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Journal of Modern Optics

A scheme for the simultaneous measurement of atomic position and momentum

W. L. Power, S. M. Tan, Martin Wilkens

Published Nov 1, 1997• 17 citations
Publisher

Abstract

We propose an optical scheme for the simultaneous measurement of the position and momentum of a single atom. The scheme involves the coupling of the atom of two light fields with different spatial and polarization characteristics. The proposed technique is closely related to the Arthurs-Kelly measurement scheme; the principal difference is that in the present case the values of the position and momentum are inferred from phase shifts in electromagnetic fields rather than from shifts in the position of a pointer.

DOI
10.1080/09500349708231904

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We propose an optical scheme for the simultaneous measurement of the position and momentum of a single atom. The scheme involves the coupling of the atom of two light fields with different spatial and polarization characteristics. The proposed technique is closely related to the Arthurs-Kelly measurement scheme; the principal difference is that in the present case the values of the position and momentum are inferred from phase shifts in electromagnetic fields rather than from shifts in the position of a pointer.
The prototype of a simultaneous measurement of position and momentum of a quantum particle was originally introduced by Arthurs and Kelly in 1965. It relies on coupling a quantum particle to two additional systems, which serve as measurement pointers. In this contribution, we start from the question for optimal pointers for such a setup. This allows us to identify a particular readout time with the help of a suitably chosen preparation energy of the measurement apparatus. Hence we find a natural connection between energy and time in a simultaneous measurement concept. Finally, our analysis sheds new light onto the distinctive role played by coherent states in quantum physics.
In two recent papers,1,2 it was shown how the interaction of a two-level atom with a standing laser field allows for a QND measurement of the atomic position using a quadrature component of the laser field as a meter. Here we demonstrate a complementary scheme in which a quadrature component of a propagating laser wave acts as the meter for a QND measurement of the atomic momentum. The measurement is based on a detectable phase shift of the cavity mode that results from the atom-field interaction. The principal characteristics of the measurement scheme are presented. We demonstrate how a selective version of the measurement scheme may be used to tailor an atomic wave function and to turn the measurement scheme into a "virtual beam handler."
The prototype for a simultaneous measurement of two conjugate variables was originally introduced by Arthurs and Kelly in 1965. It relies on coupling the quantum particle to be probed to two additional systems, which serve as measurement pointers. In this contribution we investigate an extended scheme to measure position and momentum of a massive particle. By considering an explictly time-dependent coupling we can quantify the timing of the measurement. We investigate how the noise resulting from such a measurement process depends on the interaction strength and the size of the pointers. In particular, we focus on the question of which measurement timing minimizes the corresponding uncertainty product.
We consider atoms traversing a cavity filled with an optical field. When the atoms are well detuned from the optical resonance the output momentum distribution of the atoms is found to be a sensitive probe of the photon statistics of the light field. Near resonance spontaneous emission smears the diffractive peaks. We obtain a good fit to the experimental data of Gould et at. (1991). As the atoms pass through the optical field they impart a position-dependent phase shift to the field. By making a quadrature phase measurement on the optical field a position measurement of the atom is achieved. We show that it is possible to prepare the atom in a 'contractive state' which beats the standard quantum limit for position measurements.