In these notes, based on a mini-course held at the summer school "Methods and Models of Kinetic Theory" that took place in Porto Ercole in June 2016, we review some of the recent developments in the derivation of effective evolution equations starting from many-body quantum mechanics. We discuss the derivation of the Hartree equation in the bosonic mean-field limit, of the Gross-Pitaevskii equation describing the dynamics of initially trapped Bose-Einstein condensates and of the Hartree-Fock equation for fermions in a joint mean-field and semiclassical limit.
Using a new method [9] it is possible to derive mean field equations from the microscopic N-body Schrodinger evolution of interacting particles without using BBGKY hierarchies. Recently this method was used to derive the Hartree equation for singular interactions [4] and the Gross Pitaevskii equation without positivity condition on the interaction [10] where one had to restrict the scaling behaviour of the interaction. In this paper more general scalings shall be considered assuming positivity of the interaction.
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.
In these notes, we review some recent mathematical results concerning the derivation of effective evolution equations from many body quantum mechanics. In particular, we discuss the emergence of the Hartree equation in the so-called mean field regime (for example, for systems of gravitating bosons), and we show that the Gross-Pitaevskii equation approximates the dynamics of initially trapped Bose-Einstein condensates. We explain how effective evolution equations can be derived, on the one hand, by analyzing the so called BBGKY hierarchy, describing the time-evolution of reduced density matrices, and, on the other hand, by studying the dynamics of coherent initial states in a Fock-space representation of the many body system.