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Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR)

Scheme for directly observing the noncommutativity of the position and momentum operators with interference

Jong Chan Lee, Yong‐Su Kim, Young-Sik Ra, Hyang‐Tag Lim, Yoon-Ho Kim

Published Jun 1, 2013• 0 citations
Publisher

Abstract

In this paper, we propose and analyze an experimental scheme for directly observing the noncommutativity of the position and the momentum operators using single-photon quantum interference.
DOI
10.1109/cleopr.2013.6600361

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The expectation value of the product of two noncommuting observables (POTNO) can be used to test the commutation relation and dig out the correlation between two observables. However, since a POTNO is generally a non-Hermitian operator, its average cannot be measured directly according to standard quantum mechanics. Recently, a theoretical scheme has been proposed to obtain POTNOs, but the observables are limited to projection operators. In this paper, we present a scheme that employs weak measurement to directly measure the expectation value of the POTNO, where the observables are arbitrary operators. Our scheme transforms the measurement of this expectation value into the measurements of several single observable weak values. Experimentally we demonstrate our scheme by measuring the expectation value of the product of two Pauli operators and verify their commutation and anticommutation relations in a trapped-ion system. Moreover, our scheme is applicable to the cases with more than two observables.
We present here what we consider a new picture of quantum mechanics with the position and momentum observables as coordinates of the usual quantum phase space of a single particle, which also serves as the model of the physical space. To minimize the mathematics involved, we stick here to the Hilbert space picture of the phase space. We argue that a quantum observable should be seen as taking noncommutative values each of which can equivalently be represented by an infinite number of real numbers. The six noncommutative values of the position and momentum observables hence serve as an alternative system of coordinates for the Hilbert space. The values can, at least in principle, be experimentally determined. This can be seen as a complete resolution of the Einstein–Bohr debate that Einstein would probably be happy with. What we have is a solid noncommutative geometrical picture of the physical space, or spacetime, beyond the Newtonian and Einsteinian framework that is sure to be relevant to Nature directly coupled with the idea that each physical quantity should better be seen as having a value beyond what can be represented by a number. We finish by sketching some implications of the results for the physics and mathematics of quantum spacetime in general.