Mean-field Hartree theory is a central tool for mapping interacting many-body dynamics into an effective nonlinear one-particle evolution. This approximation has been also employed 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, where the nonlinear equation is mapped into a many-body Schr\"odinger equation, which may be non-Hermitian. In this application, the mean-field approximation can be seen as a linearization method, and at the end of the algorithm, the obtained linear system is solved using the Harrow-Hassidim-Lloyd (HHL) algorithm. The non-Hermitian mean-field Hartree approximation is far from being completely understood, and no rigorous results justify its applications. Instead, it is usually presented as a heuristic extension of the Hermitian case. In the first part of this thesis, we show that this approximation can fail. We consider the most general bosonic mean-field Hamiltonian with two-body interactions and we prove that, if the Hamiltonian is not Hermitian, the Hartree equation does not capture the correct second time derivative of the one-particle marginal state at
t=0 in the limit of infinitely many particles. We construct a series of examples that we can solve analytically or numerically, which support the failure of the Hartree approximation and give further insight into the new phenomena that arise in non-Hermitian physics. Furthermore, we consider the mean-field evolution of open quantum systems and we show how to use the mean-field approximation of open quantum systems as a linearization method of nonlinear differential equations. In the second part of this thesis, we study the mean-field dynamics of a Hermitian system of
N interacting bosons starting from an initially condensed state. For a broad class of mean-field Hamiltonians, including models with arbitrary bounded interactions and models with unbounded interaction potentials, we prove that the probability of having
n particles outside the condensate decays exponentially in
n for any finite evolution time. Our results strengthen previously known bounds that provide only polynomial control on the probability of having
n excitations. Finally, we consider the HHL algorithm. Several circuit-level variants have been proposed to simplify its implementation, including constructions in which the clock register is factorized via Hadamard gates to reduce circuit depth and entanglement. We provide a rigorous analysis of the convergence of the output towards the ideal solution, and we show that this commonly deployed variant can fundamentally alter the asymptotic convergence of the algorithm. This phenomenon is not a finite-size artifact and compromises the correctness guarantees of the factorized-clock implementation. We provide an analytic characterization of the underlying mechanism and propose a minimal and experimentally feasible modification that suffices to restore convergence while preserving the practical advantages of the simplified circuit. Our results clarify an overlooked correctness issue in a widely used HHL implementation pattern and delineate the conditions under which simplified realizations faithfully reproduce the intended algorithmic behavior.
The mean-field approximations of many-boson dynamics are known to be effective in many physical relevant situations. The mathematical justifications of such approximations rely generally on specific considerations which depend too much on the model and on the initial states of the system which are required to be well-prepared. In this article, using the method of Wigner measures, we prove in a fairly complete generality the accuracy of the mean-field approximation. Roughly speaking, we show that the dynamics of a many-boson system are well approximated, in the limit of a large number of particles, by a one particle mean-field equation if the following general principles are satisfied:
∙ The Hamiltonian is in a mean-field regime (i.e.: The interaction and the free energy parts are of the same order with respect to the number of particles).
∙ The interaction is relatively compact with respect to a one particle and it is dominated by the free energy part.
∙ There exists at most one weak solution for the mean-field equation for each initial condition.The convergence towards the mean field limit is described in terms of Wigner (probability) measures and it holds for any initial quantum states with a finite free energy average. The main novelty of this article lies in the use of fine properties of uniqueness for the Liouville equations in infinite dimensional spaces.
Mean-field limits for bosonic systems explain how an effective one-particle nonlinear equationemerges from an N-body Schrödinger dynamics. In the standard mean-field setting, themicroscopic Hamiltonian contains a scaled pair interaction, and the limiting dynamics is aHartree equation. This paper considers a closely related but structurally different situation: inaddition to the usual pair interaction, the microscopic system carries a phase drive which isgenerated by the evolving N-body state itself.The distinction is important. The phase field considered here is not a prescribed timedependent potential and is not imported from the limiting Hartree trajectory. At finite N, it iscomputed from the normalized one-particle marginal density
This thesis is about the derivation of effective mean field equations and their next-order corrections starting from nonrelativistic many-body quantum theory. Mean field equations provide an approximate ansatz for the description of interacting many-particle systems. In this ansatz the interaction between the particles is replaced by a self-consistent external potential leading to an effective one-body description of the many-particle system. Next-order corrections provide an approximation which goes one step further and tries to capture also subleading effects that are not resolved by the mean field ansatz. We present mathematical proofs for the validity of such effective theories for different models that are motivated, e.g., from the theory of ultracold atoms (the bosonic Hartree equation and the corresponding Bogoliubov theory) and from plasma physics (the motion of a tracer particle through a degenerate and dense electron gas). Starting from a many-body Schrödinger equation, our goal is to show that the solutions converge in a particular limit to the solutions to an effective mean field equation and its next-order corrections. After a short introduction and a summary in Chapter one, we present the main part of this work in three self-contained chapters. In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.e., for a coupling constant g_N=1/N. It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective one-particle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of Bose-Einstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single one-particle quantum state. Our emphasis lies in the description of the few particles that fluctuate around the Bose-Einstein condensate. We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the N-particle space. This result allows us to prove several other assertions. Among other things, it is used to derive the Hartree equation with optimal speed of convergence 1/N for initial states that are close to ground states of interacting systems and also to prove convergence of the N-particle solution towards a time evolution obtained from the Bogoliubov Hamiltonian on Fock space. Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Here, we derive a novel estimate for low energy eigenfunctions stating that the probability for finding
l particles out of their total number N not in the condensate is exponentially small in the number
l. Using this bound, we then prove that the ground state wave function of the microscopic model satisfies certain quasifree type properties. The exponential decay is moreover used to provide an alternative proof for the validity of Bogoliubov's approximation for the low-lying energy eigenvalues. Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energy-momentum dispersion relation which is linear for small momentum. The linearity of the effective dispersion relation is an essential ingredient for the explanation of the superfluid character of the Bose gas. In Chapter four we study the time evolution of a single particle coupled through a pair potential to a dense and homogeneous ideal Fermi gas in two spatial dimensions. This type of model is well known in plasma physics where it is used to describe the energy loss of ions moving through a dense and degenerate electron gas. We analyze the model for a coupling parameter g=1 and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift. To leading order, this energy shift is given by the spatially homogeneous mean field potential produced by the gas particles, whereas at next-to-leading order, one has to consider an additional correction to the mean field energy which is due to so-called recollision processes.
In this paper we study the dynamics of fermionic mixed states in the mean-field regime. We consider initial states that are close to quasi-free states and prove that, under suitable assumptions on the initial data and on the many-body interaction, the quantum evolution of such initial data is well approximated by a suitable quasi-free state. In particular, we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent Hartree-Fock equation. Our result holds for all times and gives effective estimates on the rate of convergence of the many-body dynamics towards the Hartree-Fock evolution.
We study the dynamics of interacting fermionic systems, in the mean-field regime. We consider initial states which are close to quasi-free states and prove that, under suitable assumptions on the inital data and on the many-body interaction, the quantum evolution of the system is approximated by a time-dependent quasi-free state. In particular we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent Hartree-Fock equation. Our theorems allow to describe the dynamics of both pure states (zero temperature states) and mixed states (positive temperature states). Our results hold for all times, and give effective estimates on the rate of convergence towards the Hartree-Fock evolution. The results on pure states are based on joint works with N. Benedikter and B. Schlein, [5, 6]; while those on mixed states are based on a joint work with N. Benedikter, V. Jaksic, C. Saffirio and B. Schlein, [7].