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Physical Review A

Nonlocal momentum transfer inwelcher Wegmeasurements

Howard M. Wiseman, F. E. Harrison, Matthew J. Collett, S. M. Tan, Daniel F. Walls, Rowan Killip

Published Jul 1, 1997• 90 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.56.55

Related papers

Bohmian analysis of momentum transfer inwelcher Wegmeasurements

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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.''

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

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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.

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In optical experiments involving a single photon that takes alternative paths through an optical system and ultimately interferes with itself (e.g., Young's double-slit experiment, Mach-Zehnder interferometer, Sagnac interferometer), there exist fundamental connections between the linear and angular momenta of the photon on the one hand, and the ability of an observer to determine the photon's path through the system on the other hand. This paper examines the arguments that relate the photon momenta (through the Heisenberg uncertainty principle) to the "which path" (German: welcher Weg) question at the heart of quantum mechanics. We show that the linear momenta imparted to apertures or mirrors, or the angular momenta picked up by strategically placed wave-plates in a system, could lead to an identification of the photon's path only at the expense of destroying the corresponding interference effects. We also describe a thought experiment involving the scattering of a circularly-polarized photon from a pair of small particles kept at a fixed distance from one another. The exchange of angular momentum between the photon and the scattering particle in this instance appears to provide the "which path" information that must, of necessity, wipe out the corresponding interference fringes, although the fringe-wipe-out mechanism does not seem to involve the uncertainty principle in any obvious way.

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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 \ensuremathπ\ensuremath\Elzxh/2d\ensuremath{\pi}\ensuremath{\Elzxh}/2d\ensuremathπ\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.
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-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.
Two-slit interference experiment with a which-way detector has been a topic of intense debate. Scientific community is divided on the question whether the particle receives a momentum kick because of the process of which-way measurement. It is shown here that the same experiment can be viewed in two different ways, depending on which basis of the which-way detector states one chooses to look at. In one view, the loss of interference arises due to the entanglement of the two paths of the particle with two orthogonal states of the which-way detector. In another view, the loss of interference can be interpreted as arising from random momentum kicks of magnitude h/2dh/2dh/2d received by the particle, ddd being the slit separation. The same scenario is shown to hold for a three-slit interference experiment. The random momentum kicks for the three-slit case are of two kinds, of magnitude ±h/3d\pm h/3d±h/3d. The analysis is also generalized to the case of n-slit interference. The two alternate views are described by the same quantum state, and hence are completely equivalent. The concept of "local" versus "nonlocal" kicks, much discussed in the literature, is not needed here.
Making a "which-way" measurement (WWM) to identify which slit a particle goes through in a double-slit apparatus will reduce the visibility of interference fringes. There has been a long-standing controversy over whether this can be attributed to an uncontrollable momentum transfer. Here, by reconstructing the Bohmian trajectories of single photons, we experimentally obtain the distribution of momentum change. For our WWM, the change we see is not a momentum kick that occurs at the point of the WWM, but rather one that nonclassically accumulates during the propagation of the photons. We further confirm a quantitative relation between the loss of visibility consequent on a WWM and the total (late-time) momentum disturbance. Our results emphasize the role of the Bohmian momentum in giving an intuitive picture of wave-particle duality and complementarity.
In optical experiments involving a single photon that takes alternative paths through an optical system and ultimately interferes with itself (e.g., Young's double-slit experiment, Mach-Zehnder interferometer, Sagnac interferometer), there exist fundamental connections between the linear and angular momenta of the photon on the one hand, and the ability of an observer to determine the photon's path through the system on the other hand. This paper examines the arguments that relate the photon momenta (through the Heisenberg uncertainty principle) to the "which path" (German: welcher Weg) question at the heart of quantum mechanics. We show that the linear momenta imparted to apertures or mirrors, or the angular momenta picked up by strategically placed wave-plates in a system, could lead to an identification of the photon's path only at the expense of destroying the corresponding interference effects. We also describe a thought experiment involving the scattering of a circularly-polarized photon from a pair of small particles kept at a fixed distance from one another. The exchange of angular momentum between the photon and the scattering particle in this instance appears to provide the "which path" information that must, of necessity, wipe out the corresponding interference fringes, although the fringe-wipe-out mechanism does not seem to involve the uncertainty principle in any obvious way.