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International Workshop on Computational Electronics

Dissipative quantum transport using one-particle time-dependent (conditional) wave functions

Xavier Oriols, Zhen Zhan, Enrique Colomés, Damiano Marian

Published Sep 1, 2015• 3 citations
Publisher

Abstract

An effective single-particle Schrodinger equation to include dissipation into quantum devices is presented. This effective equation is fully understood in the context of Bohmian mechanics, a theory of particles and waves, where it is possible to define unambiguously the wave function of a subsystem, the so-called conditional wave function. In particular the change in energy and momentum of an electron when interacting with a phonon is presented, both theoretically and numerically. This work is a first step to include dissipation into the fully-quantum simulator BITLLES.

DOI
10.1109/iwce.2015.7301948

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An effective single-particle Schrodinger equation to include dissipation into quantum devices is presented. This effective equation is fully understood in the context of Bohmian mechanics, a theory of particles and waves, where it is possible to define unambiguously the wave function of a subsystem, the so-called conditional wave function. In particular the change in energy and momentum of an electron when interacting with a phonon is presented, both theoretically and numerically. This work is a first step to include dissipation into the fully-quantum simulator BITLLES.
Without access to the full quantum state, modeling quantum transport in mesoscopic systems requires dealing with a limited number of degrees of freedom. In this work, we analyze the possibility of modeling the perturbation induced by non-simulated degrees of freedom on the simulated ones as a transition between single-particle pure states. First, we show that Bohmian conditional wave functions (BCWFs) allow for a rigorous discussion of the dynamics of electrons inside open quantum systems in terms of single-particle time-dependent pure states, either under Markovian or non-Markovian conditions. Second, we discuss the practical application of the method for modeling light-matter interaction phenomena in a resonant tunneling device, where a single photon interacts with a single electron. Third, we emphasize the importance of interpreting such a scattering mechanism as a transition between initial and final single-particle BCWF with well-defined central energies (rather than with well-defined central momenta).
When talking about noise in quantum devices two issues must be faced: how to model the evolution of an electronic system with scattering and how this noise is practically computed in a quantum device simulator. In the present paper, we address both problems from a practical and computational point of view. In particular, as the electronic quantum subsystem is an open (normally far from equilibrium) system, we use the notion of conditional wave function, the wave function of a subsystem in Bohmian mechanics, an alternative version of quantum mechanics which along with the wave function posited definite positions for the particles. This allows us to define an effective equation in several physical situations, ranging from the simple tunneling barrier to the interaction with a bath of phonons. Finally, we present how this development can be used to compute quantum noise in a quantum device simulator.
Without access to the full quantum state, modeling dissipation in an open system requires approximations. The physical soundness of such approximations relies on using realistic microscopic models of dissipation that satisfy completely positive dynamical maps. Here we present an approach based on the use of the Bohmian conditional wave function that, by construction, ensures a completely positive dynamical map for either Markovian or non-Markovian scenarios while allowing the implementation of realistic dissipation sources. Our approach is applied to compute the current-voltage characteristic of a resonant tunneling device with a parabolic-band structure, including electron-lattice interactions. A stochastic Schr\"odinger equation is solved for the conditional wave function of each simulated electron. We also extend our approach to (graphenelike) materials with a linear band structure using Bohmian conditional spinors for a stochastic Dirac equation.
It is often argued that measurable predictions of Bohmian mechanics cannot be distinguished from those of a theory with arbitrarily modified particle velocities satisfying the same equivariance equation. By considering the wave function of a closed system in a state with definite total energy, we argue that a distinction in measurable predictions is possible. Even though such a wave function is time-independent, the conditional wave function for a subsystem depends on time through the time-dependent particle trajectories not belonging to the subsystem. If these trajectories can be approximated by classical trajectories, then the conditional wave function can be approximated by a wave function which satisfies Schrodinger equation in a classical time-dependent potential, which is in good agreement with observations. However, such an approximation cannot be justified for particle velocities significantly deviating from the Bohmian ones, implying that Bohmian velocities are observationally preferred.
We propose and demonstrate a quantum circuitbased framework for simulating quantum transport phenomena governed by the time-dependent Schrödinger equation for open quantum systems. While the theoretical formulation is compatible with large-scale classical simulations, its implementation as a quantum algorithm offers a promising path toward quantumaccelerated modeling of nanoscale devices. A key challenge in such simulations lies in incorporating dissipation and particle injection processes arising from coupling to external reservoirs. In this work, we present quantum circuit constructions that explicitly encode these effects using controlled rotations, measurementinduced decoherence, and reset operations. By comparing the results of our quantum simulations with those obtained from classical numerical solutions of the open-system Schrödinger equation, we confirm that the proposed quantum model can reproduce both transient dynamics and steady-state behavior. These results validate the physical fidelity of the approach and highlight the potential of quantum computing in electronic transport simulations.
With the aim of manufacturing faster and smaller devices, the electronic industry is today entering into the nanoscale and the high frequency regimes. In this particular scenario, the dynamics of the electron charge becomes affected by quantum mechanical laws, both, for its spatial or temporal description. We have recently shown that Bohmian trajectories allow a direct treatment of the time-dependent many-particle interaction among electrons with an accuracy comparable to Density Functional Theory techniques. In addition, Bohmian mechanics, by combining wave functions and trajectories, provides a very simple description on how to describe multi-time measurements in quantum scenarios. Using the previous formalism, in this work we present a general purpose time-dependent 3D quantum electron transport simulator named BITLLES (Bohmian Interacting Transport in large low-dimensional Electronic Structures) especially indicated for AC, transients and noise predictions. As a numerical example of its capabilities, we compute the full electrical characteristics (DC, High frequency and fluctuations) of a Resonant Tunneling Diode.