We shall present a new strategy for handling mean field limits of quantum mechanical systems. The new method is simple and effective. It is simple, because it translates the idea behind the mean field description of a many particle quantum system directly into a mathematical algorithm. It is effective because the strategy yields with lesser effort better results than previously achieved. As an instructional example we treat a simple model for the time dependent Hartree equation which we derive under more general conditions than what has been considered so far. Other mean field scalings leading e.g. to the Gross-Pitaevskii equation can also be treated.We shall present a new strategy for handling mean field limits of quantum mechanical systems. The new method is simple and effective. It is simple, because it translates the idea behind the mean field description of a many particle quantum system directly into a mathematical algorithm. It is effective because the strategy yields with lesser effort better results than previously achieved. As an instructional example we treat a simple model for the time dependent Hartree equation which we derive under more general conditions than what has been considered so far. Other mean field scalings leading e.g. to the Gross-Pitaevskii equation can also be treated.

arXiv:0907.4464Jul 26, 20097 citations

Mean-Field Limit and Semiclassical Expansion of a Quantum Particle System

Federica Pezzotti, Mario Pulvirenti

arXiv:0810.1387Annales Henri PoincaréMar 1, 200928 citations

Semiclassical quantization for a bosonic atom-molecule conversion system

Eva-Maria Graefe, Maria Graney, Alexander Rush

We consider a simple quantum model of atom-molecule conversion where bosonic atoms can combine into diatomic molecules and vice versa. The many-particle system can be expressed in terms of the generators of a deformed $\text{su}(2)$ algebra, and the mean-field dynamics takes place on a deformed version of the Bloch sphere, a teardrop shaped surface with a cusp singularity. We analyze the mean-field and many-particle correspondence, which shows typical features of quantum-classical correspondence. We demonstrate that semiclassical methods can be employed to recover full many-particle features from the mean-field description in cold atom systems with atom-molecule conversion, and we derive an analytic expression for the many-particle density of states in the limit of large particle numbers.We consider a simple quantum model of atom-molecule conversion where bosonic atoms can combine into diatomic molecules and vice versa. The many-particle system can be expressed in terms of the generators of a deformed $\text{su}(2)$ algebra, and the mean-field dynamics takes place on a deformed version of the Bloch sphere, a teardrop shaped surface with a cusp singularity. We analyze the mean-field and many-particle correspondence, which shows typical features of quantum-classical correspondence. We demonstrate that semiclassical methods can be employed to recover full many-particle features from the mean-field description in cold atom systems with atom-molecule conversion, and we derive an analytic expression for the many-particle density of states in the limit of large particle numbers.

arXiv:1505.03351Physical Review AJul 27, 20156 citations

Mean-Field Regime for Bosonic Systems

Niels Benedikter, Marcello Porta, Benjamin Schlein

SpringerBriefs in mathematical physicsOct 6, 20151 citation

Landau–Lifshitz damping from Lindbladian dissipation in quantum magnets

Götz Silvester Uhrig

Abstract As of now, the phenomenological classical Landau–Lifshitz (LL) damping of magnetic order is linked to a quantum theory of dissipation of Lindbladian form only in sophisticated systems via complicated calculations in certain limits. This is unsatisfactory for the booming research on magnetic dynamics. Here, it is shown that LL dynamics can be systematically derived from Lindbladian dynamics in a rather simple way using a local mean-field theory for weak external fields. The necessary assumptions are (i) the validity of a local mean-field approach, (ii) a local dissipator, and (iii) a very rapid bath adapting to the non-equilibrium H ( t ) instantaneously. The derivation also extends the LL dynamics beyond the orientation m → / | m → | to the length | m → | of the magnetization, thereby transcending the classical description.Abstract As of now, the phenomenological classical Landau–Lifshitz (LL) damping of magnetic order is linked to a quantum theory of dissipation of Lindbladian form only in sophisticated systems via complicated calculations in certain limits. This is unsatisfactory for the booming research on magnetic dynamics. Here, it is shown that LL dynamics can be systematically derived from Lindbladian dynamics in a rather simple way using a local mean-field theory for weak external fields. The necessary assumptions are (i) the validity of a local mean-field approach, (ii) a local dissipator, and (iii) a very rapid bath adapting to the non-equilibrium H ( t ) instantaneously. The derivation also extends the LL dynamics beyond the orientation m → / | m → | to the length | m → | of the magnetization, thereby transcending the classical description.

arXiv:2406.10613New Journal of PhysicsOct 1, 20252 citations

\text{su}(2)

algebra, and the mean-field dynamics takes place on a deformed version of the Bloch sphere, a teardrop shaped surface with a cusp singularity. We analyze the mean-field and many-particle correspondence, which shows typical features of quantum-classical correspondence. We demonstrate that semiclassical methods can be employed to recover full many-particle features from the mean-field description in cold atom systems with atom-molecule conversion, and we derive an analytic expression for the many-particle density of states in the limit of large particle numbers.