Failure of the mean-field Hartree approximation for a bosonic many-body system with non-Hermitian Hamiltonian

TL;DR

This paper demonstrates that the mean-field Hartree approximation can fail for bosonic many-body systems governed by non-Hermitian Hamiltonians, especially in models with anti-Hermitian interactions, leading to phenomena like finite-time state transitions.

Contribution

It analytically shows the failure of the Hartree approximation in a specific non-Hermitian bosonic model and highlights conditions where the approximation does not hold.

Findings

The limit of the one-particle marginal state does not match the Hartree evolution.

Existence of initial conditions leading to finite-time transition to mixed states.

Challenges the validity of mean-field models for non-Hermitian quantum systems.

Abstract

Mean-field Hartree theory is a central tool for reducing interacting many-body dynamics to an effective nonlinear one-particle evolution. This approximation has been employed also when the Hamiltonian that governs the many-body dynamics is not Hermitian. Indeed, non-Hermitian Hamiltonians model particle gain/loss or the evolution of open quantum systems between consecutive quantum jumps. Furthermore, the validity of the Hartree approximation for generic non-Hermitian Hamiltonians lies at the basis of a quantum algorithm for nonlinear differential equations. In this work, we show that this approximation can fail. We analytically solve a model of $N$ bosonic qubits with two-body interactions generated by a purely anti-Hermitian Hamiltonian, determine an analytic expression for the limit for $N\to\infty$ of the one-particle marginal state and show that such a limit does not agree with the solution of the non-Hermitian Hartree evolution equation. We further show that there exists an initial condition such that the exact one-particle marginal state undergoes a finite-time transition to a mixed state, a phenomenon that is completely absent in the case of Hermitian Hamiltonians. Our findings challenge the validity of the mean-field Hartree approximation for non-Hermitian Hamiltonians, and call for additional conditions for the validity of the mean-field regime to model the dynamics of particle gain and loss and the open-system dynamics in bosonic many-body systems.