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arXiv:2106.04709 · Physical Review A

Optical lattice for a tripodlike atomic level structure

Piotr Kubala, Jakub Zakrzewski, Mateusz Łącki

Published Nov 10, 2021• 7 citations
arXivHTMLPDF

Abstract

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-$\frac{1}{2}$-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 ${J}_{1}, {J}_{2}$, and ${J}_{3}$ 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.

DOI
10.1103/physreva.104.053312

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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 \ensuremathΛ\mathrm{\ensuremath{\Lambda}}\ensuremathΛ 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 \ensuremathΛ\mathrm{\ensuremath{\Lambda}}\ensuremathΛ 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.
We analyze a tripod atom light coupling scheme characterized by two dark states playing the role of quasi-spin states. It is demonstrated that by properly configuring the coupling laser fields, one can create a lattice with spin-dependent sub-wavelength barriers. This allows to flexibly alter the atomic motion ranging from atomic dynamics in the effective brick-wall type lattice to free motion of atoms in one dark state and a tight binding lattice with a twice smaller periodicity for atoms in the other dark state. Between the two regimes, the spectrum undergoes significant changes controlled by the laser fields. The tripod lattice can be produced using current experimental techniques. The use of the tripod scheme to create a lattice of degenerate dark states opens new possibilities for spin ordering and symmetry breaking.
The electric field intensity of a standing-wave laser field is periodic in space. Due to the ac Stark effect, this gives rise to a spatially periodic potential acting on an atom, as explained in Chapter 4 (see, e.g., Eq. (4.31)). This is the physical principle behind the generation of optical lattices. By superimposing a number of different laser beams it is possible to generate potentials which are periodic in one, two or three dimensions. The suggestion that standing light waves may be used to confine the motion of atoms dates back to 1968 and is due to Letokhov. The first experimental realization of an optical lattice was achieved in 1987 for a classical gas of cesium atoms. The study of atoms in such potentials has many different facets. At the simplest level, it is possible to study the energy band structure of atoms moving in these potentials and to explore experimentally a number of effects that are difficult to observe for electrons in the periodic lattice of a solid. Interactions between atoms introduce qualitatively new effects. Within mean-field theory, which applies when the number of atoms in the vicinity of a single minimum of the potential is sufficiently large, one finds that interatomic interactions give rise to novel features in the band structure. These include multivaluedness of the energy for a given band and states possessing a periodicity different from that of the optical lattice.
Optical lattices are typically created via the ac Stark shift and are limited by diffraction to periodicities $\ensuremath{\ge}\ensuremath{\lambda}/2$, where $\ensuremath{\lambda}$ is the wavelength of light used to create them. Lattices with smaller periodicities may be useful for many-body physics with cold atoms and can be generated by stroboscopic application of a phase-shifted lattice with subwavelength features. Here we demonstrate a $\ensuremath{\lambda}/4$-spaced lattice by stroboscopically applying optical Kronig-Penney-like potentials which are generated using spatially dependent dark states. We directly probe the periodicity of the $\ensuremath{\lambda}/4$-spaced lattice by measuring the average probability density of the atoms loaded into the ground band of the lattice. We measure lifetimes of atoms in this lattice and discuss the mechanisms that limit the applicability of this stroboscopic approach.
We report on the experimental realization of a conservative optical lattice for cold atoms with a subwavelength spatial structure. The potential is based on the nonlinear optical response of three-level atoms in laser-dressed dark states, which is not constrained by the diffraction limit of the light generating the potential. The lattice consists of a one-dimensional array of ultranarrow barriers with widths less than 10 nm, well below the wavelength of the lattice light, physically realizing a Kronig-Penney potential. We study the band structure and dissipation of this lattice and find good agreement with theoretical predictions. Even on resonance, the observed lifetimes of atoms trapped in the lattice are as long as 44 ms, nearly 10^{5} times the excited state lifetime, and could be further improved with more laser intensity. The potential is readily generalizable to higher dimensions and different geometries, allowing, for example, nearly perfect box traps, narrow tunnel junctions for atomtronics applications, and dynamically generated lattices with subwavelength spacings.
We propose a scheme to implement an optical kagome lattice for ultracold atoms with controllable s-wave interactions between nearest neighbor sites and a gauge potential. The atoms occupy three different internal atomic levels with electromagnetically induced coupling between the levels. We show that by appropriately shifting the triangular lattice potentials, experienced by atoms in different levels, the kagome lattice can be realized using only two standing waves, generating a highly frustrated quantum system for the atoms.
Summary form only given. It has been predicted that topological potentials play a significant role in gray optical lattices. However, so far there has been no direct experimental demonstration of their existence. Unlike light-shift potentials, which, in the case of low saturation, scale as the laser intensity, topological (or gauge) potentials are intensity independent and solely depend on the topology of the lattice, which is determined by the detuning, beam geometry and magnetic fields. In our experiment, /sup 87/Rb atoms are collected in a standard magneto-optic trap and loaded into a 1D optical lattice. The lattice is formed by counterpropagating, orthogonally polarized laser beams. The minima of the trapping potential form at the antinodes of the standing waves and, as a result of optical pumping, the wells are alternately occupied. When an atom tunnels from one type of well to another, it must then flip its magnetic spin state. The gauge potential is associated with the energy needed for this change. In this gray lattice, the potential wells are shallow due to the weak interaction with the light field. Shallow potentials are necessary for tunneling. We use two schemes to observe tunneling. In the first, we use an auxiliary weak laser pulse to selectively depopulate one of the two types of wells. We observe tunneling by measuring the population in each type of well as a function of time. The second way we study tunneling is to create atomic wave-packets within the wells and to observe their motion as they oscillate. The wave-packets are created by diabatically displacing the trapped atoms in either position or momentum space. Tunneling occurs when the wave-packets run against barriers provided by the maxima of the trapping potential.