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arXiv:1403.7979 · Physical Review A

U(3) artificial gauge fields for cold atoms

Yuxin Hu, Christian Miniatura, David Wilkowski, Benoît Grémaud

Published Aug 1, 2014• 17 citations
arXivHTMLPDF

Abstract

We propose to generate an artificial non-Abelian U(3) gauge field by using a two-tripod scheme, namely, two tripod configurations sharing a common ground-state level and driven by resonant one-photon transitions. Using an appropriate combination of four Laguerre-Gauss and two Hermite-Gauss laser beams, we are able to produce a U(3) monopole and a U(3) spin-orbit coupling for both alkali-metal and alkaline-earth-metal atoms. This two-tripod scheme could open the way to the study of interacting spinor condensates subjected to U(3) monopoles.

DOI
10.1103/physreva.90.023601

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We propose to generate an artificial non-Abelian U(3) gauge field by using a two-tripod scheme, namely, two tripod configurations sharing a common ground-state level and driven by resonant one-photon transitions. Using an appropriate combination of four Laguerre-Gauss and two Hermite-Gauss laser beams, we are able to produce a U(3) monopole and a U(3) spin-orbit coupling for both alkali-metal and alkaline-earth-metal atoms. This two-tripod scheme could open the way to the study of interacting spinor condensates subjected to U(3) monopoles.
The adiabatic motion of ultra-cold, multi-level atoms in spatially varying laser fields creating a dark state can give rise to effective gauge potentials. We show that these potentials lead to effective magnetic fields if the light fields possess a relative orbital angular momentum. This allows to study quantum-Hall like effects in ultra-cold atomic gases in various geometries. If the atom-light interaction creates several degenerate adiabatic eigenstates, the associated gauge potentials are non-Abelian. A pair of such degenerate dark states emerges e.g. if laser fields couple three internal states of an atom to a fourth common one under pairwise two--photon-resonance conditions. For this so-called tripod scheme we derive general conditions for truly non-Abelian gauge potentials and discuss special examples. In particular we show that using orthogonal laser beams with orbital angular momentum an effective magnetic field can be generated that has a monopole component.
We consider the simulation of non-abelian gauge potentials in ultracold atom systems with atom-field interaction in the ΛΛΛ configuration where two internal states of an atom are coupled to a third common one with a detuning. We find the simulated non-abelian gauge potentials can have the same structures as those simulated in the tripod configuration if we parameterize Rabi frequencies properly, which means we can design spin-orbit coupling simulation schemes based on those proposed in the tripod configuration. We show the simulated spin-orbit coupling in the ΛΛΛ configuration can only be of a form similar to pxσyp_{x}σ_{y}px​σy​ even when the Rabi frequencies are not much smaller than the detuning.
We show that the adiabatic motion of ultracold, multilevel atoms in spatially varying laser fields can give rise to effective non-Abelian gauge fields if degenerate adiabatic eigenstates of the atom-laser interaction exist. A pair of such degenerate dark states emerges, e.g., if laser fields couple three internal states of an atom to a fourth common one under pairwise two-photon-resonance conditions. For this so-called tripod scheme we derive general conditions for truly non-Abelian gauge potentials and discuss special examples. In particular we show that using orthogonal laser beams with orbital angular momentum an effective magnetic field can be generated that has a monopole component.
We show that $N\ensuremath{-}1$ degenerate dark states can be generated by coupling NNN-fold degenerate ground states and a common excited state with NNN laser fields. Interferences between light waves with different frequencies can produce laser fields with time-dependent amplitudes, which can induce not only U(N)\mathrm{U}(N)U(N) non-Abelian vector fields but also the scalar ones for the adiabatic motion of atoms in such laser fields. As an example, a time-periodic gauge potential is produced by applying specific laser fields to a tripod system. Some features of the Landau levels and the ground-state phase diagram of a rotating Bose-Einstein condensate for a concrete gauge field are also discussed.
It is known that gauge potentials/spin-orbit couplings can be simulated for ultracold atoms through their interaction with laser fields. Tripod and Lambda [Formula: see text] are two typical configurations of atom–field interaction in the simulations. We find that there is a similarity between the two configurations if we parametrize Rabi frequencies properly. Based on the similarity, we can transform a gauge potential/spin-orbit coupling simulation scheme in the tripod configuration into a scheme in the [Formula: see text] configuration.
This article is a report of Projet bibliographique of M1 at École Normale Supérieure. In this article we reviewed the historical developments in artificial gauge fields and spin-orbit couplings in cold atom systems. We resorted to origins of literatures to trace the ideas of the developments. For pedagogical purposes, we tried to work out examples carefully and clearly, to verified the validity of various approximations and arguments in detail, and to give clear physical and mathematical pictures of the problems that we discussed. The first part of this article introduced the fundamental concepts of Berry phase and Jaynes-Cummings model. The second part reviewed two schemes to generate artificial gauge fields with N-pod scheme in cold atom systems. The first one is based on dressed-atom picture which provide a method to generate non-Abelian gauge fields with dark states. The second one is about rotating scheme which is achieved earlier historically. Non-Abelian gauge field inevitably leads to spin-orbit coupling. We reviewed some developments in achieve spin-orbital coupling theoretically and experimentally. The fourth part was devoted to recently developed idea of optical flux lattice that provides a possibility to reach the strongly correlated regime in cold atom systems. We developed a geometrical interpretation based on Cooper's theory. Some useful formulae and their proofs were listed in the Appendix.