Twisted photons

PROGRESS ARTICLE Twisted photons The orbital angular momentum of light represents a fundamentally new optical degree of freedom. Unlike linear momentum, or spin angular momentum, which is associated with the polarization of light, orbital angular momentum arises as a subtler and more complex consequence of the spatial distribution of the intensity and phase of an optical field — even down to the single photon limit. Consequently, researchers have only begun to appreciate its implications for our understanding of the many ways in which light and matter can interact, or its practical potential for quantum information applications. This article reviews some of the landmark advances in the study and use of the orbital angular momentum of photons, and in particular its potential for realizing high-dimensional quantum spaces. GABRIEL MOLINA-TERRIZA1,2, JUAN P. TORRES1,3* AND LLUIS TORNER1,3 1 ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain 2 ICREA- Institució Catalana de Recerca i Estudis Avançats, 08010, Barcelona, Spain 3 Department of Signal Theory and Communications, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain *e-mail: juan.perez@icfo.es he simplest class of light ield that can carry orbital angular momentum (OAM) is an optical vortex. An optical vortex is a beam of light whose phase varies in a corkscrew-like manner along the beam’s direction of propagation1,2. he OAM carried by such a ield enables it to trap and rotate colloid particles and even living cells, and to act as a so-called ‘optical spanner’ for use in ields as diverse as biophysics3–5, micromechanics6 or microluidics7. he OAM might also be used in superhigh-density optical data storage8, for imaging and metrology9–12, or in freespace communications13. he fact that individual photons can carry OAM presents the most exciting practical possibilities for using OAM in the quantum domain. Photons, of course, play a prominent role in many quantum information processing technologies14,15, but most make use of only a subset of a photon’s characteristics. Typically, these involve exploiting the quantum superposition of a pair of orthogonal states in a two-dimensional Hilbert space — such as those of light’s linear or circular polarization. he use of the OAM of light, however, opens the possibility of going beyond such twodimensional thinking. CLASSICAL AND QUANTUM OAM he quantum state of a photon can be described by a multipole expansion of electromagnetic waves with a well-deined value of the energy ħω, parity and the total angular momentum (polarization or spin, and OAM), given by the corresponding quantum eigenvalue l(l + 1)ħ and a well-deined projection of nature physics | VOL 3 | MAY 2007 | www.nature.com/naturephysics the angular momentum in a ixed direction (say z), given by the eigenvalue mħ (ref. 16). his decomposition is analogous to the more familiar decomposition of a light beam (classical or quantum mechanical) in terms of a series of plane waves. In general, the spin and orbital contributions cannot be considered separately17, but in the small-angle (paraxial approximation) limit, both contributions can be measured and manipulated independently. herefore paraxial quantum optics is the most convenient context in which to treat the OAM of light as a quantum resource. In this regard, the OAM of light is a useful description of the spatial degree of freedom of light, the continuous nature of which means it exists within an inherently ininite dimensional Hilbert space. Moreover, for a given application, the number of efective dimensions of the Hilbert space can be readily tailored as required18, as can that other ininite-dimensional degree of freedom of light, its frequency19. It is surprisingly simple to generate, control, ilter and detect the OAM states of light experimentally. Allen and co-workers20 showed that paraxial Laguerre–Gauss (LG) laser beams carry a well-deined orbital angular momentum associated with their spiral wave fronts. Such LG beams, illustrated in Fig. 1, are characterized by two integer indices, p and m. he index m, determines the dependence of the modes on the azimuthal phase, φ, which takes the form exp (imφ), and with each mode carrying an OAM of mħ per photon. Laguerre– Gauss modes form a complete Hilbert basis and can thus be used to represent the spatial quantum photon states within the paraxial regime. In this regime, the LG modes are eigenmodes of the quantum mechanical orbital angular momentum operator, Lz|m,p〉 = mħ|m,p〉. Photons represented by a single LG mode, |Ψ〉 = |m,p〉, are in a quantum state with a well-deined value of the orbital angular momentum (mħ). State vectors that are not represented by a pure LG mode, so that |Ψ〉 = ΣmpCmp|m,p〉, correspond to photons in a superposition state. Controlling OAM state superpositions opens the door to the generation and manipulation of multidimensional quantum states, with an arbitrarily large number of dimensions21. More speciically, the use of multidimensional states enables the exploration of deeper quantum features and might guide the elucidation of proof-of-principle capacity-increased quantum information processing schemes. 305 PROGRESS ARTICLE Coincidence detection Hologram Beam preparation & Monomodefibre Crystal mp = –1 mp = 0 mp = 1 1.0 0.8 0.6 0.4 0.2 0 0 m1 1 2 Figure 1 Properties of light with orbital angular momentum. a,b, The typical transverse intensity pattern of a light beam with orbital angular momentum. a is a theoretical plot, and b corresponds to the experimentally obtained image of a light beam with OAM produced with a computer-generated hologram. The light beam exhibits a dark spot in the centre, and a ring-like intensity profile. c, The phase of the beam twists around the central dark spot, producing a staircase-like phase wavefront. d, Such a spiralling phase means that the local momentum of the beam mimics the velocity pattern of a tornado or vortex fluid, a similarity that causes these singular spots to be named optical vortices. To visualize such a spiral phase, we use the interference of the light beam with OAM and with a vorticity-free plane wave propagating at a slightly different angle. e, The typical interference pattern obtained for m = 1, as revealed by the characteristic fork-like structure. Current technology ofers several diferent approaches for generating and controlling OAM states. Appropriately designed spiral phase plates can be used to produce the required phase distribution for generating or detecting a vortex beam. Computer-generated holograms are particularly important22,23 in the context of both classical and quantum optics. A suitable combination of astigmatic optical elements can also be used to generate light with OAM24. Properly designed quantum OAM superpositions can be generated by using light vortex pancakes made of a certain distribution of singlecharge screw dislocations nested into a gaussian host21. In this area, spatial light modulators are becoming an increasingly useful tool, as they enable complex spatial phase and amplitude light patterns to be generated and modiied in a prompt and eicient manner. One technique consists of generating light ields with arbitrary superpositions of OAM states through the coherent transfer of the mechanical OAM of atoms to the light ield25. In this scheme, the mechanical OAM of atoms is controlled with a spatially varying external magnetic ield that determines the quantum phase of the atomic spin. HIGH-DIMENSIONAL ENTANGLEMENT Of particular current interest is the generation of paired photons entangled in OAM. Entanglement is an inherently quantum mechanical phenomenon with no analogue in classical physics. Spontaneous parametric down-conversion (SPDC), the process by which two low-frequency photons (signal and idler) are generated from a single high-frequency photon that belongs to an intense pump laser, when it interacts with a nonlinear crystal, 306 2 0 1 –2 –1 0 1 2 –2 –1 2 0 1 0 1 2 –2 –1 2 0 1 m2 Figure 2 Observation of orbital angular momentum correlations with single photons. a, Illustration of the experimental configuration used to detect the quantum correlations in OAM of paired photons generated in an SPDC. b, Experimental data demonstrating that the OAM of the pump beam (mp) is transferred to the sum of OAM of the generated photons (m1 and m2). In this particular case, the state of the down-converted photons is a coherent quantum superposition of all the different possibilities for the OAM state of the photons fulfilling the condition mp = m1 + m2. Reprinted from ref. 27. is a reliable source for generating entangled pairs of photons. Such photon pairs not only can be polarization entangled, but can also exhibit OAM entanglement26. In a breakthrough 2001 experiment that irst measured OAM at the single photon level, Mair and co-workers demonstrated the existence of quantum OAM correlations between pairs of photons generated by SPDC27. In this experiment, the key results of which are shown in Fig. 2, a combination of computer-generated phase holograms, singlemode ibres, and single-photon-counting module detectors was used to detect speciic OAM quantum states. Appropriately designed phase holograms can perform nearly arbitrary transformations between diferent sets of OAM superposition states28–30. A single-mode ibre projects the incoming photon into the fundamental mode of the ibre, a nearly gaussian mode with m = p = 0. Ater this experiment, alternative schemes to detect the OAM of single photons have been proposed, like the use of concatenated interferometers where the introduction of OAMdependent phase-shits discriminate desired OAM modes31. Determining the OAM spectra of downconverted photons is crucial in this endeavour, as all quantum information applications are based on the availability and use of speciic quantum states. To this end, there are two diferent ways to go about describing the processes that generate these spectra. At its most fundamental, this involves recognizing that the conservation of angular momentum means that the contributions of not only the electromagnetic ield, but of the electronic spins and orbitals, and even the atomic lattice, of the nonlinear optical crystal in which SPDC takes place must be simultaneously considered32. his requires that the whole geometry of the down-conversion process be taken into account in order to include azimuthal variations of the nonlinear coeicient26 and the phase-matching conditions33. Conversely, however, all relevant experiments reported so far have involved collecting (and using) only a small angular section of the full downconversion light cone that is produced in this process. nature physics | VOL 3 | MAY 2007 | www.nature.com/naturephysics PROGRESS ARTICLE 14,000 Mode detectors m = –1 01 2 Measurement of orbital angular momentum eigenstates Source m = +1 m = –1 Preparation of the superposition m = –1 m = +1 01 2 Mode Detectors 16 10,000 12 8,000 8 6,000 4 4,000 0 2.70 2.75 2.80 2.85 2.90 2.95 © 2002 APS m = +1 Number of measurements m = +1 m = –1 20 12,000 2,000 0 2.0 2.22 0.4 2.62 Bell parameter 0.8 Figure 3 Bell experiments with OAM modes. a, Experimental set-up used to test the validity of a Bell inequality with paired photons generated in an SPDC and correlated in OAM. In this experiment, OAM quantum states belonging to a three-dimensional Hilbert space were used. b, Experimental results obtained by projecting each of the photons of the pair into superpositions of well-defined OAM states. For pairs of photons classically correlated, the Bell parameter should never exceed two. Several of the measurements provide a value of the Bell parameter larger than the classical limit. Reprinted with permission from ref. 47. In other words, the wave vectors of the signal and idler photons belong to a narrow bundle around the corresponding central wave vectors, making it unnecessary to consider the behaviour of the whole system in total. In this scenario, the selection rule18,34 mp = m1 + m2 (where mp is the optical vortex winding number of the pump beam, and m1 and m2 are the winding numbers of the modes into which the quantum states of the signal and idler photons are projected, respectively) only holds under restricted conditions for the photon emission angle and the strength of the Poynting vector walk-of35. Nevertheless, by using a wide-enough pump beam, the non-collinear and Poynting vector walk-of can be rendered negligible27,36, which makes the previous selection rule approximately valid. In contrast, settings that rely on highly focused pump beams clearly reveal the efects caused by non-collinear SPDC geometries37. Non-collinear SPDC introduces ellipticity into the spatial mode function of the downconverted photons38,39, which allows for the detection of photons with mp ≠ m1 + m2. his efect can be dramatically enhanced in highly non-collinear conigurations, especially in SPDC conigurations where the entangled photons counterpropagate40. Besides choosing the crystal and the pumping geometry, there are other ways to control the spiral bandwidth18, or, equivalently, the amount of entanglement of the OAM correlated photons. One strategy relies on the proper manipulation of the spatial proile of the classical beam that pumps the nonlinear crystal41. Complex spatial proiles with, for example, a superposition of diferent OAM modes, modify the corresponding quantum state of the generated photons. An alternative strategy is based on the proper preparation of the downconverting crystal, namely, on spatial quantum-state manipulation by transverse quasi-phase-matching engineering42. he manipulation of the weights of diferent OAM states can also be realized once the downconverted photons have been generated, by using suitable iltering processes43. he complete characterization and control of the amount of entanglement of a pure state is provided by the Schmidt decomposition technique44, which reveals how the photons are really paired, as well as how many modes are involved45. Under the correct symmetry conditions, the Schmidt modes can correspond directly to individual OAM modes, though these are not generally Laguerre– Gauss modes. he direct manipulation of the Schmidt modes, the true information eigenstates, as well as the detection of the amount nature physics | VOL 3 | MAY 2007 | www.nature.com/naturephysics of spatial entanglement46, is a signiicant experimental challenge that is yet to be solved. APPLICATIONS Recent years have seen several ground-breaking demonstrations of the ability to generate quantum states with an arbitrary number of dimensions (in particular, more than two) through the use of the OAM of photons. One example, illustrated in Fig. 3, is the demonstration of the violation of a two-photon, three-dimensional Bell inequality47. he degree of violation of a Bell inequality is a signature of the quantum correlations of distant physical systems. No classical isolated systems can violate a Bell inequality. Consequently, they lie at the heart of many quantum applications and are widely used to distinguish quantum correlations from those correlations produced by classical processes. he observations reported in ref. 47 demonstrated experimentally that photons were actually entangled through the OAM degree of freedom. Additional experiments in nearly collinear geometries have conirmed this result with tomographic measurements of the OAM content of the two-photon states generated by SPDC30,48. Entangling systems in higher-dimensional states is important for a variety of both fundamental and practical reasons. For example, noise can rapidly degrade the quantum correlations that exist between the components of a quantum system. Several theoretical studies suggest, however, that by increasing the dimensionality of the entangled states of a system, its non-classical correlations can be made more robust to the presence of noise and other deleterious environmental efects49,50. Moreover, higher-dimensional states could have other unique and potentially useful features, such as a stronger violation of Bell inequalities for certain states when compared with the correspondingly maximally entangled state51. Such predictions are awaiting experimental conirmation. he practical potential of higher-dimensional quantum systems was clearly illustrated in the observations reported in ref. 52. In this work, a quantum protocol, known as ‘quantum coin tossing’, was implemented, where two parties share codiied information (the result of a coin toss) that can be retrieved by a posteriori manipulations, but cannot be deciphered before a determined unveiling time, thus allowing the toss to be secret until the parties have bet on the result. 307 PROGRESS ARTICLE © 2005 APS 5 his kind of protocol is of most of interest for applications where two partners wish to undertake a secure transaction, but do not fully trust each other. In this particular protocol, the use of qutrits (quantum states in three dimensions) ofers an advantage over classical communication, and also over two-dimensional quantum systems (see Fig. 4). In the experimental demonstration of the protocol, the possibility of success of one party when trying to cheat was tested, clearly showing that the use of higher-dimensional Hilbert spaces severely restrict the possibilities of cheating in this protocol. Quantum coin tossing is one example of a class of protocols where the power of higher-dimensional spaces becomes apparent. Another recent application of the quantum OAM of light is the generation of hyper-entangled quantum states53. By making use of entanglement in several degrees of freedom, it is possible to generate quantum states in ultra-high-dimensional spaces. In their experiment, Barreiro and co-workers were able to generate an entangled two-photon state in a 22 × 22 × 32 dimensional system, using polarization, the timeenergy degree of freedom and OAM. Indeed, because the OAM Hilbert space and energy–time are both ininite-dimensional systems, it seems possible that even larger Hilbert spaces could be examined, encoding higher-dimensional qudits (quantum states in d dimensions). Closing the so called ‘detection loophole’ of Bell inequalities, such as those caused by losses of single-photon detectors54, is just one of the potential uses to which these hyper-entangled states might be put. Because of its higher dimensionality, the OAM of light can ofer higher information-density coding and a higher margin of security55,56. he use of OAM states can lead to schemes that generalize the Bennet–Brassard quantum key distribution protocol41, which makes use of N-dimensional systems and M mutually unbiased 308 3 2 1 θ 0 –1 –2 0 1 2 3 4 Angle light (rad.) 5 6 © 2006 APS Figure 4 Is cheating any harder in the quantum world? This picture depicts the experimental results obtained in two realizations of a quantum coin-tossing protocol performed with OAM states, where each time 256 (16 × 16) photons where transmitted. In this protocol, two communicating parties, Alice and Bob (of whom Alice is the sender and Bob is the receiver), have to agree whether a certain bit is 1 (heads) or 0 (tails). Within each image, each small square represents the detection of a photon with a certain OAM state. Black and white squares indicate that both parties agreed on which OAM state the photon was carrying, and thereby represents ‘success’ of the protocol. Red squares indicate events where the parties did not agree on the state of the photon, and represent a ‘failure’ of the protocol. In the absence of noise, disagreements can only come from at least one of the parties cheating on the protocol. a, Results of an experiment performed in which both partners are regarded as honest. b, Result of an experiment in which one of the parties (Alice) is cheating, which leads to a considerable increase of ‘failures’ in the protocol. In the absence of noise, the use of qutrits allow Bob to detect Alice cheating in more cases than qubits would permit. These protocols provide a measure of the reliability of communication partners. Reprinted with permission from ref. 52. Angle atoms (rad.) 4 Figure 5 The OAM of light can be used as a tool to manipulate and control quantum states of matter. a, Absorption image of a BEC cloud that has undergone a Raman transition driven by two counter-propagating LG (p = 0, m = 1) and gaussian (p = 0, m = 0) laser beams. b, Interference of two rotating states of the BEC with OAM ħ and −ħ, which has an average OAM of zero. c, Interference of rotating and non-rotating states, showing the correspondingly displaced hole. d, Theoretical simulation corresponding to b. e, Theoretical simulation corresponding to c. f, Experimental data demonstrating that the information encoded into the OAM of the light beams is coherently transferred to rotational states of the BEC. The graph shows the measured phase of the atomic interference between rotating and nonrotating atomic states, as given by the angle of the hole, as a function of the relative phase difference of the Raman beams that drive the atomic transitions in the BEC. Reprinted with permission from ref. 61. bases. he use of higher values of N and M provides better security than obtainable with qubits and two bases. Recently, a quantum cryptographic scheme for key distribution based on qutrits coded in OAM has been demonstrated57. ADDRESSING ATOMS One area of great interest is in the use of the OAM of light to control systems of cold atoms, with the ultimate goal of transferring information encoded in the spatial degrees of freedom of light to those within the quantum variables of an atomic ensemble. Several recent experiments have conirmed the potential of this approach. For example, light with OAM can create difraction gratings in clouds of cold atoms that mimic single OAM states58,59 as well as superpositions of several OAM states60. Another recent landmark was the demonstration of the use of OAM to generate arbitrary superposition of atomic rotational states in a Bose–Einstein condensate (BEC)61, as shown in Fig. 5. In this experiment, the coherent transfer of information encoded on the OAM of light beams into superpositions of rotational states of a BEC of sodium atoms was observed. he experiment illustrates the potential of OAM as an enabling tool to generate and to control multidimensional quantum states of matter, and therefore adds to the existing techniques for the full control of atoms. nature physics | VOL 3 | MAY 2007 | www.nature.com/naturephysics PROGRESS ARTICLE Not only is it possible to transfer OAM information from light to matter, but also to entangle this information62, which opens up further possibilities for creating a qualitatively new class of quantum light–matter entangled states. In particular, OAM quantum states can also be used to modify atomic properties. For example, the Doppler line broadening of an atomic transition could show a contribution due to the helicoidal wave front of the beam, in the form of a ‘rotational frequency shit’, as follows from the recently observed broadening of a Hanle electromagnetically induced transparency coherence resonance on rubidium atoms when using LG light beams63. Light with OAM can also be used to induce the spin-Hall efect in atoms, which leads to the generation of pure spin currents64, with no massive charge current. In this approach, neutral atoms with diferent spins experience opposite efective magnetic ields that are mediated by optical ields with OAM. Further exploration of this efect could open up new possibilities for controlling spin currents, which are at the heart of information devices based on spin states (spintronics). OUTLOOK Photon states encoded in OAM also provide important opportunities in near-ield phenomena and related techniques. his is illustrated, for example, by the recent experimental demonstration that the entanglement in OAM of a pair of photons can be coherently transmitted by surface plasmons that propagate in arrays of subwavelength holes65. he observations suggest that OAM states may be used as to probe the spatial properties of such nanohole structures in particular, and in quantum nanometrology in general. On the other hand, plasmon–polariton, as collective wave excitations, might be considered as quantum carriers of OAM encoded information in suitable quantum circuits. 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Acknowledgements This work has been partially supported by the Generalitat de Catalunya, by the European Commission under the integrated project Qubit Applications (QAP) funded by the IST directorate (Contract No. 015848), and by the Ministerio de Educacion y Ciencia/Government of Spain (Consolider Ingenio 2010 QIOT CSD2006-00019, FIS2004-03556, and TEC2005-07815). Competing financial interests The authors declare no competing financial interests. nature physics | VOL 3 | MAY 2007 | www.nature.com/naturephysics