S Tamate et al 2009 New J. Phys. 11 093025 doi:10.1088/1367-2630/11/9/093025
Geometrical aspects of weak measurements and quantum erasers
S Tamate1,3, H Kobayashi1, T Nakanishi1,2, K Sugiyama1,2 and M Kitano1,2
Show affiliationsWe investigate the mechanism of weak measurement by using an interferometric framework. In order to appropriately elucidate the interference effect that occurs in weak measurement, we introduce an interferometer for particles with internal degrees of freedom. It serves as a framework common to quantum eraser and weak measurement. We demonstrate that the geometric phase, particularly the Pancharatnam phase, results from the post-selection of the internal state, and thereby the interference pattern is changed. It is revealed that the extraordinary displacement of the probe wavepackets in weak measurement is achieved owing to the Pancharatnam phase associated with post-selection.
A correction was made to equation (15) on 30 September 2009
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GENERAL SCIENTIFIC SUMMARY
Introduction and background. Quantum measurement with post-selection, such as quantum eraser or weak measurement, exhibits various interesting phenomena. The weak value, which is the result of the weak measurement, can be much larger than the expectation value of the measured observable under suitable conditions. Weak measurement is now utilized for measuring minute quantum effects. In this study, we investigated the underlying geometric structure of post-selected quantum systems and the mechanism by which such a large weak value can be obtained.Main results. We showed that the geometric phase plays a central role in post-selected quantum systems by discussing the analogy between the quantum eraser and the weak measurement. When post-selection is implemented, the geometric phase of the measured system is transferred to the probe system; this transfer affects the interference of the probe state. The geometric phase is proportional to the area enclosed by the geodesic paths on the Bloch sphere. We demonstrated that the unboundedly large weak value stems from the singularity of the geometric phase (see figure) and that the required condition for the weak measurement results from the nonlinearity of the geometric phase.
Wider implications. We have found the underlying relations among the three basic concepts in quantum mechanics, namely, geometric phase, quantum eraser, and weak measurement. We hope that our study sheds new light on the foundation of quantum mechanics and that it allows the development of valuable applications.
Figure. Geometric phase on the Bloch sphere. The area of the geodesic triangle, which is proportional to the geometric phase, can change drastically, even when one of the vertices is shifted slightly from B to B'. This feature of the geometric phase leads to the unboundedly large weak value. - PACS
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03.65.Ta Foundations of quantum mechanics; measurement theory
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- Dates
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Issue 9 (September 2009)
Received 28 May 2009
Published 16 September 2009
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Geometrical aspects of weak measurements and quantum erasers
S Tamate et al 2009 New J. Phys. 11 093025
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