Dynamics of Quantum Systems with Interacting Particles

Alireza Rafiyi

We consider a system of N identical quantum particles coupled to an environment. The coupling is scaled in a mean-field way, and particles interact indirectly through the environment. There is no direct interaction between particles. The time evolution of the mean-field-scaled particles is called the mean-field dynamics and the limit of the evolution as N → ∞ is called the mean-field limit. The study of mean-field limits is an active topic of research in many-body quantum theory. Previously, almost all research in this area has been carried out for directly interacting particles without interaction with an environment. In this thesis, we prove that under a suitable condition on the initial system states the mean-field limit in the above indirectly interacting particle system does exist. The condition is satisfied in particular for spin ½ (qubit) systems. Assuming this condition, we show that the system dynamics is the free dynamics in the mean-field limit N → ∞, and we find the first and second correction terms. We show that the particles have a collective effect on the dynamics of the environment (reservoir) which we derive analytically. We give examples to illustrate our results.

V. M. Akulin

Abed Alsalam Abu Moise

We consider the dynamics of fixed size subsystems of an open quantum system, in \nwhich N particles interact via a common quantum noise (reservoir). We show that \ncorrelations among the particles and between the particles and the reservoir, which \nare brought about through the interaction for finite N, vanish completely in the high \ncomplexity limit N → ∞. We investigate the effect of the particle system on the \nreservoir, which itself is a large quantum system. For each fixed time, we find the \nexplicit construction of a Hilbert space representation of the asymptotic (N → ∞) \nreservoir state and analyze the relation between those representations at different \ntimes.

Alexey Vyacheslavovich Gorshkov

Friedemann Queißer, Ralf Schützhold

Via the hierarchy of correlations, we study strongly interacting fermions on a lattice and derive a quantum Boltzmann equation describing their relaxation dynamics. Interestingly, in the strong-coupling limit, collisions between particles and holes dominate over particle-particle and hole-hole collisions -- in stark contrast to weakly interacting systems. Therefore, the relaxation towards equilibrium strongly depends on what kind of excitations (particles or holes or both) are present.