Experimental Implementation of the Optical Fractional Fourier Transform in the Time-Frequency Domain

The fractional Fourier transform (FrFT), a fundamental operation in physics that corresponds to a rotation of phase space by any angle, is also an indispensable tool employed in digital signal processing for noise reduction. Processing of optical signals in their time-frequency degree of freedom bypasses the digitization step and presents an opportunity to enhance many protocols in quantum and classical communication, sensing, and computing. In this Letter, we present the experimental realization of the fractional Fourier transform in the time-frequency domain using an atomic quantum-optical memory system with processing capabilities. Our scheme performs the operation by imposing programmable interleaved spectral and temporal phases. We have verified the FrFT by analyses of chroncyclic Wigner functions measured via a shot-noise limited homodyne detector. Our results hold prospects for achieving temporal-mode sorting, processing, and superresolved parameter estimation.

Figure 1

(a) Ideational scheme of the evolution of CWF due to temporal lenses $f_t$ and frequency lens $f_{\omega }$. (b) Relevant ${}^{87}\mathrm{Rb}$ energy level configuration. (c) Experimental setup used to perform FrFT, based on ultracold atoms in a magneto-optical trap. The frequency lens is implemented using a spatial light modulator (SLM) by imprinting an intensity pattern on ac-Stark (acS) beam. The acS beam then illuminates atoms, applying a spatial quadratic phase. The custom shot noise-limited differential photodiode (DPD) detects the beating between the signal and reference. (d) Experimental sequence for performing FrFT of two-pulse state. HP and ZP are, respectively, hyperfine pumping and Zeeman pumping.

Figure 2

CWFs of two-pulse state rotated by angles $\phi \in \{0,(\pi /3),(\pi /2),(2\pi /3)\}$. The upper row presents the experimental data. The lower row presents the results obtained from numerical simulations, taking into account experimental limitations.

Figure 3

(a) Histogram of the fidelity for measured Hermite-Gaussian modes (measured) projected onto modes (numerical) for $\phi =(\pi /4)$. The color map represents the phase of the overlap. (b) Histogram of the fidelity for measured Hermite-Gaussian modes (measured) projected onto modes (numerical) for $\phi =(2\pi /3)$. The color map represents the phase of the overlap. (c) CWFs of measured subsequent Hermite-Gaussian modes for $n\in [0,10]\cap \mathbb{Z}$. The upper row represents $\phi =0$ and lower $\phi =(\pi /4)$.

Figure 4

(a) The phase of the overlap of subsequent Hermite-Gaussian modes for $n\in [0,10]\cap \mathbb{Z}$ with fitted functions. Error bars correspond to $5\sigma$. Shaded regions represent the fitted functions with $\pm \sigma$ of the fitted parameters. Each line is shifted by 0.75 rad to clarify the plot. (b) Histogram presenting phases of overlaps of Hermite-Gaussian modes $n\in [0,10]\cap \mathbb{Z}$ and $\phi =(2\pi /3)$ with their numerical equivalents (the same mode, center, and width). The height of each bar is proportional to number of events i.e. measured phase of overlap is in a bin range. The bars are color coded by each mode.