Non-Abelian geometric potentials and spin-orbit coupling for periodically driven systems

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 $\widehat{V}\left(\mathbf{r}\right)$ and a fast oscillating periodic function $f\left(\omega t\right)$ 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 $\hslash \omega$. 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 $\mathbf{F}$ and all three components of the orbital angular momentum (OAM) $\mathbf{L}$. In particular, the spherically symmetric monopole-type synthetic magnetic field $\mathbf{B}\propto \mathbf{r}$ generates the 3D SOC of the $\mathbf{L}·\mathbf{F}$ form, which resembles the fine-structure interaction of hydrogen atom. However, the strength of the SOC here goes as $1/r^2$ for larger distances, instead of $1/r^3$ 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.

Figure 1

Radial dependence of the SOC energy ${\omega }_{\mathrm{SOC}}\left(r\right)$ given by Eq. (35) for the sinusoidal driving (26) for which $〈cos\left(2r\mathcal{F}/r_0\right)〉={\mathcal{J}}_0\left(2r/r_0\right)$. The distance is measured in the units of $r_0$ and the frequency is measured in the units of the SOC frequency ${\omega }_0={\omega }_{\mathrm{SOC}}\left(0\right)$ given by Eq. (44).

Figure 2

Dependence of eigenenergies $E_{n_r,j,l}$ on $j$ for $l=j\mp 1/2$ and up to five lowest radial quantum numbers $n_r$. The harmonic trapping potential (43) is added. Panel (a) corresponds to a softer trap with $\eta =0.5$; panel (b) corresponds to a tighter trap with $\eta =2$. The spectrum is calculated for the sinusoidal (26) driving for which $〈cos\left(2r\mathcal{F}/r_0\right)〉={\mathcal{J}}_0\left(2r/r_0\right)$.

Figure 3

Energy difference $\mathrm{\Delta }E=E_{j=\frac{1}{2},l=1}-E_{j=\frac{1}{2},l=0}$ vs the radius $r_{*}$ of the additional hard core potential for an atom in a harmonic trapping potential (43) with $\eta =0.5$ [panel (a)] and $\eta =2$ [panel (b)]. The calculations are done for the sinusoidal driving (26) for which $〈cos\left(2r\mathcal{F}/r_0\right)〉={\mathcal{J}}_0\left(2r/r_0\right)$.