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.
An explicit expression is derived for the statistical description of small quantum systems, which are relatively weakly and directly coupled to only small parts of their environments. The derived expression has a canonical form, but is given by a renormalized self-Hamiltonian of the studied system, which appropriately takes into account the influence of the system-environment interaction. In the case that the system has a narrow spectrum and the environment is sufficiently large, the modification to the self-Hamiltonian usually has a mean-field feature, given by an environmental average of the interaction Hamiltonian. In other cases, the modification may be beyond the mean-field approximation.
We show that when a quantum system is coupled to an environment in a mean field way, then its effective dynamics is governed by a unitary group with a time-dependent Hamiltonian. The time-dependent modification of the bare system Hamiltonian is given by an explicit term involving the reservoir state. We show that entanglement within the system state is not changed during the dynamics. Our results hold for arbitrary strengths of the system-environment coupling, and for finite or infinite dimensional systems. As an application we show that the qualitative dynamical features of an
N-body system can be altered drastically by the contact with the environment. For instance, bound states can turn into scattering states and vice-versa.