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Modern Physics Letters B

Tripod and Lambda configurations in simulation of gauge potentials for ultracold atoms

Ming-Yong Ye, Xiu‐Min Lin

Published Mar 7, 2016• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0217984916500585

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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 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 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 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.
Standard optical potentials use an off-resonant laser standing-wave-induced ac Stark shift. In a recent development [\L{}\k{a}cki et al., Phys. Rev. Lett. 117, 233001 (2016)] a three-level scheme in a $\mathrm{\ensuremath{\Lambda}}$ configuration coupled coherently by resonant laser fields was introduced, leading to an effective lattice with subwavelength potential peaks. Here, as an extension of that work, a four-level atomic setup in the tripod configuration is used to create spin-12\frac{1}{2}21​-like two-dimensional dark space with one-dimensional motion and the presence of external gauge fields. Most interestingly for possible applications, the lifetime for a dark subspace motion is up to two orders of magnitude longer than for a similar $\mathrm{\ensuremath{\Lambda}}$ system. The model is quite flexible, leading to lattices with significant nearest, next-nearest, or next-to-next-nearest hopping amplitudes J1,J2{J}_{1}, {J}_{2}J1​,J2​, and J3{J}_{3}J3​ offering intriguing possibilities to study, e.g., frustrated systems. The characteristic Wannier functions lead also to a type of intersite interaction not realizable in typical optical lattices.
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 study laser induced spin-orbit (SO) coupling in cold atom systems where lasers couple three internal states to a pair of excited states, in a double tripod topology. Proper choice of laser amplitudes and phases produces a Hamiltonian with a doubly degenerate ground state separated from the remaining "excited" eigenstates by gaps determined by the Rabi frequencies of the atom-light coupling. After eliminating the excited states with a Born- Oppenheimer approximation, the Hamiltonian of the remaining two states includes Dresselhaus (or equivalently Rashba) SO coupling. Unlike earlier proposals, here the SO coupled states are the two lowest energy "dressed" spin states and are thus immune to collisional relaxation. Finally, we discuss a specific implementation of our system using Raman transitions between different hyperfine states within the electronic ground state manifold of nuclear spin I = 3/2 alkali atoms.