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arXiv:1207.5369

Synthetic spin-orbit coupling in ultracold ΛΛΛ-type atoms

Ming-Yong Ye, Xiu‐Min Lin

Published Jul 23, 2012• 0 citations
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Abstract

We consider the simulation of non-abelian gauge potentials in ultracold atom systems with atom-field interaction in the $Λ$ 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 $Λ$ configuration can only be of a form similar to $p_{x}σ_{y}$ even when the Rabi frequencies are not much smaller than the detuning.

DOI
10.48550/arxiv.1207.5369

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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.
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.
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 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.
We show that the dipole–dipole interaction between two Rydberg atoms can lead to substantial Abelian and non-Abelian gauge fields acting on the relative motion of the two atoms. We demonstrate how the gauge fields can be evaluated by numerical techniques. In the case of adiabatic motion in a single internal state, we show that the gauge fields give rise to a magnetic field that results in a Zeeman splitting of the rotational states. In particular, the ground state of a molecular potential well is given by the first excited rotational state. We find that our system realizes a synthetic spin–orbit coupling where the relative atomic motion couples to two internal two-atom states. The associated gauge fields are non-Abelian.
Motivated by the recent experimental success in realizing synthetic spin-orbit coupling in ultracold atomic systems, we consider N-component atoms coupled to a non-Abelian SU(N) gauge field. More specifically, we focus on the case, referred to here as "SU(3) spin-orbit-coupling," where the internal states of three-component atoms are coupled to their momenta via a matrix structure that involves the Gell-Mann matrices (in contrast to the Pauli matrices in conventional SU(2) spin-orbit-coupled systems). It is shown that the SU(3) spin-orbit-coupling gives rise to qualitatively different phenomena and in particular we find that even a homogeneous SU(3) field on a simple square lattice enables a topologically nontrivial state to exist, while such SU(2) systems always have trivial topology. In deriving this result, we first establish an equivalence between the Hofstadter model with a 1/N Abelian flux per plaquette and a homogeneous SU(N) non-Abelian model. The former is known to have a topological spectrum for N>2, which is thus inherited by the latter. It is explicitly verified by an exact calculation for N=3, where we develop and use a new algebraic method to calculate topological indices in the SU(3) case. Finally, we consider a strip geometry and establish the existence of three gapless edge states-the hallmark feature of such an SU(3) topological insulator.
We demonstrate the emergence of the non-Abelian geometric potentials and thus the three-dimensional (3D) spin-orbit coupling (SOC) for ultracold atoms without using the laser beams. This is achieved by subjecting an atom to a periodic perturbation which is the product of a position-dependent Hermitian operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}\left(\mathbf{r}\right)$ and a fast oscillating periodic function $f\left(\ensuremath{\omega}t\right)$ with a zero average. To have a significant spin-orbit coupling, we analyze a situation where the characteristic energy of the periodic driving is not necessarily small compared to the driving energy $\ensuremath{\hbar}\ensuremath{\omega}$. Applying a unitary transformation to eliminate the original periodic perturbation, we arrive at a non-Abelian (noncommuting) vector potential term describing the 3D SOC. The general formalism is illustrated by analyzing the motion of an atom in a spatially inhomogeneous magnetic field oscillating in time. A cylindrically symmetric magnetic field provides the SOC involving the coupling between the spin F\mathbf{F}F and all three components of the orbital angular momentum (OAM) L\mathbf{L}L. In particular, the spherically symmetric monopole-type synthetic magnetic field $\mathbf{B}\ensuremath{\propto}\mathbf{r}$ generates the 3D SOC of the $\mathbf{L}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{F}$ form, which resembles the fine-structure interaction of hydrogen atom. However, the strength of the SOC here goes as 1/r21/{r}^{2}1/r2 for larger distances, instead of 1/r31/{r}^{3}1/r3 as in atomic fine structure. Such a longer-ranged SOC significantly affects not only the lower states of the trapped atom, but also the higher ones. Furthermore, by properly tailoring the external trapping potential, the ground state of the system can occur at finite OAM, while the ground state of hydrogen atom has zero OAM.