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 -symmetric (lemon, star) or -symmetric (saddle, spiral) -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 -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
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 [1–4]. Therefore, tabletop experiments in singular optics  can be used to study optical fields with properties analogous to those of black holes  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 [9–11], skyrmions [12–14], monopoles [15,16], and even synthetic fields [17,18] and spin-orbit coupling . 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 [20–22], 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 -symmetric singularities  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 -point singularities in the polarization field of an optical beam [27–29]. -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 [30–32]. 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 .
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 [36–38]. 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 .
In this work, the two-photon Raman transition couples the atomic spin states and . Consider the three-level lambda system in Fig. 1(b) with two incident square pulses with Rabi frequencies and . For a large single-photon detuning , 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
Because the pulses are square, the Rabi frequencies are constant over the interaction time and the equation can be integrated directly to give with43]. For plane waves with equal Rabi frequencies, , and Eq. (2) reduces to the case of the Raman waveplate . However, for beams with nonuniform intensity and phase profiles, , , , and hence become spatially dependent.
With the quantization defined along the direction with a magnetic field, the spinor wavefunction is naturally described in the eigenbasis of the spin operator in the direction, . However, writing the BEC wavefunction in the eigenbasis of the 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 after the Raman interaction is given in the spin basis by
Here, the condensate phase is
Consider two ways of classifying -points: spin lines and the -point index, [27,28,45]. Spin lines represent the major axis orientation of the local spin ellipse and thereby provide a fingerprint of the singularity . The number of radial spin lines that terminate at the singularity serves as a method of classification. For -symmetric singularities the number of radial lines is either 1 (lemon) or 3 (star), and for the -symmetric case, the number is either 4 (saddle) or infinite (spiral).
3. EXPERIMENTAL TECHNIQUE
Technical details of our Raman technique can be found in . Briefly, a cigar-shaped condensate of atoms with aspect ratio is created in the state 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 () and Laguerre–Gaussian () beams detuned from couple and . The two beams propagate along the quantization () axis, which is also the axis of cylindrical symmetry. A portion of the initial state is transferred to 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 surrounded by a quantized vortex in .
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 state. This means that, during the Raman process, some of the population is transferred back to and acquires OAM of . The result is a coreless vortex with circulation in the state surrounding a nonrotating 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 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 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  (see Fig. 2). The images are taken 26 ms after the magnetic trap turns off when the atomic cloud expands to 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 is a function of the azimuthal coordinate 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.
4. RESULTS AND DISCUSSION
The Raman beams are coherent superpositions of orthogonally polarized Gaussian and Laguerre–Gaussian beams; that is, they possess -symmetric singularities [27–29,45,47,48] 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 —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 -point singularities. For , we create a -symmetric coreless vortex. In a spin-1 system, such a vortex is a half-quantum or Alice state [25,26,52–54]. By changing the handedness of the vortex, we can generate star [Fig. 3(a)] or lemon [Fig. 3(b)] -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 [27–29,47,55]. Similarly, for , the quantum field is -symmetric exhibiting spiral [Fig. 3(c)] and saddle [Fig. 3(d)] -points [29,45,48], and changing the handedness of the Laguerre–Gaussian beam changes the resulting singularity between these two.
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 -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 around the -points in Figs. 4(a) and 4(b) and by in Figs. 4(c) and 4(d), meaning the first two are -symmetric and have and the second two are -symmetric with . The sign of 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 -point indices of (star) and (saddle), respectively. The -points in Figs. 4(b) and 4(c) have positive indices of 1/2 (lemon) and 1 (spiral), respectively.
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 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 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 . 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  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 , 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 .
In conclusion, we have created -symmetric (lemon, star) and -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 .
Atomic analogs of states on a higher-order Poincaré sphere  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 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 , including those that are driven by spin ordering and other spinor dynamics . 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 . 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 [67–70]. 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.
National Science Foundation (NSF) (PHY-1313539); National Aeronautics and Space Administration (NASA) (JPL-1504801); Defense Advanced Research Projects Agency (DARPA).
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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