Detecting Chiral Edge States in the Hofstadter Optical Lattice

Abstract

We propose a realistic scheme to detect topological edge states in an optical lattice subjected to a synthetic magnetic field, based on a generalization of Bragg spectroscopy sensitive to angular momentum. We demonstrate that using a well-designed laser probe, the Bragg spectra provide an unambiguous signature of the topological edge states that establishes their chiral nature. This signature is present for a variety of boundaries, from a hard wall to a smooth harmonic potential added on top of the optical lattice. Experimentally, the Bragg signal should be very weak. To make it detectable, we introduce a "shelving method", based on Raman transitions, which transfers angular momentum and changes the internal atomic state simultaneously. This scheme allows to detect the weak signal from the selected edge states on a dark background, and drastically improves the detectivity. It also leads to the possibility to directly visualize the topological edge states, using in situ imaging, offering a unique and instructive view on topological insulating phases. [2] have been realized for ultra cold atoms using suitably arranged lasers that couple different internal states [3]. This opens the path to the experimental investigation of topological phases, such as quantum Hall (QH) states, topological insulators and superconductors, in a clean and highly controllable environment [4,5]. Topological phases currently attract the attention of the scientific community for their remarkable properties, such as dissipationless transport and quantized conductivities [6,7]. In this context, the recent experimental realization of a staggered magnetic field in a 2D optical lattice, exploiting laser-induced gauge potentials, constitutes an important step in the field [8] (cf. also [9,10]). In the near future, large uniform magnetic flux should be reachable using related proposals [9][10][11][12], allowing optical-lattice experiments to explore the Hofstadter model [13]. The latter is the simplest tight-binding lattice model exhibiting topological transport properties: well-separated energy bands are associated to non-trivial topological invariants, the Chern numbers, leading to a quantized Hall conductivity when the Fermi energy is located in the bulk gaps [14,15]. These transport properties are directly related to the existence of chiral edge states: while bulk excitations remain inert, these gapless states carry current along the edge of the system. According to the bulk-edge correspondence [16,17], the Chern numbers characterizing the bulk bands determine the number of edge excitations and their chirality, which protects the edge transport against small perturbations.In view of the experimental progress [8][9][10], an important issue is to identify observables that provide unambiguous signatures of topological phases in a cold-atom framework [18][19][20][21][22][23][24][25][26][27]. This is a crucial topic from an experimental point of view: Measuring the Hall conductivity is more difficult than for solid-state systems due to the absenc...

Topological phase transitions driven by non-Abelian gauge potentials in optical square lattices

“…In order to avoid the influence of both the interference pattern, strongly dependent on the energy of the eigenstates, and the vertical edge modes, we averaged the density of the wave functions along the x direction in the central region of the lattice, far from the vertical edges. Analogously to the case of topological insulators of spinless fermions [31,32], by increasing γ the edge states become more and more localized. A similar behavior characterizes also the other phase where the presence of two different edge modes makes the edge density broader.…”

“…Some of them are based on the observation of the edge modes [31,32,39], which in our model change both in number and in direction of propagation at the transition points, providing direct evidence for the topological transition. Other viable techniques rely on the possibility of detecting the dynamic response of the bulk under an external force or on other bulk properties [33][34][35][36].…”

“…Therefore the techniques presented in Refs. [31,32] can be applied to distinguish the two topological phases by measuring their respective edge chiralities.…”

“…Conductance measurements are however challenging in atomic gases [30], and other tools can be adopted to detect the presence of edge modes [31,32]. Apart from edge mode detection, bulk properties can also be exploited to investigate topological phases.…”

Realizing Fractional Chern Insulators in Dipolar Spin Systems

“…For example, the spectral function can be measured at finite energy and momentum using two-photon Bragg spectroscopy, providing direct information regarding fractionalization [30,31]. On the edge, one should observe gapless chiral Luttinger liquid behavior, while in the bulk, the response should exhibit a gap to the multi-quasiparticle continuum.…”