In numerical studies of the dynamics of unbound quantum mechanical systems, absorbing boundary conditions are frequently applied. Although this certainly provides a useful tool in facilitating the description of the system, its applications to systems consisting of more than one particle are problematic. This is due to the fact that all information about the system is lost upon the absorption of one particle; a formalism based solely on the Schrödinger equation is not able to describe the remainder of the system as particles are lost. Here we demonstrate how the dynamics of a quantum system with a given number of identical fermions may be described in a manner which allows for particle loss. A consistent formalism which incorporates the evolution of sub-systems with a reduced number of particles is constructed through the Lindblad equation. Specifically, the transition from an N -particle system to an ( N − 1)-particle system due to a complex absorbing potential is achieved by relating the Lindblad operators to annihilation operators. The method allows for a straight forward interpretation of how many constituent particles have left the system after interaction. We illustrate the formalism using one-dimensional two-particle model problems.
In numerical studies of the dynamics of unbound quantum mechanical systems, absorbing boundary conditions are frequently applied. Although this certainly provides a useful tool in facilitating the description of the system, its applications to systems consisting of more than one particle are problematic. This is due to the fact that all information about the system is lost upon the absorption of one particle; a formalism based solely on the Schrödinger equation is not able to describe the remainder of the system as particles are lost. Here we demonstrate how the dynamics of a quantum system with a given number of identical fermions may be described in a manner which allows for particle loss. A consistent formalism which incorporates the evolution of sub-systems with a reduced number of particles is constructed through the Lindblad equation. Specifically, the transition from an N -particle system to an ( N − 1)-particle system due to a complex absorbing potential is achieved by relating the Lindblad operators to annihilation operators. The method allows for a straight forward interpretation of how many constituent particles have left the system after interaction. We illustrate the formalism using one-dimensional two-particle model problems.
In quantum physics, the theoretical study of unbound many-body systems is typically quite demanding, owing to the combination of their large spatial extension and the so-called curse of dimensionality. Often, such systems are studied on truncated numerical domains---at the cost of information. Here we present methods for calculating differential probabilities for unbound particles which are subject to a complex absorbing potential. In addition to attenuating outgoing waves, the absorber is also used to probe them by projection onto single-particle scattering states, thus rendering the calculation of multiparticle scattering states superfluous. Within formalism based on the Lindblad equation, singly differential spectra from subsequent absorptions are obtained by resolving the dynamics of the remaining particles after the first absorption. While the framework generalizes naturally to any number of particles, explicit, compact, and intuitive expressions for the differential probability distributions are derived for the two-particle case. The applicability of the method is illustrated by numerical examples involving two-particle model-systems. These examples, which address scattering and photoionization, demonstrate how energy distributions of unbound particles may be determined on numerical domains considerably smaller than the actual extension of the system.
Absorbing boundary conditions in the form of a complex absorbing potential are routinely introduced in the Schr\"odinger equation to limit the computational domain or to study reactive scattering events using the multiconfigurational time-dependent Hartree (MCTDH) method. However, it is known that a pure wave-function description does not allow the modeling and propagation of the remnants of a system of which some parts are removed by the absorbing boundary. It was recently shown [S. Selst\o{} and S. Kvaal, J. Phys. B: At. Mol. Opt. Phys. 43, 065004 (2010)] that a master equation of Lindblad form was necessary for such a description. We formulate a MCTDH method for this master equation, usable for any quantum system composed of any mixture of species. The formulation is a strict generalization of pure-state propagation using standard MCTDH for identical particles and mixtures. We demonstrate the formulation with a numerical experiment.
Abstract. Quantum dynamics, typically expressed in the form of a time-dependent Schrödinger equation with a Hermitian Hamiltonian, is a natural application for quantum computing. However, when simulating quantum dynamics that involves the emission of electrons, it is necessary to use artificial boundary conditions (ABCs) to confine the computation within a fixed domain. The introduction of ABCs alters the Hamiltonian structure of the dynamics, and existing quantum algorithms cannot be directly applied since the evolution is no longer unitary. The current paper utilizes a recently introduced Schrödingerization method that converts non-Hermitian dynamics into a Schrödinger form for the artificial boundary problems [S. Jin, N. Liu, and Y. Yu, Quantum Simulation of Partial Differential Equations via Schrödingerisation, preprint, arXiv:2212.13969 , 2022], [S. Jin, N. Liu, and Y. Yu, Phys. Rev. A, 108 (2023), 032603]. We implement this method for three types of ABCs, including the complex absorbing potential technique, perfectly matched layer methods, and Dirichlet-to-Neumann approach. We analyze the query complexity of these algorithms and perform numerical experiments to demonstrate the validity of this approach. This helps to bridge the gap between available quantum algorithms and computational models for quantum dynamics in unbounded domains.
Complex absorbing potentials are frequently employed in quantum calculations. One of the advantages that such absorbers provide is the ability to attenuate outgoing waves in simulations of unbound systems, thus allowing for truncating the numerical domain. Here we argue that the absorber may also be used to probe outgoing waves so that physical information about absorbed particles may be retained. Moreover, under certain conditions, the physical information extracted via the absorber is subject to loss in coherence, as is also the case when collapsing the wave function upon measurement. Both these aspects demonstrate clearly how, and when, the effect of introducing a complex absorbing potential corresponds to that of a detector.
In quantum mechanics, among the proposals for how to compute the probability distribution of detection time and place along a given spatial surface or in a given volume, two leading candidates involve absorbing boundary conditions (ABCs) and complex absorbing potentials (CAPs) for the Schrödinger equation. Here, we extend earlier results on how the two are related. It was previously shown that ABCs arise as a limiting case of CAPs, but for that a (reflecting) Neumann boundary condition was assumed at the end of the region with the CAP. Here, we obtain ABCs in a different limit using a Dirichlet boundary condition at the end of the CAP region, which seems more practically relevant and realizable.