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

TL;DR

This work demonstrates that the mean-field Hartree approximation can fail for non-Hermitian bosonic dynamics, even in an exactly solvable two-body anti-Hermitian model. Using an N-body system with $A^{(N)} = \frac{i}{N-1}\sum_{i<j} Z_i Z_j$, the authors compute the exact $N$-to-$1$ marginals and show that the $N\to\infty$ limit of the one-particle marginal is typically pure but does not coincide with the non-Hermitian Hartree evolution; in the balanced initial case, the limit becomes mixed after a finite critical time $t_c=1/2$. They derive the correct large-$N$ effective equation, distinguished from Hartree by explicit time dependence and matching Hartree only at $t=0$, and they provide numerical evidence for the predicted mixing and convergence rates. The results constrain the applicability of non-Hermitian mean-field reductions and suggest additional conditions or new reduced dynamics are necessary for modeling loss and open-system bosonic dynamics in practice.

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.

Figures (4)

• Figure 1: Plot of $f_t$ as a function of $x$, as defined in \ref{['eq:f']}, for different instances of initial condition and time.
• Figure 2: Linear entropy of the one-particle marginal states of the solution of the $N$-particle equation. Solid lines show the numerical computation of the linear entropy. We fit the tails of the curves (last 20%) with the function $a N^{b}$. The fit is plotted in dashed lines and the values of the exponents $b$ are written in the legends.
• Figure 3: Numerical computation of the infidelity of the one-particle marginal state with respect to the solution of the Hartree equation. Dashed lines corresponds to the limit of the infidelity for $N\to\infty$ using the results of \ref{['sec:single_maxima']}.
• Figure 4: Numerical computation of the infidelity of the one-particle marginal state with respect to its limit for $N\to\infty$. The infidelity scales as $\mathcal{O}(N^{-1})$ for $N\to\infty$ for almost all initial conditions, and scales as $\mathcal{O}(N^{-2})$ for the initial conditions such that $|\varphi_0|^2=|\varphi_1|^2=\frac{1}{2}$ and $t>\frac{1}{2}$.