Sorting Photons by Radial Quantum Number

Abstract

The Laguerre-Gaussian (LG) modes constitute a complete basis set for representing the transverse structure of a paraxial photon field in free space. Earlier workers have shown how to construct a device for sorting a photon according to its azimuthal LG mode index, which describes the orbital angular momentum (OAM) carried by the field. In this paper we propose and demonstrate a mode sorter based on the fractional Fourier transform to efficiently decompose the optical field according to its radial profile. We experimentally characterize the performance of our implementation by separating individual radial modes as well as superposition states. The reported scheme can, in principle, achieve unit efficiency and thus can be suitable for applications that involve quantum states of light. This approach can be readily combined with existing OAM mode sorters to provide a complete characterization of the transverse profile of the optical field. DOI: 10.1103/PhysRevLett.119.263602 In recent years, the transverse structure of optical photons has been established as a resource for storing and communicating quantum information [1]. In contrast to the twodimensional Hilbert space of polarization, it takes an unbounded Hilbert space to provide a mathematical representation for the transverse structure of the optical field. The large information capacity of structured photons has been recently utilized to enhance quantum key distribution [2][3][4][5] and a multitude of other applications [6][7][8][9][10]. The orbital angular momentum (OAM) modes have become increasingly popular for implementing multidimensional quantum states due to the relative ease in generation [11], manipulation [12], and characterization of these modes [13,14].Although the OAM modes provide a basis set for representing the azimuthal structure of photons, they cannot completely span the entire transverse state space, which encompasses an extra (radial) degree of freedom. The Laguerre-Gaussian (LG) mode functions provide a basis to fully represent the spatial structure of the transverse field [15][16][17]. These modes are characterized by two numbers, the radial mode index p ∈ f0; 1; 2; …g and the azimuthal mode index l ∈ f0; AE1; AE2; …g. While the azimuthal number l is well studied due to its association with the OAM of light [16]; the radial index p has so far remained relatively unexplored. The quantum coherence of photons in a superposition of orthogonal radial modes has been recently demonstrated in the context of quantum communication and high-dimensional entanglement [17][18][19]. The radial LG modes also hold a number of promising features, and have been studied in the contexts of self-healing [20], super-resolution [21], and hyperbolic momentum charge [22]. Despite the growing theoretical interest in utilizing the radial structure of photons, the experimental realizations have thus far been impeded because of the difficulty of measuring these modes.The initial step in characterization of the radial degree of freedom of light is to find a radial ...

Quantum features in the orthogonality of optical modes for structured and plane-wave light

“…However, the radial index p has to be included in the mode summation, for generality and completeness of the modal decomposition [14]. Indeed, there has been a recent surge in interest in the radial term, and it has been sought to substantiate the role p can play in informatics [15][16][17]. Of course, l and p both feature in the detailed form of the LaguerreGaussian radial distribution:…”

Spin-valley polarized quantum anomalous Hall effect and a valley-controlled half-metal in bilayer graphene

“…Remarkably, we discover that if both voltage and light are applied, the system can be in a spin-valley polarized QAH (SVP-QAH) insulator state, for which the subsystem of one spin is in a QVH state and that of the other spin is in a QAH state. To our knowledge, this state has not been previously described in the literature and it is distinct from the single-valley spin-polarized QAH states [26,29] and the QAH states with no spin or valley polarizations [13-17, 21-23, 28].…”

“…The QAH state in BLG has been suggested to be induced by Rashba interaction and internal magnetization [24] or by strong circularly polarized light [25][26][27], differently from the mechanism of Landau-level quantization in quantum Hall effect. Most recently, the valley-polarized QAH state possessing the properties of both the QVH and QAH states has been proposed in other 2D systems [28,29] similar to BLG. Interestingly, this QAH state can possess edge states which are not complectly chiral [28].…”

“…, and then C = −C υ = C S = 2 is achieved. To our knowledge, this SVP-QAH insulator is a new type of a QAH state, which differs obviously from the single-valley spin-polarized QAH [26,29], spin-unpolarized QAH [9,[12][13][14]28], and valley-unpolarized QAH [9,22,34] states discovered previously.…”

Entanglement generation via diffraction

“…[41]. This entanglement can be harnessed using the vast array of classical techniques for controlling structured light [42], including recent results for mode sorting without postselection [43][44][45]. The notion of quantummechanical transformations at apertures has been probed in Ref.…”