Seeing Topological Order in Time-of-Flight Measurements

Abstract

In this Letter, we provide a general methodology to directly measure topological order in cold atom systems. As an application, we propose the realization of a characteristic topological model, introduced by Haldane, using optical lattices loaded with fermionic atoms in two internal states. We demonstrate that time-of-flight measurements directly reveal the topological order of the system in the form of momentumspace Skyrmions. DOI: 10.1103/PhysRevLett.107.235301 PACS numbers: 67.85.Àd, 03.65.Vf Different phases of matter can be distinguished by their symmetries. This information is usually captured by locally measurable order parameters that summarize the essential properties of the phase. Topological insulators are materials with symmetries that depend on the topology of the energy eigenstates of the system [1]. These materials are of interest because they give rise to robust spin transport effects with potential applications ranging from sensitive detectors to quantum computation [2,3]. However, direct observation and measurement of topological order has been up to now impossible due to its nonlocal character. Instead, experiments have relied so far on indirect manifestations of this order, such as edge states and the quantization of conductivity.Ultracold atoms facilitate the implementation of artificial gauge fields [4]. Here, we distinguish proposals that generate continuous fields [5], such as the recent experiment by Lin et al. [6], from those that rely on optical lattices and engineering of hopping [7]. We will concentrate on the latter, introducing a method based on standard time-of-flight (TOF) measurements that can identify a topological character in the quantum state of the system. Our starting point is a possible implementation of Haldane's model using fermionic atoms in two internal states. The topological nature of its ground state is witnessed by the Chern number. This number counts the times the ground state, written as a spinor, wraps around the sphere, as a function of momentum. We demonstrate that TOF measurements reconstruct the Chern number in a way which is robust against the presence of external perturbations or state preparation. Our method can be adapted to other quantum simulations of topological order in optical lattices [8][9][10][11][12][13][14][15][16][17], as many already use internal degrees of freedom of the atoms to encode the order.One common mechanism for the appearance of topological order is based on the topology of the eigenstate manifolds. Consider a real-space lattice whose unit cell has d quantum degrees of freedom-position of the particle, spin, etc. In the case of the quantum Hall effect, the energy bands are separated from each other and the material becomes an insulator for appropriate Fermi energies, E F . In a real setup, with finite boundaries, the sample can have a quantized nonzero conductivity given by the topological invariant m xy ¼ e 2 =h P E m <E F m , which is a signature itself of topological order [18]. The transport is then supported by ''edge'' s...

Chiral superfluidity with p-wave symmetry from an interacting s-wave atomic Fermi gas

“…Besides detecting edge properties, the bulk Chern number can also be used to demonstrate the topological nature of our proposed chiral p-wave superfluidity. For the detection of Chern numbers, one may adopt the existing proposals either from time-of-flight measurement [40][41][42][43] or from Bloch oscillation technique 44,45 .…”

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

“…Such a control has been recently used in an experimental implementation of honeycomb lattices for ultracold fermions [6]. Several schemes have been discussed in the literature [7,8] with the aim of engineering topologically non-trivial twodimensional systems on honeycomb lattices, which naturally reproduce well-known paradigmatic models such as the Haldane model [9]. The discussed setups rely on the laser implementation of tight-binding models on honeycombs having both nearest-neighbor and next-nearestneighbor hoppings: these hoppings simulate the presence of a staggered magnetic field which breaks time-reversal symmetry and opens band gaps.…”

“…In particular, in Ref. [7] it has been shown that time of flight measurements, resolved with respect to the different spin species, allow us to reconstruct the spin polarization of the lowest band in the first Brillouin zone. From this polarization one can calculate a spin winding number which, for the Chern insulators related to the Haldane model, directly provides the Chern number of the occupied band, and thus the number of protected edge states [8].…”

“…Several techniques have been proposed so far to relate the Chern number of topological insulators in ultracold gases to time of flight images [7,8,38,39]. In particular, in Ref.…”

Valley-dependent gauge fields for ultracold atoms in square optical superlattices

“…For this purpose, a practical way is to modulate the atomic hopping parameters in a honeycomb OL by using the synthetic gauge poten- * Electronic address: slzhunju@163.com tials [14] or the laser-assisted tunneling (LAT) [15,16], following the schemes proposed in Refs. [17][18][19]. However, the LAT technique has not yet been demonstrated in honeycomb OLs, but in square optical (super) lattices [20][21][22].…”