Dirac-, Rashba-, and Weyl-type spin-orbit couplings: Toward experimental realization in ultracold atoms

Abstract

We propose a hierarchy set of minimal optical Raman lattice schemes toward the experimental realization of spin-orbit (SO) couplings of various types for ultracold atoms, including two-dimensional (2D) Dirac type of C4 symmetry, 2D Rashba and 3D Weyl types. These schemes are well accessible with current cold-atom technology, and in particular, a long-lived Bose-Einstein condensation of the 2D Dirac SO coupling has been experimentally proved. The generation of 2D Rashba and 3D Weyl types has an exquisite request that two sources of laser beams have distinct frequencies of factor-two difference. Surprisingly, we find that 133 Cs atoms provide an ideal candidate for the realization. A common and essential feature is the absence of any fine-tuning and phase-locking in the realization, and the resulted SO coupled ultracold atoms have a long lifetime. These schemes essentially improve over the current experimental accessibility and controllability, and open new experimental platforms to study high-dimensional SO physics and novel topological phases with ultracold atoms.Topological phase of matter has become a mainstream of research in condensed matter physics. Recent outstanding examples include topological insulators, which have been predicted and experimentally discovered in two-dimensional (2D) and 3D materials [1, 2], topological semimetals, which exhibit linear dispersion around nodes termed the Dirac or Weyl points, were found recently in a number of materials like Cd 3 As 2 [3, 4] and TaAs [5,6], and topological superconductors [7-10], which host exotic zero-energy states called Majorana modes and have attracted considerable experimental efforts [11][12][13][14][15][16][17]. In these matters, the spin-orbit (SO) interactions, manifesting Dirac, Rashba and Weyl types, play essential roles in driving the phases to be topologically nontrivial.Very recently, the experimental realizations of 2D SO couplings have been respectively reported in 87 Rb Bose-Einstein condensate (BEC) [18] and 40 K degenerate Fermi gases [19,20], where nontrivial band topology and Dirac point are observed. The reports signify an important achievement in the field of quantum simulator with ultracold atoms and optical lattice, opening a great deal of possibilities to explore novel quantum states beyond natural conditions [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Nevertheless, the current experiments suffer challenges in controllability, since the realizations rely on the blue-detuned optical lattices [18,27] or delicate manipulation of resonant Raman couplings between multiple ground states [19,20]. To be broadly applicable to studying exotic topological phases mentioned above, the SO coupled systems should exhibit the following criteria, the high controllability and high stability with long lifetime, which however cannot be satisfied in the current experiments.In this Letter, we propose a hierarchy set of min-" # % , % ( , ( %, %( (% (, | ↓ ⟩ | ↑ ⟩ Atom Cloud | " ⟩ | # ⟩ (a) (b) DB DB Δ Mirror FIG. 1: (a) Generic setup fo...

Observing Topological Charges and Dynamical Bulk-Surface Correspondence with Ultracold Atoms

“…In the tight-binding limit and only considering s-bands, the Hamiltonian (S2) takes the form in the momentum space H = q Ψ † q HΨ q [25,31], where Ψ q = (c q↑ , c q↓ ) and H (i) = [m z − 2t 0 (cos q x a + cos q y a)]σ z + 2t so sin q x aσ y + 2t so sin q y aσ x (S10)…”

“…Experimental setup.-Our experiment is based on a 2D QAH model [30] with C 4 symmetry proposed in theory [31] and realized for ultracold 87 Rb bosons trapped in a square optical Raman lattice [14,15,32]. The laser beam E x (E y ) with wavelength of λ = 787nm is incident from x (y) direction and reflected by mirror.…”

“…, and E yz =ẑ |E yz | e iϕ2 cos(k 0 y − ϕ 2 ), from which the scalar and vector potentials are generated and give the lattice and Raman potentials, respectively [15,31]. Here k 0 = 2π/λ and ϕ 1 (ϕ 2 ) is the relative phase between E xz (E yz ) and E xy (E yx ).…”

“…For the setting with (ϕ 1 , ϕ 2 ) = (π/2, π/2), the former (or latter) potential ∝ sin(k 0 x) cos(k 0 y) (or sin(k 0 y) cos(k 0 x)) is symmetric with respect to every lattice site in y (or x) direction, but antisymmetric in x (or y) direction [33]. This is the configuration of 2D optical Raman lattice to realize QAH model [14,15,31,32], in which the nearestneighbor spin-flip hopping is induced by each Raman potential along the antisymmetric direction, while onsite spin-flip transition is forbidden. We then reach the Hamiltonian (S10) with m x = m y = 0.…”

Characterizing topological phases by quantum quenches: A general theory

“…On the other hand, being a clean environment and of full controllability [12], ultracold atoms provide an ideal platform to realize novel topological models and explore exotic topological physics [13][14][15][16][17][18][19][20][21][22][23]. In particular, compared with solid state systems, realization in ultra-dence, the non-equilibrium characterization theory [33] provides new schemes with explicit advantages over equilibrium schemes to measure the bulk topology of equilibrium topological quantum phases.…”