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Chapter 1
Spin-orbit Coupling and Topological Phases for Ultracold Atoms
Long Zhang and Xiong-Jun Liu∗
International Center for Quantum Materials, School of Physics,
Peking University, Beijing 100871, China
Collaborative Innovation Center of Quantum Matter,
Beijing 100871, China
Cold atoms with laser-induced spin-orbit (SO) interactions provide promising platforms to explore novel quantum physics, in particular the exotic topological phases, beyond natural conditions of solids. The past several years have witnessed important progresses in both theory and experiment in the study of SO coupling and novel quantum states for ultracold atoms. Here we review the physics of the SO coupled quantum gases, focusing on the latest theoretical and experimental progresses of realizing SO couplings beyond one-dimension (1D), and the further investigation of novel topological quantum phases in such systems, including the topological insulating phases and topological superfluids. A pedagogical introduction to the SO coupling for ultracold atoms and topological quantum phases is presented. We show that the so-called optical Raman lattice schemes, which combine the creation of the conventional optical lattice and Raman lattice with topological stability, can provide minimal methods with high experimental feasibility to realize 1D to 3D SO couplings. The optical Raman lattices exhibit novel intrinsic symmetries, which enable the natural realization of topological phases belonging to different symmetry classes, with the topology being detectable through minimal measurement strategies. Furthermore, we discuss the realization of novel superfluid phases for SO coupled ultrocold fermions. In particular, we introduce how the non-Abelian Majorana modes emerge in the SO coupled superfluid phases which can be topologically nontrivial or trivial, for which a few fundamental theorems are presented and discussed. The experimental schemes for achieving non-Abelian superfluid phases are given. Finally, we point out the future important issues in this rapidly growing research field.
1.1.Introduction
1.A.Why study spin-orbit coupling?
Spin-orbit (SO) coupling is a relativistic quantum mechanics effect which characterizes the interaction between the spin and orbital degrees of freedom of electrons when moving in an external electric field. Due to the special relativity, the electron experiences a magnetic field in the rest frame, which is proportional to the electron velocity and couples to its spin by the magnetic dipole interaction, rendering the SO coupling with the following form
where σ is the spin, V(r) is the external electric potential experienced by the electron, p is the electron’s momentum, and λso denotes the SO coefficient. In atomic physics the SO coupling is responsible for the fine structure splitting of the optical spectroscopy. In solid state physics the SO interaction of Bloch electrons exhibits several effective forms by taking into account the crystal symmetries and local orbitals around Fermi energy, and can strongly affect the band structure of the system. The typical types of the SO coupling includes the Rashba and Dresselhaus terms, which are due to the structure inversion asymmetry and bulk inversion asymmetry of the materials, respectively, and Luttinger term which describes the SO coupling for the valence hole bands.1,2 The study of SO coupling for electrons has generated very important research fields in the recent years, including spintronics,3 topological insulator,4,5 and topological superconductors (SCs),6 etc. These studies bring about completely new understanding of the effects of the SO coupling in the condensed matter physics and material science.
Spintronics. In a system with SO coupling one can manipulate the electron spins indirectly by controlling the orbital degree of freedom. This lies in the heart of the study of semiconductor spintronics, with the spin degree of freedom of the electron being exploited for improved functionality. From the basic form of the SO interaction, one can see that the components of the spin, momentum, and electric field which couple together are perpendicular to each other:
. This implies that applying an external electric field may dynamically drive the electron at opposite spin states to move oppositely in the real space. In particular, consider a Rashba SO
coupled system with t...
