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arXiv:1210.6487 · New Journal of Physics

Factorization of numbers with Gauss sums: II. Suggestions for implementation with chirped laser pulses

W. Merkel, Sabine Wölk, Wolfgang P. Schleich, Ilya Sh. Averbukh, B. Girard, G. G. Paulus

Published Oct 7, 2011• 12 citations
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Abstract

We propose three implementations of the Gauss sum factorization schemes discussed in part I of this series (Wölk et al 2011 New J. Phys. 13 103007): (i) a two-photon transition in a multi-level ladder system induced by a chirped laser pulse, (ii) a chirped one-photon transition in a two-level atom with a periodically modulated excited state and (iii) a linearly chirped one-photon transition driven by a sequence of ultrashort pulses. For each of these quantum systems, we show that the excitation probability amplitude is given by an appropriate Gauss sum. We provide rules on how to encode the number N to be factored in our system and how to identify the factors of N in the fluorescence signal of the excited state.

DOI
10.1088/1367-2630/13/10/103008

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We propose three implementations of the Gauss sum factorization schemes discussed in part I of this series (Wölk et al 2011 New J. Phys. 13 103007): (i) a two-photon transition in a multi-level ladder system induced by a chirped laser pulse, (ii) a chirped one-photon transition in a two-level atom with a periodically modulated excited state and (iii) a linearly chirped one-photon transition driven by a sequence of ultrashort pulses. For each of these quantum systems, we show that the excitation probability amplitude is given by an appropriate Gauss sum. We provide rules on how to encode the number N to be factored in our system and how to identify the factors of N in the fluorescence signal of the excited state.
Abstract The periodicity properties of Gauss sums allow us to factor integer numbers. We show that the excitation probability amplitudes of appropriate quantum systems interacting with specific laser fields are determined by Gauss sums. The resulting probabilities are experimentally accessible by measuring the fluorescence from this level. In particular, we discuss a two‐photon transition in a ladder system driven by a chirped laser pulse. In addition, we consider two realizations of laser driven one‐photon transitions. For each quantum system we demonstrate the power of this factorization scheme using numerical examples.
We report on a new implementation of the factorisation of numbers using Gauss sums which improves tremendously the efficiency to eliminate all "ghost" factors. We show that by choosing randomly the terms in the Gauss sum, the required number of terms varies as ln N instead of . As an illustration, we present experimental results obtained by interfering thirty ultrashort laser pulses where we factorise 1340333404807. This new approach is totally general and can be implemented for all the experiments based on the Gauss sum.
Several physics-based algorithms for factorizing large numbers were recently presented. A notable recent algorthm by Schleich et al. uses Gauss sums for distinguishing between factors and nonfactors. We demonstrate two NMR techniques that evaluate Gauss sums and thus implement their algorithm. The first one is based on differential excitation of a single spin magnetization by a cascade of rf pulses. The second method is based on spatial averaging and selective refocusing of magnetization for Gauss sums corresponding to factors. All factors of 16 637 and 52 882 363 are successfully obtained.