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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

The sense in which a "weak measurement" of a spin-½ particle's spin component yields a value 100

Ian Duck, P. M. Stevenson, E. C. G. Sudarshan

Published Sep 15, 1989• 449 citations
Publisher

Abstract

We give a critical discussion of a recent Letter of Aharonov, Albert, and Vaidman. Although their work contains several flaws, their main point is valid: namely, that there is a sense in which a certain "weak measurement" procedure yields values outside the eigenvalue spectrum. Our analysis requires no approximations and helps to clarify the physics behind the effect. We describe an optical analog of the experiment and discuss the conditions necessary to realize the effect experimentally.

DOI
10.1103/physrevd.40.2112

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We give a critical discussion of a recent Letter of Aharonov, Albert, and Vaidman. Although their work contains several flaws, their main point is valid: namely, that there is a sense in which a certain "weak measurement" procedure yields values outside the eigenvalue spectrum. Our analysis requires no approximations and helps to clarify the physics behind the effect. We describe an optical analog of the experiment and discuss the conditions necessary to realize the effect experimentally.
The notion of the weak value of a quantum mechanical observable has recently been introduced by Aharonov et al., hereafter referred to as AAV. Their analysis led to the controversial result that the weak values for an observable can lie well outside its associated eigenvalue spectrum. We have performed an experiment similar to one proposed by Knight et al. and demonstrated that weak values can be observed and that they exhibit the properties described by AAV within the regime of validity defined by Duck et al. It is shown that these properties are produced by interference phenomena that result from the loss of ‘welcher Weg’ information. We have also identified an intrinsic ‘weak energy’ which appears in the equation of motion for the weak value of an observable and have shown that it can be non–vanishing only for systems which are both preselected and postselected. It is noted that such a ‘weak spin–rotation coupling energy’ can be associated with our experiment and that it is manifested in our experimental data. We suggest that preselection and postselection measurement techniques could be useful for the creation and study of physical phenomena that might be otherwise unobservable.
Abstract Is it possible that a measurement of a spin component of a spin-1/2 particle yields the value 100? In 1988 Aharonov, Albert and Vaidman argued that upon pre- and postselection of particular spin states, weakening the coupling of a standard measurement procedure ensures this paradoxical result1. This theoretical prediction, called weak value, was realised in numerous experiments2–9, but its meaning remains very controversial10–19, since its “anomalous” nature, i.e., the possibility to exceed the eigenvalue spectrum, as well as its “quantumness” are debated20–22. We address these questions by presenting the first experiment measuring anomalous weak values with just a single click, without the need for statistical averaging. The measurement uncertainty is significantly smaller than the gap between the measured weak value and the nearest eigenvalue. Beyond clarifying the meaning of weak values, demonstrating their non-statistical, single-particle nature, this result represents a breakthrough in understanding the foundations of quantum measurement, showing unprecedented measurement capability for further applications of weak values to quantum photonics.
The notion of the weak value of a quantum mechanical observable has recently been introduced by Aharonov et al. (Phys. Rev A 41,11 (1990)), hereafter referred to as AAV. We have performed an experiment similar to one proposed by Knight et al. (1990) and demonstrated that weak values can be observed and that they exhibit the properties described by AAV within the regime of validity. We suggest that preselection and postselection measurement techniques could be useful for study of physical phenomena that might be otherwise unobservable.
The first experimental realization of a measurement of the weak value of a variable, a concept recently introduced by Aharonov, Albert and Vaidman, is presented. Weak measurements (measurements of a weak value) address the situation in which the separation of the eigenvalues caused by a weakly interacting measurement device is small compared to the width of the distribution of the individual eigenvalues. By appropriate choice of the pre- and post-selected state it is possible that the overlapping eigenvalues will interfere producing a value outside the range of eigenvalues. We demonstrate that the weak value is a practical method of amplifying and resolving the separation between overlapping eigenvalues in an optical experiment proposed by Duck, Stevenson and Sudarshan. In this experiment a birefringent crystal spatially separates two linear polarization components of a Gaussian laser beam by a distance much smaller than the beam waist.
The weak value of an operator is the average result of measuring that operator on a quantum system given specified initial and final states. The overlap between these states appears in the denominator of the weak value, and thus if that overlap is small, the weak value can be arbitrarily large. In 1988, Albert, Aharonov, and Vaidman proposed a weak Stern-Gerlach experiment which would be able to measure a so-called anomalous weak value, a measurement result which lies out- side of the eigenvalue spectrum of the operator being measured. Though anomalous weak values have been measured in numerous optical systems, here we present the first such realization of this experiment as originally proposed. By placing a Bose-Einstein condensate of 87Rb in a magnetic gradient and performing an unlikely postselection, we measure an effective magnification of a weak magnetic momentum kick (~ 8 μμμm/s) by a factor of 14. We also demonstrate that increasing the momentum kick decreases the "weakness" of the measurement and reduces the amplification.
We present the first realization of a measurement of a ``weak value,'' a concept recently introduced by Aharonov, Albert, and Vaidman (AAV). Our experiment uses a birefringent crystal to separate the two linear-polarization components of a laser beam by a distance small compared to the laser-beam waist. This ``weak measurement'' is followed by a strong measurement which translates the centroid of the beam by a distance far larger than the birefringence-induced separation. In addition, we present data corresponding to orthogonal initial and final states, for which the weak value is not defined. This interference effect may have application in the amplification and detection of weak effects.