Wigner Distribution of Twisted Photons

Abstract

We present the first experimental characterization of the azimuthal Wigner distribution of a photon. Our protocol fully characterizes the transverse structure of a photon in conjugate bases of orbital angular momentum (OAM) and azimuthal angle (ANG). We provide a test of our protocol by characterizing pure superpositions and incoherent mixtures of OAM modes in a seven-dimensional space. The time required for performing measurements in our scheme scales only linearly with the dimension size of the state under investigation. This time scaling makes our technique suitable for quantum information applications involving a large number of OAM states.Ever since its introduction in 1932 [1], the Wigner distribution has been widely applied in different fields of study ranging from statistical mechanics, and optics [2] in physics to more applied fields such as electrical engineering and even seismology [3]. In physics, the Wigner distribution has been utilized to bring the machinery of phase-space statistical mechanics into study of quantum physics [4]. Wigner distribution provides a comprehensive characterization of the system, and as a quasiprobability distribution the negativity of the Wigner distribution signals a wave-like behavior [5,6].The orbital angular momentum (OAM) of single photons has lately been identified as a valuable platform for realizing multilevel quantum systems [7,8]. The discrete nature of OAM makes it attractive for encoding quantum [9] and classical information [10]. The ongoing research suggests that there is no fundamental limit to the maximum value of OAM that a photon can carry. In a recent experiment, quantum entanglement was demonstrated between states differing by 600 in their value of OAM [11]. The full characterization of a quantum state in the Hilbert space of OAM poses a serious experimental challenge.A large body of previous research has enabled efficient and accurate projective measurements of light's OAM [8,[12][13][14][15][16]. Quantum mechanically, a pure state in the Hilbert space of OAM is described by a discrete state vector. Thus, the probability distribution provided by projective measurements along with the knowledge of relative phase between the different OAM components found by interferometry adequately described a pure state [17]. Nevertheless, pure states are only a restricted set of physical states, because the vast majority of conceivable states are mixed states [18]. The most general description of a quantum state requires knowledge of its density matrix, which can be found through use of standard quantum state tomography [19,20]. However, quantum state tomography in the OAM basis requires the capability to perform projective measurements on arbitrary superpositions of two or more OAM eigenstates [21], a task that remains challenging due to technical limitations such as variations in the efficiency of measuring different OAM modes and the cross-talk between neighboring modes [22].In this article, we propose and demonstrate a method for obtaining the Wigner distributio...

On Wigner function of a vortex electron

“…This is all the more so for wave packets with the non-Gaussian spatial profiles, such as a vortex state with an orbital angular momentum (OAM) ℓ [6], an Airy beam [7,8], and their different generalizations. Although there are several works where the Wigner functions of such packets are derived (see, for instance, [9-12]) and even studied experimentally for twisted photons [11,13], the problem of invariance of these functions under the Lorentz transformations for massive wave packets (first of all for vortex electrons, as they have recently been obtained experimentally [6]) remains unsolved.One of the reasons why it has not been done for vortex electrons yet is that the very wave functions describing these states (the so-called Bessel beams and the Laguerre-Gaussian packets [6]) are usually written in terms of the non-invariant quantities and the transformation properties of these functions remain unclear. As noted by , this lack of explicit invariance per se does not violate correct transformation properties of the observables.…”

Performance analysis of d -dimensional quantum cryptography under state-dependent diffraction

“…Beams with an azimuthal phase dependence exp(iℓθ) carry an OAM of ℓh per photon, where ℓ is the integer OAM quantum number. After the breakthrough work by Allen et al in 1992 [9], the properties and applications of OAM have been studied in both classical and quantum regimes [10][11][12][13][14].…”

Sorting Photons by Radial Quantum Number

“…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].…”