Rabi Oscillations in Landau-Quantized Graphene

We investigate the relation between the canonical model of quantum optics, the Jaynes-Cummings Hamiltonian and Dirac fermions in quantizing magnetic field. We demonstrate that Rabi oscillations are observable in the optical response of graphene, providing us with a transparent picture about the structure of optical transitions. While the longitudinal conductivity reveals chaotic Rabi oscillations, the Hall component measures coherent ones. This opens up the possibility of investigating a microscopic model of a few quantum objects in a macroscopic experiment with tunable parameters.

Figure 1

A small fragment of the honeycomb lattice is shown, the filled red and empty black circles denote the two sublattices.

Figure 2

The real-time evolution of $C_{xx}(t)$ is shown at $T=0$, taking both valley and spin degeneracies into account. We introduced a cutoff $D$, and the number of levels is measured as $D=V\sqrt{N}$, corresponding to different magnetic field strengths. The left/right panels show $N=10000$ (blue)/$N=100$ (red) with $D=V\sqrt{N}$ for $(\mu ,\Delta )/V\sqrt{N}=(0,0)$ (a),(e), (0.4,0) (b),(f), (0.4,0.2) (c),(g), and (0,0.2) (d),(h). These structures correspond to thermal field induced random oscillation in quantum optics [26].

Figure 3

The real-time evolution of $C_{xx}(t)$ is shown for long times for $T=0$, $(\mu ,\Delta )=(0,0)$. We introduced a cutoff $D$, and the number of levels is measured as $D=V\sqrt{N}$, corresponding to different magnetic field strengths with $N=10000$ (blue) and $N=100$ (red). Collapse and revival shows up with time similarly to the thermal field Jaynes-Cummings model. The revivals gradually get wider and overlap. The presence of thermal revivals are related to the finite average boson number in the Jaynes-Cummings model, which translate to a finite cutoff in the Dirac case. As opposed to Fig. 2, these revivals at long times are caused by the finite cutoff.

Figure 4

The real-time evolution of $C_{xy}(t)$ is shown for $T=0$. The explicit value of the number of levels ($N$) does not influence the resulting pattern. The chemical potential varied as $\mu /V=1.2$ (a),(d), 3.2 (b),(e) and 10 (c),(f) with $\Delta =0$ (left panel, blue) and $\Delta =2V$ (right panel, red), and the frequency of the envelope function is $v_F^{2}eB/\mu$, the cyclotron frequency of massless Dirac fermions. For $T=0$ and $\Delta >\mu$, $\mathrm{Im}{C}_{xy}(t)=0$. Note the different horizontal scales.