**Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing**

Eleonora Nagali and Fabio Sciarrino *Dipartimento di Fisica, Sapienza Universitá di Roma Italy*

#### **1. Introduction**

74 Advanced Photonic Sciences

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More than a century ago, pioneering works carried out by Poynting (1909) and other physicists have provided the evidence of the validity of Marxwell equations for an electromagnetic field, showing how a beam of light carries energy and momentum, both in the linear and angular components. In particular the angular momentum of light, related to the generator of rotations in quantum mechanics, has been typically associated with its polarization, and more specifically with its circular polarization components. An optical beam traveling in the positive direction of the *z* axis that is circularly polarized, carries a *z*-component angular momentum content *σ* = ±*h*¯ per photon, which is positive if the circular polarization is left-handed and negative if it is right-handed. This angular momentum content is not just a formal property, but a very concrete one that can have significant mechanical effects as for example an observable induced rotation by absorption to a material particle Beth (1936); Friese et al. (1998)

At the same time, calculations of angular momentum for a free electromagnetic beam gave rise to a *second* contribution not related to photon spin, to which was attributed the name of *orbital angular momentum* (OAM). Unlike the spinorial angular momentum (SAM), considered as the intrinsic part of angular momentum since it does not depend on the specific reference frame, the orbital component is associated to the transverse spatial structure of the wavefront. More precisely, this angular momentum appears when the beam wavefront acquires a helicoidal structure, or equivalently, its field spatial dependence contains a helical phase factor having the form *eimϕ*, where *ϕ* is the azimuthal phase of the position vector *r* around the beam axis *z* and *m* is any integer, positive or negative, providing the direction and the "velocity" of the phase spiraling along the beam direction. It can be shown that in this case the optical beam carries an angular momentum along its axis *z* equal to *mh*¯ per photon, in addition to the polarization one *σ*. When *m* is nonzero, the helical phase factor imposes the existence of an optical vortex at the center of the beam where the light intensity vanishes. Although the fundamental concept of orbital angular momentum associated with a light field was already known since the early forties, the research on the orbital angular momentum of light has begun only in 1992, with the appearance of a seminal publication carried out by Allen et al. (1992). In this work, Allen et al. demonstrated experimentally that a particular set of solutions of Helmoltz equation in paraxial approximation, the Laguerre-Gauss modes (LG), carry a fixed amount of orbital angular momentum. Moreover, such beam could be

there may be a significant advantage in introducing the use of higher dimensional systems for encoding and manipulating the quantum information. Such *d*-level quantum systems, or *qudits*, provide a natural extension of qubits that has been shown to be suitable for prospective applications such as quantum cryptography and computation Cerf et al. (2002); Lanyon et al. (2009). In this framework the orbital angular momentum, defined in an infinitely dimensional Hilbert space, provides a natural choice for implementing *qudits* encoded in a single photon state (see Franke-Arnold et al. (2008); Molina-Terriza, Torres & Torner (2007)). This can be an important practical advantage, as it allows increasing the information content per photon, and this, in turn, may cut down substantially the noise and losses arising from the imperfect generation and detection efficiency, by reducing the total number of photons needed in a given process. Moreover, the combined use of different degrees of freedom of a photon, such as OAM and spin, enables the implementation of entirely new quantum tasks, as shown by

Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 77

Since the seminal paper of 1992 a large effort has been spent to develop optical tools able to manipulate and control efficaciously the orbital angular momentum degree of freedom. Up to now, by observing the transfer of OAM to matter, some devices have been adopted in order to generate/analyse LG modes (computer generated holograms, spatial light modulators), or manipulate the OAM analogously to what is commonly carried out through waveplates on polarization (cylindrical lenses mode converters). Despite these successes, the optical tools for controlling the OAM quantum states remain rather of limited use: a wider and more practical control of the OAM resource somehow analogous to that currently possible for the

Here we present a brief introduction on orbital angular momentum in quantum optics, describing the main devices adopted in order to achieve an efficient manipulation. Furthermore, we describe some experiments that have been carried out by adopting an optical device, the *q-plate*, able to couple the spinorial and orbital contributions of the angular

Light beams carry energy and momentum, the latter both in its linear and angular components. Thus, when an electromagnetic field interacts with matter, an exchange of energy and momentum occurs, manifested in interesting mechanical effects or in changing of the beam properties. Here we will focus on light angular momentum, composed by a spinorial and orbital component. The manipulation of spinorial angular momentum (SAM), commonly associated to the polarization of light, is largely diffused in all the different field of physics. On the other hand, the orbital angular momentum (OAM) of light did not go through the same diffusion for many years, since after seminal works of Poynting (1909) and Beth (1936), only in 1992 Allen et al. (1992) demonstrated that particular solutions of the Helmoltz equation in the paraxial regime exhibit an azimuthal phase structure typical of beams possessing OAM. Such beams, as for example the Laguerre-Gauss beams, can be experimentally manipulated and thus offer a valuable resource in quantum information, where the possibility to exploit the infinite-dimensionality of OAM opens interesting perspectives. Here we present theoretically the main elements that characterize, both in classical and quantum regimes, the orbital angular

Aolita & Walborn (2007); Barreiro et al. (2005; 2008)

polarization degree of freedom is by the way under progress.

momentum of light by exploiting the properties of liquid crystals.

**2. The orbital angular momentum of light**

generated experimentally in the laboratory by manipulating gaussian beams emerging from a laser cavity. Interestingly, even though it was largely diffused to indicate the "single-photon contribution" to the whole value of OAM carried by the beam, the first experimental test on the OAM as an individual property of single photons has been carried out only in 2001 by Mair et al. (2001)

For all the reasons listed above, the orbital angular momentum is considered a recently discovered photonic degree of freedom. In general, only the global angular momentum is associated to an observable quantity, however in the paraxial approximation, both SAM and OAM can be manipulated and measured separately. Indeed the OAM of light can be exchanged with matter, thus opening new perspectives in several fields of classical and quantum physics as well as in biology. In contrast to SAM, which couples only with the material local anisotropy (birefringence), OAM couples mainly with material inhomogeneities characterized by a rotational asymmetry around the beam axis. This coupling may be considered a negative feature when OAM is considered for communication purposes, as it makes it very sensitive to turbulence or other sources of noise Paterson (2005), but it becomes an useful property when OAM is adopted as a tool for sensitively probing the properties of a given medium as considered in several recent works (see Molina-Terriza, Rebane, Torres, Torner & Carrasco (2007); Torner et al. (2005)). The use of OAM for probing can lead to microscopic imaging with a spatial resolution that is higher than the Rayleigh limit Tamburini et al. (2006) and, when OAM fields are used in combination with suitable fluorescence methods (e.g., the stimulated emission depletion), they enable new methods of far-field microscopy with theoretically unlimited resolution Harke et al. (2006); Hell (2009). Optimal spatially structured light beams have also been considered as tools to cage/uncage specific molecules for accurate and rapid biological imaging Shoham et al. (2005). Some of these approaches may have relevant applications in the imaging of biological tissues, e.g. for diagnostic or research purposes. Finally the characteristic doughnut profile of the intensity pattern of a LG mode allows an efficient ion and atom trapping with low scattering and hence heating of the atom, useful for atom optics and BECs purposes Andersen et al. (2006).

Beyond all these applications, it has been recognized that the orbital angular momentum has a great potential for quantum photonics, in particular regarding quantum information protocols implemented through quantum optics techniques. Quantum information (QI) is based on the combination of classical information theory and quantum mechanics. In the last few decades, the development of this new field has opened far-reaching prospects both for fundamental physics, such as the capability of a full coherent control of quantum systems, as well as in technological applications, most significantly in the communication field. In particular, quantum optics has enabled the implementation of a variety of quantum information protocols. The fundamental unit of information in QI theory is a two-level system, the quantum bit or qubit. Exploiting the features of quantum states, however, it has been proven that qubits allows the transfer of more information than the one encoded in a classical boolean alphabet and, at the same time, the quantumness of qubit systems ensure high level of security in communication processing. In this context, the information encoding based on two-dimensional system can be experimentally implemented by exploiting degrees of freedom of single photons as, for example, the polarization. Up to now, several quantum information protocols have been successfully implemented, thanks to a notable control on the polarization degree of freedom achieved through different efficient devices. However 2 Will-be-set-by-IN-TECH

generated experimentally in the laboratory by manipulating gaussian beams emerging from a laser cavity. Interestingly, even though it was largely diffused to indicate the "single-photon contribution" to the whole value of OAM carried by the beam, the first experimental test on the OAM as an individual property of single photons has been carried out only in 2001 by

For all the reasons listed above, the orbital angular momentum is considered a recently discovered photonic degree of freedom. In general, only the global angular momentum is associated to an observable quantity, however in the paraxial approximation, both SAM and OAM can be manipulated and measured separately. Indeed the OAM of light can be exchanged with matter, thus opening new perspectives in several fields of classical and quantum physics as well as in biology. In contrast to SAM, which couples only with the material local anisotropy (birefringence), OAM couples mainly with material inhomogeneities characterized by a rotational asymmetry around the beam axis. This coupling may be considered a negative feature when OAM is considered for communication purposes, as it makes it very sensitive to turbulence or other sources of noise Paterson (2005), but it becomes an useful property when OAM is adopted as a tool for sensitively probing the properties of a given medium as considered in several recent works (see Molina-Terriza, Rebane, Torres, Torner & Carrasco (2007); Torner et al. (2005)). The use of OAM for probing can lead to microscopic imaging with a spatial resolution that is higher than the Rayleigh limit Tamburini et al. (2006) and, when OAM fields are used in combination with suitable fluorescence methods (e.g., the stimulated emission depletion), they enable new methods of far-field microscopy with theoretically unlimited resolution Harke et al. (2006); Hell (2009). Optimal spatially structured light beams have also been considered as tools to cage/uncage specific molecules for accurate and rapid biological imaging Shoham et al. (2005). Some of these approaches may have relevant applications in the imaging of biological tissues, e.g. for diagnostic or research purposes. Finally the characteristic doughnut profile of the intensity pattern of a LG mode allows an efficient ion and atom trapping with low scattering and hence

heating of the atom, useful for atom optics and BECs purposes Andersen et al. (2006).

Beyond all these applications, it has been recognized that the orbital angular momentum has a great potential for quantum photonics, in particular regarding quantum information protocols implemented through quantum optics techniques. Quantum information (QI) is based on the combination of classical information theory and quantum mechanics. In the last few decades, the development of this new field has opened far-reaching prospects both for fundamental physics, such as the capability of a full coherent control of quantum systems, as well as in technological applications, most significantly in the communication field. In particular, quantum optics has enabled the implementation of a variety of quantum information protocols. The fundamental unit of information in QI theory is a two-level system, the quantum bit or qubit. Exploiting the features of quantum states, however, it has been proven that qubits allows the transfer of more information than the one encoded in a classical boolean alphabet and, at the same time, the quantumness of qubit systems ensure high level of security in communication processing. In this context, the information encoding based on two-dimensional system can be experimentally implemented by exploiting degrees of freedom of single photons as, for example, the polarization. Up to now, several quantum information protocols have been successfully implemented, thanks to a notable control on the polarization degree of freedom achieved through different efficient devices. However

Mair et al. (2001)

there may be a significant advantage in introducing the use of higher dimensional systems for encoding and manipulating the quantum information. Such *d*-level quantum systems, or *qudits*, provide a natural extension of qubits that has been shown to be suitable for prospective applications such as quantum cryptography and computation Cerf et al. (2002); Lanyon et al. (2009). In this framework the orbital angular momentum, defined in an infinitely dimensional Hilbert space, provides a natural choice for implementing *qudits* encoded in a single photon state (see Franke-Arnold et al. (2008); Molina-Terriza, Torres & Torner (2007)). This can be an important practical advantage, as it allows increasing the information content per photon, and this, in turn, may cut down substantially the noise and losses arising from the imperfect generation and detection efficiency, by reducing the total number of photons needed in a given process. Moreover, the combined use of different degrees of freedom of a photon, such as OAM and spin, enables the implementation of entirely new quantum tasks, as shown by Aolita & Walborn (2007); Barreiro et al. (2005; 2008)

Since the seminal paper of 1992 a large effort has been spent to develop optical tools able to manipulate and control efficaciously the orbital angular momentum degree of freedom. Up to now, by observing the transfer of OAM to matter, some devices have been adopted in order to generate/analyse LG modes (computer generated holograms, spatial light modulators), or manipulate the OAM analogously to what is commonly carried out through waveplates on polarization (cylindrical lenses mode converters). Despite these successes, the optical tools for controlling the OAM quantum states remain rather of limited use: a wider and more practical control of the OAM resource somehow analogous to that currently possible for the polarization degree of freedom is by the way under progress.

Here we present a brief introduction on orbital angular momentum in quantum optics, describing the main devices adopted in order to achieve an efficient manipulation. Furthermore, we describe some experiments that have been carried out by adopting an optical device, the *q-plate*, able to couple the spinorial and orbital contributions of the angular momentum of light by exploiting the properties of liquid crystals.
