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Simulating Twistronics Without a Twist

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Simulating Twistronics Without a Twist PHYSICAL REVIEW LETTERS 125, 030504 (2020) Simulating Twistronics without a Twist Tymoteusz Salamon ,1 Alessio Celi ,2 Ravindra W. Chhajlany ,3 Ir´en´ee Fr´erot ,1,4 Maciej Lewenstein,1,5 Leticia Tarruell ,1 and Debraj Rakshit 1,4 1ICFO—Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 2Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain 3Faculty of Physics, Adam Mickiewicz University, 61614 Poznan, Poland 4Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany 5Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig Lluis Companys 23, ES-08010 Barcelona, Spain (Received 8 February 2020; accepted 22 June 2020; published 14 July 2020) Rotational misalignment or twisting of two monolayers of graphene strongly influences its electronic properties. Structurally, twisting leads to large periodic supercell structures, which in turn can support intriguing strongly correlated behavior. Here, we propose a highly tunable scheme to synthetically emulate twisted bilayer systems with ultracold atoms trapped in an optical lattice. In our scheme, neither a physical bilayer nor twist is directly realized. Instead, two synthetic layers are produced exploiting coherently coupled internal atomic states, and a supercell structure is generated via a spatially dependent Raman coupling. To illustrate this concept, we focus on a synthetic square bilayer lattice and show that it leads to tunable quasiflatbands and Dirac cone spectra under certain magic supercell periodicities. The appearance of these features are explained using a perturbative analysis. Our proposal can be implemented using available state-of-the-art experimental techniques, and opens the route toward the controlled study of strongly correlated flatband accompanied by hybridization physics akin to magic angle bilayer graphene in cold atom quantum simulators. DOI: 10.1103/PhysRevLett.125.030504 Novel routes to band engineering in lattice systems often [22–26]. The geometrical moir´e patterns physically lead to fundamental questions and new material function- induce spatially varying interlayer couplings that are alities. Different schemes of stacking two-dimensional behind the strong modification of the band structure. layers have emerged as a fruitful way of modifying material As in artificial graphene systems [27], emulating this properties through the design of supercell structures and physics beyond materials research may allow identifying opened the field of so-called Van der Waals materials [1].In key minimal ingredients that give rise to the phenom- particular, twisted bilayer graphene (TWBLG) has emerged enology of TWBLG while also providing additional as a tunable experimental platform hosting flatband physics microscopic control. Photonic systems are particularly and strongly correlated phenomena, such as possibly suited for exploring this physics at the single-particle unconventional superconductivity, magnetism, and other level. Very recently, single-particle transport in tunable exotic phases [2–5]. This has inspired much theoretical photonic moir´e lattices has been experimentally studied debate around the origin of the electronic properties of [28], where two dimensional localization of light and TWBLG [6–21]. localization-delocalization have been experimentally The interesting correlated phenomenology is apparently demonstrated. Ultracold atoms in optical lattices related to moir´e patterns around small twist angles, the so- [29,30] are also the most promising platform for exper- called magic angles, which lead to band flattening or imentally exploring the corresponding emerging many- effective mass reduction already at the single-particle level body phenomena. The experimental realization of artifi- cial graphene geometries [31,32], lattice geometries dis- playing flatbands like Kagome [33] and Lieb [34,35],or quasicrystal structures [36–38], provides the building Published by the American Physical Society under the terms of blocks for such exploration. the Creative Commons Attribution 4.0 International license. One obvious approach for studying twisted bilayer Further distribution of this work must maintain attribution to graphene physics with ultracold atoms is to directly the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck implement twisted bilayers using two intertwined optical Society. lattices, as recently proposed in Ref. [39]. Schemes for 0031-9007=20=125(3)=030504(7) 030504-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 125, 030504 (2020) simulating other bilayer heterostructures have also been (a) put forward [40]. This direct strategy poses significant experimental challenges, as it is difficult to stabilize the two layers at relative small angles and simultaneously achieve a sufficiently large lattice containing several supercells. Here, we propose an alternative scheme that builds on the concept of synthetic dimensions, i.e., reinterpreting the coherent Raman coupling between spin states of an atom as controllable tunneling along (b) an artificial extra dimension [41–43]. For this, we note that the key effect of physical twisting is to induce spatially modulated interlayer tunnelings across the lattice. In our scheme, the modulated interlayer tunneling patterns are proposed to be directly imprinted on the lattice via spatial control of the Raman couplings, thus, amounting to twisting the system without a physical twist. This leads to the creation of supercells with (c) controllable sizes and shapes. By adjusting the strength, phase, and spatial periodicity of Ω, we show that an exemplary bilayer square lattice system supports a broad range of band structures. In particular, magic values of the periodicity result in the appearance of (quasi) flat- bands as well as Dirac cone spectra. Although we focus in our illustrative example on a particular spatial modu- lation, general interlayer coupling patterns can be exper- imentally induced including quasiperiodic or moir´e-like patterns. Our proposal can be realized with fermionic two-electron atoms, such as strontium or ytterbium, using available experimental techniques. Concept.—We consider a two-dimensional Fermi gas with four internal states, indicated here generically as spin states fm; σg¼1=2. The system is loaded into a spin- d FIG. 1. Synthetic bilayer structure with a supercell. (a) Real- independent square optical lattice of lattice spacing , space potential of the synthetic bilayer. Each plane corresponds to x y which lies in the - plane and is characterized by a real one spin state m ¼1=2 (orange, green), which experiences a tunneling amplitude t. We select two states to play the role square lattice potential (tunneling t) and is connected to the other of the electron spin σ ¼ ↑, and the other two of electron layer by a spatially dependent and complex coupling Ωðx; yÞ spin σ ¼ ↓. In addition, spin states corresponding to (vertical red lines of variable width). A top view of the lattice the same σ are coupled in pairs. We label them by the indicating the unit cell of the system containing 2 × 8 sites for l l 4d index m, and make them play the role of a synthetic x ¼ y ¼ is shown (black line). (b) Sketch of the first layer dimension. Since m ¼1=2, we obtain a bilayer Brillouin zone, indicating the position of the high-symmetry structure of synthetic layer tunneling given by the coher- points, and three dimensional view of the energy spectrum in the vicinity of E −Ω0 1 − α for Ω0α=h 20t with α 0.2 and ent coupling. In order to obtain a lattice geometry with a ¼ ð Þ ¼ ¼ γ ¼ 0. It has two quasiflatbands intersecting a Dirac point. Note tunable supercell, we choose the amplitude of the syn- that a simple square lattice supports neither flatbands nor Dirac thetic layer tunneling to be spatially modulated according cones. (c) Proposed experimental realization. Top: Two retrore- to Ωðx; yÞ¼Ω0f1 − α½1 þ cos ð2πx=lxÞ cos ð2πy=lyÞg. flected optical lattice beams (green) create the square lattice. Two Here, lx (ly) is its periodicity along the x (y) Raman beams of opening angle θ (red) produce complex axis. The synthetic tunneling also induces a Peierls synthetic tunneling between the two layers. One “modulation phase γ · r, where γ ¼ γðxˆ þ yˆÞ and r ¼ xxˆ þ yyˆ.This laser” with a spatially varying intensity distribution (blue) mimics the effect of a magnetic flux that pierces the modulates the amplitude of the Raman coupling. Bottom: laser system perpendicularly to the synthetic layer beams involved in the synthetic bilayer coupling scheme. The dimension [42]. As depicted in Fig. 1(a),thecomplete single-photon detuning of the Raman beams (red arrows) is spatially modulated with respect to its initial value Δ0=2π ∼ scheme represents a synthetic spinful bilayer structure 75 MHz using a laser beam blue detuned with respect to the Θ lx;ly 3 3 subjected to a magnetic field, denoted as ð Þ. P1 → S1 transition (blue arrow). It produces a light shift of The Hamiltonian of the system is given by maximal amplitude 2δ=2π ∼ 30 MHz. 030504-2 PHYSICAL REVIEW LETTERS 125, 030504 (2020) X H H H −t a† r dxˆ a† r dyˆ Ω r −iγ r a† r a r : :; ¼ in þ inter ¼ ½ m;σð þ Þþ m;σð þ Þþ ð Þ expð · Þ mþ1;σð Þ m;σð ÞþH c ð1Þ r;m;σ where we distinguish the in-layer and the interlayer (θ ¼ 0°), which yields γ ¼ 0.8 (mod 2π). However, other tunnelings. magnetic fluxes can be easily realized by adjusting the To diagonalize it, we combine a gauge transformation value of θ. Pand a Fourier transform such that am;σðrÞ¼ To implement a periodic modulation of the Raman q exp ½iðq · r þ mγ · rÞam;σðqÞ. Here, q is the momen- coupling amplitude on the scale of several lattice sites, tum conjugatedP to r. The Hamiltonian can then be rewritten which is the key ingredient of our scheme, we propose to as H ¼ q Hq, where the dimension of Hq is set by the exploit a periodic potential created by a laser close-detuned from the excited state to excited state spatial periodicity of the synthetic tunneling.
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