Driven quantum tunneling and pair creation with graphene Landau levels

Driven tunneling between graphene Landau levels is theoretically linked to the process of pair creation from vacuum, a prediction of quantum electrodynamics (QED). Landau levels are created by the presence of a strong, constant, quantizing magnetic field perpendicular to a graphene monolayer. Following the formal analogy between QED and the description of low-energy excitations in graphene, solutions of the fully interacting Dirac equation are used to compute electron-hole pair creation driven by a circularly or linearly polarized field. This is achieved via the coupled channel method, a numerical scheme for the solution of the time-dependent Dirac equation in the presence of bound states. The case of a monochromatic driving field is first considered, followed by the more realistic case of a pulsed excitation. We show that the pulse duration yields an experimental control parameter over the maximal pair yield. Orders of magnitude of the pair yield are given for experimentally achievable magnetic fields and laser intensities weak enough to preserve the Landau level structure.

Figure 1

Landau levels (LLs) in graphene in the presence of a quantizing magnetic field. LL energies are ${\epsilon }_{\lambda n}\propto \lambda \sqrt{Bn}$, where $n$ is non-negative integer. In the case of neutral graphene, all holelike states are filled (solid lines), and the zero-energy LL is half-filled (dashed-dotted line). The Dirac cone describing the linear relationship between energy and momentum in the case of free fermions is shown for convenience. Throughout this paper, it will be assumed that an incident laser excitation drives the transition between LLs $-1$ and 2, as indicated by the white arrow.

Figure 2

Allowed transitions with frequency $\omega ={\omega }_c(\sqrt{2}+1)$ and their respective handedness. Transitions from LL indices $n\to n+1$ are associated with the absorption of RHP photons, and transitions from $n\to n-1$ with the absorption of LHP photons [31].

Figure 3

Time evolution of the population of LL 2 for ${\mathrm{\Omega }}_R=0.207{\omega }_c$, corresponding to $E=v_{F}B$. For comparison, a lower Rabi frequency $\left({\mathrm{\Omega }}_R=0.067{\omega }_c\right)$ is also shown. The monochromatic laser field is tuned to the transition between levels $-1\to 2$. The closed-form solution obtained from a two-level approximation with zero detuning $\left(\mathrm{\Delta }=0\right)$ is also shown.

Figure 4

(Top) Dependence of the normalized pair yield $\overline{N}$ on the Rabi frequency for a monochromatic driving field. The monochromatic laser field is tuned to the transition between levels $-1\to 2$. The lower bound of the vertical axis corresponds to $E=v_{F}B({\mathrm{\Omega }}_{\mathrm{R}}={\omega }_c^2/2\omega$). (Bottom) Same result excluding pairs produced in the resonant upper LL with energy $\epsilon /{\omega }_c=\sqrt{2}$.

Figure 5

Dependence of the pair yield on the driving pulse duration. The location of the maxima predicted by the two-level approximation, i.e., Eqs. (B7) and (C9), are indicated by dashed lines (odd integer values of ${\mathrm{\Omega }}_{\mathrm{R}}T/2\pi$).

Figure 6

(a) LL dynamics with a pulsed field with $T/T_0=36.33$, corresponding to the square marker on Fig. 5. This corresponds to a value which maximizes the population of LL 2 after the passage of the pulse, thus maximizing the pair yield. The markers indi-cate the multilevel numerical result obtained via CCM, and the blue curve indicates the approximate two-level solution. (b) Driving pulse profile.

Figure 7

(a) LL dynamics with a pulsed field with $T/T_0=24.0$, corresponding to the diamond marker on Fig. 5. This corresponds to a local minimum of the population of LL 2 after the passage of the pulse. The markers indicate the multilevel numerical result obtained via CCM, and the blue curve indicates the approximate two-level solution. (b) Driving pulse profile.

Figure 8

(Top) Dependence of the normalized pair yield $\overline{N}$ on the Rabi frequency for a linearly polarized monochromatic driving field. The monochromatic laser field drives the transition between levels $-1\to 2$ and the transition $-2\to 1$. The lower bound of the vertical axis corresponds to $E=v_{F}B({\mathrm{\Omega }}_{\mathrm{R}}={\omega }_c^2/2\omega$). (Bottom) Same result excluding pairs produced in the resonant upper LLs with energies $\epsilon /{\omega }_c=1$ and $\epsilon /{\omega }_c=\sqrt{2}$.

Figure 9

Dependence of the pair yield on the driving pulse duration (linear polarization). The location of the maxima predicted by the two-level approximation, i.e., Eqs. (B7) and (C9), are indicated by dashed lines (odd integer values of ${\mathrm{\Omega }}_{\mathrm{R}}T/2\pi$).