Measuring the Second Chern Number from Nonadiabatic Effects

The geometry and topology of quantum systems have deep connections to quantum dynamics. In this Letter, I show how to measure the non-Abelian Berry curvature and its related topological invariant, the second Chern number, using dynamical techniques. The second Chern number is the defining topological characteristic of the four-dimensional generalization of the quantum Hall effect and has relevance in systems from three-dimensional topological insulators to Yang-Mills field theory. I illustrate its measurement using the simple example of a spin-$3/2$ particle in an electric quadrupole field. I show how one can dynamically measure diagonal components of the Berry curvature in an overcomplete basis of the degenerate ground state space and use this to extract the full non-Abelian Berry curvature. I also show that one can accomplish the same ideas by stochastically averaging over random initial states in the degenerate ground state manifold. Finally, I show how this system can be manufactured and the topological invariant measured in a variety of realistic systems, from superconducting qubits to trapped ions and cold atoms.

Figure 1

Measuring second Chern number dynamically. (a) General setup where measurement is possible. $N$ degenerate ground states are separated by a nonzero gap from the excited states. The ground states must remain degenerate and gapped from the excited states, but with no restrictions on the excited states. (b) Illustration of the ramping protocol to find one component $F_{\mu \nu }^{jj}$ of the non-Abelian Berry curvature at measurement point $\mathit{\varphi }$.

Figure 2

Experimental realizations of four-level systems where the second Chern number may be measured: (a) atomic hyperfine levels, (b) bound states of an artificial atom, (c) a particle hopping in a 4-site lattice. While all sites (levels) must be coupled to realize arbitrary $4\times 4$ Hamiltonians, only the indicated drives (hoppings) are necessary in the presence of symmetry about the ${\lambda }_0$ axis. (d) Second Chern number measured dynamically for a spin-$3/2$ in an electric quadrupole field, as described in the text, either without utilizing symmetry (blue points) or with symmetry (red points). The deviation from quantization near the transition at ${\mathrm{\Lambda }}_0=1$ is due to using finite time protocols; I ramp from the north pole to the measurement point in time $t=100$ then ramp for measurement with $v=0.01$ and $t_m=100$. The inset shows how as ${\mathrm{\Lambda }}_0{H}_0$ is added, the 4-sphere shifts away from the origin until eventually the (fourfold) degeneracy at $\mathit{\lambda }=\mathbf{0}$ is no longer enclosed.