Whether the loss of visibility due to measurement in two-slit interference patterns can always be attributed to momentum transfer depends upon how one defines the last term. The momentum transfer to a quantum particle can be determined unambiguously if it was initially in a momentum eigenstate, but that is not the state relevant to a two-slit experiment. A sensible answer for the two-slit problem was obtained by Wiseman et al. [Phys. Rev. A 56, 944 (1997)] using the formalism of the Wigner function. Here I show that a more general answer can be obtained using the Bohmian formulation of quantum mechanics, in which particles have a definite position and momentum at all times. By following all possible trajectories of the particle it is possible to calculate the probability distribution ${P}_{\mathrm{total}}^{B}{(p}^{\ensuremath{'}})$ for it to receive a momentum transfer ${p}^{\ensuremath{'}}$ as a result of the measurement. Furthermore, the 1-norm of this distribution obeys the relation $〈|{p}^{\ensuremath{'}}|{〉}_{\mathrm{total}}^{B}d>~2\ensuremath{\Elzxh}(1\ensuremath{-}V)/\ensuremath{\pi},$ where
V is the visibility of the interference pattern and
d is the slit separation. This confirms that the momentum transfer in a welcher Weg (which path) experiment is in accord with the uncertainty principle. Like the Wignerian analysis, the Bohmian analysis clearly distinguishes between a local momentum transfer (which occurs even if the particle is localized at a single slit) and a nonlocal momentum transfer (which occurs only if the particle is delocalized, at both slits). In the Bohmian analysis the former occurs at the time of the measurement, while the latter occurs after the measurement, and is a consequence of Bohm's ``quantum potential.''
The momentum changes caused by position measurements are a central feature of wave-particle duality. Here we investigate two cases - localization by a single slit, and which-way detection in the double-slit interference experiment - and examine in detail the associated momentum changes. Particular attention is given to the transfer of momentum between particle and detector, and the recoil of the measuring device. We find that single-slit diffraction relies on a form of `interaction-free' scattering, and that an ideal which-way measurement can be made without any back-reaction on the detector.
A ``which-path'' (welcher Weg) measurement necessarily destroys the fringes in a double-slit interference experiment. We show that in all instances one may attribute this destruction to a disturbance of the particle's momentum by an amount equal to at least $\ensuremath{\pi}\ensuremath{\Elzxh}/2d$, where
d is the slit separation, in accordance with the uncertainty principle. However, this momentum transfer need not be local; that is, it need not act at either of the slits through which the particle passes. For well-known welcher Weg measurements such as Einstein's recoiling slit and Feynman's light microscope, the disturbance can be understood in terms of random classical momentum kicks that act locally. In some recent proposals, including that by Scully, Englert, and Walther [Nature (London) 351, 111 (1991)], the momentum transfer is of a peculiarly quantum, nonlocal nature. In this paper we introduce a formalism based on the Wigner function, as this describes both the local and nonlocal momentum transfer caused by any welcher Weg measurement. We show that for some examples, such as that of Scully, Englert, and Walther, there is no momentum disturbance at the slits even though the nonlocal momentum disturbance is sufficient to destroy the interference pattern. Finally, we discuss the experimental signatures of nonlocal versus local momentum transfer and demonstrate a strong similarity to the nonlocality of the Aharonov-Bohm effect.
Young’s one-dimension-double-slit experiment represents the basic mystery of quantum world and was extended to which-way-1D-double-slit experiment. To explore the mystery, 2D-cross-double-slit apparatuses have been proposed, which consists of more than one double-slits intersecting to each other. In this article, we report which-way-2D-cross-double-slit experiments by detecting which slit of a double-slit of 2D-cross-double-slit apparatus a photon would pass through. The experimental results show that photons passing through the detected double-slit behave as particle, while photons passing through other undetected double-slit(s) distribute as wave in the same experiment with the same apparatus. Namely the particle nature and wave distribution coexist, which violates Bohr’s complementarity principle. We suggest that photon’s “particle nature” is intrinsic, while the term “wave nature” represents a wave-like distribution, which is a consequence of movement of photon as particle. It is a challenge to interpret mathematically those observations consistently.
Photons in a two-path interferometer best embody wave-particle duality (WPD), which is a core concept of quantum theory. So far, the WPD relation is commonly written as
V2+D2≤1, where
V is the interference fringe visibility and
D is path distinguishability, i.e., the distinguishability of which path a photon passed. This inequality is saturated only when the which-way marker (WWM), which embodies which-path information (WPI) via an internal degree of freedom of photons, such as polarization, is in a pure state. For mixed-state WWM, conventionally defined distinguishability underestimates the amount of WPI and thus does not saturate the WPD relation. Here, we introduce a generalized measure of distinguishability
D that properly quantifies the WPI and saturates the WPD relation for all pure- and mixed-state WWM within a purification-based framework. To this end, mixed-state WWM is treated as a result of entanglement formation between the WWM and an external degree of freedom, e.g., environment, and
D is defined so that it incorporates the total WPI shared between the WWM and the environment. We show that
D thus defined is experimentally quantifiable, independently of
V, without access to the environment. We experimentally evaluate
V and
D using true single photons generated in the completely mixed (unpolarized) state, and thus verify the saturated WPD relation.
Abstract The recent state-of-the-art double-slit experiments with single electrons and single photons seem to emphasize a contradictable dilemma concerning the ontological physical reality in quantum physics. Because of the importance of this problem, we propose and perform another modified laser-beam asymmetrical double-slit experiment with a constraint of single-photon interference. In the results, a Feynman condition with a mobile closing mask allows assessing the complementary contributions of photons passing through one or another slit. Under the assumption of applicable self-interference, a ‘which-way’ phenomenon is observed quantitatively in the vicinity of the main minima with low-intense interference fringes, demonstrating a possible solution to this problem. The result would encourage similar asymmetrical double-slit experiments with truly single photons or single electrons for consistent and direct observation of the wave-particle duality.