Absorbing boundary conditions for dynamical many-body quantum systems

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 is problematic. This is due to the fact that all information about the system is lost upon absorption of one particle; a formalism based solely on the Scrh{"o}dinger 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.

💡 Research Summary

The paper addresses a long‑standing difficulty in the numerical simulation of unbound many‑body quantum systems: while complex absorbing potentials (CAPs) are routinely used to emulate open boundaries for single‑particle dynamics, their naïve inclusion in a Schrödinger equation for several interacting particles leads to the unphysical loss of the entire wavefunction as soon as any particle is absorbed. To overcome this, the authors embed the problem in the formalism of open quantum systems and employ the Lindblad master equation.

The key idea is to identify the Lindblad jump operators with the product of the square root of the local CAP strength and the fermionic annihilation operators:
(L_j = \sqrt{\Gamma(\mathbf{r}_j)},c_j).
Here (c_j) removes a particle at position (\mathbf{r}_j) and (\Gamma(\mathbf{r})) encodes the spatial profile of the absorbing potential. Substituting these operators into the Lindblad equation yields a dynamics that consists of the usual unitary evolution generated by the many‑body Hamiltonian (H) plus a non‑unitary term that both reduces the trace of the (N)-particle density matrix (\rho^{(N)}) and feeds the lost probability into the ((N-1))-particle density matrix (\rho^{(N-1)}). Consequently, when a particle is absorbed, the system’s description automatically “drops” to a lower‑particle sector while preserving the correct quantum correlations among the remaining particles.

To demonstrate the practicality of the method, the authors consider two one‑dimensional model systems. The first is a quantum‑dot‑like setup with two interacting fermions confined in a finite region; the second describes two free particles that are accelerated by an external electric field. In both cases a CAP is placed near the computational boundary. Time‑dependent simulations track the trace of (\rho^{(N)}) and (\rho^{(N-1)}), the spatial charge density, and the energy expectation values. The results show that the first particle is smoothly removed at a rate dictated by the CAP strength, while the second particle’s wavefunction is automatically renormalized and continues to evolve under the same Hamiltonian, now in the ((N-1))-particle Hilbert space. Importantly, the total probability is conserved across the combined set of density matrices, and the physical observables (e.g., charge flow, kinetic energy) behave as expected for an open system.

The authors also discuss numerical aspects. Because the Lindblad equation is a linear differential equation for the density matrix, standard time‑propagation schemes (e.g., split‑operator or Runge‑Kutta) can be applied without instability. The method scales naturally to larger particle numbers, provided the appropriate many‑body basis is used, and it can be combined with existing electronic‑structure techniques such as time‑dependent density‑functional theory (TDDFT).

Beyond fermionic systems, the formalism is readily extensible to bosons or mixed statistics, and to situations where both particle loss and gain occur (e.g., photo‑ionization with subsequent electron capture). The paper therefore offers a unified, physically consistent framework for incorporating absorbing boundaries into many‑body quantum dynamics, opening the door to accurate simulations of electron detachment, ionization, and other processes where particle number changes during the evolution.

In summary, by linking CAPs to Lindblad jump operators, the authors provide a mathematically rigorous and computationally feasible way to describe particle loss in dynamical many‑body quantum systems, preserving the integrity of the remaining subsystem and enabling direct measurement of how many particles have left the system at any given time.