhibit correlations even in the absence of inter-particle interaction, due to the exchange (anti)symmetry of the many-particle wavefunction. Two fermions obey the Pauli principle and anti-bunch, whereas two bosons favor bunched, doubly occupied states. Here, we show that the collective interference of three or more particles leads to a much more diverse behavior than expected from the boson-fermion dichotomy known from quantum statistical mechanics. The emerging complexity of many-particle interference is tamed by a simple law for the strict suppression of events in the Be…
We analyze an interferometric complementarity between one- and two-particle interference in the general case: $V_{i}^{2}+V_{12}^{2}\leq 1$ $(i=1$, $2)$, and further examine the relation among one-particle interference visibility $V_{i}$, two-particle interference visibility $V_{12}$ and the predication $P_{i}$ of the path of a single particle. An equality $V_{i}^{2}+V_{12}^{2}+P_{i}^{2}=1$ $(i=1$, $2)$ is achieved for any pure two-particle source, which implies the condition of the complementarity relation to reach the upper bound and its relation to another interferometric complementarity bet…
In Lorentz-violating electrodynamics a steady current (and similarly a static charge) generates both static magnetic and electric fields. These induced fields, acting on interfering particles, change the interference pattern. We find that particle interference experiments are sensitive to small Lorentz violating effects, and thus they can be used to improve current bounds on some Lorentz-violating parameters.
We show how robust statistical features of a many-particle quantum state's two-point correlations after transmission through a multi-mode random scatterer can be used as a sensitive probe of the injected particles' mutual indistinguishability. This generalises Hong-Ou-Mandel interference as a diagnostic tool for many-particle transmission signals across multi-mode random scatterers. Furthermore, we show how, from such statistical features of the many-particle interference pattern, information can be deduced on the temporal structure of the many-particle input state, by inspection of the many-p…
We propose a simple and realizable method using a two-particle interferometer for the experimental measurement of pairwise entanglement, assuming some prior knowledge about the quantum state. The basic idea is that the properties of the density matrix can be revealed by the single- and two-particle interference patterns. The scheme can easily be implemented with polarized entangled photons.
In this article a question of the ghost spinors influence to the quantum particles interference is investigated. The interaction between spinors and ghost spinors are considered. Furthermore the conditions of zero-point energy-momentum tensor in private cases are found. Also we consider a question about the experimental test of Deutsch shadow pacticles existense.
We analyze the predictions of an event-based corpuscular model for interference in the case of two-beam interference experiments in which the two sources are alternatingly blocked. We show that such experiments may be used to test specific predictions of the corpuscular model.
We conceive an all-optical representation of the dynamics of two distinct types of interacting bosons in a double well by an array of evanescently coupled photonic waveguides. Many-particle interference effects are probed for various interaction strengths by changing the relative abundance of the particle species and can be readily identified by monitoring the propagation of the light intensity across the waveguide array. In particular, we show that finite inter-particle interaction strengths reduce the many-particle interference contrast by dephasing. A general description of the many-particl…
Reflection of a microscopic particle from a mesoscopic/macroscopic `mirror' generates two-body correlated interference from the incident and reflected particle substates and their associated mirror substates. The microscopic momentum exchanged generates two mirror substates which interfere to produce fringes which do not vanish as the mirror mass increases. The small displacement between these mirror states can yield negligible environmental decoherence times. Mirror coherence lengths impose constraints on the extent of this interference, which are mitigated using interference of the two-body …
The spatial coherence of an atomic wavepacket can be detected in the scattered photons, even when the center-of-mass motion is in the quantum coherent superposition of two distant, non-overlapping wave packets. Spatial coherence manifests itself in the power spectrum of the emitted photons, whose spectral components can exhibit interference fringes as a function of the emission angle. The contrast and the phase of this interference pattern provide information about the quantum state of the center of mass of the scattering atom.
In a general, multi-mode scattering setup, we show how the permutation symmetry of a many-particle input state determines those scattering unitaries which exhibit strictly suppressed many-particle transition events. We formulate purely algebraic suppression laws that identify these events and show that the many-particle interference at their origin is robust under weak disorder and imperfect indistinguishability of the interfering particles. Finally, we demonstrate that all suppression laws so far described in the literature are embedded in the general framework that we here introduce.
The wave-particle duality of massive objects is a cornerstone of quantum physics and a key property of many modern tools such as electron microscopy, neutron diffraction or atom interferometry. Here we report on the first experimental demonstration of quantum interference lithography with complex molecules. Molecular matter-wave interference patterns are deposited onto a reconstructed Si(111) 7x7 surface and imaged using scanning tunneling microscopy. Thereby both the particle and the quantum wave character of the molecules can be visualized in one and the same image. This new approach to nano…
The dynamics of bosons in generic multimode systems, such as Bose-Hubbard models, is not only determined by interactions among the particles, but also by their mutual indistinguishability manifested in many-particle interference. We introduce a measure of indistinguishability for Fock states of bosons whose mutual distinguishability is controlled by an internal degree of freedom. We demonstrate how this measure emerges both in the non-interacting and interacting evolution of observables. In particular, we find an unambiguous relationship between our measure and the variance of single-particle …
It is well known that in a two-slit interference experiment, acquiring which-path information about the particle, leads to a degrading of the interference. It is argued that path-information has a meaning only when one can umabiguously tell which slit the particle went through. Using this idea, two duality relations are derived for the general case where the two paths may not be equally probable, and the two slits may be of unequal widths. These duality relations, which are inequalities in general, saturate for all pure states. Earlier known results are recovered in suitable limit.
Multimode two-particle systems show interference effects in one-particle detections when both particles have common modes. We explore the possibility of extending the usual concepts of distinguishability and visibility to these types of systems. Distinguishability will refer now to the balance between common and different modes of a two-particle system, instead of the standard definition concerning available alternatives for a one-particle system. On the other hand, the usual concept of visibility is not suitable for our problem and must be replaced with that of contrast, measuring the ratio o…
The compatibility of standard and Bohmian quantum mechanics has recently been challenged in the context of two-particle interference, both from a theoretical and an experimental point of view. We analyze different setups proposed and derive corresponding exact forms for Bohmian equations of motion. The equations are then solved numerically, and shown to reproduce standard quantum-mechanical results.
Very much like the ubiquitous quantum interference of a single particle with itself, quantum interference of two independent, but indistinguishable, particles is also possible. This interference is a direct result of quantum exchange statistics, however, it is observed only in the joint probability to find the particles in two separated detectors. Here we report the first observation of such interference fringes between two independent and non-interacting electrons in an interferometer proposed by Yurke et al. and Samuelsson et al. Our experiment resembles the "Hanbury Brown and Twiss" (HBT) e…
If an unstable particle used in a two-path interference experiment decays before reaching a detector, which-way information becomes available which reduces the detected interference fringe visibility ${\cal V}$. Here we argue that even when an unstable particle does {\em not} decay while in the interferometer, {\em a priori} which-way information is still available in the form of path predictability ${\cal P}$ which depends on the particle's decay rate $Γ$. We further demonstrate that in a matter-wave Mach-Zehnder interferometer using an excited atom with an appropriately tuned cavity, ${\cal …
When a Bose-Einstein condensate is divided into two parts, that are subsequently released and overlap, interference fringes are observed. We show here that this interference is typical, in the sense that most wave functions of the condensate, randomly sampled out of a suitable ensemble, display interference. We make no hypothesis of decoherence between the two parts of the condensates.
By using an exact analytical non-Hermitian approach in terms of resonance (quasinormal) states we express the decaying wave function as the sum of exponential and nonexponential decaying solutions to the time-dependent Schrödinger equation. We show that the exponential-nonexponential transition of decay at long times represents physically a process of interference in time domain of the decaying particle with itself.
A sufficient condition for entanglement in two-mode continuous systems is constructed based on interference visibility and the uncertainty of the total particle number. The observables to be measured (particle numbers and particle number variances) are relatively easily accessible experimentally. The method may be used to detect entanglement in light fields or in Bose-Einstein condensates. In contrast to the standard approach based on entanglement witnesses, the condition is expressed in terms of an inequality which is nonlinear in expectation values.
It is well known that in classical optics, the visibility of interference, in a two-beam light interference, is related to the optical coherence of the two beams. A wave-particle duality relation can be derived using this mutual coherence. The issue of wave-particle duality in classical optics is analyzed here, in the more general context of multipath interference. New definitions of interference visibility and path distinguishability have been introduced, which lead to a duality relation for multipath interference. The visibility is shown to be related to a new multi-point optical coherence f…
We describe a general method of realizing entanglement witnesses in terms of the interference pattern of a single quantum probe. After outlining the principle, we discuss specific realizations both with electrons in mesoscopic Aharonov-Bohm rings and with photons in standard Young's double-slit or coherent-backscattering interferometers.
A pair of coherent femtosecond pulse excitations applied to a molecule with strong electron-phonon coupling creates a coherent superposition of a low momentum and a high momentum wavepacket in the vibrational states of both the excited state and the ground state of the coherent transition. As the excited state is accelerated further, interference between the high momentum ground state contribution and the low momentum excited state contribution causes a photon echo. This photon echo is a direct consequence of quantum interference between separate vibrational trajectories and can therefore prov…
Several distinct classes of unitary mode transformations have been known to exhibit the strict suppression of a large set of transmission events, as a consequence of totally destructive many-particle interference. In another work [Dittel et al., Phys. Rev. Lett. 120, 240404 (2018)] we unite these cases by identifying a general class of unitary matrices which exhibit such interferences. Here, we provide a detailed theoretical analysis that substantially expands on all aspects of this generalisation: We prove the suppression laws put forward in our other paper, establish how they interrelate wit…
Quantum interference between two distinct vibrational trajectories induced by two pulse femtosecond excitation in molecules is shown to result in a photon echo, providing direct evidence of the cat state superposition of Gaussian vibrational wavepackets.
The recent development of dynamic single-electron sources makes it possible to observe and manipulate the quantum properties of individual charge carriers in mesoscopic circuits. Here, we investigate multi-particle effects in an electronic Mach-Zehnder interferometer driven by dynamic voltage pulses. To this end, we employ a Floquet scattering formalism to evaluate the interference current and the visibility in the outputs of the interferometer. An injected multi-particle state can be described by its first-order correlation function, which we decompose into a sum of elementary correlation fun…
