Researchers from the Faculty of Physics at the University of Warsaw, in collaboration with experts from the QOT Center for Quantum Optical Technologies, have created an innovative technology that allows performing fractional Fourier transform of optical pulses using quantum memory.

This achievement is unique at the global level, as the team was the first to present an experimental application of the aforementioned transformation in this type of system. The results of the research were published in the prestigious journal *Physical review letters*. In their work, the students tested the implementation of a fractional Fourier transform using a double optical pulse, also known as the “Schrödinger’s cat” condition.

#### Pulse spectrum and time distribution

Waves, like light, have their own characteristic properties – the duration of the pulse and its frequency (corresponding, in the case of light, to its color). It turns out that these properties are related to each other through a process called the Fourier transform, which makes it possible to switch from describing a wave in time to describing its spectrum in frequencies.

The fractional Fourier transform is a generalization of the Fourier transform that allows a partial transition from a description of a wave in time to a description in frequency. Intuitively, it can be understood as a rotation of a distribution (e.g., Wigner time toroid function) of the studied signal at a given angle in the time-frequency domain.

Transforms of this kind have turned out to be exceptionally useful in designing special spectral and temporal filters to eliminate noise and enable the creation of algorithms that make it possible to use the quantum nature of light to distinguish pulses of different frequencies more accurately than with conventional methods. This is particularly important in spectroscopy, which helps study the chemical properties of matter, and in telecommunications, which requires the transmission and processing of information with high accuracy and speed.

#### Lenses and Fourier transform?

An ordinary glass lens is able to focus a beam of monochromatic light falling on it to approximately a single point (focus). Changing the angle of incidence of light on the lens changes the focus position. This allows us to convert angles of incidence into positions, obtaining a Fourier transform analogy, in the space of directions and positions. Classical diffraction grating-based spectrographs use this effect to convert wavelength information of light into positions, allowing us to distinguish between spectral lines.

#### Time and frequency lenses

Similar to a glass lens, time-frequency lenses allow the pulse duration to be transformed into its spectral distribution, or effectively, to perform a Fourier transform in space-frequency time. Correct selection of the powers of these lenses makes it possible to perform a fractional Fourier transform. In the case of optical pulses, the action of time-frequency lensing corresponds to the application of quadratic phases to the signal.

To process the signal, the researchers used a quantum memory – or more precisely a memory equipped with quantum light processing capabilities – based on a cloud of rubidium atoms placed in a magneto-optical trap. The atoms were cooled to a temperature tens of millions of degrees above absolute zero. The memory is placed in a variable magnetic field, allowing components of different frequencies to be stored in different parts of the cloud. The pulse was subjected to a time lens during writing and reading, and a frequency lens during storage.

The device developed at the University of Wisconsin allows such lenses to be implemented over a very wide range of parameters and in a programmable manner. The double pulse is very prone to decoherence, hence it is often compared to Schrödinger’s famous cat – a microscopic superposition of your dead and alive universe, which is almost impossible to achieve experimentally. However, the team was able to perform precise operations on these fragile double-pulse cases.

The publication was the result of work in the Quantum Optical Devices Laboratory and the Quantum Memory Laboratory at the Center for Quantum Optical Technologies with the participation of two master’s students: Stanislav Korzyna and Marcin Jastrzebski, two undergraduates Bartosz Neult and Jan Novosielski, and Dr. Mateusz Mazlanek, and the heads of the laboratory, Dr. Michal Barniak and Professor Wojciech Vasilevski. For the results described, Bartosz Neolt was also awarded a Presentation Grant during the recent DAMOP conference in Spokane, Washington.

Before direct application in communications, the method must first be mapped to wavelengths and other parameter ranges. However, the partial Fourier transform can be critical for optical receivers in modern networks, including optical satellite links. A quantum light processor developed at the University of Wisconsin makes it possible to find and test such new protocols in an efficient way.

References: “Experimental Implementation of the Optical Fractional Fourier Transform in the Time-Frequency Domain” by Bartosz Neulte, Marcin Jastrzebski, Stanisław Korzyna, Jan Novosilski, Wojciech Vasilewski, Mateusz Mazelanek, and Michal Barniak, 12 June 2023, Available here. *Physical review letters*.

doi: 10.1103/PhysRevLett.130.240801

The project “Quantum Optical Technologies” (MAB/2018/4) is implemented within the International Research Agendas Program of the Polish Science Foundation co-financed by the European Union under the European Regional Development Fund.

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