Absorption and injection models for open time-dependent quantum systems

F. L. Traversa, Z. Zhan, and X. Oriols

Phys. Rev. E 90, 023304 – Published 11 August 2014

Abstract

In the time-dependent simulation of pure states dealing with transport in open quantum systems, the initial state is located outside of the active region of interest. Using the superposition principle and the analytical knowledge of the free time evolution of such a state outside the active region, together with absorbing layers and remapping, a model for a very significant reduction of the computational burden associated with the numerical simulation of open time-dependent quantum systems is presented. The model is specially suited to study (many-particle and high-frequency effects) quantum transport, but it can also be applied to any other research field where the initial time-dependent pure state is located outside of the active region. From numerical simulations of open quantum systems described by the (effective mass) Schrödinger and (atomistic) tight-binding equations, a reduction of the computational burden of about two orders of magnitude for each spatial dimension of the domain with a negligible error is presented.

Received 27 May 2014

DOI:https://doi.org/10.1103/PhysRevE.90.023304

Authors & Affiliations

F. L. Traversa^*, Z. Zhan^†, and X. Oriols^‡

Departament d'Enginyeria Electrònica, Universitat Autònoma de Barcelona, 08193-Bellaterra (Barcelona), Spain

^*fabio.traversa@polito.it
^†zhenzhanh@gmail.com
^‡xavier.oriols@uab.es

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Vol. 90, Iss. 2 — August 2014

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Figure 1
Spatial simulation domains. (a) Infinite domain: the full domain $Ω$ is a large enough domain to avoid interactions with boundaries during the simulation time. (b) Absorbing layers: reduced domain $δ Ω_{out} \cup δ Ω_{inj} \cup Ω^{'} \cup δ Ω_{out}$ using standard absorbing algorithms, where $δ Ω_{inj}$ is the layer for the injection, i.e., the layer that includes the wave function at the initial time, $Ω^{'}$ is the interaction box, and $δ Ω_{out}$ at both sides are the absorbing layers. (c) Absorbing layers plus analytical injection: further reduced domain $δ Ω_{out}^{'} \cup Ω^{'} \cup δ Ω_{out}^{'}$ presented in this work; it has no need for an injection layer, and it employs smaller absorbing layers $δ Ω_{out}^{'} ≪ δ Ω_{out}$ obtained exploiting a change of coordinates (remapping).
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Figure 2
Simulation of a Gaussian wave packet of a free particle with $E = ℏ^{2} k_{x}^{2} / (2 m^{*}) = 0.01$ –1 eV equally spaced into 10 values, $σ_{0} = 25 / \sqrt{2}$ nm, and initially centered in $x_{0} = - 70$ nm and $m^{*} = 0.2 m_{0}$ , where $m_{0}$ is the free-electron mass. The simulation parameters are $Δ t = 0.01$ fs, $Δ x = 0.2$ nm, the full simulation space ranges from $- 800$ to 800 nm, and the simulation is stopped when $\int_{- 800 nm}^{50 nm} {| ψ_{0} |}^{2} d x < 10^{- 10}$ . The dashed blue line is $log [{∥ ψ_{G_{num}} (t) - ψ_{G} (t) ∥}^{2}]$ , and the solid black line is $log ({∥ ψ_{G_{num}} (t) - ψ_{an} (t) ∥}^{2})$ , where $ψ_{G_{num}}$ is the numerical solution for the Gaussian free particle, $ψ_{G}$ is the analytical solution (9), and $ψ_{an}$ is the analytical solution given by (19).
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Figure 3
Simulation of a Gaussian wave packet with energy $E = 0.1$ eV, and the other parameters as in Fig. 2. The solid black line is the absorbed wave packet with $L = 100$ nm, and the dashed red line is the wave packet given by (9).
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Figure 4
Simulation of a Gaussian wave packet of a free particle with the same parameters as in Fig. 2. The simulation parameters are as follows: the full simulation space ranges from $- 800$ to 800 nm, $a = - 800$ nm, $b = 50$ nm, and the length $L$ , taken as $10 λ$ , is the dashed black line referring to the right $y$ axis. The solid lines are ${log}_{10} (ɛ_{abs})$ .
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Figure 5
Simulation of a Gaussian wave packet of a free particle with the same parameters as in Fig. 2. The simulation parameters are as follows: $E = 0.1$ eV, the full simulation space ranges from $- 800$ to 800 nm, $a = - 800$ nm, and $b = 50$ nm. The dotted black line referring to the right $y$ axis is the norm $\int_{- 800 nm}^{50 nm} {| ψ_{G} |}^{2} d x$ . The solid lines are ${log}_{10} (ɛ_{rem})$ and the dashed lines are ${log}_{10} (ɛ_{cut})$ .
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Figure 6
Simulation of a Gaussian wave packet with the same parameters as in Fig. 2. The simulation is carried out for the three energies $E = 0.01$ , $0.1$ , and 1 eV, and the wave packet interacts with a potential barrier centered at $27.5$ nm with a width of 5 nm and heights of $0.015$ , $0.0825$ , and $0.93$ eV, respectively. The different barrier heights guarantee that the transmitted and reflected waves after interaction have the same norm for each energy, guaranteeing that the absorbing and remapping algorithms work equally well on both sides of $Ω^{'}$ . The full simulation space ranges from $- 800$ to 800 nm, $Ω^{'}$ from $a = 0$ nm to $b = 50$ nm, the absorbing layers have $L_{a} = 20$ nm, $L = 274$ , $86.8$ , and $27.4$ nm (corresponding to double the width of $δ Ω_{out}$ ), and $L_{eff} = 18.9$ , $16.4$ , and $10.5$ nm (corresponding to the width of $δ Ω_{out}^{'}$ ), respectively, depending on the initial wave-packet energy, and finally the absorption exponent is $m = 5$ . The dashed blue line is $log (ɛ_{inj})$ and the solid red line is $log (ɛ_{ar})$ , and they refer to the left $y$ axis. The dotted black line is the evolution of the norm $\int_{a}^{b} {| ψ_{num} |}^{2} d x$ and the dashed-dotted green line is the evolution of the norm $\int_{a - L_{a}}^{a} c^{- 1} (z) | Ψ_{new} |^{2} d z + \int_{a}^{b} | Ψ_{new} |^{2} d x + \int_{b}^{b + L_{a}} c^{- 1} (z) {| Ψ_{new} |}^{2} d z$ , and it refers to the right $y$ axis.
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