Geometrical Rabi oscillations and Landau-Zener transitions in non-Abelian systems

Topological phases of matter became a new standard to classify quantum systems in many cases, yet key quantities like the quantum geometric tensor providing local information about topological properties are still experimentally hard to access. In non-Abelian systems this accessibility to geometric properties can be even more restrictive due to the degeneracy of the states. We propose universal protocols to determine quantum geometric properties in non-Abelian systems. First, we show that for a weak resonant driving of the local parameters the coherent Rabi oscillations are related to the quantum geometric tensor. Second, we derive that in a Landau-Zener-like transition the final probability of an avoided energy crossing is proportional to elements of the non-Abelian quantum geometric tensor. Our schemes suggest a way to prepare eigenstates of the quantum metric, a task that is difficult otherwise in a degenerate subspace.

Figure 1

Schematic of the interaction between the states of the two energy levels by a modulation of a single external parameter in a non-Abelian system in an arbitrary basis (a) and in the basis of the eigenstates of the QGT (b).

Figure 2

Protocol for preparing the system in the eigenstates of the QGT starting from an arbitrary state in the ground state. The states evolve for the time $T=n(\pi /{\mathrm{\Omega }}_1^s)=(m+1/2)(\pi /{\mathrm{\Omega }}_2^s)$ yielding an even Rabi cycle for the first pair and an odd Rabi cycle for the second pair due to the different Rabi frequencies ${\mathrm{\Omega }}_1^s\ne {\mathrm{\Omega }}_2^s$ for each pair.

Figure 3

Schematic for a Landau-Zener transition for a pair $|{\tilde{\psi }}_{\nu }^{\pm }\rangle$ for a single modulation of the external parameter (or similar $|{\overline{\psi }}_{\nu }^{\pm }\rangle$ for a two-parameter modulation). Starting from a state far away from the avoided crossing the effective two-state system is linearly driven through this crossing proportional to the QGT ${\left[{\tilde{Q}}_{jj}\right]}_{\nu \nu }$. The final probability $P_{\mathrm{LZ}}$ of remaining in the initial state is thus given by the eigenvalue of the QGT ${\left[{\tilde{Q}}_{jj}\right]}_{\nu \nu }$ (or ${[{\overline{Q}}_{jj}^{\pm }+{\overline{Q}}_{kk}^{\pm }+e^{\pm i\varphi }{\overline{Q}}_{jk}^{\pm }+e^{\mp i\varphi }{\overline{Q}}_{kj}^{\pm }]}_{\nu \nu }$ for the two-parameter modulation).