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arXiv:1601.00327 · Journal of Modern Optics

Raman fingerprints on the Bloch sphere of a spinor Bose–Einstein condensate

Justin T. Schultz, Azure Hansen, Joseph D. Murphree, Maitreyi Jayaseelan, N. P. Bigelow

Published Feb 1, 2016• 9 citations
arXivHTMLPDF

Abstract

We explore the geometric interpretation of a diabatic, two-photon Raman process as a rotation on the Bloch sphere for a pseudo-spin-1/2 system. The spin state of a spin-1/2 quantum system can be described by a point on the surface of the Bloch sphere, and its evolution during a Raman pulse is a trajectory on the sphere determined by properties of the optical beams: the pulse area, the relative intensities and phases and the relative frequencies. We experimentally demonstrate key features of this model with a 87Rb spinor Bose–Einstein condensate, which allows us to examine spatially dependent signatures of the Raman beams. The two-photon detuning allows us to precisely control the spin density and imprinted relative phase profiles, as we show with a coreless vortex. With this comprehensive understanding and intuitive geometric interpretation, we use the Raman process to create and tailor as well as study and characterize exotic topological spin textures in spinor BECs.

DOI
10.1080/09500340.2016.1139204

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We explore the geometric interpretation of a diabatic, two-photon Raman process as a rotation on the Bloch sphere for a pseudo-spin-1/2 system. The spin state of a spin-1/2 quantum system can be described by a point on the surface of the Bloch sphere, and its evolution during a Raman pulse is a trajectory on the sphere determined by properties of the optical beams: the pulse area, the relative intensities and phases and the relative frequencies. We experimentally demonstrate key features of this model with a 87Rb spinor Bose–Einstein condensate, which allows us to examine spatially dependent signatures of the Raman beams. The two-photon detuning allows us to precisely control the spin density and imprinted relative phase profiles, as we show with a coreless vortex. With this comprehensive understanding and intuitive geometric interpretation, we use the Raman process to create and tailor as well as study and characterize exotic topological spin textures in spinor BECs.
Abstract A coherent two-photon optical Raman interaction in a pseudo-spin-1/2 Bose–Einstein condensate (BEC) serves as a q -plate for atoms, converting spin to orbital angular momentum. This Raman q -plate has a singular pattern in its polarization distribution in analogy to the singular birefringent q -plates used in singular optics. The vortex winding direction and magnitude as well as the final spin state of the BEC depend on the initial spin state and the topology of the optical Raman q -plate beams. Drawing on the mathematical and geometric foundations of singular optics, we derive the equivalent Jones matrix for this Raman q -plate and use it to create and characterize atomic spin singularities in the BEC that are analogous to optical C -point singularities in polarization. By tuning the optical Raman parameters, we can generate a coreless vortex spin texture which contains every possible superposition in a two-state system. We identify this spin texture as a full-Bloch BEC since every point on the Bloch sphere is represented at some point in the cross section of the atomic cloud. This spin–orbit interaction and the spin textures it generates may allow for the observation of interesting geometric phases in matter waves and lead to schemes for topological quantum computation with spinor BECs.
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.