Workshop on "Macroscopic Limits of Quantum Systems"

We start by presenting a short summary of examples of effective dynamics in quantum theory. We then study more closely the effective quantum dynamics of systems interacting with a long queue of independent probes, one after another, which are then subject to projective measurement. This leads us to develop a theory of indirect measurements of time-independent quantities (non-demolition measurements). Next, the theory of indirect measurements of time-dependent quantities is outlined, and a new family of diffusion processes — quantum jump processes — is described. Some open problems are proposed.

The dynamics of a particle coupled to a dense and homogeneous ideal Fermi gas in two spatial dimensions is studied. We analyze the model for coupling parameter g=1 (i.e., not in the weak coupling regime), 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 which is given at leading order by the spatially homogeneous mean field potential of the gas particles. Here, the mean field approximation turns out to be accurate although the fluctuations of the potential around its mean value can be arbitrarily large. Our result is in contrast to a dense bosonic gas in which the free motion of a tracer particle would be disturbed already on a very short time scale. The proof is based on the use of strong phase cancellations in the deviations of the microscopic dynamics from the mean field time evolution.

The polaron model of H. Fröhlich describes an electron coupled to the quantized longitudinal optical modes of a polar crystal. In the strong-coupling limit one expects that the phonon modes may be treated classically, which leads to a coupled Schrödinger-Poisson system with memory. For the effective dynamics of the electron this amounts to a nonlinear and non-local Schrödinger equation. We use the Dirac-Frenkel variational principle to derive the Schrödinger-Poisson system from the Fröhlich model and we present new results on the accuracy of their solutions for describing the motion of Fröhlich polarons in the strong-coupling limit. Our main result extends to N-polaron systems.

I shall report on a novel multi-scale technique to study many-body quantum systems where the total number of particles is kept fixed. The method is based on the Feshbach-Schur map and the scales are represented by occupation numbers of particle states. Consider an interacting Bose gas at zero temperature, constrained to a finite box and in the mean field limiting regime, where the N gas particles interact through a pair potential of positive type and with an ultraviolet cut-off. The (nonzero) Fourier components of the potential are assumed to be sufficiently large with respect to the corresponding kinetic energies of the modes. For this system, we provide a convergent expansion of the ground state of the Hamiltonian in terms of the bare operators. In the limit \(N\to\infty\) the expansion is up to any desired accuracy.

We consider systems of \(N\) bosons interacting through a repulsive potential \(N^{3\beta-1} V (N^\beta x)\) that scales with N. For \(\beta = 1\), we recover the well-known Gross-Pitaevskii regime. We present new techniques that allow us to prove the convergence towards the time-dependent Gross-Pitaevskii equation with optimal rate. Furthermore, we explain how , for small potentials, this approach can be used to show complete Bose-Einstein condensation (with a uniform bound on the number of excitations), for the ground state and, more generally, for states with small excitation energy. For \(\beta < 1\), the same method can be used to establish the validity of Bogoliubov theory for the low-lying excitation spectrum. This talk is based on joint works with C. Boccato, C. Brennecke and S. Cenatiempo.

In my talk I will review concepts related to bosonic quadratic operators and their connection with mean-field limits. In particular, I will provide general conditions under which such operators can be diagonalized by Bogoliubov transformations. Our results cover the case when quantum systems have infinite degrees of freedom and the associated one-body kinetic and paring operators are unbounded. This is joint work with Phan Thành Nam and Jan Philip Solovej.

We present the linear two-body gap-equation in the presence of external macroscopic fields, in particular, a constant magnetic field, and derive the critical temperature s dependence on the fields strength. We further comment on possible relations to high-Tc-superconductors.

Jellium is one of the simplest, yet very rich, Coulomb systems in statistical mechanics. It has been shown to arise naturally in mean-field limits for confined classical systems. On the other hand, the Uniform Electron Gas (UEG) is a cornerstone of Density Functional Theory in chemistry and material science and it arises naturally in the limit of slowly varying densities. This gas, which is defined by the property that its density is exactly constant over the whole space, is usually assumed to be the same as Jellium. In the talk I will review some elementary properties of the UEG and mention several open problems, some of which are important for practical DFT calculations.

We will consider the many-body evolution of initially confined fermions in a joint mean-field and semiclassical scaling, focusing on the case of Coulomb interaction. We will show that, for initial states close to Slater determinants and under some conditions on the solution of the time-dependent Hatree- Fock equation, the many-body evolution converges towards the Hartree-Fock dynamics. This is a joint work with M.Porta, S.Rademacher and B.Schlein.

The adiabatic theorem is a central tool in quantum mechanics. It has also been used and applied very much in the many-body setting to describe eg crossing of quantum phase transition point (kibble-zurek mechanism), preparation of quantum states, quantum computation, etcetc... Nonetheless, what is commonly known as the 'adiabatic theorem', is not applicable to many-body systems, unless one would take the driving vanishing with volume. We point out this fact and we prove an alternative adiabatic theorem that is applicable to many-body systems. As a mathematical bonus, this yields a proof of the validity of linear response for gapped ground states. This is joint with sven bachmann and martin fraas.

In this talk I will discuss some recent results about singularity formation and long time asymptotics for two kinetic equations containing cubic nonlinearities. These equations are the Nordheim’s equation for bosons and the kinetic equation for Weak Turbulence associated to the Nonlinear Schrödinger Equation. The solutions of these equations can yield singularity formation in finite time for homogeneous particle distributions. In the case of Nordheim equation the singularities are related to the formation of Bose-Einstein condensates. Issues like nonuniqueness of the solutions of this equation and self-similar behaviour for long times of the solutions of the Weak Turbulence equation will be also addressed.

Weakly interacting quantum fluids allow for a natural kinetic theory description which takes into account the fermionic or bosonic nature of the interacting particles. In the simplest cases, one arrives at the Boltzmann-Nordheim, aka Uehling-Uhlenbeck, equations for the reduced density matrix of the fluid. We discuss here two related topics: the kinetic theory of the fermionic Hubbard model, in which conservation of total spin results in an additional Vlasov-type term in the Boltzmann equation, and the relation between kinetic theory and thermalization. In particular, we compare recent simulations of an anharmonic chain with very slow thermalization to the corresponding kinetic description and to the "pre-thermalization" observed in certain quantum systems.

I first briefly review a well known procedure for an approximate block decomposition of perturbed periodic Hamiltonians. If the periodic system is time-reversal symmetric, the block subspaces are spanned by localised (composite) Wannier functions and each block of the Hamiltonian can be nicely represented by an effective Hamiltonian obtained through Peierls substitution. If the periodic Hamiltonian is not time-reversal symmetric, as in the presence of a constant magnetic field or in Chern insulators, generically no localised (composite) Wannier functions exist and the geometry of the underlying Bloch bundles, certain complex vector bundles over the torus, becomes relevant for the construction of effective Hamiltonians in the spirit of Peierls substitution. The recent results are from joint work with Silvia Freund, Domenico Monaco, Gianluca Panati, and Adriano Pisante, and originate from not so recent anymore work with Herbert Spohn.