Scattered Wave Packet Formalism with Markovian Outgoing Wave Boundary Conditions for Open Quantum Systems

The scattered wave formalism is developed for a quantum subsystem interacting with the external environment through open boundaries. The total wave function is divided into incident and scattered components and Markovian outgoing wave boundary conditions are applied to the scattered wave function. This formalism significantly reduces the computational effort relative to other methods which rely on Green functions and memory kernels. The method is applied to one-dimensional barrier scattering and to a three-dimensional model for the field effect transistor.

Figure 1

(a) The incident, scattered, and total wave functions obtained from the scattered wave formalism (●). The wave functions obtained from the integration of the TDSE using a large spatial grid (curves) and from the non-Markovian TDSE (□) with the Eckart barrier in arbitrary units (dotted line) at $t=4$. (b) The probability flux at the boundaries obtained from the scattered wave formalism (●) and from the integration of the TDSE using a large spatial grid (curves).

Figure 2

Potential energy diagram in the channel region for the transistor model. The potential energy is shown for the vertical ($x,y$) slice at the middle of the channel along the $y$ direction. The left side connects to the source electrode from which the incident wave packet is launched, the right side leads to the sink electrode, and the gate occupies the region along the $x$ direction between 20 and 30 nm.

Figure 3

Isosurface plots for the scattered wave packet at three times: (a) $T=175\mathrm{fs}$, (b) $T=225\mathrm{fs}$, (c) $T=300\mathrm{fs}$. Surfaces are drawn for positive [dark gray (red)] and negative [light gray (cyan)] values of the real part of this wave function. These isosurfaces are plotted for the values $\pm 0.05\gamma$ where $\gamma$ is a scale factor defined in the text.

Figure 4

Flux vectors for a wave packet propagating in a 2D region are shown at $t=0$ and $t=100\mathrm{fs}$. The initial packet is launched toward the step potential, which begins at $z=25\mathrm{nm}$. At the later time, the contour map of real ($\Psi$) and the flux map show that the packet has partially transmitted the transparent boundaries along the top and right edges. The ratio method applied along the normal vectors to the two boundary surfaces accurately describes propagation through these surfaces.