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arXiv:1204.5304 · Physical Review A

Revisiting Bohr's principle of complementarity with a quantum device

Jian‐Shun Tang, Yu-Long Li, Chuan‐Feng Li, Guang‐Can Guo

Published Jul 22, 2013• 38 citations
arXivHTMLPDF

Abstract

Bohr's principle of complementarity (BPC) is the cornerstone of quantum mechanics. According to this principle, the total wavelike and particlelike information of a particle is limited by the Englert-Greenberger (EG) duality relation. Here, by introducing a quantum detecting device into the experiment, we find that the limit of the EG duality relation is exceeded because of the interference between the wave and particle properties of the photon. A generalized EG duality relation is further developed. Our work provides a generalization of BPC and gives new insights into quantum mechanics.

DOI
10.1103/physreva.88.014103

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In an interesting series of articles published few years ago Afshar and coworkers [1,2] reported an optical experiment in which they claimed to refute the well known N. Bohr principle of complementarity [3, 4]. Obviously this result, if justified, would constitute a serious attack against the orthodox interpretation of quantum mechanics (known as the Copenhagen interpretation). This work stirred much debate in different journals (see for examples references [5–12]). We think however that there are still some important misunderstandings concerning the interpretation of this experiment. In a preprint written originally in 2004 [5] (and following some early discussions with Afshar) we claimed already that the interpretation by Afshar et al. can be easily stated if we stay as close as possible from the texts written by Bohr. The aim of the present article (which was initially written in 2005 to precise a bit the thought developed in [5]) is to comment the interpretation discussed in [1]. 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Later in the discussion we will try to precise this definition but for the moment it is enough to illustrate the concepts by examples Consider for instance the well known Young double-pinholes interference experiment made with photons. The discrete nature of light precludes the simultaneous observation of a same photon in the aperture plane and in the interference pattern: the photon cannot be absorbed twice. This is already a trivial manifestation of the principle of Bohr. Here it implies that the two statistical patterns associated with the wave in the aperture plane and its Fourier (i. e., momentum) transform require necessarily different photons for their recording. It is in that sense that each experiment excludes and completes reciprocally the other. In the case considered before the photon is absorbed during the first detection (this clearly precludes any other detection). However even a non-destructive solution for detection implying entanglement with other quantum systems has a radical effect of the same nature: the complementarity principle is still valid. For example, during their debate Bohr and Einstein [3] discussed an ideal which-way experiment in which the recoil of the slits is correlated to the motion of the photon. Momentum conservation added to arguments based on the uncertainty relations are sufficient to explain how such entanglement photon-slits can erase fringes [15–19]. It is also important for the present discussion to remind that the principle of complementarity has a perfidious consequence on the experimental meaning of trajectory and path followed by a particle. Indeed the unavoidable interactions existing between photons and detectors imply that a trajectory existing independently of any measurement process cannot be unambiguously defined. This sounds even like a tragedy when we consider once again the two-holes experiment. Indeed for Bohr this kind of experiments shows definitely the essential element of ambiguity which is involved in ascribing conventional physical attributes to quantum systems. Intuitively (i. e., from the point of view of classical particle dynamic) one would expect that a photon detected in the focal plane of the lens must have crossed only one of the hole 1 or 2 before to reach its final destination. However, if this is true, one can not intuitively understand how the presence of the second hole (through which the photon evidently did not go) forces the photon to participate to an interference pattern (which obviously needs an influence coming from both holes). Explanations to solve this paradox have been proposed by de Broglie, Bohm, and others using concepts such as empty waves or quantum potentials [20, 21]. However all these explanations are in agreement with Bohr principle (since they fully reproduce quantum predictions) and can not be experimentally distinguished. Bohr and Heisenberg proposed for all needed purposes a much more pragmatic and simpler answer: don’t bother, the complementarity principle precludes the simultaneous observation of a photon trajectory and of an interference pattern. For Bohr [3]: This point is of great logical consequence, since it is only the circumstance that we are presented with a choice of either tracing the path

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Kazufumi Sakai

The principle of complementarity is the foundation of quantum mechanics; its correctness has been verified by several studies. At present, the Englert-Greenberger duality relation is used for quantitative evaluations. We fabricated a new double slit experimental apparatus capable of simultaneously measuring the visibility and path-distinguishability, and measured the wave and particle properties. We thus obtained results in disagreement with the principle of complementarity.

•Apr 1, 2018•4 citations
Bohr's principle of complementarity (BPC) is the cornerstone of quantum mechanics. According to this principle, the total wavelike and particlelike information of a particle is limited by the Englert-Greenberger (EG) duality relation. Here, by introducing a quantum detecting device into the experiment, we find that the limit of the EG duality relation is exceeded because of the interference between the wave and particle properties of the photon. A generalized EG duality relation is further developed. Our work provides a generalization of BPC and gives new insights into quantum mechanics.
An experimental test of the quantum complementarity principle based on single neutral atoms trapped in a blue detuned bottle trap was here performed. A Ramsey interferometer was used to assess the wavelike behavior or particlelike behavior with second $\ensuremath{\pi}/2$ rotation on or off. The wavelike behavior or particlelike behavior is characterized by the visibility VVV of the interference or the predictability PPP of which-path information, respectively. The measured results fulfill the complementarity relation ${P}^{2}+{V}^{2}\ensuremath{\le}1$. Imbalance losses were deliberately introduced to the system and we find the complementarity relation is then formally ``violated.'' All the experimental results can be completely explained theoretically by quantum mechanics without considering the interference between wave and particle behaviors. This observation complements existing information concerning Bohr's complementarity principle based on wave-particle duality of a massive quantum system.
One of the most remarkable features of quantum physics is that attributes of quantum objects, such as the wavelike and particlelike behaviors of single photons, can be complementary in the sense that they are equally real but cannot be observed simultaneously. Quantum measurements, serving as windows providing views into the abstract edifice of quantum theory, are basic tools for manifesting the intrinsic behaviors of quantum objects. However, a quantitative formulation of complementarity that highlights its manifestations in general measurement scenarios remains elusive. Here we develop a general framework for demonstrating quantum complementarity in the form of information exclusion relations (IERs), which incorporates the wave-particle duality relations as particular examples. Moreover, we explore the applications of our theory in entanglement detection and elucidate that our IERs lead to an extended form of entropic uncertainty relations, providing insights into the connection between quantum complementarity and the preparation uncertainty.
We propose a theoretical framework demonstrating that the traditional interpretation of Bohr’s complementarity principle, asserting mutual exclusivity of wave and particle signatures, arises not from a fundamental physical prohibition but from the historical restriction to projective (von Neumann) measurements. Using positive operator valued measures (POVMs), the Lindblad master equation, and quantum Fisher information, we establish that simultaneous access to interference visibility V and path distinguishability D is mathematically permitted within the Englert–Greenberger bound D² + V² ≤ 1. Five experimental architectures are proposed: a photovoltaic detection screen (baseline), an asymmetric photovoltaic slit apparatus, an optomechanical mirror configuration, a quantum memory delayed emission scheme, and a room temperature telecom fiber duality disc mapper. Three closed form metrological identities are derived: flat Fubini–Study geometry of the Englert circle (F_Q(θ) = 1), reciprocity bounds √N σ(V̂) ≥ D and √N σ(D̂) ≥ V with the estimation cost sum rule, and autonomous duality flow equations with Gudermannian (tanh/sech) solutions. We term the resulting reframing the “duality illusion”: complementarity is a constraint on joint measurement precision, not a prohibition against joint measurement. This is a purely theoretical proposal. No experiments have been performed; all numerical values are predictions. Keywords: wave particle duality, complementarity, quantum measurement, POVM, Lindblad equation, quantum Fisher information, Englert–Greenberger inequality, optomechanics, quantum dots
One of the milestones of quantum mechanics is Bohr's complementarity principle. It states that a single quantum can exhibit a particlelike ororor a wavelike behavior, but never both at the same time. These are mutually exclusive and complementary aspects of the quantum system. This means that we need distinct experimental arrangements in order to measure the particle or the wave nature of a physical system. One of the most known representations of this principle is the single-photon Mach-Zehnder interferometer. When the interferometer is closed an interference pattern is observed (wave aspect of the quantum) while if it is open, the quantum behaves like a particle. Here, using a molecular quantum information processor and employing nuclear magnetic resonant (NMR) techniques, we analyze the quantum version of this principle by means of an interferometer that is in a quantum superposition of being closed and open, and confirm that we can indeed measure both aspects of the system with the same experimental apparatus. More specifically, we observe with a single apparatus the interference between the particle and the wave aspects of a quantum system.
In an interesting series of articles published few years ago Afshar and coworkers [1,2] reported an optical experiment in which they claimed to refute the well known N. Bohr principle of complementarity [3, 4]. Obviously this result, if justified, would constitute a serious attack against the orthodox interpretation of quantum mechanics (known as the Copenhagen interpretation). This work stirred much debate in different journals (see for examples references [5–12]). We think however that there are still some important misunderstandings concerning the interpretation of this experiment. In a preprint written originally in 2004 [5] (and following some early discussions with Afshar) we claimed already that the interpretation by Afshar et al. can be easily stated if we stay as close as possible from the texts written by Bohr. The aim of the present article (which was initially written in 2005 to precise a bit the thought developed in [5]) is to comment the interpretation discussed in [1]. We will in the following analyze the meaning of Bohr principle and show that far from disproving its content the experiment [1] is actually a complete confirmation of its general validity. The difficulties associated with the understanding of this principle are not new and actually complementarity created troubles even in Einstein mind [3] so that we are here in good company. To summarize a bit emphatically Bohr’s complementarity we here remind that this principle states that if one of a pair of non commuting observables of a quantum object is known for sure, then information about the second (complementary) is lost [3, 4, 15, 16]. This can be equivalently expressed as a kind of duality between different descriptions of the quantum system associated with different experimental arrangements which mutually exclude each other (read in particular [3,4]). Later in the discussion we will try to precise this definition but for the moment it is enough to illustrate the concepts by examples Consider for instance the well known Young double-pinholes interference experiment made with photons. The discrete nature of light precludes the simultaneous observation of a same photon in the aperture plane and in the interference pattern: the photon cannot be absorbed twice. This is already a trivial manifestation of the principle of Bohr. Here it implies that the two statistical patterns associated with the wave in the aperture plane and its Fourier (i. e., momentum) transform require necessarily different photons for their recording. It is in that sense that each experiment excludes and completes reciprocally the other. In the case considered before the photon is absorbed during the first detection (this clearly precludes any other detection). However even a non-destructive solution for detection implying entanglement with other quantum systems has a radical effect of the same nature: the complementarity principle is still valid. For example, during their debate Bohr and Einstein [3] discussed an ideal which-way experiment in which the recoil of the slits is correlated to the motion of the photon. Momentum conservation added to arguments based on the uncertainty relations are sufficient to explain how such entanglement photon-slits can erase fringes [15–19]. It is also important for the present discussion to remind that the principle of complementarity has a perfidious consequence on the experimental meaning of trajectory and path followed by a particle. Indeed the unavoidable interactions existing between photons and detectors imply that a trajectory existing independently of any measurement process cannot be unambiguously defined. This sounds even like a tragedy when we consider once again the two-holes experiment. Indeed for Bohr this kind of experiments shows definitely the essential element of ambiguity which is involved in ascribing conventional physical attributes to quantum systems. Intuitively (i. e., from the point of view of classical particle dynamic) one would expect that a photon detected in the focal plane of the lens must have crossed only one of the hole 1 or 2 before to reach its final destination. However, if this is true, one can not intuitively understand how the presence of the second hole (through which the photon evidently did not go) forces the photon to participate to an interference pattern (which obviously needs an influence coming from both holes). Explanations to solve this paradox have been proposed by de Broglie, Bohm, and others using concepts such as empty waves or quantum potentials [20, 21]. However all these explanations are in agreement with Bohr principle (since they fully reproduce quantum predictions) and can not be experimentally distinguished. Bohr and Heisenberg proposed for all needed purposes a much more pragmatic and simpler answer: don’t bother, the complementarity principle precludes the simultaneous observation of a photon trajectory and of an interference pattern. For Bohr [3]: This point is of great logical consequence, since it is only the circumstance that we are presented with a choice of either tracing the path