No abstract

No abstract

When a gauge potential is constructed from the Berry phase, there is an associated scalar potential coming with the gauge potential. In this paper, we investigate the impact of this scalar potential on cold atoms by constructing an artificial gauge potential in a cold-atomic system. We interpret the scalar potential as the coupling between the atom and the mass of the non-Abelian part of the gauge potential. We demonstrate that the gauge potential can produce spin-orbit coupling and that the scalar potential will suppress the spin Hall currents which are generated by spin-orbit coupling. We also discuss the observation of these phenomena in a real experiment.

The topological force and torque are investigated in the systems with spin-orbit coupling. It is demonstrated that the topological force and torque appears as a pure relativistic quantum effect in an electromagnetic field. The origin of both topological force and torque is the Zitterbewegung effect. Considering nonlinear behaviors of spin-orbit coupling, we address possible novel phenomena driven by the topological forces.PACS numbers: 03.65. Pm, 72.10.Bg, 71.70.Ej Recently, the spin-orbit coupling has become an interesting topic due to the spin Hall effect [1][2][3][4][5]. It provides an efficient route to generate and control quantum spin state electrically. Generally speaking, the spin-orbit coupling arises as relativistic quantum effect from the Dirac equation, and describes the interaction of the electron spin, momentum and electromagnetic field. In the system with spin-orbit coupling, the semiclassical equations of electron were studied recently, and some novel effects have been found [6][7][8][9][10][11][12][13][14][15]. However, in the systems with spin-orbit coupling, there remain some questions to be answered. As well known, the non-relativistic approximation of Dirac equation can be obtained from Foldy-Wouthuysen transformation. In the transformation, there exists a gauge potential in momentum space [15,16], so one question is, what is the position operator in semiclassical equations? Since the position operator is the space-time parameter, its definition is significant. If we are uncapable of defining it correctly, we couldn't have the complete understanding for spin-orbit coupling.In this paper, we investigate the gauge field of position operator in momentum space, and its related effects. Based on the dynamic continuity equation, we derive the quantum force and torque which contain two parts. The conventional part has the same form as in the classic electrodynamics, and the topological part originates from the spin-orbit coupling and other terms from relativistic quantum correction. We notice that the topological part has a close relation to the Zitterbewegung effect. For a two-dimensional system in a magnetic field, we propose that the topological force and torque can reveal more complicated phenomena. Topological velocity The Dirac equation of electron with the wave function Ψ = (ϕ, χ)T reads i ∂ ∂t Ψ = HΨ, with the Hamiltonian H = cα · π + βmc 2 + V (x), where π = P − e c A, α and β are the 4 × 4 Dirac matrices, V (x) = eφ is a scalar potential, and A is a vector potential for a magnetic field, B = ∇ × A. m and e are the electron mass and charge, and c is the speed of light. To reveal the spin-orbit coupling in Dirac equation, we perform the Foldy-Wouthuysen transformation Ψ ′ = U (π)Ψ, where U (π) = e iS , it is a unitary transformation [17,18]. Choosing S = −i(βα · π/2mc), and substituting it into the Dirac equation, we obtain the transformed Hamiltonian(1) The scalar potential becomes V (D) with covariant derivative defined by D = i ∂ p + A, with the pure gauge potential. By defining the c...