The Hamiltonian of mean force is a widely used concept to describe the modification of the usual canonical Gibbs state for a quantum system whose coupling strength with the thermal bath is non-negligible. Here, we perturbatively derive general approximate expressions for the Hamiltonians of mean force in the weak-coupling approximation and in the high-temperature one. We numerically analyze the accuracy of the corresponding expressions and show that the precision of the Bloch–Redfield equantum master equation can be improved if we replace the original system Hamiltonian by the Hamiltonian of mean force.
Mean-field Hartree theory is a central tool for reducing many-body dynamics to an effective one-particle evolution. This approximation has been employed when the Hamiltonian governing 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.
First, we consider the most general bosonic mean-field Hamiltonian with two-body interactions and we prove that, for non-Hermitian Hamiltonian, the Hartree equation does not capture the correct second time derivative of the one-particle marginal state at
t=0 in the thermodynamic limit. Then, we analytically solve a model of
N bosonic qubits with a purely anti-Hermitian Hamiltonian, compute the limit for
N→∞ of the one-particle marginal state and show that it does not agree with the solution of the non-Hermitian Hartree 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. Finally, we simulate a non-Hermitian Hamiltonian with real spectrum and show that, also in this case, there are initial conditions for which the Hartree equation does not reproduce the correct one-particle marginal state in the limit
N→∞.
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.
A four-level system coupled with lasers is analyzed in terms of dressed-atom model by introducing auto-ionization decay which leads to a non-Hermitian Hamiltonian. The population of the system obtained by means of solving the secular equation of the Hamiltonian is found in good agreement with that from numerical solution of time-dependent Schrdinger equation. This result suggests that non-Hermitian Hamiltonian can be introduced into the framework of quantum mechanics , and the imaginary part of the corresponding eigenvalue represents the populati on decaying of systems.