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Physical Review Letters

Quantum-Trajectory Approach to Time-Dependent Transport in Mesoscopic Systems with Electron-Electron Interactions

Xavier Oriols

Published Feb 7, 2007• 133 citations
Publisher

Abstract

It is proved that many-particle Bohm trajectories can be computed from single-particle time-dependent Schrödinger equations. From this result, a practical algorithm for the computation of transport properties of many-electron systems with exchange and Coulomb correlations is derived. As a test, two-particle Bohm trajectories in a tunneling scenario are compared to exact results with an excellent agreement. The algorithm opens the path for implementing a many-particle quantum transport (Monte Carlo) simulator, beyond the Fermi liquid paradigm.

DOI
10.1103/physrevlett.98.066803

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It is proved that many-particle Bohm trajectories can be computed from single-particle time-dependent Schrödinger equations. From this result, a practical algorithm for the computation of transport properties of many-electron systems with exchange and Coulomb correlations is derived. As a test, two-particle Bohm trajectories in a tunneling scenario are compared to exact results with an excellent agreement. The algorithm opens the path for implementing a many-particle quantum transport (Monte Carlo) simulator, beyond the Fermi liquid paradigm.
Following Oriols (2007 Phys. Rev. Lett. 98 066803), an algorithm to deal with the exchange interaction in non-separable quantum systems is presented. The algorithm can be applied to fermions or bosons and, by construction, it exactly ensures that any observable is totally independent of the interchange of particles. It is based on the use of conditional Bohmian wave functions which are solutions of single-particle pseudo-Schrödinger equations. The exchange symmetry is directly defined by demanding symmetry properties of the quantum trajectories in the configuration space with a universal algorithm, rather than through a particular exchange-correlation functional introduced into the single-particle pseudo-Schrödinger equation. It requires the computation of N(2) conditional wave functions to deal with N identical particles. For separable Hamiltonians, the algorithm reduces to the standard Slater determinant for fermions (or permanent for bosons). A numerical test for a two-particle system, where exact solutions for non-separable Hamiltonians are computationally accessible, is presented. The numerical viability of the algorithm for quantum electron transport (in a far-from-equilibrium time-dependent open system) is demonstrated by computing the current and fluctuations in a nano-resistor, with exchange and Coulomb interactions among electrons.
A recent proposal for modeling time-dependent quantum electron transport with Coulomb and exchange correlations using quantum (Bohm) trajectories (Oriols 2007 Phys. Rev. Lett. 98 066803) is extended towards the computation of the total (particle plus displacement) current in mesoscopic devices. In particular, two different methods for the practical computation of the total current are compared. The first method computes the particle and the displacement currents from the rate of Bohm particles crossing a particular surface and the time-dependent variations of the electric field there. The second method uses the Ramo–Shockley theorem to compute the total current on that surface from the knowledge of the Bohm particle dynamics in a 3D volume and the time-dependent variations of the electric field on the boundaries of that volume. From a computational point of view, it is shown that both methods achieve local current conservation, but the second is preferred because it is free from 'spurious' peaks. A numerical example, a Bohm trajectory crossing a double-barrier tunneling structure, is presented, supporting the conclusions.
With the aim of manufacturing faster and smaller devices, the electronic industry is today approaching both the nano and the picosecond scales. In this particular scenario, electron dynamics becomes strongly correlated both in space and time. We have recently shown that Bohmian trajectories allow a direct treatment of the time-dependent many-particle interaction among electrons with accuracy comparable to Density Functional Theory techniques. In this work we present a general purpose time-dependent 3D quantum electron transport simulator based on Bohmian trajectories that we call BITLLES. As a numerical example of its capabilities, we compute the full electrical characteristics (DC, High frequency and fluctuations) of a Resonant Tunneling Diode.
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.
A prominent tool to study the dynamics of open quantum systems is the reduced density matrix. Yet, approaching open quantum systems by means of state vectors has well known computational advantages. In this respect, the physical meaning of the so-called conditional states in Markovian and non-Markovian scenarios has been a topic of recent debate in the construction of stochastic Schrödinger equations. We shed light on this discussion by acknowledging the Bohmian conditional wavefunction (linked to the corresponding Bohmian trajectory) as the proper mathematical object to represent, in terms of state vectors, an arbitrary subset of degrees of freedom. As an example of the practical utility of these states, we present a time-dependent quantum Monte Carlo algorithm to describe electron transport in open quantum systems under general (Markovian or non-Markovian) conditions. By making the most of trajectory-based and wavefunction methods, the resulting simulation technique extends to the quantum regime, the computational capabilities that the Monte Carlo solution of the Boltzmann transport equation offers for semi-classical electron devices.
Abstract The direct solution of the many‐particle Schrödinger equation is computationally inaccessible for more than very few electrons. In order to surpass this limitation, one of the authors [X. Oriols, Phys. Rev. Lett. 2007, 98 (066803)] has recently proposed a new model to study electron‐electron correlations from Bohm trajectories associated to time‐dependent wave‐packets solutions of pseudo single‐particle Schrödinger equations. In the aforementioned paper only the orbital exchange interaction is considered assuming that all electrons have the same spin orientation. Then, the many‐particle wave function is a complex Slater determinant of the single‐particle wave‐packets. In the present work the previous formalism is extended to study many‐particle wave functions where the electrons have different spin orientations. The main difficulty to treat N different electron spin orientations with time‐dependent wave‐packets is that one must study all the possible N!N! products of permutations among spin states. To overcome this computationally inaccessible problem, in this article the total wave function is treated as a separated product of two many‐particle wave functions, the first with spin up and the second with spin down. In order to numerically justify this approximation, the Bohm velocity in different antisymmetric total wave‐function scenarios is computed. The computational results confirms the accurate validity of our approximation under a large number of cases (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)