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Optics Express

Fractional Fourier processing of quantum light

Yifan Sun, Ran Tao, Xiangdong Zhang

Published Jan 6, 2014• 8 citations
Publisher

Abstract

We have extended Fourier transform of quantum light to a fractional Fourier processing, and demonstrated that a classical optical fractional Fourier processor can be used for the shaping of quantum correlations between two or more photons. Comparing the present method with that of Fourier processing, we find that fractional Fourier processing for quantum light possesses many advantages. Based on such a method, not only quantum correlations can be shaped more rich, but also the initial states can be easily identified. Moreover, the twisted phase information can be recovered and quantum states are easily controlled in performing quantum information experiments. Our findings open up new avenues for the manipulation of correlations between photons in optical quantum information processing.

DOI
10.1364/oe.22.000727

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We have extended Fourier transform of quantum light to a fractional Fourier processing, and demonstrated that a classical optical fractional Fourier processor can be used for the shaping of quantum correlations between two or more photons. Comparing the present method with that of Fourier processing, we find that fractional Fourier processing for quantum light possesses many advantages. Based on such a method, not only quantum correlations can be shaped more rich, but also the initial states can be easily identified. Moreover, the twisted phase information can be recovered and quantum states are easily controlled in performing quantum information experiments. Our findings open up new avenues for the manipulation of correlations between photons in optical quantum information processing.
It is shown that a classical optical Fourier processor can be used for the shaping of quantum correlations between two or more photons, and the class of filter functions applicable in the multiphoton Fourier space is identified. This concept is experimentally demonstrated by using two types of periodic phase Fourier filters to manipulate the quantum correlations between path-entangled photon pairs. As many more types of filter functions are applicable, this opens up an alternative avenue for the manipulation of correlations between photons, a key ingredient in optical quantum information processing.
The light's image is the primary source of information carrier in nature. Indeed, a single photon's image possesses a vast information capacity that can be harnessed for quantum information processing. Our scheme for implementing quantum information processing via universal multiport processors employs a class of quantum Fourier optical systems composed of spatial phase modulators and 4f-processors with phase-only pupils having a characteristic periodicity that reduces the number of optical resources quadratically as compared to other conventional path encoding techniques. In particular, this paper employs quantum Fourier optics to implement some key quantum logical gates that can be instrumental in optical quantum computations. For instance, we demonstrate the principle by implementing the single-qubit Hadamard and the two-qubit controlled-NOT gates via simulation and optimization techniques. Due to various advantages of the proposed scheme, including the large information capacity of the photon wavefront, a quadratically reduced number of optical resources compared with other conventional path encoding techniques, and dynamic programmability, the proposed scheme has the potential to be an essential contribution to linear optical quantum computing and optical quantum signal processing.
All-quantum signal processing techniques are at the core of the successful advancement of most information-based quantum technologies. This article develops coherent and comprehensive methodologies and mathematical models to describe Fourier optical signal processing in full quantum terms for any input quantum state of light. We begin this article by introducing a spatially 2-D quantum state of a photon, associated with its wavefront and expressible as a 2-D creation operator. Then, by breaking down the Fourier optical processing apparatus into its key components, we strive to acquire the quantum unitary transformation or the input/output quantum relation of the 2-D creation operators. Subsequently, we take advantage of the above results to develop and obtain the quantum analogous of a few essential Fourier optical apparatuses, such as quantum convolution via a 4f-processing system and a quantum 4f-processing system with periodic pupils. Moreover, due to the importance and widespread use of optical pulse shaping in various optical communications and optical sciences fields, we also present an analogous system in full quantum terms, namely quantum pulse shaping with an 8f-processing system. Finally, we apply our results to two extreme examples of the quantum state of light. One is based on a coherent (Glauber) state and the other on a single-photon number (Fock) state for each of the above optical systems. We believe the schemes and mathematical models developed in this article can impact many areas of quantum optical signal processing, quantum holography, quantum communications, quantum radars and multiple-input/multiple-output antennas, and many more applications in quantum imaging, quantum computations, and quantum machine learning algorithms.
Light processing plays a central role in quantum communication, since optical quantum networks that will allow for secure quantum communication [1] and linking quantum computers [2] are of significant scientific and engineering interest [3], [4]. In this work, we present the experimental implementation of optical Fractional Fourier Transform. The protocol is realised in gradient echo memory. Hermite-Gaussian modes of light are encoded in rubidium atoms with the coupling impulse linearly chirped in frequency - this corresponds to the temporal lens [5]. Then the ac-Stark beam is applied to the atoms which leads to them acquiring the spectral phase corresponding to the propagation through a spectral lens. This operation results in the rotation of the Wigner function of the input state [6] and is exactly the Fractional Fourier Transform (FrFT) of the input function. The Hermite-Gaussian modes of light are the eigenfunctions of FrFT.
The light's image is the primary source of information carrier in nature. Indeed, a single photon's image possesses a vast information capacity that can be harnessed for quantum information processing. Our scheme for implementing quantum information processing on discretized photon wavefront via universal multiport processors employs a class of quantum Fourier optical systems composed of spatial phase modulators and 4f-processors with phase-only pupils having a characteristic periodicity that reduces the number of optical resources quadratically as compared to other conventional path encoding techniques. In particular, this paper employs quantum Fourier optics to implement some key quantum logical gates that can be instrumental in optical quantum computations. For instance, we demonstrate the principle by implementing the single-qubit Hadamard and the two-qubit controlled-NOT gates via simulation and optimization techniques. Due to various advantages of the proposed scheme, including the large information capacity of the photon wavefront, a quadratically reduced number of optical resources compared with other conventional path encoding techniques, and dynamic programmability, the proposed scheme has the potential to be an essential contribution to linear optical quantum computing and optical quantum signal processing.