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Measuring the weak value in an optical experiment

Nicholas W. M. Ritchie

Published Jan 1, 1991• 0 citations
OpenAlex

Abstract

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.

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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.
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.
Abstract "Weak measurements" are measurements in which the coupling between the measuring device and the observable to be measured is so weak that the eigenvalues of the observable are not resolved. Under certain circumstances the corresponding eigenfunctions can be made to interfere, producing a measurement result which is outside the allowed range of the observable's eigenvalues. We present the first measurement of this so-called "weak value" using an optical experiment. In our experiment, the small displacement between the two orthogonally polarized components of a laser beam passed through a birefringence crystal is measured. We use a numerical simulation to show that this phenomenon may be practical for detecting and amplifying small effects.
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.
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 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.
A simple formula to read out the weak value from the wave function of the measuring device after the postselection with the initial Gaussian profile is proposed. We apply this formula for the weak value to the classical experiment of the realization of the weak measurement by the optical polarization and obtain the weak value for any pre- and post-selections. This formula automatically includes the interference effect which is necessary to yields the weak value as an outcome of the weak measurement.