Quantum metric and wave packets at exceptional points in non-Hermitian systems

The usual concepts of topological physics, such as the Berry curvature, are not always relevant for non-Hermitian systems. We show that another object, the quantum metric, which often plays a secondary role in Hermitian systems, becomes a crucial quantity near exceptional points in non-Hermitian systems, where it diverges in a way that fully controls the description of wave-packet trajectories. The quantum metric behavior is responsible for a constant acceleration with a fixed direction, and for a nonvanishing constant velocity with a controllable direction. Both contributions are independent of the wave-packet size.

Figure 1

(a) Dispersion of a diabolical point (conical intersection). (b) Transformation of a diabolical point into two exceptional points. (c) Zoom on a single exceptional point at the origin. The false color shows the imaginary part of the energy.

Figure 2

Circularly polarized WP at an exceptional point. (a), (b) The probability density distribution ${\left|\psi (x,y,t)\right|}^2$ for a circularly polarized initial wave packet of ${\sigma }_r=64\mu \mathrm{m}$ at (a) $t=0$ and (b) $t=5$ ps. The center of mass is shown with a white cross and its trajectory is shown with a magenta line. (c) The center-of-mass position as a function of time. (d) The center-of-mass velocity as a function of time, demonstrating constant acceleration. Solid, dashed, and dashed-dotted lines correspond to different WP sizes.

Figure 3

(a) The gradient of the imaginary part of the energy $\nabla \text{Im}E$ for one dispersion branch. The arrows show how the center of mass of the WP moves in the reciprocal space due to the amplification of its components. The linearly polarized WP projection of the branch is marked with a red circle. (b) The group velocity map $\nabla \text{Re}E$ with the trajectory of the center of mass in red, and the corresponding group velocities highlighted in blue.

Figure 4

Linearly polarized WP. (a), (b) The probability density distribution ${\left|\psi (x,y,t)\right|}^2$ for an initial wave packet of ${\sigma }_r=64\mu \mathrm{m}$ at (a) $t=0$ and (b) $t=10$ ps. The center of mass is shown with a white cross and its trajectory is shown with a magenta line. (c) Polarization-dependent center-of-mass trajectories. (d) The $x$ projection of the H-polarized WP velocity over time [solid curve: same parameters as in (a); dashed curve: increased $a$ (gain/losses); dashed-dotted curve: increased celerity $\alpha$].

Figure 5

The center-of-mass wave vector as a function of time, obtained from numerical simulations for a linearly polarized wave packet: (a) $k_x\left(t\right)$ (parabolic growth), (b) $k_y\left(t\right)$ (linear growth).