Abstract

Spinor Bose–Einstein condensates (BECs) and singular optical systems have both recently served as sandboxes to create and study analogs of phenomena from other fields of physics that are otherwise difficult to create and control experimentally. Interfacing singular optics and spinor BECs allows us to take advantage of and build on the foundations of singular optics to create and describe complex spin textures in BECs that serve as analogs of other systems. Here, the complete BEC wavefunctions are precisely engineered via a two-photon Raman interaction to contain π-symmetric (lemon, star) or 2π-symmetric (saddle, spiral) C-point singularities. The optical Raman beams are singular optical beams that contain these singularities and transfer them to the condensate, thereby creating vector-vortex spin textures—the spinor counterparts to scalar vortices—in pseudo-spin-1/2 BECs. With a version of atom-optic polarimetry, we can measure the Stokes parameters of the atomic cloud and characterize the singularities by the patterns present in their ellipse fields or by the C-point index. In the low density limit, these spin textures are analogs of optical vector vortices and should have dynamics driven by a matter-wave Gouy phase. With precise tuning of Raman beam parameters, we can create full Bloch BECs that contain every possible superposition between two states in the atomic cloud. Full Bloch BECs are similar to topologically stable magnetic skyrmions such as those created in thin metal films and nanowires, which may prove useful for atom-spintronics and topological quantum processes.

© 2016 Optical Society of America

1. INTRODUCTION

Singularities are ubiquitous in science, and studying them helps to make connections between branches of physics and enables analogies between very different physical systems. Singular optics focuses on the topology of optical fields that contain singularities in their phase, polarization, or both [14]. Therefore, tabletop experiments in singular optics [5] can be used to study optical fields with properties analogous to those of black holes [6] and chaos [7,8]. However, because photons do not interact without a mediating nonlinear medium, emulating the evolution of systems from other fields of physics with singular optics is limited.

Another vehicle for creating analogs from other branches of physics is the spinor Bose–Einstein condensate (BEC). External magnetic, rf, and optical fields have been used to sculpt the wavefunction of spinor BECs to create coreless vortices [911], skyrmions [1214], monopoles [15,16], and even synthetic fields [17,18] and spin-orbit coupling [19]. Atoms can have higher dimensional spin manifolds than light, adding a richness to the classes of textures and analogs that can be created and studied. For example, 1/3 quantized vortices in spin-2 BECs exhibit non-Abelian braiding statistics [2022], which are of interest as topological quantum computing protocols [23,24]. Some of the more complex topological structures in BECs are expected to decay into half-quantum vortices [13,25] containing π-symmetric singularities [26] and are in some ways analogous to the vector vortices of singular optics. Much of the language used to classify and explain singularities in optics has been adopted from studies of singularities in condensed matter systems and liquid crystals. Extending this language to spinor BECs establishes a common ground for characterizing and discussing the behavior of these singularities.

We link the mathematical analysis of singular optics and the physics of spinor BECs to develop descriptions of ultracold atomic systems that exploit intuitive visualizations from singular optics. We precisely engineer and characterize singularities in pseudo-spin-1/2 BECs analogous to optical C-point singularities in the polarization field of an optical beam [2729]. C-points are umbilic points where the orientation of the polarization ellipse is undefined—that is, points of circular polarization. The state of polarization of an optical field can be represented on the Poincaré sphere, with antipodal points representing orthogonal polarizations. In atomic physics, the Bloch sphere [Fig. 1(a)] replaces the Poincaré sphere to geometrically represent the state of a spin-1/2 system [3032]. Right- and left-hand circular polarization states are spin eigenstates of light with spin angular momentum ± per photon [33,34]; therefore, magnetic spin (Zeeman) states are their atomic analog. When multiple spin states are populated in a BEC, its wavefunction becomes a vector—or spinor—with its elements representing the spin state amplitudes. The electric field of polarized light is described by a Jones vector, which is also a spinor in the right-/left-hand circular basis. Therefore in this spin angular momentum setting, manipulating the spin state amplitudes in an atomic system is analogous to changing the polarization of an optical beam, and atomic spin textures are analogous to spatially varying polarization patterns in optical vector-vortex beams [29].

 

Fig. 1. (a) Bloch sphere. (b) Three-level lambda system.

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We use a coherent, optical, two-photon Raman interaction to engineer spatially dependent wavefunction amplitudes and phases and therefore manipulate the superfluid phases, superfluid velocities, spin orientations, and relative phases between spin components. Unlike other spin texture generation techniques that rely on magnetic imprinting or spontaneous decay of more complicated structures [10,15,26,35], the variety of excitations and nonequilibrium spin textures is limited only by the diversity of complex optical beams that can be created [3638]. Due to the ability to control the relative phases of the spin components, this technique also serves as a method of characterizing complex spin textures, such as half-quantum vortices in spin-1 systems [26].

2. THEORY

In this work, the two-photon Raman transition couples the atomic spin states |ψ and |ψ. Consider the three-level lambda system in Fig. 1(b) with two incident square pulses with Rabi frequencies ΩA and ΩB. For a large single-photon detuning Δ, the excited state can be adiabatically eliminated [39,40] reducing the lambda system to a pseudo-spin-1/2 system. Neglecting density-dependent interactions, the time evolution of the spin state amplitudes can be described [41,42] in the rotating wave approximation via

ψ⃗t=i4Δ[|ΩA|2ΩA*ΩBΩAΩB*|ΩB|2]ψ⃗.

Because the pulses are square, the Rabi frequencies are constant over the interaction time and the equation can be integrated directly to give ψ⃗(t)=exp(iΩt/2)M(t)ψ⃗(0) with

M(t)=cosΩt2I+isinΩt2P(2α,ϕAB),
where Ω=(|ΩA|2+|ΩB|2)/4Δ is the generalized Rabi frequency, I is the 2×2 identity matrix, and tanα=|ΩA|/|ΩB|. In general, the Rabi frequencies are complex, Ωj=|Ωj|eiϕj, and their relative phase ϕAB=ϕAϕB is the relative phase of the electric fields of the Raman lasers. The matrix
P(2α,ϕAB)=[cos2αsin2αeiϕABsin2αeiϕABcos2α]
is a generalized version of the pseudo-rotation matrix for polarization [43]. For plane waves with equal Rabi frequencies, α=π/4, and Eq. (2) reduces to the case of the Raman waveplate [44]. However, for beams with nonuniform intensity and phase profiles, Ω, α, ϕAB, and hence M become spatially dependent.

With the quantization defined along the z direction with a magnetic field, the spinor wavefunction is naturally described in the eigenbasis of the spin operator in the z direction, S^z. However, writing the BEC wavefunction in the eigenbasis of the S^y operator gives us a Cartesian (horizontal/vertical) basis allowing us to draw maps of the local spin ellipses, which provide an avenue to classify not only wavefunction singularities but also the overall spin texture of the atomic cloud. For plane wave Raman beams, the wavefunction of the BEC with number density n after the Raman interaction is given in the spin basis by

ψ⃗(t)=n(ψψ)neiΦ2(|ψ|eiϕ2|ψ|eiϕ2).

Here, the condensate phase is

Φ=arctan(tanΩt2cos2α)+π2ϕAB+Ωt2,
and ϕ=ΦΩt/2. The angle ϕ/2 represents the angle of the major axis of spin ellipses as well as the orientation of the spinor. C-points are locations where the spin ellipse is a circle; that is, they are umbilic points where the orientation of the spin ellipse is undefined.

Consider two ways of classifying C-points: spin lines and the C-point index, IC [27,28,45]. Spin lines represent the major axis orientation of the local spin ellipse and thereby provide a fingerprint of the singularity [29]. The number of radial spin lines that terminate at the singularity serves as a method of classification. For π-symmetric singularities the number of radial lines is either 1 (lemon) or 3 (star), and for the 2π-symmetric case, the number is either 4 (saddle) or infinite (spiral).

The C-point (disinclination) index, which is related to the Hopf–Poincaré index for vector fields [45], is given by [28,45]

IC=14πϕ(r)·dr,
where the contour around the C-point is taken in the right-handed (counterclockwise) direction. For π-symmetric and 2π-symmetric singularities, the index takes on values of IC=±1/2 (lemon, star) and IC=±1 (spiral, saddle), respectively. The sign of the index depends on whether the spin lines rotate in the same (IC>0) or the opposite (IC<0) direction to that of the contour.

3. EXPERIMENTAL TECHNIQUE

Technical details of our Raman technique can be found in [46]. Briefly, a cigar-shaped Rb87 condensate of 6.5×106 atoms with aspect ratio 9 is created in the state |ψ|F=2,mf=2 in a magnetic trap. After the trap turns off, the BEC expands for 9 ms until it is approximately a 50 μm diameter sphere and density-dependent interactions are negligible. Two simultaneous 5 μs square pulses of copropagating orthogonally polarized Gaussian (ΩB) and Laguerre–Gaussian (ΩA) beams detuned Δ=440MHz from |e|F=1,mf=1 couple |ψ and |ψ|F=2,mf=0. The two beams propagate along the quantization (z) axis, which is also the axis of cylindrical symmetry. A portion of the initial state |ψ is transferred to |ψ and acquires orbital angular momentum (OAM) of with corresponding to the winding number, or topological charge, of the Laguerre–Gaussian beam. The resulting spin texture is a coreless vortex with a nonrotating core in |ψ surrounded by a quantized vortex in |ψ.

We can add to the spin texture control by applying a 150 μs rf pulse before the Raman interaction to coherently transfer the BEC to the |ψ state. This means that, during the Raman process, some of the population is transferred back to |ψ and acquires OAM of . The result is a coreless vortex with circulation in the |ψ state surrounding a nonrotating |ψ core. Therefore, by changing the handedness of the Laguerre–Gaussian beam and using coherent rf transfer, we can create a vortex of either handedness in either spin state.

The Raman process creates 3D spin textures in the atomic cloud. Just as the transverse intensity, phase, and polarization properties of the optical Raman beams determine the resulting transverse spin state of the BEC, the longitudinal properties of the beams affect how the resulting BEC spin state changes between transverse planes at different z values. In our case, the Raman beams are collimated such that the optical Gouy phase is negligible over the length of the cloud and the atomic spin textures created are approximately the same at each transverse plane, making our spin textures effectively 2D. We image the atoms with a resonant beam propagating along the z axis, producing a signal that depends on the column density at each transverse location. In this way, we effectively sum the contributions of the atomic density in each transverse plane to produce a single 2D image.

Using atomic interference, we create 2D maps of the Stokes parameters and reconstruct the spinor wavefunction. The Stern–Gerlach effect and a two-photon Raman interaction provide a polarizing beamsplitter and waveplate for the BEC, respectively, which enable us to perform the equivalent polarimetry on the atoms [44] (see Fig. 2). The images are taken 26 ms after the magnetic trap turns off when the atomic cloud expands to 500μm in diameter and allows the atomic spin states to sufficiently separate due to the Stern–Gerlach magnetic field gradient. This technique gives us access to not only the individual spin state populations but also their relative phase. Since the Raman beams are composed of Gaussian and Laguerre–Gaussian optical modes, the relative phase ϕAB is a function of the azimuthal coordinate φ in the cross section of the beams. Therefore singularities in the phase of the Laguerre–Gaussian beam become singularities in the relative phase of the atomic spin states.

 

Fig. 2. Maps of the Stokes parameters. (a) Experiment and (b) theory maps of the Stokes parameters of a coreless vortex where =2 in the spin state containing the vortex. Maps for each of the Stokes parameters Si are experimentally obtained using an atom-optic polarimetry technique [44] on essentially identical coreless vortices. S1 and S2 can be used to reconstruct the relative phase of the spin states. Examples of Stokes maps for =1 coreless vortices appear in [44]. With this information, we create spin ellipse maps to classify the singularities. These Stokes parameters correspond to a spiral singularity with C-point index IC=1. The Stokes parameter S0 would correspond to the total density of the cloud; in this case, the vortex state (|ψ) dominates at the edge of the cloud, so the shape of the spin density (S3) gives a good indication of the overall atomic density.

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4. RESULTS AND DISCUSSION

The Raman beams are coherent superpositions of orthogonally polarized Gaussian and Laguerre–Gaussian beams; that is, they possess π-symmetric singularities [2729,45,47,48] and imprint these singularities to the BEC through the Raman process. The understanding of these results can be further reinforced by noting that the matrix in Eq. (2) can be interpreted as the Jones matrix of either the stress-engineered optic of [49,50] or a q-plate that converts spin angular momentum into OAM [51]—both of which have been used to create spatially varying polarization patterns in optical beams.

Figure 3 shows maps of the spin ellipses and spin lines to help classify the C-point singularities. For ||=1, we create a π-symmetric coreless vortex. In a spin-1 system, such a vortex is a half-quantum or Alice state [25,26,5254]. By changing the handedness of the vortex, we can generate star [Fig. 3(a)] or lemon [Fig. 3(b)] C-points as can be seen from the fingerprint created by the spin lines as well as the number of radial spin lines that terminate at the singularity [2729,47,55]. Similarly, for ||=2, the quantum field is 2π-symmetric exhibiting spiral [Fig. 3(c)] and saddle [Fig. 3(d)] C-points [29,45,48], and changing the handedness of the Laguerre–Gaussian beam changes the resulting singularity between these two.

 

Fig. 3. Maps of the local spin ellipses. The orientation of the major axis of the ellipses, ϕ/2, is found from maps of the Stokes parameters S1 and S2. The ellipticity is determined by S3, and the ellipses are red and blue for right- and left-handed precession of the transverse components of the spin vector, respectively. When the population of the spin states is equal, the spin ellipse becomes a line (drawn in green). Spin lines show how the orientation of the major axes of the spin ellipses changes, revealing a fingerprint of the (a) star, (b) lemon, (c) spiral, and (d) saddle singularities.

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Although the overall shape of the spin lines clearly identifies the singularity, it can be difficult to draw the specific radial spin lines that terminate at the singularity with a limited number of data points. It is therefore convenient to classify these singularities according to their C-point index. In Fig. 4, the condensate phase and spin lines are shown along with the local spinor, which is represented as an arrow on a contour encircling the singularity. The local spinor rotates by π around the C-points in Figs. 4(a) and 4(b) and by 2π in Figs. 4(c) and 4(d), meaning the first two are π-symmetric and have |IC|=1/2 and the second two are 2π-symmetric with |IC|=1. The sign of IC is determined by looking at the direction that the spinor arrow rotates as we travel along the contour in a right-handed (counterclockwise) direction. Both Figs. 4(a) and 4(d) rotate in a left-handed (clockwise) sense and therefore have negative C-point indices of 1/2 (star) and 1 (saddle), respectively. The C-points in Figs. 4(b) and 4(c) have positive indices of 1/2 (lemon) and 1 (spiral), respectively.

 

Fig. 4. Characterizing singularities by the C-point index. Spin lines representing the local orientation of the major axis of the spin ellipse are plotted over the condensate phase. The singularities here are the same as those in Fig. 3 with a narrowed plot range focused on the singularities. The local spinors in the xy plane are parallel to the major axes of the spin ellipses, and rotate by π and 2π around the singularity for π- and 2π-symmetric singularities corresponding to an |IC| of 1/2 [(a), (b)] and 1 [(c), (d)], respectively. The sign of IC is determined by the direction of rotation of the spinor as the contour is followed in a counterclockwise (positive) direction. Both (a) and (d) have negative indices because the spinors rotate clockwise, while in (b) and (c) the spinors rotate counterclockwise corresponding to positive indices. In (c) and (d) the extra singularities near the exterior are artifacts from the edge of the atomic cloud and imperfect separation of the spin components during the Stern–Gerlach operation and therefore do not correspond to actual singularities in the BEC.

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Using coherent rf transfer to change the initial atomic spin state before the Raman process changes not only the spin state in which the vortex appears, but also the handedness of the vortex. As a result the type of singularity does not change when the initial spin state for the Raman process changes; the only change is the overall handedness of the spin ellipses in the spin texture. Therefore, the type of singularity created does not depend on the initial atomic state.

In Figs. 4(c) and 4(d), the experiment appears to show that a doubly quantized atomic vortex breaks into two singly charged vortices due to the spatial separation of the phase singularities much like higher-order optical singularities break up in the presence of a nonlinear medium. In our case, the evolution of the BEC is effectively linear because the expansion of the atomic cloud has greatly reduced the density and thus the density-dependent interactions are negligible. The singularity appears to split due to the imaging process. When the Raman beams are not perfectly aligned with the imaging axis, the singularity changes position in each transverse plane. The imaging process integrates along the z axis and washes out the singularity in the center of the cloud. The misalignment of the Raman beams is apparent in the elliptical shape of S3 in Fig. 2.

For the Gaussian and Laguerre–Gaussian Raman beam configuration described here, the Rabi frequencies and relative phases are spatially dependent. The ellipticity of the spin ellipses changes spatially with the variation in the spin state populations. The phase, besides varying azimuthally due to the phase winding of the Laguerre–Gaussian beam, also varies radially due to the AC Stark shift from the beams’ spatial intensity dependence [46]. This can be seen directly in the plots of the condensate phase in Fig. 4 or in the radial change of the orientation of the major axes of the spin ellipses in Fig. 3.

The optical versions of these singularities have been studied in full Poincaré beams, which have been created both using the stress birefringence of a stress-engineered optical element [50] and by coherently superimposing orthogonally polarized beams with different amounts of OAM [29,56]. Full Poincaré beams have spatially dependent polarization such that every point on the Poincaré sphere is represented in the cross section of the beam. With careful control of the Raman beam parameters [57], our spin textures can fully cover the Bloch sphere and therefore contain all possible superpositions and relative phases of the two states in a single atomic cloud [58].

5. CONCLUSION

In conclusion, we have created π-symmetric (lemon, star) and 2π-symmetric (spiral, saddle) singularities in a pseudo-spin-1/2 BEC. The singularities were transferred from an optical system to the atomic system by a coherent Raman process. Using atom-optic polarimetry, we create 2D maps of the atomic Stokes parameters and reconstruct the spinor wavefunction. In a Cartesian basis, we can draw maps of the spin ellipses and classify the topological singularities in the context of singular optics morphology [59].

Atomic analogs of states on a higher-order Poincaré sphere [60] can be created by engineering spinor wavefunctions that contain OAM in both spin states. Because atoms have higher dimensional spin manifolds and have interactions at higher densities, there is potential to extend these studies into realms where optical analogs do not exist. The Raman imprinting process we describe can be used to create these spin singularities in dense trapped condensates. Because the optical pulses can be <5μs in duration, the spin texture imprinting is faster than the spinor dynamics and our mathematical description holds for short pulses even in the presence of density-dependent interactions. Studying the evolution of 2D maps of the Stokes parameters of full Bloch BECs in trap will allow for observations of singularity dynamics [61], including those that are driven by spin ordering and other spinor dynamics [26]. In the diffuse regime, we expect that the Stokes map evolution will be driven by the matter-wave Gouy phase whose optical counterpart is responsible for the evolution of the Stokes parameters in optical vector-vortex beams [62]. This technique can therefore permit direct measurements of the Gouy phase in matter waves [63,64] and other geometric matter-wave phases such as the Pancharatnam–Berry phase [65,66].

Creating BECs that cover the Bloch sphere has the potential to simplify fidelity tests of quantum protocols as all possible state combinations can be tested simultaneously. The spin textures discussed here have similar potential utility in quantum computation schemes to vector vortices in optical beams [6770]. The spinor wavefunctions of these BECs are similar to those of magnetic skyrmions, which are being used in spintronics circuits because they are topologically stable particle-like objects [71,72]. Skyrmion analogs in either single spinor BECs or arrays of skyrmions in optical lattices can be created and manipulated optically through the Raman process and therefore may prove useful as qubits in quantum computation with nontrivial topological spin states.

Funding

National Science Foundation (NSF) (PHY-1313539); National Aeronautics and Space Administration (NASA) (JPL-1504801); Defense Advanced Research Projects Agency (DARPA).

Acknowledgment

We thank M. A. Alonso, M. Bhattacharya, J. Ruostekoski, M. O. Borgh, and J. D. Murphree for useful and stimulating discussions. A. Hansen and J. T. Schultz contributed equally to this project.

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51. F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51, C1–C6 (2012). [CrossRef]  

52. T. Ohmi and K. Machida, “Bose-Einstein condensation with internal degrees of freedom in alkali atom gases,” J. Phys. Soc. Jpn. 67, 1822–1825 (1998). [CrossRef]  

53. Y. Kawaguchi and M. Ueda, “Symmetry classification of spinor Bose-Einstein condensates,” Phys. Rev. A 84, 053616 (2011). [CrossRef]  

54. M. O. Borgh and J. Ruostekoski, “Topological interface engineering and defect crossing in ultracold atomic gases,” Phys. Rev. Lett. 109, 015302 (2012). [CrossRef]  

55. J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983). [CrossRef]  

56. E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light,” Appl. Opt. 51, 2925–2934 (2012). [CrossRef]  

57. J. T. Schultz, A. Hansen, J. D. Murphree, M. Jayaseelan, and N. P. Bigelow, “Raman fingerprints on the Bloch sphere of a spinor Bose-Einstein condensate,” J. Mod. Opt. 63, 1–9 (2016). [CrossRef]  

58. A. Hansen, J. T. Schultz, and N. P. Bigelow, “Full Bloch Bose-Einstein condensates,” in Frontiers in Optics 2012/Laser Science XXVIII, OSA Technical Digest (online) (Optical Society of America, 2012), paper LTu1I.2.

59. M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008). [CrossRef]  

60. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011). [CrossRef]  

61. D. Lopez-Mago, B. Perez-Garcia, A. Yepiz, R. I. Hernandez-Aranda, and J. C. Gutiérrez-Vega, “Dynamics of polarization singularities in composite optical vortices,” J. Opt. 15, 044028 (2013). [CrossRef]  

62. G. M. Philip, V. Kumar, G. Milione, and N. K. Viswanathan, “Manifestation of the Gouy phase in vector-vortex beams,” Opt. Lett. 37, 2667–2669 (2012). [CrossRef]  

63. I. G. da Paz, P. L. Saldanha, M. C. Nemes, and J. G. Peixoto de Faria, “Experimental proposal for measuring the Gouy phase of matter waves,” New J. Phys. 13, 125005 (2011). [CrossRef]  

64. A. Hansen, J. T. Schultz, and N. P. Bigelow, “Measuring the Gouy phase of matter waves using full Bloch Bose–Einstein condensates,” in The Rochester Conferences on Coherence and Quantum Optics and the Quantum Information and Measurement Meeting, OSA Technical Digest (Optical Society of America, 2013), paper M6.64.

65. G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012). [CrossRef]  

66. V. Kumar and N. K. Viswanathan, “The Pancharatnam-Berry phase in polarization singular beams,” J. Opt. 15, 044026 (2013). [CrossRef]  

67. K. T. Kapale and J. P. Dowling, “Vortex phase qubit: generating arbitrary, counterrotating, coherent superpositions in Bose–Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95, 173601 (2005). [CrossRef]  

68. C. Vo, S. Riedl, S. Baur, G. Rempe, and S. Dürr, “Coherent logic gate for light pulses based on storage in a Bose–Einstein condensate,” Phys. Rev. Lett. 109, 263602 (2012). [CrossRef]  

69. V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015). [CrossRef]  

70. R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Quantum entanglement of complex photon polarization patterns in vector beams,” Phys. Rev. A 89, 060301(R) (2014). [CrossRef]  

71. N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, “Writing and deleting single magnetic skyrmions,” Science 341, 636–639 (2013). [CrossRef]  

72. X. Zhang, M. Ezawa, and Y. Zhou, “Magnetic skyrmion logic gates: conversion, duplication and merging of skyrmions,” Sci. Rep. 5, 9400 (2015). [CrossRef]  

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    [Crossref]
  70. R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Quantum entanglement of complex photon polarization patterns in vector beams,” Phys. Rev. A 89, 060301(R) (2014).
    [Crossref]
  71. N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, “Writing and deleting single magnetic skyrmions,” Science 341, 636–639 (2013).
    [Crossref]
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    [Crossref]

2016 (1)

J. T. Schultz, A. Hansen, J. D. Murphree, M. Jayaseelan, and N. P. Bigelow, “Raman fingerprints on the Bloch sphere of a spinor Bose-Einstein condensate,” J. Mod. Opt. 63, 1–9 (2016).
[Crossref]

2015 (4)

V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
[Crossref]

X. Zhang, M. Ezawa, and Y. Zhou, “Magnetic skyrmion logic gates: conversion, duplication and merging of skyrmions,” Sci. Rep. 5, 9400 (2015).
[Crossref]

T. Mawson, G. Ruben, and T. Simula, “Route to non-Abelian quantum turbulence in spinor Bose-Einstein condensates,” Phys. Rev. A 91, 063630 (2015).
[Crossref]

S. W. Seo, S. Kang, W. J. Kwon, and Y. Shin, “Half-quantum vortices in an antiferromagnetic spinor Bose-Einstein condensate,” Phys. Rev. Lett. 115, 015301 (2015).
[Crossref]

2014 (6)

V. Kumar and N. K. Viswanathan, “Topological structures in vector-vortex beam fields,” J. Opt. Soc. Am. B 31, A40–A45 (2014).
[Crossref]

Editorial, “The power of analogies,” Nat. Photonics 8, 1 (2014).

M. W. Ray, E. Ruokokoski, S. Kandel, M. Möttönen, and D. S. Hall, “Observation of Dirac monopoles in a synthetic magnetic field,” Nature 505, 657–660 (2014).
[Crossref]

R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Quantum entanglement of complex photon polarization patterns in vector beams,” Phys. Rev. A 89, 060301(R) (2014).
[Crossref]

J. T. Schultz, A. Hansen, and N. P. Bigelow, “A Raman waveplate for spinor Bose-Einstein condensates,” Opt. Lett. 39, 4271–4273 (2014).
[Crossref]

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801R (2014).
[Crossref]

2013 (6)

D. Lopez-Mago, B. Perez-Garcia, A. Yepiz, R. I. Hernandez-Aranda, and J. C. Gutiérrez-Vega, “Dynamics of polarization singularities in composite optical vortices,” J. Opt. 15, 044028 (2013).
[Crossref]

N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, “Writing and deleting single magnetic skyrmions,” Science 341, 636–639 (2013).
[Crossref]

V. Kumar and N. K. Viswanathan, “The Pancharatnam-Berry phase in polarization singular beams,” J. Opt. 15, 044026 (2013).
[Crossref]

V. P. Lukin, O. V. Angelski, L. A. Bolbasova, E. A. Kopylov, M. V. Tuev, and V. V. Nosov, “Comparison of singular-optical and statistical approaches for investigations of turbulence tasks,” Proc. SPIE 9066, 906612 (2013).
[Crossref]

J. Choi, S. Kang, S. W. Seo, W. J. Kwon, and Y. Shin, “Observation of a geometric Hall effect in a spinor Bose-Einstein condensate with a skyrmion spin texture,” Phys. Rev. Lett. 111, 245301 (2013).
[Crossref]

R. D. Ramkhalawon, T. G. Brown, and M. A. Alonso, “Imaging the polarization of a light field,” Opt. Express 21, 4106–4115 (2013).
[Crossref]

2012 (8)

O. V. Angelsky, P. V. Polyanskii, and C. V. Felde, “The emerging field of correlation optics,” Opt. Photon. News 23(4), 25–29 (2012).
[Crossref]

J. Choi, W. J. Kwon, and Y. Shin, “Observation of topologically stable 2D skyrmions in an antiferromagnetic spinor Bose-Einstein condensate,” Phys. Rev. Lett. 108, 035301 (2012).
[Crossref]

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

C. Vo, S. Riedl, S. Baur, G. Rempe, and S. Dürr, “Coherent logic gate for light pulses based on storage in a Bose–Einstein condensate,” Phys. Rev. Lett. 109, 263602 (2012).
[Crossref]

G. M. Philip, V. Kumar, G. Milione, and N. K. Viswanathan, “Manifestation of the Gouy phase in vector-vortex beams,” Opt. Lett. 37, 2667–2669 (2012).
[Crossref]

M. O. Borgh and J. Ruostekoski, “Topological interface engineering and defect crossing in ultracold atomic gases,” Phys. Rev. Lett. 109, 015302 (2012).
[Crossref]

E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light,” Appl. Opt. 51, 2925–2934 (2012).
[Crossref]

F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51, C1–C6 (2012).
[Crossref]

2011 (6)

Y. Kawaguchi and M. Ueda, “Symmetry classification of spinor Bose-Einstein condensates,” Phys. Rev. A 84, 053616 (2011).
[Crossref]

I. G. da Paz, P. L. Saldanha, M. C. Nemes, and J. G. Peixoto de Faria, “Experimental proposal for measuring the Gouy phase of matter waves,” New J. Phys. 13, 125005 (2011).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg, “Colloquium: artificial gauge potential for neutral atoms,” Rev. Mod. Phys. 83, 1523–1543 (2011).
[Crossref]

Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, “Spin-orbit-coupled Bose-Einstein condensates,” Nature 471, 83–86 (2011).
[Crossref]

F. Tamburini, B. Thidé, G. Molina-Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. 7, 195–197 (2011).
[Crossref]

2010 (1)

2009 (6)

K. C. Wright, L. S. Leslie, A. Hansen, and N. P. Bigelow, “Sculpting the vortex state of a spinor BEC,” Phys. Rev. Lett. 102, 030405 (2009).
[Crossref]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[Crossref]

J. A. M. Huhtamäki, T. P. Simula, M. Kobayashi, and K. Machida, “Stable fractional vortices in the cyclic states of Bose-Einstein condensates,” Phys. Rev. A 80, 051601(R) (2009).
[Crossref]

Y.-J. Lin, R. L. Compton, K. Jiménez-García, J. V. Porto, and I. B. Spielman, “Synthetic magnetic fields for ultracold neutral atoms,” Nature 462, 628–632 (2009).
[Crossref]

L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch, and N. P. Bigelow, “Creation and detection of skyrmions in a Bose-Einstein condensate,” Phys. Rev. Lett. 103, 250401 (2009).
[Crossref]

M. Kobayashi, Y. Kawaguchi, M. Nitta, and M. Ueda, “Collision dynamics and rung formation of non-Abelian vortices,” Phys. Rev. Lett. 103, 115301 (2009).
[Crossref]

2008 (4)

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[Crossref]

K. C. Wright, L. S. Leslie, and N. P. Bigelow, “Raman coupling of Zeeman sublevels in an alkali-metal Bose-Einstein condensate,” Phys. Rev. A 78, 053412 (2008).
[Crossref]

K. C. Wright, L. S. Leslie, and N. P. Bigelow, “Optical control of the internal and external angular momentum of a Bose-Einstein condensate,” Phys. Rev. A 77, 041601 (2008).
[Crossref]

M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
[Crossref]

2007 (5)

A. Y. Bekshaeva and M. S. Soskin, “Transverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332–348 (2007).
[Crossref]

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt. 46, 61–66 (2007).
[Crossref]

G. W. Semenoff and F. Zhou, “Discrete symmetries and 1/3-quantum vortices in condensates of F = 2 cold atoms,” Phys. Rev. Lett. 98, 100401 (2007).
[Crossref]

E. Brion, L. H. Pedersen, and K. Mølmer, “Adiabatic elimination in a lambda system,” J. Phys. A 40, 1033–1043 (2007).
[Crossref]

A. Ardavan, “Exploiting the Poincaré-Bloch symmetry to design high-fidelity broadband composite linear retarders,” New J. Phys. 9, 24–32 (2007).
[Crossref]

2006 (1)

L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, “Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose condensate,” Nature 443, 312–315 (2006).
[Crossref]

2005 (2)

M. P. Fewell, “Adiabatic elimination, the rotating-wave approximation and two-photon transitions,” Opt. Commun. 253, 125–137 (2005).
[Crossref]

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: generating arbitrary, counterrotating, coherent superpositions in Bose–Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95, 173601 (2005).
[Crossref]

2003 (3)

J. Ruostekoski and J. R. Anglin, “Monopole core instability and Alice rings in spinor Bose-Einstein condensates,” Phys. Rev. Lett. 91, 190402 (2003).
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C. M. Savage and J. Ruostekoski, “Dirac monopoles and dipoles in ferromagnetic spinor Bose-Einstein condensates,” Phys. Rev. A 68, 043604 (2003).
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A. E. Leanhardt, Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle, “Coreless vortex formation in a spinor Bose-Einstein condensate,” Phys. Rev. Lett. 90, 140403 (2003).
[Crossref]

2002 (3)

L.-M. Kuang, J.-H. Li, and B. Hu, “Polarization and decoherence in a two-component Bose-Einstein condensate,” J. Opt. B 4, 295–299 (2002).
[Crossref]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[Crossref]

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
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2001 (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
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1999 (1)

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose-Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
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1998 (1)

T. Ohmi and K. Machida, “Bose-Einstein condensation with internal degrees of freedom in alkali atom gases,” J. Phys. Soc. Jpn. 67, 1822–1825 (1998).
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1983 (1)

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).
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1977 (1)

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
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1936 (2)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
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A. H. S. Holbourn, “Angular momentum of circularly polarized light,” Nature 137, 31 (1936).
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Alfano, R. R.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
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G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
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Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

Alonso, M. A.

Anderson, B. P.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose-Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
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Angelski, O. V.

V. P. Lukin, O. V. Angelski, L. A. Bolbasova, E. A. Kopylov, M. V. Tuev, and V. V. Nosov, “Comparison of singular-optical and statistical approaches for investigations of turbulence tasks,” Proc. SPIE 9066, 906612 (2013).
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Angelsky, O. V.

O. V. Angelsky, P. V. Polyanskii, and C. V. Felde, “The emerging field of correlation optics,” Opt. Photon. News 23(4), 25–29 (2012).
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Anglin, J. R.

J. Ruostekoski and J. R. Anglin, “Monopole core instability and Alice rings in spinor Bose-Einstein condensates,” Phys. Rev. Lett. 91, 190402 (2003).
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Anzolin, G.

F. Tamburini, B. Thidé, G. Molina-Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. 7, 195–197 (2011).
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Ardavan, A.

A. Ardavan, “Exploiting the Poincaré-Bloch symmetry to design high-fidelity broadband composite linear retarders,” New J. Phys. 9, 24–32 (2007).
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Arnold, C.

V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
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Baur, S.

C. Vo, S. Riedl, S. Baur, G. Rempe, and S. Dürr, “Coherent logic gate for light pulses based on storage in a Bose–Einstein condensate,” Phys. Rev. Lett. 109, 263602 (2012).
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Beckley, A. M.

Bekshaeva, A. Y.

A. Y. Bekshaeva and M. S. Soskin, “Transverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332–348 (2007).
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Berry, M. V.

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
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Beth, R. A.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
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Bickel, J. E.

N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, “Writing and deleting single magnetic skyrmions,” Science 341, 636–639 (2013).
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Bigelow, N. P.

J. T. Schultz, A. Hansen, J. D. Murphree, M. Jayaseelan, and N. P. Bigelow, “Raman fingerprints on the Bloch sphere of a spinor Bose-Einstein condensate,” J. Mod. Opt. 63, 1–9 (2016).
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J. T. Schultz, A. Hansen, and N. P. Bigelow, “A Raman waveplate for spinor Bose-Einstein condensates,” Opt. Lett. 39, 4271–4273 (2014).
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K. C. Wright, L. S. Leslie, A. Hansen, and N. P. Bigelow, “Sculpting the vortex state of a spinor BEC,” Phys. Rev. Lett. 102, 030405 (2009).
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L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch, and N. P. Bigelow, “Creation and detection of skyrmions in a Bose-Einstein condensate,” Phys. Rev. Lett. 103, 250401 (2009).
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K. C. Wright, L. S. Leslie, and N. P. Bigelow, “Optical control of the internal and external angular momentum of a Bose-Einstein condensate,” Phys. Rev. A 77, 041601 (2008).
[Crossref]

K. C. Wright, L. S. Leslie, and N. P. Bigelow, “Raman coupling of Zeeman sublevels in an alkali-metal Bose-Einstein condensate,” Phys. Rev. A 78, 053412 (2008).
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A. Hansen, J. T. Schultz, and N. P. Bigelow, “Full Bloch Bose-Einstein condensates,” in Frontiers in Optics 2012/Laser Science XXVIII, OSA Technical Digest (online) (Optical Society of America, 2012), paper LTu1I.2.

A. Hansen, J. T. Schultz, and N. P. Bigelow, “Measuring the Gouy phase of matter waves using full Bloch Bose–Einstein condensates,” in The Rochester Conferences on Coherence and Quantum Optics and the Quantum Information and Measurement Meeting, OSA Technical Digest (Optical Society of America, 2013), paper M6.64.

Bolbasova, L. A.

V. P. Lukin, O. V. Angelski, L. A. Bolbasova, E. A. Kopylov, M. V. Tuev, and V. V. Nosov, “Comparison of singular-optical and statistical approaches for investigations of turbulence tasks,” Proc. SPIE 9066, 906612 (2013).
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M. O. Borgh and J. Ruostekoski, “Topological interface engineering and defect crossing in ultracold atomic gases,” Phys. Rev. Lett. 109, 015302 (2012).
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E. Brion, L. H. Pedersen, and K. Mølmer, “Adiabatic elimination in a lambda system,” J. Phys. A 40, 1033–1043 (2007).
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Brown, T. G.

Cardano, F.

Choi, J.

J. Choi, S. Kang, S. W. Seo, W. J. Kwon, and Y. Shin, “Observation of a geometric Hall effect in a spinor Bose-Einstein condensate with a skyrmion spin texture,” Phys. Rev. Lett. 111, 245301 (2013).
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J. Choi, W. J. Kwon, and Y. Shin, “Observation of topologically stable 2D skyrmions in an antiferromagnetic spinor Bose-Einstein condensate,” Phys. Rev. Lett. 108, 035301 (2012).
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Compton, R. L.

Y.-J. Lin, R. L. Compton, K. Jiménez-García, J. V. Porto, and I. B. Spielman, “Synthetic magnetic fields for ultracold neutral atoms,” Nature 462, 628–632 (2009).
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Cornell, E. A.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose-Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
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D’Ambrosio, V.

V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
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I. G. da Paz, P. L. Saldanha, M. C. Nemes, and J. G. Peixoto de Faria, “Experimental proposal for measuring the Gouy phase of matter waves,” New J. Phys. 13, 125005 (2011).
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J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg, “Colloquium: artificial gauge potential for neutral atoms,” Rev. Mod. Phys. 83, 1523–1543 (2011).
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Das Sarma, S.

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
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de Lisio, C.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
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M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
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M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
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Deutsch, B. M.

L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch, and N. P. Bigelow, “Creation and detection of skyrmions in a Bose-Einstein condensate,” Phys. Rev. Lett. 103, 250401 (2009).
[Crossref]

Dowling, J. P.

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: generating arbitrary, counterrotating, coherent superpositions in Bose–Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95, 173601 (2005).
[Crossref]

Dürr, S.

C. Vo, S. Riedl, S. Baur, G. Rempe, and S. Dürr, “Coherent logic gate for light pulses based on storage in a Bose–Einstein condensate,” Phys. Rev. Lett. 109, 263602 (2012).
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Eberly, J. H.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

Evans, S.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
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X. Zhang, M. Ezawa, and Y. Zhou, “Magnetic skyrmion logic gates: conversion, duplication and merging of skyrmions,” Sci. Rep. 5, 9400 (2015).
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Felde, C. V.

O. V. Angelsky, P. V. Polyanskii, and C. V. Felde, “The emerging field of correlation optics,” Opt. Photon. News 23(4), 25–29 (2012).
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Fewell, M. P.

M. P. Fewell, “Adiabatic elimination, the rotating-wave approximation and two-photon transitions,” Opt. Commun. 253, 125–137 (2005).
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R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Quantum entanglement of complex photon polarization patterns in vector beams,” Phys. Rev. A 89, 060301(R) (2014).
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Freedman, M.

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
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Freund, I.

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
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E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801R (2014).
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E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light,” Appl. Opt. 51, 2925–2934 (2012).
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J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg, “Colloquium: artificial gauge potential for neutral atoms,” Rev. Mod. Phys. 83, 1523–1543 (2011).
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Gutiérrez-Vega, J. C.

D. Lopez-Mago, B. Perez-Garcia, A. Yepiz, R. I. Hernandez-Aranda, and J. C. Gutiérrez-Vega, “Dynamics of polarization singularities in composite optical vortices,” J. Opt. 15, 044028 (2013).
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Haljan, P. C.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose-Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
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Hall, D. S.

M. W. Ray, E. Ruokokoski, S. Kandel, M. Möttönen, and D. S. Hall, “Observation of Dirac monopoles in a synthetic magnetic field,” Nature 505, 657–660 (2014).
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M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose-Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
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Hannay, J. H.

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
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Hanneken, C.

N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, “Writing and deleting single magnetic skyrmions,” Science 341, 636–639 (2013).
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Hansen, A.

J. T. Schultz, A. Hansen, J. D. Murphree, M. Jayaseelan, and N. P. Bigelow, “Raman fingerprints on the Bloch sphere of a spinor Bose-Einstein condensate,” J. Mod. Opt. 63, 1–9 (2016).
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J. T. Schultz, A. Hansen, and N. P. Bigelow, “A Raman waveplate for spinor Bose-Einstein condensates,” Opt. Lett. 39, 4271–4273 (2014).
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K. C. Wright, L. S. Leslie, A. Hansen, and N. P. Bigelow, “Sculpting the vortex state of a spinor BEC,” Phys. Rev. Lett. 102, 030405 (2009).
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L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch, and N. P. Bigelow, “Creation and detection of skyrmions in a Bose-Einstein condensate,” Phys. Rev. Lett. 103, 250401 (2009).
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A. Hansen, J. T. Schultz, and N. P. Bigelow, “Full Bloch Bose-Einstein condensates,” in Frontiers in Optics 2012/Laser Science XXVIII, OSA Technical Digest (online) (Optical Society of America, 2012), paper LTu1I.2.

A. Hansen, J. T. Schultz, and N. P. Bigelow, “Measuring the Gouy phase of matter waves using full Bloch Bose–Einstein condensates,” in The Rochester Conferences on Coherence and Quantum Optics and the Quantum Information and Measurement Meeting, OSA Technical Digest (Optical Society of America, 2013), paper M6.64.

Hernandez-Aranda, R. I.

D. Lopez-Mago, B. Perez-Garcia, A. Yepiz, R. I. Hernandez-Aranda, and J. C. Gutiérrez-Vega, “Dynamics of polarization singularities in composite optical vortices,” J. Opt. 15, 044028 (2013).
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Higbie, J. M.

L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, “Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose condensate,” Nature 443, 312–315 (2006).
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Holbourn, A. H. S.

A. H. S. Holbourn, “Angular momentum of circularly polarized light,” Nature 137, 31 (1936).
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L.-M. Kuang, J.-H. Li, and B. Hu, “Polarization and decoherence in a two-component Bose-Einstein condensate,” J. Opt. B 4, 295–299 (2002).
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J. A. M. Huhtamäki, T. P. Simula, M. Kobayashi, and K. Machida, “Stable fractional vortices in the cyclic states of Bose-Einstein condensates,” Phys. Rev. A 80, 051601(R) (2009).
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Jayaseelan, M.

J. T. Schultz, A. Hansen, J. D. Murphree, M. Jayaseelan, and N. P. Bigelow, “Raman fingerprints on the Bloch sphere of a spinor Bose-Einstein condensate,” J. Mod. Opt. 63, 1–9 (2016).
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Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, “Spin-orbit-coupled Bose-Einstein condensates,” Nature 471, 83–86 (2011).
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Y.-J. Lin, R. L. Compton, K. Jiménez-García, J. V. Porto, and I. B. Spielman, “Synthetic magnetic fields for ultracold neutral atoms,” Nature 462, 628–632 (2009).
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Juzeliunas, G.

J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg, “Colloquium: artificial gauge potential for neutral atoms,” Rev. Mod. Phys. 83, 1523–1543 (2011).
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Kandel, S.

M. W. Ray, E. Ruokokoski, S. Kandel, M. Möttönen, and D. S. Hall, “Observation of Dirac monopoles in a synthetic magnetic field,” Nature 505, 657–660 (2014).
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Kang, S.

S. W. Seo, S. Kang, W. J. Kwon, and Y. Shin, “Half-quantum vortices in an antiferromagnetic spinor Bose-Einstein condensate,” Phys. Rev. Lett. 115, 015301 (2015).
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J. Choi, S. Kang, S. W. Seo, W. J. Kwon, and Y. Shin, “Observation of a geometric Hall effect in a spinor Bose-Einstein condensate with a skyrmion spin texture,” Phys. Rev. Lett. 111, 245301 (2013).
[Crossref]

Kapale, K. T.

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: generating arbitrary, counterrotating, coherent superpositions in Bose–Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95, 173601 (2005).
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M. Kobayashi, Y. Kawaguchi, M. Nitta, and M. Ueda, “Collision dynamics and rung formation of non-Abelian vortices,” Phys. Rev. Lett. 103, 115301 (2009).
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A. E. Leanhardt, Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle, “Coreless vortex formation in a spinor Bose-Einstein condensate,” Phys. Rev. Lett. 90, 140403 (2003).
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Khadka, S.

Kielpinski, D.

A. E. Leanhardt, Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle, “Coreless vortex formation in a spinor Bose-Einstein condensate,” Phys. Rev. Lett. 90, 140403 (2003).
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Kobayashi, M.

J. A. M. Huhtamäki, T. P. Simula, M. Kobayashi, and K. Machida, “Stable fractional vortices in the cyclic states of Bose-Einstein condensates,” Phys. Rev. A 80, 051601(R) (2009).
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M. Kobayashi, Y. Kawaguchi, M. Nitta, and M. Ueda, “Collision dynamics and rung formation of non-Abelian vortices,” Phys. Rev. Lett. 103, 115301 (2009).
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Kopylov, E. A.

V. P. Lukin, O. V. Angelski, L. A. Bolbasova, E. A. Kopylov, M. V. Tuev, and V. V. Nosov, “Comparison of singular-optical and statistical approaches for investigations of turbulence tasks,” Proc. SPIE 9066, 906612 (2013).
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Kuang, L.-M.

L.-M. Kuang, J.-H. Li, and B. Hu, “Polarization and decoherence in a two-component Bose-Einstein condensate,” J. Opt. B 4, 295–299 (2002).
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Kubetzka, A.

N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, “Writing and deleting single magnetic skyrmions,” Science 341, 636–639 (2013).
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V. Kumar and N. K. Viswanathan, “Topological structures in vector-vortex beam fields,” J. Opt. Soc. Am. B 31, A40–A45 (2014).
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E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801R (2014).
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V. Kumar and N. K. Viswanathan, “The Pancharatnam-Berry phase in polarization singular beams,” J. Opt. 15, 044026 (2013).
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G. M. Philip, V. Kumar, G. Milione, and N. K. Viswanathan, “Manifestation of the Gouy phase in vector-vortex beams,” Opt. Lett. 37, 2667–2669 (2012).
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S. W. Seo, S. Kang, W. J. Kwon, and Y. Shin, “Half-quantum vortices in an antiferromagnetic spinor Bose-Einstein condensate,” Phys. Rev. Lett. 115, 015301 (2015).
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J. Choi, S. Kang, S. W. Seo, W. J. Kwon, and Y. Shin, “Observation of a geometric Hall effect in a spinor Bose-Einstein condensate with a skyrmion spin texture,” Phys. Rev. Lett. 111, 245301 (2013).
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J. Choi, W. J. Kwon, and Y. Shin, “Observation of topologically stable 2D skyrmions in an antiferromagnetic spinor Bose-Einstein condensate,” Phys. Rev. Lett. 108, 035301 (2012).
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R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Quantum entanglement of complex photon polarization patterns in vector beams,” Phys. Rev. A 89, 060301(R) (2014).
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Laurat, J.

V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
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Appl. Opt. (3)

J. Mod. Opt. (1)

J. T. Schultz, A. Hansen, J. D. Murphree, M. Jayaseelan, and N. P. Bigelow, “Raman fingerprints on the Bloch sphere of a spinor Bose-Einstein condensate,” J. Mod. Opt. 63, 1–9 (2016).
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J. Opt. (2)

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J. Opt. B (1)

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J. Opt. Soc. Am. B (1)

J. Phys. A (2)

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J. Phys. Soc. Jpn. (1)

T. Ohmi and K. Machida, “Bose-Einstein condensation with internal degrees of freedom in alkali atom gases,” J. Phys. Soc. Jpn. 67, 1822–1825 (1998).
[Crossref]

Nat. Commun. (1)

V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6, 7706 (2015).
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Nat. Photonics (1)

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Figures (4)

Fig. 1.
Fig. 1. (a) Bloch sphere. (b) Three-level lambda system.
Fig. 2.
Fig. 2. Maps of the Stokes parameters. (a) Experiment and (b) theory maps of the Stokes parameters of a coreless vortex where = 2 in the spin state containing the vortex. Maps for each of the Stokes parameters S i are experimentally obtained using an atom-optic polarimetry technique [44] on essentially identical coreless vortices. S 1 and S 2 can be used to reconstruct the relative phase of the spin states. Examples of Stokes maps for = 1 coreless vortices appear in [44]. With this information, we create spin ellipse maps to classify the singularities. These Stokes parameters correspond to a spiral singularity with C -point index I C = 1 . The Stokes parameter S 0 would correspond to the total density of the cloud; in this case, the vortex state ( | ψ ) dominates at the edge of the cloud, so the shape of the spin density ( S 3 ) gives a good indication of the overall atomic density.
Fig. 3.
Fig. 3. Maps of the local spin ellipses. The orientation of the major axis of the ellipses, ϕ / 2 , is found from maps of the Stokes parameters S 1 and S 2 . The ellipticity is determined by S 3 , and the ellipses are red and blue for right- and left-handed precession of the transverse components of the spin vector, respectively. When the population of the spin states is equal, the spin ellipse becomes a line (drawn in green). Spin lines show how the orientation of the major axes of the spin ellipses changes, revealing a fingerprint of the (a) star, (b) lemon, (c) spiral, and (d) saddle singularities.
Fig. 4.
Fig. 4. Characterizing singularities by the C -point index. Spin lines representing the local orientation of the major axis of the spin ellipse are plotted over the condensate phase. The singularities here are the same as those in Fig. 3 with a narrowed plot range focused on the singularities. The local spinors in the x y plane are parallel to the major axes of the spin ellipses, and rotate by π and 2 π around the singularity for π - and 2 π -symmetric singularities corresponding to an | I C | of 1/2 [(a), (b)] and 1 [(c), (d)], respectively. The sign of I C is determined by the direction of rotation of the spinor as the contour is followed in a counterclockwise (positive) direction. Both (a) and (d) have negative indices because the spinors rotate clockwise, while in (b) and (c) the spinors rotate counterclockwise corresponding to positive indices. In (c) and (d) the extra singularities near the exterior are artifacts from the edge of the atomic cloud and imperfect separation of the spin components during the Stern–Gerlach operation and therefore do not correspond to actual singularities in the BEC.

Equations (6)

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ψ⃗ t = i 4 Δ [ | Ω A | 2 Ω A * Ω B Ω A Ω B * | Ω B | 2 ] ψ⃗ .
M ( t ) = cos Ω t 2 I + i sin Ω t 2 P ( 2 α , ϕ A B ) ,
P ( 2 α , ϕ A B ) = [ cos 2 α sin 2 α e i ϕ A B sin 2 α e i ϕ A B cos 2 α ]
ψ⃗ ( t ) = n ( ψ ψ ) n e i Φ 2 ( | ψ | e i ϕ 2 | ψ | e i ϕ 2 ) .
Φ = arctan ( tan Ω t 2 cos 2 α ) + π 2 ϕ A B + Ω t 2 ,
I C = 1 4 π ϕ ( r ) · d r ,

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