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Institute of Physics eBooks

Entanglement of the orbital angular momentum states of photons

Alois Mair, Alipasha Vaziri, Gregor Weihs, Anton Zeilinger

Published Nov 26, 2004• 86 citations
Publisher

Abstract

Entanglement contains one of the most interesting features of quantum mechanics, often named quantum non-locality [1,2].This means entangled states are not separable regardless of the spatial separation of their components.Measurement results on one particle of a two-particle entangled state define the state of the other particle instantaneously with neither particle enjoying its own well-defined state before the measurement.So far experimental confirmation of entanglement has been restricted to qubits, i.e. two-state quantum systems including recent realization of three-[3, 4] and four-qubit [5,6] entanglements.Yet, an ever increasing body of theoretical work calls for entanglement in quantum system of higher dimensions [7,8].For photons one is restricted to qubits as long as the entanglement is realized using the photons polarization.Here we report the first realization of entanglement exploiting the orbital angular momentum of photons,

DOI
10.1887/0750309016/b1142c50

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Introduction In this chapter, we will be concerned with entangled states of light. Historically, quantum states of light, i.e., photons, have played a dominant role in the experimental study of quantum entanglement. For instance, the first Bell tests and the first realizations of quantum teleportation were performed using two-dimensional polarization-entangled twophoton states [3]. From an experimental point of view these states are readily prepared and manipulated using conventional polarization optics. Recently, however, both theoretical and experimental efforts have shifted towards entanglement in more than two modes , which is often referred to as high-dimensional entanglement. This is much richer than two-dimensional entanglement, as is, for instance, reflected in stronger violations of various measures of nonlocality which reveal the complexity of high-dimensional Hilbert spaces [4–7]. Also, from an applications perspective, it promises a larger channel capacity [8, 9] and improved security for quantum communication [10, 11]. There are several ways to implement high-dimensional two-photon entanglement. One can exploit the temporal [12], frequency [13], or spatial [14–18] degrees of freedom of the optical field. A promising way to implement high-dimensional two-photon entanglement is to use the orbital-angular-momentum (OAM) degree of freedom of light. Light beams that carry OAM have a helical wavefront that spirals around the optical axis during propagation [19]. The winding number m , which characterizes the OAM eigenmodes, can adopt any integer value between −∞ and ∞; since all coaxial OAM modes are orthogonal they can thus span a very high-dimensional Hilbert space.

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The angular momentum of light is a useful resource for many applications. In specific physical architectures it can be considered as the sum of two independent terms, the spin and the orbital components, in analogy to particle systems. The spin angular momentum is related to the polarization of the optical beam, that is the direction of the oscillating electric field, whereas the orbital angular momentum is associated with the spatial distribution of the field. Being independent, spin and orbital angular momenta have been discovered and explored in separate contexts for many years, while only recently it has been considered the possibility to address both quantities on the same beam (or individual photons). The interaction between these two quantities gives rise to complex structures of the electromagnetic field, or to the so called classical entanglement in the domain of single photons. The research presented in this work aimed to show that combining spin and orbital angular momenta in light beams or single photons may be a useful tool for a variety of applications, with particular interest to the case of architectures characterized by spin-orbit interaction. This concept was made concrete through the design and the realization of several experiments, in the framework of singular optics, foundations of quantum mechanics, quantum information theory and quantum simulation.

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We investigate two-photon entangled states using two important degrees of freedom of the electromagnetic field, namely orbital angular momentum (OAM) and spin angular momentum. For photons propagating in the same direction we apply the idea of entanglement duality and develop schemes to do entanglement sorting based on either OAM or polarization. In each case the entanglement is tested using appropriate witnesses. We finally present generalizations of these ideas to three- and four-photon entangled states.

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Orbital angular momentum (OAM) is one of the most recently discovered properties of light, and it is only in the past decade its quantum properties have been the subject of experimental investigations and have found applications. Unlike polarization, which is only bidimensional, orbital angular momentum provides, with relative ease, unprecedented access to a theoretically unbounded discrete state space. The process of spontaneous parametric down-conversion has long been used as a source of two-photon states that can be entangled in several degrees of freedom, including OAM. In this thesis, the properties of the natural OAM spectrum associated with the entangled states produced by parametric down-conversion were investigated. Chapters 2 and 3 describe the production and detection of tunable high-dimensional OAM entanglement in a down-conversion system. By tuning the phase-matching conditions and improving the detection stage, a substantial increase in the half-width of the OAM correlation spectrum was observed. The conjugate variable of OAM, angular position, was also considered when examining high-dimensional states entangled in OAM. In order to efficiently determine their dimension, high-dimensional entangled states were probed by implementing a technique based on phase masks composed of multiple angular sectors, as opposed to narrow single-sector analysers. Presented in chapter 4, this technique allows the measurements of tight angular correlations while maintaining high optical throughput. The states so produced were then used for a number of applications centred around the concept of mutually unbiased bases. One can define sets of mutually unbiased bases for arbitrary subspaces of the OAM state space. Two bases are mutually unbiased if the measurement of a state in one basis provides no information about the state as described in the other basis. Complete measurements in mutually unbiased bases of high-dimensional OAM spaces are presented in chapter 5. Measurements in sets of mutually unbiased bases are integral to quantum science and can be used in a variety of protocols that fully exploit the large size of the OAM state space; we describe their use in efficient quantum state tomography, quantum key distribution and entanglement detection. Caution is however necessary when dealing with state spaces embedded in higher-dimensional spaces, such as that provided by OAM. Experimental tests of Bell-type inequalities allow us to rule out local hidden variable theories in the description of quantum correlations. Correlations inconsistent with the states observed, or even with quantum mechanics, known as super-quantum correlations, have however been recorded previously in experiments that fail to comply with the fair-sampling conditions. Chapter 6 describes an experiment that uses a particular choice of transverse spatial modes for which super-quantum correlations persist even if the detection is made perfectly efficient. The sets of modes carrying OAM allow a complete description of the transverse field. The ability to control and combine additional degrees of freedom provides the possibility for richer varieties of entanglement and can make quantum protocols more powerful and versatile. One such property of light, associated with transverse modes possessing radial nodes in the field distribution, can be accessed within the same type of experimental apparatus used for OAM. In chapter 7, the radial degree of freedom is explored, together with OAM, in the context of Hong-Ou-Mandel interference.

•Jan 1, 2014•0 citations
Entanglement contains one of the most interesting features of quantum mechanics, often named quantum non-locality [1,2].This means entangled states are not separable regardless of the spatial separation of their components.Measurement results on one particle of a two-particle entangled state define the state of the other particle instantaneously with neither particle enjoying its own well-defined state before the measurement.So far experimental confirmation of entanglement has been restricted to qubits, i.e. two-state quantum systems including recent realization of three-[3, 4] and four-qubit [5,6] entanglements.Yet, an ever increasing body of theoretical work calls for entanglement in quantum system of higher dimensions [7,8].For photons one is restricted to qubits as long as the entanglement is realized using the photons polarization.Here we report the first realization of entanglement exploiting the orbital angular momentum of photons,
Introduction In this chapter, we will be concerned with entangled states of light. Historically, quantum states of light, i.e., photons, have played a dominant role in the experimental study of quantum entanglement. For instance, the first Bell tests and the first realizations of quantum teleportation were performed using two-dimensional polarization-entangled twophoton states [3]. From an experimental point of view these states are readily prepared and manipulated using conventional polarization optics. Recently, however, both theoretical and experimental efforts have shifted towards entanglement in more than two modes , which is often referred to as high-dimensional entanglement. This is much richer than two-dimensional entanglement, as is, for instance, reflected in stronger violations of various measures of nonlocality which reveal the complexity of high-dimensional Hilbert spaces [4–7]. Also, from an applications perspective, it promises a larger channel capacity [8, 9] and improved security for quantum communication [10, 11]. There are several ways to implement high-dimensional two-photon entanglement. One can exploit the temporal [12], frequency [13], or spatial [14–18] degrees of freedom of the optical field. A promising way to implement high-dimensional two-photon entanglement is to use the orbital-angular-momentum (OAM) degree of freedom of light. Light beams that carry OAM have a helical wavefront that spirals around the optical axis during propagation [19]. The winding number m , which characterizes the OAM eigenmodes, can adopt any integer value between −∞ and ∞; since all coaxial OAM modes are orthogonal they can thus span a very high-dimensional Hilbert space.
Orbital angular momentum (OAM) of single photons represents a new optical degree of freedom for the entanglement of photons [1,2]. One physical realization of OAM carrying light beams are the so called Laguerre-Gaussian modes of light with the required helical phase structure. One big advantage over the well known polarization degree of freedom is the possibility of realizing entanglement between two photons with very high quantum numbers. However, the creation of photonic OAM entanglement using the widely used spontaneous parametric downconversion process is limited by the strongly reduced efficiency for higher momenta [3]. We have realized a novel method to create entanglement between two photons with a very high difference in their OAM quantum number and momentum respectively. Furthermore we have explored hybrid entanglement of photons in the OAM and polarization degree of freedom.
The angular momentum of light is a useful resource for many applications. In specific physical architectures it can be considered as the sum of two independent terms, the spin and the orbital components, in analogy to particle systems. The spin angular momentum is related to the polarization of the optical beam, that is the direction of the oscillating electric field, whereas the orbital angular momentum is associated with the spatial distribution of the field. Being independent, spin and orbital angular momenta have been discovered and explored in separate contexts for many years, while only recently it has been considered the possibility to address both quantities on the same beam (or individual photons). The interaction between these two quantities gives rise to complex structures of the electromagnetic field, or to the so called classical entanglement in the domain of single photons. The research presented in this work aimed to show that combining spin and orbital angular momenta in light beams or single photons may be a useful tool for a variety of applications, with particular interest to the case of architectures characterized by spin-orbit interaction. This concept was made concrete through the design and the realization of several experiments, in the framework of singular optics, foundations of quantum mechanics, quantum information theory and quantum simulation.
We investigate two-photon entangled states using two important degrees of freedom of the electromagnetic field, namely orbital angular momentum (OAM) and spin angular momentum. For photons propagating in the same direction we apply the idea of entanglement duality and develop schemes to do entanglement sorting based on either OAM or polarization. In each case the entanglement is tested using appropriate witnesses. We finally present generalizations of these ideas to three- and four-photon entangled states.
Orbital angular momentum (OAM) is one of the most recently discovered properties of light, and it is only in the past decade its quantum properties have been the subject of experimental investigations and have found applications. Unlike polarization, which is only bidimensional, orbital angular momentum provides, with relative ease, unprecedented access to a theoretically unbounded discrete state space. The process of spontaneous parametric down-conversion has long been used as a source of two-photon states that can be entangled in several degrees of freedom, including OAM. In this thesis, the properties of the natural OAM spectrum associated with the entangled states produced by parametric down-conversion were investigated. Chapters 2 and 3 describe the production and detection of tunable high-dimensional OAM entanglement in a down-conversion system. By tuning the phase-matching conditions and improving the detection stage, a substantial increase in the half-width of the OAM correlation spectrum was observed. The conjugate variable of OAM, angular position, was also considered when examining high-dimensional states entangled in OAM. In order to efficiently determine their dimension, high-dimensional entangled states were probed by implementing a technique based on phase masks composed of multiple angular sectors, as opposed to narrow single-sector analysers. Presented in chapter 4, this technique allows the measurements of tight angular correlations while maintaining high optical throughput. The states so produced were then used for a number of applications centred around the concept of mutually unbiased bases. One can define sets of mutually unbiased bases for arbitrary subspaces of the OAM state space. Two bases are mutually unbiased if the measurement of a state in one basis provides no information about the state as described in the other basis. Complete measurements in mutually unbiased bases of high-dimensional OAM spaces are presented in chapter 5. Measurements in sets of mutually unbiased bases are integral to quantum science and can be used in a variety of protocols that fully exploit the large size of the OAM state space; we describe their use in efficient quantum state tomography, quantum key distribution and entanglement detection. Caution is however necessary when dealing with state spaces embedded in higher-dimensional spaces, such as that provided by OAM. Experimental tests of Bell-type inequalities allow us to rule out local hidden variable theories in the description of quantum correlations. Correlations inconsistent with the states observed, or even with quantum mechanics, known as super-quantum correlations, have however been recorded previously in experiments that fail to comply with the fair-sampling conditions. Chapter 6 describes an experiment that uses a particular choice of transverse spatial modes for which super-quantum correlations persist even if the detection is made perfectly efficient. The sets of modes carrying OAM allow a complete description of the transverse field. The ability to control and combine additional degrees of freedom provides the possibility for richer varieties of entanglement and can make quantum protocols more powerful and versatile. One such property of light, associated with transverse modes possessing radial nodes in the field distribution, can be accessed within the same type of experimental apparatus used for OAM. In chapter 7, the radial degree of freedom is explored, together with OAM, in the context of Hong-Ou-Mandel interference.