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 $d$-dimensional Fourier transform acting on single photons is known that uses path encoding and requires $O(d{log}_{2}d)$ optical elements. In this paper, we present an alternative design that uses orbital angular momentum as a carrier of information and needs only $O(\sqrt{d}{log}_{2}d)$ 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 factorization. An implementation scheme of the d-dimensional Fourier transform acting on single photons is known that uses path encoding and requires O(dlog2d) 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) 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 d-dimensional Fourier transform acting on single photons is known that uses the path encoding and requires 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(dlogd) 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.
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.
High-dimensional quantum computing has emerged as a recent alternative to conventional qubit-based approaches, offering advantages such as an increased information density and enhanced robustness against noise. In this work, we explore the use of the orbital angular momentum (OAM) of light as a degree of freedom in which high-dimensional quantum gates can be implemented. We present a theoretical and experimental study of structured light beams carrying OAM, with emphasis on Laguerre–Gauss modes, as a natural basis for high-dimensional quantum computing. We then describe the generation of OAM beams using computer-generated holograms and spatial light modulators, detailing practical methods for encoding arbitrary phase profiles. Building on this framework, we analyze the implementation of high-dimensional quantum logic gates, focusing on the generalized Pauli Z_d and X_d operations. Optical setups based on Dove prisms, Sagnac interferometers, spiral phase plates, and parity sorters are proposed and theoretically validated. Finally, we report experimental results demonstrating the generation of Laguerre–Gauss beams, the transformation properties of Dove prisms, and the operation of a parity sorter, which constitutes a building block for realizing high-dimensional quantum gates. These results establish a solid foundation for future implementations of high-dimensional quantum information processing using the orbital angular momentum of light.
Quantum finite automata (QFA) are basic computational devices that make binary decisions using quantum operations. They are known to be exponentially memory efficient compared to their classical counterparts. Here, we demonstrate an experimental implementation of multi-qubit QFAs using the orbital angular momentum (OAM) of single photons. We implement different high-dimensional QFAs encoded on a single photon, where multiple qubits operate in parallel without the need for complicated multi-partite operations. Using two to eight OAM quantum states to implement up to four parallel qubits, we show that a high-dimensional QFA is able to detect the prime numbers 5 and 11 while outperforming classical finite automata in terms of the required memory. Our work benefits from the ease of encoding, manipulating, and deciphering multi-qubit states encoded in the OAM degree of freedom of single photons, demonstrating the advantages structured photons provide for complex quantum information tasks.
Quantum memory is a key component of quantum networks. A successful development in enabling the distribution of quantum information requires the storage of single-photon light. Usually quantum information is encoded using a 2D space, spanned, for example, by the orthogonal polarization states of a photon. In this case, each photon carries a quantum bit of information. If the photon could be encoded in a high-dimensional space instead, the information carried by each photon could be increased from a bit to |log2d| bits, where d is the dimension of the Hilbert space in which the photon is described. The channel capacity of the network and the transmission efficiency would then be greatly enlarged [1]. Moreover, storing high-dimensional states in quantum memory can reduce its sensitivity to memory coherence time [2], leading to significant improvements in storage capacity [3]. It is thus no surprise that many research groups are actively working on different physical systems for storing light encoded using a high-dimensional space. The degrees of freedom for position, linear momentum, time-bin or orbital angular momentum (OAM) of a photon can all be considered as a high-dimensional space for encoding. The photon's OAM has the most potential for encoding [4]. Electromagnetic waves have both spin and OAM. Spin angular momentum produces polarization. Photons with OAM can be regarded as helices with left and right handedness twisted to varying degrees (Fig. 1). Information is encoded using this handedness and the amount of twist. Because of the inherent infinite degrees of freedom for OAM, a photon can in principle carry an arbitrarily large amount of information, and hence the eagerness to store light encoded in OAM space. Although some work on this has been reported for different physical systems, what is lacking is a way of storing and releasing at some later time a true single photon, including its shape and structure [5–7]. Constructing this basic element of quantum memory has been a major challenge. The wavefront of an OAM light is a helical structure (left and center), with an optical vortex at its center in the intensity distribution of light (right). The vortex of light can carry non-zero OAM of any order m. OAM states of a different order are orthogonal. A group at the Key Laboratory of Quantum Information, Chinese Academy of Science, at the University of Science and Technology of China finally resolved this problem [8]. They reported on the first experimental realization of a true single-photon-carrying OAM stored via electromagnetically induced transparency (EIT) in a cold atomic ensemble (Fig. 2). With such a suitable candidate, this work indicates a means of constructing high-dimensional quantum memory. After this, similar results were reported by Laurat's group at Laboratoire Kastler Brossel, at Université Pierre et Marie Curie in France [9], where an extremely weak laser at single-photon level carrying OAM was also stored via EIT in a cold atomic ensemble. Ding et al. [8] first prepared two cold 85Rb atomic clouds in two magnetic-optical traps (MOT) using laser cooling and trapping techniques. They generated non-classical correlated photon pairs using spontaneous four-wave mixing in an atomic cloud. One photon of each pair was used as a trigger; the other mapped and stored in a second atomic cloud via EIT. Each photon to be stored carried one OAM unit. After a programed storage time, the photon was retrieved. What they achieved during storage included (1) proof that the non-classical correlation between the trigger photon and the retrieved photon is retained by demonstrating a strong violation of the Cauchy–Schwarz inequality, (2) a demonstration that the single-photon property of the signal photon is maintained by performing the Hanbury–Brown and Twiss experiment on the trigger photon, (3) proof that the spatial structure of the photon is also very well preserved by showing a high similarity between input and output, and (4) evidence, gathered by Sagnac interferometry, that the coherence of a single photon is also preserved. A heralded single photon is generated through spontaneous four-wave mixing in MOT 1 (a); this single photon carrying OAM is then stored through EIT in MOT 2 (b); (c) shows an image of the photon carrying OAM. Experimental measurements in the transverse position of the image carried by the photon before and after storage are presented in (d) and (e), respectively; (f) shows the simplified experimental setup for storage of a superposition of OAM. Interference patterns taken at different angle settings of the half-wave plate are shown in (g). The left column represents the experimental angle settings of the half-wave and the quarter-wave plates, the middle column gives the experimental results, and the right column gives results from theoretical simulations; (h) and (i) present the interference patterns of the input signal photon and the retrieved signal with plate angle, respectively. This is a significant breakthrough towards the realization of a quantum internet and, potentially, quantum computing. This work addresses the ability to generate, store, retrieve on demand an arbitrary single-photon-level photonic state encoded in OAM space. It signifies an important step in the development of practical quantum-enabled technologies, opens up the possibility for realizing high-dimensional quantum memories based on this degree of freedom of light, and will have a large impact in a broad physical community, especially in the quantum information and quantum optics communities. The Technology Review website of MIT comments on their work very positively with an article entitled ‘First Quantum Memory That Records the Shape of a Single Photon Unveiled in China’. Further, Nature Photonics selected this work as a research highlight. Clearly, memory that can preserve the spatial structure of photons will make the internet far more flexible and capable. Of course, there are still many basic problems remaining that need to be solved to build the future quantum internet, one of which is the construction of quantum memory exploiting entanglement. This derives from the fact that a quantum network needs the distribution of quantum-entangled photons over channels between different nodes [10]. A high-dimensional quantum internet needs to store OAM-entangled states and perform entanglement swapping to efficiently extend the achievable communication distances. While Ding et al. [8] reported the storage of light with OAM at the single-photon level, the photon was, however, still encoded in a 2D space. Therefore, whether it is possible to construct a high-dimensional quantum memory remains an open issue that needs to be investigated further.
The orbital angular momentum (OAM) carried by twisted photons provides a promising playground for high-dimensional quantum information processing. While Bell-state measurement is the cornerstone for various quantum information applications, the deterministic discrimination of the complete high-dimensional Bell states with linear optics remains relatively unexplored in the OAM state space. Here, we demonstrate a theoretical scheme for the complete four-dimensional OAM Bell-state measurement by using the single-photon hyperentangled state analyzer, in which the auxiliary two-dimensional polarization entanglement and two-dimensional path entanglement are utilized. Our scheme offers an alternative route toward enhancing the channel capacity in quantum communication and increasing the robustness against deleterious noise in practical experiments with twisted photons.