Quantum state transfer in diffractive and refractive media

Abstract

High-dimensional, discrete quantum states, also known as qudits, offer several advantages over their two- dimensional counterpart (qubits). For example, in quantum communication, the use of qudits not only increases the amount of information encoded into a single carrier, but also enhances the security of quantum key distribution (QKD) protocols. In particular, in entanglement-based QKD the intervention of an eavesdropper can be excluded by the violation of a Bell-type inequality, which is the more violated the larger the dimensionality of the employed states. Spanning a discrete infinite-dimensional Hilbert space, the orbital angular momentum (OAM) of light can be used to realize such high-dimensional quantum systems. Moreover, the non-linear process of spontaneous parametric down conversion naturally provides biphoton states entangled in a large number of OAM modes. However, the defining feature of photons carrying OAM, namely their helical phase front, is strongly perturbed by disturbances along the light propagation path. In this thesis, we study how different perturbations influence OAM entanglement, and how we can counteract their detrimental effects. We first consider the entanglement losses induced by diffraction on obstructions or apertures. In this setting, we derive an analytical expression for the entanglement losses depending only on the overlap between the diffracted images of the OAM modes used to encode the entanglement. With the help of this expression, we investigate the role of the radial structure of the encoding modes in mitigating entanglement losses due to diffraction. In particular, we show that entanglement encoded in multi-ring Bessel-Gaussian modes is more resilient against diffraction than that encoded into single ring Laguerre-Gaussian modes. Moreover, using the uncertainty relation for angular position and angular momentum, we demonstrate that the entanglement of a biphoton state, with well defined OAM, past a "cake-slice" aperture is a universal function of the product of the angular uncertainty defined by the aperture and of the OAM of the encoding modes. The use of OAM entanglement in free-space quantum communication is severely limited by phase distortions introduced by random refractive index fluctuations due to atmospheric turbulence. In this dissertation, we discuss the efficiency of adaptive optics (AO) in mitigating turbulence-induced OAM entanglement losses, for a vast range of atmospheric conditions. In particular, we show that the stronger Bell correlations available in higher dimensions are nullified by their faster turbulence-induced decay. In contrast, AO corrections allow to restore non-locality, and thus the security of entanglement-based quantum communication, even for high-dimensional states under moderate turbulence. In many practical situations, high-dimensional states are encoded in a subspace of a large (eventually infinite-dimensional) Hilbert space. Moreover, even though the dynamical evolution of the system couples the encoding subspace to the complement of the Hilbert space, the evolved states are often projected (truncated) into a smaller subspace, because of experimental restrictions such as limited detector size or resolution. With the help of numerically generated random unitary matrices, we obtain an expression for truncation-induced entanglement losses as a function of the dimensions of the total, encoding and truncation Hilbert spaces. Finally, we present a first endeavour towards a fully quantum description of light propagation through atmospheric turbulence. In particular, we derive a quantized Hamiltonian that governs the behaviour of monochromatic paraxial photons in a random, non-homogeneous dielectric medium. Our Hamiltonian is expressed in terms of creation and annihilation operators of photons populating OAM modes, where the turbulence-induced mixing of several different modes is described by an appropriate coupling term.