Factoring and Fourier transformation with a Mach-Zehnder interferometer

The scheme of Clauser and Dowling (Phys. Rev. A 53, 4587 (1996)) for factoring N by means of an N -slit interference experiment is translated into an experiment with a single Mach-Zehnder interferometer. With dispersive phase shifters the ratio of coherence length to wavelength limits the numbers that can be factored. A conservative estimate permits N ≈ 10. It is furthermore shown, that sine and cosine Fourier coefficients of a real periodic function can be obtained with such an interferometer. PACS: 03.65.Bz, 06.50.Mk, 07.60.Ly Recently Clauser and Dowling (CD) have shown that factors of an integer N can be determined by simply measuring the peaks of the intensity distribution on the screen behind a Young’s N -slit arrangement [1]. This device is distinct from the currently much debated proposals for quantum computation, because it does not need the entanglement of several quantal degrees of freedom. Therefore it is very immune against decoherence and could readily be implemented with present technology. The drawback is that it will not exhibit the potential increase in computational power characteristic of entanglement. Nevertheless, the work of CD indicates that single particle interference arrangements have useful applications beyond physical measurements. The purpose of this note is to point out that the CD-proposal can be translated into an experiment with only a single Mach-Zehnder interferometer. This will enhance the flexibility of this calculational device. A further point is that a Mach-Zehnder interferomter can 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 chosen central region on the detection screen behind an N -slit arrangement, all intensity peaks are equal if, and only if, the quantity n ≡ λR/a is a factor of N , where λ is the wavelength of the incident radiation, a is the center-to-center distance of the slits, and R is the distance between the slits and the screen. Furthermore, the Fraunhofer limit is assumed (R ≫ a≫ λ). Different values of n can be tested by adjusting any of the parameters, for instance λ. The probability amplitude caused by slit i at a point r on the screen is given by ψi(r). The probability that the particle will hit this point is therefore I(r) = ∣