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arXiv:1306.2606 · Physical Review Letters

Magnetically Generated Spin-Orbit Coupling for Ultracold Atoms

Brandon Anderson, Ian B. Spielman, Gediminas Juzeliūnas

Published Sep 18, 2013• 192 citations
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Abstract

We present a new technique for producing two- and three-dimensional Rashba-type spin-orbit couplings for ultracold atoms without involving light. The method relies on a sequence of pulsed inhomogeneous magnetic fields imprinting suitable phase gradients on the atoms. For sufficiently short pulse durations, the time-averaged Hamiltonian well approximates the Rashba Hamiltonian. Higher order corrections to the energy spectrum are calculated exactly for spin-1/2 and perturbatively for higher spins. The pulse sequence does not modify the form of rotationally symmetric atom-atom interactions. Finally, we present a straightforward implementation of this pulse sequence on an atom chip.

DOI
10.1103/physrevlett.111.125301

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We present a new technique for producing two- and three-dimensional Rashba-type spin-orbit couplings for ultracold atoms without involving light. The method relies on a sequence of pulsed inhomogeneous magnetic fields imprinting suitable phase gradients on the atoms. For sufficiently short pulse durations, the time-averaged Hamiltonian well approximates the Rashba Hamiltonian. Higher order corrections to the energy spectrum are calculated exactly for spin-1/2 and perturbatively for higher spins. The pulse sequence does not modify the form of rotationally symmetric atom-atom interactions. Finally, we present a straightforward implementation of this pulse sequence on an atom chip.
We discuss a general scheme for creating atomic spin-orbit coupling (SOC) such as the Rashba or Dresselhaus types using magnetic-field-gradient pulses. In contrast to conventional schemes based on adiabatic center-of-mass motion with atomic internal states restricted to a dressed-state subspace, our scheme works for the complete subspace of a hyperfine-spin manifold by utilizing the coupling between the atomic magnetic moment and external magnetic fields. A space-dependent pulsed magnetic field acts as an internal-state-dependent impulse, thereby coupling the atomic internal spin with its orbital center-of-mass motion, as in the Einstein--de Haas effect. This effective coupling can be dynamically manipulated to synthesize SOC of any type (Rashba, Dresselhaus, or any linear combination thereof). Our scheme can be realized with most experimental setups of ultracold atoms and is especially suited for atoms with zero nuclear spins.
The spin-orbit coupling (SOC) affecting the center of mass of ultracold atoms can be simulated using a properly chosen periodic sequence of magnetic pulses. Yet such a method is generally accompanied by micromotion, which hinders precise control of atomic dynamics, thus complicating practical applications. Here we show how to bypass the micromotion emerging in the magnetically induced SOC by properly switching on and off the oscillating magnetic fields at the initial and final times. We consider the exact dynamics of the system and demonstrate that the overall dynamics can be immune to the micromotion. The exact dynamics is shown to agree well with the evolution of the system described by the slowly changing effective Floquet Hamiltonian including the SOC term. The agreement is shown to be best when the phase of the periodic driving takes a specific value for which the effect of the spin-orbit coupling is maximum.
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.
Spin-orbit coupling (SOC) can give rise to interesting physics, from spin Hall to topological insulators, in condensed matter systems. Recently, this topical area has extended into atomic quantum gases in searching for artificial or synthetic gauge potentials. The prospects of tunable interaction and quantum state control promote neutral atoms as nature's quantum emulators for SOC. Lin et al. recently demonstrated a special form of the SOC ${k}_{x}{\ensuremath{\sigma}}_{y}$, which they interpret as an equal superposition of Rashba and Dresselhaus couplings in Bose-condensed atoms [Lin, Jim\'enez-Garc\'{\i}a, and Spielman, Nature (London) 471, 83 (2011)]. This work reports a method of implementing arbitrary forms of SOC by switching between two pairs of Raman laser pulses like the method used by Lin et al. While one pair affects ${k}_{x}{\ensuremath{\sigma}}_{y}$ for some time, a second pair creates ${k}_{y}{\ensuremath{\sigma}}_{y}$ at other times with Raman pulses from different directions and a subsequent spin rotation into $\ifmmode\pm\else\textpm\fi{}{k}_{y}{\ensuremath{\sigma}}_{x}$. With a sufficient number of pulses, the effective actions from different durations are small and accumulate in the same exponent despite the fact that ${k}_{x}{\ensuremath{\sigma}}_{y}$ and $\ifmmode\pm\else\textpm\fi{}{k}_{y}{\ensuremath{\sigma}}_{x}$ do not commute. Our scheme involves no added complication and can be demonstrated within current experiments. It applies equally to bosonic or fermionic atoms.
We propose a hierarchy set of minimal optical Raman lattice schemes to pave the way for experimental realization of high-dimensional spin-orbit (SO) couplings for ultracold atoms, including two-dimensional (2D) Dirac type, 2D Rashba type, and three-dimensional (3D) Weyl type. The proposed Dirac-type SO coupling exhibits precisely controllable high symmetry, for which a large topological phase region is predicted. The generation of 2D Rashba and 3D Weyl types requires that two sources of laser beams have distinct frequencies of factor 2 difference. Surprisingly, we find that 133Cs^{133}\mathrm{Cs}133Cs atoms provide an ideal candidate for the realization. A common and essential feature is of high controllability and absent of any fine-tuning in the realization, and the resulting SO coupled ultracold atoms have a long lifetime. In particular, a long-lived topological Bose gas of 2D Dirac SO coupling has been proved in the follow-up experiment. These schemes essentially improve over the current experimental accessibility and controllability, and open a realistic way to explore novel high-dimensional SO physics, particularly quantum many-body physics and quantum far-from-equilibrium dynamics with novel topology for ultracold atoms.
We describe a new class of atom-laser coupling schemes which lead to spin-orbit-coupled Hamiltonians for ultracold neutral atoms. By properly setting the optical phases, a pair of degenerate pseudospin (a linear combination of internal atomic) states emerge as the lowest-energy eigenstates in the spectrum and are thus immune to collisionally induced decay. These schemes use NNN cyclically coupled ground or metastable internal states. We focus on two situations: a three-level case and a four-level case, where the latter adds a controllable Dresselhaus contribution. We describe an implementation of the four-level scheme for 87Rb{}^{87}\text{Rb}87Rb and analyze its sensitivity to typical laboratory noise sources. Last, we argue that the Rashba Hamiltonian applies only in the large intensity limit since any laser coupling scheme will produce terms nonlinear in momentum that decline with intensity.