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Mean-field dynamics in a class of open quantum systems

Alireza Rafiyi

Published Dec 1, 2017• 0 citations
OpenAlex

Abstract

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.

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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.
In this thesis, we consider N quantum particles coupled to collective thermal quantum environments. The coupling is energy conserving and scaled in the mean field way. There is no direct interaction between the particles, they only interact via the common reservoir. It is well known that an initially disentangled state of the N particles will remain disentangled at times in the limit N -> [infinity]. In this thesis, we evaluate the η-body reduced density matrix (tracing over the reservoirs and the N - η remaining particles). We identify the main disentangled part of the reduced density matrix and obtain the first order correction term in 1/N. We show that this correction term is entangled. We also estimate the speed of convergence of the reduced density matrix as N -> [infinity]. Our model is exactly solvable and it is not based on numerical approximation.
We consider quantum particles coupled to local and collective thermal quantum environments. The coupling is energy conserving, and the collective coupling is scaled in the mean-field way. There is no direct interaction between the particles. We show that an initially factorized state of the particles remains factorized at all times, in the limit of large particle number. Each single-particle factor evolves according to an explicit, nonlinear, dissipative and time-dependent Hartree–Lindblad equation. The model is exactly solvable; we do not make any weak coupling or any Markovian approximations, and our results are mathematically rigorous.
The dynamics of a quantum system with a large number NNN of identical bosonic particles interacting by means of weak two-body potentials can be simplified by using mean-field equations in which all interactions to any one body have been replaced with an average or effective interaction in the mean-field limit N→∞N \rightarrow \inftyN→∞. In order to show these mean-field equations are accurate, one needs to show convergence of the quantum NNN-body dynamics to these equations in the mean-field limit. Previous results on convergence in the mean field limit have been derived for certain initial conditions in the case of one species of bosonic particles, but no results have yet been shown for multi-species. In this thesis, we look at a quantum bosonic system with two species of particles. For this system, we derive a formula for the rate of convergence in the mean-field limit in the case of an initial coherent state, and we also show convergence in the mean-field limit for the case of an initial factorized state. The analysis for two species can then be extended to multiple species.
We consider a large number N of quantum particles coupled via a mean field interaction to another quantum system (reservoir). Our main result is an expansion for the averages of observables, both of the particles and of the reservoir, in inverse powers of N . The analysis is based directly on the Dyson series expansion of the propagator. We analyse the dynamics, in the limit N → ∞ , of observables of a fixed number n of particles, of extensive particle observables and their fluctuations, as well as of reservoir observables. We illustrate our results on the infinite mode Dicke model and on various energy-conserving models.
We examine different nonlinear open quantum systems and calculate the non-Markovianity based on the distinguishability between two density matrices. We show that for a single spin (qubit) coupled to a bosonic field the non-Markovianity depends on the spectral density function and can take on any number N with 0 ≤ N ≤ 1, meaning the dynamics can be Markovian or highly non-Markovian. For the main result we consider a system of Ntot identical spins coupled to an environment in the mean-field way. Each spin is coupled to a local and the common reservoir. There are only indirect interactions between the spins through the common reservoir. In limit Ntot → 1 the subsystem consisting of a fixed set of n particles reduces to a factorized state in which each factor is a single spin evolving according to a nonlinear Hartree-Lindblad equation which is exactly solvable. For non-stationary initial spin states the non-Markovianity diverges. In this instance we can never approximate the dynamics of the quantum system by a Markovian master equation and this implies that in the mean field models, memory effects are significant.
We consider a class of open quantum many-body Lindblad dynamics characterized by an all-to-all coupling Hamiltonian and by dissipation featuring collective ``state-dependent" rates. The latter encodes local incoherent transitions that depend on average properties of the system. This type of open quantum dynamics can be seen as a generalization of classical (mean-field) stochastic Markov dynamics, in which transitions depend on the instantaneous configuration of the system, to the quantum domain. We study the time evolution in the limit of infinitely large systems, and we demonstrate the exactness of the mean-field equations for the dynamics of average operators. We further derive the effective dynamical generator governing the time evolution of (quasi-)local operators. Our results allow for a rigorous and systematic investigation of the impact of quantum effects on paradigmatic classical models, such as quantum generalized Hopfield associative memories or (mean-field) kinetically-constrained models.