Implementation of quantum and classical discrete fractional Fourier transforms

Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools.

for a pair of distinguishable photons after propagating through a -lattice. The photon density at the output is shown on the right side of the map.

Supplementary Note 1
As additional classical experiments we examine a well-known example of a FT pair, the top-hat function whose FT is a sinc function. This input state is intentionally chosen to illustrate the difference between the FT and the DFrFT when the continuous limit is not met. The input fields are prepared by tailoring laser light at a wavelength of 632 nm using a SLM (Holoeye Pluto VIS).
In order to be able to shape the input field at will, we modulate amplitude and phase. Since this SLM allows for phase modulation only, the amplitude modulation is realized by imprinting an extra phase grating onto the SLM. The first diffraction order can thus be modulated in amplitude as well (1). An initial Gaussian beam at a wavelength of 632 nm is expanded to homogeneously illuminate the display of the SLM. The reflected beam is Fourier transformed by a spherical lens with 300mm focal length. The first diffraction order is isolated by a slit aperture 300mm after the lens. The so prepared field distribution is scaled down to micrometer size by means of a 4f configuration involving a 250mm lens and a 20x microscope objective (Olympus Plan Achromat). For the sake of maximizing the efficiency of coupling the free space electric field distribution into the discrete array of single-mode waveguides, the beams are prepared as arrays  Figure 1B). This is achieved by tilting the array by an angle of approximately 0.01°. This very small tilt is not significantly influencing the input intensity distribution which is created by the 4f-setup. As can be seen, the ramping phase in the input field does not result in a pure spatial displacement at ⁄ . Again there are deviations between the DFrFT and the shifted sinc function, meaning that the continuous limit is not achieved.

Supplementary Note 2
In order to assess the statistical consistency of the results in Fig. 4  In order to compare the results to a classical, incoherent counterpart, we have performed an additional simulation of the correlation function when the lattice is illuminated with distinguishable photons at the lattice edges. After injection into channels and , the correlation function of a two-photon wavefunction with distinguishable particles reads | | | | . The simulation result is shown below. Such correlation map does not show any signature of quantum interference since time-evolved states of this kind can always be factorized.
As a result, the building-up of non-classical correlations is prevented. Evidently, the correlation maps obtained either with the product state or the entangled state in the main text (Fig. 4) show a checkerboard pattern, as states are suppressed due to destructive interference between the different paths each photon can take. This is in strong contrast to the simulation presented here.