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arXiv:0709.0613 · Physical Review A

Complementarity and uncertainty relations for matter-wave interferometry

Karl-Peter Marzlin, Barry C. Sanders, P. L. Knight

Published Dec 9, 2008• 8 citations
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Abstract

We establish a rigorous quantitative connection between (i) the interferometric duality relation for which-way information and fringe visibility and (ii) Heisenberg's uncertainty relation for position and modular momentum. We apply our theory to atom interferometry, wherein spontaneously emitted photons provide which-way information and unambiguously resolve the challenge posed by the metamaterial ``perfect lens'' to complementarity and to the Heisenberg-Bohr interpretation of the Heisenberg microscope thought experiment.

DOI
10.1103/physreva.78.062107

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We establish a rigorous quantitative connection between (i) the interferometric duality relation for which-way information and fringe visibility and (ii) Heisenberg's uncertainty relation for position and modular momentum. We apply our theory to atom interferometry, wherein spontaneously emitted photons provide which-way information and unambiguously resolve the challenge posed by the metamaterial ``perfect lens'' to complementarity and to the Heisenberg-Bohr interpretation of the Heisenberg microscope thought experiment.
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Wave and particle properties of a quantum object cannot be observed simultaneously. In particular, the fringe visibility in an interferometer is limited by the amount of which-way information which can be obtained. This limit is set by the recently discovered duality relation. So far, all derivations of the duality relation are independent of Heisenberg’s uncertainty relation. Here we demonstrate that it is alternatively possible to derive the duality relation in the form of an uncertainty relation for some suitably chosen observables.
We revisit the connection between the complementarity and uncertainty principles of quantum mechanics within the framework of Mach–Zehnder interferometry. We focus our attention on the trade-off relation between complementary path information and fringe visibility. This relation is equivalent to the uncertainty relation of Schrödinger and Robertson for a suitably chosen pair of observables. We show that it is equivalent as well to the uncertainty inequality provided by Landau and Pollak. We also study the relationship of this trade-off relation with a family of entropic uncertainty relations based on Rényi entropies. There is no equivalence in this case, but the different values of the entropic parameter do define regimes that provides us with a tool to discriminate between non-trivial states of minimum uncertainty. The existence of such regimes agrees with previous results of Luis (2011 Phys. Rev. A 84 034101), although their meaning was not sufficiently clear. We discuss the origin of these regimes with the intention of gaining a deeper understanding of entropic measures.
We revisit the connection between the complementarity and uncertainty principles of quantum mechanics within the framework of Mach?Zehnder interferometry. We focus our attention on the trade-off relation between complementary path information and fringe visibility. This relation is equivalent to the uncertainty relation of Schrödinger and Robertson for a suitably chosen pair of observables. We show that it is equivalent as well to the uncertainty inequality provided by Landau and Pollak. We also study the relationship of this trade-off relation with a family of entropic uncertainty relations based on Rényi entropies. There is no equivalence in this case, but the different values of the entropic parameter do define regimes that provides us with a tool to discriminate between non-trivial states of minimum uncertainty. The existence of such regimes agrees with previous results of Luis (2011 Phys. Rev. A 84 034101), although their meaning was not sufficiently clear. We discuss the origin of these regimes with the intention of gaining a deeper understanding of entropic measures.
The exact analytical values of the position and momentum information entropies are calculated for the single-slit and double-slit diffraction experiments. In both cases, the product of the exponentials of the entropies is strictly greater than the lower bound $\ensuremath{\pi}e\ensuremath{\Elzxh}$ given by the optimal entropic uncertainty relation for position and momentum, which implies that the single-slit and double-slit configurations are not minimum-uncertainty states. The results obtained show that the position-momentum entropic uncertainty relation provides a rigorous quantitative expression for the uncertainty principle in these experiments, unlike the Heisenberg inequality for standard deviations. However, it is also shown that, in the double-slit experiment, wave-particle duality cannot be derived from the entropic uncertainty relation.