Extracting the Chern Number from the Dynamics of a Fermi Gas: Implementing a Quantum Hall Bar for Cold Atoms

Abstract

We propose a scheme to measure the quantized Hall conductivity of an ultracold Fermi gas initially prepared in a topological Chern insulating phase and driven by a constant force. We show that the time evolution of the center of mass, after releasing the cloud, provides a direct and clear signature of the topologically invariant Chern number. We discuss the validity of this scheme, highlighting the importance of driving the system with a sufficiently strong force to displace the cloud over measurable distances while avoiding band-mixing effects. The unusual shapes of the driven atomic cloud are qualitatively discussed in terms of a semiclassical approach.

An Aharonov-Bohm interferometer for determining Bloch band topology

“…A similar π flux also plays a crucial role in the nuclear dynamics of molecules featuring conical intersections of energy surfaces [2]. Our direct detection of the paradigmatic π flux demonstrates the capability to reveal even singular Berry flux fea- tures that are not observable by alternative techniques based on transport measurements [7,8,[17][18][19] and thereby paves the way to full topological characterization of optical lattice systems [18][19][20][21][22][23][24][25]. The effect of Berry curvature in our interferometer is analogous to the Aharonov-Bohm effect, where an electron wavepacket is split into two parts that encircle a given area arXiv:1407.5635v1 [cond-mat.quant-gas] 21 Jul 2014 The duration of the interferometer sequence is 2τ = 1.6 ms for all measurements.…”

“…The Berry flux density (Berry curvature) is indeed essential to the characterization of an energy band and determines its topological invariants. However, mapping out the geometric structure of an energy band [7][8][9] has remained a major unresolved challenge for experiments.Here, we demonstrate a versatile technique for measuring geometric phases in reciprocal space using spin-echo interferometry with ultracold atoms [9, 10]. In contrast to typical solid state experiments, where all geometric effects are averaged over the Fermi sea, the use of a Bose-Einstein condensate (BEC) enables measurements with high momentum resolution.…”

“…The Berry flux density (Berry curvature) is indeed essential to the characterization of an energy band and determines its topological invariants. However, mapping out the geometric structure of an energy band [7][8][9] has remained a major unresolved challenge for experiments.…”

Interferometric detection of Chern numbers of Haldane model on bosonic optical lattices

“…Therefore, nowadays, measuring the Hall conductance is technically unrealistic for cold atoms in optical lattices. On the other hand, alternative strategies for the detecting the CN include measuring the time evolution of the center of mass [22], the gapless edge modes [23][24][25], the pumped charge [26], the Landau-Zener-Stückelberg tunneling [27] and the bulk Chern number from Berry's curvature over the BZ [28][29][30]. However, these methods either rely on the weak experimental signals or depend on complicated manipulations/measurements on the whole bulk band.…”

Artificial magnetic fields in momentum space in spin-orbit-coupled systems

“…The Berry curvature is a geometrical property of an energy band which can be viewed as an artificial magnetic field in a general effective quantum Hamiltonian where the roles of momentum and position are reversed [1][2][3][4][5][6]. This has important physical consequences in the anomalous Hall effect [7][8][9], in the collective modes of an ultracold atomic gas [10,11] and in the semiclassical dynamics of a wave packet [12][13][14][15][16][17].…”