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arXiv:2010.02676 · Physical Review A

Absorption and analysis of unbound quantum particles one by one

Sølve Selstø

Published Jan 27, 2021• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.103.012812

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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.
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.
The aim of this work is to provide a new conceptually and practically useful method of obtaining the time development and particularly the decay of quantum states initially localized on a domain with dynamically open boundaries. A projection operator derivation and a Markoffian approximation is used to obtain an effective Hamiltonian capable of describing the dynamics on the open domain. The basic algebra is implemented for one dimensional systems and a series of test calculations presented which serve to verify the validity of the approximations and illustrate the accuracy of the new procedure. The absorbing boundary method developed here bears some resemblance to the optical potential methods and the Feshbach projection scheme used in scattering theory and can be applied to dissociation and ionization processes as well as to scattering processes.
We address the question of how to compute the probability distribution of the time at which a detector clicks in the situation of nnn nonrelativistic quantum particles in a volume $\mathrm{\ensuremath{\Omega}}\ensuremath{\subset}{\mathbb{R}}^{3}$ in physical space and detectors placed along the boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Omega}}$ of $\mathrm{\ensuremath{\Omega}}$. We previously [Tumulka, Ann. Phys. (NY) 442, 168910 (2022)] argued in favor of a rule for the one-particle case that involves a Schr\"odinger equation with an absorbing boundary condition on $\ensuremath{\partial}\mathrm{\ensuremath{\Omega}}$ introduced by Werner; we call this rule the ``absorbing boundary rule.'' Here, we describe the natural extension of the absorbing boundary rule to the nnn-particle case. A key element of this extension is that, upon a detection event, the wave function gets collapsed by inserting the detected position, at the time of detection, into the wave function, thus yielding a wave function of $n\ensuremath{-}1$ particles. We also describe an extension of the absorbing boundary rule to the case of moving detectors.
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.
Nonrelativistic quantum chemistry is based on the Schrödinger equation in continuous, unbounded space. Computational approaches on classical and quantum computers often truncate the spatial domain and introduce a finite-dimensional basis, reducing the continuum problem to a finite-dimensional one. There is therefore a fundamental question: to what extent does this approximation capture the original continuum problem? In this work, we develop a general Sobolev-Galerkin framework for estimating spatial discretization errors in quantum simulation with unbounded Hamiltonians. We apply this framework to many-body Coulomb systems and establish both periodization and Galerkin error estimates for wave-function dynamics and ground-state energies. For dynamics, the finite-basis projection error of the exact wave function does not capture the full basis-discretization error: an additional dynamical projection error enters our bound at a different rate, and we introduce an inner-product argument to control this contribution. Interestingly, the spatial convergence rates for wave-function dynamics and ground-state energies differ quantitatively. For ground-state energies, we establish a sharp M−1M^{-1}M−1 convergence rate at fixed cell size, where MMM is the number of one-particle basis functions. We also show that a uniform M−1M^{-1}M−1 convergence rate is impossible for the corresponding real-time wave-function error with general H2H^2H2 initial data. The analysis relies on many-body Sobolev estimates with explicit polynomial dependence on system size. Combining these estimates with state-of-the-art quantum algorithms, we obtain, to our knowledge, the first rigorous end-to-end complexity bounds for quantum simulation of molecular and electronic dynamics with all spatial errors explicitly controlled. The number of basis functions is determined directly from the continuum problem, without regularization.
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.