Artificial magnetic fields in momentum space in spin-orbit-coupled systems

Abstract

The Berry curvature is a geometrical property of an energy band which can act as a momentumspace magnetic field in the effective Hamiltonian of a wide-range of systems. We apply the effective Hamiltonian to a spin-1/2 particle in two dimensions with spin-orbit coupling, a Zeeman field and an additional harmonic trap. Depending on the parameter regime, we show how this system can be described in momentum space as either a Fock-Darwin Hamiltonian or a one-dimensional ring pierced by a magnetic flux. With this perspective, we interpret important single-particle properties, and identify analogue magnetic phenomena in momentum space. Finally we discuss the extension of this work to higher spin systems, as well as experimental applications in ultracold atomic gases and photonic systems.

Quantum Hall effect in momentum space

“…This expression shows that the Niu-Thouless-Wu formulation of the quantum Hall effect applies also to momentum space. Note that generally speaking there can be a momentum-space current even in the absence of the artificial electric field due to the first term of (22), which gives the persistent current in momentum space [19]. Such a persistent current can also exist in the real-space quantum Hall effect; an analogous expression for the real-space Hall current including the persistent current can be found, for example, in [20].…”

“…Since the insertion of a flux corresponds to a shift in the trap center, we consider adiabatically moving the trap center along the x direction according to a generic smooth function x 0 (t) (a fully analogous analysis holds for the motion of the trap center in the y direction). The current operator can be defined in momentum space in analogy to the real-space current operator (3) as [19]…”

“…The momentum-space current derived above is an experimentally observable quantity. In fact, since r = i∇ p +A 0 from the minimal coupling in momentum space, we have from (19)…”

Phase-space curvature in spin-orbit-coupled ultracold atomic systems

“…where jln is the Levi-Civita symbol. Effects of the momentum-space curvature have already been extensively studied theoretically [63,64].…”

Momentum-space Landau levels in driven-dissipative cavity arrays

“…The Berry curvature, Ω n (p) = ∇ × A n,n (p), is then like a momentum-space magnetic field. For certain models, this analogy leads to a clear analytical understanding of singleparticle dynamics [19,[26][27][28]. In particular, we will focus on the small-flux limit of the Harper-Hofstadter Hamiltonian [29,30], which is a model that has recently been realized in a multitude of experimental configurations, ranging from ultracold gases [31][32][33][34], solid state superlattices [35,36] and silicon photonics [37] to classical systems such as coupled pendula [38] and oscillating circuits [39].…”