For two-path interferometers, the which-path predictability $\mathcal{P}$ and the fringe visibility $\mathcal{V}$ are familiar quantities that are much used to talk about wave-particle duality in a quantitative way. We discuss several candidates that suggest themselves as generalizations $P$ of $ mathcal{P}$ for multi-path interferometers, and treat the case of three paths in considerable detail. To each choice for the \emph{path knowledge} $P$, the \emph{interference strength} $V$ -- the corresponding generalization of $\mathcal{V}$ -- is found by a natural, operational procedure. In experimental terms it amounts to finding those equal-weight superpositions of the path amplitudes which maximize $P$ for the emerging intensities. Mathematically speaking, one needs to identify a certain optimal one among the Fourier transforms of the state of the interfering quantum object. Wave-particle duality is manifest, inasmuch as P=1 implies V=0 and V=1 implies P=0, whatever definition is chosen. The possible values of the pair $(P,V)$ are restricted to an area with corners at $(P,V)=(0,0)$, $(P,V)=(1,0)$, and $(P,V)=(0,1)$, with the shape of the border line from $(1,0)$ to $(0,1)$ depending on the particular choice for $P$ and the induced definition of $V$.
We present an architecture to investigate wave-particle duality in $N$-path interferometers on a universal quantum computer involving as low as $2\log N$ qubits and develop a measurement scheme which allows the efficient extraction of quantifiers of interference visibility and which-path information. We implement our algorithms for interferometers with up to $N=16$ paths in proof-of-principle experiments on a noisy intermediate-scale quantum (NISQ) device using down to $\mathcal{O}(\log N)$ gates and despite increasing noise consistently observe a complementary behavior between interference vi…
An interferometer - no matter how clever the design - cannot reveal both the wave and particle behavior of a quantum system. This fundamental idea has been captured by inequalities, so-called wave-particle duality relations (WPDRs), that upper bound the sum of the fringe visibility (wave behavior) and path distinguishability (particle behavior). Another fundamental idea is Heisenberg's uncertainty principle, stating that some pairs of observables cannot be known simultaneously. Recent work has unified these two principles for two-path interferometers. Here we extend this unification to $n$-pat…
The Mach-Zehnder interferometric setup quantitatively characterizing the wave-particle duality implements in fact a joint measurement of two unsharp observables. We present a necessary and sufficient condition for such a pair of unsharp observables to be jointly measurable. The condition is shown to be equivalent to a duality inequality, which for the optimal strategy of extracting the which-path information is more stringent than the Jaeger-Shimony-Vaidman-Englert inequality.
A formalism have been recently derived [J. Martinez-Linares and D. Harmin, quantum-ph/0306057] allowing one to separate different sources of which-way information contributing to the total distinguishability D of the ways in a two-way interferometer. Here we apply the formalism to a Quantum Optical Ramsey Interferometer where both sources, the a-priori predictability of the ways P and the quantum "Quality" Q of the which-way detector, stems from the same physical interaction. We show that the formalism is able to separate both sources of which-way information. Moreover, it is shown that Q succ…
We investigate quantitatively the wave-particle duality in a general Mach-Zehnder interferometer setup with an asymmetric beam splitter. The asymmetric beam splitter introduces additional a priori which-path knowledge, which is different for a particle detected at one output port of the interferometer and a particle detected at the other. Accordingly, the fringe visibilities of the interference patterns emerging at the two output ports are also different. Hence, in sharp contrast with the symmetric case, here we should concentrate on one output port and distinguish two possible paths taken by …
Wave-particle duality of photons with losses in the Mach-Zehnder interferometer (MZI) is investigated experimentally and theoretically. The experiment is done with the standard MZI with the beam splitter or the beam merger being continuously varied. The losses are deliberately introduced either inside the MZI (the two arms between the beam splitter and beam mergers) or outside the MZI (after the beam merger). It is proved that the unbalanced losses have great influence on the predictability $P$ (particle nature) and visibility $V$ (wave nature). For the former case the duality inequality holds…
It is well known that the Mach-Zender interferometer exhibits a trade-off between the a priori which-path knowledge and the visibility of its interference pattern. This trade-off is expressed by the inequality $\mathcal{P}^2 + \mathcal{V}^2 \leq 1$, constraining the predictability $\mathcal{P}$ and visibility $\mathcal{V}$ of the interferometer. In this paper we extend the Mach-Zender scheme to a setup where the central phase shifter is substituted by a generic unitary operator. We find that the sum $\mathcal{P}^2 + \mathcal{V}^2$ is in general no longer upper bounded by $1$, and that there ex…
Wave-particle duality and complementarity principle stand at the conceptual core of quantum theory in its orthodox Copenhagen interpretation. They imply that the wave behavior and particle behavior of quantum objects are mutually exclusive to each other in experimental observation. Here we make a systematic analysis using the elementary methodology of quantum mechanics upon Young`s two-slit interferometer and Mach-Zehnder two-arm interferometer with the focus placed on how to measure the interference pattern (wave nature) and which-way information (particle nature) of quantum objects. We desig…
We derive different relations quantifying duality in a generic two-way interferometer. These relations set different upper bounds to the visibility V of the fringes measured at the output port of the interferometer. A hierarchy of inequalities is presented which exhibits the influence of the availability to the experimenter of different sources of which-way information contributing to the total distinguishability D of the ways. For mixed states and unbalanced interferometers an inequality is derived, V^2+ Xi^2 \leq 1, which can be more stringent than the one associated with the distinguishabil…
Complementarity lies at the heart of conceptual foundation of orthodox quantum mechanics. The wave-particle duality makes it impossible to tell which slit each particle passes through and still observe an interference pattern in a Young's double-slit experiment. In this paper, this duality is appraised under the standard formulation of quantum mechanics for atom interferometers. It is found that the internal freedoms like electronic states of an atom can be used to detect the which-path information of each atom while uphold the interference pattern of atoms when the center-of-mass motion of at…
Thought experiments based on the double-slit interferometer had a crucial role to develop ideas concerning the wave-particle duality and the Bohr's complementarity principle. Ideally, a slit with a sufficiently low mass recoils due to the passage of the photon. This motion denounces the path taken by the light and suppresses any attempt to observe an interference pattern. In real life, however, available which-way information in such a setup is significantly impaired by the typical magnitudes of photons and slits, making the verification of the effect almost impossible. Here, we extend this di…
The wave-particle duality demonstrates a competition relation between wave and particle behavior for a particle going through an interferometer. This duality can be formulated as an inequality, which upper bounds the sum of interference visibility and path information. However, if the particle is entangled with a quantum memory, then the bound may decrease. Here, we find the duality relation between coherence and path information for a particle going through a multipath interferometer in the presence of a quantum memory, offering an upper bound on the duality relation which is directly connect…
We give operational meaning to wave-particle duality in terms of discrimination games. Duality arises as a constraint on the probability of winning these games. The games are played with the aid of an n-port interferometer, and involve 3 parties, Alice and Bob, who cooperate, and the House, who supervises the game. In one game called ways they attempt to determine the path of a particle in the interferometer. In another, called phases, they attempt to determine which set of known phases have been applied to the different paths. The House determines which game is to be played by flipping a coin…
Trace-distance-based complementarity relations in a multipath interferometer
School of Mathematics, Nanjing University of Aeronautics and Astronautics（南京航空航天大学数学学院）；School of Science, Beijing Information Science and Technology University（北京信息科技大学理学院）；Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院量子信息中心）
In a recent paper [Phys. Rev. {\bf A64}, 042113 (2001)] S. Dürr proposed an interesting multibeam generalization of the quantitative formulation of interferometric wave-particle duality, discovered by Englert for two-beam interferometers. The proposed generalization is an inequality that relates a generalized measure of the fringe visibility, to certain measures of the maximum amount of which-way knowledge that can be stored in a which-way detector. We construct an explicit example where, with three beams in a pure state, the scheme proposed by Dürr leads to the possibility of an ideal which-w…
A polarized photon with well-defined orbital angular momentum that emerges from a Mach-Zehnder interferometer (MZI) is shown to circumvent wave-particle duality. Its polarization-resolved detection probability forms a non-sinusoidal interferometric pattern. For certain phase differences between the MZI arms, this pattern yields both reliable which-path information and high phase-sensitivity.
We study the wave-particle duality in a general Mach-Zehnder interferometer with an asymmetric beam splitter from the viewpoint of quantum information theory. The correlations(including the classical correlation and the quantum correlation) between the particle and the which-path detector are derived when they are in pure state or mixed state at the output of Mach-Zehnder interferometer. It is found that the fringe visibility and the correlations are effected by the asymmetric beam splitter and the input state of the particle. The complementary relations between the fringe visibility and the c…
Classical wisdom of wave-particle duality says that it is impossible to observe simultaneously the wave and particle nature of microscopic object. Mathematically the principle requests that the interference visibility V and which-path distinguishability D satisfy an orthodox limit of square(V)+square(D)<=1. This work presents a new wave-particle duality test experiment with single photon in a modified Mach-Zehnder interferometer and convincingly show the possibility of breaking the limit. The key element of the interferometer is a weakly-scattering total-internal reflection prism surface, whic…
Wave-particle duality and entanglement are two fundamental characteristics of quantum mechanics. All previous works on experimental investigations in wave{particle properties of single photons (or single particles in general) show that a well-defined interferometer setting determines a well-defined property of single photons. Here we take a conceptual step forward and control the wave-particle property of single photons with quantum entanglement. By doing so, we experimentally test the complementarity principle in a scenario, in which the setting of the interferometer is not defined at any ins…
We show a wave-particle duality of light and its complementary relation in the context of the intensity interference measured by intensity correlation measurement, especially for the case of the second-order intensity interference observed in the Hong-Ou-Mandel interferometer. Different from the complementary relation appearing in the interference based on the phase coherence like in the Young's double-slit interferometer, the complementary relation in the intensity interference has a gap between classical and nonclassical lights. This reveals a new nonclassical nature of light where both wave…
We discuss the concept of an all-optical and ionizing matter-wave interferometer in the time domain. The proposed setup aims at testing the wave nature of highly massive clusters and molecules, and it will enable new precision experiments with a broad class of atoms, using the same laser system. The propagating particles are illuminated by three pulses of a standing ultraviolet laser beam, which detaches an electron via efficient single photon-absorption. Optical gratings may have periods as small as 80 nm, leading to wide diffraction angles for cold atoms and to compact setups even for very m…
We investigate wave-particle duality in a symmetric two-way interferometer with a which-way detector. We find that it is important to state wether the interfering object or the which-way detector is read out first. In case that the interfering object is read out first, we discover that it is possible to use the information about its state to increase the distinguishability to a value which violates the widely accepted bounds set by an inequality introduced by Jaeger et al. (PRA 51, 54) and Englert (PRL 77, 2154).
Interferometers capture a basic mystery of quantum mechanics: a single particle can exhibit wave behavior, yet that wave behavior disappears when one tries to determine the particle's path inside the interferometer. This idea has been formulated quantitively as an inequality, e.g., by Englert and Jaeger, Shimony, and Vaidman, which upper bounds the sum of the interference visibility and the path distinguishability. Such wave-particle duality relations (WPDRs) are often thought to be conceptually inequivalent to Heisenberg's uncertainty principle, although this has been debated. Here we show th…
The notion of wave-particle duality may be quantified by the inequality V^2+K^2 <=1, relating interference fringe visibility V and path knowledge K. With a single-photon interferometer in which polarization is used to label the paths, we have investigated the relation for various situations, including pure, mixed, and partially-mixed input states. A quantum eraser scheme has been realized that recovers interference fringes even when no which-way information is available to erase.
We propose an experiment that permits observation of the de Broglie two-photon wave packet behavior for a pair of photons, using a Mach-Zehnder interferometer. It is based on the use of pulsed lasers to generate pairs of photons via spontaneous parametric down-conversion and the post-selection of events. It differs from previous realizations by the use of a third time-correlated photon to engineer the state of the photons. The same technique can give us which-path information via an ``interaction-free'' experiment and can be used in other experiments on the foundations of quantum mechanics rel…
We have studied the properties of the non-classical behavior of atoms in a double-slit interferometer. An indication of this behavior for metastable helium was reported by Kurtsiefer, Pfau and Mlynek [Nature 386, 150 (1997)] showing distinctive negative values of the Wigner function, which was reconstructed from the measured diffraction data. Our approach to explain this non-classical behavior is based on the de Broglie-Bohm-Vigier-Selleri understanding of the wave-particle duality and compatible statistical interpretation of the atomic wave function. It follows from the results that the atomi…
Many delayed-choice experiments based on Mach-Zehnder interferometers (MZI) have been thought and made to address the fundamental problem of wave-particle duality. Conventional wisdoms long hold that by inserting or removing the second beam splitter (BS2) in a controllable way, microscopic particles (photons, electrons, etc.) transporting within the MZI can lie in the quantum superposition of the wave and particle state as ψ=a_wψ_wave+a_pψ_particle. Here we present an alternative interpretation to these delayed-choice experiments. We notice that as all composite devices of MZI including BS2 ar…
Duality games are a way of looking at wave-particle duality. In these games. Alice and Bob together are playing against the House. The House specifies, at random, which of two sub-games Alice and Bob will play. One game, Ways, requires that they obtain path information about a particle going through an $N$-path interferometer and the other, Phases, requires that they obtain phase information. In general, because of wave-particle duality, Alice and Bob cannot always win the overall game. However, if the required amount of path and phase information is not too great, for example specifying a set…
The aim of this paper is to revisit the implications of complementarity when we inject into a Mach Zehnder interferometer particles with internal structure, prepared in special translational-internal entangled (TIE) states. This correlation causes the path distinguishability to be phase dependent in contrast to the standard case. We show that such a TIE state permits us to detect small phase shifts along with almost perfect path distinguishability, beyond the constraints imposed by complementarity on simultaneous which-way and which-phase measurements for standard cases (when distinguishabilit…
Wave particle duality, also called complementarity, is deeply rooted in the heart of quantum theory. It is fully exemplified in the famous Wheeler's delayed choice experiment where the choice of the wave nature (ability to interfere) or the particle like behavior (path distinguishability) is introduced a posteriori. We perform here a delayed choice experiment in a Mach-Zehnder interferometer, using a classical laser beam and twisted light in a given mode. We entangle the polarization and the twisted internal degrees of freedom, with the which-path-information external degree of freedom of the …
To test the principle of complementarity and wave-particle duality quantitatively, we need a quantum composite system that can be controlled by experimental parameters. Here, we demonstrate that a double-path interferometer consisting of two parametric downconversion crystals seeded by coherent idler fields, where the generated coherent signal photons are used for quantum interference and the conjugate idler fields are used for which-path detectors with controllable fidelity, is useful for elucidating the quantitative complementarity. We show that the source purity $μ_s$ is tightly bounded by …
Induced Coherence in Quantum and Classical Interferometry: Origin and Control
The Institute of Mathematical Sciences（马蒂马拉科学院）；Homi Bhabha National Institute（霍米·巴巴国立学院）；Institute of Science, Banaras Hindu University（贝拿勒斯印度教大学理学院）；Centre for Quantum Science and Technology, Siksha ‘O’ Anusandhan（锡克沙'O'安苏丹量子科学与技术中心）；Indian Institute of Science Education and Research (IISER), Kolkata（加尔各答印度科学教育与研究学院）
For any ideal two-path interferometer it is shown that the wave-particle duality of quantum mechanics implies Heisenberg's uncertainty relation and vice versa. It is conjectured that complementarity and uncertainty are two aspects of the same general principle.
The quantum "mystery which cannot go away" (in Feynman's words) of wave-particle duality is illustrated in a striking way by Wheeler's delayed-choice GedankenExperiment. In this experiment, the configuration of a two-path interferometer is chosen after a single-photon pulse has entered it : either the interferometer is \textit{closed} (\textit{i.e.} the two paths are recombined) and the interference is observed, or the interferometer remains \textit{open} and the path followed by the photon is measured. We report an almost ideal realization of that GedankenExperiment, where the light pulses ar…
This Festschrift in honour of J. A. de Azcarraga gives an introduction to the concept of duality, i.e., to the relativity of the notion of a quantum, in the context of the quantum mechanics of a finite number of degrees of freedom. Although the concept of duality arises in string and M--theory, Vafa has argued that it should also have a counterpart in quantum mechanics, before moving on to second quantisation, fields, strings and branes. We illustrate our analysis with the case when classical phase space is complex projective space, but our conclusions can be generalised to other complex, symp…
From its very beginning quantum theory has been revealing extraordinary and counter-intuitive phenomena, such as wave-particle duality, Schrödinger cats and quantum non-locality. In the study of quantum measurement, a process involving pre- and postselection of quantum ensembles in combination with a weak interaction was found to yield unexpected outcomes. This scheme, usually referred to as "weak measurements", can not only be used as an amplification technique and for minimal disturbing measurements, but also for the exploration of quantum paradoxes. Recently the quantum Cheshire Cat has att…
It is shown that the concept of elementary resonator in the theory of thermal radiation implies the indivisible connection between particles (photons) and electromagnetic waves. This wave-particle duality covers both the Wien and Rayleigh-Jeans regions of spectrum.
Wheeler has strikingly illustrated the wave-particle duality by the delayed-choice thought experiment, in which the configuration of a 2-path interferometer is chosen after a single-photon light-pulsed has entered it. We present a quantitative theoretical analysis of an experimental realization of Wheeler's proposal.
Particle-wave duality enables the construction of interferometers for matter waves, which complement optical interferometers in precision measurement devices. This requires the development of atom-optics analogs to beam splitters, phase shifters, and recombiners. Integrating these elements into a single device has been a long-standing goal. Here we demonstrate a full Mach-Zehnder sequence with trapped Bose-Einstein condensates (BECs) confined on an atom chip. Particle interactions in our BEC matter waves lead to a non-linearity, absent in photon optics. We exploit it to generate a non-classica…
Grangier, Roger and Aspect (GRA) performed a beam-splitter experiment to demonstrate the particle behaviour of light and a Mach-Zehnder interferometer experiment to demonstrate the wave behaviour of light. The distinguishing feature of these experiments is the use of a gating system to produce near ideal single photon states. With the demonstration of both wave and particle behaviour (in two mutually exclusive experiments) they claim to have demonstrated the dual particle-wave behaviour of light and hence to have confirmed Bohr's principle of complementarity. The demonstration of the wave beha…
I argue that quantum optical experiments that purport to refute Bohr's principle of complementarity (BPC) fail in their aim. Some of these experiments try to refute complementarity by refuting the so called particle-wave duality relations, which evolved from the Wootters-Zureck reformulation of BPC (WZPC). I therefore consider it important for my forgoing arguments to first recall the essential tenets of BPC, and to clearly separate BPC from WZPC, which I will argue is a direct contradiction of BPC. This leads to a need to consider the meaning of particle-wave duality relations and to question…
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 …
A textbook interpretation of quantum physics is that quantum objects can be described in a particle or a wave picture, depending on the operations and measurements performed. Beyond this widely held believe, we demonstrate in this contribution that neither the wave nor the particle description is sufficient to predict the outcomes of quantum-optical experiments. To show this, we derive correlation-based criteria that have to be satisfied when either particles or waves are fed into our interferometer. Using squeezed light, it is then confirmed that measured correlations are incompatible with ei…
By applying the basic concept of the density matrix in an open quantum system and modification of quantum mechanics, we derive Kossakowski-Lindblad equation and different properties of this equation are reviewed. Next, a pedagogical approach is used to present Ramsey's trick for linear modification of the quantum mechanics. We discuss how an open quantum mechanics or its modification changes the fraction of excited states in Ramsey Interferometers.
Wave-particle duality is certainly one of the most curious concepts of contemporary physics, which ascribes mutually exclusive behaviors to quantum systems that cannot be observed simultaneously. In the context of two-path interferometers, these two behaviors are usually described in terms of the visibility of interference fringes and the path distinguishability. Here, we use quantum information-theoretic tools to derive quantifiers of these two properties, which account for the combined influence of path probability and polarization, and demonstrate that they satisfy a complementarity relatio…
The quantum object is in general considered as displaying both wave and particle nature. By particle is understood an item localized in a very small volume of the space, and which cannot be simultaneously in two disjoint regions of the space. By wave, to the contrary, is understood a distributed item, occupying in some cases two or more disjoint regions of the space. The quantum formalism did not explain until today the so called "collapse" of the wave function, i.e. the shrinking of the wave function to one small region of the space, when a macroscopic object is encountered. This seems to hap…
The techniques of power recycling and signal recycling have proven as key concepts to increase the sensitivity of large-scale gravitational wave detectors by independent resonant enhancement of light power and signal sidebands within the interferometer. Developing the latter concept further, twin signal recycling was proposed as an alternative to conventional detuned signal recycling. Twin signal recycling features the narrow-band sensitivity gain of conventional detuned signal recycling but furthermore facilitates the injection of squeezed states of light, increases the detector sensitivity o…
Feynman contended that the double-slit experiment contained the `only mystery' in quantum mechanics. The mystery was that electrons traverse the interferometer as waves, but are detected as particles. This note was motivated by the question whether single electrons can be detected as waves. It suggests a double-slit interferometry experiment with atoms of noble gases in which it may be possible to detect an individual atom as a probability wave, using a detector which can execute two different types of simple harmonic motion: as a simple pendulum, and as a torsion pendulum. In the experiment, …
The sensitivity of light and matter-wave interferometers to rotations is based on the Sagnac effect and increases with the area enclosed by the interferometer. In the case of light, the latter can be enlarged by forming multiple fibre loops, whereas the equivalent for matter-wave interferometers remains an experimental challenge. We present a concept for a multi-loop atom interferometer with a scalable area formed by light pulses. Our method will offer sensitivities as high as $2\cdot10^{-11}$ rad/s at 1 s in combination with the respective long-term stability as required for Earth rotation mo…
Wave-particle duality epitomizes the counterintuitive character of quantum physics. A striking illustration is the quantum delay-choice experiment, which is based on Wheeler's classic delayed-choice gedanken experiment, but with the addition of a quantum-controlled device enabling wave-to-particle transitions. Here we realize a quantum delayed-choice experiment in which we control the wave and the particle states of photons and in particular the phase between them, thus directly establishing the created quantum nature of the wave-particle. We generate three-photon entangled states and inject o…
