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Applied Physics Letters

Issues in general quantum transport with complex potentials

David K. Ferry, J. R. Barker

Published Jan 25, 1999• 23 citations
Publisher

Abstract

Complex potentials have been used previously to simulate dissipative processes in quantum simulations of transport and scattering. A general formulation is presented which shows how such non-Hermitian terms can lead to changes in the dynamics of transport operators, and to the decay of hydrodynamic quantities such as phase, energy, and momentum.

DOI
10.1063/1.123152

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Complex potentials have been used previously to simulate dissipative processes in quantum simulations of transport and scattering. A general formulation is presented which shows how such non-Hermitian terms can lead to changes in the dynamics of transport operators, and to the decay of hydrodynamic quantities such as phase, energy, and momentum.
The simulation of quantum transport in a realistic, many-particle system is a nontrivial problem with no quantitatively satisfactory solution. While real-time propagation has the potential to overcome the shortcomings of conventional transport methods, this approach is prone to finite size effects that are associated with modeling an open system on a closed spatial domain. Using a master equation framework, we exploit an equivalence between the superoperators coupling an open system to external particle reservoirs and non-Hermitian terms defined at the periphery of a quantum device. By taking the mean-field limit, the equation of motion for the single-particle reduced density matrix becomes equivalent to real-time time-dependent density functional theory in the presence of imaginary source and sink potentials. This method may be used to converge nonequilibrium steady states for a many-body quantum system using a previously reported constraint algorithm.
We have previously shown that the flow in open quantum dots may be described in terms of meandering open orbits which separate regions of trapped flows around vortices. Furthermore the topology of the flow follows from a simple geometrical construction in which the classical ballistic trajectories are first computed and then all crossings are replaced by hyperbolic flow separation and for which all trapped flows form quantised closed orbits. In the present paper we extend this picture to a computational methodology for developing quantum hydrodynamic descriptions of flows in the presence of quantised vortices by extension from classical hydrodynamics. For coherent transport we again encounter the Bohm quantum potential but the singularity structure is predetermined by applying a fictitious repulsion which prevents trajectories and hence streamlines crossing. This leads to vortex producing terms in the hydrodynamic equations. The discussion is completed by including the effects of non-hermitian potentials into the Hamiltonian in order to simulate dissipation and phase breaking flows.
Describing open quantum systems in terms of effective non-Hermitian Hamiltonians gives rise to nonunitary time evolution. In this paper we study the impact of nonunitary dynamics on the emergent hydrodynamics in quantum systems with a global conservation law. To this end we demonstrate how linear-response correlation functions can be generalized and interpreted in the case of non-Hermitian systems. Moreover we show that dynamical quantum typicality provides an efficient numerical approach to evaluate such correlation functions, even though the nonunitary dynamics leads to subtleties that are absent in the Hermitian case. As a point of reference for our analysis, we consider the Hermitian spin-1/21/21/2 XXZ chain, whose high-temperature transport properties have been characterized extensively in recent years. Here we explore the resulting hydrodynamics for different non-Hermitian perturbations of the XXZ chain. We also discuss the role of integrability by studying the complex energy-level statistics of the non-Hermitian quantum models.