Factoring and Fourier transformation with a Mach-Zehnder interferometer

AbstractThe scheme of Clauser and Dowling (Phys. Rev. A53, 4587 (1996)) for factoringN by means of an N-slit interference experiment is translated into an experimentwith a single Mach-Zehnder interferometer. With dispersive phase shifters the ra-tio of coherence length to wavelength limits the numbers that can be factored. Aconservative estimate permits N ≈ 10 7 . It is furthermore shown, that sine andcosine Fourier coefficients of a real periodic function can be obtained with such aninterferometer. PACS: 03.65.Bz, 06.50.Mk, 07.60.LyRecently Clauser and Dowling (CD) have shown that factors of an integer Ncan bedetermined by simply measuring the peaks of the intensity distribution on the screenbehind a Young’s N-slit arrangement [1]. This device is distinct from the currently muchdebated proposals for quantum computation, because it does not need the entanglementof several quantal degrees of freedom. Therefore it is very immune against decoherenceand could readily be implemented with present technology. The drawback is that it willnot exhibit the potential increase in computational power characteristic of entanglement.Nevertheless, the work of CD indicates that single particle interference arrangements haveuseful applications beyond physical measurements.The purpose of this note is to point out that the CD-proposal can be translated intoan experiment with only a single Mach-Zehnder interferometer. This will enhance theflexibility ofthis calculational device. A further point isthat a Mach-Zehnder interferomtercan also perform other computations, in particular cosine and sine Fourier transformations.Let us first focus on the work of CD. This proposal shows, that in a suitably chosencentral region on the detection screen behind an N-slit arrangement, all intensity peaks areequal if, and only if, the quantity n≡ λR/a