Cold atoms with laser-induced spin-orbit (SO) interactions provide intriguing new platforms to explore novel quantum physics beyond natural conditions of solids. Recent experiments demonstrated the one-dimensional (1D) SO coupling for boson and fermion gases. However, realization of 2D SO interaction, a much more important task, remains very challenging.Here we propose and experimentally realize, for the first time, 2D SO coupling and topological band with 87 Rb degenerate gas through a minimal optical Raman lattice scheme, without relying on phase locking or fine tuning of optical potentials. A controllable crossover between 2D and 1D SO couplings is studied, and the SO effects and nontrivial band topology are observed by measuring the atomic cloud distribution and spin texture in the momentum 1 arXiv:1511.08170v1 [cond-mat.quant-gas] 24 Nov 2015space. Our realization of 2D SO coupling with advantages of small heating and topological stability opens a broad avenue in cold atoms to study exotic quantum phases, including the highly-sought-after topological superfluid phases.Spin-orbit (SO) interaction of an electron is a relativistic quantum mechanic effect, which characterizes the coupling between motion and spin of the electron when moving in an electric field. In the rest frame the electron experiences a magnetic field which is proportional to the electron velocity and couples to its spin by the magnetic dipole interaction, rendering the SO coupling. The SO interaction plays essential roles in many novel quantum states of solids. The recent outstanding examples include the topological insulators, which have been predicted and experimentally discovered in two-dimensional (2D) and 3D materials 1, 2 , and the topological superconductors 3, 4 , which host exotic zero-energy states called Majorana fermions 5,6 and still necessitate rigorous experimental verification. For topological insulators, the strong SO interaction leads to the so-called band inversion mechanism which drives a topological phase transition in such systems 7,8 . In superconductors, a triplet p-wave pairing is generically resulted when SO coupling is present, for which the superconductivity can be topologically nontrivial under proper conditions 9 .Recently, considerable interests have been drawn in emulating SO effects and topological phases with cold atoms, mostly driven by the fact that cold atoms can offer extremely clean platforms with full controllability to explore such exotic physics. In cold atoms the synthetic SO interaction can be generated by Raman coupling schemes which flip atom spins and transfer momentum where 1 is the 2 × 2 unit matrix, σ x,y,z are Pauli matrices acting on the spins, m is mass of an atom, V latt denotes the lattice potential in the x-z plane, M x,y are periodic Raman coupling potentials, and m z represents a tunable Zeeman field. Atoms can hop between nearest-neighboring sites due to lattice potential as well as the Raman coupling terms. Note that V latt is spin-independent and can induce hopping which conserves ...

Roton-type excitations usually emerge from strong correlations or long-range interactions, as in superfluid helium or dipolar ultracold atoms. However, in a weakly short-range interacting quantum gas, the recently synthesized spin-orbit (SO) coupling can lead to various unconventional phases of superfluidity and give rise to an excitation spectrum of roton-maxon character. Using Bragg spectroscopy, we study a SO-coupled Bose-Einstein condensate of ^{87}Rb atoms and show that the excitation spectrum in a "magnetized" phase clearly possesses a two-branch and roton-maxon structure. As Raman coupling strength Ω is decreased, a roton-mode softening is observed, as a precursor of the phase transition to a stripe phase that spontaneously breaks spatially translational symmetry. The measured roton gaps agree well with theoretical calculations. Furthermore, we determine sound velocities both in the magnetized and in the nonmagnetized phases, and a phonon-mode softening is observed around the phase transition in between. The validity of the f-sum rule is examined.