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arXiv:quant-ph/0310081 · Journal of Optics B Quantum and Semiclassical Optics

The uncertainty relation in which-way experiments: how to observe directly the momentum transfer using weak values

J L Garretson, Howard M. Wiseman, Damian T. Pope, David Thomas Pegg

Published May 29, 2004• 21 citations
arXivHTMLPDF

Abstract

A which-way measurement destroys the twin-slit interference pattern. Bohr argued that this can be attributed to the Heisenberg uncertainty relation: distinguishing between two slits a distance s apart gives the particle a random momentum transfer of order h / s . This was accepted for more than 60 years, until Scully, Englert and Walther (SEW) proposed a which-way scheme that, they claimed, entailed no momentum transfer. Storey, Tan, Collett and Walls (STCW), on the other hand, proved a theorem that, they claimed, showed that Bohr was right. This work reviews and extends a recent proposal (Wiseman 2003 Phys. Lett. A 311 285) to resolve the issue using a weak-valued probability distribution for momentum transfer, . We show that must be nonzero for some . However, its moments can be identically zero, such as in the experiment proposed by SEW. This is possible because is not necessarily positive definite. Nevertheless, it is measurable experimentally in a way understandable to a classical physicist. The new results in this paper include the following. We introduce a new measure of spread for : half the length of the unit-confidence interval. We conjecture that it is never less than h /4 s , and find numerically that it is approximately h /1.59 s for an idealized version of the SEW scheme with infinitely narrow slits. For this example, the moments of , and of the momentum distributions, are undefined unless a process of apodization is used. However, we show that by considering successively smoother initial wavefunctions, successively more moments of both and the momentum distributions become defined. For this example the moments of are zero, and these moments are equal to the changes in the moments of the momentum distribution. We prove that this relation also holds for schemes in which the moments of are nonzero, but it holds only for the first two moments . We also compare these moments to the moments of two other momentum-transfer distributions that have previously been considered, and with the moments of (which is defined in the Heisenberg picture). We find agreement between all of these, but again only for the first two moments. Our results reconcile the seemingly opposing views of SEW and STCW.

DOI
10.1088/1464-4266/6/6/008

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A which-way measurement destroys the twin-slit interference pattern. Bohr argued that this can be attributed to the Heisenberg uncertainty relation: distinguishing between two slits a distance s apart gives the particle a random momentum transfer of order h / s . This was accepted for more than 60 years, until Scully, Englert and Walther (SEW) proposed a which-way scheme that, they claimed, entailed no momentum transfer. Storey, Tan, Collett and Walls (STCW), on the other hand, proved a theorem that, they claimed, showed that Bohr was right. This work reviews and extends a recent proposal (Wiseman 2003 Phys. Lett. A 311 285) to resolve the issue using a weak-valued probability distribution for momentum transfer, . We show that must be nonzero for some . However, its moments can be identically zero, such as in the experiment proposed by SEW. This is possible because is not necessarily positive definite. Nevertheless, it is measurable experimentally in a way understandable to a classical physicist. The new results in this paper include the following. We introduce a new measure of spread for : half the length of the unit-confidence interval. We conjecture that it is never less than h /4 s , and find numerically that it is approximately h /1.59 s for an idealized version of the SEW scheme with infinitely narrow slits. For this example, the moments of , and of the momentum distributions, are undefined unless a process of apodization is used. However, we show that by considering successively smoother initial wavefunctions, successively more moments of both and the momentum distributions become defined. For this example the moments of are zero, and these moments are equal to the changes in the moments of the momentum distribution. We prove that this relation also holds for schemes in which the moments of are nonzero, but it holds only for the first two moments . We also compare these moments to the moments of two other momentum-transfer distributions that have previously been considered, and with the moments of (which is defined in the Heisenberg picture). We find agreement between all of these, but again only for the first two moments. Our results reconcile the seemingly opposing views of SEW and STCW.
A which-way measurement in Young's double-slit will destroy the interference pattern. Bohr claimed this complementarity between wave- and particle-behaviour is enforced by Heisenberg's uncertainty principle: distinguishing two positions at a distance s apart transfers a random momentum q ∼ ℏ/ s to the particle. This claim has been subject to debate: Scully et al (1991 Nature 351 111) asserted that in some situations interference can be destroyed with no momentum transfer, while Storey et al (1994 Nature 367 626) asserted that Bohr's stance is always valid. We address this issue using the experimental technique of weak measurement. We measure a distribution for q that spreads well beyond [−ℏ/ s , ℏ/ s ], but nevertheless has a variance consistent with zero. This weak-valued momentum-transfer distribution P wv ( q ) thus reflects both sides of the debate.
Is the destruction of interference by a which-way measurement due to a random momentum transfer wpagthbar/s\\wp\\agt\\hbar/swpagthbar/s, with sss the slit separation? The weak-valued probability distribution Prmwv(wp)P_{\\rm wv}(\\wp)Prmwv​(wp), which is {\\em directly observable}, provides a subtle answer. Prmwv(wp)P_{\\rm wv}(\\wp)Prmwv​(wp) cannot have support on the interval [−hbar/s,hbar/s][-\\hbar/s,\\hbar/s][−hbar/s,hbar/s]. Nevertheless, its moments can be identically zero.
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 VVV is the visibility of the interference pattern and ddd 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.''
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 ddd 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.
A model of the Einstein–Bohr recoiling slit experiment is formulated in a fully quantum theoretical setting. In this model, the state and dynamics of a movable wall that has two slits in it, as well as the state of a particle incoming to the two slits, are described by quantum mechanics. Using this model, we analyzed complementarity between exhibiting an interference pattern and distinguishing the particle path. Comparing the Kennard–Robertson type and the Ozawa-type uncertainty relations, we conclude that the uncertainty relation involved in the double-slit experiment is not the Ozawa-type uncertainty relation but the Kennard-type uncertainty relation of the position and the momentum of the double-slit wall. A possible experiment to test the complementarity relation is suggested. It is also argued that various phenomena which occur at the interface of a quantum system and a classical system, including distinguishability, interference, decoherence, quantum eraser, and weak value, can be understood as aspects of entanglement. Quanta 2015; 4: 1–9.
Summary form only given. We say that the quantum observables "position" and "momentum" are "complementary" because the precise knowledge of the position (momentum) implies that all possible outcomes of measuring the momentum (position) are equally probable. In two-slit experiments, the common understanding is that the position-momentum uncertainty relation makes it impossible to determine which slit the photon (or electron) passes through without at the same time disturbing the object enough to destroy the interference pattern. In 1982, Scully et al. suggested a way around this position-momentum uncertainty obstacle and proposed a quantum eraser to obtain which-path or particle-like information without scattering or otherwise introducing large uncontrolled phase factors to disturb the interference. Since 1982, quantum has been claimed in several experiments; however, the original proposed ideas have not been fully demonstrated. We demonstrate them here.