Klein tunneling in driven-dissipative photonic graphene

We theoretically investigate Klein tunneling processes in photonic artificial graphene. Klein tunneling is a phenomenon in which a particle with Dirac dispersion going through a potential step shows a characteristic angle- and energy-dependent transmission. We consider a generic photonic system consisting of a honeycomb-shaped array of sites with losses, illuminated by coherent monochromatic light. We show how the transmission and reflection coefficients can be obtained from the steady-state field profile of the driven-dissipative system. Despite the presence of photonic losses, we recover the main scattering features predicted by the general theory of Klein tunneling. Signatures of negative refraction and the orientation dependence of the intervalley scattering are also highlighted. Our results will stimulate the experimental study of intricate transport phenomena using driven-dissipative photonic simulators.

Figure 1

Incident, reflected, and transmitted waves in a honeycomb lattice with a step. The thick vertical line represents the step edge; the lattice sites on the right of the thick vertical line have an additional potential energy $V$ compared to those on the left of the line. The direction of the full arrows indicates the direction of the momentum measured from a Dirac point, while the open ones indicate the group velocity. The parameters chosen in this figure correspond to a negative refraction case.

Figure 2

Schematic illustration of the Klein tunneling. The cones represent the Dirac dispersion before ($x<0$) and after ($x>0$) the step. (a) When the step height is smaller than the energy of the incident particle. The transmitted particle is in the upper band and therefore the transmission is particlelike. (b) When the step height is larger than the energy of the incident particle. The transmitted particle is in the lower band and therefore the transmission is holelike.

Figure 3

(a) First Brillouin zone of a honeycomb lattice denoted by a hexagon in momentum space, whose vertices are the Dirac points. The solid circles, ${\mathbf{K}}_1, {\mathbf{K}}_2$, and ${\mathbf{K}}_3$, are the Dirac points in the same valley which we mainly explore in this paper. The hollow circles are the Dirac points in the other valley. (b)–(d) The momentum space distribution of the steady-state emission of photons when the spatially tightly focused pump fields are concentrated around the different Dirac points ${\mathbf{K}}_1, {\mathbf{K}}_2$, and ${\mathbf{K}}_3$ and the emission is detected around the same respective Dirac points. The black circles are the iso-energy surfaces corresponding to the pump energy. The width of the pumping field is $\sigma =5a$.

Figure 4

(a) Transmittivity $T$ as a function of $k_y$. The line is a theoretical curve from (5). The dots (blue) are calculated from the simulation with the reflection coefficient ${\left|r\right|}^2$ estimated by summing the signals at $x<0$ around ${\mathbf{K}}_1, {\mathbf{K}}_2$, and ${\mathbf{K}}_3$, the brown squares are the transmittivity calculated by only using signals emitted around ${\mathbf{K}}_2$, and the red crosses are the simulation where the reflection coefficient ${\left|r\right|}^2$ is estimated from the signal at $x<0$ after subtracting the no-step distribution to isolate the reflection signal. (b)–(d) Cut of the momentum distribution for $k_y=-0.05/a$ as a function of $k_x$ around ${\mathbf{K}}_1, {\mathbf{K}}_2$, and ${\mathbf{K}}_3$, respectively. The vertical axis is dimensionless photon counts, and we use the same scale for (b)–(d). The solid (blue) lines are the spatial Fourier transform of the field in the $x<0$ region, and the dashed (green) lines are the spatial Fourier transform of the field in the $x>0$ region. The dotted (red) lines are the spatial Fourier transform of the $x<0$ region after subtracting the no-step distribution.

Figure 5

Momentum-space distribution of the steady-state emission of photons around Dirac points calculated for the reflected signal, where the incident signal is subtracted to isolate the reflected signal. (a) and (b) When the step is vertical located at $x=0$, and (c) and (d) when the step is horizontal located at $y=0$. In (a) and (c), the emission around the equivalent Dirac points ${\mathbf{K}}_1, {\mathbf{K}}_2$, and ${\mathbf{K}}_3$ are summed, whereas in (b) and (d), the emission around the Dirac points in the other valley is summed. The simulation is done on a very large lattice with a size of $1000\times 1000$. The central momentum of the pump beam is at ${\mathbf{K}}_1$ with the spot size of $\sigma =5a$. The pump is located at 100 lattice sites away from the center of the system in (a) and (b) the horizontal direction, and (c) and (d) the vertical direction.

Figure 6

Field intensity profiles under a spatially wide pump. Real space (a) and momentum space around ${\mathbf{K}}_1$ (b) distribution of the photon emission for no-step potential. (c) Real-space profile for a high potential step $V=0.6J>\omega$. (d) Real-space profile after subtracting the no-step amplitude in the $x<0$ region. In all panels, the pump energy is $\omega =0.3J$ and the loss is $\gamma =0.02J$. The other parameters are given in the text.

Figure 7

Momentum-space distribution of the steady-state emission of photon fields for (a) ${\varphi }_i=0$ ($k_y=0$) and (b) ${\varphi }_i={18}^{\circ }$ [$k_y=0.2sin\left({18}^{\circ }\right)/a$] as a function of $k_x$, measured from $K_{1,x}$. The vertical axes of (a) and (b) are plotted with the same scale. The solid line is the Fourier transform at $x<0$, and the dashed line is that at $x>0$. The transmission rates when (c) the incident angle (in degrees) is changed for a fixed value of $V=0.4J$ and (d) the height of the step is changed for a fixed value of ${\varphi }_i={30}^{\circ }$. The dots are calculated numerically, and the solid lines are the theoretical prediction of (4) and (5). The calculations are done on a lattice with $200\times 200$ unit cells with $\gamma /J=0.02$.

Figure 8

(a) Cuts of the steady-state momentum distribution for (a) ${\varphi }_i=0^{\circ }$ and (b) ${\varphi }_i={18}^{\circ }$ ($k_y=0.4sin\left({18}^{\circ }\right)/a$) as a function of $k_x$, measured from $K_{1,x}$. The same scale is used for the vertical axes for both plots. The solid lines are the spatial Fourier transforms in the $x<0$ region, and the dashed lines are the ones in the $x>0$ region. The transmission rates when (c) the incident angle is changed for a fixed $V=0.9J$ and (d) the height of the step is changed for a fixed ${\varphi }_i={30}^{\circ }$. The dots are calculated numerically, and the solid lines are the analytical predictions of (4) and (5). The calculations are done on experimentally realistic lattice parameters from [18]: a lattice of $20\times 20$ unit cells with $\gamma /J=0.1$.