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arXiv:2004.12788 · Physical Review Applied

Fourier Transform of the Orbital Angular Momentum of a Single Photon

Jaroslav Kysela, Xiaoqin Gao, Borivoje Dakić

Published Sep 14, 2020• 15 citations
arXivHTMLPDF

Abstract

Optical networks implementing single-qudit quantum gates may exhibit superior properties to those for qubits as each optical element in the network can work in parallel on many optical modes simultaneously. We present a class of such networks that implements in a deterministic and efficient way the quantum Fourier transform (QFT) in an arbitrarily high dimension. These networks redistribute the initial quantum state into the orbital angular momentum (OAM) and path degrees of freedom and offer two modes of operation. Either the OAM-only QFT can be implemented, which uses the path as an internal auxiliary degree of freedom, or the path-only QFT is implemented, which uses the OAM as the auxiliary degree of freedom. The resources for both schemes scale linearly $O(d)$ with the dimension $d$ of the system, beating the best known bounds for the path-encoded QFT. While the QFT of the orbital-angular-momentum states of single photons has been applied in a multitude of experiments, these schemes require specially designed elements with nontrivial phase profiles. In contrast, we propose a different approach that utilizes only conventional optical elements.

DOI
10.1103/physrevapplied.14.034036

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The Fourier transform proves indispensable in the processing of classical information as well as in the quantum domain, where it finds many applications ranging from state reconstruction to prime factorization. An implementation scheme of the ddd-dimensional Fourier transform acting on single photons is known that uses path encoding and requires O(dlog2d)O(d{log}_{2}d)O(dlog2​d) optical elements. In this paper, we present an alternative design that uses orbital angular momentum as a carrier of information and needs only O(dlog2d)O(\sqrt{d}{log}_{2}d)O(d​log2​d) elements, rendering the path-encoded design inefficient. The advantageous scaling and the fact that our approach uses only conventional optical elements allows for the implementation of a 256-dimensional Fourier transform with existing technology. Improvements of our design as well as explicit setups for low dimensions are also presented.
The Fourier transform proves indispensable in the processing of classical information as well as in the quantum domain, where it finds many applications ranging from state reconstruction to prime factoring. An implementation scheme of the ddd-dimensional Fourier transform acting on single photons is known that uses the path encoding and requires O(dlog⁡d)O(d \log d)O(dlogd) optical elements. In this paper we present an alternative design that uses the orbital angular momentum as a carrier of information and needs only O(dlog⁡d)O(\sqrt{d}\log d)O(d​logd) elements, rendering the path-encoded design inefficient. The advantageous scaling and the fact that our approach uses only conventional optical elements allows for the implementation of a 256-dimensional Fourier transform with the existing technology. Improvements to our design, as well as explicit setups for low dimensions, are also presented.
Abstract A simple argument is presented that explicitly shows how to construct an arbitrary quantum gate acting on orbital angular momentum (OAM) of single photons. The scheme can be applied to implement subspace multiplexing, where a single high-dimensional OAM qudit represents effectively a stack of multiple independent lower-dimensional qudits. A special subclass of unitaries composed of single-photon controlled gates is studied in detail and notable examples of the general approach are discussed. The generalization of the simple argument leads to the parallelization scheme, which results in the savings of resources. The presented schemes utilize only conventional optical elements and apply not only to single photons but also to classical light.
Inside computer networks, different information processing tasks are necessary to deliver the user data efficiently. This processing can also be done in the quantum domain. We present simple optical quantum networks where the orbital angular momentum (OAM) of a single photon is used as an ancillary degree of freedom which controls decisions at the network level. Linear optical elements are enough to provide important network primitives such as multiplexing and routing. First we show how to build a simple multiplexer and demultiplexer which combine photonic qubits and separate them again at the receiver. We also give two different self-routing networks where the OAM of an input photon is enough to make it find its desired destination.
Structured light carrying Orbital Angular Momentum (OAM) enables high-dimensional photonic quantum information processing. In this work, we propose and theoretically demonstrate a four-dimensional controlledNOT (CNOT) quantum gate implemented using the radial (p) and azimuthal (l) degrees of freedom of Laguerre–Gaussian modes. Quantum information is encoded within a single photon, with the radial mode acting as the control qubit and the OAM mode as the target qubit. The gate is realized using fully linear and unitary optical transformations based on Fourier transform based radial mode sorting and OAM inversion within the architecture. This approach enables deterministic two-qubit logic in an expanded Hilbert space, providing a scalable pathway toward high-dimensional quantum gates using structured light.
Abstract High‐dimensional quantum systems offer many advantages over low‐dimensional quantum systems. Meanwhile, unitary transformations on quantum states are important parts in various quantum information tasks, whereas they become technically infeasible as the dimensionality increases. The photonic orbital angular momentum (OAM), which is inherent in the transverse spatial mode of photons, offers a natural carrier to encode information in high‐dimensional spaces. However, it is even more challenging to realize arbitrary unitary transformations on the photonic OAM states. Here, by combining the path and OAM degrees of freedom of a single photon, an efficient scheme to realize arbitrary unitary transformations on the path‐OAM coupled quantum states is proposed. The proposal reduces the number of required interferometers by approximately one quarter compared with previous works, while maintaining the symmetric structure. It is shown that by using OAM‐to‐path interfaces, this scheme can be utilized to realize arbitrary unitary transformations on the OAM states of photons. This work facilitates the development of high‐dimension quantum state transformations and opens a new door to the manipulation of the photonic OAM states.